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open dielectric cavities have attracted a large interest in recent years due to their numerous and potentially important applications @xcite . from a theoretical point of view , the crucial difference between dielectric cavities and much more investigated case of closed quantum billiards @xcite is that in the latter the spectrum is discrete but in the former it is continuous . indeed , the main subject of investigations in open systems is not the true spectrum but the spectrum of resonances defined as poles of the scattering @xmath0-matrix ( see e.g. @xcite ) . the wavelength of electromagnetic field is usually much smaller than any characteristic cavity size ( except its height ) and semiclassical techniques are useful and adequate for a theoretical approach to such objects . it is well known that the trace formulas are a very powerful tool in the semiclassical description of closed systems , see e.g. @xcite . therefore , the generalization of trace formulas to different open systems , in particular to dielectric cavities , is of importance . the trace formula for resonances with transverse magnetic ( tm ) polarization in two - dimensional ( 2d ) dielectric cavities has been developed in @xcite and shown to agree well with the experiments and numerical calculations @xcite . this paper is devoted to the construction of the trace formula for 2d dielectric cavities but for transverse electric ( te ) polarization . due to different boundary conditions the case of te modes differs in many aspects from tm modes . in particular , a special treatment is required for the resonances related to brewster s angle @xcite at which the fresnel reflection coefficient vanishes . our main result is the asymptotic formula in the semiclassical ( aka short wave length ) regime for the average number of te resonances for a 2d dielectric cavity with refraction index @xmath1 , area @xmath2 and perimeter @xmath3 @xmath4 here @xmath5 is the mean number of resonances ( defined below ) whose real part is less than @xmath6 , the coefficient @xmath7 is given by the expression @xmath8 and @xmath9 is the fresnel reflection coefficient for the scattering on a straight dielectric interface at imaginary momentum @xmath10 the plan of the paper is the following . in sec . [ general ] the main equations describing the te modes are reminded . in sec . [ circle ] the circular cavity is briefly reviewed : an exact quantization condition is derived , which allows a direct semiclassical treatment . in sec . [ sectweyl ] the first two weyl terms for the resonance counting function are derived . it is important to notice that , for te modes , one can have total transmission of a ray when the incidence angle is equal to brewster s angle . this leads to a special set of resonances , which are counted separately in sec . [ additional ] . section [ oscillating ] is devoted to a brief derivation of the oscillating part of the resonance density . in sec . [ numerics ] our obtained formulae are shown to agree well with numerical computation for cavities of different shapes . in appendix [ krein ] another method of deriving the weyl series for te polarization based on krein s spectral shift formula is presented . to describe a dielectric cavity correctly one should solve the @xmath11-dimensional maxwell equations . in many applications the transverse height of a cavity , say along the @xmath12 axis , is much smaller than any other cavity dimensions . in such situation the @xmath11-dimensional problem in a reasonable approximation can be reduced to two 2d scalar problems ( for each polarization of the field ) following the so - called effective index approximation , see e.g. @xcite for more details . in the simplest setting , when one ignores the dependence of the effective index on frequency , such 2d approximation consists in using the maxwell equations for an infinite cylinder . it is well known @xcite that in this geometry the maxwell equations are reduced to two scalar helmholtz equations inside and outside the cavity @xmath13 where @xmath1 is the refractive index of the cavity , @xmath14 indicates the interior of the dielectric cavity , and @xmath15 for the tm polarization and @xmath16 for the te polarization . helmholtz equations ( [ equations ] ) have to be completed by the boundary conditions . the field , @xmath17 , is continuous across the cavity boundary and its normal derivatives along both sides of the boundary are related for two polarizations as below @xcite @xmath18 open cavities have no true discrete spectrum . instead , we are interested in the discrete resonance spectrum , which is defined as the ( complex ) poles of the @xmath0-matrix for the scattering on a cavity ( see e.g. @xcite ) . it is well known that the positions of the resonances can be determined directly by the solution of the problem and by imposing the outgoing boundary conditions at infinity @xmath19 the set ( [ equations])-([infinity ] ) admit complex eigen - values @xmath6 with im@xmath20 , which are the resonances of the dielectric cavity and are the main object of this paper . our goal is to count such resonances for the te polarization in the semiclassical regime . this will provide us with the analogue of weyl s law derived for closed systems , see e.g. @xcite . the circular dielectric cavity is the only finite 2d cavity , which permits an analytical solution . let @xmath21 be the radius of such a cavity . writing @xmath22 inside the cavity and @xmath23 outside the cavity , it is plain to check , that in order to fulfill the boundary conditions , it is necessary that @xmath6 is determined from the equation @xmath24 with @xmath25 and @xmath26 \label{s_m}\ ] ] where @xmath27 ( resp . @xmath28 ) denotes the bessel function ( resp . the hankel function of the first kind ) . here and below the prime indicates the derivative with respect to the argument . factor @xmath29 in is introduced for further convenience . using @xmath30 the equation @xmath24 can be rewritten in the form @xmath31 where @xmath32 and @xmath33 in the semiclassical limit , @xmath34 , the asymptotic formula for the hankel function @xcite ( @xmath35 ) gives @xmath36 \label{hankel}\ ] ] where @xmath37 in this way one obtains @xmath38 and @xmath39 where @xmath40 is the standard te fresnel coefficient for the scattering on an infinite dielectric interface @xmath41 the above formulas mean that in the semiclassical limit , eq . takes the form @xmath42 or @xmath43 with integer @xmath44 . in fact , this equation is valid in the semiclassical limit for closed and open circular cavities with other boundary conditions as well . the only difference is that , instead of the fresnel reflection coefficient @xmath40 , it is necessary to use the reflection coefficient for the problem under consideration . for example , for closed billiards , @xmath45 and for neumann ( resp . dirichlet ) boundary conditions @xmath46 in ( [ rscl ] ) equals to @xmath47 ( resp . @xmath48 ) . for open dielectric circular cavity with the tm polarization @xmath49 , where @xmath50 is the usual fresnel reflection coefficient for the tm modes @xcite @xmath51 semiclassical formulas like eq . are convenient to obtain the average number of eigenvalues and resonances for closed and open systems with different boundary conditions . let us consider first the simplest case of a closed billiard with neumann boundary conditions for which @xmath52 . in the semiclassical regime the eigenvalues for this model are determined from eq . which reads @xmath53 where @xmath54 is defined in and @xmath55 is an integer . therefore , for fixed @xmath56 , the number of eigenvalues less than @xmath29 is @xmath57 $ ] where @xmath58 $ ] stands for the integer part and @xmath59 @xmath47 is added as the integer @xmath60 in starts with @xmath61 but @xmath62 $ ] has to begin with @xmath47 . summing over all @xmath56 leads to the total number of eigenvalues less than @xmath29 , usually called the counting function . this sum is finite as the asymptotics is valid when @xmath63 . finally @xmath64 } \big [ n_m(x)\big ] \ .\ ] ] the averaged number of levels is determined from the equation @xmath65 with a needed precision one can substitute the summation over @xmath56 by an integral and , consequently , the averaged number of eigenvalues for a circular billiard with neumann boundary conditions can be approximated as follows @xmath66 using the formula @xmath67 one gets @xmath68 as for the circle the area is @xmath69 and the perimeter is @xmath70 , these results can be rewritten in the standard form @xcite @xmath71 with @xmath72 . for dirichlet boundary conditions similar arguments show that @xmath73 in is substituted by @xmath74 and @xmath75 , as it should be @xcite . for open cavities eq . ( [ semiclassical ] ) gives complex solutions ( resonances ) @xmath76 with negative imaginary part , @xmath77 . in the semiclassical limit for all investigated cases one has @xmath78 . separating the imaginary and real parts in and using that @xmath79 one gets that in the first order in @xmath80 the real part of the resonance position , @xmath81 ( or @xmath82 ) , is determined from the real equation similar to @xmath83 where @xmath84 is the argument of the reflection coefficient @xmath85 in the same approximation the imaginary part of the resonance position , @xmath80 is @xmath86 this semiclassical approximation is quite good even for not too large @xmath56 as indicated in fig . [ m_23_app ] . the above arguments demonstrate that the total number of resonances can be calculated from the real equation . as in one concludes that the mean number of resonances with real part @xmath87 less than @xmath29 is given by the expression @xmath88 with @xmath89 consider first the case of tm modes . the reflection coefficient in this case is given by and one has @xmath90 therefore @xmath91\mathrm{d}m\nonumber\\ & + & 2\int_{0}^{nx}\left [ \frac{1}{\pi } \delta_{\mathrm{tm}}\left ( \frac{m}{x}\right ) + \frac{1}{4 } \right ] \mathrm{d}m = \frac{n^2}{4}x^2+\frac{n}{2}x\nonumber\\ & -&\frac{2x}{\pi}\int_{1}^n\arctan \left ( \frac{\sqrt{t^2 - 1}}{\sqrt{n^2-t^2}}\right ) \mathrm{d}t\ .\end{aligned}\ ] ] by integration by parts and contour deformation it is easy to check that @xmath92 where @xmath93 is the same as but for pure imaginary argument @xmath94 finally , these considerations lead to the expression similar to @xmath95 where @xmath96 which agrees with the result in @xcite obtained by a different method . consider now te modes . in this case the reflection coefficient is given by and its argument is @xmath97 where @xmath98 corresponds to the zero of the te reflection coefficient ( brewster s angle ) , @xmath99 , @xmath100 using these values we get @xmath101 where @xmath102 is given by the following expression @xmath103 similar to one can prove that @xmath104 where , as above , @xmath105 is the te reflection coefficient analytically continued to imaginary @xmath106 @xmath107 combining all terms together we obtain that @xmath108 the first two terms are the same as for tm modes but with te reflection coefficient . the last term is the new one related to the change of the sign of the te reflection coefficient . higher order terms in weyl s expansions and are not yet calculated so we prefer to use a conservative estimate of them as @xmath109 though all numerical checks suggest that for smooth boundary cavities it is @xmath110 . formula is the correct description for the resonances whose real part of the eigen - momentum @xmath6 corresponds to non - zero reflection coefficient ( i.e. @xmath111 ) . this is due to the fact that when the reflection coefficient is zero its phase is not defined . for te modes there is a special branch of resonances for which semiclassically the real part does obey @xmath112 . the existence of such additional resonances were first discussed in a different context in @xcite . the approximate positions of these resonances can be calculated as follows . assume that the resonances have a large imaginary part . as @xmath113 tends to zero when @xmath114 one can approximate eq . by @xmath115 from the asymptotic it follows that @xmath116 using this expression one concludes that the solution of the equation @xmath117 has the form @xmath118 this approximation is better for large @xmath1 when the imaginary part is large but it gives reasonable results even for @xmath1 of the order of @xmath47 . in practice one may use as the initial value for any root search algorithm ( cf . [ m_23_app ] ) . from it follows that the ratio @xmath119 tends to @xmath98 defined in so these resonances are not taken explicitly into account in eq . ( [ r_te ] ) . their number can be estimated as follows . the discussed resonances correspond to waves propagating along the boundary whose direction forms an angle with the normal exactly equal to brewster s angle @xmath120 if the length of the boundary is @xmath3 , the possible values for the momenta of such states in the semiclassical limit are @xmath121 with integer @xmath122 . therefore , the number of additional resonances related with brewster s angle is @xmath123 comparing it with eq . we conclude that the second term in the weyl expansion for the averaged number of resonances for te polarization is the following @xmath124 where the plus sign is used when the above additional resonances are taken into account and the minus sign corresponds to the case when these resonances are ignored . for small values of @xmath1 the additional resonances are mixed with other resonances and their separation seems artificial . for large @xmath1 the additional branch of resonances is well separated from the main body of resonances and one can decide not to take them into account . in such a case , the minus sign has to be used in ( see below section [ numerics ] ) . when the cavity remains invariant under a group of symmetry it is often convenient to split resonances according to their symmetry representations . for reflection symmetries it is equivalent to consider a smaller cavity where along parts of the boundary one has to impose either dirichlet or neumann boundary conditions . in this case the total boundary contribution to the average counting function @xmath125 is given by the general formula @xmath126 here @xmath127 and @xmath128 are the lengths of the boundary parts with respectively neumann and dirichlet boundary conditions and @xmath129 is the length of the true dielectric interface . it is this formula , which will be used in section [ numerics ] for dielectric cavities in the shape of a square and a stadium . the quantization conditions or permit also to obtain the resonance trace formula for a circular dielectric cavity . let @xmath130 be resonance eigen - momenta . define the density of resonances as follows @xmath131 in general , if @xmath132 are the zeros of a certain function @xmath133 which has no other singularities then the density of these zeros formally is given by the following expression @xmath134 in the semiclassical limit @xmath135 it is sufficient to consider the semiclassical formula i.e. @xmath136 and @xmath137 a more careful discussion is performed in appendix [ krein ] . in such a manner one gets @xmath138 here @xmath139 is the fresnel reflection coefficient for te polarization . the further steps are as usual , see e.g. @xcite . using the poisson summation formula @xmath140 the expression becomes @xmath141 where the action is @xmath142 when @xmath143 the dominant contribution to the integral is due to saddle point solutions @xmath144 determined from the equation @xmath145 . it is plain that @xmath146 with @xmath147 . this saddle point corresponds geometrically to a periodic orbit of the circle in the shape of regular polygon with @xmath148 vertices going around the center @xmath149 times . expanding the action @xmath150 around the saddle point one gets @xmath151 here @xmath152 is the classical length of the periodic orbit determined by @xmath153 and @xmath148 . in the end one gets the trace formula for the resonances of the circular dielectric cavity in the form @xmath154 } + \mathrm{c.c.}\ ] ] where @xmath155 is the area occupied by a given periodic orbit family , @xmath156 is the fresnel reflection coefficient for the te scattering with an angle equal to the reflection angle , @xmath157 , for the given periodic orbit . repeating the arguments presented in @xcite we argue that in general the oscillating part of the resonance trace formula in the strong semiclassical limit has the form of the sum over all classical periodic orbits @xmath158 where contribution of an individual orbit depends on the orbit considered * for an isolated primitive periodic orbit @xmath60 repeated @xmath148 times @xmath159 where @xmath160 , @xmath161 are , respectively , the length , the monodromy matrix , the maslov index , and the total te fresnel reflection coefficient for the chosen primitive periodic orbit . * for a primitive periodic orbit family @xmath162 where @xmath163 is the area covered by one periodic orbit family , @xmath164 is the mean value of the te fresnel reflection coefficient averaged over a periodic orbit family . the only difference with corresponding results derived in @xcite is that the te reflection coefficient is used instead of the tm coefficient . the numerical calculations of the resonance spectrum for the te modes of the circular dielectric cavity is presented in fig . [ circle_n_1.5_2_3 ] . notice that when the cavity refraction index @xmath1 increases the additional branch of the resonances ( [ tilde_x ] ) separates more and more from the main part of the spectrum . a ) b ) c ) in fig . [ circle_fit_n_1.5_2_3 ] we plot the difference between the function counting the numerically computed resonances resonances with a real part less than @xmath6 ( with radius @xmath165 ) and the best fit to it of the form , see , @xmath166 where @xmath167 and @xmath168 are fitting parameters . a ) b ) c ) for @xmath169 and @xmath170 we consider all resonances including the additional branch . for @xmath171 this branch is quite far from the other resonances ( cf . [ circle_n_1.5_2_3 ] c ) ) and it is natural not to include it in the counting . the fitted values of the parameters for these three cases are the following @xmath172 the term @xmath167 has to be compared with the theoretical prediction which follows from ( used with plus sign for @xmath169 and @xmath170 , and with minus sign for @xmath171 ) @xmath173 the agreement with our numerical calculations is very good . in fig . [ d_l_circle ] the fourier transform of the resonance density for the circular dielectric cavity with different values of the refractive index is displayed . as expected from the trace formula , this quantity has peaks at the length of classical periodic orbits of the circle . notice especially that the triangular orbit is not confined for @xmath169 . hence the fresnel reflection coefficient is small and induces damping , which can be clearly seen in fig . [ d_l_circle ] a ) . as the index grows it is also shown that the contribution of short - period orbits become closer and closer to the one of the closed billiard . a ) b ) c ) in fig . [ square_fig ] a ) we present the spectrum of the te resonances for the square cavity of side @xmath174 with @xmath175 symmetry along the diagonals . for such cavity the fit function similar to ( [ nfitcirc ] ) is @xmath176 and the best fit gives , see fig . [ square_fig ] b ) , @xmath177 the theoretical prediction for this symmetry class is obtained from @xmath178 which agrees well with the numerical calculations . a ) b ) finally the same procedure was done for the dielectric stadium consisted of two half - circles of radius @xmath21 connected by a rectangle with sides @xmath179 and @xmath180 where @xmath181 called the aspect ratio of the stadium . the calculations were restricted to the symmetry class such that the associated function vanishes along both symmetry axis of the stadium , which is again called @xmath182 symmetry class . the resonance spectrum is presented in fig . [ stadium_fig ] a ) . a ) b ) the fit function is now @xmath183 where the aspect ratio @xmath181 has been taken to @xmath47 in the numerical calculations . the best fit gives , see fig . [ stadium_fig ] b ) , @xmath184 which agrees well with the prediction for this symmetry class : @xmath185 ( cf . ) . trace formulas are the main tool of the semiclassical description of multi - dimensional quantum problems . for closed systems the trace formulas relate two objects : quantum density of discrete states and a sum over classical periodic orbits @xmath186 where @xmath187 is the classical action over a periodic orbit and @xmath188 is the mean density of eigen - energies , averaged over a small window around @xmath189 . for 2d billiards with area @xmath2 and perimeter @xmath3 this averaged density of states is @xmath190 where @xmath72 for the neumann boundary conditions and @xmath75 for the dirichlet ones . for open quantum models the true eigen - energy spectrum is continuous and the main object of interest is the discrete spectrum of resonances defined as the poles of the @xmath0-matrix in the complex plane : @xmath191 with @xmath192 and @xmath193 real . the real part of the resonance energy , @xmath192 , gives the position of the resonance while its imaginary part , @xmath193 , determines the resonance width . the analogue of the trace formula for open systems has the form similar to @xmath194 in ref . @xcite such type of formula has been obtained for a 2d dielectric cavity with transverse magnetic polarization of the field . here we derive the trace formula for a 2d dielectric cavity but with boundary conditions corresponding to the transverse electric polarization of the electromagnetic field . as expected , the oscillating part of this trace formula is given by the usual periodic orbits weighted in the leading semiclassical order by the fresnel coefficient corresponding to te reflection on the cavity boundary , . our main result is the expression for the average resonance density of a dielectric cavity with area @xmath2 , perimeter @xmath3 , and refraction index @xmath1 @xmath195 where @xmath196 and @xmath197 is the fresnel reflection coefficient for the te polarization at imaginary momentum @xmath10 the plus - minus sign in front of the last term in is connected with the existence for the te modes of an additional series of resonances related to brewster s angle . as these resonances have large imaginary parts , they may be included or not in the counting function . for small values of @xmath1 additional resonances are mixed with other resonances and their separation is artificial . in this case the plus sign has to be used . for large @xmath1 the branch of additional resonances is well separated from the body of resonances and it is natural to ignore them . it corresponds to the minus sign in . the results of this paper together with ref . @xcite demonstrate that semiclassical trace formulas can be derived and applied for open dielectric cavities in a close similarity with closed billiards . further investigations of trace formulas for other physical open systems is of considerable interest . it is a pleasure to thank martin sieber for fruitful discussions and stefan bittner for providing numerical data for the dielectric circle with @xmath171 . the purpose of this appendix is to present another derivation of the number of resonances in a circular dielectric cavity based on the krein spectral shift formula @xcite . the true eigen - energy spectrum for an open system is continuous and , consequently , the density of states for open quantum systems is infinite . nevertheless , the difference between the density of states with a cavity and the density of state without the cavity is finite and is given by the krein formula @xmath198 where @xmath199 is the @xmath0-matrix for the scattering on the cavity . this formula is general and can be used for any type of short - range potential . we apply it for a scattering on a circular dielectric cavity . it is easy to check that the @xmath0-matrix for the the scattering on 2d circular cavity with te boundary conditions is diagonal in the polar coordinates and @xmath200 where @xmath201 is given by and @xmath202 differs from @xmath201 by changing @xmath28 to @xmath203 : @xmath204\ .\ ] ] from properties of the bessel functions @xcite it is straightforward to show that @xmath205 using the equality @xmath30 , this expression can be rewritten in the form @xmath206}\ ] ] where @xmath207 and @xmath46 are defined in and respectively , and @xmath208 expanding this expression into series of @xmath207 gives @xmath209 where @xmath210 and @xmath211 } { \pi n^2 x h_m^{(1)}(nx)h_m^{(2)}(nx ) c_m(x ) d_m(x ) } \ .\ ] ] with @xmath212 in the semiclassical limit @xmath213 the above formulae are simplified by using the asymptotic of the hankel function @xmath214 consider first the smooth term . from the identity @xmath215 it is straightforward to check that @xmath216\label{q_m}\\ & - & \frac{x}{2 } r_{\mathrm{te}}\big ( \frac{m}{x}\big ) \left [ \frac{n^2}{n^2x^2-m^2}-\frac{1}{x^2-m^2 } \right ] \nonumber \end{aligned}\ ] ] where @xmath217 is the fresnel reflection coefficient for the te polarization given by . the difference between the density of state with a cavity and the one without the cavity averaged over an energy interval such that periodic orbit terms are small can be calculated from @xmath218 @xmath219 changing the summation over @xmath56 to the integration and turning the integration contour in the second term in in the complex plane to avoid poles , @xmath220 leads to @xmath221 \nonumber\\ & + & \frac{r x}{4\pi k } \int_{-\infty}^{\infty}\mathrm{d}t r_{\mathrm{te}}\big ( -{\mathrm{i}}\frac{t}{x}\big ) \big [ \frac{n^2}{n^2x^2+t^2}-\frac{1}{x^2+t^2 } \big ] \ . \nonumber\end{aligned}\ ] ] rescaling integration variables one gets @xmath222 \nonumber\end{aligned}\ ] ] where @xmath69 and @xmath70 are the area and the perimeter of a circular cavity . this formula differs form the averaged total number of resonances and . this is the consequence of the fact discussed in @xcite for the case of tm modes that the @xmath0-matrix for the scattering on a cavity has an additional phase ( and additional zeros ) connected with the outside scattering on the impenetrable cavity . the form of this additional @xmath0-matrix may be argued as follows . it is known that when a wave from outside the cavity scatters on a cavity it reflects with the reflection coefficient which differs by the sign from the reflection coefficient from inside the cavity ( this is a consequence of current conservation ) . for the te polarization the fresnel reflection coefficient for a scattering from a medium with the refraction index @xmath47 on another medium with the refraction index @xmath1 is @xmath223 where @xmath40 is given by . in semiclassical region accessible in outside scattering , @xmath224 , the reflection coefficient @xmath225 is real and the effective reflection coefficient corresponding to the scattering on impenetrable cavity equals to the sign of @xmath225 ( cf . ) @xmath226 where @xmath227 . the reflection coefficient equals to @xmath48 ( resp . @xmath228 ) corresponding to the scattering with dirichlet ( resp . neumann ) conditions on the cavity boundary . for a circular cavity the @xmath0-matrices with dirichlet and neumann boundary conditions are well known ( see e.g. @xcite ) @xmath229 the additional @xmath0-matrix for the te scattering is thus formally @xmath230 where @xmath231 $ ] . to find the total phase of this additional @xmath0-matrix one can proceed as follows . to the leading order in the semiclassical limit @xmath34 the dirichlet and neumann @xmath0-matrices can be calculated from . it gives @xmath232 where @xmath54 is given by . it means that @xmath233 differs from @xmath234 only by its sign , which is another manifestation of the opposite sign of the reflection coefficient . therefore one can rewrite expression as follows @xmath235 where @xmath236 is the full @xmath0-matrix for the scattering on a cavity with the dirichlet boundary condition . the @xmath237 sign in the exponent reflects the ambiguity of the phase , @xmath238 . the calculation of the mean density of states related with @xmath239-matrix is straightforward ( see e.g. @xcite ) @xmath240 and finally from and the krein formula one finds that the change of the density of states due to the additional @xmath0-matrix is @xmath241 the total density of resonances is thus the difference between and . in the end one gets eqs . and . the ambiguity in the phase of the additional @xmath0-matrix corresponds to the the possibility to include resonances related with brewster s angle in the weyl formula or not which has been discussed in section [ additional ] . 99 k. vahala , ed . , _ optical microcavities _ ( world scientific press , 2004 ) . a. b. matsko , _ practical applications of microresonators in optics and photonics _ , ( crc press , taylor and francis group , 2009 ) . r. balian , c. bloch , ann . phys . * 60 * , 401 ( 1970 ) ; ann . phys . * 64 * , 271 ( 1971 ) ; ann . phys . * 69 * , 76 ( 1972 ) . m. gutzwiller , _ chaos in classical and quantum mechanics _ , ( springer - 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the construction of the semiclassical trace formula for the resonances with the transverse electric ( te ) polarization for two - dimensional dielectric cavities is discussed .
special attention is given to the derivation of the two first terms of weyl s series for the average number of such resonances .
the obtained formulas agree well with numerical calculations for dielectric cavities of different shapes .
| 9,561 | 90 |
theoretical investigation of hadron production in heavy - ion collisions at high energies is usually separated into different camps , characterized by the regions of transverse momenta @xmath0 of the produced hadrons . at low @xmath0 statistical hadronization and hydrodynamical models are generally used @xcite , whereas at high @xmath0 jet production and parton fragmentation with suitable consideration of medium effects in perturbative qcd are the central themes @xcite . the two approaches have been studied essentially independent of each other with credible success in interpreting the data , since their dynamics are decoupled at the energies investigated . the situation may have changed at the cern large hadron collider ( lhc ) , where pb - pb collisions have been carried out at @xmath1 tev , resulting in thousands of soft hadrons on the one hand , and multiple hard jets on the other . minijets that are copiously produced at intermediate @xmath0 can fragment into soft partons with multiplicities so high that their effects on the hadronization of all partons created in the soft sector can not be ignored . it is the aim of this paper to investigate what those effects are and to offer an explanation of the observed hadronic spectra of all species and for all @xmath0 measured up to 20 gev / c . hard parton scattering and hydrodynamical flow are processes that involve very different time scales . it would be hard to incorporate them into a unified formalism that describes all aspects of the system , including thermalization time , initial configuration , fluid nature of the medium , its quenching effect on the hard protons , the creation of shower partons , and the hadronization of all partons at the end of the whole process . our attempt here is far from being so ambitious . we focus only on the @xmath0 dependencies of the hadrons produced from 0.5 to 20 gev in a formalism that can be valid throughout that range , provided that we use some model inputs for the thermal component of the low-@xmath0 behavior to supplement the hard component that can be calculated at high @xmath0 . we use quark recombination to treat hadronization , applied uniformly at all @xmath0 . in treating the degradation of momenta of hard and semihard partons we shall adjust some parameters to fit the high-@xmath0 data . since we aim to confront the @xmath0 spectra of all observed hadrons , @xmath2 , @xmath3 , @xmath4 and @xmath5 , the system is highly constrained . the primary feature of this study is to quantify the effect of hard and semihard jets on the soft sector . what we find is that the soft partons generated by the hard partons are so much more at lhc , compared to the situation at rhic , that any treatment without including that aspect of the problem would be incomplete . our investigation of produced hadrons with various contents of strangeness also reveals contrasting features of heavy - ion physics not commonly addressed . whereas hard scattering of gluons and light quarks can readily occur at high energies , jet fragmentation into multi - strange hadrons like @xmath5 and @xmath4 is rare even at lhc . but the production of @xmath5 relative to @xmath6 grows exponentially with @xmath0 even to the highest @xmath0 measured , the data for which will be exhibited explicitly in the next section . surely , one can not expect @xmath5 to be easily produced at @xmath7 gev / c by jet fragmentation . an explanation of the observed phenomenon must be an integral part of a description of the production mechanism of all hadrons . to give a description of the experimental motivation for our study , we show in sec . ii several pieces of data presented in novel ways so as to emphasize the problems that have not been commonly discussed . it will become clear that the hadronization problem at lhc is drastically different from that at rhic . in the framework of the recombination models @xcite in which the partons just before hadronization are categorized into thermal ( t ) and shower ( s ) partons , that difference at lhc can be succinctly stated in the form that s is much greater than t at low @xmath0 for light quarks , but not strange quarks . such a statement has no phenomenological consequence unless the hadronization of those quarks is treated by recombination . we do not consider here other features of heavy - ion collisions besides @xmath0 distributions , most notably the azimuthal dependence in non - central collision . conventional description of elliptic flow does not consider the effects of jets . we shall treat that subject separately , after our concern about the shower partons establishes a footing in the general terrain of heavy - ion physics . to clarify the nature of our approach it is necessary to contrast it from the standard model based on hydrodynamics . if hard and semihard partons produced in high - energy - energy nuclear collisions are important in their effects on soft particles , then one should recognize that their in - medium radiated daughter partons take some time to thermalize , much longer than the rapid equilibration time ( @xmath8 fm / c ) usually assumed in hydro calculations . a hard parton produced near the center of the medium in central collisions would take about 6 fm / c to reach the surface . thus rapid thermalization is not realistic if minijets are important , as we shall show that they are at lhc . as a consequence , we can not make use of hydro results in our approach , nor can hydro results be used to censure our calculations . for example , the thermal parton that we consider is not to be identified with any of the fluid constituents in the hydro medium . also , in the hydro treatment @xmath9 is identified with elliptic flow , but it is only a possible , not a necessary , explanation . other explanations are also possible ; see , for example , refs . in this paper we consider only central collisions and establish the importance of shower partons at low momenta . it is suggested that a reader withhold comparison with hydro treatment until the main points advanced here can be made . this paper is organized as follows : in sec . ii we show experimental features that motivate this investigation . section iii describes the general formulation of our approach to the problem . shower parton distributions are discussed in detail in sec . iv with emphasis on how the degradation of parton momenta is treated . with those partons shown to be dominant , the recombination of shower partons from nearby jets becomes a possibility that is considered in sec . v. with all the basic inputs on partons at hand we then proceed to the determination of the transverse - momentum distributions of @xmath10 and @xmath11 in sec . multi - strange hyperons and meson are treated in sec . vii with detail equations given in the appendices . section viii contains our conclusion . we show first some data from lhc that can be taken to suggest something unusual about the usual observables . compared to the data at rhic energies and below , it seems that simple extrapolation to pb - pb collisions at 2.76 tev is likely to miss some new physics . the charged - particle multiplicity density averaged over @xmath12 for 0 - 5% central collisions is shown in fig . 1 as a function of collision energy @xmath13 @xcite . what is notable is that a straight line can be drawn through all the points in the semilog plot from @xmath14 gev to 200 gev , but at 2.76 tev the lhc data point deviates significantly from the straight - line extrapolation . a power - law fit can be made to connect the rhic and lhc points for @xmath15 gev , as shown in @xcite , resulting in the behavior @xmath16 , but that would overlook the distinctive feature of the lhc point . the dramatic increase above the logarithmic dependence suggests the onset of new physics . another difference between lhc and rhic is the dependence on @xmath0 . from the @xmath0 distributions measured at the two energies , 2.76 and 0.2 tev , we can calculate their ratio . when the data points are not in the same @xmath0 bin , we make lagrangian interpolation between adjacent bins in the rhic data @xcite to match the lhc bin @xcite . the result for pion is shown by the solid ( black ) line in fig . 2 . note the exponential increase by two orders of magnitude as @xmath0 is increased up to 10 gev / c . similar increases are noted for @xmath6 and @xmath5 up to @xmath17 gev / c . the ratios are all around 2 for @xmath18 gev / c , consistent with what we see in fig . 1 where the lhc / rhic ratio of the multiplicity densities at mid - rapidity per participant - pair is @xmath19 . of course , most of the particles contributing to that ratio are pions with @xmath20 gev / c . but for @xmath21 gev / c , there are abundantly more particles produced at lhc than at rhic . it is not unexpected that more high-@xmath0 particles are produced at higher collision energy . the question is what effects do the hard scatterings of partons have on the production of intermediate-@xmath0 hadrons at @xmath22 gev / c . furthermore , it is reasonable to ask whether the physics at low @xmath0 can be treated by hydrodynamics as at rhic , totally decoupled from the physics at high @xmath0 . if jets are copiously produced in order to account for a factor of @xmath23 at @xmath24 gev / c in fig . 2 , why would their fragmentation products not populate the low-@xmath0 region below 2 gev / c ? our knowledge on fragmentation functions derived by leptonic collisions tells us that the distribution of hadronic products increased monotonically with decreasing momentum fraction @xcite . denoting @xmath25 , the ratios @xmath26 vs @xmath0 are shown for @xmath27 . the data are from @xcite . ] finally , we show another plot of data from lhc that is thought provoking . from the @xmath0 distributions of @xmath6 and @xmath5 measured by alice @xcite , we plot their ratio vs @xmath0 as shown by the solid ( black ) line in fig . the general trend is an exponential rise up to the highest available @xmath0 with an increase of a factor of 10 . the conventional understanding of hadrons produced at @xmath28 gev / c is by the fragmentation of hard scattered gluons or light - quarks . however , @xmath29-quark jets are highly suppressed ; moreover , even if an @xmath29 quark is produced at high @xmath0 , its fragmentation into @xmath5 is even more suppressed . to our knowledge it has never been measured , let alone at @xmath7 gev / c . figure 3 shows that the @xmath30 ratio at rhic in dashed ( red ) line also grows exponentially until @xmath31 gev / c and then decreases slowly . the phenomena at both energies are clearly calling for an explanation . vs @xmath0 for central collision at rhic and lhc . the data are from @xcite . ] to calculate the @xmath0 distribution at mid - rapidity for all hadrons , we use the same formalism as described earlier for au - au collisions at 200 gev @xcite and for pb - pb collisions at 2.76 tev @xcite , i.e. , the recombination of thermal and shower partons . we shall use an improved version of the treatment of momentum degradation @xcite and adjust the degradation parameters to fit the lhc data over a wider range of @xmath0 . as a consequence , the study in ref . @xcite for @xmath32 gev / c is superseded because the inclusion of harder jets up to 30 gev / c with less momentum degradation results in a profusion of soft shower partons . furthermore , we shall include also the production of multi - strange hadrons . the high density of shower partons introduces another complication , which is the recombination of partons from different , but adjacent , jets . although that component turns out not to be dominant , its effect is not negligible and must be calculated so as to ascertain its magnitude . the basic framework that describes the recombination of thermal and shower partons at midrapidity is straightforward @xcite @xmath33 f_{q_1q_2q_3}(p_1,p_2,p_3 ) { r}_{q_1q_2q_3}^b(p_1,p_2,p_3,p_t ) \label{32}\end{aligned}\ ] ] the essence is in the details of what the symbols mean . the lhs of eqs . ( [ 31 ] ) and ( [ 32 ] ) are the invariant @xmath0 distributions of meson and baryon , respectively , averaged over @xmath34 at midrapidity and over all @xmath4 . they appear as invariant s in the 1d momentum space , but are derived from the invariant in 3d as follows : @xmath35 with @xmath36 being a narrow interval at @xmath37 , say from @xmath38 to @xmath39 . thus our formalism here is not framed to address the global properties of the nuclear collisions , such as total charge multiplicity or long - range correlation . the parton momenta @xmath40 are the transverse momenta ( with the subscript @xmath41 omitted ) of the coalescing quarks . @xmath42 and @xmath43 are the recombination functions ( rfs ) for meson and baryon , respectively . the central issue in the formalism is the determination of the parton distribution @xmath44 and @xmath45 just before hadronization . because we intend to treat hadron produced in as wide a range in @xmath0 as experimental data on identified particles are available ( for pion up to 20 gev / c ) , we must consider partons that are produced in soft , semihard and hard scatterings . we group them into two classes of partons , thermal ( @xmath46 ) and shower ( @xmath47 ) , and use @xmath48 and @xmath49 to denote their invariant distributions in @xmath40 . taking into account the recombination of different types of partons , we thus have @xmath50 we do not commit ourselves to any specific hydrodynamical description of the soft partons , a position that is made more reasonable when , as will be seen below , low-@xmath0 hadrons can be strongly influenced by shower partons at lhc , thus rendering hydro approach to be inadequate even at low @xmath0 . it dose not mean that we do not recognize the picture that the hot and dense medium created in heavy - ion collision expands . we leave open the issues concerning equilibration time , viscosity , freeze - out dynamics , etc . , since undetermined parameters can not be adjusted to fit the data at low @xmath0 when the effects of shower partons can not be ignored . more specifically , we are concerned with the copious production of hard and semihard jets , whose initiating partons can take up to @xmath51 fm / c to reach the surface , depending on where they are created in the overlapping nuclei and which directions they move in the transverse plane . they can radiate gluons along their in - medium trajectories , and those gluons would take a long time to thermalize with the soft partons in the medium , longer than the short thermalization time @xmath8 fm / c assumed in hydrodynamical treatment . the effects of such hard and semihard partons may be ignored in the soft region if they are rarely produced , as at lower collision energies . but at lhc they are important and can not be neglected . if the basic tenets of hydro are not reliable , the notion of what is thermal must be liberated from the constraints of hydrodynamics . the shower partons generated in the medium interact with the bulk partons , and can not be distinguished from the latter by the time the density of all soft partons is low enough for hadronization . they are all referred to here as thermal partons in the final stage of the quark matter as they move out of the deconfinement phase . the shower partons that we consider are the fragmentation products of the hard and semihard partons that emerge from the surface after momentum degradation . they are distinguished from the thermal partons that are in their environment . those are the @xmath48 and @xmath49 in eqs . ( [ 33 ] ) and ( [ 34 ] ) . we use a simple exponential form to represent the thermal parton distribution @xmath52 with the dimensionless prefactor @xmath53 necessary to yield pure exponential behavior for the pion distribution @xmath54 arising from @xmath55 recombination only , as observed at rhic @xcite . thus @xmath56 has the dimension of inverse momentum . the values of the parameters @xmath56 and @xmath41 wll be discussed below . when shower partons are important at low @xmath0 , then @xmath57 and @xmath58 components need to be included . nevertheless , we retain the form of @xmath59 in eq . ( [ 35 ] ) for the thermal component . the shower parton distribution after integration over jet momentum @xmath60 and summed over all jets is @xmath61 where @xmath62 is the distribution of hard or semihard parton of type @xmath63 at the medium surface after momentum degradation while transversing the medium but before fragmentation . @xmath62 was introduced previously for collisions at rhic for any centrality @xcite , but will be modified below to suit our description of the physics at lhc . @xmath64 is the unintegrated shower - parton distribution ( spd ) in a jet of type @xmath63 fragmentation into a parton of type @xmath65 with momentum fraction @xmath66 . it is determined from the fragmentation function ( ff ) on the basis that hadrons in a jet are formed by recombination of the shower partons in the jet @xcite . in particular , the recombination of a quark @xmath65 with an antiquark @xmath67 in a jet of type @xmath63 forms a pion , for which the ff is @xmath68 . the numerical form for @xmath69 can therefore be calculated from the data on @xmath70 and the rf for pion . the rfs were introduced a long time ago @xcite and have been applied successfully to many collision processes @xcite . here for brevity we give only the rfs for pion and proton , leaving other hadrons to be specified later as the cases arise , @xmath71 where @xmath72 , @xmath73 , @xmath74 , and @xmath75^{-1 } , \label{39}\end{aligned}\ ] ] @xmath76 being the beta function . as a note of affirmation , we recall that with these rfs used in eqs . ( [ 31 ] ) and ( [ 32 ] ) , and considering only the @xmath77 ( @xmath78 ) component for pion ( proton ) , we have been able to fit the pion and proton spectra for @xmath79 gev / c in au - au collisions at 200 gev @xcite with a common value of the inverse slope in eq . ( [ 35 ] ) @xcite . for @xmath18 gev / c there is resonance contribution that eq . ( [ 31 ] ) does not account for , while for @xmath21 gev / c shower parton contributions invalidate the approximation of @xmath80 and @xmath81 by @xmath82 and @xmath78 , respectively . in the @xmath79 gev / c interval one may find the excellent agreement with data surprising , when only the exponential form of eq . ( [ 35 ] ) is used for both pion and proton , since the proton data for @xmath83 is not exponential . however , it is precisely because of the momentum dependence in @xmath84 in eq . ( [ 38 ] ) and the fact that @xmath85 in eq . ( [ 32 ] ) is the transverse mass @xmath86 at @xmath87 that renders @xmath83 to deviate from pure exponential . the phenomenological success there gives strong support to the recombination model . as we shall see below , the situation of dominance by @xmath55 and @xmath88 recombination changes when the collision energy is increased tenfold , whereby @xmath57 and @xmath89 can no longer be neglected . thus the essence of this work is to calculate the effects of the shower partons at low and intermediate @xmath0 region in collisions at lhc . focusing on the shower partons , we see in eq . ( [ 36 ] ) that @xmath62 is the distribution to be determined for collisions at lhc , since @xmath90 is the spd outside the nucleon medium and is independent of the collision system ; it has been determined previously from ffs in vacuum @xcite . at any particular impact parameter @xmath91 , @xmath92 is the average over azimuthal angle @xmath4 of @xmath93 , which has three essential parts @xcite @xmath94 where @xmath95 is the parton density in the phase space @xmath96 at the point of creation , @xmath97 being the initial momentum of the hard or semihard parton @xmath63 , and @xmath98 is the probability for the parton @xmath63 to have a dynamical path length @xmath99 at @xmath4 and @xmath91 . the two parts are connected by @xmath100 @xmath101 which is the momentum degradation function , relating the initial parton momentum @xmath97 to the final momentum @xmath102 at the medium surface by an exponential decay in @xmath99 , the length that carries all the geometrical and dynamical information of the process through @xmath98 . the details of calculating @xmath98 are given in ref . @xcite and summarized in the appendices in ref . we shall recall the essence below in order to re - parametrize it for suitable use at lhc . first , we need to state why we describe momentum degradation in the way outlined above without adopting the results obtained by pqcd in the literature . because we intend to calculate the @xmath0 s of all hadrons from 1 to 20 gev / c , we need to let @xmath102 in eq . ( [ 36 ] ) be integrated from low values in order for the shower partons to have their momenta be as low as 0.5 gev / c . in practice , @xmath102 is integrated from 2 to 30 gev / c . low - order perturbative qcd is not reliable for virtuality less than 8 gev / c , so the major portion of the contribution to the shower partons in the soft region can not make use of the established theory . furthermore , the usual calculation based on dglap evolution equation is on medium modification of the fragmentation function , while we need shower parton for the purpose of recombination . the dependence on the medium is usually described in terms of entropy density and local flow velocity , which are hydrodynamical quantities tuned to fit low-@xmath0 data , which are exactly what we attempt to reproduce in addition to intermediate-@xmath0 data independent of fluid dynamics . for these reasons we use a phenomenological procedure that has been shown to generate the azimuthal and @xmath0 dependencies of @xmath103 at rhic @xcite and can readily be extended to higher energy , as we now proceed to do . the initial momentum distributions have been determined in ref . @xcite for au - au collisions at 200 gev and pb - pb collisions at 5.5 tev . they are parametrized in the form @xmath104 we make logarithmic interpolations of the parameters between the two energies for @xmath105 , @xmath106 and @xmath107 and obtain for @xmath1 tev the parameters shown in table i with @xmath108 . 0.2 in .parameters for @xmath95 in eq . ( [ 43 ] ) . [ cols="^,^,^,^,^,^,^",options="header " , ] with @xmath62 now known explicitly , we can proceed to the calculation of @xmath109 in eq . ( [ 36 ] ) . the spds @xmath90 are derived in refs . @xcite and summarized in @xcite . since the fragmentation of hard and semihard partons into shower partons takes place outside the medium in our treatment , the structure of spds is independent of the collision energy . thus @xmath109 at lhc differs from that at rhic only because @xmath62 is now enhanced , not because of any changes in @xmath90 . while @xmath63 in eq . ( [ 36 ] ) is summed over all parton types listed in table ii , @xmath65 will only be @xmath110 , @xmath111 , @xmath29 and their antiquarks because in our formalism of recombination gluons do not directly participate in hadronization . they are always converted to @xmath112 pairs first , which dress themselves before becoming the constituent quarks of the produced hadrons @xcite . the conversion of gluons to @xmath112 pairs are referred to as enhancing the sea for hadronization at large rapidity @xcite . here at large @xmath0 the same concept of gluon conversion applies , except that instead of enhancing the sea each @xmath102 and @xmath113 can participate in forming a hadron , but in single - particle inclusive distribution only the leading partons with large momentum fractions are considered in the calculation . before showing the result from calculating @xmath109 , we note that in using eq . ( [ 36 ] ) in practice , apart from @xmath102 being integrated from @xmath114 to 30 gev / c , as mentioned earlier , the spd @xmath90 is made to deviate from the scaling form @xmath64 by our insertion of a cutoff factor @xmath115 @xmath116 where @xmath117 such a factor is necessary to render the shower partons meaningful in the soft region , for otherwise the ir divergent ff , @xmath118 , as @xmath119 , would lead to unrealistically large @xmath120 . this point is discussed in appendix c of ref . @xcite , where @xmath115 is denoted by @xmath121 . the value of @xmath122 in eq . ( [ 411 ] ) is chosen so that we can obtain a good fit of the proton spectrum at low @xmath0 , as will be shown in sec . the situation here for lhc is different from that at rhic , where the shower parton are less important than the thermal partons at low @xmath123 , so the precise value of @xmath122 is not significant . at lhc @xmath124 is dominant throughout all @xmath123 so without the cutoff @xmath122 the divergence of @xmath120 as @xmath125 would lead to unrealistically large hadronic for @xmath18 gev / c . by relinquishing our claim for any reliability of our model predictions in the region @xmath18 gev / c , we find that what we can calculate at @xmath126 gev / c is insensitive to the precise value of @xmath122 . we use @xmath127 gev / c just to fit the proton spectrum at @xmath18 gev / c . note that we use the proton distribution as the guide , not pion , because there are resonance and other contributions to the pion distribution at very low @xmath0 . the details will become more clear when the mathematical expressions for recombination are shown explicitly below . is depicted by the dashed ( blue ) line for @xmath128 gev . shower parton distribution @xmath129 is shown in solid ( red ) line with low-@xmath130 cutoff . ] substituting eqs . ( [ 410 ] ) and ( [ 411 ] ) into ( [ 36 ] ) , we obtain the invariant shower - parton distribution @xmath124 after integrating over @xmath102 and summing over all initiating partons @xmath63 . for @xmath131 , it is shown in fig . 5 by the solid ( red ) line , plotted against @xmath123 but labeled as @xmath130 , since it is to be compared to the thermal parton distribution @xmath59 in the same figure . for @xmath59 we use eq . ( [ 35 ] ) with parameters @xmath56 and @xmath41 essentially the same as at rhic , the details of which will discussed in sec . the @xmath59 distribution is shown by the dashed ( blue ) line in fig . evidently , @xmath132 dominates over @xmath59 for all @xmath133 gev / c . hereafter , for the sake of brevity we omit the superscript of quark type @xmath65 in @xmath134 , as we routinely do for @xmath59 , when no confusion is likely to ensue . this is the most remarkable feature about the parton distribution at lhc . although we can not show the phenomenology based on these distribution until later , the dominance of @xmath132 is so important that it reorients our thinking about hadron production at low and intermediate @xmath0 from this point of our discussion onward . in essence , minijets are so copiously produced at lhc that their effects at low @xmath0 can not be ignored , thus posing a substantive question on the meaningfulness of any hydrodynamical study without taking minijets into account . to place fig . 5 in the proper context , we show the ratio @xmath135 by the solid line in fig . it is substantially above 1 for @xmath133 gev / c . for comparison the ratio for the partons at rhic is shown by the dashed line in the same figure . some aspects of the shower partons at rhic are discussed in appendix a. we see in fig . 6(a ) that @xmath135 at lhc is significantly larger than that at rhic . whereas the latter does not exceed 1 until @xmath130 is above 2 gev / c , the former is almost always greater than 1 . since @xmath136 is the same in both , the ratio of @xmath135 at lhc to that at rhic is just @xmath137 , which is shown in fig . 6(b ) , exhibiting a factor of 7 even at @xmath138 gev / c . it is therefore reasonable to draw a connection between the enhancement of shower partons and the increase of average multiplicity in fig . 1 in going from rhic to lhc energies . for lhc and rhic at 0 - 5% centrality . ( b ) the ratio of shower - parton distribution at lhc to that at rhic.,title="fig:",scaledwidth=100.0%][fig6a ] for lhc and rhic at 0 - 5% centrality . ( b ) the ratio of shower - parton distribution at lhc to that at rhic.,title="fig:",scaledwidth=100.0%][fig6b ] before we embark on the actual task of computing the inclusive distributions , we discuss an issue that should arise upon examining fig . we see in that figure that @xmath139 is larger than @xmath136 for all @xmath133 gev / c so one would expect the last terms @xmath140 and @xmath141 in eqs . ( [ 33 ] ) and ( [ 34 ] ) to be more important . however , those equations display only the schematic structure of the various components , and are adequate only as a general layout for use in eqs . ( [ 31 ] ) and ( [ 32 ] ) . kinematic constraints on the shower - parton momenta that will be shown in detail in the next section result in the contribution from @xmath140 and @xmath141 terms to be dominant only in the large @xmath0 region . there is another type of shower - parton recombination that has not been discussed above ; that is the subject of our consideration in this section . in refs . @xcite where @xmath58 recombination is considered , the shower partons arise from the same jet . ( the same applies to @xmath142 for baryons as well , but will not be reiterated . ) such a term is equivalent to fragmentation , since it is from the ff , @xmath143 , that the spds are derived in the first place @xcite . in view of the dominance of @xmath144 over @xmath145 , it is reasonable to expect the integral of @xmath146 to be larger than @xmath147 when convoluted with the same rf , @xmath148 . at this point it is important for us to be more explicit with indices and distinguish one - jet and two - jet recombinations , which we shall denote by @xmath149 and @xmath150 , respectively . in fig . 7 we show the diagrams in the transverse plane for three types of recombination : ( a ) @xmath57 , ( b ) @xmath149 and ( c ) @xmath150 . in the notation of eq . ( [ 42 ] ) , @xmath97 is the momentum of the hard or semihard parton at creation , and @xmath102 is the momentum at the medium surface . the thick red vectors have the dual role of representing the jet momentum in the medium and the degradation effect described by @xmath100 . the thinner red lines outside the medium are the semihard partons @xmath151 , which can emit shower partons represented by the thinnest red lines denoted by @xmath152 . the blue dashed arrows are thermal partons . recombination is represented by a large black blob with the outgoing open arrow depicting the produced pion . we emphasize that the shower parton lines are inclusive in the sense that only the ones contributing to the formation of the observed hadron are shown . in particular , a gluon generates a cluster of partons which can not all be depicted . thus quark types and baryon numbers can not be recognized from the schematic diagrams . furthermore , the lengths and angles of the vectors are not drawn to scale due to the limitation in presenting the figures clearly , and should not be taken literally . note that in fig . 7(a ) and ( b ) the hard or semihard partons are labeled by @xmath63 , while in ( c ) the two partons are labeled by @xmath63 and @xmath153 . therein lies the essential point that @xmath57 and @xmath154 each involves only one jet of type @xmath63 , while @xmath150 involves two jets of types @xmath63 and @xmath153 . thus for @xmath155 and @xmath154 there is only one hard scattering contained in @xmath62 , while for @xmath156 there are two hard scatterings contained separately in @xmath157 . more explicitly , but leaving out integration over @xmath102 and summation over @xmath63 for now ( with full expression to be shown in the next section ) , we have @xmath158 while for @xmath159 we need to retain the @xmath4 variable in @xmath160 before it is averaged over @xmath4 in eq . ( [ 48 ] ) : @xmath161 \bar f_i(q_1 , \phi_1)\bar f_{i'}(q_2 , \phi_2){{s}_i^{q}(p_1 , q_1)\rm { s}_{i'}^{\bar q}(p_2 , q_2)}{\bf r}_{\gamma}^{\pi}(p_1 , \phi_1 , p_2 , \phi_2 , p_t , \phi ) . \label{53}\end{aligned}\ ] ] because there are two initiating hard partons @xmath63 and @xmath153 we need to integrate over their respective azimuthal angels @xmath162 and @xmath163 , allowing the rf @xmath164 to play the role of restricting @xmath162 and @xmath163 to be really equal for the coalescence process to take place . non - parallel partons have large relative momentum transverse to @xmath165 , which should not exceed the binding energy of the constituents of the hadron that it is to be formed . that is different from large relative longitudinal momentum parallel to @xmath166 because in the parton model the momentum fractions of partons in a hadron can vary from 0 to 1 . the azimuthal angles @xmath162 and @xmath163 may be given by a gaussian distribution in @xmath167 with an appropriate width . however , since @xmath162 and @xmath163 are integrated over in eq . ( [ 53 ] ) , it is simpler to adopt a factorizable form that requires the partons to be parallel but with a suitable normalization factor @xmath168 that we can estimate , i.e. , @xmath169 where @xmath168 is the probability that two parallel partons can recombine . since the partons are emitted from the medium at early times , we may consider the emitting system as being a thin almond - shaped overlap region viewed from its side in the same transverse plane at midrapidity as where the pion is detected . for 0 - 5% centrality the almond is almost circular . the partons at @xmath170 are parallel , but can be emitted at any distance from the center of the circle . looking at the emitting source edgewise , it is essentially a one - dimensional system of width approximately 10 fm , which is slightly less than @xmath171 since high - density partons are not likely to be emitted tangentially from the edges . the two parallel partons should be separated by a distance not greater than the diameter of a pion ( @xmath172 fm ) , given that the jets have some width . thus our estimate for @xmath168 is the ratio @xmath173 . we do not see that any more elaborate analysis of the coalescence process can provide a more transparent description of @xmath174 . applying eq . ( 24 ) to ( 23 ) we obtain upon averaging over @xmath4 @xmath175 by comparing this equation with eq . ( 22 ) we see that the 2j contribution has an extra factor of @xmath176 with @xmath123 ranging from 0 to @xmath177 . on the other hand , the symmetrization of the two shower - parton product in the 1j contribution , when expressed in terms of momentum fractions @xmath178 , reveals the ranges @xmath179 , and @xmath180 in the two terms @xmath181 . \label{56}\end{aligned}\ ] ] thus , when two shower partons are in the same jet , the sum of their momenta , @xmath182 , can not exceed the jet momentum @xmath102 . that is the kinematical restriction mentioned in the beginning of this section , and corresponds to the familiar condition that @xmath183 in the ff @xmath184 in eq . ( 22 ) . since the large-@xmath102 dependence of @xmath62 is power - law behaved , as given explicitly in eq . ( [ 49 ] ) , the @xmath185 component dominates at high @xmath0 , where the components involving the thermal partons ( i.e. @xmath55 and @xmath186 ) are damped due to the exponential behavior of @xmath145 . the @xmath150 component involves @xmath187 and @xmath188 in eq . ( [ 55 ] ) so it is suppressed compared to @xmath149 , but by how much requires explicit calculation . to take multi - jet recombination into account for the production of proton , we show more explicitly the terms in eq . ( [ 34 ] ) , but still symbolically , @xmath189}^{2j } + { \cal ( sss)}^{3j } \label{57}\end{aligned}\ ] ] except for the first term that does not involve any @xmath47 , the other six terms are depicted by the six figures in fig . 8 , respectively . the first three figures have only 1-jet and are conventional . figure 8 ( d ) corresponds to eq . ( [ 55 ] ) plus one thermal parton , so the equation for it is @xmath190 the last two figures can easily be obtained by straightforward generalization @xmath191\hat f_i(q_1 ) \large\{s_i^q(p_1 , q_1 ) , s_{i}^{q'}(p_2 , q_1)\large\}\nonumber \\ & & \times \hat f_{i'}(q_2 ) s_{i'}^{q''}(p_3 , q_2)r^p(p_1 , p_2 , p_3 , p_t ) , \label{59 } \\ ( { \widehat{\mathcal { sss}}})^{3j}&=&\gamma^2\int\left[\prod\limits_{a=1}^3\frac{dp_a}{p_a}\hat f_{i_a}(q_a ) s_{i_a}^{q_a}(p_a , q_a)\right]r^p(p_1 , p_2 , p_3 , p_t ) . \label{510}\end{aligned}\ ] ] three - jet recombination is highly suppressed and will be neglected in the following . we now calculate the @xmath0 distributions of @xmath10 and @xmath11 produced at @xmath192 and for 0 - 5% centrality in pb - pb collisions at 2.76 tev . they are based on the essential points discussed in the preceding sections , some of which have previously been applied to collisions at rhic @xcite . now we consider lhc without changing the basic formalism . although we have studied the @xmath0 spectra at lhc before @xcite , it was , however , for a limited range of @xmath0 ( @xmath193 gev / c ) and was based on a simple assumption about momentum degradation , which we have subsequently found to be unrealistic as the @xmath0 range is extended to above 10 gev / c . our present treatment of momentum degradation , discussed in sec . iv , enables us below to reproduce the data up to @xmath194 gev / c , thus superseding the earlier parametrizations in @xcite . nevertheless , we stress by repeating that the basic equations are the same , as summarized in @xcite , except that a new @xmath195 is to be adjusted to fit the data . to be specific we consider the production of @xmath196 @xmath197 , \label{62 } \\ { dn^{{ss}^{1j}}_{\pi}\over p_tdp_t } & = & { 1\over p_t } \int { dq\over q^2 } \sum_i \hat{f}_i(q)d^{\pi}_i(p_t , q ) , \label{63}\\ { dn_{\pi}^{{ss}^{2j}}\over p_tdp_t } & = & { \gamma\over p_t^3 } \int_0^{\pt } dp_1 { \cal s}^{u}(p_1 ) { \cal s}^{\bar d}(p_t - p_1 ) . \label{64 } \end{aligned}\ ] ] while pion mass is neglected above , proton mass is certainly not negligible , so @xmath198 in eq . ( [ 32 ] ) becomes the transverse mass @xmath199 for @xmath200 . with the rf given in eq . ( [ 38 ] ) , we have @xmath201 where @xmath202 , @xmath203 and @xmath107 being given after eq . ( [ 38 ] ) , and @xmath204 @xmath205 @xmath206 where @xmath207 in eq . ( [ 67 ] ) is @xmath208 equations ( [ 66])-([68 ] ) correspond to fig . 8(a)-(c ) . for 2-jet contributions in fig . 8(d ) and ( e ) we have @xmath209 @xmath210 the above equations describe the production of pion and proton in the recombination model for hadronization at the final stage of the nuclear collision process where the medium density is low . since thermal partons represent the properties of the bulk medium at hadronization irrespective of the initiating system , we use for the normalization factor @xmath56 and inverse slope @xmath41 in eq . ( [ 35 ] ) the same values as at rhic @xcite @xmath211 to justify the use of these values for collisions at lhc , we recall first that in our treatment of hadronization the thermal distributions @xmath212 is not what can be derived from hydro studies . at rhic it is determined by fitting the pion distribution at @xmath213 gev / c . using eqs . ( [ 35 ] ) and ( [ 37 ] ) in ( [ 31 ] ) one obtains ( [ 61 ] ) for tt recombination only , which yields the values of @xmath56 and @xmath41 in eq . ( [ 612 ] ) in order to reproduce the pion data at low @xmath0 @xcite , as can be seen in fig . 17 in appendix a below . as mentioned earlier in sec . iii , the thermal partons include the soft partons generated by hard and semihard partons as they traverse the medium and have thermalized with the bulk partons by the end of the deconfined phase . when those thermal partons are dilute enough and be ready for confinement through recombination , their local properties are no longer sensitive to the collisional system in which the medium is created initially . the concept is consistent with the notion of universal hadrosynthesis where statistical study of hadron ratios has found universality independent of collision energy , analogous to water vapor condensing at 100@xmath214c independent of how hot it has previously been . @xmath56 and @xmath41 are local measures that carry no information of the global properties , such as rapidity range and overall multiplicities , which depend on the collision energy . the s we study are at mid - rapidity , so the increase of total multiplicity due largely to the broadening of the rapidity plateau is not of concern here . our interest is in the increase of @xmath215 which we claim is related to the increase of @xmath216 by demonstrating that the observed spectra can be reproduced in the rm . the thermal @xmath217 was determined at rhic for low @xmath130 where @xmath218 is negligible ; that same @xmath217 is now used at lhc . in appendix b it is shown that the use of any values of @xmath56 and @xmath41 different from eq . ( [ 612 ] ) fails to reproduce the data at all @xmath0 . we remark , parenthetically , that the value of @xmath56 above corresponds very well to the formula in ref . @xcite that gives the centrality dependence @xmath219 wherein we use @xmath220 for 0 - 5% in pb - pb collisions @xcite . it is reasonable to question why @xmath56 should remain the same as at rhic , when more partons are produced at lhc , even though @xmath41 is the same at hadronization . our answer is that our formalism is inadequate to treat accurately the hadron formation at very low @xmath0 for @xmath18 gev / c . the values of @xmath56 and @xmath41 in eq . ( [ 612 ] ) are used for calculating the spectra for @xmath126 gev / c . at lower @xmath0 our pion distribution is lower than the data , which is undoubtedly related to the extra low-@xmath130 partons created at lhc that we can not easily include in our parametrization . besides , there are resonance contribution to the pion spectrum that we have not counted for . we recall that in order to tame the soft shower parton distributions from minijets we need to introduce a cut - off parameter @xmath122 in the spd @xmath90 in eq . ( [ 49 ] ) . the value of @xmath122 is determined mainly by keeping the proton distribution under bound for @xmath18 gev / c , since pions have resonance and other contributions mentioned above that are not included in eqs . ( [ 61])-([63 ] ) . nevertheless , the dependence on @xmath122 is not sensitive ; its value at 0.5 gev / c is essentially chosen as a reasonable value . such a cutoff in the shower parton @xmath221 for @xmath222 gev / c can not affect the outcome of the dominant tts contribution in the @xmath223 gev / c ( to be seen in fig . 10 below ) because at small @xmath224 we see in eq . ( [ 66 ] ) that @xmath225 must be greater than 0.5 gev / c so the integral is suppressed by the exponential factor @xmath226 in the integrand . the other parameters @xmath227 and @xmath228 in eq . ( [ 47 ] ) for the @xmath102-dependent gluon degradation factor @xmath229 are crucial in our attempt to find a good fit of both @xmath230 and @xmath6 distributions at all @xmath0 up to 20 gev / c . that makes good sense in physics since the degradation of hard- and semihard - parton momenta is the central theme of heavy - ion physics at lhc . our study here reveals how important minijets are in explaining the hadron spectra at all @xmath0 observed . with the choice @xmath231 we calculate the pion distribution for @xmath232 gev / c and obtain the different components shown in fig . 9 by different line types , although only the region @xmath126 gev / c is reliable . their sum in black - cross line agrees with data from alice @xcite very well for @xmath126 gev / c . the solid black line includes what we can not calculate and is put in by hand to raise the distribution to fit the data at @xmath18 gev / c . we note that @xmath57 is larger than @xmath55 for @xmath126 gev / c . the total goes below the data points at @xmath233 gev / c . some further adjustment of @xmath229 at very high @xmath102 can repair that deficiency by raising @xmath234 there , but that much fine tuning is not our interest here since our focus is on the interplay among the different components at low and intermediate @xmath0 . the 2-jet component @xmath235 is too small to be significant ; nevertheless , it is interesting to observe that @xmath235 has very nearly the same magnitude as @xmath234 at @xmath236 gev / c . that is not the situation at rhic , as can be seen in fig . 17 in appendix a , where @xmath235 is much less than @xmath234 at all @xmath0 . the difference owes its origin to the relative sizes of @xmath144 shown in fig . 6(a ) and ( b ) . since recombination is dominated by shower partons in the dense region , i.e. , at low @xmath130 , two such partons from nearby jets can contribute as much as from a single jet . data are from @xcite for centrality 0 - 5% . ] data are from @xcite for centrality 0 - 5% . ] without changing any parameter we calculate the proton distribution that is shown in fig . . it also agrees with the data @xcite extremely well . note that @xmath89 , @xmath237 , @xmath238 and @xmath239 components are all of similar magnitudes at @xmath240 gev / c ; together they lift the total to meet the data points . that is a feature that is unique among the hadronization models . as with the pion distribution , @xmath89 is larger than @xmath88 for @xmath126 gev / c , demonstrating again that the soft shower partons play an important role at low @xmath0 . furthermore , one sees that @xmath241 around @xmath31 gev / c just as @xmath242 for pions , although they are all much less than @xmath89 and @xmath57 , respectively . with the results shown in fig . 9 and 10 we regard our main objective as having been accomplished . it is non - trivial to reproduce the data in such a wide range of @xmath0 and it is remarkable that the main input that is adjustable is just the momentum degradation factor @xmath229 in eq . ( [ 47 ] ) . what we have obtained for @xmath227 and @xmath228 in eq . ( [ 614 ] ) are good not only for @xmath230 and @xmath6 distributions , but also for all other particles , as we shall show below . thus the result strongly supports the assertion that minijet production plays the dominate role in the structure of hadronic spectra . the corresponding shower partons have been exhibited already in fig . 5 together with discussions on their dominance over thermal partons for nearly all @xmath130 . proceeding to the production of strange particles , we use the same formalism as for pion and proton , except that @xmath29 quark being more massive than the light quarks requires separate attention . for the thermal @xmath29 quarks we use the same distribution as in eq . ( [ 35 ] ) @xmath244 but with a different inverse slope @xmath245 , which is the only parameter we adjust to fit the data . since the @xmath29 quark mass , @xmath246 , does not appear explicitly in eq . ( [ 615 ] ) , and also since @xmath245 may be regarded as an effective temperature at the time of hadronization , the fluid velocity may raise @xmath245 above @xmath41 ( for light quarks ) . the @xmath29 shower parton distribution @xmath247 is as given in eq . ( [ 36 ] ) with the unintegrated spd @xmath64 determined from the ffs into @xmath243 and @xmath11 @xcite . the degradation of @xmath29-quark momentum is taken to be the same as others , i.e. , @xmath248 . with the rf for kaon given in ref . @xcite we have for the @xmath249 distributions @xmath250 , \label{617 } \\ { dn^{{ss}^{1j}}_{k}\over p_tdp_t } & = & { 1\over m^k_t } \int { dq\over q^2 } \sum_i \hat{f}_i(q)d^{k}_i(p_t , q ) , \label{618}\\ { dn_{k}^{{ss}^{2j}}\over p_tdp_t } & = & { 12\gamma\over m_t^kp_t^5 } \int_0^{\pt } dp_1 p_1(\pt - p_1)^2 { \cal s}^{u}(p_1 ) { \cal s}^{\bar s}(p_t - p_1 ) . \label{619 } \end{aligned}\ ] ] with @xmath245 being the only adjustable parameter we obtain for @xmath251 the distribution shown in fig . 11 . evidently , the data from alice @xcite are well reproduced . the value of @xmath245 is slightly higher than @xmath41 in eq . ( [ 612 ] ) . as it is with pions , the @xmath57 components is greater than @xmath55 for @xmath252 gev / c . although @xmath253 is suppressed relative to @xmath254 , the @xmath255 recombination sustains the @xmath57 component . however , @xmath256 is clearly much lower than that for pion in fig . 9 at low @xmath0 . note that @xmath257 is again very close to @xmath256 at @xmath258 gev / c . for @xmath11 production we use eq . ( [ 615 ] ) again for the thermal @xmath29 quarks , but allow @xmath245 to be different from the value in eq . ( [ 620 ] ) . appendix c contains the explicit distributions of the various components . with the choice @xmath259 we obtain the result shown in fig . the data @xcite are reproduced very well . the physics is clearly very much the same as for @xmath260 and @xmath243 . the value of @xmath261 is higher because @xmath262 is higher , although how the thermal partons depend on the quark mass is not specified explicitly . we stress that the momentum degradation parameters have not been adjusted so the hard parton and minijet distributions @xmath62 are the same as described in sec . iv , independent of the hadrons produced . thus the recombination model has enabled us to calculate the spectra of all strange and non - strange hadron at all @xmath0 in a universal formalism . data are from @xcite for centrality 0 - 5% . ] produced in pb - pb collision at @xmath1 tev . data are from @xcite for centrality 0 - 5% . ] we complete our investigation of hadron production by considering @xmath3 , @xmath5 and @xmath4 . apart from different quark contents of those particles , the physics of hadronization through recombination is the same as before . since they can not be used either as target on beam particles , their wave functions in terms of momentum fractions of constituent quarks are not known as firmly as we do with @xmath263 and @xmath6 . furthermore , there is the question of the probability for more than one strange quark to find one another to recombine . as the system expands , the plasma gets out of chemical equilibrium first as the temperature is lowered because @xmath264 and @xmath265 processes become less frequent than their reverses on account of @xmath266 . thus the density of @xmath29 quarks becomes lower . the language used above is that of the conventional interpretation of the expanding medium getting out of chemical equilibrium . we need not subscribe to the details of that description , while still adhering to the qualitative physical picture of the system that has general validity . thus we proceed in the same manner as we have for @xmath230 and @xmath6 . for a single @xmath29 quark to hadronize at late time there are abundant light quarks in the neighborhood to form @xmath243 and @xmath11 with . however , for multi - strange hadron to form , the probability of @xmath267 , @xmath268 or @xmath269 to be in close proximity of one another at late time is reduced , when the density of @xmath29 quark is lower than that of light quarks . if at earlier time @xmath3 , @xmath5 and @xmath4 are formed at higher density , their survival in the medium is suppressed due to their dissociation through interaction with the plasma that is still active . thus in either case the rate of multi - strange hadron production is lower . we can not predict that rate in the recombination model , so an adjustable parameter will be used to fit the overall normalization ; that is in addition to the inverse slope @xmath245 , since each particle has it own hadronization time and mass effect on the effective temperature . on the other hand , the density of shower partons arising from hard and semihard partons is independent of the final hadrons formed , so we can still use our formalism to calculate the various components of the @xmath0 distributions . the detail equations for @xmath3 and @xmath5 formations are given in appendices d and e , respectively . the only free parameters we use in each case are @xmath270 and @xmath245 . for best fit we obtain @xmath271 @xmath272 the results are shown in figs . 13 and 14 , reproducing the data very well . there are , however , some differences in the strengths of different components , even though the shower partons are the same in all cases . produced in pb - pb collision at @xmath1 tev . data are from @xcite for centrality 0 - 10% . ] produced in pb - pb collision at @xmath1 tev . data are from @xcite for centrality 0 - 10% . ] what is most noticeable about the @xmath3 distributions is that the @xmath273 component dominates the whole spectrum for @xmath126 gev / c and that @xmath274 and @xmath275 components are much lower . the relative strengths of those components are unlike the situation with proton and @xmath11 . whereas the @xmath47 in @xmath273 can be non - strange , @xmath274 must have at least one @xmath29 in the @xmath58 , and @xmath275 must have two @xmath29 quarks . since @xmath276 is suppressed compared to @xmath277 , the ordering of @xmath273 , @xmath274 and @xmath275 is evident in fig . 13 . moreover , @xmath278 and @xmath279 have roughly the same magnitude ; so also do @xmath280 and @xmath281 . for @xmath5 production shown in fig . 14 , similar remarks about the ordering of the various components can be made as for @xmath3 . one notable difference is that this time even @xmath273 is suppressed relative to @xmath282 . that is because every coalescing quark for @xmath5 must be strange , so @xmath276 in @xmath273 lowers its magnitude relative to @xmath282 . herein lies a very interesting point that was noticed several years ago even in rhic data @xcite the @xmath0 distribution of @xmath5 is exponential ( apart from the prefactor @xmath283 in eq . ( [ d1 ] ) ) without any power - law up - bending at high @xmath0 . it means that @xmath5 is produced thermally even at @xmath17 gev / c without any contribution from parton fragmentation , which is the usual mechanism considered in pqcd . neither can hydrodynamics be applied to particle production at such high @xmath0 . in recombination each @xmath29 quark need only be at @xmath213 gev / c on the average . our thermal partons at @xmath284 gev / c imply that @xmath5 is formed earlier than other hyperons . in fact , it is of interest to exhibit the dependence of @xmath245 on the number @xmath285 of @xmath29-quark content of the hyperons . figure 15 shows that there is a linear increase from @xmath11 to @xmath5 , and therefore non - linear if plotted against the hyperon masses , since @xmath286 on the number @xmath285 of strange quarks in hyperons . ] a comparison between figs . 10 and 14 reveals the drastic difference in the compositions of the various components contributing to @xmath6 and @xmath5 . for @xmath6 the all thermal ttt component is unimportant compared to tts , tss and sss , while for @xmath5 ttt is the only dominant component . if we were to compare only the ttt components in @xmath6 and @xmath5 , then their ratio @xmath287 would be exponentially rising in @xmath0 . using @xmath128 gev and @xmath284 gev , that ratio rises by 3 orders of magnitude if only the exponential factors are considered with neglect of the multiplicative factors . in reality , as we have seen in fig . 3 , the ratio for lhc rises by only a factor of 10 . the reason is , of course , the dominance of the @xmath102 shower partons in the production of proton , as evident from fig . 10 , where fragmentation is not important until @xmath288 gev / c . on the other hand , the @xmath29 shower partons are unimportant for the production of @xmath5 , which can adequately be described by the exponential behavior of ttt alone . in fig . 3 we have noted the difference between lhc and rhic in the @xmath0 dependencies of @xmath30 . while @xmath5 production at rhic is also mainly ttt and thus exponential @xcite , fig . 18 in appendix a shows that the @xmath0 distribution for proton at rhic has a transition from ttt to tts in the region @xmath289 gev / c . that accounts for the saturation of @xmath30 in that region in fig . that transition is absent in fig . 10 for lhc , hence no saturation seen for lhc in fig . 3 . all these inter - related phenomena can be traced to the simple source , namely : @xmath102 shower partons are abundant at lhc , not @xmath29 shower partons . lastly , we consider the production of @xmath4 , for which the equations are given in appendix f. since no light quarks are involved in the formation of both @xmath5 and @xmath4 , we use the same value of @xmath245 for both , i.e. , @xmath284 gev . by varying @xmath290 only for the overall normalization , we obtain the result shown in fig . 16 for @xmath291 . the underlying components are very similar to those for @xmath5 , namely : tt dominates over ts , while ss ( whether 1j or 2j ) is nearly 2 orders of magnitudes farther down . produced in pb - pb collision at @xmath1 tev . data are from @xcite for centrality 0 - 10% . ] the small value of @xmath292 is an indication of quarkonium suppression after @xmath4 is formed at a time much earlier than @xmath230 , when the density of @xmath29 ( and @xmath293 ) is higher . as is the case with @xmath294 suppression , @xmath4 experiences the effects of dissociation by the plasma as it traverses the remaining portion of the medium before it completely hadronizes . the value of @xmath292 depends on aspects of the process that are not included in the formalism discussed in this paper , and therefore can not be predicted . the same remarks can be made for the formation of @xmath3 and @xmath5 , for which @xmath295 and @xmath296 are quite small in eqs . ( [ 71 ] ) and ( [ 72 ] ) . we have made a thorough study of the production of all identified hadrons in pb - pb collisions at lhc in a formalism that displays all the components of thermal- and shower - parton recombination . the degradation of momenta of hard and semihard partons is treated in a way that uses two free parameters , which are determined by fitting the high-@xmath0 distribution of the pion . the resultant shower - parton distributions of @xmath102 and @xmath29 quarks are then used to calculate the spectra of all hadrons ( @xmath2 , @xmath3 , @xmath5 and @xmath4 ) . they agree well with the data for all @xmath0 up to 20 gev / c . the description not only establishes a consistent scheme for treating the hadronization process of a quark - gluon plasma at lhc , but also points out the importance of the effects of minijets on the pion and proton distributions at low and intermediate @xmath0 yet not at all on the @xmath4 and @xmath5 distributions on the other end of the spectrum in strangeness content . the dominance of light shower partons over the thermal partons in nearly the whole range of parton momenta is an observation we make on the basis of an adopted form of the thermal parton distribution @xmath145 . while the shower parton distribution @xmath144 can be calculated , we have no dynamical scheme to calculate @xmath145 , which is at the final stage of the evolution of the dense medium , dilute enough to enter into the confinement process . since hadronization is insensitive to the initial process in which the dense medium is created , we have used the @xmath145 determined at rhic , where thermal partons dominate the low-@xmath0 region of all particles produced . the use of that @xmath145 for our treatment at lhc is justified by the fact that alice data @xmath230 and @xmath6 distributions at low @xmath0 are well reproduced by our results in which @xmath155 and @xmath273 components dominate . any more ( or less ) thermal partons would not have resulted in satisfactory fits of the low-@xmath0 data , since the density of soft shower partons is constrained by the fragmentation of hard and semihard jets . it is therefore meaningful to compare @xmath144 with @xmath145 and arrive at the conclusion that there are far more soft shower partons than thermal parton at lhc . it then follows that any theoretical treatment of hadrons produced at low @xmath0 would be incomplete without taking the effects of minijets into account . in particular , the parameters in the hydrodynamical formalism can not be determined by phenomenology in the soft sector without including also the soft partons from minijets . it may be of interest to mention here that there is a phenomenological two - component model , in which the hard component exerts a strong influence on the production of pions in the low-@xmath0 region to the extent that the validity of hydrodynamical treatment of soft hadrons is questioned @xcite . although the physical basis for that observation may share some common ground with what we have found here ( despite the very different languages and concepts used ) , it should be emphasized that our shower partons are dominant only at lhc , whereas ref . @xcite contends the importance of the hard component in the soft region even at rhic . the dominance of @xmath132 over @xmath59 for production of @xmath230 and @xmath6 does not apply to @xmath4 and @xmath5 . the @xmath29 quarks in the shower are suppressed , so @xmath155 and @xmath273 are lower than @xmath297 and @xmath282 , respectively . the other particles ( @xmath298 and @xmath3 ) with less strangeness contents are in the intermediate situation . the recombination of thermal partons as the mechanism for the production of @xmath4 and @xmath5 is therefore a satisfactory explanation for their @xmath0 distributions up to 6.5 gev / c that is too high for hydrodynamics and too abundantly produced for fragmentation . a serious consequence of our conclusion about shower partons dominating over thermal partons is its implication on azimuthal anisotropy in non - central collisions . the usual explanation is that the azimuthal harmonics are due to the flow effects of the fluctuations of the initial configuration of the collision system . if , however , the non - flow effects such as minijets are important , the fluid treatment would be inadequate on the one hand , and our approach is in need of suitable treatment to be convincing on the other . for au - au collisions at 200 gev , we have shown that the azimuthal harmonics can be obtained by taking into account the azimuthal dependence of minijet and the related ridge effect @xcite . now for pb - pb collisions at 2.76 tev we have only investigated the case of central collisions here . to extend the study to non - central collisions is , of course , the natural problem to pursue next . how minijets influence the azimuthal asymmetry will undoubtedly be a major area of investigation . the consideration described here represents only the first , but significant , step toward understanding the physics of hadronization at lhc . this work was supported , in part , by the nsfc of china under grant no . 11205106 and by the u. s. department of energy under grant no . de - fg02 - 96er40972 . although the problem of hadron production at rhic has been extensively studied previously @xcite , we have made progressive improvement on the treatment of momentum degradation . in order to make sensible comparison between lhc and rhic results , we recalculated here the pion and proton distributions at rhic , using the same description of the effects of energy loss on the shower partons , as has been done in sec . data are from @xcite for centrality 0 - 10% . ] data are from @xcite for centrality 0 - 10% . ] the basic difference between what we do now and what was done in ref . @xcite is that @xmath229 is @xmath102 dependent as given in eq . ( [ 47 ] ) . keeping @xmath128 gev as in eq . ( [ 612 ] ) , as well as in ref . @xcite , we vary @xmath227 to find the best fit of the @xmath230 distribution in au - au collisions at 200 gev for 0 - 10% centrality , with @xmath299 gev / c fixed , as in eq . ( [ 614 ] ) . the initial parton distribution @xmath95 are as given in ref . @xcite , and the recombination equation are the same as those in sec . vi . with @xmath300 , we obtain the results shown in fig . 17 for pion and fig . 18 for proton , which are evidently very good . comparing fig . 17 to the pion distribution at lhc in fig . 9 , one can see the drastic difference in @xmath57 relative to @xmath55 between the two cases . at rhic @xmath57 crosses @xmath55 at @xmath301 gev / c , whereas at lhc it occurs at @xmath302 gev / c . the latter is a consequence of @xmath303 for @xmath133 gev / c , shown in fig . 5 . in contrast , at rhic that cross - over does not occur until @xmath304 gev / c , as shown in fig . 19 . the ratio of @xmath305 is already previewed in fig . 6(a ) . thus at rhic @xmath132 is a factor of 7 lower than that at lhc for @xmath306 gev / c . the low density of shower partons makes the hydrodynamical treatment of thermal partons to be sensible without concern for minijets , which is not the case at lhc . for au - au collisions at @xmath307 gev is depicted by the dashed ( blue ) line for @xmath128 gev , while the shower parton distribution @xmath129 is shown by the solid ( red ) line with low-@xmath130 cutoff . ] the thermal parton is given in eq . ( [ 35 ] ) and the parameters @xmath56 and @xmath41 are given in eq . ( [ 612 ] ) . section iv - a contains extensive discussion on why the thermal @xmath217 remains the same at lhc as it is at rhic . in short , at late time when the bulk system is ready for hadronization its local properties at midrapidity are insensitive to its early history , except in very low @xmath130 region ( @xmath308 gev / c ) where the enhanced thermal partons due to the energy lost by the semihard partons to the medium becoming even more enhanced at lhc . in this appendix we show that different sets of higher values of @xmath56 and @xmath41 lead to s of @xmath230 and @xmath6 that are unacceptable for @xmath126 gev / c . all equations we use to calculate the pion and proton spectra are as before , namely : eq . ( [ 35 ] ) for @xmath217 , ( [ 36 ] ) for @xmath309 , ( [ 61])-([64 ] ) for @xmath310 , and ( [ 65])-([611 ] ) for @xmath311 . the only changes are in the parameters @xmath56 and @xmath41 . for our demonstration here , we use the four combinations of @xmath312 and 30 gev / c@xmath313 , and @xmath128 and 0.4 gev / c . the results are shown in figs . 20 and 21 . the solid black lines are the ones corresponding to the universal values of @xmath56 and @xmath41 given in eq . ( [ 612 ] ) . the other three lines are for larger values of either @xmath56 , or @xmath41 , or both . evidently , all of them far exceed the data in the range shown and must be rejected . we have not exhibited the components @xmath314 etc . for each case for the sake of clarity ; however , it is obvious that the thermal - shower recombination raises the contribution at intermediate significantly above the data when @xmath217 is increased . we have not changed @xmath315 so ss and sss terms are not affected and remain the only dominant terms when is high enough . our conclusion is therefore that with the shower parton @xmath315 fixed by the phenomenology at high , only the thermal parton described by @xmath56 and @xmath41 given in eqs . ( [ 35 ] ) and ( [ 612 ] ) can reproduce the spectra of @xmath230 and @xmath6 for @xmath126 gev / c . and @xmath41 . ] and @xmath41 . ] the @xmath0 distribution of @xmath11 is very similar to that of proton except for the replacement of a @xmath110 quark by an @xmath29 quark . the thermal and shower parton distributions for @xmath29 are different from those for @xmath110 , and the rf for @xmath11 is different from that for @xmath6 . for @xmath316 we use the same form as eq . ( [ 615 ] ) , but allow @xmath245 to be adjustable . @xmath247 is the same as used for @xmath243 production in sec . the rf for @xmath11 has the same form as eq . ( [ 38 ] ) for proton but with @xmath317 and @xmath318 in a problem on strange particle production at rhic considered in ref . we simply list the equation below for the various components . @xmath319 @xmath320 @xmath321 @xmath322 @xmath323 @xmath324 the statistical factor is @xmath325 , and the prefactor from rf is @xmath326^{-1}$ ] . the corresponding ffs , @xmath327 , are given by akk @xcite by fitting the data at next - leading - order ( nlo ) . for the recombination of @xmath328 to form @xmath3 we make the simplifying assumption that the rf is proportional to @xmath329-functions , i.e. , @xmath330 . then it is straightforward to write the distributions @xmath331 @xmath332 @xmath333 @xmath334 @xmath335 @xmath336 where @xmath337 with the rf for @xmath5 assumed to be @xmath330 , as it is for @xmath3 , the distributions for the different components are simplest of all baryons , since all constituent quarks are the same . we have @xmath338 @xmath339 @xmath340 @xmath341 @xmath342 @xmath343 apart from the prefactor that involves @xmath344 , the @xmath282 term is a pure exponential . if it is dominant , then the @xmath0 dependence of eq . ( [ d1 ] ) is a direct test of the validity of our description of @xmath5 production . as it is for @xmath5 , the distributions for @xmath4 is simple when the rf is taken to be @xmath345 for @xmath269 recombination . one gets a. adare _ et al . _ ( phenix collaboration ) , phys . * 101 * , 232301 ( 2008 ) ; phys . c * 87 * , 034911 ( 2013 ) ; phys . c * 88 * , 024906(2013 ) . s. s. adler , _ et al _ ( phenix collaboration ) , phys . lett * 91 * , 072301(2003 ) ; phys . c * 69 * , 034909(2004 ) .
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the production of all identified hadrons at the cern large hadron collider ( lhc ) is studied with emphasis on the @xmath0 distributions up to 20 gev / c in central collisions . in the framework of the recombination model
we find that the shower partons ( due to the fragmentation of semihard partons ) play an important role in the formation of hadrons in the low- and intermediate-@xmath0 regions .
parameters that control the energy loss of minijets are determined by fitting the upper half of the @xmath0 range of the pion distribution .
the resultant soft shower partons are then found to dominate over the thermal partons in the non - strange sector , but not in the strange sector .
since the data on the @xmath0 spectra of all observed hadrons are well reproduced , there is no way out of the implication that any alternative dynamical model on particle production would be incomplete if it does not consider the effects of minijets even at very low @xmath0 .
hydrodynamics that relies on rapid equilibration without accounting for the delayed thermalization effects of the hard and semihard partons copiously produced at lhc is an example of such models .
the difference between the densities of shower partons produced at lhc and at bnl relativistic heavy - ion collider ( rhic ) is quantified and discussed .
| 21,932 | 345 |
during the first stages of star formation , highly collimated jets from new born stars influence the physical structure of the hosting cloud by sweeping up material , compressing and accelerating the surrounding environment . the propagation of high velocity outflows generates shock fronts triggering endothermic chemical reactions and ice grain mantle sublimation or sputtering . at a distance of 250 pc ( looney et al . 2007 ) , the chemically rich l1157 bipolar outflow ( bachiller & prez gutirrez 1997 , hereafter bp97 , bachiller et al . 2001 ) is an ideal laboratory to observe the effects of such shocks on the gas chemistry . l1157 is driven by a low - mass ( @xmath2 4 @xmath14 ) class 0 protostar l1157-mm and it is associated with several blue - shifted ( b0 , b1 , b2 ) and red - shifted ( r0 , r1 , r2 ) shocks at different ages ( see fig . [ maps]top panel ) , and seen in both co ( gueth et al . 1996 , 1998 ) , and ir h@xmath0 ( e.g. neufeld et al . 1994 , nisini et al . these shocks ( see fig . [ maps]bottom panel ) , when mapped with interferometers , reveal a clumpy bow structure ( e.g. tafalla & bachiller 1995 ; benedettini et al . 2007 ; codella et al . 2009 ) at the apex of different molecular cavities , corresponding to different mass loss episodes ( gueth et al . 1996 ) . both interferometer and single - dish surveys confirm that the l1157 outflow is well traced by molecules thought to be released off from the dust mantles such as h@xmath0co , ch@xmath15oh , h@xmath0o , and nh@xmath15 ( e.g. codella et al . 2010 , lefloch et al . 2010 , vasta et al . 2012 ) as well as by the refractory grain cores such as sio ( e.g. nisini et al . 2007 ; gusdorf et al . 2008 ) . the abundance of these neutral molecules are enhanced , and the emission shows broad wings ( up to 2030 km s@xmath16 ) . on the contrary , diazenylium ( n@xmath0h@xmath17 ) , usually used as tracer of cold prestellar cores ( e.g. caselli et al . 2002 ) , shows a completely different behaviour . single - dish ( iram 30-m ) and interferometric ( iram pdb , sma , carma ) observations indicate that n@xmath0h@xmath17 traces only the central condensation l1157-mm through narrow ( 0.41.0 km s@xmath16 ) emission and it has not been observed , to date , towards the outflow component ( bachiller et al . 2001 , chiang et al . 2010 , tobin et al . 2011 , 2012 , 2013 , yamaguchi et al . the interferometric maps show that the narrow n@xmath0h@xmath17 line traces the protostellar envelope elongated along a direction perpendicular to the outflow axis ( i.e. along a hypothetical disk ) . however , by analysing their iram pdb data , tobin et al . ( 2011 ) concluded that although the overall n@xmath0h@xmath17 velocity structure is unaffected by the outflow , the morphology of the sligthly blue - shifted emission ( @xmath18@xmath19@xmath20@xmath21 @xmath22 0.8 km s@xmath16 ) outlines the outflow cavity walls in the inner 20@xmath730@xmath7 protostellar environment . tobin et al . ( 2011 ) proposed that such emission is due either to outflow entrainment or to a hypothetical shock near the driving protostar . the same suggestion is found in the atca n@xmath0h@xmath17(10 ) image of the protostellar core cg30 by chen et al . ( 2008 ) . on the other hand , j@xmath23rgensen et al . ( 2004 ) investigated with bima the protostellar binary ngc1333-iras2a - b at 3 mm showing that the spatial distribution of n@xmath0h@xmath17 peaks towards the nearby starless core iras2c , and is missing in the outflows . therefore , it is still under debate what role , if any , n@xmath0h@xmath17 is playing in a shocked gas scenario : is the n@xmath0h@xmath17 emission observed by tobin et al . ( 2011 ) and that marks the cavity opened up by the outflow due to just enhanced gas column density or really associated with a shock ? such question is important , given that n@xmath0h@xmath17 is considered a standard molecular tracer of cold and quiescent prestellar environments ( e.g. tafalla et al . 2006 ) . in order to uniquely answer these questions it is essential to study a region @xmath24 associated with a protostar , as the young ( 2000 years ; gueth et al . 1996 ) , and bright bow - shock l1157-b1 , located at @xmath2 69@xmath7 ( @xmath2 0.1 pc , see fig . [ maps ] ) from the protostar . as part of the herschel key program chess ( chemical herschel surveys of star forming regions ; ceccarelli et al . 2010 ) , l1157-b1 is currently being investigated with a spectral survey in the @xmath280@xmath25350 ghz interval using the iram 30-m telescope ( lefloch et al . in preparation ) , and in the @xmath2500@xmath252000 ghz range using the herschel hifi instrument ( de graauw et al . we present here the first unambiguous detection of n@xmath0h@xmath17 emission towards a protostellar shock : the observed broad emission has been modeled using a simple pseudo - time dependent chemical model , showing how n@xmath0h@xmath17 can be used to shed light on the chemical history of the pre - shock gas . the n@xmath0h@xmath17(1@xmath260 ) line at 93173.76 mhz,@xmath27 = 2,31,2 . ] was observed towards l1157-b1 with the iram 30-m telescope at pico veleta ( spain ) . the pointed coordinates were @xmath28 = 20@xmath29 39@xmath30 10@xmath312 , @xmath32 = + 68@xmath33 01@xmath34 10@xmath355 , i.e. at @xmath36 = + 25@xmath356 and @xmath37 = 63@xmath355 from the driving protostar . the iram survey was performed during several runs in 2011 and 2012 , using the broad - band emir receivers and the fts spectrometer in its 200 khz resolution mode , corresponding to a velocity resolution of 0.6 km s@xmath16 at 93.2 ghz . the main - beam efficiency ( @xmath38 ) was 0.75 , while the hpbw is 26@xmath7 . all the spectra are reported here in units of main beam temperature ( t@xmath39 ) . figure [ n2h+ ] shows the n@xmath0h@xmath17(1@xmath250 ) spectrum : thanks to the high sensitivity of the iram - emir receiver ( r.m.s . = 2 mk after smoothing the spectrum to 1.3 km s@xmath16 ) , we are able to detect the three main groups of hyperfine components of the @xmath3 = 1@xmath260 transition . the integrated intensity is 327@xmath4014 mk km s@xmath16 . the n@xmath0h@xmath17 emission in l1157-b1 was hidden in the noise of the bp97 spectrum , which has 1@xmath41 rms of 20 mk , definitely larger than that of the present dataset ( 2 mk ) . n@xmath0h@xmath17 is a linear molecular ion in a stable closed - shell @xmath42@xmath43 configuration . the dominant hyperfine interactions are those between the molecular electric field gradient and the electric quadrupole moments of the two nitrogen nuclei ( e.g. caselli et al . 1995 ) , producing a splitting of the j = 10 line into 15 hyperfine components , characterised by the corresponding quantum numbers @xmath44 and @xmath27 ( e.g. pagani et al . 2009 ) . to fit the n@xmath0h@xmath1 spectrum , we first assumed a unique velocity component and used gildas - class90 , which gives the best fit ( reported in table 1 ) of the hyperfine components ( see the blue line in fig . [ n2h+]middle panel ) . the sum of the opacity at the central velocities of all the hyperfine components @xmath45@xmath46 is 0.1@xmath400.9 . although the opacity is not well determined the fit indicates @xmath45@xmath46 @xmath47 1 , thus suggesting optically thin emission . fits fixing @xmath46 to larger values never gave better results . cccccc & & & & & + & & & & & + + 34(2 ) & 2 & + 1.3(0.1 ) & 4.3(0.2 ) & 0.1(0.9 ) & 2.47.8 10@xmath8 + + 26(2 ) & 2 & + 1.8(0.1 ) & 2.6(0.1 ) & 0.2(0.2 ) & 2.48.0 10@xmath8 + 14(2 ) & 2 & 1.1(0.4 ) & 5.9(0.9 ) & 0.1(0.1 ) & 0.41.3 10@xmath8 + @xmath48 the spectrum has been centered at the frequency of the main hyperfine component @xmath49,@xmath27 = 2,31,2 ( 93173.76 ) . frequencies have been extracted from the cologne database or molecular spectroscopy ( mller et al . 2005 ) . see also pagani et al . @xmath50 at a spectral resolution of 1.3 km s@xmath16 . @xmath51 assuming a t@xmath52 = 20 - 80 k and a source size of 20@xmath725@xmath7 ( see text ) . + the peak lsr velocity ( + 1.3 km s@xmath16 ) of the n@xmath0h@xmath17 profile is sligthly blue - shifted with respect to the ambient velocity ( + 2.6 km s@xmath16 , bp97 ) . the linewidth ( 4.3 km s@xmath16 ) is also considerably larger than what observed by bp97 and tobin et al . ( 2013 ) towards the driving protostar l1157-mm ( 0.60.8 km s@xmath16 ) . this is clearly shown in figure [ n2h+ ] , where we report the n@xmath0h@xmath17(1@xmath250 ) line ( see the red histogram in the upper panel ) recently observed towards l1157-mm in the framework of the asai iram 30-m large program ( pi : r. bachiller & b. lefloch ) . the n@xmath0h@xmath17 profile from the b1 shock is definitely broader and more blue - shifted that what observed towards the l1157-mm protostar , indicating a different origin . note also that the weak , but not blended , @xmath49,@xmath27 = 0,11,2 line at @xmath2 8 km s@xmath16 from the main hyperfine component clearly shows blue - shifted emission . o emission at 1669 ghz ( nisini et al . offsets are with respect to the l1157-mm sources ( black star ) , at coordinates : @xmath53 = 20@xmath29 39@xmath30 06@xmath312 , @xmath32 = + 68@xmath33 02@xmath34 16@xmath350 . magenta contours refer to the sio(3 - 2 ) iram 30-m map reported by bachiller et al . the labels indicate the main blue- and red - shifted knots . circles are for the iram 30-m hpbw at the n@xmath0h@xmath17(1@xmath250 ) frequency ( 26@xmath7 ) , centred at the driving l1157-mm protostar ( observed by bp97 and tobin et al . 2013 ) , and at @xmath36 = + 25@xmath356 and @xmath37 = 63@xmath355 from the driving protostar ( present observations , see black triangles and coordinates reported in sect . bottom panel _ : the l1157-b1 bow shock as traced using the ch@xmath15cn(87 ) @xmath54 = 0,1,2 emission at 3 mm , observed with the iram pdb interferometer ( codella et al . 2009).,width=264 ] the best fit of fig . [ n2h+ ] shows a non - negligible residual ( @xmath2 3@xmath41 ; see bottom panel ) at about 4.0 km s@xmath16 , which suggests non - gaussian emission from gas at high blue - shifted velocity . indeed a definitely more satisfactory fit can be obtained by assuming two blue - shifted gaussian components ( see the magenta lines in fig . [ n2h+ ] and table 1 ) : ( i ) a line centered at + 1.8 km s@xmath16 with fwhm = 2.6 km s@xmath16 , plus ( ii ) a broader ( 5.9 km s@xmath16 ) line peaking at 1.1 km s@xmath16 ( dashed and dot - dashed magenta lines in fig . [ n2h+ ] , respectively ) . in summary , despite the complexity due to the hyperfine components , this clearly shows that a single - gaussian component is insufficient to reproduce the n@xmath0h@xmath17(10 ) profile towards the b1 shock , and one needs to invoke additional broad blue - shifted emission . the present observation thus reports the first detection of n@xmath0h@xmath17 emission towards a low - mass outflow , definitely far from the protostellar environment . the line profiles in l1157-b1 , as in other molecular shock spots , have a relatively complex structure where several excitation components are visible . disentangling such components is not an easy task . in l1157-b1 , the recent co multi - line analysis by lefloch et al . ( 2012 ) indicates that the line profiles are composed by a linear combination of exponential curves @xmath55(@xmath19 ) = @xmath55(0 ) exp(@xmath18@xmath19/@xmath56@xmath57 ) , independently of the co transition considered . the three velocity components correspond to three different physical components : ( 1 ) a small ( @xmath2 7@xmath710@xmath7 ) dissociative j - type shock called @xmath58 ( identified where the line intensity is @xmath59 exp(@xmath18@xmath19/12.5@xmath57 ) ) dominating at the highest velocities ( @xmath47 20 km s@xmath16 ) , ( 2 ) the outflow cavity walls , @xmath60 ( @xmath59 exp(@xmath18@xmath19/4.4@xmath21 ) ) , with size @xmath47 20@xmath7 , and ( 3 ) the larger ( @xmath2 25@xmath7 ) outflow cavity created by the older bow shock l1157-b2 , @xmath61 ( @xmath59 exp(@xmath21@xmath19/2.5@xmath21 ) ) dominating at velocities close to the systemic one ( @xmath19 @xmath4 2 km s@xmath16 ) . each component shows the same slope at all @xmath3 , but different relative intensities . the higher is the line excitation the brighter is the @xmath58 component . on the contrary , @xmath61 is observed only towards the low@xmath3 ( @xmath47 3 ) co lines . figure [ compa ] compares the n@xmath0h@xmath17(10 ) line with other line profiles observed towards l1157-b1 ( lefloch et al . 2010 , codella et al . 2010 , 2012 ) : ( i ) the co(1615 ) at 1841.3 ghz observed with herschel - hifi as an example of a spectrum where the @xmath58 component is clearly dominating the line profile ; ( ii ) the co(32 ) profile , _ as observed towards l1157-b2 _ , representing a pure @xmath61 profile , without the @xmath58 and @xmath60 components observed towards l1157-b1 ; ( iii ) the nh@xmath15(1@xmath62@xmath260@xmath62 ) transition , showing a profile well reproduced by the @xmath60 component alone . the n@xmath0h@xmath17 line profile , despite the blending between hyperfine components , seems to exclude the extremely high - velocity emission associated with the @xmath58 component , being consistent with the @xmath60 and @xmath61 ones . in conclusions , n@xmath0h@xmath1 is associated either with the b1 outflow cavity ( with @xmath63 @xmath13 70 k and @xmath64 @xmath4 10@xmath5 @xmath6 , according to the lvg co analysis by lefloch et al . 2012 ) and/or with the older and colder b2 cavity ( @xmath2 20 k , @xmath4 6 @xmath10 10@xmath12 @xmath6 ) . h@xmath17(1@xmath250 ) line ( black histogram ; in t@xmath65 scale ) observed in l1157-b1 with the iram 30-m antenna . the red histogram refers to the n@xmath0h@xmath17(1@xmath250 ) spectrum ( scaled for a direct comparison ) as observed towards l1157-mm with the iram 30-m antenna in the framework of the asai iram large program ( pi : r. bachiller & b. lefloch ) . the vertical dashed line indicates the ambient lsr velocity ( + 2.6 km s@xmath16 , from bp97 ) . the vertical seven blue lines stand for the 15 hyperfine components of the n@xmath0h@xmath17(1@xmath250 ) pattern ( several of them spectrally unresolved at the present frequency resolution ; see pagani et al . we centered the spectrum at the frequency of the main hyperfine component @xmath66,@xmath27 = 2,31,2 ( 93173.76 mhz ) . _ middle panel : _ analysis of the n@xmath0h@xmath17(1@xmath250 ) profile . the blue line shows the best fit ( fwhm = 4.3 km s@xmath16 ) assuming a single gaussian component . the magenta solid line shows the best fit using two gaussian components ( dashed magenta : fwhm = 2.6 km s@xmath16 ; dot - dashed magenta : fwhm = 5.9 km s@xmath16 ) in order to minimise the residual . the corresponding residuals are reported in the _ bottom panel _ : the single component approach gives a 3@xmath41 ( rms = 2 mk ) residual.,width=302 ] h@xmath17(1@xmath260 ) fit ( black ; see fig . 2 ) with typical profiles of the @xmath58 , @xmath60 , and @xmath61 components ( from codella et al . 2010 , and lefloch et al . 2012 , see text ) . co(1615 ) represents @xmath58 ( 1841.3 ghz , red , decreased by a factor 6.4 for a direct comparison ) , while nh@xmath15(1@xmath62@xmath260@xmath62 ) ( 572.5 ghz , blue , decreased by a factor 4.5 ) is for @xmath60 . in addition , we report the co(32 ) spectra for @xmath61 ( magenta , 345.8 ghz , decreased by a factor 269.0 ) observed by lefloch et al . ( 2012 ) towards the l1157-b2 position , tracing a cavity older than the l1157-b1 one , and created by a previous wind ejection ( gueth et al . the spectra have been smoothed to a common spectral resolution of 1.3 km s@xmath16.,width=302 ] the low excitation n@xmath0h@xmath17(1@xmath260 ) transition ( @xmath67 = 5 k ) has a critical density of @xmath2 10@xmath5 @xmath6 ( e.g. friesen et al . the line emission is thus expected to be close to lte conditions at the densities of the @xmath60 and @xmath61 gas components . following the results of the lvg analysis by lefloch et al . ( 2012 ) , we assume a @xmath63 between 20 and 70 k and an emitting size of 20@xmath725@xmath7 . the n@xmath0h@xmath17 total column density is then well constrained @xmath68(n@xmath0h@xmath1 ) = ( 28 ) @xmath10 10@xmath8 @xmath9 . using the source - averaged column density @xmath68(co ) = 1 @xmath10 10@xmath69 @xmath9 ( found for both @xmath60 and @xmath61 by lefloch et al . 2012 ) , and assuming [ co]/[h@xmath0]=10@xmath71 , we can derive the n@xmath0h@xmath1 abundance : @xmath72(n@xmath0h@xmath17 ) = 28 @xmath10 10@xmath11 . a lower abundance , between 4 @xmath10 10@xmath73 and @xmath2 10@xmath11 , is derived for the weaker emission at higher velocity , represented by the velocity component peaking at 1.1 km s@xmath16 ( see table 1 ) . these values are consistent with what found towards the l1157-mm protostar by bp97 ( 4 @xmath10 10@xmath11 ) using the iram 30-m antenna . on the other hand , chiang et al . ( 2010 ) measured lower values ( 36 @xmath10 10@xmath73 ) towards l1157-mm using the carma array , possibly due to interefometric filtering . similar values have been also found in co depleted prestellar cores and dense protostellar envelopes ( @xmath2 10@xmath7310@xmath11 ; see e.g. caselli et al . 2002 , tafalla et al . 2004 , 2006 , maret et al . 2007 , chen et al . 2007 , 2008 ) . this value represents an estimate of the abundance of the gas in the outflow cavities and will be used for a comparison with the outputs predicted by our models . h@xmath1 abundance , with respect to h@xmath0 , versus the h@xmath0 density at different times : from @xmath74 yr ( the age of l1157-b1 ) to @xmath75 yr . the dashed blue box gives the observed value with the 1 @xmath41 uncertainty ( see text ) . the gas is at a temperature of 70 k , but the curve is identical in the range 20 to 70 k. the cosmic ionisation rate is 1@xmath76 s@xmath16.,width=321 ] to understand the origin of the observed n@xmath0h@xmath1 , we compared its measured abundance with the n@xmath0h@xmath1 abundance predicted by a simple pseudo - time dependent model . we used the publicy available astrochem code . the code follows the evolution of the chemical composition of a gas cloud initially in the diffuse state and with fixed temperature and density . a simple gas - grain interaction due to freeze - out , thermal , and photo - desorption , has been considered . in these calculations we assumed a nitrogen elemental abundance equal to @xmath77 ( with respect to h nuclei ) , carbon and oxygen equal to @xmath78 and @xmath79 respectively , grain size of 0.1 @xmath80 m , and cosmic ionisation rates @xmath81 in the @xmath82@xmath83 s@xmath16 range ( e.g. dalgarno 2006 ; padovani et al . 2009 ) . figure [ model ] shows the predicted n@xmath0h@xmath1 abundance as a function of the volume density at different evolutionary times , from @xmath74 yr ( the age of l1157-b1 ) to @xmath75 yr . the chemistry of n@xmath0h@xmath1 is relatively simple : it is formed by the reaction of the h@xmath84 ( created by the cosmic rate ionisation of h@xmath0 ) and destroyed by the reaction of co ( or electrons in case of co depletion ) . therefore , the larger the density the lower is the h@xmath84 abundance , and consequently @xmath72(n@xmath0h@xmath1 ) . the comparison of the measured and predicted n@xmath0h@xmath1 abundances yields an important conclusion : the observed n@xmath0h@xmath1 abundance is perfectly matched by a model of cold , quiescent , and relatively old ( @xmath4 10@xmath12 yr ) gas and does not require the intervent of a shock . the age of the shock in l1157-b1 is around 2000 yr ( gueth et al . 1996 ) ; hence fig . [ model ] shows that n@xmath0h@xmath1 was present before the shock occurred , and it is consistent with a pre - shock h@xmath0 density of @xmath22 5 @xmath10 10@xmath85 @xmath6 . in addition , given that @xmath72(e ) @xmath59 @xmath64@xmath86 ( e.g. mckee 1989 ) , we can @xmath87 that the lower @xmath72(n@xmath0h@xmath1 ) abundance ( by a factor @xmath13 56 ) measured at the highest velocities indicates a density gradient in the shocked gas in the cavity . in other words , the n@xmath0h@xmath1 emitting at higher velocities could trace gas with @xmath64 about one order of magnitude higher than that of the gas at velocities closer to the systemic one . ) of n@xmath0h@xmath1 , h@xmath15@xmath1 , and n@xmath0 can vary as a function of distance ( see text).,width=321 ] in addition , to verify whether the detected n@xmath0h@xmath1 molecules were pre - existing to the shock , we used a shock model of l1157-b1 reported by viti et al . ( 2011 ) , who coupled the chemical model ucl_chem with a parametric shock model ( jimnez - serra et al . ucl_chem is a gas - grain chemical code which first simulates the formation of high - density clumps from an atomic diffuse cloud , and then follows their chemical evolution when subjected to the passage of a c - type shock . full details of the code can be found in viti et al . ( 2004 , 2011 ) . we updated the grid of models from viti et al . ( 2011 ) varying the cosmic ray ionisation rate @xmath81 ( which of course directly influences the behavour of ions ) in the 10@xmath8810@xmath89 s@xmath16 range . figure [ ucl ] reports an example of a ucl_chem shock model assuming @xmath81 = 10@xmath89 s@xmath16 and a pre - shock density of 10@xmath90 @xmath6 . we confirm that n@xmath0h@xmath1 is indeed formed in the gas phase and that the passage of a shock , with the subsequent release of n@xmath0 into the gas , does not yield an increase in the n@xmath0h@xmath1 abundance . this is consistent with the lack of signatures of very high - velocity associated with the @xmath58 component in the n@xmath0h@xmath1(10 ) profile . on the contrary , the passage of a shock does decrease the n@xmath0h@xmath1 abundance by about 12 orders of magnitude , depending on the pre - shock conditions and velocity of the shock . this allows us to further constrain the pre - shock density to @xmath2 10@xmath12 @xmath6 ( see fig . [ model ] ) in order to mantain the observed abundance once the outflow cavities have been compressed to @xmath91 10@xmath5 @xmath6 . a value of @xmath81 ( @xmath2 10@xmath89 s@xmath16 ) helps to achieve and mantain a high n@xmath0h@xmath1 abundance . a pre - shock density of @xmath2 10@xmath12 @xmath6 is consistent with the results suggested by the study of deuteration in l1157-b1 ( codella et al . 2012b ) where it was found that the most likely scenario is that of a a gas passing through a pre - shock phase with @xmath64 @xmath47 4 @xmath10 10@xmath12 @xmath6 , during which formaldehyde and methanol ices are formed . we present the first detection of diazenylium towards outflowing gas far from the driven low - mass protostar . we found evidence that n@xmath0h@xmath1(10 ) emission observed towards the l1157-b1 shock originates from the dense ( @xmath4 10@xmath5 @xmath6 ) gas associated with the cavities opened , and accelerated by the prototellar wind . the line width ( @xmath4 4 km s@xmath16 ) is significantly broader than the n@xmath0h@xmath1 line widths previously observed towards the driving protostar l1157-mm ( @xmath22 1 km s@xmath16 ) , as well as than the typical line widths observed in quiescent regions , probably as a result of the energy injection from the sweeping outflow . the estimated n@xmath0h@xmath1 abundance is ( 28 ) @xmath10 10@xmath11 , which can be reproduced by a model of quiescent gas evolved for more than 10@xmath12 yr ( i.e. older than the shock kinematical age , 2000 yr ) . in other words , n@xmath0h@xmath1 can be considered a fossil record of the pre - shock phase , when the gas density was @xmath2 10@xmath12 @xmath6 . modelling of c - shocks confirms that @xmath72(n@xmath0h@xmath1 ) is not enhanced by the passage of the shock . the present n@xmath0h@xmath1 detection is the result of the increase of its column density due to the compression ( by a factor @xmath2 10 ) of swept - up material , and not to its relative abundance . c. codella , c. ceccarelli , b. lefloch , and s. viti acknowledge the financial support from the cost action cm0805 `` the chemical cosmos '' . the italian authors gratefully acknowledge funding from italian space agency ( asi ) through the contract i/005/011/0 , which also supports the fellowships of g. busquet and a. gmez - ruiz . c. ceccarelli and b. lefloch acknowledge funding from the french space agency cnes and the national research agency funded project forcom , anr-08-blan-0225 . s. viti acknowledges support from the [ european community s ] seventh framework programme [ fp7/2007 - 2013 ] under grant agreement n@xmath92 238258 . * references * + bachiller r. , perz gutirrez m. , kumar m. s. n. , et al . , 2001 , a&a 372 , 899 + bachiller r. , & perz gutirrez m. , 1999 , apj 487 , l93 ( bp97 ) + benedettini m. , viti s. , codella c. , et al . , 2007 , mnras 381 , 1127 + caselli p. , myers p.c . , thaddeus p. , 1995 , apj 455 , l77 + caselli p. , benson p.j . , myers p. c. , tafalla m. , 2002 , apj 572 , 238 + ceccarelli c. , bacmann a. , boogert a. , et al . , 2010 , a&a 521 , l22 + chen x. , launhardt r. , bourke t.l . , henning th . , barnes p.j . , 2008 , apj 683 , 862 + chen x. , launhardt r. , henning th . , 2007 , apj 669 , 1058 + chiang h .- f . , looney l.w . , tobin j.j . , & hartmann l. , 2010 , apj 709 , 470 + codella c. , benedettini m. , beltrn m.t . , et al . 2009 , a&a 507 , l25 + codella c. , lefloch b. , ceccarelli c. , et al . , 2010 , a&a 518 , l112 + codella c. , ceccarelli c. , bottinelli s. , et al . , 2012a , apj 744 , l164 + codella c. , ceccarelli c. , lefloch b. , et al . , 2012b , apj 757 , l9 + dalgarno a. , 2006 , proceedings of the national academy of science 103 , 12269 + friesen r.k . , di francesco j. , shirley y.l . , myers p.c . , 2009 , apj 697 , 1457 + de graauw th . , helmich f.p . , phillips t.g . , et al . , 2010 , a&a 518 , l6 + gusdorf a. , pineau des for@xmath93ts g. , cabrit s. , flower d.r . , 2008 , a&a 490 , 695 + gueth f. , guilloteau s. , & bachiller r. , 1996 , a&a 307 , 891 + gueth f. , guilloteau s. , & bachiller r. , 1998 , a&a 333 , 287 + jimnez - serra i. , caselli p. , martn - pintado j. , hartquist t.w . , 2008 , a&a 482 , 549 + j@xmath23rgensen j.k . , hogerheijde m.r . , van dishoeck e.f . , blake g.a . , schier f.l . , 2004 , a&a 413 , 993 + lefloch b. , cabrit s. , codella c. , et al . , 2010 , a&a 518 , l113 + lefloch b. , cabrit s. , busquet g. , et al . , 2012 , apj 757 , l25 + looney l. w. , tobin j. , & kwon w. , 2007 , apj 670 , l131 + maret s. , bergin e.a . , & lada c.j . , 2007 , apj 670 , l25 + mckee , c.f . , 1989 , apj 345 , 782 + mller h.s.p . , schier f.l . , stutzki j. , winnewisser g. , 2005 , j.mol.struct . 742 , 215 + neufeld d.a . , & green s. , 1994 , apj 432 , 158 + nisini b. , codella c. , giannini t. , et al . , 2007 , a&a 462 , 163 + nisini b. , giannini t. , neufeld d.a . , et al . , 2010a , apj 724 , 69 + nisini b. , benedettini m. , codella c. , et al . , 2010b , a&a 518 , l12 + padovani m. , galli d. , & glassgold a.e . , 2009 , a&a 501 , 619 + pagani l. , daniel f. , & dubernet m l . , 2009 , a&a 494 , 719 + tafalla m. , & bachiller r. , 1995 , apj 443 , l37 + tafalla m. , myers p.c . , caselli p. , walmsley c.m . , 2004 , a&a 416 , 191 + tafalla m. , santiago - garca j. , myers p.c . , et al . , 2006 , a&a 455 , 577 + tobin j.j . , hartmann l. , chiang h .- f . , et al . , 2011 , apj 740 , 45 + tobin j.j . , hartmann l. , bergin e. , et al . , 2012 , apj 748 , 16 + tobin j.j . , bergin e. , hartmann l. , et al . , 2013 , apj 765 , 18 + wilson t.l . , & rood r. , 1994 , ara&a 32 , 191 + vasta , m. , codella , c. , lorenzani , a. , et al . , 2012 , a&a 537 , a98 + viti s. , collongs m.p . , dever j.w . , mccoustra m.r.s . , williams d.a . , 2004 , mnras 354 , 1141 + viti s. , jimnez - serra i. , yates j.a . , et al . , 2011 , apj 740 , l3 yamaguchi t. , takano s. , watanabe y. , et al . , 2012 , pasp 64 , 105
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we present the first detection of n@xmath0h@xmath1 towards a low - mass protostellar outflow , namely the l1157-b1 shock , at @xmath2 0.1 pc from the protostellar cocoon .
the detection was obtained with the iram 30-m antenna .
we observed emission at 93 ghz due to the @xmath3 = 10 hyperfine lines .
the analysis of the emission coupled with the hifi chess multiline co observations leads to the conclusion that the observed n@xmath0h@xmath1(10 ) line originates from the dense ( @xmath4 10@xmath5 @xmath6 ) gas associated with the large ( 20@xmath725@xmath7 ) cavities opened by the protostellar wind .
we find a n@xmath0h@xmath1 column density of few 10@xmath8 @xmath9 corresponding to an abundance of ( 28 ) @xmath10 10@xmath11 .
the n@xmath0h@xmath1 abundance can be matched by a model of quiescent gas evolved for more than 10@xmath12 yr , i.e. for more than the shock kinematical age ( @xmath13 2000 yr ) .
modelling of c - shocks confirms that the abundance of n@xmath0h@xmath1 is not increased by the passage of the shock . in summary , n@xmath0h@xmath1 is a fossil record of the pre - shock gas , formed when the density of the gas was around 10@xmath12 @xmath6 , and then further compressed and accelerated by the shock .
| 11,262 | 437 |
_ ab initio _ methods for polymers and crystals come more and more into focus of quantum chemists and solid state physicists @xcite . as most _ ab initio _ treatments of such extended systems rely on incomplete one - particle basis sets , we consider it timely to investigate the basis set convergence of hartree - fock and correlation energies in _ periodic _ systems . _ molecular _ hartree - fock energies are well known to converge exponentially , e.g. ref . @xcite , towards the basis set limit , but molecular correlation energies turn out to converge only with the third power of the highest angular momentum employed in the basis sets @xcite . the convergence properties of the hartree - fock and the correlation energy can be exploited to extrapolate hartree - fock @xcite and correlation energies @xcite towards the basis set limit . as only standard methods of quantum chemistry are required , basis set extrapolation of correlation energies provides an interesting alternative over the specialised , explicitly correlated ( r12 ) methods , which directly yield near basis set limit wave functions and correlation energies but have a high computational demand @xcite . especially well suited in conjunction with extrapolation schemes , are the correlation consistent basis sets @xcite cc - pvxz @xcite , aug - cc - pvxz @xcite and d - aug - cc - pvxz @xcite , x = d , t , q , 5 , 6 which are hierarchical series of basis sets of increasing quality . our study elucidates the performance of basis set extrapolation for hartree - fock and correlation energies in _ infinite _ periodic systems , the hydrogen bonded bent chains ( hf)@xmath0 and ( hcl)@xmath0 which are representatives for strong and weak hydrogen bonds @xcite and require a very accurate description by a large one particle basis to reliably determine their binding energies per monomer . hartree - fock energies of the infinite chains are obtained by periodic calculations @xcite whereas their correlation energy is calculated utilising stoll s incremental scheme @xcite which has been successfully applied to various semiconductors @xcite , ionic crystals @xcite , rare gas crystals @xcite and polymers @xcite . hartree - fock energies turn out to converge rapidly with increasing basis set quality towards the basis set limit . however , the actual convergence behaviour has only empirically been determined , ref . @xcite ( and refs . therein ) , to depend both on the number of basis functions and on the highest angular momentum in basis sets . the cardinal number @xmath2 of correlation consistent basis sets is related to both quantities , and hartree - fock energies follow @xmath3 with @xmath4 and @xmath5 being the hartree - fock basis set limit while the hartree - fock energy obtained with a basis set @xmath2 is denoted by @xmath6 . correlation energies converge differently ; the partial wave analysis of the correlation energy of the helium atom @xcite facilitates to derive the relation @xcite @xmath7 where @xmath8 is the basis set limit correlation energy and @xmath9 represents the correlation energy obtained with basis set @xmath2 ( in our case , @xmath2 is equal to the highest angular momentum of basis functions in the basis set ) . ( [ eq : atomicx3series ] ) is derived for the asymptotic behaviour , i.e. large @xmath2 , of the correlation energy , assuming basis sets of highest angular momentum @xmath2 , being centred around a single point in space . the basis sets are supposed to be complete for all angular momenta @xmath10 and are required to be complete with respect to their radial part @xcite . however , a simple two - point fit based on eq . ( [ eq : atomicx3series ] ) , which involves the correlation energies of two basis sets @xmath2 and @xmath11 , turns out to yield highly accurate molecular binding energies @xcite . the extrapolation scheme for correlation energies of park , huh and lee @xcite is a more flexible basis set extrapolation which we consider to cope slightly better with the increasing radial and angular completeness of hierarchical basis set series . park harness [ eq : atomicxgamma ] e_corr^chain ( ) & = & + _ x , y & = & , with the underlying assumption that the basis set convergence rate @xmath12 is the same for a monomer and an infinite chain formed by many monomers . @xmath12 is the ratio of the absolute error in the correlation energy of the monomer described by two different basis sets @xmath2 and @xmath11 . if the electronic structure of a monomer does not change substantially upon chain formation , a given basis set represents both the monomer and the infinite chain equally well . in ( hf)@xmath0 . circles and squares represent @xmath13 of the cc - pvxz and aug - cc - pvxz basis sets where open and closed symbols denote bare and cp corrected hartree - fock binding energies . the straight line results from two nearly coinciding lines which indicate the extrapolated hartree - fock binding energies , the upper and the lower line referring to the cc - pvxz and the aug - cc - pvxz basis sets . the crosses indicate the mean of the cp corrected and the corresponding bare hartree - fock binding energies . ] basis set extrapolation of hartree - fock and correlation energies shall now be used to obtain accurate binding energies of ( hf)@xmath0 and ( hcl)@xmath0 chains . both ( hf)@xmath0 and ( hcl)@xmath0 form zig - zag chains where in both cases the unit cell consists of two monomers . details concerning the employed experimental geometries can be found in refs . @xcite . in ( hcl)@xmath0 . symbols are chosen as in fig . [ fig : hf_rhf_basis ] . the upper and the lower straight lines now refer to the aug - cc - pvxz and the cc - pvxz basis sets , in reverse order compared to fig . [ fig : hf_rhf_basis ] . ] the hartree - fock binding energies per monomer , @xmath14 , @xmath15 had to be removed from the basis sets . ] are plotted for ( hf)@xmath0 and ( hcl)@xmath0 in figs . [ fig : hf_rhf_basis ] and [ fig : hcl_rhf_basis ] . the basis set superposition error ( bsse ) is removed beyond microhartree accuracy by counterpoise correction ( cp ) @xcite , where a monomer is additionally surrounded by the basis functions of eight neighbouring monomers . for each series of correlation consistent basis sets , there is an upper curve for the cp corrected binding energies and a corresponding lower curve giving the bare binding energies without cp correction of the monomer energies . both curves converge unsystematically towards the hartree - fock basis set limit , especially , they generally converge not monotonic . the deviation of the lower curve from the upper curve of the same basis set series yields an estimate of the error of the approximation introduced by the finite basis sets as this deviation is the size of the bsse @xcite and an estimate of the incompleteness of a one - particle basis set . it is very small , @xmath16 for ( hf)@xmath0 and @xmath17 for ( hcl)@xmath0 , utilising cc - pv6z basis sets . nevertheless , we would like to elucidate whether , in case of an infinite chain , the hartree - fock energies follow eq . ( [ eq : rhfconv ] ) as well , i.e. whether the packing in infinite periodic systems has an unexpected impact on hartree - fock basis set convergence . tab . [ tab : rhfextra ] gives hartree - fock energies at the basis set limit for the monomers and the infinite chains as obtained by a least squares fit using eq . ( [ eq : rhfconv ] ) which turns out to describe the hartree - fock energies , underlying figs . [ fig : hf_rhf_basis ] and [ fig : hcl_rhf_basis ] , excellently . a three - point fit based on eq . ( [ eq : rhfconv ] ) to the hartree - fock energies , obtained with three basis sets @xmath2 , @xmath11 and @xmath18 , also yields convincing results that converge rapidly with the quality of the three basis sets used . halkier @xcite found for the basis set convergence of the hartree - fock binding energy of several hydrogen - bonded complexes , including ( hf)@xmath19 and ( hcl)@xmath19 , that the mean of the bare and the counterpoise corrected hartree - fock binding energies frequently provides a decent extrapolation to the basis set limit . this behaviour of the mean hartree - fock binding energy is solely observed for the aug - cc - pvxz series for ( hf)@xmath0 . .basis set extrapolated hartree - fock binding energies per monomer @xmath20 of ( hf)@xmath0 and ( hcl)@xmath0 obtained by least squares fits to eq . ( [ eq : rhfconv ] ) of their hartree - fock energies for the cc - pvxz ( @xmath21d , , 6 ) and aug - cc - pvxz ( @xmath21d , , 5 for ( hf)@xmath0 and @xmath21d , , 6 for ( hcl)@xmath0 ) series of basis sets . all data are given in millihartree . [ cols=">,^,^,^ " , ] in our study we focus on the basis set convergence of the hartree - fock and the correlation energy in the _ infinite _ chains ( hf)@xmath0 and ( hcl)@xmath0 . especially in hydrogen bonded systems , the binding energy per monomer is very small and very accurate calculations are required . our data shows that the error of the binding energy , employing the cc - pv5z basis set , leads to a deviation of the bare binding energy per monomer of ( hf)@xmath0 from the cp corrected one by @xmath22 and by @xmath23 for ( hcl)@xmath0 which is too large to be satisfactory . to reduce the error , we extrapolate hartree - fock @xcite and correlation energies @xcite where two different extrapolation schemes are studied for the correlation energy . the accuracy of the resulting correlation contribution to the binding energies in tab . [ tab : bindtotal ] is estimated by the deviation of the extrapolated binding energies for the aug - cc - pvxz series from the ones for the cc - pvxz series , where the largest basis sets of the respective series [ t q for aug - cc - pvxz and q5 for cc - pvxz ] are utilised as they could be shown to yield the best results . the deviation is @xmath24 or @xmath25 for ( hf)@xmath0 and @xmath26 or @xmath27 for ( hcl)@xmath0 depending on the extrapolation method employed , i.e. @xmath28 @xcite or park @xcite . we would like to point out further , that eqs . ( [ eq : atomicx3 ] ) and ( [ eq : atomicxgamma ] ) facilitate to extrapolate the individual energy increments occurring in the decomposition of the correlation energy in terms of the incremental scheme @xcite , separately . this is advantageous for treating , e.g. , hydrogen bonds in larger molecules and facilitates a more accurate treatment of different atoms or fragments in crystals . basis sets were obtained from the extensible computational chemistry environment basis set database , version 02/25/04 , as developed and distributed by the molecular science computing facility , environmental and molecular sciences laboratory which is part of the pacific northwest laboratory , p.o . box 999 , richland , washington 99352 , usa , and funded by the u.s . department of energy . the pacific northwest laboratory is a multi - program laboratory operated by battelle memorial institute for the u.s . department of energy under contract de - ac06 - 76rlo 1830 . contact david feller or karen schuchardt for further information . r. d. amos , a. bernhardsson , a. berning , p. celani , d. l. cooper , m. j. o. deegan , a. j. dobbyn , f. eckert , c. hampel , g. hetzer , p. j. knowles , t. korona , r. lindh , a. w. lloyd , s. j. mcnicholas , f. r. manby , w. meyer , m. e. mura , a. nicklass , p. palmieri , r. pitzer , g. rauhut , m. schtz , u. schumann , h. stoll , a. j. stone , r. tarroni , t. thorsteinsson , h .- j . werner , molpro , a package of _ ab initio _ programs designed by h .- j . werner and p. j. knowles , version 2002.6 ( 2002 ) .
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basis set convergence of the hartree - fock and the correlation energy is examined for the hydrogen bonded _
infinite _ bent chains ( hf)@xmath0 and ( hcl)@xmath0 . we employ series of correlation consistent basis sets up to quintuple @xmath1 quality together with a coupled cluster method ( ccsd ) to describe electron correlation on _ ab initio _ level .
the hartree - fock energy converges rapidly with increasing basis set quality whereas the correlation energy is found to be slowly convergent for the same series of basis sets .
we study basis set extrapolation for ( hf)@xmath0 and ( hcl)@xmath0 and show that it substantially enhances the accuracy of both the hartree - fock and the correlation energy in _ extended _ systems .
, electron correlation , ab initio calculations , hydrogen fluoride , hydrogen chloride , infinite bent chains , basis set convergence , basis set extrapolation 31.15.ar , 31.25.qm , 71.15.nc , 71.20.ps
| 3,810 | 278 |
the hadronic final state of collisions at 200 gev may provide a reference for other high - energy nuclear collisions at the relativistic heavy ion collider ( rhic ) and large hadron collider ( lhc ) . claims for novel physics at higher energies or in , or collision systems should be based on an accurate and self - consistent phenomenology for conventional processes at 200 gev . however , current theoretical and experimental descriptions of high - energy collisions appear to be incomplete . several unresolved aspects of collisions are notable : ( a ) the role of collision centrality in relation to the low-@xmath3 gluon transverse structure of the proton @xcite , ( b ) the nature and systematics of the _ underlying event _ ( ue ) defined as complementary to contributions from an event - wise - triggered high - energy dijet @xcite , ( c ) the systematics of _ minimum - bias _ ( mb ) dijet ( minijet ) production manifested in spectra and angular correlations @xcite and ( d ) possible existence and phenomenology of a _ nonjet azimuth quadrupole _ component in 2d angular correlations previously studied in collisions ( as quantity @xmath4 ) @xcite , especially in connection with a claimed same - side ridge observed in lhc angular correlations @xcite . a more detailed discussion of those issues is presented in sec . [ issues ] . in the present study we establish a more complete mathematical model for phenomenology based on the @xmath5 dependence of single - particle ( sp ) @xmath6 spectra and -integral @xmath2 densities and -integral 2d angular correlations . we confront several issues : is there any connection between @xmath5 and centrality ? is centrality a relevant concept ? a nonjet ( nj ) quadrupole component in collisions is the complement to jet - related and projectile - fragment correlations . is there an equivalent phenomenon in collisions , and what might a nj quadrupole component reveal about centrality or ue structure ? the phenomenological model should offer a conceptual context with two manifestations : ( a ) as a mathematical framework to represent data systematics efficiently , and ( b ) as a theoretical framework to provide physical interpretation of model elements via comparisons between data structures and qcd theory . preliminary responses to such questions were presented in ref . they are supplemented here by new sp density and 2d angular - correlation measurements . we emphasize the @xmath5 dependence of angular correlations from collisions , extending the two - component model ( tcm ) to include a nj quadrupole component previously extrapolated from measurements in collisions @xcite and now obtained directly from 2d angular correlations . we establish @xmath5-dependent phenomenology for soft ( projectile proton dissociation ) , hard ( parton fragmentation to mb dijets ) and nj quadrupole components in a _ three_-component model and explore possible correspondence with centrality , ue structure , dijet production and the partonic structure of projectile protons . we also present a tcm for the systematics of hadron densities on pseudorapidity @xmath2 . this article is arranged as follows : section [ issues ] summarizes open issues for collisions . section [ methods ] reviews analysis methods for two - particle correlations . section [ pptcm ] describes a two - component model for hadron production in collisions . section [ ppangcorr ] presents measured 2d angular correlations for 200 gev collisions . section [ modelfits ] summarizes the parametric results of 2d model fits to those correlation data . section [ jetcorr1 ] describes jet - related data systematics . section [ njcorr ] describes nonjet data systematics . section [ etadensity ] presents a two - component model for @xmath2 densities and the @xmath2-acceptance dependence of transverse - rapidity spectra . section [ syserr ] discusses systematic uncertainties . section [ ridgecms ] reviews same - side `` ridge '' properties and a proposed mechanism . sections [ disc ] and [ summ ] present discussion and summary . we present a summary of issues introduced in sec . i including centrality in relation to a conjectured underlying event , manifestations of mb dijets in spectra and correlations and existence and interpretation of a nj quadrupole component of 2d angular correlations . item ( a ) of sec . [ intro ] relates to interpretations of deep - inelastic scattering ( dis ) data to indicate that low-@xmath3 gluons are concentrated within a transverse region of the proton substantially smaller than its overall size . it is argued that a high-@xmath6 dijet trigger may select more - central collisions with greater soft - hadron production @xcite . the soft ( nonjet ) multiplicity increase should be observed most clearly within a narrow azimuth _ transverse region _ ( tr ) centered at @xmath8 and thought to _ exclude contributions from the triggered jets _ centered at 0 and @xmath9 . item ( b ) relates to measurements of charge multiplicity @xmath10 within the tr vs trigger condition @xmath11 and @xmath12 spectra employed to characterize the ue @xcite . substantial increase of @xmath10 with higher @xmath11 relative to a minimum - bias or non - single - diffractive ( nsd ) value is interpreted to reveal novel contributions to the ue , including _ multiple parton interactions _ ( mpi ) corresponding to a high rate of dijet production @xcite . monte carlo collision models such as pythia @xcite are tuned to accommodate such results @xcite . in ref . @xcite items ( a ) and ( b ) were considered in the context of a two - component ( soft+hard ) model ( tcm ) of hadron production as manifested in yields and spectra . it was observed that imposing a trigger condition on events does lead to selection for _ hard events _ ( containing at least one dijet ) but that the soft component of the selected events is not significantly different from a mb event sample , in contrast to expectations from ref . @xcite that increased dijet frequency should correspond to more - central collisions and therefore to a larger soft component from low-@xmath3 gluons . since apparently determines dijet rates directly @xcite it might also control centrality , but ref . @xcite concluded that further correlation measurements are required to explore that possibility . the present study responds with the dependence of mb dijet correlations and nj quadrupole systematics that speak to the issue of centrality and ue systematics . item ( c ) relates to the role of mb dijets in yields , spectra and various types of two - particle correlations . the contribution of mb dijets ( minijets @xcite ) to sp spectra was established in refs . @xcite , and the contribution of minijets to 2d angular correlations was identified in refs . however , further effort is required to establish a complete and self - consistent description of mb dijets in and yields , spectra and correlations . in ref . @xcite item ( c ) was addressed with 2d model fits applied to angular correlations from collisions at 62 and 200 gev to isolate several correlation components , including structures attributed to mb dijets and a nj quadrupole , with emphasis on the former in that study . the systematics of two components ( soft + hard ) are consistent with the tcm . the dijet ( hard - component ) trend on centrality exhibits a _ sharp transition _ near 50% fractional cross section below which collisions appear to be simple linear superpositions of binary collisions ( transparency ) and above which _ quantitative _ changes in the dijet component appear , but not in the nj quadrupole component @xcite . the mb dijet interpretation has been questioned variously , for more - central collisions @xcite or for all nuclear collisions @xcite . we wish to confirm the role of mb dijets as such via a self - consistent description of _ and _ collisions based on qcd theory . although a tcm for yields and spectra vs has been established @xcite the systematics of mb dijet production in collisions is incomplete . mb 2d angular correlations for collisions have been decomposed into soft and hard components via a single cut ( at 0.5 gev / c ) @xcite , but a tcm for angular correlations vs has not been available . in ref . @xcite mb jet - related correlation structure vs centrality was related quantitatively to spectrum hard components ( dijets ) to establish a direct link of both data formats with pqcd predictions . in the present study we carry out a similar analysis of vs trends . we also extend the tcm established on marginal to the 2d @xmath13 system to determine the distribution of minijets on the full sp momentum space . the extension to @xmath2 may provide further evidence that a mb dijet interpretation of the inferred tcm hard component is _ necessary _ as a distinct element of hadron production . item ( d ) relates to the possibility of a significant amplitude for a unique azimuth quadrupole in collisions . ( the nj quadrupole should be distinguished from the quadrupole component of a _ jet - related _ 2d peak projected onto 1d azimuth . ) measurements of a nj quadrupole component of angular correlations in collisions ( conventionally represented by parameter @xmath4 ) are found to be consistent with a simple universal trend on centrality and collision energy extrapolating to a nonzero value for collisions @xcite . the extrapolation is consistent with a qcd - theory prediction for @xmath4 in collisions @xcite . the nj quadrupole may be related to a same - side `` ridge '' reported in collisions at 7 tev ( with special cuts on and imposed ) @xcite . it has been suggested that the same - side ridge arises from the same mechanism proposed for collisions based on collective motion ( flows ) coupled to initial - state collision geometry . systematics of a possible nj quadrupole in collisions have thus emerged as an important new topic . reference @xcite considered extrapolation of nj quadrupole centrality systematics in 200 gev collisions to collisions , and further extrapolation to lhc energies based on measured rhic energy dependence . in that scenario the same - side ridge observed in lhc collisions corresponds to one lobe of the nj quadrupole . the other lobe is obscured by the presence of a dominant away - side ( as ) 1d jet peak . quantitative correspondence was observed in ref . @xcite suggesting that the nj quadrupole may play a significant role in collisions , but no direct quadrupole measurements existed . this study offers a response to that issue . measurement of nj quadrupole trends may shed light on the question of centrality [ item ( a ) ] by analogy with quadrupole systematics wherein the nj quadrupole measured by a _ per - particle _ variable first increases rapidly with centrality and then falls sharply toward zero with decreasing eccentricity , as described by a glauber model based on the eikonal approximation . since dijet production vs in collisions suggests that the eikonal approximation is not valid for that system @xcite the nj quadrupole trend could provide a critical test of the eikonal assumption for collisions . we review technical aspects of two - particle angular - correlation analysis methods applied to collisions at the rhic . further method details appear in refs . @xcite . high - energy nuclear collisions produce final - state hadrons as a distribution within cylindrical 3d momentum space @xmath14 , where @xmath6 is transverse momentum , @xmath2 is pseudorapidity and @xmath15 is azimuth angle . transverse mass is @xmath16 with hadron mass @xmath17 . pseudorapidity is @xmath18 $ ] ( @xmath19 is polar angle relative to collision axis @xmath20 ) , and @xmath21 near @xmath22 . to improve visual access to low-@xmath6 structure and simplify description of the -spectrum hard component ( defined below ) we present spectra on transverse rapidity @xmath23 $ ] . for unidentified hadrons @xmath1 , with pion mass assumed ( about 80% of hadrons ) , serves as a regularized logarithmic @xmath6 measure . a typical detector acceptance @xmath24 gev / c corresponds to @xmath25 . correlations are observed in two - particle momentum space @xmath26 . autocorrelation _ on angular subspace @xmath27 ( where @xmath28 or @xmath15 ) is derived by averaging pair density @xmath29 along diagonals on @xmath27 parallel to the sum axis @xmath30 @xcite . the averaged pair density @xmath31 on defined _ difference variable _ @xmath32 is then an autocorrelation . the notation @xmath33 rather than @xmath34 for difference variables is adopted to conform with mathematical notation conventions and to retain @xmath34 as a measure of a detector acceptance on parameter @xmath3 . for correlation structure approximately independent of @xmath35 over some limited acceptance @xmath34 ( stationarity , typical over @xmath36 azimuth and within some limited pseudorapidity acceptance @xmath37 ) angular correlations remain undistorted ( no information is lost in the projection by averaging ) . -integral 2d angular autocorrelations are thus lossless projections of 6d two - particle momentum space onto angle difference axes @xmath38 . the @xmath39 axis is divided into _ same - side _ ( ss , @xmath40 ) and _ away - side _ ( as , @xmath41 ) intervals . for collisions between two composite projectiles the collision final state ( fs ) may depend on the transverse separation of the collision partners ( impact parameter @xmath42 ) and the phase - space distribution of constituents within each projectile , collectively the initial - state ( is ) geometry . we wish to determine how the is geometry relates to an observable derived from fs hadrons and how the is influences fs hadron yields , spectra and correlations . for a - a collisions the projectile constituents are nucleons @xmath43 all sharing a common lab velocity ( modulo fermi motion ) and distributed over a nuclear volume . based on a glauber model of collisions ( assuming the eikonal approximation ) nucleons are classified as participants ( total number @xmath44 ) or spectators , and the mean number of binary encounters @xmath45 is estimated . the relation @xmath46 is a consequence of the eikonal approximation . parameters @xmath44 and @xmath45 , depending on impact parameter @xmath42 , are in turn related to macroscopic fs observable within some angular acceptance via the mb cross - section distribution @xmath47 . for collisions the constituents are partons distributed on the transverse configuration space of projectile protons _ and _ on longitudinal - momentum fraction @xmath3 ( fraction of proton momentum carried by a parton ) . one could apply a similar glauber approach to is geometry , including assumed eikonal approximation as in the description ( e.g. default pythia @xcite ) . as noted in the introduction , it is conjectured that imposing a dijet trigger should favor more - central collisions and therefore a substantial increase in soft - hadron production from low-@xmath3 gluons @xcite . however , some aspects of collision data appear to be inconsistent with such a description , specifically parton transverse position and a impact parameter . nevertheless , fs measures for number of participant low-@xmath3 partons and their binary encounters may be relevant and experimentally accessible @xcite . @xmath49 represents a basic pair density on 6d pair momentum space . the event - ensemble - averaged pair density @xmath50 derived from sibling pairs ( pairs drawn from single events ) includes the correlation structures to be measured . @xmath51 is a density of mixed pairs drawn from different but similar events . @xmath52 denotes a minimally - correlated reference - pair density derived from ( a ) a mixed - pair density or ( b ) a cartesian product of sp angular densities @xmath53 via a factorization assumption . differential correlation structure is determined by comparing a sibling - pair density to a reference - pair density in the form of difference @xmath54 representing a correlated - pair density or _ covariance _ density . _ per - particle _ measure @xmath55 has the form of pearson s normalized covariance @xcite wherein the numerator is a covariance and the denominator is approximately the geometric mean of marginal variances . in the poisson limit a marginal variance may correspond to @xmath56 . since @xmath57 it follows that the geometric mean of variances is given by @xmath58 and the normalized covariance density is a per - particle measure @xcite . the number of final - state charged hadrons @xmath59 in the denominator can be seen as a place holder . other particle degrees of freedom may be more appropriate for various physical mechanisms ( e.g. number of participant nucleons in collisions , number of participant low-@xmath3 partons in collisions ) as described below . we define @xmath60 where pair ratio @xmath61 cancels instrumental effects . that per - particle measure is not based on a physical model . in some analyses a correlation amplitude is defined as @xmath62 with @xmath63 @xcite , but such an amplitude then relies on a specific detector acceptance , is not `` portable . '' to assess the relation of data to is geometry we convert per - charged - hadron model - fit results to @xmath64 $ ] , the quantity in square brackets representing the _ number of correlated pairs _ within the detector acceptance . for this analysis we assume the soft - component density @xmath65 is an estimator for is @xmath66 ( low-@xmath3 parton participants ) . given the simplified notation @xmath67 we plot @xmath68 to convert `` per - particle '' from fs hadrons to is low-@xmath3 partons and obtain a more interpretable per - particle measure . correlations on two - particle momentum space @xmath69 can be factorized into distributions on 2d transverse - momentum space @xmath70 or transverse - rapidity space @xmath71 @xcite and on 4d angle space @xmath72 reducible with negligible information loss to autocorrelations on difference variables @xmath38 @xcite . in this study we focus on minimum - bias ( @xmath1-integral ) 2d angular correlations . each of the several features appearing in 2d angular correlations ( a correlation _ component _ ) can be modeled within acceptance @xmath37 by a simple functional form ( a model _ element _ ) , including 1d and 2d gaussians and azimuth sinusoids uniform on @xmath73 . the cosine elements @xmath74 represent _ cylindrical multipoles _ with pole number @xmath75 , e.g. , dipole , quadrupole and sextupole for @xmath76 . angular correlations can be formed separately for like - sign ( ls ) and unlike - sign ( us ) charge combinations , as well as for charge - independent ( ci = ls + us ) and charge - dependent ( cd = ls @xmath77 us ) combinations @xcite . the azimuth quadrupole ( @xmath78 fourier ) component is a prominent feature of angular correlations , represented there by symbol @xmath79 . the mean value is nominally relative to an estimated reaction plane @xcite . @xmath4 data are conventionally interpreted to represent elliptic flow , a hydrodynamic ( hydro ) response to is asymmetry in non - central collisions @xcite . if 2d angular correlations are projected onto 1d azimuth _ any _ resulting distribution can be expressed exactly in terms of a fourier series . the density of correlated pairs is then [ nf ] ( _ ) & = & |_sib - _ ref = v_0 ^ 2 + 2_m=1^ v_m^2 ( m _ ) , defining the _ power - spectrum _ elements @xmath80 of autocorrelation density @xmath81 . the corresponding _ per - pair _ correlation measure is the ratio & = & v_0 ^ 2 + 2_m=1^v_m^2 ( m _ ) . some fourier amplitudes from analysis of 1d azimuth projections may include contributions from more than one mechanism . for example , @xmath4 data from conventional 1d analysis may include contributions from jet - related ( `` nonflow '' ) as well as nj ( `` flow '' ) mechanisms @xcite . in contrast , a complete model of 2d angular correlations _ with @xmath73-dependent elements _ permits isolation of several production mechanisms including a nj quadrupole component @xcite . fourier components from 2d correlation analysis are denoted by @xmath82 or @xmath83 , and only the @xmath84 and 2 ( dipole and quadrupole ) fourier terms are _ required _ by 2d data histograms ( see sec . [ ppangcorr ] ) @xcite . for measure @xmath55 data derived from model fits to 2d angular correlations the quadrupole component is denoted by @xmath85 since the factorized reference density is @xmath86 . in the present study of 2d angular correlations we admit the possibility that a significant nj quadrupole component may persist in high - energy collisions ( not necessarily of hydro origin ) and retain the corresponding model element in the 2d data model function eq . ( [ modelfunc ] ) . the measured -integral nj quadrupole data for collisions are represented above 13 gev by @xcite [ loglog ] a_q\{}(b , ) & & |_0(b ) v_2 ^ 2\{}(b , ) + & = & c r ( ) n_bin(b ) _ 2,opt^2(b ) , where @xmath87 , the energy - dependence factor is @xmath88 , @xmath45 is the estimated number of binary encounters in the glauber model , and @xmath89 is the @xmath78 overlap eccentricity assuming a continuous ( optical - model ) nuclear - matter distribution . equation ( [ loglog ] ) describes measured @xmath1-integral azimuth quadrupole data in heavy ion collisions for all centralities down to collisions and energies above @xmath90 gev and represents factorization of energy and centrality dependence for the nj quadrupole . the 2d quadrupole data are also consistent with @xmath91 @xcite , a centrality trend that , modulo the is eccentricity , _ increases much faster than the dijet production rate_. a non - zero value @xmath92 from eq . ( [ loglog ] ) extrapolated to collisions agrees with a qcd color - dipole prediction @xcite . as one aspect of the present correlation study we confirm extrapolation of the nj quadrupole centrality trend to collisions and determine the dependence of @xmath93 . nj quadrupole systematics may help clarify the concept of centrality : is an is eccentricity relevant for collisions ; if so how does it vary with ? the two - component ( soft+hard ) model ( tcm ) of hadron production in high energy nuclear collisions has been reviewed in refs . @xcite for collisions and refs . @xcite for collisions . the tcm serves first as a mathematical framework for data description and then , after comparisons with theory , as a basis for physical interpretation of data systematics . the tcm has been interpreted to represent two main sources of final - state hadrons : longitudinal projectile - nucleon dissociation ( soft ) and large - angle - scattered ( transverse ) parton fragmentation ( hard ) . in collisions the two processes scale respectively proportional to @xmath44 ( participant nucleons @xmath43 ) and @xmath45 ( binary encounters ) . analogous scalings for collisions were considered in ref . @xcite . the ( soft + hard ) tcm accurately describes most fs hadron yield and spectrum systematics @xcite , whereas there is no significant manifestation of the nj quadrupole in yields and spectra @xcite . in contrast , the nj quadrupole plays a major role in 2d angular correlations and is measurable as such even for collisions ( per this study ) . the tcm previously applied to yields and spectra must therefore be extended to include the nj quadrupole as a third component of all high - energy nuclear collisions . an effective tcm should be complete and self - consistent , capable of describing all aspects of data from any collision system . the joint single - charged - particle 2d ( azimuth integral ) density on @xmath1 and @xmath2 is denoted by @xmath95 . the @xmath2-averaged ( over @xmath37 ) spectrum is @xmath96 . the @xmath1-integral mean angular density is @xmath97 averaged over acceptance @xmath37 ( @xmath2 averages are considered in more detail in sec . [ etadensity ] ) . the @xmath5 dependence of @xmath1 spectra was determined in ref . @xcite , and mb spectra were decomposed into soft and hard components according to the tcm . in collisions soft and hard spectrum components have fixed forms but their relative admixture varies with @xmath5 @xcite . the relation of the hard component to pqcd theory was established in ref . @xcite . in collisions the soft component retains its fixed form but the hard - component form changes substantially with centrality , reflecting quantitative modification of jet formation @xcite . identification of the hard component with jets in and more - peripheral collisions is supported by data systematics and comparisons with pqcd theory @xcite . in more - central collisions a jet interpretation for the tcm hard component has been questioned @xcite . the two - component decomposition of @xmath1 spectra conditional on uncorrected @xmath98 integrated over angular acceptance @xmath37 within @xmath36 azimuth is denoted by @xcite [ ppspec ] |_0(y_t;n_ch ) & = & s(y_t;n_ch ) + h(y_t;n_ch ) + & = & |_s(n_ch ) s_0(y_t ) + |_h(n_ch ) h_0(y_t ) , where @xmath99 and @xmath100 are corresponding @xmath2-averaged soft and hard components with corrected @xmath101 ( see sec . [ etadensity ] ) . the inferred soft and hard @xmath1 spectrum shapes [ unit normal @xmath102 and @xmath103 are independent of @xmath98 and are just as defined in refs . @xcite . the fixed hard - component spectrum shape @xmath104 ( gaussian plus power - law tail ) is predicted quantitatively by measured fragmentation functions convoluted with a measured 200 gev minimum - bias jet spectrum @xcite . figure [ fig1a ] ( left ) shows @xmath1 spectra for several @xmath98 classes . the spectra ( uncorrected for tracking inefficiencies ) are normalized by corrected - yield soft component @xmath105 . a common @xmath1-dependent inefficiency function is introduced for comparison of this analysis with corrected spectra in ref . @xcite , indicated below @xmath106 by the ratio of the two dash - dotted curves representing uncorrected @xmath107 and corrected @xmath102 soft - component models . the data spectra are represented by spline curves rather than individual points to emphasize systematic variation with @xmath98 . left : normalized spectra for six multiplicity classes of 200 gev collisions ( @xmath108 see table [ multclass ] ) . @xmath102 is the soft - component model function for corrected ( upper dash - dotted ) and uncorrected ( lower dash - dotted ) data . @xmath105 is the soft - component multiplicity assuming @xmath109 ( see text ) . the spectra are averaged over acceptance @xmath110 . right : spectrum hard components in the form @xmath111 from eq . ( [ ppspec ] ) compared to hard - component model function @xmath112 ( dashed ) . bars and carets are omitted from the figure labels . , title="fig:",width=158,height=153 ] left : normalized spectra for six multiplicity classes of 200 gev collisions ( @xmath108 see table [ multclass ] ) . @xmath102 is the soft - component model function for corrected ( upper dash - dotted ) and uncorrected ( lower dash - dotted ) data . @xmath105 is the soft - component multiplicity assuming @xmath109 ( see text ) . the spectra are averaged over acceptance @xmath110 . right : spectrum hard components in the form @xmath111 from eq . ( [ ppspec ] ) compared to hard - component model function @xmath112 ( dashed ) . bars and carets are omitted from the figure labels . , title="fig:",width=158,height=153 ] figure [ fig1a ] ( right ) shows normalized spectra from the left panel in the form @xmath113 / \bar \rho_s \approx \alpha \hat h_0(y_t)$ ] with @xmath114 . from ref . @xcite we infer @xmath115 . given that empirical relation and @xmath116 as simultaneous equations we can obtain @xmath117 , @xmath105 and @xmath118 for any @xmath98 and @xmath37 ( see details in sec . [ etayt ] ) . note that although the data hard - component shapes for the lowest two @xmath98 values deviate significantly from the @xmath112 model the integrals on remain close to the value @xmath119 . the amplitude 0.33 of unit - normal @xmath120 corresponds to the maximum of the dashed curve @xmath121 . these spectrum results are consistent with those from ref . @xcite with @xmath122 ( see sec . [ etadep ] ) . in ref . @xcite the tcm spectrum hard - component yield @xmath123 within @xmath122 was observed to vary as @xmath124 , with uncorrected @xmath125 . a refined analysis provided the more precise density relation @xmath126 . as noted , the tcm soft - component density @xmath105 is then defined by the simultaneous equations @xmath127 , and @xmath128 for some @xmath129 , consistent with pqcd plus ref . @xcite . in ref . @xcite @xmath105 is associated with the number of _ participant low-@xmath3 partons _ ( gluons ) and dijet production , proportional to the number of participant - parton binary encounters , is then observed to scale @xmath130 . based on a dijet interpretation for the hard component we define @xmath131 , where @xmath132 is the dijet frequency per collision and per unit @xmath2 , @xmath133 $ ] is the average fraction of a dijet appearing in acceptance @xmath37 , and @xmath134 is the mb mean dijet fragment multiplicity in @xmath135 . dijet fraction @xmath136 should be distinguished from initial - state eccentricity @xmath137 associated with the nj quadrupole . we also distinguish among number of dijets , number of jets , dijet mean fragment multiplicity and jet mean fragment multiplicity ( their values integrated over @xmath135 vs within some limited detector acceptance @xmath37 ) . the definition of @xmath118 above separates factors @xmath138 $ ] and @xmath139 that were combined in previous studies . the @xmath139 values estimated here are thus approximately a factor 2 ( i.e. @xmath140 ) larger than previous estimates @xcite . for 200 gev nsd collisions with @xmath141 and dijet mean fragment multiplicity @xmath142 derived from measured jet systematics the inferred jet frequency @xmath143 is inferred from spectrum data within @xmath122 . that value can be compared with the pqcd prediction @xmath144 for 200 gev collisions @xcite based on a measured jet spectrum and nsd cross section corresponding to mean - value parton distribution functions . thus , the observed nsd spectrum hard - component yield @xmath123 @xcite is quantitatively consistent with measured dijet systematics derived from event - wise jet reconstruction @xcite . if a non - nsd event sample with arbitrary mean @xmath98 is selected we employ empirical trends consistent with ref . @xcite and having their own pqcd implications , as discussed in ref . we assume for the present study that the dijet frequency varies with soft multiplicity as [ nj1 ] f(n_ch ) & & 0.027 ^2 with @xmath145 for 200 gev collisions . we define @xmath146 as the _ dijet number _ within some angular acceptance @xmath37 . for the @xmath98 range considered in this study the fraction of hard hadrons @xmath147 is not more than about 15% . the final state is never dominated by hard processes but mb dijet production does play a major role , especially for @xmath148 ( @xmath149 gev / c ) where _ most jet fragments appear_. mb two - particle correlations have been studied extensively for nsd collisions @xcite and collisions @xcite . a correspondence between jet - related correlations and @xmath1-spectrum hard components has been established quantitatively in refs . angular - correlation structure is consistent with extrapolation of the centrality systematics of angular correlations from collisions @xcite . both correlations on transverse rapidity @xmath150 and 2d angular correlations on @xmath38 from collisions are described by the tcm . @xmath150 correlations for 200 gev collisions are fully consistent with the sp spectrum results described above and in ref . the hard component corresponds ( when projected onto 1d @xmath1 ) to the hard component of eq . ( [ ppspec ] ) . figure [ ppcorr ] ( left panel ) shows @xmath150 correlations for 200 gev approximately nsd collisions . the logarithmic interval @xmath151 $ ] corresponds to @xmath152 $ ] gev / c . the two peak features correspond to tcm soft and hard components . the 2d hard component with mode near = 2.7 ( @xmath153 gev / c ) corresponds quantitatively to the 1d sp spectrum hard component modeled by @xmath104 in ref . the soft component ( us pairs only ) is consistent with longitudinal fragmentation ( dissociation ) of projectile nucleons manifesting local charge conservation . ( color online ) ( a ) minimum - bias correlated - pair density on 2d transverse - rapidity space @xmath150 from 200 gev collisions showing soft ( smaller ) and hard ( larger ) components as peak structures . ( b ) correlated - pair density on 2d angular difference space @xmath38 . hadrons are selected with @xmath154 gev / c ( @xmath155 ) . nevertheless , features expected for dijets are observed : ( i ) same - side 2d peak representing intrajet correlations and ( ii ) away - side 1d peak on azimuth representing interjet ( back - to - back jet ) correlations @xcite . , title="fig:",width=158,height=134 ] ( -85,92 ) * ( a ) * ( color online ) ( a ) minimum - bias correlated - pair density on 2d transverse - rapidity space @xmath150 from 200 gev collisions showing soft ( smaller ) and hard ( larger ) components as peak structures . ( b ) correlated - pair density on 2d angular difference space @xmath38 . hadrons are selected with @xmath154 gev / c ( @xmath155 ) . nevertheless , features expected for dijets are observed : ( i ) same - side 2d peak representing intrajet correlations and ( ii ) away - side 1d peak on azimuth representing interjet ( back - to - back jet ) correlations @xcite . , title="fig:",width=158,height=134 ] ( -85,92 ) * ( b ) * figure [ ppcorr ] ( right panel ) shows 2d angular correlations on difference variables @xmath38 . the hadron values for that plot are constrained to lie near 0.6 gev / c ( just above = 2 ) , corresponding to the saddle between soft and hard peaks in the left panel . although the hadron is very low the structures expected for jet angular correlations are still clearly evident : a ss 2d peak at the origin representing intrajet correlations and a 1d peak on azimuth at @xmath156 corresponding to interjet correlations between back - to - back jet pairs . the volume of the ss 2d peak corresponds quantitatively to the hard component of the total hadron yield inferred from @xmath1 spectrum data and to pqcd calculations @xcite . the soft component , a narrow 1d gaussian on @xmath73 including only us charge pairs , is excluded by the @xmath157 gev / c cut @xcite . angular correlation systematics have been compared to the qcd monte carlo pythia @xcite , and general qualitative agreement is observed @xcite . the correlation measure @xmath55 proportional to the number of correlated pairs per final - state hadron @xcite is analogous to ratio @xmath158 given @xmath159 correlated - pair number . in the present study we extend the results by measuring systematic variations of 2d angular correlations with parameter @xmath98 . ( -120,115 ) * ( a ) * ( -120,115 ) * ( b ) * ( -120,115 ) * ( c ) * + ( -120,115 ) * ( d ) * ( -120,115 ) * ( e ) * ( -120,115 ) * ( f ) * two - particle angular correlations are obtained with the same basic methods as employed in refs . @xcite . data for this analysis were obtained from a mb sample of collisions at @xmath160 gev . charged particles were detected with the star time projection chamber ( tpc ) . the acceptance was @xmath36 azimuth , pseudorapidity @xmath161 , and @xmath24 gev / c . the _ observed _ ( uncorrected ) charge multiplicity within the @xmath2 acceptance is denoted by @xmath98 , whereas the efficiency - corrected and @xmath6-extrapolated _ true _ event multiplicity in the acceptance is denoted by @xmath5 with corrected mean angular density @xmath162 within acceptance @xmath37 . seven event classes indexed by the observed charged - particle multiplicity are defined in table [ multclass ] . the range of corrected particle density @xmath163 is approximately 2 - 20 particles per unit @xmath2 . this analysis is based on 6 million ( m ) events , compared to 3 m events for the @xmath6-spectrum study in ref . @xcite . .multiplicity classes based on the observed ( uncorrected ) multiplicity @xmath98 falling within acceptance @xmath161 or @xmath110 . the efficiency - corrected density is @xmath162 . event numbers are given in millions ( m = @xmath164 ) . tcm parameters include @xmath109 and @xmath165 . [ cols="^,^,^,^,^,^,^,^",options="header " , ] alteration of the ss 2d peak is also notable . visually the peak appears to narrow dramatically on @xmath39 with increasing . however , 2d model fits reveal that the fitted peak width decreases only slightly . the _ apparent _ narrowing is due to superposition of the nj quadrupole component onto the 2d peak structure . the change from left to right panel is dominated by the ten - fold increase of curvature ratio @xmath166 , as indicated in fig . [ curvature ] ( right ) . in ref . @xcite the systematics of 2d angular correlations derived from 62 and 200 gev collisions were extrapolated first to peripheral collisions ( as a proxy for nsd collisions ) and then to 7 tev for comparisons with cms data . the question posed : are ss 2d peak , as dipole and nj quadrupole systematics at and below 200 gev consistent with those at 7 tev and especially with the emergence of a ss `` ridge '' structure for certain conditions imposed at that energy ? as to estimates , at 200 gev the nsd values of @xmath167 and @xmath168 from the present study are numerically consistent with the extrapolation to collisions . the value of @xmath169 for 200 gev collisions was overestimated by a factor 2 by addition of a conjectured quadrupole contribution from the as 1d peak modeled as a 1d gaussian . according to the present study the as peak for collisions is actually well described by a single dipole element . it was demonstrated that dijet production at 7 tev is consistent with extrapolation from 200 gev using factor @xmath170 ( derived from comparison of 62 and 200 gev data ) interpreted to describe the increase of participant low-@xmath3 gluons with increasing collision energy . that factor applies to the _ per - particle _ ss 2d peak amplitude @xmath167 ( intrajet correlations ) , whereas increase of as dipole amplitude @xmath168 ( jet - jet correlations ) is considerably less ( consistent with no amplitude increase from 62 to 200 gev @xcite . nj quadrupole measurements at rhic suggest that @xmath169 also increases by factor 2.3 . as to multiplicity trends for collisions , in the present study the corrected charge density for multiplicity class @xmath171 is @xmath172 , 5.5 times the mb value 2.5 , whereas at 7 tev the cms @xmath173 multiplicity class corresponds to corrected @xmath174 , 4.8 times the mb value 5.8 . in this study the @xmath171 class corresponds to measured four - fold increase of @xmath168 and @xmath167 and fifteen - fold increase of @xmath169 over their mb values . as to responses to cuts we note that about half of all mb jet fragments appear below the mode of the 200 gev spectrum hard component at 1 gev / c @xcite . in contrast , the mode of @xmath175 on is close to 3 gev / c @xcite . the cms cut @xmath176 $ ] gev / c is effectively a lower limit at 1 gev / c , since the hadron spectrum falls rapidly with . we thus expect that a lower limit imposed at 1 gev / c should reduce the as dipole substantially more than the nj quadrupole . we have reduced @xmath168 by factor 1/2 and @xmath169 by factor 2/3 to estimate the effect of cuts . table [ table ] summarizes the various estimates . the first two rows report results from the present study and the curvature ratios shown in fig . [ curvature ] . as noted in the text the 7 tev mb values are obtained by multiplying @xmath167 and @xmath169 by 2.3 and @xmath168 by 1.4 . the values for @xmath177 are obtained with factors 4 and 15 applied as for the 200 gev values for @xmath171 ( ignoring the small difference in ratios to mb multiplicities between rhic and lhc energies ) . the effect of the cut is estimated by factors 1/2 and 2/3 as noted above . results from the present study describe the reported cms 7 tev 2d angular correlations quantitatively and are generally consistent with ref . @xcite but also provide insight into the physical origins of the reported ss `` ridge . '' the large collision - energy increase combined with imposed @xmath6 and multiplicity cuts increases the nj quadrupole amplitude eight - fold relative to the as 1d jet peak , changing the ss curvature sign and producing an apparent ss ridge . in effect , the ss azimuth curvature functions as a comparator , switching from valley to ridge as one amplitude increases relative to another . a quantitative curvature change is transformed to a qualitative shape change ( mis)interpreted as emergence of a novel phenomenon at a higher energy . several open issues for high - energy collisions were summarized in sec . [ issues ] : ( a ) the role of collision centrality , ( b ) the definition and nature of the underlying event or ue , ( c ) the systematics of mb dijet production and ( d ) confirmed existence and characteristics of a nj quadrupole component in angular correlations . we return to those points in light of results from this study . the measured hard components of spectra , @xmath2 densities and 2d angular correlations presented in this study complete a unified experimental and theoretical picture of mb dijet production ( no cuts ) established previously for collisions @xcite . there were no previous measurement of a nj quadrupole component for collisions . the combined dijet and quadrupole results from the present study convey important implications for claims of collectivity ( flows ) , centrality , ue studies and the mechanism of the cms ridge . the term `` collective '' or `` collectivity '' ( e.g. as recently applied to small collision systems at the lhc ) is ambiguous , since jet formation is a form of `` collectivity '' as is the nj quadrupole whatever its production mechanism . introducing the term `` collectivity '' as synonymous with `` flow '' may produce confusion . there are certainly collective aspects of collisions although it is unlikely that hydrodynamic flow ( in the sense of fluid motion resulting from particle rescattering ) plays a role . dijet production and the nj quadrupole amplitude follow characteristic trends on @xmath105 precisely over a large range of amplitudes ( 100-fold for dijets , 1000-fold for quadrupole as correlated - pair numbers ) while the underlying particle ( participant - gluon ) density varies 10-fold . how can a small collision system with extremely low particle density support a hydrodynamic phenomenon that conspires to follow the same trends over such a large density interval ? the notion of centrality ( impact parameter ) for collisions is ambiguous in principle . concerning centrality several questions arise : what does `` is geometry '' mean for collisions ? is an impact parameter relevant ? how are total , triggered dijets , transverse low-@xmath3 parton ( gluon ) density and centrality correlated ? how do those factors relate to measured ensemble - mean pdfs , _ event - wise _ participant - parton distributions on @xmath3 and initial momentum transfer ? there are certainly large event - wise fluctuations in soft - hadron and dijet production , but whether those correspond to fluctuations of is transverse geometry or some other collision aspect is in question . the need for comprehensive study of the dependence of angular correlations in relation to centrality was one motivation for the present study . results from this study support two arguments against a major role for a centrality concept : ( a ) dijet production scales as @xmath178 but the eikonal approximation implies binary - collision scaling as @xmath179 ( as for collisions ) . the observed dijet trend is consistent with encounters between all possible participant - gluon pairs in each collision , not a smaller subset determined by an impact parameter . ( b ) the nj quadrupole amplitude scales as @xmath180 over a large range consistent with _ part _ of the @xmath181 trend observed for collisions , but there is no significant reduction with a decreasing eccentricity . the combined trends suggest that is geometry is not a determining factor for either phenomenon . instead , the event - wise depth of penetration on momentum fraction @xmath3 of the projectile wave function may be the main source of variation for soft , hard and quadrupole components . as noted in sec . [ issues ] ue studies rely on several assumptions : ( a ) concentration of low-@xmath3 gluons at small radius in the proton ( inferred from dis data ) provides a correlation among centrality , soft hadron production and dijet production , ( b ) the conventionally - defined tr on azimuth includes no contribution from a triggered dijet and ( c ) multiple - parton interactions ( mpi ) may occur in jet - triggered collisions . the integrated tr yield denoted by @xmath10 is observed to increase with increasing jet - trigger condition and is interpreted to represent a soft background increasing with centrality . reference @xcite addressed part of that narrative with simulations based on the tcm for hadron production from collisions . it concluded that most dijets ( what would result from an applied trigger at lower ) emerge from _ low_-multiplicity collisions . spectrum studies already indicated that higher - multiplicity collisions do produce dijets at higher rates but are few in number and so contribute only a small fraction of the -triggered event population . a @xmath182 condition can not significantly alter the soft component or collision centrality ( if relevant ) . the present study adds the following new information : ( a ) 2d angular correlations confirm a strong contribution from mb dijets _ within the tr_. ( b ) the dijet production trend @xmath130 suggests that the eikonal approximation is invalid and that centrality is not a useful concept for collisions . ( c ) monotonic increase of the nj quadrupole @xmath183 over a large range also suggests that centrality , as manifested in a varying is eccentricity , is not a useful concept . those factors confirm the conclusions of ref . @xcite and lead to the following scenario for @xmath10 variation with a trigger : as the trigger condition is increased from zero the integrated spectrum soft component increases @xmath10 from zero to a plateau on @xmath11 . the hard - component ( jet ) contribution to @xmath10 is similarly integrated up to a plateau . @xmath10 thus has both soft and hard components exhibiting plateau structures slightly displaced from one another on @xmath11 . almost all events satisfying an increased trigger condition contain a dijet ( are hard events ) but remain characteristic of a mb population with smaller soft multiplicity , not the expected more - central population with larger soft multiplicity . given the observed -dependent structure of 2d angular correlations we arrive at three conclusions : ( a ) all dijets include a large - angle base that strongly overlaps the tr . that base dominates minimum - bias jets but may persist within all higher - energy ( e.g. -triggered ) dijets . ( b ) the region with a minimal dijet contribution that might suffice for ue studies is defined by @xmath184 and @xmath185 ( see fig . [ quadcomp ] ) . an immediate example of novel ue structure that might be discovered there is provided by the cms `` ridge , '' a manifestation of the nj quadrupole that was not expected in collisions . ( c ) the likelihood of multiple dijet production in -triggered events ( which retain a low soft multiplicity as noted ) is small whereas the likelihood of multiple dijets in high - multiplicity events approaches unity . the usual interpretation of @xmath10 trends in terms of mpi may be misleading . arguments against the tcm have been presented since commencement of rhic operation . it has been noted that the hijing monte carlo @xcite ( based on pythia @xcite ) fails to describe rhic and lhc data . that failure as been expressed as `` too slow increase '' of hadron production with centrality and energy @xcite . hijing is assumed to represent the tcm and its failure is then confused with failure of the tcm itself , of which hijing is only a specific theory implementation . the problems with hijing are traceable to the eikonal - model assumption included in default pythia @xcite . such arguments typically rely on data from a centrality range covering only the more - central 40 - 50% of the cross section @xcite . the critical centrality region extending from or collisions to the _ sharp transition _ in jet formation @xcite is not considered . alternative models that seem to describe the more - central data are actually falsified by more - peripheral data @xcite . an alternative argument is based on assuming that tcm agreement with data is accidental , that a _ constituent - quark _ model ( soft only , excluding jets ) is more fundamental and describes data as well @xcite , but that argument is questionable @xcite . the tcm invoked in this study is based on previous analysis of spectrum @xmath5 and centrality dependence @xcite , fluctuations and correlations @xcite , transverse - rapidity @xmath150 correlations @xcite and minimum - bias 2d number angular correlations @xcite . in each case model elements were determined quantitatively by systematic analysis without regard to physical mechanisms . only after the tcm was so established were connections with theory and physical interpretations introduced . in the present study we extend the tcm to describe the dependence of 200 gev @xmath2 densities and 2d angular correlations . in the latter we observe for the first time a significant nj quadrupole component and its dependence as a novel nonjet phenomenon within the ue . we find that the extended tcm remains fully self - consistent and provides accurate and efficient representation of a large body of data . we report measurements of the charge - multiplicity dependence of single - particle ( sp ) densities on transverse rapidity ( as spectra ) and pseudorapidity @xmath2 and 2d angular correlations on @xmath186 from 200 gev collisions . the sp densities are described accurately by a two - component ( soft + hard ) model ( tcm ) of hadron production . the inferred -spectrum tcm is consistent with a previous study . the result for @xmath2 densities newly reveals the distribution on @xmath2 of minimum - bias ( mb ) jet fragments . 2d angular correlations are fitted with a multi - element fit model previously applied to data from 62 and 200 gev collisions . fit residuals are consistent with statistical uncertainties in all cases . trends for several 2d correlation model parameters are simply expressed in terms of tcm soft - component multiplicity @xmath187 or mean density @xmath188 ( @xmath37 is a detector acceptance ) . correlated - pair numbers for soft component ( projectile dissociation ) scale @xmath189 , for hard component ( dijet production ) scale @xmath130 and for nonjet ( nj ) quadrupole scale @xmath183 . the nj quadrupole amplitude is quite significant for higher - multiplicity collisions . the dijet production trend is inconsistent with an eikonal approximation for collisions ( which would require dijets @xmath190 ) , and the monotonically - increasing nj quadrupole trend is inconsistent with an initial - state eccentricity determined by impact parameter . the two trends combined suggest that centrality is not a useful concept for collisions . fluctuations may instead depend on the event - wise depth of penetration on momentum fraction @xmath3 of the projectile wave functions and in turn on the number of participant low-@xmath3 partons . 2d angular - correlation data are in conflict with assumptions relating to the underlying event ( ue , the complement to a triggered dijet ) . the azimuth transverse region ( tr ) bracketing @xmath191 is assumed to contain no contribution from a triggered dijet , but minimum - bias dijets are observed to make a strong contribution there . the region with minimal jet contribution is defined by @xmath184 near the azimuth origin that excludes the same - side 2d jet peak and most of the away - side 1d jet peak . the presence of a significant nj quadrupole component and its multiplicity trend have several implications : ( a ) initial - state transverse geometry does not appear to be a useful concept for collisions as noted above , ( b ) the appearance of a nj quadrupole component in a small system with negligible particle density contradicts the concept of a hydro phenomenon based on particle rescattering and large energy / particle density gradients and ( c ) the same - side `` ridge '' observed in collisions at the large hadron collider ( lhc ) , interpreted by some to suggest `` collectivity '' ( flows ) in small systems , results from an interplay of the jet - related away - side 1d peak and the nj quadrupole that together determine the curvature on azimuth near the origin . when that curvature transitions from positive to negative ( depending on collision energy and other applied cuts ) a same - side `` ridge '' appears . in a hydro narrative the nj quadrupole component interpreted as elliptic flow should represent azimuth modulation of radial flow detected as a modification of sp spectra . but no corresponding modification is observed in spectra despite precise differential analysis . t. a. trainor , d. t. kettler , d. j. prindle and r. l. ray , j. phys . g * 42 * , no . 2 , 025102 ( 2015 ) . d. t. kettler , d. j. prindle and t. a. trainor , phys . c * 91 * , 064910 ( 2015 ) . d. t. kettler ( star collaboration ) , eur . j. c * 62 * , 175 ( 2009 ) . d. kettler ( star collaboration ) , j. phys . . ser . * 270 * , 012058 ( 2011 ) . t. a. trainor and d. t. kettler , phys . c * 84 * , 024910 ( 2011 ) . t. a. trainor , phys . d * 87 * , no . 5 , 054005 ( 2013 ) . t. a. trainor , mod . a * 23 * , 569 ( 2008 ) . t. a. trainor , int . j. mod . e * 17 * , 1499 ( 2008 ) . t. a. trainor , phys . c * 80 * , 044901 ( 2009 ) . g. agakishiev , _ et al . _ ( star collaboration ) , phys . c * 86 * , 064902 ( 2012 ) . t. a. trainor and d. t. kettler , phys . c * 83 * , 034903 ( 2011 ) . s. gavin , l. mclerran and g. moschelli , phys . c * 79 * , 051902 ( 2009 ) . t. a. trainor , phys . c * 91 * , no . 4 , 044905 ( 2015 ) . b. z. kopeliovich , a. h. rezaeian and i. schmidt , phys . d * 78 * , 114009 ( 2008 ) . cms collaboration , jhep * 1009 * , 091 ( 2010 ) . t. a. trainor , phys . c * 90 * , no . 2 , 024909 ( 2014 ) t. a. trainor , phys . c * 92 * , no . 2 , 024915 ( 2015 ) . j. adams _ et al . _ ( star collaboration ) , phys . c * 73 * , 064907 ( 2006 ) . j. adams _ et al . _ ( star collaboration ) , phys . b * 634 * , 347 ( 2006 ) . t. a. trainor , r. j. porter and d. j. prindle , j. phys . g * 31 * , 809 ( 2005 ) . r. j. porter and t. a. trainor ( star collaboration ) , acta phys . b * 36 * , 353 ( 2005 ) . t. a. trainor , phys . d * 89 * , no . 9 , 094011 ( 2014 ) . t. a. trainor , j. phys . g * 37 * , 085004 ( 2010 ) . t. a. trainor , phys . c * 81 * , 014905 ( 2010 ) . ya . i. azimov , yu . l. dokshitzer , v. a. khoze , s. i. troyan , z. phys . c * 27 * , 65 ( 1985 ) , z. phys . c * 31 * , 213 ( 1986 ) . g. wolschin , phys . c * 91 * , 014905 ( 2015 ) . t. a. trainor , j. phys . g * 40 * , 055104 ( 2013 ) . t. a. trainor , d. j. prindle and r. l. ray , phys . c * 86 * , 064905 ( 2012 ) . m. b. de kock , h. c. eggers and t. a. trainor , phys . c * 92 * , no . 3 , 034908 ( 2015 ) . b. i. abelev _ et al . _ ( star collaboration ) , phys . c * 80 * , 064912 ( 2009 ) . t. a. trainor , j. phys . g * 39 * , 095102 ( 2012 ) m. gyulassy and l. mclerran , nucl . a * 750 * , 30 ( 2005 ) . b. b. back _ et al . _ ( phobos collaboration ) , phys . b * 578 * , 297 ( 2004 ) . d. kharzeev and e. levin , phys . b * 523 * , 79 ( 2001 ) . s. s. adler _ et al . _ ( phenix collaboration ) , phys . c * 89 * , no . 4 , 044905 ( 2014 ) .
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an established phenomenology and theoretical interpretation of @xmath0-@xmath0 collision data at lower collision energies should provide a reference for @xmath0-@xmath0 and other collision systems at higher energies , against which claims of novel physics may be tested .
the description of @xmath0-@xmath0 collisions at the relativistic heavy ion collider ( rhic ) has remained incomplete even as claims for collectivity and other novelties in data from smaller systems at the large hadron collider ( lhc ) have emerged recently . in this study
we report the charge - multiplicity dependence of two - dimensional ( 2d ) angular correlations and of single - particle ( sp ) densities on transverse rapidity @xmath1 and pseudorapidity @xmath2 from 200 gev @xmath0-@xmath0 collisions .
we define a comprehensive and self - consistent two - component ( soft + hard ) model ( tcm ) for hadron production and report a significant @xmath0-@xmath0 nonjet ( nj ) quadrupole component as a third ( angular - correlation ) component .
our results have implications for @xmath0-@xmath0 centrality , the underlying event ( ue ) , collectivity in small systems and the existence of flows in high - energy nuclear collisions .
| 16,935 | 350 |
since the early years of quantum mechanics @xcite the principle of quantum superposition has been recognized to play a prominent role in the theory and its applications . the destruction and preservation of these superpositions of quantum states occupy a central place in issues such as the quantum - to - classical transition @xcite and potential technological applications in quantum information , computation and cryptography @xcite . from a physical standpoint the loss of coherence in quantum systems is rooted on the pervasive action of the environment upon the system . this environmental action has received a careful mathematical treatment ( cf . @xcite and multiple references therein ) going from a constructive approach based on disregarding the degrees of freedom of the environment due to their lack of control by the experimenter ( `` tracing - out '' methods ) to an axiomatic approach based on the initial setting of physically motivated axioms to derive an appropiate evolution ( master ) equation for the system @xcite . + most of these master equations ( me s hereafter ) satisfy the markov approximation ( semigroup condition ) and can be put into the lindblad form : @xmath0+\frac{1}{2}\sum_{j}\left\{[v_{j}\rho(t),v_{j}^{\dagger}]+[v_{j},\rho(t)v_{j}^{\dagger}\right\}\ ] ] where @xmath1 is the hamiltonian of the system and @xmath2 are operators ( so - called lindblad operators ) containing the effect of the environment upon the system . indeed in the axiomatic approach the markov approximation is posed as an initial hypothesis @xcite , thus rendering highly difficult a generalization to nonmarkovian situations . + in this work we develop a novel attempt to derive me s both in the markovian and the nonmarkovian regimes using stochastic methods @xcite jointly with well - known operator techniques commonly used in quantum mechanics @xcite . the main idea consists of building _ random _ evolution operators ( evolution operators with one or several stochastic parameters in it ) which contains the decohering effect of the environment and then taking the stochastic expectaction value with respect to this ( uncontrollable ) randomness . the paper is organized as follows . in section [ lindasrand ] we state and prove our main ( though still somewhat partial ) result , namely that any lindblad - type me , either markovian or nonmarkovian , with selfadjoint lindblad operators can be understood as an averaged random unitary evolution . in section [ analcons ] we discuss some first mathematical consequences of this result such as a very fast method to solve me s provided the unitary solutions are known ; we illustrate this by solving the phase - damping me for the multiphoton resonant jaynes - cummings model in the rotating - wave approximation @xcite ( section [ soldampjcm ] ) . we then comment in section [ nonmarkevol ] two immediate consequences , namely both markovian and nonmarkovian regimes are attainable under the same mathematical formalism and the lindbladian structure with selfadjoint lindblad operators is shown to have an origin independent of the markov approximation . in section [ intrdecoh ] we show how the flexibility of the mathematical language employed can easily generalize some intrinsic decoherence models present in the literature @xcite . in section [ rabiqed ] we discuss the previous main result under a more physical spirit by proposing a slight generalization of the jaynes - cummings model ( section [ stojcm ] ) , comparing this proposal with experimental results in optical cavities experiments ( section [ qed ] ) and finally ( section [ iondecay ] ) showing how the proposed formalim can account for reported exponential decays of rabi oscillations in ion traps . we include in section [ discuss ] some important comments regarding a brief comparison with existing models of stochastic evolution in hilbert space , the possibility of intrinsic decoherence phenomena and future prospects . conclusions and a short appendix close the paper . the main result whose consequences are to be discussed below is the following : _ every lindblad evolution with selfadjoint lindblad operators can be understood as an averaged random unitary evolution_. we will analyse this proposition in detail . the objective is to reproduce the lindblad equation . ] @xmath3+\frac{1}{2}\sum_{i=1}^{n}\{[v_{i}\rho(t),v_{i}]+[v_{i},\rho(t)v_{i}]\}\nonumber\\ \label{lindeq}&=&-i[h,\rho(t)]-\frac{1}{2}\sum_{i=1}^{n}[v_{i},[v_{i},\rho(t)]]\end{aligned}\ ] ] by adequately modifying chosen parameters in the original evolution operator . for simplicity let us start by considering the case @xmath4 . we will first study the case where the hamiltonian @xmath1 and the ( selfadjoint ) lindblad operator @xmath5 commute . it is very convenient to introduce the following notation . the commutator between an operator @xmath6 and @xmath7 will be denoted by @xmath8\equiv[g , x]$ ] . thus the von neumann - liouville operator will be @xmath9 , where @xmath1 denotes the hamiltonian ( @xmath10 ) . then the lindblad equation with @xmath4 can be arrived at by 1 . adding a stochastic term @xmath11 to the argument of the evolution operator : + @xmath12 + where @xmath13 denotes standard real brownian motion @xcite . 2 . taking the stochastic average with respect to @xmath13 in the density operator deduced from @xmath14 : + @xmath15\ ] ] + where @xmath16 denotes the expectation value with respect to the probability measure of @xmath13 . the proof of this result is nearly immediate . taking advantage of the commutativity of @xmath1 and @xmath5 and making use of theorem 3 in @xcite ( cf . appendix ; relation ) we may write for the density operator : @xmath17[\rho(0)]\ ] ] thus all we have to do is to calculate the expectation value of the random superoperator @xmath18 . developing the exponential into a power series and recalling @xcite @xmath19=\frac{(2n)!}{2^{n}n!}t^{n}$ ] if @xmath20 is even and @xmath19=0 $ ] otherwise , we arrive at @xmath21\nonumber\\ & = & \exp(-t\mathcal{l}-\frac{t}{2}\mathcal{c}_{v}^{2})[\rho(0)]\end{aligned}\ ] ] which produces the desired master equation : @xmath22-\frac{1}{2}[v,[v,\rho(t)]]\ ] ] when the hamiltonian @xmath1 and the lindblad operator @xmath5 do not commute , the previous method is not suitable , since the calculation of the expectation value can not be performed in the same way . a way to circumvent this problem is to proceed in the same way as before but in the heisenberg picture . thus let @xmath23 be the original evolution operator in the schrdinger picture . the corresponding evolution operator in the heisenberg picture will trivially be @xmath24 . as before we proceed in steps : 1 . [ nonc1 ] add a stochastic term to the argument of the evolution operator : @xmath25 + where @xmath26 denotes time - ordering and @xmath27 is the operator @xmath5 in the heisenberg representation . [ nonc2 ] take the stochastic average with respect to @xmath13 in the corresponding density operator : + @xmath28\ ] ] 3 . [ nonc3 ] finally to arrive at change to the schrdinger representation . a comment should be made . the stochastic term added to the original evolution operator in is a natural generalization of the one added in . the ito integration now appears as a consequence of the time dependence of the operator to be added : @xmath27 . note that when @xmath29=0 $ ] , @xmath30 and the stochastic term reduces to @xmath31 as before + now to perform the previous tasks is a bit more involved . firstly combining relation and the @xmath32operation , the step [ nonc1 ] can be carried over : @xmath33=\nonumber\\ \label{expvalnonc}&=&\mathcal{t}\mathbb{e}[e^{-i\int_{0}^{t}\mathcal{c}_{v_{h}(s)}d\mathcal{b}_{s}}][\rho(0)]\end{aligned}\ ] ] the expectation value can be evaluated by resorting to functional techniques @xcite . recall that the characteristic functional of a stochastic process @xmath34 is defined as @xmath35=\mathbb{e}\left[\exp\left(i\int k(t)\chi_{t}dt\right)\right]\ ] ] where @xmath36 is an arbitrary real - valued function . in particular , for white noise @xmath37 ( cf . @xcite ) @xmath38&=&\mathbb{e}[\exp\left(i\int k(s)d\mathcal{b}_{s}\right)]=\nonumber\\ & = & e^{-\frac{1}{2}\int_{0}^{t}k^{2}(s)ds}\end{aligned}\ ] ] where @xmath39 , @xmath40 and @xmath41=\delta(t - s)$ ] have been used . from this it is then clear that can be written as @xmath42\ ] ] back to the schrdinger picture , the master equation derived from is @xmath22-\frac{1}{2}[v,[v,\rho(t)]]\ ] ] the generalization to many lindblad operators is elementary : all we have to do is to use the n - dimensional standard real brownian motion @xcite @xmath43 . the strategy is the same . the first consequences one can derive from the previous result are of analytical fashion . as an immediate aplication we will show how the jaynes - cummings model with phase damping in the rotating - wave approximation can be solved provided we know the solution to the original jaynes - cummings model . as a second consequence we will discuss how the previous result can be generalized to nonmarkovian situations , thus providing a common language for both markovian and nonmarkovian evolutions . finally it is shown how existing intrinsic decoherence models are naturally generalized using this formalism . the jaynes - cummings model ( jcm hereafter ) @xcite shows an undoubtable relevance in the study of quantum systems in different fields such as quantum optics , nuclear magnetic resonance or particle physics . it is an exactly solvable model which allows us to study specifically quantum properties of nature such as electromagnetic field quantization or periodic collapses and revivals in atomic population . the jcm describes the evolution of a two - level quantum system ( the atom ) interacting with a mode of the electromagnetic field under certain approximations ( rotating wave approximation , dipole approximation , etc . @xcite for details ) . usually in normal experimental conditions this will be an idealization and the environment should be taken into account , the effect of which can be very appropiately treated introducing a phase - damping term @xcite . thus the master equation for this system will read @xmath44-\frac{\gamma}{2}[h,[h,\rho(t)]]\ ] ] where @xmath1 is the hamiltonian of the system and @xmath45 is a damping constant . here we will show how can be very easily solved when @xmath1 is the resonant multiphoton jc hamiltonian ( cf . @xcite for an alternative approach ) , i.e. when @xmath46 where @xmath47 denotes the frequency of the field mode , @xmath48 is the atomic transition frequency , @xmath49 is the atom - field coupling constant , @xmath50 and @xmath51 are the mode creation and annihilation operators respectively , @xmath52 is the atomic - inversion operator and @xmath53 are the atomic `` raising '' and `` lowering '' operators . an exact resonance is assumed , thus @xmath54 . + we will focus in two quantities of relevant physical meaning , namely the atomic inversion @xmath55 $ ] and the photon number distribution at time @xmath56 : @xmath57 . to compare with methods found in the literature @xcite we will restrict to the case in which initially the atom is in its excited state @xmath58 and the electromagnetic field is in a coherent state @xmath59 , with @xmath60 . the unitary evolution ( @xmath61 ) provides the following expressions for these quantities : @xmath62\\ p_{n}(t)&=&|q_{n}|^2\cos^{2}\left[\lambda t\sqrt{\frac{(n+m)!}{n!}}\right]+\nonumber\\ & + & |q_{n - m}|^{2}\sin^{2}\left[\lambda t\sqrt{\frac{(n+m)!}{n!}}\right]\end{aligned}\ ] ] the objective is to calculate these same quantities when the phase damping term is present in , i.e. when @xmath63 . we will make use of the result proved in the previous section and note that the equation can be obtained by adding a stochastic term to the original evolution operator and then performing the stochastic average . in our case , @xmath64 , which obviously commutes with the hamiltonian , thus we are in the first case . the original evolution operator is promoted to @xmath65 equivalently we may think that it is @xmath56 which is promoted @xmath66 . thus to arrive at the desired `` phase - damped '' expressions @xmath67 and @xmath68 all we have to do is to add a stochastic term to the time variable @xmath56 and then perform the average : @xmath69\right]\nonumber\\ & & \\ p_{n}^{pd}(t)&=&\mathbb{e}\left[|q_{n}|^2\cos^{2}\left[\lambda(t+\gamma^{1/2}\mathcal{b}_{t})\sqrt{\frac{(n+m)!}{n!}}\right]\right.+\nonumber\\ & + & \left.|q_{n - m}|^{2}\sin^{2}\left[\lambda(t+\gamma^{1/2}\mathcal{b}_{t})\sqrt{\frac{(n+m)!}{n!}}\right]\right]\nonumber\\\end{aligned}\ ] ] using the linearity property of the expectation value and recalling the moments of the standard real brownian motion ( cf . above and @xcite ) , the previous calculations can be carried over elementarily using ( see appendix [ appendix ] ) : @xmath70&=&e^{-2\gamma\lambda^{2}t\frac{(n+m)!}{n!}}\times\nonumber\\ & \times&\cos\left[2\lambda t\sqrt{\frac{(n+m)!}{n!}}\right]\nonumber\\ & & \end{aligned}\ ] ] hence @xmath71\nonumber\\ & & \\ p_{n}^{pd}(t)&=&\frac{1}{2}|q_{n}|^{2}\left\{1+e^{-2\gamma\lambda^{2}t\frac{(n+m)!}{n!}}\right\}\cos\left[2\lambda t\sqrt{\frac{(n+m)!}{n!}}\right]+\nonumber\\ & + & \frac{1}{2}|q_{n - m}|^{2}\left\{1-e^{-2\gamma\lambda^{2}t\frac{(n+m)!}{n!}}\right\}\cos\left[2\lambda t\sqrt{\frac{(n+m)!}{n!}}\right]\nonumber\\\end{aligned}\ ] ] which exactly coincides with equations ( 41 ) and ( 43 ) in @xcite and eqs . ( 3.26 ) and ( 3.27 ) in @xcite for @xmath72 . we encourage the reader to compare this method with those used in @xcite . + obviously this formalism can also be used to solve the equation with any arbitrary hamiltonian provided we already know the solution when @xmath61 . a second consequence of the formalism depicted above is its immediate generalization to nonmarkovian situations . the result in section [ lindasrand ] can be readily generalized to the following : _ any lindbladian master equation , whether markovian or nonmarkovian , but with selfadjoint lindblad operators can be obtained as the stochastic average of a random unitary evolution_. the generalization stems out from the single fact that whereas in the markovian regime we necessarily have to add a stochastic term of the form @xmath73 , in the nonmarkovian case this restriction drops out and then we may add a term like @xmath74 , where @xmath75 is an arbitrary real - valued function which encodes e.g. the time response of the environment to the system evolution . is a stochastic process . in this case , since we are interested in physical properties which are obtained after averaging , this does not suppose any actual gain . ] under these circumstances , the previous procedure ( for simplicity s sake we will only care about the commuting case ; the noncommuting case is similar ) drives us to @xmath76[\rho(0)]\ ] ] now developing the exponential again into a power series , calculating the expectation value of each term with some elementary ito calculus and resumming the series , one arrives at @xmath22-\frac{\gamma(t)}{2}[v,[v,\rho(t)]]\ ] ] where @xmath77 . this is clearly a lindbladian nonmarkovian master equation . the extension to more than one lindblad operator is again trivial . this result casts some light into the origin of the lindbladian structure of master equations with selfadjoint lindblad operators , independently of their markovian or nonmarkovian character , something beyond reach of the original axiomatic approach of @xcite . + in this sense the result proven here generalizes previous derivation of lindblad evolution using stochastic calculus @xcite by dropping out the semigroup condition . note that this generalization allows us to conclude that since @xmath77 the decoherence process is irreversible , i.e. no coherence can be recovered within the domain of validity of the phase - damping me as an evolution equation for the quantum system . + the time dependence of the decoherence factor also suggests a classification of different kinds of environments depending on the rate at which the environment decoheres the system ( cf . it remains open what the physical conditions should be to have the different decoherence factors . a third advantage appears as a natural generalization of intrinsic decoherence models already present in the literature @xcite . these two models propose an intrinsic mechanism of decoherence based on the random nature of time evolution ( we will not enter into the discussion of the physical justification of this hypothesis see original references for discussion , we will only show how they can be generalized ) , which basically drives us to the evolution equation . the starting hypotheses ( apart from the random nature of time evolution and the usual representation of a quantum system by a density operator ) are a specific probability distribution @xcite or a semigroup condition ( markovianity ) @xcite for the time evolution . as a result we obtain a _ nondissipative markovian _ master equation in both cases . + the formalism presented here dispenses with any of these specific conditions , something which allows us to obtain more general master equations , i.e. both markovian or nonmarkovian and dissipative or nondissipative . + the result comes from the combination of ito calculus and the spectral representation theorem for unitary operators @xcite . let @xmath78 be an unitary evolution operator . by means of the spectral decomposition theorem @xcite it can be written as @xmath79 where @xmath80 denotes the spectral measure of the evolution operator . now we perform the stochastic promotion as before by substituting @xmath81 where @xmath82 is real stochastic process . then ( see @xcite for details ) 1 . the markovian nondissipative master equation appearing in @xcite and @xcite is obtained if @xmath83 . but note that now the lindblad operators are not fixed by any initial assumption . if e.g. @xmath84 with correlation function @xmath85 we arrive at a lindblad equation + @xmath3-\nonumber\\ & -&\frac{\gamma}{2}\left(h^{2}\rho(t)+\rho(t)h^{2}-2he^{-\tau^2\mathcal{c}_{h}^{2}}[\rho(t)]h\right)\nonumber\\\end{aligned}\ ] ] + which is clearly different from the phase - damping master equation . thus even restricting ourselves to the same range of assumptions ( markovianity and nondissipation ) we can obtain more master equations . 2 . a nonmarkovian nondissipative master equation like e.g. + @xmath22-\frac{\lambda(t)}{2}[h,[h,\rho(t)]]\ ] ] + is obtained if @xmath86 and @xmath87 . more general equations can be readily obtained by appropiately combining the correlation properties of @xmath88 and the time dependency of @xmath89 . this allows these models to be used to explain a wider range of phenomena than that originally considered . in previous sections we have developed a method to adequately modify the original evolution operator of a quantum system to finally arrive at a lindbladian master equation . now we find legitimate to proceed the other way around , i.e. what physical predictions are derived from the assumption that a parameter in the evolution operator of a quantum system is random ? to be concrete we will focus upon two different physical systems , namely a rydberg atom in an optical cavity and a linear rf ( paul ) ion trap . we will confront the previous hypothesis with experimental results . the jcm model describes the interaction between an atom and the electromagnetic field under very special conditions @xcite . different generalizations have been proposed to take the model closer to experimental reality keeping its solvability . among these one can find the inclusion of dissipation ( often modelled by coupling the field oscillator to a reservoir of external modes ) and/or damping ( as a consequence of spontaneous emission ) , multi - atom , multi - level atom , generalized - interaction and multiple - mode generalizations ( see @xcite for references ) . + here we want to introduce a novel proposal , which states that jcm predictions can be rendered more realistic by noticing that the coupling constant @xmath49 between the atom and the field mode should have a stochastic part which contains part of the effects of the approximations assumed in constructing the original model . since these effects are not under control , to make physical predictions we must average on the introduced random parameters . to illustrate the idea let us consider the original jcm with hamiltonian @xmath90 , where @xmath47 denotes the frequency of the field mode , @xmath51 and @xmath50 their corresponding creation and destruction operators , @xmath48 the frequency difference between the two energy atomic levels , @xmath52 the atomic population operator , @xmath49 the atom - field coupling constant and @xmath53 the energy raising / lowering atomic operators . we claim that the evolution stemming out from @xmath1 should be modified by inserting a random part @xmath34 , where @xmath91 is a real stochastic process which contains the departure from the original ideal situation . the connection with the previous formalism is established by noticing that the evolution operator ( in interaction picture ) must then be : @xmath92=\nonumber\\ & = & \exp\left[-i(\lambda t+\int_{0}^{t}\chi_{s}ds)h^{int}\right]\end{aligned}\ ] ] where @xmath93 is the interaction hamiltonian and @xmath94 is the interaction hamiltonian in interaction picture ( for simplicity s shake exact resonance has been assumed @xmath95 ) . now the expression @xmath96 is a real stochastic process itself which can always be expressed in the form @xcite @xmath97+\int_{0}^{t}v_{s}d\mathcal{b}_{s}\ ] ] where @xmath98 is a real stochastic process uniquely determined by @xmath99 . then the density operator in interaction picture will then be given by @xmath100+v_{s}d\mathcal{b}_{s}\right)\mathcal{c}_{h^{int}_{i}(s)}\right)ds\right][\rho(0)]\ ] ] instead of giving the general form of the expectation value in ( which will be difficult to obtain in full generality ) , we will propose some physical choices based on @xmath34 . if @xmath101 , i.e. the original deterministic evolution is randomly perturbed by a white noise coming from a stochastic perturbation of the coupling constant , then @xmath102 and reduces to @xmath103\nonumber\\ & = & \mathbb{e}[\exp\left(-i\left(\lambda t+\gamma^{1/2}\mathcal{b}_{t}\right)\mathcal{c}_{h_{i}}\right)][\rho(0)]\end{aligned}\ ] ] which yields a density operator in schrdinger picture given by @xmath104[\rho(0)]\ ] ] this relationship means that to obtain the physical predictions of this proposal , all one has to do is to make the sustitution @xmath105 in the original expressions and calculate the expectation value . + note that this proposal allows us to embrace nonmarkovian ( though lindbladian ) situations with little extra effort , e.g. by claiming that the random perturbation is time - dependent @xmath106 . more general options are also possible . the particular choice for @xmath34 relies upon the specific system under study . notice also that the generalization proposed here is compatible with the ones quoted above , i.e. one may combine both type of generalizations . let us consider the situation depicted in @xcite , which appears as the first direct ( in time domain ) experimental evidence of field quantization . the system consists of a rydberg atom in a high - q optical cavity with the atom initially excited and the electromagnetic field in a coherent state @xmath107 . this system is accurately described using the jcm , thus rabi oscillations are expected and concordantly experimentally measured . the theoretical prediction for the probability @xmath108 to find the atom at a time @xmath56 in its ground state is @xmath109 . however an exponential damping in these oscillations are detected ( see @xcite for details ) . the stochastic jcm accounts for this damping ( see fig . [ qedrabi ] ) assuming @xmath101 which produces @xmath110 the physical interpretation under this assumption is rather clear : the ideal coupling assumed in the original jcm does not hold any longer and departures from this ideality should be considered . darks counts and decoherence caused by collisions with background gas have been considered as candidates to explain the damping behaviour and even more radical proposals appear in the literature @xcite . except for the latter which reveals an original intrinsic process , all of them resort to external agents . here note that we do not need to do so , since the departure from ideal atom - field mode coupling can be justified within the domain of the jcm assumptions themselves , i.e. the assumption of coupling between a unique field mode and two levels of the atom can be relaxed by adding in a natural way a random background in this coupling . this decay is also present in the case of arbitrary initial conditions . if @xmath111 is the orthodox prediction , then the previous recipe drives us to @xmath112 where @xmath113 depends on the actual initial conditions of both the atom and field mode . note that not only can any actual exponential decay ( with arbitrary time dependency ) be obtained with the substitution @xmath114 , but also possible changes in the argument of the cosine function could be accounted for by making @xmath115 . see appendix [ appendix ] for details . + in this way we have obtained a decohering system ( oscillations coming from quantum superpositions are progressively supressed ) without necessarily resorting to the action of the environment and keeping quantum principles untouched ( see discussion in section [ discuss ] later on ) . this opens new possibilities to discuss possible sources of decoherence . the previous experimental supression of quantum coherence has also been detected in a linear paul ion trap @xcite . the physical situation is formally similar to that of the rydberg atom coupled to a field mode : the laser field is operated upon the trap in such a way that it can be considered that only two internal energy levels of the ions are coupled to the center - of - mass ( com ) mode of the set of ions ( see @xcite ) . in the dipole and rotating - wave approximations , the interaction hamiltonian ( in interaction picture ) is @xmath116\right)+\textrm{h.c.}\ ] ] where @xmath117 is the lamb - dicke parameter ( @xmath118 and @xmath119 ) , @xmath53 denote the raising / lowering operators for the internal levels , @xmath51 and @xmath50 denote the destruction and creation operators for the com mode , @xmath120 denotes the frequency of the harmonic trap for the com , @xmath121 with @xmath47 the frequency of the laser mode and @xmath48 denotes the difference between the two internal energy levels of the ions . + the statevector can then be written as @xmath122 where @xmath123 and @xmath124 denote the ( time - independent ) internal and motional eigenstates . in the conditions of interest , i.e. in resonant transitions ( @xmath125 with @xmath126 integers ) , the coefficients @xmath127 satisfy the equations @xcite @xmath128 where @xmath129 is given by @xmath130 ( @xmath131 and @xmath132 is the operator for the com motion ) . from these one can predict the well - known rabi oscillations of the system . for concreteness shake let us focus upon the first blue sideband case , i.e. when @xmath133 . if the trap is prepared in the initial state @xmath134 , then the probability of finding a single ion in the @xmath135 state at time @xmath56 is @xmath136 . however experimentally an exponential decay is obtained . as before , one can argue that ideality should be restricted and both a substitution @xmath137 and the corresponding averaging should be performed on @xmath138 . this would drives us to the relation @xmath139 where @xmath140 . this way the exponential decay would have been obtained . physically this recipe can be justified by taking into account intensity fluctuations in the laser field ( see @xcite notice that some generality is gained with respect to this work ) . + however experimental data for the com initially in an arbitrary state and the ion in the ground state @xmath135 are better fit by @xmath141\ ] ] where @xmath113 is an @xmath20-dependent quantity which relies upon the initial conditions of the ion s motion and @xmath142 is a phenomenologically decoherence rate @xcite . the peculiar exponent @xmath143 in @xmath144 renders the previous physical explanation insufficient . more involved schemes to account for this exponent can already be found in the literature @xcite . here we propose a new one based on the previously introduced random evolution schemes . + the main problem attains the peculiar @xmath20-dependency of the argument of the exponential decaying function . in the mathematical realm the necessary flexibility comes from a combination of stochastic calculus and the spectral theorem for the evolution operator @xcite and in the physical one from realizing that not all energy levels of the com mode can be equally affected by a stochastic perturbation . this idea , in a different context in which the trap is coupled to a boson reservoir to account for the detected decoherence , has already been paid attention @xcite . + let s start by considering the spectral decomposition @xcite of the evolution operator generated by the hamiltonian when the laser is tuned to the first blue sideband , i.e. when the hamiltonian is given by @xmath145\ ] ] then the evolution operator will be decomposed as follows : @xmath146 where @xmath147 , @xmath148 ( @xmath149 ) and @xmath150 denote the eigenvalues and eigenstates of respectively and @xmath151 is the projector - valued measure associated to . the stochastic promotion is performed by substituting @xmath152 in and calculating @xmath153 $ ] . here note that the different energy levels are perturbed in a distinct fashion determined both by the deterministic functions @xmath154 and the standard real brownian motions @xmath155 . the latter show correlation properties expressed by the functions @xmath156 : @xmath157 the density operator in interaction picture will then be given by @xmath158p_{n}^{m_{z}}\rho(0)p_{m}^{m_{z}^{'}}\ ] ] the expectation value in can be calculated with the same techniques as before ( cf . also appendix [ appendix ] ) and drives us to : @xmath159 where @xmath160 with @xmath161 . the expression already contains the necessary ingredients to arrive at the detected behaviour , since if the ion trap is initially set in a fock state for the com mode and the ground state for the internal levels , i.e. @xmath162 , then the probability @xmath163 in this scheme is @xmath164\nonumber\\\end{aligned}\ ] ] before making physical assumptions let us notice that since the brownian motions are standard , @xmath165 for all @xmath20 ( the brackets mean that both superscripts must be equal ) and thus @xmath166 for all @xmath20 . now we pose the most important physical hypothesis , namely the stochastic perturbation depends exclusively upon the energy level of the com mode ( at least up to the order of detection we are nowadays capable ) . as a first consequence we then can claim that @xmath167 for all @xmath20 , and then @xmath168 and @xmath166 and also @xmath169 , hence reduces to @xmath170=\nonumber\\ & = & \frac{1}{2}\left[1+e^{-\frac{\lambda_{n+1,n+1}^{+,-}(t)}{2}}\cos(2\eta\omega t\sqrt{n+1})\right]\end{aligned}\ ] ] second since for fixed @xmath20 the internal levels are equally affected , we can also write @xmath171 for each @xmath20 . finally instead of discussing upon absolute energy values , it is physically more reasonable to talk about energy differences and we propose that the stochastic perturbations be introduced in such a way as to have @xmath172 where @xmath173 is an arbitrary exponent , @xmath174 a constant and where we have assumed for simplicity that the random perturbation is a white noise . note however that it is possible to use more general expressions . notice the different behaviour of the added term for each distinct subspace of constant com energy in agreement with the physical hypothesis assumed above . under these hypotheses @xmath175 and after elementary calculations finally reduces to @xmath176 this expression shows a clear resemblance to @xmath177 written above . we believe that both @xmath174 and @xmath173 depends sensitively upon the particular physical system under study . + for completeness we also include the expression for @xmath163 when the com mode has an initial state with diagonal density - matrix elements @xmath113 : @xmath178 this expression has the same structure as the experimental ones shown in @xcite . the whole scheme can obviously be applied to the carrier , first red sideband and successive excitation too . we include in figs . [ fockgraph ] , [ thermalgraph ] and [ cohergraph ] the predictions in the orthodox and the above formalism in the cases when the com mode is in a fock state , thermal state and coherent state and the internal state is the ground state . the use of stochastic methods in hilbert space is of course not new ( cf . e.g. @xcite ) . the idea of representing open quantum systems by means of stochastic processes already appeared in the literature some years ago @xcite and it has been widely used in quantum optics @xcite and in the foundations of quantum mechanics @xcite . here we pursue the line initiated in @xcite stepping forward by randomizing not just the ( thus stochastic ) state vector of the open quantum system , but its evolution operator . we find at least three reasons to do that . firstly when one write a random evolution equation for the state vector ( thus an ito stochastic differential equation ) an extra term must be included , namely the ito correction . consider for example the following evolution @xmath179 where the operator @xmath180 commutes with the hamiltonian @xmath1 ( just for simplicity ) . from a physical point of view we find little intuitive the origin of the ito correction term , which however appears in a natural way by applying ito s formula to the evolution operator with the stochastic modification @xmath181 . secondly the use of random evolution operators emphasizes the idea that it is the evolution which is random and there is nothing random about the hamiltonian ( and thus the energy levels of the system ) , something which may misleadingly be understood from equation . finally the use of operators rather than just state vectors opens the possibility of trying to employ group - representation techniques @xcite and thus of rooting the random nature of the evolution upon possible stochastic symmetries . + this proposal is not intended to solve the so - called _ macroobjectification _ problem ( better known as the measurement problem ) by means of a random dynamical reduction process . indeed it can be readily shown that the state driven by a random evolution operator @xmath182 is never reduced in clear contrast to these models ( see @xcite ) . for the case of the previous random evolution operator this can be readily proven . let @xmath181 be the random evolution operator of a quantum system ( @xmath183=0 $ ] for simplicity ) . then the ito differential equation for the state vector is . now to check whether this evolution produces dynamical state - reduction or not it is sufficient to study the stochastic process ( cf . @xcite ) @xmath184 where @xmath185 . since @xmath186 and @xmath187 , the random evolution is unitary almost surely and then @xmath188 thus there is no actual state - vector reduction process around the eigenvectors of @xmath180 . this of course differs radically for the behaviour of @xmath189 in stochastic state vector reduction models , where the evolution equation is typically written as @xcite @xmath190 the nonlinear terms appear as a consequence of normalization conditions @xcite and play no significant role in the reduction process . note the singular difference between eqs . and : despite the fact that both of them produce the same master equation ( already noted in @xcite ) , only the second one ensures a reduction process taking place and this is because of the @xmath191 factor appearing with the wiener differential @xmath192 . it is an open question whether there exists a physical process or not introducing this phase factor in the evolution equation for @xmath189 . + the great utility of stochastic processes to account for the decoherence suffered by a quantum system is its versatility to also account for possible intrinsic decohering effects . by this we mean not a fundamental modification of quantum principles , as e.g. in @xcite but just the idea that when a quantum system is described ( e.g. an atom interacting with the electromagnetic field ) some approximations must necessarily be made to be able to analytically handle with it and we claim that some of these approximations may hidingly induce decohering effects upon the approximated model . in this sense this decoherence can be called _ intrinsic _ since no environmental effect is taken into account . undoubtedly this does not deny in any way the possibility of having a system decohered by its environment . notice the special relevance of such a hypothetical effect in quantum systems designed to implement quantum - computational and quantum - information - processing tasks , since they usually possess certain degree of complexity which forces us to seek for adequate approximations to describe them . a possible relationship of this intrinsic decoherence with scalability of quantum - informational and quantum computational systems would also have important consequences to find robust mechanisms to process information in a quantum way . + besides possible new physical interpretations supporting the use of stochastic processes in quantum mechanics , its utility to solve certain me as shown above justifies the search to extend the main result reported here . currently the way to drop the condition of selfadjointness of lindblad operators is under study . + finally a mathematical remark should be made regarding the proof of the previous result . one may wonder why the same procedure as in the commuting case , i.e. adding a stochastic term @xmath193 , is not used in the noncommuting case . the reason jointly rests upon the ito s formula and the lack of some derivatives in the case of noncommuting operators . formula @xcite basically states that if the real stochastic process @xmath194 satisfies the ito sde @xmath195 , then the real stochastic process defined as @xmath196 satisfies the ito sde given by @xmath197 this means that both the first and second partial derivatives of @xmath198 must exist to be able to apply this formula . if the stochastic process is operator - valued as e.g. @xmath199 with @xmath29\neq 0 $ ] , then ito s formula can not be directly applied since the partial derivates of @xmath200 can not be found . this difficulty has been circumvented by changing to the heisenberg picture before introducing the stochastic modifications . however it would be desirable to have an ito s formula for operator - valued stochastic processes valid both for commuting and noncommuting cases . we have proven how any lindblad evolution with selfadjoint lindblad operators can be understood as an averaged random evolution operator . the proof included here allows us to extend the previous result to nonmarkovian situations , though keeping the lindbladian structure of the master equations , and as a result we have provided a straightforward method to solve this kind of master equations . the conjunction of stochastic methods and the spectral representation theorem has also allowed us to generalize intrinsic decoherence models already present in the literature . the mathematical versatility of stochastic methods has also permitted us to propose a generalization to the jaynes - cummings model , and compare its predictions with experimental results in cavity qed as well as in ion traps . we have argued that the main physical advantage stems from the possibility of studying decohering effects not necessarily rooted on the environmental action upon the system . a comparison with dynamical collapse models reveals that the random unitary operators used here do not produce any kind of state vector reduction . one of us ( d.s . ) acknowledges the support of madrid education council under grant bocam 20 - 08 - 1999 . though they are elementary we include some useful relations in order to render the text self - contained . first we will show how the moments of arbitrary order @xmath20 of the stochastic process @xmath201 @xmath202 a real function are calculated . let us denote the stochastic process and their @xmath20th moments respectively as @xmath203 and @xmath204^{n}$ ] . it is evident that @xmath205 . @xmath194 satisfies the stochastic differential equation @xmath206 . applying ito s formula @xcite to @xmath207 with @xmath208 and taking expectation values one readily arrives at
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it is shown how any lindbladian evolution with selfadjoint lindblad operators , either markovian or nonmarkovian , can be understood as an averaged random unitary evolution .
both mathematical and physical consequences are analyzed .
first a simple and fast method to solve this kind of master equations is suggested and particularly illustrated with the phase - damped master equation for the multiphoton resonant jaynes - cummings model in the rotating - wave approximation . a generalization to some intrinsic decoherence models present in the literature
is included .
under the same philosophy a proposal to generalize the jaynes - cummings model is suggested whose predictions are in accordance with experimental results in cavity qed and in ion traps . a comparison with stochastic dynamical collapse models
is also included .
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the recent discovery of superconductivity in sodium cobalt oxide compound intercalated water molecules , na@xmath0coo@xmath1@xmath2h@xmath1o,@xcite trigged intense attentions and stimulated lots of discussions@xcite . the superconductivity induced in the planer structure of coo@xmath1 is similar with that in the cuo@xmath1 plane of cuprates@xcite . however , the underlaying triangular lattice of the co atoms is fundamentally different from the square lattice of the cu atoms in cuprates because the antiferromagnetic interactions on the triangular lattice are frustrated . the carrier density in the sodium cobalt oxide can be tuned by the na concentration . by changing the sodium doping , a rich phase diagram appears and the superconductivity occurs@xcite in the doping regime @xmath3 . furthermore , the study in co - nmr and co - nqr found that the spin - lattice relaxation rate at the critical temperature ( @xmath4 ) shows no coherent peak and follows a power below @xmath4 , hiniting an unconventional superconducting phase@xcite . the node of the superconducting gap is confirmed by the specific - heat measurements@xcite and also by the muon spin relaxation experiments@xcite . however , the symmetry of the cooper pairs remains unknown at present . in order to identify the pairing symmetry , the measurement of spin susceptibility in the superconducting state through the knight shift is helpful@xcite . the measurements of the powder samples show that the knight shifts along the @xmath5-axis do not decrease below @xmath4 , raising the possibility of spin - triplet superconducting state@xcite . on the other hand , recent measurements on the single - crystal samples@xcite show that the knight shift decreases below @xmath4 along the @xmath6- and @xmath5- axes , which suggests for the spin - singlet pairing instead . from the study of the normal - state fermi surface topology by the angle - resolved photoemission spectroscopy@xcite and the mn doping effects@xcite , it also seems to support the singlet superconducting state . thus , the pairing symmetry of superconductivity in na@xmath0coo@xmath1@xmath2h@xmath1o compounds remains controversial at the point of writing . there are also theoretical efforts to pin down the pairing symmetry of the gap function in na@xmath0coo@xmath1@xcite . the underlaying triangular lattice is proposed to host the resonating - valence - bond ( rvb ) state for an unconventional superconductor@xcite . base on the rvb picture , theoretical investigations on the @xmath7-@xmath8 model@xcite favor the @xmath9 symmetry . however , within the third - order perturbative expansions , a stable @xmath10-wave pairing is found in the hubbard model@xcite with repulsive on - site interaction . the same conclusion is reached from the theoretical study on the single - band extended hubbard model within random phase approximations@xcite . furthermore , recent discovery of the hubbard - heisenberg model on the half - filled anisotropic triangular lattice show that varying the frustration @xmath11 changes the spatial anisotropy of the spin correlations and leads to transitions of the pairing symmetries of the superconducting oder parameter@xcite . taking different routes for theoretical investigations , other groups demonstrate the possibility of the @xmath12 pairing@xcite . in addition , starting from the fluctuation - exchange approximations , the triplet @xmath10-wave and @xmath13-wave pairings are favored on the triangular lattice@xcite . with the same approximations , solving the linearized liashberg equation@xcite leads to dominant pairing in the spin - triplet @xmath10-wave sector . therefore , the pairing symmetry also posts a challenging task for theoretical understanding from the microscopic perspective . while it is important to determine the pairing symmetry from microscopic approaches , it is equally crucial to develop phenomenological theories so that one can extract the pairing symmetry from the experimental data@xcite such as the andreev bound states@xcite near the edges of the superconductors . note that the andreev edge state@xcite in a superconductor is tied up with the pairing symmetry in the bulk . in addition , recent breakthroughs in the fourier - transformed scanning tunneling spectroscopy ( ft - sts ) experiments@xcite allow further insight into the edge states with momentum resolutions . in these experiments , not only the spatial profile of the local density of states ( ldos ) can be measured , the peaks of the ldos in the momentum space can also be determined by appropriate fourier analysis of the experimental data . in a letter published by one of the authors@xcite , a theoretical approach was developed to compute the momentum - resolved ldos for the andreev edge state in sodium cobalt oxide with @xmath10-wave pairing symmetry . the exponential decay away from the boundary can be compared with the experiments directly , while the dependence upon the transverse momentum ( along the edge where the system is translational invariant ) can be seem in fourier space through scattering processes . here , we elaborate and extend the previous work by considering gap functions of @xmath13- , @xmath14- and @xmath10-pairing at both zigzag and flat edges and predict the position of the sharp peaks that can be observed in ft - sts experiments . .existence of andreev edge state at zigzag and flat edges and its implication for pairing symmetry . [ cols="^,^,^",options="header " , ] we start with the two dimensional ( 2d ) bogoliubov - de gennes hamiltonian and map the semi - infinite triangular lattice to a collection of one - dimensional ( 1d ) chains , labeled by the transverse momentum along the boundary . due to the hidden structure of these effective 1d models , the aes can be categorized into the positive and negative witten parity states@xcite in supersymmetric ( susy ) algebra . for readers no familiar with the witten parity and the susy algebra , we have included a brief introduction in appendix a. by computing the witten parity states constrained by the boundary conditions , the ldos with specific transverse momentum is obtained . furthermore , we can predict the hot spots in ft - sts by spotting all momentum differences between sharp peaks in the ldos . our results show that the existence of aes sensitively depends on the pairing symmetry and the edge topology and can thus be used as a good indicator of the underlying pairing symmetry . the existence of the aes for different pairing symmetries and edge topologies are summarized in table . i. finally , following an elegant gauge argument devised by oshikawa@xcite , we also find that the phase diagram for the aes crucially depends on the nodal points on the fermi surface where the pairing amplitude vanishes . the rest of paper is organized as the followings . in sec . ii , we introduce the 2d bogoliubov - de gennes hamiltonian for a triangular lattice with the zigzag boundary topology . by transforming the hamiltonian into supersymmetric form and use the generalized bloch state , the ldos of aes is obtained . in sec . iii , in the same spirit and method , we will compute the ldos of aes for the flat edge . we will discuss about the gauge argument of phase diagram and draw a conclusion in sec . to accommodate different pairing symmetries within one theoretical framework , it is convenient to start from the bogoliubov - de gennes ( bdg ) hamiltonian@xcite , @xmath15,\end{aligned}\ ] ] where only the nearest - neighbor hopping and pairing are included . because the particle - hole symmetry is absent in the triangular lattice , the sign of the hopping amplitude @xmath7 is crucial . recent experiments@xcite suggest that the maximum of the band occurs at the @xmath16 point , which implies @xmath17 . the pairing amplitudes are either symmetric @xmath18 or antisymmetric @xmath19 depending on the cooper pairs are spin singlets or triplets . in this paper , we will discuss two natural boundary topologies of a triangular lattice zigzag and flat edges , as showed in fig . [ pszf ] and fig . [ psfd ] respectively . the conventions for the spatial coordinates and also the pairing symmetries can be found in the figures as well . for instance , the zigzag edge is chosen to lie in the @xmath20-axis and the flat edge along the @xmath21-axis in our convention . we start with the zigzag edge first , by cutting the infinite triangular lattice along the @xmath20-axis . note that the semi - infinite lattice is still translational invariant along the boundary and thus can be mapped onto a collection of semi - inifinite 1d chains , carrying definite transverse momentum @xmath22 after partial fourier transformation . one important subtlety about partial fourier transformation is the folding of brillouin zone . the conventional hexagonal shape must be reshaped into appropriate rectangular one so that the summations over @xmath23 and @xmath22 are decoupled@xcite . for the zigzag edge , the reconstruct rectangle brillouin zone is shown in the bottom of figs . [ pszf ] , [ pszdxy ] and [ pszpx ] . after the partial fourier transformation , the hamiltonian for the collection of the effective 1d chains along @xmath21-direction is @xmath24 with @xmath25 and @xmath13 , which are denoted as @xmath10- , @xmath14- and @xmath13-wave pairing respectively . here we introduce the nambu basis @xmath26 $ ] and the semi - infinite matrix for the hopping term of semi - infinite 1d chains , @xmath27 with the effective hopping amplitude @xmath28 and @xmath29 . the momentum dependence of the matrix elements is a consequence of the partial fourier transformation . the pairing potential @xmath30 with different symmetries will be studied in details in the following subsections . we start with the aes of @xmath10-wave pairing symmetry at zigzag edge . the @xmath10-wave symmetry carries angular momentum @xmath31 and thus corresponds to spin - triplet pairing required by fermi statistics . it implies that the pairing potential is antisymmetric , @xmath32 . taking the tight - binding approximation , the pairing potential is rather simple @xmath33 with the relative angle @xmath34 where @xmath35 is an integer . the sign convention for different bond orientations is fixed in fig . we can solve for the nodal lines by setting the gap function to zero , @xmath36\sin(k_x/2)=0 $ ] . these nodal line are drawn in the reshaped brillouin zone in fig . [ pszf ] . at different fillings ( chemical potentials ) , the nodal points are the intersections of the fermi surface contour and the nodal lines . these nodal points turn out to be the key for determining the structure of the phase diagrams for the aes . the presence of the open boundary complicates the story and we need to write down the pairing potential in the coordinate space . after some algebra , the semi - infinite matrix @xmath37 of eq . ( [ bdgz ] ) takes the form , @xmath38 with @xmath39 . a simple unitary transformation brings the hamiltonian into susy form described in appendix a. the effective hamiltonian@xcite in canonical susy notation is @xmath40 where the matrix @xmath41 takes the general form @xmath42 although we concentrate on the @xmath10-wave symmetry in this section , the derivations of the matrix elements of @xmath41 are completely general and work for different pairing symmetries . -wave symmetry at the zigzag edge of a triangular lattice . the sign convention of the pairing potential is shown in the shaded hexagon . the bottom figure represents the fermi surface in the reshaped brillouin zone for the zigzag edge . the nodal lines of the @xmath10-wave gap function are shown in blue lines and the nodal points are the intersections of the fermi surface contour and the nodal lines . , width=234 ] for current case , @xmath43 for @xmath10-wave pairing , the new basis for the susy form is @xmath44,\end{aligned}\ ] ] and the matrix elements of @xmath45 are @xmath46 it will become clear later that @xmath47 and @xmath48 are the crucial for the existence of the edge states . for the hamiltonian in eq . ( [ diracz ] ) , the zero - energy states are nodal " , i.e. half of the components in the spinor vanish , and can be classified by the so - called susy parity ( see appendix a ) , @xmath49 it is straightforward to show that the harper equations become decoupled for the zero - energy states and simplify a bit . the solution with positive witten parity @xmath50 is annihilated by @xmath51 , i.e. it belongs to the null space of the operator . similarly , the solution with negative witten parity @xmath52 spans the null space of the operator @xmath53 . it is worth emphasizing that bring the hamiltonian into the susy form simplifies the algebra and allows analytic calculations for aes as derived here . to include the open boundary condition , the edge state can be constructed by the generalized bloch theorem@xcite . taking states with negative witten parity as a working example , one can construct an edge state from appropriate linear combinations of the zero - energy modes , @xmath54 . since the zero energy modes satisfy @xmath55 , @xmath56 is a solution of the following characteristic equation , @xmath57 it is clear that the algebraic equation gives four solutions of @xmath56 for the given chemical potential and the transverse momentum . however , not all solutions are allowed . for the infinite lattice , the wave function must remain finite at infinities , @xmath58 and @xmath59 . it implies that only @xmath60 solutions are allowed . these are the plane - wave solutions with real momentum defined as @xmath61 . however , for an open boundary with zigzag shape , the boundary conditions change to @xmath62 thus , @xmath63 is required to keep the wave function finite which is less strict than the @xmath60 criterion for translational invariant systems . however , we have additional two boundaries conditions at @xmath64 , the edge state does not always exist , unless we have enough @xmath65 zero - modes to construct the edge states . momentum dependence of @xmath66 for the generalized bloch states with the @xmath10-wave symmetry at the zigzag edge . the lines with different colors represent four solutions of the bloch state with the parameters , @xmath67 and @xmath68 . the chemical potentials in the top - left and top - right figures are @xmath69 and @xmath70 respectively . the bottom - left and bottome - right figures are for @xmath71 and @xmath72.,title="fig : " ] momentum dependence of @xmath66 for the generalized bloch states with the @xmath10-wave symmetry at the zigzag edge . the lines with different colors represent four solutions of the bloch state with the parameters , @xmath67 and @xmath68 . the chemical potentials in the top - left and top - right figures are @xmath69 and @xmath70 respectively . the bottom - left and bottome - right figures are for @xmath71 and @xmath72.,title="fig : " ] momentum dependence of @xmath66 for the generalized bloch states with the @xmath10-wave symmetry at the zigzag edge . the lines with different colors represent four solutions of the bloch state with the parameters , @xmath67 and @xmath68 . the chemical potentials in the top - left and top - right figures are @xmath69 and @xmath70 respectively . the bottom - left and bottome - right figures are for @xmath71 and @xmath72.,title="fig : " ] momentum dependence of @xmath66 for the generalized bloch states with the @xmath10-wave symmetry at the zigzag edge . the lines with different colors represent four solutions of the bloch state with the parameters , @xmath67 and @xmath68 . the chemical potentials in the top - left and top - right figures are @xmath69 and @xmath70 respectively . the bottom - left and bottome - right figures are for @xmath71 and @xmath72.,title="fig : " ] here comes the simple counting . if all of the four solutions satisfy @xmath65 , we can construct two edge states . if three solutions are found , one edge state can be constructed . otherwise , there will be no edge state . in the case of the @xmath10-wave pairing symmetry , we plot the magnitude of the solutions @xmath73 as a function of the transverse momentum @xmath22 in fig . [ zsolution ] . the @xmath56-plot sensitively depends on the chemical potential @xmath74 . -wave pairing at different chemical potentials . the single point circle mark the nodal points without degeneracy while the double circle denote the two - fold degenerate nodal points . the green / yellow colors denote the susy parity @xmath75 and the single / double lines mean one / two - fold degenerate edge states . , width=294 ] now we would like to explain how to obtain the phase diagram for aes from the @xmath56-plots . we start with the first @xmath56-plot ( upper left ) in fig . [ zsolution ] where the chemical potential is @xmath69 and the pairing potential is @xmath76 . there are four intersections with @xmath60 dashed line . these are the nodal points . at larger momentum , the @xmath60 dashed line intersects with one solution ( orange line ) and gives rise to the nodal point . at small momentum , the dashed line intersects with two degenerate solutions ( orange and blue lines ) at the same time and corresponds to a pair of degenerate nodal points . these nodal points correspond to the single and double circles in the phase diagram . now we can proceed to determine how many edge states @xmath77 with negative witten parity can be found . near the zone boundary @xmath78 , there are two solutions ( green and blue lines ) with @xmath63 . since there are two constraints from the zigzag boundary , no edge state can be constructed . passing the nodal point , there are three solutions ( green , blue and orange lines ) and thus one edge state starts to emerge . the aes with negative witten parity is marked by yellow color in the phase diagram . moving toward to zone center , the number of desired solution reduces to one ( green line ) after passing the two - fold degenerate nodal point . thus , no edge state in presence in this regime . due to the parity symmetry in @xmath20-direction , the phase diagram is symmetric when @xmath79 . what about the edge state @xmath80 with positive witten parity ? one should repeat the derivation for the characteristic equation and look for @xmath63 solutions to construct the edge states again . however , there is some symmetry hidden in the algebraic equation and the repetition is not necessary . since the matrices @xmath81 and @xmath82 are hermitian conjugate to each other , the algebraic equation for the positive wittien parity modes can be obtained by replacing @xmath83 in eq . [ zsol1 ] . that is to say , the decaying solutions for positive witten parity can be calculated from the @xmath84 solutions in eq . [ zsol1 ] . this relation is very helpful in constructing the remaining part of the phase diagram . near the zone center in the first @xmath56-plot ( upper left ) in fig . [ zsolution ] , there are three @xmath84 solutions and correspond to one edge state with positive witten parity . in other regimes , no such edge state exists . combining the results for both witten parities , the first part of the phase diagram in fig . [ pdzf ] is obtained . -wave pairing ( top figure ) along the @xmath21-direction at different transverse momentum @xmath22 at the chemical potential @xmath69 . the bottom figure shows the spatial trend of the integrated ldos over the brillouin zone that can be measured directly from stm experiments.,title="fig:",width=264 ] -wave pairing ( top figure ) along the @xmath21-direction at different transverse momentum @xmath22 at the chemical potential @xmath69 . the bottom figure shows the spatial trend of the integrated ldos over the brillouin zone that can be measured directly from stm experiments.,title="fig : " ] -wave pairing at the chemical potential @xmath70.,title="fig:",width=264 ] -wave pairing at the chemical potential @xmath70.,title="fig : " ] -wave pairing at the chemical potential @xmath71.,title="fig : " ] -wave pairing at the chemical potential @xmath71.,title="fig : " ] since we compute the value of @xmath56 for each transverse momentum @xmath22 , the quasiparticle wave function of bloch states can be obtain straightforwardly . thus , in addition to the phase diagram , we can also compute the ldos of the edge state at specific transverse momentum @xmath22 . we can also integrate over the brillouin zone to obtain the spatial profile for ldos that can be measured directly in the stm experiments . furthermore , the momentum - resolved ldos provides additional information about the enhanced spectral weight of the quasi - particles at specific transverse momenta . thus , we can predict the evolution of the so - called hot spots " in the ft - sts experiments . let us elaborate on the physical properties of the aes now . in order to visualize these edge states better , we calculate the local density of states @xmath85 versus transverse momentum @xmath22 , as shown in the top panels of figs . [ ldos1 ] , [ ldos2 ] and [ ldos3 ] at different chemical potentials . noted that the edge states merge into the bulk at the nodal points and the weighting of the ldos is suppressed to zero . furthermore , the lattice approach reveals a much richer spatial structure in comparison with the conventional andreev equations in the continuous limit . for instance , the ldos has a strong dependence on the transverse momentum with transparent peak structures . for @xmath86 , there are three peaks separated by the nodal points and the peak positions change with the chemical potential . at @xmath87 , the outer peaks move to the boundary of brillouin zone and merge into one . therefore , for @xmath88 , there are only two peaks located at the center and the boundary of the brillouin zone and the locations of the peaks do not change with the chemical potential . further reducing the chemical potential to the regime @xmath89 , the relative weights of the peaks change but the locations remain fixed . by integrating over the brillouin zone , we can compute the spatial profile of the ldos in coordinate space as shown in the bottom panels of figs . [ ldos1 ] , [ ldos2 ] and [ ldos3 ] at different chemical potentials . on top of the decaying trend , the ldos also shows non - trivial oscillation due to quantum interferences due to different zero modes . these oscillations can only be captured faithfully in the lattice approach . for instance , at @xmath69 , the ldos at the outmost edge site is not the largest as one would naively expect so in the continuous theory . furthermore , the decay length is smaller as the chemical potential decreases . -wave symmetry near the zigzag edge . the hot spots are obtained by locating the momentum difference between large peaks in ldos . after @xmath90 , the second peak disappears since the outer peaks merge into one at the boundary of brillouin zone.,width=275 ] the momentum - resolved ldos can also help us to determine the hot spots due to quasi - particle scattering / interferences in ft - sts experiments . by fourier analysis of the stm data , the momentum transfer between quasi - particle scattering is revealed . the momentum transfer associated with the scattering process between peaks in ldos will emerge after fourier transformation . in fig . [ stmzf ] , the momentum transfers between peaks in ldos are plotted versus the chemical potential . for @xmath91 , there are three peaks giving two specific momentum transfers . for @xmath90 , there are only two peaks located at the center and the boundary of the brillouin zone . thus , the momentum transfer is always @xmath92 that is half of the brillouin zone . for easier experimental comparisons , we would also work out some other pairing symmetries explicitly . since the derivations are rather similar , we would skip the repeated parts and concentrated on the different outcomes . now we turn to the @xmath93 pairing symmetry at the zigzag edge . for @xmath93 pairing symmetry , cooper pairs form spin singlets and the gap function in the coordinate space is thus symmetric , @xmath94 . again , within the tight - binding approximations , the pairing potential is rather simple , @xmath95 , with relative angle @xmath34 where @xmath35 is an integer . the nodal lines , satisfying the constraint @xmath96 , are shown in the reconstructed brillouin zone in fig . [ pszdxy ] . the hamiltonian for the hopping is identically the same so that we do not put it down again . on the other hand , the pairing potential consists of another semi - infinite matrix @xmath97 , @xmath98 with the effective 1d pairing potential @xmath99 . note that the next nearest - neighbor pairing potentials are absent due to the nodal structure of @xmath93-wave pairing symmetry along the @xmath21-direction ( see fig . [ pszdxy ] ) . making use of @xmath100 , a unitary transformation is devised , @xmath101,\end{aligned}\ ] ] to bring the bdg hamiltonian into the susy form in eq.([diracz ] ) . although the pairing symmetry is different , the structure of the susy hamiltonian remain the same form . after some algebra , the off - diagonal components of the semi - infinite matrix @xmath102 are , @xmath103 symmetry at the zigzag edge of a triangular lattice . the bottom figure represents the reshaped brillouin zone , nodal lines and nodal points for the @xmath93 symmetric gap function with the same convention as explained in the @xmath10-wave case . , width=245 ] as mentioned before , the hidden susy in the bdg hamiltonian makes the zero - energy modes nodal for all pairing symmetries . repeating the same calculations , the @xmath56-plots are obtained at different chemical potentials . the only differences are the matrix elements @xmath104 and @xmath105 due to different pairing symmetry . by counting the decaying modes with @xmath63 , we can construct the phase diagram for aes with @xmath93 pairing symmetry as shown in fig . the phase diagram for @xmath93 symmetry appears to be much simpler since the number of nodes are reduced and the only double node lie in @xmath106 . starting from the regime @xmath107 , there exists an edge state with the positive / negative witten parity depending on the sign of the transverse momentum . when reaching @xmath87 , the nodal points move to the boundary of the reshaped brillouin zone so that the edge state exists for every transverse momentum . further reducing the chemical potential to the regime @xmath108 , the nodal points move backward to the center again . for @xmath109 , no edge state can be found . it is worth mentioning that the nodal point connecting edge states with opposite witten parities must be two - fold degenerate by simple counting . finally , we also calculated the hot spots at different chemical potentials , as shown in fig . [ stmdp ] , which can be measured in ft - sts experiment . pairing in the presence of the zigzag edge . the meanings of the labels are the same as in the @xmath10-wave case . , width=294 ] we come to the last case at the zigzag edge the @xmath110 pairing symmetry . within the tight - binding approximations , the gap function is @xmath111 , with relative angle @xmath34 where @xmath35 is an integer . the nodal lines in the momentum space , as shown in fig . [ pszpx ] , is determined by the constraint , @xmath112\sin(k_x/2)=0 $ ] . following the same steps , the semi - infinite matrix @xmath113 is @xmath114 with the effective gap potential @xmath115 . it is clear that @xmath116 . since the semi - infinite matrix @xmath113 share the same property as @xmath37 for the @xmath10-wave pairing , the same basis , eq.([fbasis ] ) , can utilized to bring the bdg hamiltonian into the susy form . after some algebra , the matrix elements of the semi - infinite matrix @xmath117 in eq.([diracz ] ) can be computed , @xmath118 following the same steps to obtain the @xmath56-plot , we can count the number of decaying modes with @xmath63 . the same construction leads to the phase diagram of aes for the @xmath110 pairing symmetry as plotted in fig . although we do not show the momentum - resolved ldos for the present case , it can be computed in a similar way as for the @xmath10-wave pairing . [ stmdp ] shows the evolution of the momentum transfer between the peaks in ldos at different chemical potentials and can be compared with the hot spots in the ft - sts measurements . symmetry at the zigzag edge of a triangular lattice . the bottom figure represents the reshaped brillouin zone , nodal lines and nodal points for the @xmath119 symmetric gap function with the same convention as explained in the @xmath10-wave case . , width=245 ] pairing in the presence of the zigzag edge . the meanings of the labels are the same as in the @xmath10-wave case.,width=294 ] as fig . [ pdzp ] shows , for the @xmath110 pairing symmetry , all edge states live in the null space of the semi - inifinite matrix @xmath120 , which are rather different from the @xmath10- and @xmath14-wave symmetries . that is to say , only aes with positive witten parity ( according to the our convention here ) exists ! the qualitative difference arises from the sign of the gap function across the open boundary . for the @xmath110 pairing symmetry , the pairing potentials at the edge sites all share the same sign . the pairing potential only changes signs when crossing the edge along the @xmath20-direction . as a result , the null space of the semi - infinite matrix @xmath117 vanishes and all edge states belong to the null space of @xmath121 instead . later , we will find that it also happens for the @xmath122 pairing symmetry at the flat edge . again , the underlying reason is that the gap function only changes signs across the open boundary of the system . or @xmath110 pairing symmetries near the zigzag edge . the hot spots are obtained by locating the momentum difference between large peaks in ldos . , width=275 ] by cutting the triangular lattice in another direction ( along the @xmath21-axis ) , we end up with a semi - infinite lattice with a flat edge as shown in figs . [ psfd ] and [ psfp ] . since the semi - infinite lattice is still translational invariant along the @xmath21-direction , the semi - infinte can be brought into the sum of the 1d chains by partial fourier transformation along the edge direction . noted that , to maintain the fermi statistics between the lattice operators , the brillouin zone must be reshaped in a different way as shown in figs . [ psfd ] and [ psfp ] . in the nambu basis , @xmath123 $ ] , the bdg hamiltonian of the @xmath124-wave pairing symmetry can be represented as , @xmath125 here @xmath126 is a semi - infinite matrix for the effective hopping in the 1d chains labeled by different momentum @xmath23 , @xmath127 with the momentum - dependent hopping amplitude @xmath128 . note that the chemical potential is renormalized , @xmath129 after the partial fourier transformation . not only the hopping matrix is different from that for the zigzag edge , the other semi - infinite matrix @xmath130 for the pairing potentials with the @xmath124-wave pairing symmetry would be different as well . in the following , we will study the phase diagrams of aes with different pairing symmetries near the flat edge in details . symmetry at the flat edge of a triangular lattice . the bottom figure represents the reshaped brillouin zone , nodal lines and nodal points for the @xmath93 symmetric gap function with the same convention as explained in the @xmath10-wave case . , width=313 ] for the @xmath93 pairing symmetry , the aes exists for both the zigzag and the flat edges . after partial fourier transformation in the @xmath21-direction , the pairing potential @xmath131 in eq.([bdgf ] ) can be explicitly worked out , @xmath132 with @xmath133 . to obtain the zero - energy states , it is convenient to bring the effective hamiltonian into the susy form as in the zigzag case , @xmath134 where @xmath124 denotes the pairing symmetry considered . for the flat edge , the semi - infinite matrix @xmath135 is simpler than that for the zigzag edge since it only has two off - diagonal rows instead of four , @xmath136 the matrix elements can be worked out explicitly from the semi - infinite matrix @xmath130 in eq . ( [ bdgf ] ) which depends on the pairing symmetry . for the @xmath93 pairing symmetry , the semi - infinite matrix satisfies @xmath137 . thus , the unitary transformation to the susy form is @xmath138.\end{aligned}\ ] ] it is straightforward to work out the matrix elements of the semi - infinie matrix @xmath139 , @xmath140 pairing in the presence of the flat edge . the meanings of the labels are the same as in the @xmath10-wave case . , width=294 ] again , the zero - energy modes exhibit the nodal structure and can be classified into two categories with opposite witten parities , @xmath141 here @xmath142 and @xmath143 belong to the null space of the semi - infinite matrices @xmath135 and @xmath144 respectively . the edge state is constructed from the generalized bloch theorem . for instance , the edge state with negative witten parity is @xmath145 , where @xmath56 satisfies , @xmath146 the above algebraic equation gives two solutions for @xmath56 . in the presence of the flat edge , the boundary conditions are slightly different , @xmath147 as before , only decaying modes with @xmath63 are allowed . but , only one constraint is required at the flat edge in contrast to the two constraints for the zigzag edge . the simplification is due to the missing matrix elements @xmath148 and @xmath149 at the flat edge which makes searching for the aes much easier here . the phase diagram for the aes with @xmath93 pairing symmetry is shown in fig . [ pdfd ] . using the bloch wave function of those edge states , we obtain the ldos for all transverse momenta @xmath23 . then , we can proceed to predict the sharp peaks in stm data after fourier analysis by finding out the momentum transfer between peaks in the ldos . the results are plotted in fig . [ stmf ] versus the chemical potential @xmath150 . symmetry at the flat edge of a triangular lattice . the bottom figure represents the reshaped brillouin zone , nodal lines and nodal points for the @xmath151 symmetric gap function with the same convention as explained in the @xmath10-wave case . ] we now continue to study the aes with the @xmath122-wave pairing symmetry at the flat edge as shown in fig . the nearest - neighbor gap amplitude of @xmath122-wave pairing takes the form , @xmath152 , with relative angle @xmath153 . the nodal lines in reshaped brilliouin zone , shown in fig . [ psfp ] , are determined by the equation @xmath154 . again , applying partial fourier transformation to the gap function , we obtain the semi - infinite matrix for the pairing potential , @xmath155 with @xmath156 . because of @xmath157 , the hamiltonian can be brought into susy form in the basis , @xmath158.\end{aligned}\ ] ] after straightforward algebra , the components of the matrix @xmath159 for the @xmath122-wave pairing are @xmath160 pairing in the presence of the flat edge . the meanings of the labels are the same as in the @xmath10-wave case . , width=294 ] or @xmath122 pairing symmetries near the flat edge . the hot spots are obtained by locating the momentum difference between large peaks in ldos . , width=275 ] the effective 1d chains for the flat edge are universal . thus , the whole discussions and calculations for the @xmath14-wave pairing with the flat edge can be applied here . substituting the matrix elements @xmath161 and @xmath162 into eq.([zsol2 ] ) and combining the boundary conditions of the flat edge , eq.([bcf ] ) , we obtain the phase diagram of aes for the @xmath122 pairing symmetry as shown in fig . [ pdfp ] . as we mentioned in previous section , the @xmath13-wave pairing potential changes sign across the edge boundary and lead to edge states with positive witten parity only . finally , using the bloch wave function of those edge states , we obtain the ldos for all transverse momenta @xmath23 . then , we can proceed to predict the sharp peaks in stm data after fourier analysis by finding out the momentum transfer between peaks in the ldos . the results are plotted in fig . [ stmf ] versus the chemical potential @xmath150 . there are simple patterns behind the phase diagrams we investigated in previous sections . for instance , the total witten parity changes by one when crossing a single nodal point while it changes by two across the double nodal point . it seems that the global structure of the phase diagram is dictated by the nodal points . these observations are indeed correct and can be explained by the continuity of @xmath56-plots . however , there is something deeper about why the nodal points are so important . in the following , we would like to make use of oshikawa s gauge argument@xcite and explain why edge states can only start / end at the nodal points . suppose we wrap up the semi - infinite lattice into tubural conformation and adiabatically thread a unit flux @xmath163 through it . the flux insetion changes the hamiltonian from @xmath164 to a different topological sector @xmath165 . if the ground state of original hamiltonian is protected by a gap , the insertion of a unit flux also transforms the ground state from @xmath166 to @xmath167 of the same energy . the flux insertion can be achieved by the constant vector potential @xmath168 with the circumference of the tube @xmath169 in the transverse direction of @xmath170 . meanwhile , the constant vector potential commutes with the transverse momentum @xmath171 that implies that the momentum remains constant in the whole adiabatic procedure , i.e. @xmath172 . before being able to compare @xmath166 to @xmath167 , we need to restore the hamiltonian to the same topological sector @xmath164 . the required large gauge transformation is @xmath173,\ ] ] where @xmath174 is the electron density at @xmath175 . now @xmath176 , so @xmath177 is a ground state of the original hamiltonian @xmath178 . the momentum of the new ground state can be evaluated straightforwardly , @xmath179)|\psi_0'\rangle=(p_0 + 2\pi n / l_d)u|\psi_0'\rangle$ ] . the total number of electrons can be separated into bulk and edge parts , @xmath180 . the momentum shift is then , @xmath181 , with @xmath182 and @xmath183 are the filling factors of the lattice and edge respectively . the area of the system is @xmath184 and the transverse size is @xmath185 . now , let us focus on the edge part . if we fill in only one edge state with @xmath186 , the momentum shift by the flux - insertion - removal trick is @xmath187 . the number of edge state then equal to the ground state degeneracy . since the gauge argument holds only when the ground state is protected by a finite gap , we can move one edge state to another between the nodal points . another interesting perspective is to relate the existence of aes to the underlying structure of the effective 1d model.@xcite the semi - infinte lattice can be mapped into effective 1d models . by choosing an appropriate unit cell , the 1d chain will contain only nearest - neighbor hopping described by the general hamiltonian @xmath188 where @xmath189 is the hopping matrix connecting nearest - neighbor cells and @xmath190 for the hopping within the cell . the matrices @xmath191 and @xmath189 are square matrices with @xmath192 rows , where @xmath192 is the number of effective lattice site in the unit cell . the semi - infinite matrix @xmath193 is the displacement operator on the effective 1d chain . we construct the edge states from the bloch states , @xmath194 , where @xmath56 satisfies @xmath195 . the boundary condition is extremely simple in this representation , @xmath196 . therefore , the number of the edge states is the dimension of the null space of of @xmath189 . if the rank of matrix @xmath189 is full , it means no edge state . in fact , the reflection symmetry with respect to the open boundary often implies that the rank of @xmath189 is full . for example , the @xmath122-wave pairing symmetry at the zigzag edge , one can find out that @xmath56 in the bloch state should satisfy eq.([zsol1 ] ) with @xmath197 and @xmath198 . that is to say , if @xmath56 is a solution , @xmath199 is also a solution . thus , except the nodal points , there are always two zero modes with @xmath200 . since there are also two constraints , we end up with no edge state . one can also check that the reflection symmetry makes the rank of the matrix @xmath189 full and thus leads to no edge state . in conclusion , we study the aes with different pairing symmetries and boundary topologies on semi - infinite triangular lattice of na@xmath0coo@xmath1@xmath2h@xmath1o . by mapping the 2d triangular lattice to the 1d counterpart , we can obtain the phase diagram and calculate the ldos of the aes at both zigzag and flat edges . surprisingly , the structure of the phase diagram crucially relies on the nodal points on the fermi surface and can be explained by an elegant gauge argument . finally , the momentum - resolved ldos allow us to predict the hot spots in fourier - transformed scanning tunneling spectroscopy experiments . we acknowledge supports from the national science council of taiwan through grants nsc-96 - 2112-m-007 - 004 and nsc-97 - 2112-m-007 - 022-my3 and also partial financial aids from the national center for theoretical sciences in taiwan . the effective hamiltonians in eqs . ( [ diracz ] ) and ( [ diracf ] ) can be described as the @xmath201 susy quantum mechanics@xcite , where @xmath202 is the number of supercharge operators . the two supercharge operators can be constructed explicitly @xmath203 one can verify that all susy algebra is satisfied . according to the definition , the susy hamiltonian is @xmath204 once we know how to diagonalize the susy hamiltonian , we can also construct the eigenstates of the supercharge operators ( our goal here ) as well . the susy algebra relates the @xmath205 ( the energy of the susy hamiltonian ) eigenstates with the opposite _ witten parities _ @xmath206 from the transformation of the witten parities , one can realize the energy spectrum of @xmath207 is symmetric about @xmath208 , i.e. @xmath209 . on the other hand , the @xmath210 states satisfy the operator equation , i.e. they live in the null space of the complex supercharge @xmath211 and @xmath212 , @xmath213 if we do find some states satisfying the above equation , it is called good susy because the @xmath210 states are annihilated by supercharge . on the other hand , if we can not find any @xmath210 state . it is often referred as bad susy since the ground state carries non - zero supercharge@xcite . however , for condensed matter systems , the good susy gives rise to the zero - energy anomaly while the bad susy actually makes the energy spectrum symmetric about the zero energy without anomaly .
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we study the andreev edge states with different pairing symmetries and boundary topologies on semi - infinite triangular lattice of na@xmath0coo@xmath1@xmath2h@xmath1o . a general mapping from the two dimensional lattice to the one dimensional tight - binding model is developed .
it is shown that the phase diagram of the andreev edge states depends on the pairing symmetry and also the boundary topology .
surprisingly , the structure of the phase diagram crucially relies on the nodal points on the fermi surface and can be explained by an elegant gauge argument .
we compute the momentum - resolved local density of states near the edge and predict the hot spots which are measurable in fourier - transformed scanning tunneling spectroscopy .
| 12,550 | 196 |
the detection of gravitational waves ( g.w . ) is one of the most fascinating and challenging subjects in physics research nowadays . besides checking the general relativity theory , the detection of this phenomenon will mark the beginning of a new phase in the comprehension of astrophysical phenomena by the use of gravitational wave astronomy . although these waves were predicted at the beginning of the century @xcite , the research on their detection only started around 1960 , with the studies of joseph weber @xcite . the major obstacle to this detection is the tiny amplitude the g.w . have @xcite . even though the more sensitive detector now operating @xcite is capable to detect amplitudes near @xmath0 , this value must be decreased by several orders of magnitude so that impulsive waves can be detected regularly . on the other hand , the discovery of pulsars with periods lying in the milliseconds range stimulated the investigations on the detection of gravitational waves of periodic origin . although these waves generally have amplitudes even smaller than those emitted by impulsive sources , periodic sources are continuously emitting gravitational waves in space and they can be detected as soon as the correct sensitivity is reached . since many of the resonant mass antennae now operating are designed to detect frequencies near 1000 hz , the millisecond pulsars will probably be detected if these antennae ever become sensitive to amplitudes @xmath1 . this value is bigger if we consider the crab pulsar ( @xmath2 ) : @xmath3 . there is a resonant mass detector with a torsional type antenna ( crab iv ) being developed by the tokyo group @xcite to detect gravitational waves emitted by the crab pulsar . this group expects to reach @xmath4 soon . the main purpose of this paper is a contribution towards the increase in sensitivity of resonant mass continuous gravitational wave detectors looking at the use of adequate filters . we study two kinds of filters , the first optimizes the signal - to - noise ratio ( snr ) , and is normally used in the detection of impulsive waves @xcite . the second filter reproduces the wave with minimum error . both filters apparently were not investigated in the continuous gravitational wave context yet . linear , stationary filters obey the relation @xmath5 @xmath6 is the impulse response function that characterizes the filter @xmath7 , @xmath8 is the input at the filter and @xmath9 is the filter output . generally @xmath10 has a useful part , @xmath11 , and an unwanted part , @xmath12 : @xmath13 . we have a similar relation for the filter output , given by @xmath14 . it is well known from noise theory@xcite that the filter @xmath15 that optimizes snr at its output represents the average value of @xmath16 . ] , @xmath17 must have the following transfer function : @xmath18 with @xmath19 @xmath20 is the instant in which the observation takes place , @xmath21 is the fourier transform of @xmath11 ( * denotes complex conjugation ) and @xmath22 is the noise power spectrum density : @xmath23 the maximum snr at the optimal filter output is given by the expression @xmath24 from ( [ 2 ] ) and ( [ 3 ] ) we conclude that a very weak signal will leave the filter when the noise is much stronger than the useful signal at the relevant frequency range . equation ( [ 2 ] ) is valid as long as @xmath21 is well behaved . for example , if @xmath11 were a strictly monochromatic wave like @xmath25 it would be difficult to build this filter since @xmath26 . in order to use the optimal filter ( [ 2 ] ) in continuous gravitational wave detectors we will describe these waves as _ quasi - monochromatic _ useful signals . it means that the waves that reach the antenna will be of the form " . ] @xmath27 the constant _ a _ is related to the signal spectral density bandwidth , @xmath28 , and the corresponding spectral density is of the form real and @xmath29 . ] @xmath30.\ ] ] the signal ( [ 6 ] ) is quasi - monochromatic whenever @xmath31 , @xmath32 being its central frequency . note that when @xmath33 we recover ( [ 5 ] ) , the monochromatic case . the continuous gravitational waves emitted by periodic sources can be regarded as quasi - monochromatic waves . the frequency of the crab pulsar , for example , which is centered near @xmath34 , has a slow down rate of @xmath35 . besides , the orbital motion of the earth causes a maximum variation of @xmath36 , and the spinning motion of earth implies a maximum variation of @xmath37@xcite . for future use in the optimal filter expression , ( [ 2 ] ) , we write the fourier transform of the quasi - monochromatic signal ( [ 6 ] ) : @xmath38 . \label{8}\ ] ] a resonant mass detector can be represented by the scheme of figure [ figure 1 ] . in this model @xmath39 represents the gravitational interaction force between the g.w . and the antenna . the two - port circuit is related to the massive antenna and the transducer , and it is described by its admittance matrix @xmath40 , which relates the force @xmath41 and the velocity @xmath42 at the input port to the current @xmath43 and the voltage @xmath44 at the output port : @xmath45 @xmath46 the transducer and the amplifier have force and velocity noise generators represented by the stochastic , stationary functions @xmath16 and @xmath47 , respectively . @xmath48 [ @xmath49 is the spectral density of @xmath16 [ @xmath47 ] . we will assume that these functions are not correlated , so that @xmath51 . ( 400,250 ) ( 0,0)(400,250 ) in this model the optimal filter follows the lock - in amplifier . in figure [ figure 2 ] the elements that precede the optimal filter in the detector are redrawn@xcite . ( 300,200 ) ( 0,0)(300,200 ) in this figure @xmath52 is the antenna effective mass , @xmath53 its elastic constant and @xmath54 the damping constant . @xmath55 represents the mechanical dissipation at the antenna , @xmath16 is associated to the transducer back - reaction on the antenna and @xmath47 represents the wideband , serial noise introduced by the amplifier . the equation of motion of the system in given by @xmath56 -f(t),\ ] ] where @xmath57 represents the displacement . however , in the calculations we will deal with the velocity @xmath58 ; at the amplifier output we have @xmath59 . in the absence of @xmath39 we can obtain the velocity noise spectral density : @xmath60 at the denominator of this expression , the minus signal is used when @xmath61 , and the plus signal is used when @xmath62 . since the thermal and the back - reaction noises are white noises , the force spectral densities generated by them obey the following nyquist relations@xcite : @xmath63 and @xmath64 . in these expressions , @xmath65 and @xmath66 are the time constants of mechanical loss at the antenna and of electrical loss at the transducer and amplifier , respectively . they are related to the energy decay time of the antenna , @xmath67 , according to the expression @xmath68 , where @xmath69 is the antenna quality factor . @xmath70 is the antenna temperature and @xmath71 is the back - reaction noise temperature . @xmath72 is the boltzmann constant . the function @xmath73 that appears in ( [ 10 ] ) represents a serial white noise introduced in the useful signal by the electrical network , and it has the following expression : @xmath74 , where @xmath75 comes from ( [ 9 ] ) . @xmath76 is the real part of the impedance at the transducer output and @xmath77 is the circuit noise temperature . we assume that the amplifier has a large but limited bandwidth , as it occurs in practice , given by @xmath78 . generally @xmath79 introducing the antenna s _ equivalent temperature _ , @xmath80 , defined by the relation @xmath81 , equation ( [ 10 ] ) becomes @xmath82 this is the complete expression for the total noise spectral density at the filter input . the velocity that the g.w . ( [ 6 ] ) generates at the antenna in the absence of the noises @xmath83 and @xmath16 has the following fourier transform : @xmath84 @xmath85 is the fourier transform of the g.w . force on the antenna , which is given by the relation@xcite @xmath86 @xmath87 is the dynamic part of the mass quadrupole tensor of the antenna . it is a matrix of constant elements which depend on the antenna s geometry and mass distribution . for our calculations we use @xmath88 , where @xmath89 is the characteristic length and @xmath90 is the density of the antenna . equation ( [ 13 ] ) is the useful signal contribution at the filter input , which corresponds to the spectral density henceforth . ] @xmath91 @xmath92 is the spectral density of the force ( [ 13a ] ) and it has the form @xmath93 using ( [ 2 ] ) , ( [ 12 ] ) and ( [ 13 ] ) and adopting @xmath94 we obtain the transfer function of the filter that optimizes snr for the model considered in the preceding section : @xmath95[1-\imath \tau_0 ( \omega -\omega_0 ) ] } } { \frac{n_p}{1+(\omega-\omega_0)^2\tau_0 ^ 2}+\frac{n_s}{1+(\omega-\omega_0)^2 \tau_a^2}}. \label{18}\ ] ] to simplify this expression we introduced the following definitions : @xmath96 , @xmath97 and @xmath98 . the maximum snr related to this filter is obtainable from ( [ 3 ] ) , ( [ 12 ] ) and ( [ 14 ] ) , and it corresponds to @xmath99 note that when @xmath100 , this expression becomes @xmath101 assuming , for simplicity , that @xmath102 , we find @xmath103 by imposing @xmath104 we obtain the following condition on the parameters of the detector : @xmath105 to illustrate the use of the filter ( [ 18 ] ) we will adopt two different realistic detectors . in both cases we will assume that @xmath102 and @xmath106 ( see equation ( [ 11 ] ) . the first of them , detector a , is designed to detect crab pulsar ( @xmath107 , @xmath108 and @xmath109 . this bandwidth arises from the slow down of the pulsar after 100 milliseconds of observation ) ; this detector has the following characteristics : @xmath110 , @xmath111@xcite , @xmath112 , @xmath113 , @xmath114 . under these conditions , ( [ 18 ] ) shows a very narrow peak centered in @xmath32 . it implies a maximum snr at its output given by @xmath115 . if the signal bandwidth is smaller ( implying a smaller observation time ) , we have @xmath116 . on the other hand , if detector a has a lower equivalent temperature we attain @xmath115 with a longer observation time . for instance , we obtain this result if @xmath117 , and @xmath118 . such bandwidth corresponds to a 10 seconds observation time of the crab pulsar s slow down . the other detector considered , detector b , is designed to detect the millisecond pulsar psr 1937 + 214@xcite ( @xmath119 , @xmath120 ) . since we did not find any information about the bandwidth of the g.w . emitted by this pulsar , we will assume that it has the value @xmath121 . this detector is characterized by @xmath110 , @xmath113 , @xmath122 , @xmath123 and @xmath124 ; these are typical values of several ultracryogenic cylindrical antennae . for detector b , ( [ 18 ] ) also shows a very narrow peak centered in @xmath32 , implying @xmath115 . like detector a , @xmath125 is greater if the signal bandwidth is smaller . after the g.w . is detected we have to determine its shape with minimum error . this can be accomplished with the help of an adequate linear filter , @xmath126 , designed to reproduce the useful signal with the greatest possible accuracy ( depending on the noise and the useful signal present at its input ) . this accuracy is characterized by the mean square error , @xmath127 , which is obtained from the instantaneous reproduction error , @xmath128 , defined by @xmath129 @xmath130 is the desired signal at the filter output . in a simple filtering process , as the one we are considering in this work , @xmath130 must be equal to the useful signal at the filter input , @xmath11 . we obtain the transfer function @xmath131 of the filter @xmath126 by imposing @xmath132 . this condition implies@xcite @xmath133 supposing there is no cross - correlation between @xmath11 and @xmath12 . the corresponding mean square error is @xmath134 from this equation it is evident that the error becomes smaller if so becomes the noise . on the other hand , if the noise is too strong ( @xmath135 ) it results @xmath136 and no signal leaves the filter . if we define the _ total _ power of a signal @xmath137 by @xmath138 , the snr at the filter input will be ) is different from ( [ 1 ] ) . this happens because we are now interested on the total spectrum of the useful signal , while in the analysis of the first kind of filter we were interested only on the maximum amplitude of this signal . ] @xmath139 at the filter output the signal @xmath137 will have the following total power : @xmath140 using this relation we can find the snr at the filter output , @xmath141 we now consider the particular case of noise spectral density given by ( [ 12 ] ) and useful signal spectral density given by ( [ 13 ] ) . in this case the filter that reproduces @xmath11 with minimum error has the following transfer function : @xmath142[1+\tau_0 ^ 2(\omega-\omega_0)^2 ] } } \right ) ^{-1}. \label{27}\ ] ] the minimum error introduced by this filter is obtained from ( [ 23 ] ) using ( [ 12 ] ) and ( [ 14 ] ) : @xmath143[1+\tau_0 ^ 2(\omega-\omega_0)^2 ] } { { \cal g}^2a } + \frac{1 } { \frac{n_p}{1+(\omega-\omega_0)^2\tau_0 ^ 2 } + \frac{n_s}{1+(\omega-\omega_0)^2\tau_a^2 } } \right ) ^{-1 } d\omega . \label{29}\ ] ] in detector a the total noise power at the filter input is @xmath144 . if we use filter ( [ 27 ] ) in this detector we obtain @xmath145 , which is @xmath146 times smaller than @xmath147 . comparing ( [ 25 ] ) and ( [ 25a ] ) for this case , we find @xmath148 and @xmath149 , so that @xmath150 . on the other hand , using filter ( [ 27 ] ) in detector b we obtain an output error of @xmath151 , which is almost @xmath152 times smaller then the input noise , @xmath153 . in this case , @xmath154 and @xmath155 , which imply @xmath156 . if the crab bandwidth were @xmath157 we would obtain @xmath158 ; this bandwidth also implies @xmath159 . we would find the same @xmath160 if the bandwidth of psr1937 + 214 were @xmath161 , corresponding to @xmath162 . we have derived expressions for the transfer functions of two kinds of filters , both designed to detect continuous monochromatic waves . the first filter , @xmath15 , optimizes snr at its output ( equation ( [ 1 ] ) ) and is important for a first detection of the wave . the second filter , @xmath126 , reproduces the wave with minimum error and should be used when we intend to know the complete shape of the wave . in the study of @xmath15 we have first analysed the detection of crab pulsar . supposing @xmath163 ( see section @xmath44 ) we concluded that this pulsar could be detected if its signal were as monochromatic as @xmath164 ; it means a sampling time of the order of 100 msec . we have also analysed a possible detection of psr 1937 + 214 and suggested a maximum limit for its bandwidth , allowing its detection by third generation resonant mass detectors . with the continuous optimization of present detectors the condition of monochromaticity of the signal becomes weaker . for example , if @xmath166 , the bandwidths of the signal can be larger than those we obtained in this paper . besides , the inequality ( [ 211 ] ) can be used as a reference to optimize resonant mass continuous gravitational detectors . note that continuous sources with high frequency , small bandwidth and high amplitude are the most favourable for detection with less improvement of the detector . on the other hand , the detector should have a small equivalent temperature and materials with high quality factor and density should be preferred for the antenna body , which should be as large as possible . thanks cnpq ( braslia - df , brazil ) and fapesp ( so paulo - sp , brazil ) for financial support , and c.o.e . thanks cnpq for partial support . we are grateful to o.d.aguiar and c.frajuca for fruitful discussions .
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we determine the transfer functions of two kinds of filters that can be used in the detection of continuous gravitational radiation .
the first one optimizes the signal - to - noise ratio , and the second reproduces the wave with minimum error .
we analyse the behaviour of these filters in connection with actual detection schemes .
| 4,874 | 69 |
the extended hubbard model with anisotropic spin exchange interactions @xcite is a conceptually simple phenomenological model for studying correlations and for a description of magnetism and other types of electron orderings in narrow band systems with easy - plane or easy - axis magnetic anisotropy . in this report we will focus on the zero - bandwidth limit of the extended hubbard model with magnetic interactions for the case of arbitrary electron density . we consider the @xmath0-@xmath1 hamiltonian of the following form : @xmath4 where @xmath0 is the on - site density interaction , @xmath5 is @xmath6-component of the intersite magnetic exchange interaction , restricts the summation to nearest neighbours . @xmath7 denotes the creation operator of an electron with spin @xmath8 at the site @xmath9 , , and . the chemical potential @xmath10 depending on the concentration of electrons is calculated from @xmath11 with and @xmath12 is the total number of lattice sites . the model ( [ row:1 ] ) can be treated as an effective model of magnetically ordered insulators . the interactions @xmath0 and @xmath1 will be assumed to include all the possible contributions and renormalizations like those coming from the strong electron - phonon coupling or from the coupling between electrons and other electronic subsystems in solid or chemical complexes . in such a general case arbitrary values and signs of @xmath0 are important to consider . we restrict ourselves to the case of positive , because of the symmetry between ferromagnetic ( ) and antiferromagnetic ( ) case for lattice consisting of two interpenetrating sublattices such as for example sc or bcc lattices . we have performed extensive study of the phase diagram of the model ( [ row:1 ] ) for arbitrary @xmath13 and @xmath10 @xcite . in the analysis we have adopted a variational approach ( va ) which treats the on - site interaction @xmath0 exactly and the intersite interaction @xmath1 within the mean - field approximation ( mfa ) . we restrict ourselves to the case of the positive @xmath1 , as it was mentioned above . let us point out that in the mfa , which does not take into account collective excitations , one obtains the same results for the model and the model , where the term is replaced with , describing interactions between @xmath14-components of spins at neighbouring sites , . in both cases the self - consistent equations have the same form , only the replacement is needed and a magnetization along the @xmath6-axis becomes a magnetization in the @xmath14-plane @xcite . for the model ( [ row:1 ] ) only the ground state phase diagram as a function of @xmath10 @xcite and special cases of half - filling ( ) @xcite and @xcite have been investigated till now . within the va the intersite interactions are decoupled within the mfa , what let us find a free energy per site @xmath15 . the condition ( [ row:2 ] ) for the electron concentration and a minimization of @xmath15 with respect to the magnetic - order parameter lead to a set of two self - consistent equations ( for homogeneous phases ) , which are solved numerically . the order parameter is defined as , where is the average magnetization in a sublattice in the direction ( @xmath16 corresponds @xmath17 here ) . if @xmath18 is non - zero the ferromagnetic phase ( f@xmath3 ) is a solution , otherwise the non - ordered phase ( no ) occurs . phase separation ( ps ) is a state in which two domains with different electron concentration exist in the system ( coexistence of two homogeneous phases ) . the free energies of the ps states are calculated from the expression : @xmath19 where @xmath20 are values of a free energy at @xmath21 corresponding to the lowest energy homogeneous solutions and is a fraction of the system with a charge density @xmath22 . we find numerically the minimum of @xmath23 with respect to @xmath22 and @xmath24 . in the model considered only ps@xmath3 state ( i. e. a coexistence of f@xmath3 and no phases ) can occur . in the paper we have used the following convention . a second ( first ) order transition is a transition between homogeneous phases with a ( dis-)continuous change of the order parameter at the transition temperature . a transition between homogeneous phase and ps state is symbolically named as a `` third order '' transition . during this transition a size of one domain in the ps state decreases continuously to zero at the transition temperature . second order transitions are denoted by solid lines on phase diagrams , dotted curves denote first order transitions and dashed lines correspond to the `` third order '' transitions . we also introduce the following denotation : for , where @xmath25 is the number of nearest neighbours . obtained phase diagrams are symmetric with respect to half - filling because of the particle - hole symmetry of the hamiltonian ( [ row:1 ] ) , so the diagrams will be presented only in the range . in the ground state the energies of homogeneous phases have the form : for no : and for f@xmath26 : if and if . comparing the energies we obtain diagram shown in fig . [ rys : gdpd ] . at the first order transition f@xmath3no takes place in the system . this transition is associated with a discontinuous disappearance of the magnetization . without consideration of ps states . the dotted line denotes discontinuous transition.,scaledwidth=45.0% ] the first derivative of the chemical potential for in the lowest energy phases is negative what implies that homogeneous phases are not stable ( except ) . finite temperature phase diagrams taking into account only homogeneous phases and plotted as a function of @xmath27 for chosen @xmath13 are shown in fig . [ rys : pdjed]a . the tricritical point @xmath28 , which is connected with a change of transition order , for is located at and @xcite . the range of the occurrence of f@xmath3 phase is reduced with decreasing @xmath13 . for and any we observe only one transition f@xmath3no with increasing temperature . in the range the @xmath27 coordinate of the remains constant , so for the transition is discontinuous . however , for in some range of there can appear a sequence of two transitions : . in fig . [ rys : pdjed]b there are shown dependencies of the transition temperature as a function of @xmath13 for chosen values of . the range of f@xmath3 stability is reduced with decreasing of . for and any @xmath13 we observe only one second order transition f@xmath3no with increasing temperature . there exist ranges of @xmath13 and , where the sequence of transitions : is present . at sufficiently low temperatures homogeneous phases are not states with the lowest free energy and there ps state can occur . on the phase diagrams , where we considered the possibility of appearance of the ps states , there is a second order line at high temperatures , separating f@xmath3 and no phases . a `` third order '' transition takes place at lower temperatures , leading to a ps into f@xmath3 and no phases . the critical point for the phase separation ( denoted as @xmath29 , a tricritical point ) lies on the second order line . phase diagrams for and are shown in fig . [ rys : pdsep ] . in the ranges of ps stability the homogeneous phases can be metastable ( if ) or unstable ( if ) . we leave a deeper analyses of meta- and unstable states to future publications . we considered a simple model for magnetically ordered insulators . it was shown that at the sufficiently low temperatures homogeneous phases do not exist and the states with phase separation are states with the lowest free energy . on phase diagrams we also observe the tricritical points , which are associated with a change of transition order ( , fig . [ rys : pdjed ] ) or are located in the place where the second order line connects with `` third order '' lines ( , fig . [ rys : pdsep ] ) . let us stress that the knowledge of the zero - bandwidth limit can be used as starting point for a perturbation expansion in powers of the hopping and as an important test for various approximate approaches ( like dynamical mfa ) analyzing the corresponding finite bandwidth models .
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a simple effective model for a description of magnetically ordered insulators is analysed .
the tight binding hamiltonian consists of the effective on - site interaction ( @xmath0 ) and intersite magnetic exchange interactions ( @xmath1 , @xmath2 ) between nearest - neighbours .
the phase diagrams of this model have been determined within the variational approach , which treats the on - site interaction term exactly and the intersite interactions within the mean - field approximation .
we show that , depending on the values of interaction parameters and the electron concentration , the system can exhibit not only homogeneous phases : ( anti-)ferromagnetic ( f@xmath3 ) and nonordered ( no ) , but also phase separated states ( ps@xmath3 : ) .
| 2,260 | 201 |
during the last decades , a new type of artificial materials , the so - called left - handed metamaterials ( lh ) , have attracted a great deal of attention . they present negative indices of refraction for some wavelengths @xcite , with considerable applications in modern optics and microelectronics @xcite . metamaterials can resolve images beyond the diffraction limit @xcite , act as an electromagnetic cloak @xcite , enhance the quantum interference @xcite or yield to slow light propagation @xcite . regarding the localization length in disordered systems , the presence of negative refraction in one - dimensional ( 1d ) disordered metamaterials strongly suppresses anderson localization @xcite . as a consequence , an unusual behavior of the localization length @xmath0 at long - wavelengths @xmath5 has been observed . et al . _ reported a sixth power dependence of @xmath0 with @xmath5 under refractive - index disorder @xcite instead of the well - known quadratic asymptotic behavior @xmath10 @xcite . recently , mogilevtsev _ et al . _ @xcite have also found a suppression of anderson localization of light in 1d disordered metamaterials combining oblique incidence and dispersion while torres - herrera _ et al . _ @xcite have developed a fourth order perturbation theory to resolve the problem of non - conventional anderson localization in bilayered periodic - on - average structures . the effects of polarization and oblique incidence on light propagation in disordered metamaterials were also studied in ref . @xcite . in this article , we calculate numerically the localization length of light @xmath0 for a one - dimensional arrangement of layers with index of refraction @xmath1 and thickness @xmath2 alternating with layers of index of refraction @xmath3 and thickness @xmath4 . in order to introduce disorder in our system , we change the position of the layer boundaries with respect to the periodic values maintaining the same values of the refraction indices @xmath1 and @xmath3 . this is the case of positional disorder , in contrast to the compositional disorder where there exist fluctuations of the index of refraction @xcite . two structures will be analyzed in detail : homogeneous stacks ( h ) , composed entirely by the traditional right - handed materials ( rh ) with positive indices of refraction , and mixed stacks ( m ) with alternating layers of left- and right- handed materials . for the sake of simplicity , the optical path in both layers will be the same , that is , the condition @xmath9 is satisfied in most of the work . these periodic - on - average bilayered photonic systems have already been studied analytically by izrailev _ these authors have developed a perturbative theory up to second order in the disorder to derive an analytical expression for the localization length for both h and m stacks . in our case , we have obtained two equations for the localization length @xmath0 as a function of the wavelength @xmath5 from our numerical results . for h stacks , a quadratic dependence of @xmath0 for long - wavelengths is found , as previously reported in the literature . on the other hand , the localization length saturates for lower values of @xmath5 . an exhaustive study of @xmath0 in the allowed and forbidden bands ( gaps ) of weakly disordered systems will be carried out . we will show that the localization length is modulated by the corresponding bands and this modulation decreases as the disorder increases . for low - disordered m stacks and wavelengths of several orders of magnitude greater than the grating period @xmath11 , the localization length @xmath0 depends linearly on @xmath5 with a slope inversely proportional to the modulus of the reflection amplitude between alternating layers . the plan of the work is as follows . in sec . ii we carry out an exhaustive description of our one - dimensional disordered system and the numerical method used in our localization length calculations . a detailed analysis of @xmath0 in the allowed bands and gaps of homogeneous stacks is performed in sec . iii where a practical expression for the localization length as a function of @xmath5 and the disorder is derived . in sec . iv we calculate @xmath0 for mixed stacks of alternating lh and rh layers . a linear dependence of the localization length at long - wavelengths is found for low - disordered m stacks . finally , we summarize our results in sec . v. let us consider a one - dimensional arrangement of layers with index of refraction @xmath1 alternating with layers of index of refraction @xmath3 . the width of each one is the sum of a fixed length @xmath12 for @xmath13 and a random contribution of zero mean and a given amplitude . the wave - numbers in layers of both types are @xmath14 , where @xmath15 is the frequency and @xmath16 the vacuum speed of light . as previously mentioned , the grating period of our system @xmath17 is defined as the sum of the average thicknesses @xmath2 and @xmath4 of the two types of layers , that is , @xmath18 . we have introduced the optical path condition @xmath19 for simplicity ( in the case of left - handed layers @xmath20 , so the absolute value has been written to consider these type of materials ) . without disorder , each layer would be limited by two boundaries @xmath21 and @xmath22 where @xmath23 is the total number of boundaries . the periodic part of the system considered is schematically represented in fig . [ fig1 ] . and thickness @xmath2 alternating with layers of index of refraction @xmath3 and thickness @xmath4 . the grating period is @xmath11 . ] in the presence of disorder , the position of the corresponding boundaries are @xmath24 except for the first and the last boundary , so as to maintain the same total length @xmath25 . the parameters @xmath26 are zero - mean independent random numbers within the interval @xmath27 $ ] . throughout all our calculations , we have chosen values of the disorder parameter @xmath28 less than @xmath2 and @xmath4 . for each @xmath25 , we calculate the transmission coefficient of our structure @xmath29 and average its logarithm , @xmath30 , over 800 disorder configurations . then , we obtain numerically the localization length @xmath0 via a linear regression of @xmath30 @xcite @xmath31 here , the angular brackets @xmath32 stand for averaging over the disorder . we choose 6 values of the total length @xmath25 to perform the linear regression of eq.([linearregr ] ) . the localization length @xmath0 is evaluated as a function of the disorder parameter @xmath28 and the frequency of the incident photon @xmath15 . we calculate the transmission coefficient of our system via the characteristic determinant method , firstly introduced by aronov _ this is an exact and non perturbative method that provides the information contained in the green function of the whole system . in our case , the characteristic determinant @xmath33 can be written as @xcite @xmath34 where the index @xmath35 runs from 1 to @xmath23 and the coefficients @xmath36 and @xmath37 can be written as @xmath38 and @xmath39 the parameters @xmath40 , which are the reflection amplitudes between media @xmath41 and @xmath35 , are given by @xmath42 where @xmath43 corresponds to the impedance of layer @xmath35 and can be be expressed for normal incidence in terms of its dielectric permittivity @xmath44 and magnetic permeability @xmath45 @xmath46 the quantity @xmath47 entering eqs . ( [ aj ] ) and ( [ bj ] ) is a phase term @xcite @xmath48.\ ] ] here @xmath49 is the wave - number in a layer with boundaries @xmath50 and @xmath51 . this recurrence relation facilitates the numerical computation of the determinant . the initial conditions are the following @xmath52 the transmission coefficient of our structure @xmath29 is given in terms of the determinant @xmath53 by @xmath54 before dealing with mixed stacks , we present results for low - disordered homogeneous systems with underlying periodicity , which has not been previously studied . in this section we perform a detailed analysis of the localization length @xmath0 in the allowed bands and in the forbidden gaps of disordered h stacks as a function of the disorder @xmath28 , the incident wavelength @xmath5 and the reflection coefficient between alternating layers @xmath55 . as it is well known , in the absence of disorder the transmission spectrum of right - handed systems presents allowed and forbidden bands whose position can be easily determined via the following dispersion relation obtained from the bloch - floquet theorem @xcite @xmath56 where @xmath57 is the block wave - vector . when the modulus of the right - hand side of eq . ( [ beta ] ) is greater than 1 , @xmath57 has to be taken as imaginary . this situation corresponds to a forbidden band . taking into account the condition @xmath7 , eq . ( [ beta ] ) reduces to @xmath58 on the other hand , when @xmath59 is equal to unity , the incident frequency @xmath15 is located at the center of the @xmath60-th allowed band , @xmath61 . after some algebra , we obtain from eq . ( [ betasimpl ] ) @xmath62 let us first consider a periodic h stack formed by 50 layers of length @xmath63 52.92 nm and index of refraction @xmath64 1.58 alternating with 49 layers of length @xmath65 39.38 nm and @xmath66 2.12 . the total size of our structure is 4.57 @xmath67 m and the reflection coefficient between alternating layers 0.05259 . fig.[fig2](a ) represents the transmission coefficient @xmath29 as a function of the frequency @xmath15 to illustrate its behavior . also shown are the center of each allowed band calculated via eq.([omegacenter ] ) . there are 99 peaks in each band so they can hardly been resolved on the scale used . moreover , in fig.[fig2](b ) the parameter @xmath59 is plotted versus the frequency @xmath15 for this periodic system . the first gap and the first allowed band have been shown for a better comprehension . and ( b ) the parameter @xmath59 versus the frequency @xmath15 for the homogeneous periodic system described in the text ( 99 layers ) . ] a systematic numerical simulation of a realistic system with 50000 layers has been carried out . the parameters are the same as in the previous example . in fig . [ fig3 ] we represent the localization length @xmath0 versus the wavelength @xmath5 for different values of the disorder parameter @xmath28 ( shown in the legend of the figure ) . the dashed line corresponds to total disorder , that is , an arrangement of layers with random boundaries and alternating indices of refraction @xmath1 and @xmath3 . several features are evident in the figure . for long - wavelengths , one observes a quadratic asymptotic behavior , as can be compared with the dotted line @xcite . an in - deep numerical analysis of the coefficient characterizing this dependence has been performed . to this aim , 20 different h stacks were considered and the following expression for the localization length was found @xmath68 where @xmath69 is the optical path across one grating period @xmath17 . all the lengths in eq.([longlochinf ] ) are expressed in units of @xmath17 . in the opposite limit of short @xmath5 , the localization length @xmath0 saturates to a constant value @xcite . our numerical results have shown that this constant is proportional to the inverse of the reflection coefficient between alternating layers @xmath70 , that is , @xmath71 izrailev _ et al . _ @xcite have developed a perturbative theory up to second order in the disorder to calculate analytically the localization length in both homogeneous and mixed stacks . this model is quite general and is valid for both quarter stack medium ( mainly considered in our work ) and systems with different optical widths . assuming uncorrelated disorder and random perturbations with the same amplitude in both layers ( the main considerations in our numerical calculations ) one can easily derive the following analytical expression for @xmath0 at long - wavelengths from izrailev s formulation @xmath72 for similar values of the layer impedances @xmath73 , the first term in eq.([izragen ] ) can be approximated by @xmath74 and @xmath75 , so eq.([izragen ] ) reduces to @xmath76 which is similar to our numerical expression eq . ( [ longlochinf ] ) . the randomness only affects partially the periodicity of the system , which manifests in the existence of bands and gaps . the localization length depends on the position in the band and on the disorder . the modulation of @xmath0 by the bands can be clearly appreciated in fig . these results are consistent with other published works on this topic @xcite . recently , mogilevtsev _ et al . _ @xcite have reported that the photonic gaps of the corresponding periodic structure are not completely destroyed by the presence of disorder while luna - acosta _ et al . _ @xcite have shown that the resonance bands survive even for relatively strong disorder and large number of cells . versus the wavelength @xmath5 for different values of the disorder parameter @xmath28 . the h stack corresponds to the arrangement represented in fig . [ fig2 ] but now 50000 layers have been considered . the dashed line stands for the total disorder case . all lengths are expressed in units of the grating period @xmath17 . ] having a close look into the first gap in fig . [ fig3 ] , one observes that the localization length is practically independent of the disorder @xmath28 . in order to visualize this effect , fig.[fig4 ] represents ( a ) the first and ( b ) the second gaps depicted in fig . [ fig3 ] . as mentioned , the dependence of @xmath0 with the disorder is almost negligible in the first gap . when the wavelength is similar to the grating period @xmath17 , the influence of the disorder is greater , as can be easily deduced from simple inspection of fig . [ fig4](b ) . versus the wavelength @xmath5 for ( a ) the first and ( b ) the second gaps depicted in fig . [ fig3 ] . ] let us now focus on the allowed bands and study in detail the behavior of the localization length in these regions . to this aim , three - dimensional ( 3d ) graphs of @xmath0 versus the wavelength @xmath5 and the disorder @xmath28 have been plotted in fig.[fig5 ] for ( a ) the first and ( b ) the third allowed bands ( see again fig . [ fig2 ] ) . all this magnitudes have been normalized to the grating period @xmath17 . the localization length @xmath0 is enhanced in a small region around the center of each allowed band . a similar result was found by hernndez - herrejn _ et al . _ @xcite who obtained a resonant effect of @xmath0 close to the band center in the kronig - penney model with weak compositional and positional disorder . this increase in the localization length is due to emergence of the fabry perot resonances associated with multiple reflections inside the layers from the interfaces @xcite . in particular , for homogeneous quarter stack systems , the fabry perot resonances arise exactly in the middle of each allowed band where @xmath57 vanishes @xcite . the saturation of @xmath0 for short - wavelengths is also appreciated in these 3d images . versus the wavelength @xmath5 and the disorder @xmath28 for ( a ) the first and ( b ) the third allowed bands . the h stack is the same as in fig . all lengths are expressed in units of the grating period @xmath17.,scaledwidth=47.0% ] up to now , h stacks with the same optical path in layers of both types have been considered , that is , arrangements verifying the condition @xmath9 in the absence of disorder . as a consequence , the transmission spectrum @xmath29 of the corresponding periodic system presented a symmetric distribution of allowed bands and gaps ( as previously shown in fig . [ fig2 ] ) . what happens in the case of a non - symmetric band distribution , that is , when the condition @xmath9 is not satisfied ? to answer this question , we have plotted the transmission coefficient @xmath29 ( fig . [ fig6](a ) ) and the parameter @xmath77 ( fig . [ fig6](b ) ) versus the frequency @xmath15 for a periodic h stack formed by 50 layers of length @xmath63 52.92 nm and index of refraction @xmath64 1.58 alternating with 49 layers of length @xmath65 28.80 nm and @xmath66 2.12 . note that the condition @xmath7 is no longer held , so the band structure is asymmetric . accordingly , the localization length @xmath0 shown in fig.[fig6](c ) presents an irregular form in the allowed and forbidden bands . as in the symmetric case , no band modulation exists for high disorders and the quadratic asymptotic behavior for long - wavelengths is also verified . moreover , the peaks in the localization length due to fabry perot resonances still can be appreciated , although they are no longer in the center of the bands @xcite . a total number of 50000 layers was considered in our localization length calculations . and ( b ) the parameter @xmath59 versus the frequency @xmath15 for the asymmetric periodic h stack described in the main text ( 99 layers ) and ( c ) the corresponding localization length @xmath0 versus the wavelength @xmath5 for different disorder parameters @xmath28 ( 50000 layers ) . once analyzed in detail the behavior of the localization length @xmath0 for homogeneous systems , let us now deal with m stacks composed of alternating lh and rh layers . in our numerical calculations we have considered a periodic m stack formed by 50 layers of length @xmath63 52.92 nm and index of refraction @xmath64 -1.58 alternating with 49 layers of length @xmath65 39.38 nm and @xmath66 2.12 . again , the condition @xmath19 has been imposed . note that this arrangement has similar parameters than the one depicted in sec . iii , but now @xmath1 is negative . this change of sign results in a severe modification of the transmission coefficient @xmath29 , as we will show immediately . for this the periodic system , fig . [ fig7 ] represents ( a ) the transmission coefficient @xmath29 and ( b ) the parameter @xmath59 versus the frequency @xmath15 of the incident light . unlike the h stack case , no allowed bands exist and practically the entire transmission spectrum is formed by gaps . a set of periodically distributed lorentzian resonances is found instead . the position of the center of each resonance is given by eq . ( [ omegacenter ] ) , that is , the center of the allowed bands in homogeneous systems . and ( b ) the parameter @xmath59 versus the frequency @xmath15 for the mixed periodic system described in the text ( 99 layers ) . ] in respect to the localization length , positional disorder was introduced as explained in sec . as previously considered , the total number of layers in our numerical calculations was 50000 and the number of disordered configurations to average the logarithm of the transmission coefficient was 800 . the result is shown in fig . [ fig8 ] where the localization length @xmath0 is represented versus the wavelength @xmath5 for different values of the disorder parameter @xmath28 . the dashed line corresponds to the total disorder case . again , for long - wavelengths a quadratic asymptotic behavior of @xmath0 is found , but now a region where the localization length is proportional to @xmath5 exists . we will turn to this point in the next figure to quantify the slope of this linear dependence . as it is noticed , the lorentzian resonances associated with multiple reflections in the layers modulate the shape of @xmath0 and this modulation decreases as the disorder increases . moreover , the saturation of the localization length for low - wavelengths can also be appreciated . as in the h stack case , the constant where @xmath0 saturates is proportional to the inverse of the reflection coefficient between alternating layers @xmath70 . versus the wavelength @xmath5 for different disorder parameters @xmath28 . the m stack corresponds to the one represented in fig . [ fig7 ] but here 50000 layers have been considered . the dashed line stands for the total disorder case . ] the linear dependence of @xmath0 with the wavelength @xmath5 has been exhaustively studied by our group to find a simple analytical expression for the localization length in this region . more than 30 different m stacks have been simulated and we have arrived at the following empirical equation @xmath78 where @xmath0 , @xmath5 and @xmath79 are expressed in units of the grating period @xmath17 . in fig . [ fig9 ] , our numerical calculations of the slope @xmath80 versus @xmath81 have been plotted for several values of @xmath79 , triangles ( 1.25 ) , squares ( 3.25 ) and circles ( 7.55 ) . the solid lines correspond to the results obtained via eq . ( [ longlocm ] ) . one notices a good degree of validity for a wide range of @xmath81 values . versus @xmath81 for several values of @xmath79 ( expressed in units of the grating period @xmath17 ) . the solid lines correspond to the results obtained via eq.([longlocm ] ) . ] finally , let us now consider an asymmetrical m stack where the condition @xmath9 is no longer satisfied . in fig.[fig10 ] we have represented ( a ) the transmission coefficient @xmath29 and ( b ) the parameter @xmath59 versus the frequency @xmath15 for a periodic m stack formed by 50 layers of length of length @xmath63 52.92 nm and index of refraction @xmath64 -1.58 alternating with 49 layers of length @xmath65 28.80 nm and @xmath66 2.12 . note the strong difference between this transmission spectrum and the symmetrical one ( see fig . [ fig7](a ) ) where a set of periodically distributed lorenztian resonances exists . despite this fact , the localization length @xmath0 shown in fig.[fig10](c ) presents a region of linear dependence with the wavelength , as in the symmetric case . however , eq.([longlocm ] ) can not be used to evaluate the localization length in this region . and ( b ) the parameter @xmath59 versus the frequency @xmath15 for the asymmetric periodic m stack described in the main text ( 99 layers ) and ( c ) the corresponding localization length @xmath0 versus the wavelength @xmath5 for different disorder parameters @xmath28 ( 50000 layers ) . we have analyzed numerically the localization length of light @xmath0 for homogeneous and mixed stacks of layers with index of refraction @xmath82 and thickness @xmath2 alternating with layers of index of refraction @xmath83 and thickness @xmath4 . the positions of the layer boundaries have been randomly shifted with respect to ordered periodic values . the refraction indices @xmath1 and @xmath3 present no disorder . for h stacks , the parabolic behavior of the localization length in the limit of long - wavelengths , previously found in purely disordered systems @xcite , has been recovered . on the other hand , the localization length @xmath0 saturates for very low values of @xmath5 . the transmission bands modulate the localization length @xmath0 and this modulation decreases with increasing disorder . moreover , the localization length is practically independent of the disorder @xmath28 at the first gap , that is , it has a very low tendency in this region . we have also characterized @xmath0 in terms of the reflection coefficient of alternating layers @xmath70 and the optical path across one grating period @xmath79 . eq.([longlochinf ] ) has been proved to be valid for a wide range of @xmath70 values , that is , from transparent to opaque h stacks . it has also been shown ( see fig . [ fig5 ] ) that the localization length @xmath0 is enhanced at the center of each allowed band . when left - handed metamaterials are introduced in our system , the localization length behavior presents some differences with respect to the traditional stacks , formed exclusively by right - handed materials . for low - disordered m stacks and wavelengths of several orders of magnitude greater than the grating period @xmath17 , the localization length @xmath0 depends linearly on @xmath5 with a slope inversely proportional to the modulus of the reflection amplitude between alternating layers @xmath81 ( see eq . ( [ longlocm ] ) ) . as in the h case , @xmath0 saturates for low - wavelengths , being this saturation constant proportional to the inverse of @xmath70 . if we take into account losses , there is an absorption term whose absorption length @xmath84 is @xcite @xmath85 where @xmath86 is an absorption coefficient . the inverse of the total decay length is the sum of the inverse of the localization length @xmath0 plus the inverse of the absorption length @xmath87 . note that @xmath84 is proportional to @xmath5 , so , for low - disordered m stacks and weak absorption metamaterials , the final expression for the localization length @xmath0 in the linear region can be written as @xmath88 in the case of both homogeneous and mixed stacks with non - symmetric band distribution , that is , when the condition @xmath89 is not satisfied , the localization length @xmath0 presents an irregular form in all the transmission spectrum . these changes in @xmath0 are more sensitives in mixed stacks than in homogeneous structures . a. yariv and p. yeh , optical waves in crystals , propagation and control of laser radiation ( wiley , new york , 1984 ) ; o. del barco , m. ortuo and v. gasparian , phys . a * 74 * , 032104 ( 2006 ) ; o. del barco and m. ortuo , phys . a * 81 * , 023833 ( 2010 ) .
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we have analyzed numerically the localization length of light @xmath0 for nearly periodic arrangements of homogeneous stacks ( formed exclusively by right - handed materials ) and mixed stacks ( with alternating right and left - handed metamaterials ) .
layers with index of refraction @xmath1 and thickness @xmath2 alternate with layers of index of refraction @xmath3 and thickness @xmath4 .
positional disorder has been considered by shifting randomly the positions of the layer boundaries with respect to periodic values . for homogeneous stacks , we have shown that the localization length is modulated by the corresponding bands and that @xmath0 is enhanced at the center of each allowed band . in the limit of long - wavelengths @xmath5 ,
the parabolic behavior previously found in purely disordered systems is recovered , whereas for @xmath6 a saturation is reached . in the case of nearly periodic mixed stacks with the condition @xmath7 , instead of bands there is a periodic arrangement of lorenztian resonances , which again reflects itself in the behavior of the localization length . for wavelengths of several orders of magnitude greater than @xmath8 ,
the localization length @xmath0 depends linearly on @xmath5 with a slope inversely proportional to the modulus of the reflection amplitude between alternating layers . when the condition @xmath9 is no longer satisfied , the transmission spectrum is very irregular and this considerably affects the localization length .
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non - hermitian operator has been introduced phenomenologically as an effective hamiltonian to fit experimental data in various fields of physics @xcite . in spite of the important role played non - hermitian operator in different branches of physics , it has not been paid due attention by the physics community until the discovery of non - hermitian hamiltonians with parity - time symmetry , which have a real spectrum @xcite . it has boosted the research on the complex extension of quantum mechanics on a fundamental level ann , jmp1,jpa1,jpa2,prl1,jmp2,jmp3,jmp4,jpa3,jpa4,jpa5 . non - hermitian hamiltonian can possess peculiar feature that has no hermitian counterpart . a typical one is the spectral singularity ( or exceptional point for finite system ) , which is a mathematic concept . it has gained a lot of attention recently @xcite , motivated by the possible physical relevance of this concept since the pioneer work ofmostafazadeh @xcite . the majority of previous works focus on the non - hermitian system arising from the complex potential , mean - field nonlinearity pra2,jpa6,ali3,pra13,prd2,prd3,prd4,prd5,prd6,prd7,prd8 as well as imaginary hopping integral @xcite . in this paper , we investigate the physical relevance of the spectral singularities for non - hermitian interacting many - particle system . the non - hermiticity arises from the imaginary interaction strength . for two - particle case , the exact solution shows that there exist a series of spectral singularities , forming a spectrum of singularity associated with the central momentum of the two particles . we consider dynamics of two bosons as well as fermions in one - dimensional system with imaginary delta interaction strength . it shows that the two - particle collision leads to amplitude - reduction of the wave function . for fermion pair , the amplitude - reduction depends on the spin configuration of two particles . remarkably , in both cases , the residual amplitude can vanish only when the relative group velocity of two single - particle gaussian wave packets with equal width reaches the magnitude of the interaction strength . this phenomenon of complete particle - pair annihilation is the direct result of the spectral singularity . we also discuss the complete annihilations of a singlet fermion pair and a maximally two - mode entangled boson pair based on the second quantization formalism . this paper is organized as follows . in section [ hamiltonian and solutions ] , we present the model hamiltonian and exact solution . in section [ dynamical signature ] , we construct the local boson pair initial state as initial state which is allowed to calculate the time evolution . based on this , we reveal the connection between the phenomenon of complete pair annihilation and the spectral singularity . in section [ second quantization representation ] , we extend our study a singlet fermion pair and a maximally two - mode entangled boson pair based on the second quantization formalism . finally , we give a summary in section [ summary ] . we start with an one - dimensional two - distinguishable particle system with imaginary delta interaction . the solution can be used to construct the eigenstates of two - fermion and boson systems . the hamiltonian has the form @xmath0 where @xmath1 and we use dimensionless units @xmath2 for simplicity . introducing new variables @xmath3 and @xmath4 , where @xmath5 we obtain the following hamiltonian @xmath6 with@xmath7here @xmath3 is the center - of - mass coordinate and @xmath4 is the relative coordinate . the hamiltonian is separated into a center - of - mass part and a relative part , and can be solvable exactly . the eigenfunctions of the center - of - mass motion @xmath8 are simply plane waves , while the hamiltonian @xmath9 is equivalent to that of a single - particle in an imaginary delta - potential , which has been exactly solved in the ref.@xcite . then the eigen functions of the original hamiltonian can be obtained and expressed as @xmath10 \right . & & \label{wf_even } \\ \left . -\frac{i\gamma } { k}\sin \left [ k\left ( x_{1}-x_{2}\right ) \right ] \text{\textrm{sign}}\left ( x_{1}-x_{2}\right ) \right\ } , & & \notag\end{aligned}\ ] ] in symmetrical form , and@xmath11 , \label{wf_odd}\]]in antisymmetrical form . the corresponding energy is @xmath12with the central and relative momenta @xmath13 . the symmetrical wavefunction @xmath14 is the spatial part wavefunction for two bosons or two fermions in singlet pair , while the antisymmetrical wavefunction @xmath15 only for two triplet fermions . before starting the investigation on dynamics of two - particle collision , we would like to point that there exist spectral singularities in the present hamiltonian . it arises from the same mechanism as that in the single - particle systems @xcite . we can see that the eigen functions with even parity and momentum @xmath16 can be expressed in the form@xmath17with energy @xmath18we note that function @xmath19 satisfies@xmath20 = 0,\]]which accords with the definition of the spectral singularity in ref . it shows that there exist a series of spectral singularities associated with energy @xmath21 for @xmath22 , which constructs a spectrum of spectral singularities . we will demonstrate in the following section that such a singularity spectrum leads to a peculiar dynamical behavior of two local boson pair or equivalently , singlet fermion pair . the emergence of the spectral singularity induces a mathematical obstruction for the calculation of the time evolution of a given initial state , since it spoils the completeness of the eigen functions and prevents the eigenmode expansion . nevertheless , the completeness of the eigen functions is not necessary for the time evolution of a state with a set of given coefficients of expansion . it does not cause any difficulty in deriving the time evolution of an initial state with arbitrary combination of the eigen functions . namely , any linear combination of function set @xmath23 or @xmath24 can be an initial state , and the time evolution of it can be obtained simply by adding the factor @xmath25 . in order to investigate the dynamical consequence of the singularity spectrum , we consider the time evolution of the initial state of the form @xmath26 where @xmath27 is the normalization factor , which will be given in the following and @xmath28 , \\ g\left ( k\right ) & = & \exp \left [ -\frac{1}{2\beta ^{2}}(k - k_{0})^{2}-ikr_{0}% \right ] .\end{aligned}\ ] ] here @xmath29 and @xmath30 is arbitrary real number . we explicitly have@xmath31\]]where@xmath32furthermore , from the identity @xmath33 \\ & & + 4\beta ^{2}\left [ \left ( x_{1}+b\right ) ^{2}+\left ( x_{2}-b\right ) ^{2}-b^{2}\right ] + \left ( \alpha ^{2}-4\beta ^{2}\right ) x_{1}x_{2 } \notag\end{aligned}\ ] ] we can see that the cross term @xmath34 vanishes if we take @xmath35 . the initial state can be written as a separable form @xmath36 \\ & & + \left ( k_{0}-\gamma \right ) \left . \left [ \varphi _ { + } \left ( x_{2}\right ) \varphi _ { -}\left ( x_{1}\right ) u\left ( x_{2}-x_{1}\right ) + \varphi _ { + } \left ( x_{1}\right ) \varphi _ { -}\left ( x_{2}\right ) u\left ( x_{1}-x_{2}\right ) \right ] \right\ } , \notag\end{aligned}\ ] ] where @xmath37 is heaviside step function and @xmath38 . \label{phi(+/-)}\ ] ] in this case , @xmath27 can be obtained as @xmath39without loss of generality we have set the initial center - of - mass coordinate @xmath40 and dropped an overall phase @xmath41 . we note that functions @xmath42 and @xmath43 represent gaussian functions with centers at @xmath44 and @xmath45 , respectively . obviously , the probability contributions of @xmath46 and @xmath47 are negligible under the condition @xmath48 . we then yield @xmath49 , \label{initial state_1}\ ] ] which represents two - boson wavepacket state with the same width , group velocity @xmath50 , and location @xmath51 . here the renormalization factor has been readily calculated by gaussian integral . so far we have construct an expected initial state without using the biothogonal basis set . the dynamics of two separated boson wavepackets can be described by the time evolution as that in the conventional quantum mechanics . it is presumable that before the bosons start to overlap they move as free particles with the center moving in their the group velocities @xmath52 and the width spreading as function of time @xmath53 . we concern the dynamic behavior after the collision . to this end , we calculate the time evolution of the given initial state , which can be expressed as @xmath54 by the similar procedure as above , we find that the evolved wave function can always be written in the separated form@xmath55where@xmath56{\frac{4\beta ^{2}}{\pi \left ( 1 + 4\beta ^{2}t^{2}\right ) } } \exp \left [ -\frac{2\beta ^{2}\left ( r - k_{0}/2\right ) ^{2}% } { 1 + 4\beta ^{2}t^{2}}+\frac{i\left ( 16\beta ^{4}r^{2}+4rk_{0}-tk_{0}^{2}\right ) } { 4 + 16\beta ^{2}t^{2}}\right ] , \]]and@xmath57 \left [ \cos \left ( kr\right ) -\frac{i\gamma } { k}\sin \left ( k\left\vert r\right\vert \right ) \right ] \exp \left ( -ik^{2}t\right ) \text{\textrm{d}}k.\]]where the normalization factor @xmath58 straightforward algebra shows that@xmath59where @xmath60 ^{2}}{2\left ( 4\beta ^{4}t^{2}+1\right ) } + i\delta _ { \pm } \right\ } , \\ \delta _ { \pm } & = & \frac{\beta ^{4}\left ( \left\vert r\right\vert \pm r_{0}\right ) ^{2}t-2k_{0}^{2}t\mp 2k_{0}\left ( \left\vert r\right\vert \pm r_{0}\right ) } { 2\left ( 4\beta ^{4}t^{2}+1\right ) } .\end{aligned}\ ] ] in the case of @xmath61 , @xmath62 the probability distribution is @xmath63 which leads the total probability under the case @xmath64 @xmath65 we can see that , after the collision the residual probability becomes a constant and vanishes when @xmath66 . it shows that when the relative group velocity of two single - particle gaussian wave packets with equal width reaches the magnitude of the interaction strength , the dynamics exhibits complete particle - pair annihilation . in order to demonstrate such dynamic behavior and verify our approximate result , the numerical method is employed to simulate the time evolution process for several typical situations . the profiles of @xmath67 are plotted in fig . 1 . we would like to point that the complete annihilation depends on the relative group velocity , which is the consequence of singularity spectrum . this enhances the probability of the pair annihilation for a cloud of bosons , which mayprovide an detection method of the spectral singularity in experiment . in this section , we will investigate the two - particle collision process from another point of view and give a more extended example . by employing the second quantization representation , the initial state in eq . ( [ initial state_1 ] ) can be expressed as the form @xmath68 , where @xmath69 @xmath70 is the creation operator for a boson in single - particle state with the wavefunction @xmath71 and @xmath72 denotes the vacuum state of the particle operator . similarly , if we consider a fermion pair , the initial state in eq . ( [ initial state_1 ] ) can be written as@xmath73where @xmath74 @xmath75 is the creation operator for a fermion in single - particle state with the wavefunction@xmath76here@xmath77are the spin part of wavefunction . we see that the initial state in eq . ( fermion pair ) is singlet pair with maximal entanglement . in contract , state @xmath78 should not lose any amplitude after collision . on the other hand , we can extend our conclusion to other types of initial state . for instance , we can construct the initial state with@xmath79 , \\ g\left ( k\right ) & = & \left ( k - k_{0}\right ) \exp \left [ -\frac{1}{2\beta ^{2}}% \left ( k - k_{0}\right ) ^{2}-ikr_{0}\right ] , \notag\end{aligned}\]]which are also local states in @xmath80 and @xmath81 spaces , respectively . in coordinate space , the above wavefunction has the from@xmath82 \notag \\ & & -\left ( k_{0}-\gamma \right ) \left [ \left ( \varphi _ { -}^{\left ( 1\right ) } \left ( x_{1}\right ) \varphi _ { + } \left ( x_{2}\right ) -\varphi _ { + } ^{\left ( 1\right ) } \left ( x_{2}\right ) \varphi _ { -}\left ( x_{1}\right ) \right ) u\left ( x_{2}-x_{1}\right ) \right . \notag \\ & & + \left . \left . \left ( x_{1}\rightleftarrows x_{2}\right ) \right ] \right\ } , \end{aligned}\]]which can be reduced to@xmath83 , \label{initial state_2}\]]under the approximation @xmath48 . here @xmath84 is the normalized constant , and @xmath85 . by the same procedure , at time @xmath86 the evolved wavefunction is@xmath87with@xmath88 } { % k_{0}\left ( 1 + 2i\beta ^{2}t\right ) ^{3/2}}\exp \left\ { -\frac{\beta ^{2}% \left [ \left\vert r\right\vert \pm \left ( r_{0}-2k_{0}t\right ) \right ] ^{2}}{% 2\left ( 4\beta ^{4}t^{2}+1\right ) } + i\delta _ { \pm } \right\ } , \\ \delta _ { \pm } & = & \frac{\beta ^{4}\left ( \left\vert r\right\vert \pm r_{0}\right ) ^{2}t-2k_{0}^{2}t\mp 2k_{0}\left ( \left\vert r\right\vert \pm r_{0}\right ) } { 2\left ( 4\beta ^{4}t^{2}+1\right ) } .\end{aligned}\]]where the normalization factor@xmath89 in the case of @xmath61 , @xmath62 the probability distribution is @xmath90 which leads the total probability@xmath65the profiles of @xmath91 are plotted in fig . 2 . we can see that the same behavior occurs in the present situation . in order to clarify the physical picture , we still employ the second quantization representation by introducing another type of boson creation operator @xmath92 @xmath70 with@xmath93then the initial state in eq . ( [ initial state_2 ] ) can be expressed as@xmath94which is maximally two - mode entangled state . in summary we identified a connection between spectral singularities and dynamical behavior for interacting many - particle system . we explored the collision process of two bosons as well as fermions in one - dimensional system with imaginary delta interaction strength based on the exact solution . we have showed that there is a singularity spectrum which leads to complete particle - pair annihilation when the relative group velocity is resonant to the magnitude of interaction strength . the result for this simple model implies that the complete particle - pair annihilation can only occur for two distinguishable bosons , maximally two - mode entangled boson pair and singlet fermions , which may predict the existence of its counterpart in the theory of particle physics .
|
motivated by the physical relevance of a spectral singularity of interacting many - particle system , we explore the dynamics of two bosons as well as fermions in one - dimensional system with imaginary delta interaction strength .
based on the exact solution , it shows that the two - particle collision leads to amplitude - reduction of the wave function . for fermion pair ,
the amplitude - reduction depends on the spin configuration of two particles . in both cases ,
the residual amplitude can vanish when the relative group velocity of two single - particle gaussian wave packets with equal width reaches the magnitude of the interaction strength , exhibiting complete particle - pair annihilation at the spectral singularity .
| 4,865 | 171 |
stellar feedback the injection of energy and momentum by stars originates at the small scales of star clusters ( @xmath01 pc ) , yet it shapes the interstellar medium ( ism ) on large scales ( @xmath11 kpc ) . at large scales , stellar feedback is necessary in order to form realistic galaxies in simulations and to account for observed galaxy properties . in the absence of feedback , baryonic matter cools rapidly and efficiently forms stars , producing an order of magnitude too much stellar mass and consuming most available gas in the galaxy ( e.g. , @xcite ) . stellar feedback prevents this `` cooling catastrophe '' by heating gas as well as removing low angular momentum baryons from galactic centers , thereby allowing only a small fraction of the baryonic budget of dark matter halos to be converted to stars . the removal of baryons may also flatten the dark matter mass profile , critical to form bulgeless dwarf galaxies ( e.g. , @xcite ) . furthermore , stellar feedback possibly drives kpc - scale galactic winds and outflows ( see @xcite for a review ) which have been frequently observed in local galaxies ( e.g. , @xcite ) as well as in galaxies at moderate to high redshift ( e.g. , @xcite ) . at the smaller scales of star clusters and giant molecular clouds ( gmcs ) , newborn stars dramatically influence their environments . observational evidence suggests that only a small fraction ( @xmath212% ) of gmc mass is converted to stars per cloud free - fall time ( e.g. , @xcite ) . this inefficiency can be attributed to stellar feedback processes of h ii regions that act to disrupt and ultimately to destroy their host clouds ( e.g. , @xcite ) . in addition to the pressure of the warm ionized h ii region gas itself , there are several other forms of stellar feedback that can drive the dynamics of h ii regions and deposit energy and momentum in the surrounding ism : the direct radiation of stars ( e.g. , @xcite ) , the dust - processed infrared radiation ( e.g. , @xcite ) , stellar winds and supernovae ( sne ; e.g. , @xcite ) , and protostellar outflows / jets ( e.g. , @xcite ) . from a theoretical perspective , sne were the first feedback mechanism to be considered as a means to remove gas from low - mass galaxies ( e.g. , @xcite ) and to prevent the cooling catastrophe ( e.g. , @xcite ) . however , resolution limitations precluded the explicit modeling of individual sne in galaxy formation simulations , so phenomenological prescriptions were employed to account for `` sub - grid '' feedback ( e.g. , @xcite ) . since then , extensive work has been done to improve and to compare these sub - grid models ( e.g. , @xcite ) . furthermore , the use of `` zoom - in '' simulations ( which can model feedback physics down to @xmath11 pc scale ) has enabled the modeling of several modes of feedback simultaneously ( e.g. , @xcite ) . while simulations are beginning to incorporate many feedback mechanisms , most observational work focuses on the effects of the individual modes . consequently , the relative contribution of these components and which processes dominate in different conditions remains uncertain . to address this issue , we recently employed multiwavelength imaging of the giant h ii region n157 ( 30 doradus ; `` 30 dor '' hereafter ) to assess the dynamical role of several stellar feedback mechanisms in driving the shell expansion @xcite . in particular , we measured the pressures associated with the different feedback modes across 441 regions to map the pressure components as a function of position ; we considered the direct radiation pressure exerted by the light from massive stars , the dust - processed radiation pressure , the warm ionized ( @xmath3 k ) gas pressure , and the hot shocked ( @xmath4 k ) gas pressure from stellar winds and sne . we found that the direct radiation pressure from massive stars dominates at distances @xmath075 pc from the central star cluster r136 , while the warm ( @xmath5 k ) ionized gas pressure dominates at larger radii . by comparison , the dust - processed radiation pressure and the hot ( @xmath4 k ) gas pressure are weak and are not dynamically important on the large scale ( although small bubbles of the hot gas can have significant pressures @xcite ; see appendix [ app : hot gas ] of this paper for a discussion on how choice of hot gas filling factor is critical when evaluating the dynamical role of hot gas ) . in this paper , we extend the methodology applied to 30 dor to a larger sample of 32 h ii regions in the large and small magellanic clouds ( lmc and smc , respectively ) , with the aim of probing how stellar feedback properties vary between sources . the organization of this paper is as follows . section [ sec : sample ] describes our lmc and smc h ii region sample and the data we have employed for our analyses . section [ sec : method ] outlines the methods we have used to assess the dynamical role of several stellar feedback mechanisms in the 32 sources . section [ sec : results ] presents the results from these analyses , and section [ sec : discussion ] explores implications of our findings related to the importance of radiation pressure ( section [ sec : radpressure ] ) , the confinement of hot gas in the h ii regions ( section [ sec : leakage ] ) and the momentum deposition of the dust - processed radiation to the warm gas ( section [ sec : dusty ] ) . finally , we summarize this work in section [ sec : summary ] . for our feedback analyses , we selected the 16 lmc and 16 smc h ii regions of @xcite , who chose sources based on their bright 24@xmath6 m and h@xmath7 emission and which are distributed throughout these galaxies . we opted to include sources based on both ir and h@xmath7 , since bright h@xmath7 emission alone is not unique to h ii regions . for example , several of the emission nebulae identified by @xcite are now known to be supernova remnants . furthermore , bright 24@xmath6 m emission arises from stochastically heated small dust grains ( i.e. , dust is heated by collisions with starlight photons : e.g. , @xcite ) , so it is well - correlated with h ii regions within the milky way and other galaxies . lccccc n4 & dem l008 & 04:52:09 & @xmath866:55:13 & 0.7 & 10.2 + n11 & dem l034 , l041 & 04:56:41 & @xmath866:27:19 & 10.0 & 145 + n30 & dem l105 , l106 & 05:13:51 & @xmath867:27:22 & 3.1 & 45.1 + n44 & dem l150 & 05:22:16 & @xmath867:57:09 & 7.1 & 103 + n48 & dem l189 & 05:25:50 & @xmath866:15:03 & 5.2 & 75.6 + n55 & dem l227 , l228 & 05:32:33 & @xmath866:27:20 & 3.6 & 52.4 + n59 & dem l241 & 05:35:24 & @xmath867:33:22 & 3.9 & 56.7 + n79 & dem l010 & 04:52:04 & @xmath869:22:34 & 4.4 & 64.0 + n105 & dem l086 & 05:09:56 & @xmath868:54:03 & 2.9 & 42.2 + n119 & dem l132 & 05:18:45 & @xmath869:14:03 & 5.9 & 85.8 + n144 & dem l199 & 05:26:38 & @xmath868:49:55 & 4.9 & 71.3 + n157 & 30 dor & 05:38:36 & @xmath869:05:33 & 6.8 & 98.9 + n160 & & 05:40:22 & @xmath869:37:35 & 5.0 & 40.0 + n180 & dem l322 , l323 & 05:48:52 & @xmath870:03:51 & 2.7 & 39.3 + n191 & dem l064 & 05:04:35 & @xmath870:54:27 & 2.1 & 30.5 + n206 & dem l221 & 05:30:38 & @xmath871:03:53 & 7.7 & 112 + dem s74 & & 00:53:14 & @xmath873:12:18 & 2.7 & 47.9 + n13 & & 00:45:23 & @xmath873:22:38 & 0.5 & 8.87 + n17 & & 00:46:41 & @xmath873:31:38 & 1.5 & 26.6 + n19 & & 00:48:23 & @xmath873:05:54 & 0.7 & 12.4 + n22 & & 00:48:09 & @xmath873:14:56 & 0.9 & 16.0 + n36 & & 00:50:26 & @xmath872:52:59 & 2.5 & 44.4 + n50 & & 00:53:26 & @xmath872:42:56 & 4.3 & 76.3 + n51 & & 00:52:40 & @xmath873:26:29 & 1.9 & 33.7 + n63 & & 00:58:17 & @xmath872:38:57 & 1.3 & 23.1 + n66 & & 00:59:06 & @xmath872:10:44 & 3.6 & 63.9 + n71 & & 01:00:59 & @xmath871:35:30 & 0.2 & 3.55 + n76 & & 01:03:32 & @xmath872:03:16 & 3.1 & 55.0 + n78 & & 01:05:18 & @xmath871:59:53 & 2.6 & 46.1 + n80 & & 01:08:13 & @xmath872:00:06 & 2.2 & 39.0 + n84 & & 01:14:56 & @xmath873:17:51 & 5.7 & 101 + n90 & & 01:29:27 & @xmath873:33:10 & 1.7 & 30.2 + [ tab : sample ] our final sample of h ii regions are listed in table [ tab : sample ] , and figures [ fig : lmcthreecolor ] and [ fig : smcthreecolor ] shows the three - color images of the lmc and smc h ii regions , respectively . we note that although our sample spans a range of parameter space ( e.g. , two orders of magnitude in radius and in ionizing photon fluxes @xmath9 ) , the h ii regions we have selected represent the brightest in the magellanic clouds in h@xmath7 and at 24 @xmath6 m . we utilize published ubv photometry of 624 lmc star clusters @xcite to assess upper limits on the cluster ages and lower limits on star cluster masses powering our sample . within the radii of the lmc h ii regions , we found 18 star clusters from the bica sample . to estimate the cluster ages , we compare the extinction - corrected ubv colors of the enclosed star clusters to the colors output from starburst99 simulations @xcite of a star cluster of @xmath10 which underwent an instantaneous burst of star formation . for this analysis , we adopt a color excess @xmath11 , the foreground reddening in the direction of the lmc @xcite . this value is almost certainly an underestimate and represents the minimum reddening toward our clusters ( for example , the reddening in r136 is @xmath12 ) and neglects local extinction . based on the clusters ubv colors , we find upper limit ages of @xmath2315 myr ; greater extinction toward the clusters would yield younger ages . additionally , we estimate the lower limit of the star cluster masses by normalizing @xmath10 by the ratio of the v - band luminosities of our clusters with those of the simulated clusters at their respective ages . we find cluster masses of @xmath2300@xmath13 . as relatively bright and evolved sources , the dynamical properties of our sample may differ from more dim h ii regions ( those powered by smaller star clusters ) and h ii regions which are much younger or older . for our analyses , we employed data at several wavelengths ; a brief description of these data is given below . throughout this paper , we assume a distance @xmath14 of 50 kpc to the lmc @xcite and of 61 kpc to the smc @xcite . to illustrate the h ii regions structure , we show the h@xmath7 emission of the 32 sources in figures [ fig : lmcthreecolor ] and [ fig : smcthreecolor ] . we used the narrow - band image ( at 6563@xmath15 , with 30@xmath15 full - width half max ) that was taken with the university of michigan / ctio 61-cm curtis schmidt telescope at ctio as part of the magellanic cloud emission line survey ( mcels : @xcite ) . the total integration time was 600 s , and the reduced image has a resolution of 2pixel@xmath16 . to estimate the h@xmath7 luminosity of our smc sources within the radii given in table [ tab : sample ] , we used the flux - calibrated , continuum - subtracted mcels data . as the flux calibrated mcels data of the lmc is not yet available , we employed the southern h@xmath7 sky survey atlas ( shassa ) , a robotic wide - angle survey of declinations @xmath17 to @xmath18 @xcite , to measure h@xmath7 luminosities of our lmc h ii regions . shassa data were taken using a ccd with a 52-mm focal length canon lens at f/1.6 . this setup enabled a large field of view ( @xmath19 ) and a spatial resolution of 47.64 pixel@xmath16 . the total integration time for the lmc exposure was @xmath2021 minutes . infrared images of the lmc were obtained through the _ spitzer _ space telescope legacy program surveying the agents of galaxy evolution ( sage : @xcite ) . the survey covered an area of @xmath27 @xmath21 7 degrees of the lmc with the infrared array camera ( irac ; @xcite ) and the multiband imaging photometer ( mips ; @xcite ) . images were taken in all bands of irac ( 3.6 , 4.5 , 5.8 , and 7.9 @xmath6 m ) and of mips ( 24 , 70 , and 160 @xmath6 m ) at two epochs in 2005 . for our analyses , we used the combined mosaics of both epochs with 1.2pixel@xmath16 in the 3.6 and 7.9 @xmath6 m irac images and 2.49pixel@xmath16 and 4.8pixel@xmath16 in the mips 24 @xmath6 m and 70 @xmath6 m , respectively . the smc was also surveyed by _ spitzer _ with the legacy program surveying the agents of galaxy evolution in the tidally stripped , low metallicity small magellanic cloud ( sage - smc : @xcite ) . this project mapped the full smc ( @xmath230 deg@xmath22 ) with irac and mips and built on the pathfinder program , the spitzer survey of the small magellanic cloud ( s@xmath23mc : @xcite ) , which surveyed the inner @xmath23 deg@xmath24 of the smc . sage - smc observations were taken at two epochs in 20072008 , and we employed the combined mosaics from both epochs ( plus the s@xmath23mc data ) . the lmc and smc were observed with the australian telescope compact array ( atca ) as part of 4.8-ghz and 8.64-ghz surveys @xcite . these programs had identical observational setups , using two array configurations that provided 19 antenna spacings , and the atca observations were combined with the parkes 64-m telescope data of @xcite to account for extended structure missed by the interferometric observations . for our analyses , we utilized the resulting atca@xmath25parkes 8.64 ghz ( 3.5-cm ) images of the lmc and smc , which had gaussian beams of fwhm 22 and an average rms noise level of 0.5 mjy beam@xmath16 . given the large extent of the lmc , _ chandra _ and _ xmm - newton _ have not observed the majority of that galaxy . thus , for our x - ray analyses of the 16 lmc h ii regions , we use archival data from _ rosat _ , the rntgen satellite . the lmc was observed via pointed observations and the all - sky survey of the rosat position sensitive proportional counter ( pspc ) over its lifetime ( e.g. , @xcite ) . the rosat pspc had modest spectral resolution ( with @xmath26 ) and spatial resolution ( @xmath225 ) over the energy range of 0.12.4 kev , with @xmath27 field of view . table [ tab : xrayobslog ] lists the archival pspc observations we utilized in our analyses of our sample . all the lmc h ii regions except for n191 were observed in pointed observations from 19911993 with exposures ranging from @xmath2400045000 s. some of these observations were presented and discussed originally in @xcite . lccc n4 & july 1993 & rp500263n00 & 12.7 + n11 & november 1992 & rp900320n00 & 17.6 + n30 & february 1992 & rp500052a01 & 8.0 + n44 & march 1992 & rp500093n00 & 8.7 + n48 & october 1991 & rp200692n00 & 44.7 + n55 & october 1991 & rp200692n00 & 44.7 + n59 & december 1993 & rp900533n00 & 1.6 + n79 & october 1993 & rp500258n00 & 12.7 + n105 & april 1992 & rp500037n00 & 6.8 + n119 & june 1993 & rp500138a02 & 14.6 + n144 & june 1993 & rp500138a02 & 14.6 + 30 dor & april 1992 & rp500131n00 & 16.0 + n160 & april 1992 & rp500131n00 & 16.0 + n180 & october 1993 & rp500259n00 & 4.0 + n191 & & & + n206 & december 1993 & rp300335n00 & 11.3 + dem s74 & november 2009 & 0601211401 & 46.8 + n13 & october 2009 & 0601211301 & 32.7 + n17 & october 2009 & 0601211301 & 32.7 + n19 & march 2007 & 0403970301 & 39.1 + n22 & october 2000 & 0110000101 & 28.0 + n36 & march 2010 & 0656780201 & 12.8 + n50 & december 2003 & 0157960201 & 14.8 + n51 & april 2007 & 0404680301 & 20.4 + n63 & october 2009 & 0601211601 & 32.3 + n66 & may 2001 & 1881 & 99.9 + n71 & june 2007 & 0501470101 & 16.1 + n76 & march 2000jan 2009 & 52 observations & 471.0 + n78 & dec 2000feb 2009 & 36 observations & 297.6 + n80 & november 2009 & 0601211901 & 31.6 + n84 & march 2006 & 0311590601 & 11.3 + n90 & april 2010 & 0602520301 & 96.3 + [ tab : xrayobslog ] the smc was surveyed by _ xmm - newton _ between may 2009 and march 2010 @xcite . we exploit data from this campaign as well as from pointed _ xmm - newton _ observations for 13 of the 16 smc h ii regions . for the other three smc sources ( n66 , n76 , and n78 ) , we use deep _ chandra _ acis - i observations . n66 was targeted in a 99.9 ks acis - i observation @xcite . n76 and n78 are in the field of a _ chandra _ calibration source , the supernova remnant 1e 0102@xmath87219 , so they have been observed repeatedly since the launch of _ chandra _ in 1999 . we searched these calibration data and merged all the observations where the _ chandra _ chip array imaged the full diameter of the sources : 52 observations for n76 , and 36 observations for n78 . we follow the same methodology as in our 30 dor pressure analysis @xcite with only a few exceptions , described below . instead of calculating spatially - resolved pressure components for the sources , we determine the average pressures integrated over the radii listed in table [ tab : sample ] . thus , these pressure components are those `` felt '' within the h ii shells . we describe the uncertainties associated with the calculation of each term in section [ sec : uncertainty ] . to select the radius of each region , we produced surface brightness profiles of their h-@xmath7 emission , and we determined the apertures which contained 90% of the total h-@xmath7 fluxes . we opted for this phenomenological definition of the radii to reduce the systematic uncertainties between sources . as seen in figures [ fig : lmcthreecolor ] and [ fig : smcthreecolor ] , the h ii regions are quite complex , and the calculations below are simple and aimed to describe the general properties of these sources . the light output by stars produces a direct radiation pressure that is associated with the photons energy and momentum . the resulting radiation pressure @xmath28 at some position within the h ii region is related to the bolometric luminosity of each star @xmath29 and the distance @xmath30 the light traveled to reach that point : @xmath31 where the summation is over all the stars in the region . the volume - averaged direct radiation pressure @xmath32 is then @xmath33 where @xmath34 is the total volume within the h ii region shell and @xmath35 is the h ii region radius . the above equation is the formal definition of radiation pressure ( i.e. , it is the trace of the radiation pressure tensor ) . we note that radiation pressure and radiation force do not always follow the same simple relationship as e.g. , gas pressure and force , where the force is the negative gradient of pressure . in particular , @xcite point out that in a relatively transparent medium ( such as the interior of an h ii region ) , it is possible for the radiation pressure to exceed the gas pressure while the local force exerted on matter by the radiation is smaller than the force exerted by gas pressure . however , at the h ii shells where the gas is optically thick to stellar radiation , radiation force and pressure follow the same relationship as gas force and pressure , and the radiation pressure defined by equation [ eq : pdir ] is the relevant quantity to consider . to obtain @xmath29 of the stars in our 30 dor analyses , we employed ubv photometry of r136 from @xcite using hst planetary camera observations , and the ground - based data of @xcite and @xcite to account for stars outside r136 . while several large - scale optical surveys of the lmc have now been done and include ubv photometry ( e.g. , @xcite ) , these data do not resolve the crowded regions of young star clusters , and they focus principally on the ( uncrowded ) field population . an alternative means to estimate the bolometric luminosities of the star clusters is using the extinction - corrected h@xmath7 luminosities of the h ii regions . from @xcite , for a stellar population that fully samples the initial mass function ( imf ) and the stellar age distribution , the bolometric luminosity @xmath36 is related to the extinction - corrected h@xmath7 luminosity @xmath37 by the expression @xmath38 . we use the shassa and mcels data to estimate the observed h@xmath7 luminosities @xmath39 within the radii listed in table [ tab : sample ] . to correct for extinction , we employ the reddening maps of the lmc and smc presented in @xcite , from the third phase of the optical gravitational lensing experiment ( ogle iii ) . these authors used observations of red clump and rr lyrae stars to derive spatially - resolved extinction estimates ( with typical subfield sizes of 4.5@xmath214.5 ) across the lmc and smc , and these data are publicly available through the german astrophysical virtual observatory ( gavo ) interface . using gavo , we obtained the mean extinction in the b- and v - bands , @xmath40 and @xmath41 , respectively . in the cases when the h ii region radii included multiple subfields of the ogle extinction measurements , we calculated the average @xmath40 and @xmath41 over that aperture . then , we use the color excess @xmath42 to compute @xmath43 , the extinction at the wavelength @xmath44 of the h@xmath7 line , given @xmath45 where @xmath46 ) is the value of the extinction curve at the wavelength of the h@xmath7 line . @xcite derive the best - fit expression for @xmath47 ) at optical wavelengths as @xmath48 where @xmath49 . we adopt the standard @xmath50 , which @xcite demonstrate to be valid in the optical for the lmc and smc , and we find @xmath51 2.362 . finally , the extinction - corrected h@xmath7 luminosity @xmath37 is @xmath52 the parameters associated with these calculations , including the intrinsic h@xmath7 luminosities and corresponding @xmath36 of the 32 h ii regions , are listed in table [ tab : extinction ] . the extinction - corrected h@xmath7 luminosities are typically 1020% greater than the observed h@xmath7 luminosities . we note that local reddening and extinction may be greater than the average values obtained in the ogle iii maps , and thus the bolometric luminosities of the star clusters may be greater . however , even if the local extinction is a factor of ten larger , the direct radiation pressure will still be dynamically insignificant , as seen in the results given in section [ sec : results ] . lccccccc n4 & 0.31 & 0.24 & 0.17 & 37.1 & 37.2 & 39.4 & 49.2 + n11 & 0.08 & 0.06 & 0.05 & 38.9 & 39.0 & 41.1 & 51.0 + n30 & 0.26 & 0.20 & 0.14 & 37.7 & 37.8 & 39.9 & 49.7 + n44 & 0.28 & 0.22 & 0.14 & 38.5 & 38.6 & 40.7 & 50.6 + n48 & 0.19 & 0.14 & 0.12 & 37.8 & 37.9 & 40.0 & 49.9 + n55 & 0.30 & 0.23 & 0.17 & 38.0 & 38.0 & 40.2 & 50.0 + n59 & 0.36 & 0.28 & 0.19 & 38.4 & 38.5 & 40.6 & 50.5 + n79 & 0.40 & 0.30 & 0.24 & 38.1 & 38.2 & 40.4 & 50.2 + n105 & 0.20 & 0.15 & 0.12 & 38.1 & 38.2 & 40.3 & 50.1 + n119 & 0.20 & 0.15 & 0.12 & 38.5 & 38.5 & 40.7 & 50.5 + n144 & 0.35 & 0.27 & 0.19 & 38.4 & 38.4 & 40.6 & 50.4 + n157 & 0.76 & 0.59 & 0.40 & 39.5 & 39.7 & 41.8 & 51.7 + n160 & 0.57 & 0.44 & 0.31 & 38.9 & 39.0 & 41.1 & 51.0 + n180 & 0.36 & 0.28 & 0.19 & 38.0 & 38.1 & 40.2 & 50.1 + n191 & 0.18 & 0.13 & 0.12 & 37.0 & 37.0 & 39.2 & 49.0 + n206 & 0.30 & 0.23 & 0.17 & 38.5 & 38.5 & 40.7 & 50.5 + dem s74 & 0.16 & 0.12 & 0.09 & 37.1 & 37.1 & 39.3 & 49.1 + n13 & 0.25 & 0.19 & 0.14 & 37.0 & 37.1 & 39.2 & 49.0 + n17 & 0.21 & 0.16 & 0.12 & 37.1 & 37.2 & 39.3 & 49.1 + n19 & 0.25 & 0.19 & 0.14 & 36.7 & 36.8 & 38.9 & 48.8 + n22 & 0.27 & 0.21 & 0.14 & 37.0 & 37.1 & 39.2 & 49.1 + n36 & 0.24 & 0.18 & 0.14 & 37.8 & 37.9 & 40.0 & 49.9 + n50 & 0.19 & 0.14 & 0.12 & 37.8 & 37.8 & 39.9 & 49.8 + n51 & 0.15 & 0.12 & 0.08 & 36.8 & 36.8 & 39.0 & 48.8 + n63 & 0.22 & 0.17 & 0.12 & 37.0 & 37.0 & 39.1 & 49.0 + n66 & 0.08 & 0.06 & 0.05 & 38.6 & 38.6 & 40.8 & 50.6 + n71 & 0.11 & 0.09 & 0.05 & 36.2 & 36.3 & 38.4 & 48.2 + n76 & 0.09 & 0.07 & 0.05 & 38.0 & 38.0 & 40.2 & 50.0 + n78 & 0.13 & 0.10 & 0.07 & 37.7 & 37.7 & 39.9 & 49.7 + n80 & 0.16 & 0.12 & 0.09 & 37.4 & 37.5 & 39.6 & 49.4 + n84 & 0.32 & 0.24 & 0.19 & 38.2 & 38.2 & 40.4 & 50.2 + n90 & 0.19 & 0.14 & 0.12 & 37.5 & 37.5 & 39.7 & 49.5 + [ tab : extinction ] one issue related to the above estimates of @xmath36 is the star formation history . while both the h@xmath7 and bolometric luminosity of an actively star - forming region are dominated by massive stars with lifetimes @xmath53 myr , the bolometric luminosity also contains a non - negligible contribution from longer - lived stars . the implication is that the ratio of h@xmath7 to bolometric luminosity of a stellar population evolves with time . the relation @xmath38 is appropriate for a population with a continuous star formation history over 100 myr , but for a nearly coeval stellar population as in our star clusters , the h@xmath7 to bolometric ratio will start out somewhat larger than kennicutt & evans value , then decline below it over a timescale of @xmath54 myr . thus , depending on the age of the stellar population , @xmath36 can be either an underestimate or an overestimate . given that our stellar sources are bright h ii regions and thus the stars are likely to be young , the latter seems more likely . we also note uncertainty related to imf sampling . stellar populations with masses below @xmath55 @xmath56 do not fully sample the imf , and this makes the h@xmath7 to bolometric luminosity ratio vary stochastically @xcite . most of our clusters are near the edge of the stochastic regime . for a randomly selected cluster , the most common effect is to lower the h@xmath7 luminosity relative to the bolometric luminosity ; the expected magnitude of the effect is a factor of @xmath57 ( e.g. , figure 7 of @xcite ) . this will tend to make our @xmath36 an underestimate by this amount . however , the real effect is likely to be smaller , because our sample is not randomly selected . for a rare subset of clusters stochasticity has no effect or actually raises the h@xmath7 to bolometric ratio compared to that of a fully - sampled imf , and since our sample is partly selected based on h@xmath7 luminosity , it is biased in favor of the inclusion of such clusters . it is not possible to model this effect quantitatively without knowing both the underlying distribution of cluster masses and the selection function used to construct the sample . thus we restrict ourselves to noting that this effect probably introduces a factor of @xmath58 level uncertainty into @xmath36 . in the remainder of this paper , we will use @xmath59 to calculate @xmath32 . the pressure of the dust - processed radiation field @xmath60 is related to the energy density of the radiation field absorbed by the dust , @xmath61 ( i.e. , assuming a steady state ) , @xmath62 we follow the same procedure of @xcite to estimate the energy density @xmath61 of the radiation absorbed by the dust in our sample . specifically , we measure the flux densities @xmath63 in the irac and mips bands and compare them to the predictions of the dust models of @xcite ( hereafter dl07 ) . the dl07 models determine the ir spectral energy distribution for a given dust content and radiation field intensity heating the dust . dl07 assume a mixture of carbonaceous grains and amorphous silicate grains that have a size distribution that reproduces the wavelength - dependent extinction in the local milky way ( see @xcite ) . in particular , polycyclic aromatic hydrocarbons ( pahs ) contribute substantial flux at @xmath2319 @xmath6 m and are observed in normal and star - forming galaxies ( e.g. , @xcite ) . to account for the different spatial resolutions of the ir images , we convolved the 3.6 , 8 , and 24 @xmath6 m images with kernels to match the point - spread function of the 70 @xmath6 m image using the convolution kernels of @xcite . then , we measured the average flux densities @xmath63 at 8 , 24 , and 70 @xmath6 m wavelengths in the apertures listed in column 5 of table [ tab : sample ] . we removed the contribution of starlight to the 8 and 24 @xmath6 m fluxes using the 3.6 @xmath6 m flux densities and the empirical relations @xmath64 where @xmath65 is the non - stellar flux at the respective wavelengths . the coefficients 0.232 and 0.032 are given in @xcite . in figure [ fig : lmc_models ] , we plot the resulting ratios @xmath66 versus @xmath67 measured for the 32 h ii regions . additionally , we plot the @xcite predictions for given values of @xmath68 , the fraction of dust mass in pahs , and @xmath69 , the dimensionless scale factor of energy density @xmath61 of radiation absorbed by the dust , where @xmath70 here , @xmath71 is the energy density of the @xmath72 ev photons in the local ism , 8.65 @xmath21 10@xmath73 erg @xmath74 @xcite . the 32 h ii regions span a factor of @xmath220 in @xmath67 , with the smc h ii regions having systematically lower @xmath67 than the lmc h ii regions . the lmc and smc sources have a similar range of a factor of @xmath26 in @xmath66 . broadly , the data follow a similar arc - like trend in the ratios as we found in the spatially - resolved regions of 30 dor @xcite . errors in our flux ratios are @xmath202.8% from a @xmath202% uncertainty in the _ spitzer _ photometry . and scaling @xmath69 of the energy density of the dust - processed radiation field ( equation [ eq : u ] ) from @xcite . the black star denotes the values for 30 dor . we interpolate the grid of predicted flux ratios to obtain @xmath75 and @xmath69 for each region listed in table [ table : pirresults ] . ] we interpolate the @xmath69-@xmath75 grid using delaunay triangulation , a technique appropriate for a non - uniform grid , to find the @xmath69 and @xmath68 values for our regions . for the points that lay outside the grid , we translated them to @xmath67 within the grid . since the y - axis ratio largely determines @xmath69 , this adjustment does not affect the pressure calculation for those sources . figure [ fig : lmc_uvspah ] plots the interpolated values of @xmath69 versus @xmath68 ; we also print the results in table [ table : pirresults ] so individual sources can be identified . we find that the @xmath69 values of the lmc and smc h ii regions span a large range , with @xmath76856 ( corresponding to @xmath777.4@xmath78 erg @xmath74 ) , and several of the smc sources have @xmath79 . the pah fractions of the smc h ii regions ( with @xmath801% ) are suppressed relative to those of the lmc h ii regions ( with @xmath811% ) . the smaller pah fractions in the low metallicity smc are consistent with the results of @xcite , who find a deficiency of pahs in the smc based on observations with the _ spitzer _ infrared spectrograph . based on pah band ratios in the irs data , these authors suggest that this deficiency arises because smc pahs are smaller and more neutral than pahs in higher metallicity galaxies . versus pah fraction @xmath75 for the 16 lmc h ii regions ( black circles ) and 16 smc h ii regions ( open squares ) , as given by the interpolation of the grid in figure [ fig : lmc_models ] . the numerical values for the two parameters are given in table [ table : pirresults ] , and the black star denotes the values for 30 dor . ] lrrr n4 & 740 & 2.1 & 500 + n11 & 230 & 3.2 & 50 + n30 & 250 & 3.4 & 60 + n44 & 230 & 2.8 & 60 + n48 & 140 & @xmath824.6 & 50 + n55 & 200 & 2.6 & 50 + n59 & 400 & 1.9 & 120 + n79 & 320 & 2.0 & 80 + n105 & 340 & 2.2 & 130 + n119 & 200 & 3.0 & 60 + n144 & 270 & 2.3 & 70 + 30 dor & 860 & 1.0 & 250 + n160 & 380 & 2.1 & 120 + n180 & 230 & 2.1 & 120 + n191 & 500 & 1.9 & 50 + n206 & 140 & 3.4 & 50 + dem s74 & 40 & 0.9 & 30 + n13 & 280 & 0.7 & 260 + n17 & 120 & 0.8 & 70 + n19 & 140 & @xmath830.5 & 160 + n22 & 740 & @xmath830.5 & 160 + n36 & 80 & @xmath830.5 & 60 + n50 & 50 & 0.7 & 20 + n51 & 140 & 0.7 & 30 + n63 & 90 & 0.7 & 60 + n66 & 380 & @xmath830.5 & 100 + n71 & 240 & @xmath830.5 & 330 + n76 & 130 & 0.6 & 70 + n78 & 570 & @xmath830.5 & 70 + n80 & 90 & 0.6 & 50 + n84 & 160 & 0.6 & 30 + n90 & 110 & 0.6 & 50 + [ table : pirresults ] finally , we employ the interpolated @xmath69 values and equations [ eq : pir ] and [ eq : u ] to estimate the dust - processed radiation pressure @xmath60 in our 32 sources . the warm ionized gas pressure is given by the ideal gas law , @xmath84 , where @xmath85 , @xmath86 , and @xmath87 are the electron , hydrogen , and helium number densities , respectively , and @xmath88 is temperature of the hii gas , which we assume to be the same for electrons and ions . if helium is singly ionized , then @xmath89 . if we adopt the temperature @xmath88 = 10@xmath90 k , then the warm gas pressure is determined by the electron number density @xmath91 . one way to estimate @xmath91 is via fine - structure line ratios in the ir ( e.g. , @xcite ) : since these lines have smaller excitation potentials than optical lines , they depend less on temperature and depend sensitively on the density @xcite . here , we estimate @xmath91 using an alternative means : by measuring the average flux density @xmath63 at 3.5 cm , where free - free emission dominates in h ii regions . for free - free emission , @xmath91 is given by eq . 5.14b of @xcite : @xmath92 where @xmath93 is the gaunt factor and @xmath14 is the distance to the sources , and @xmath34 is the volume of the sources . if we set the gaunt factor @xmath94 , we derive the densities @xmath91 listed in table [ table : pirresults ] . we find both the lmc and smc h ii regions have moderate densities , with @xmath95 22500 @xmath74 . the hot gas pressure is also given by an ideal gas law : @xmath96 , where @xmath97 is the electron density and @xmath98 is the temperature of the x - ray emitting gas . the factor of 1.9 is derived assuming that he is doubly ionized and the he mass fraction is 0.3 . furthermore , we assume that the electrons and ions have reached equipartition , so that a single temperature describes both populations . to estimate @xmath97 and @xmath98 , we model the bremsstrahlung emission at x - ray wavelengths of our sources using pointed _ rosat _ pspc observations ( for the lmc sources ) and _ chandra _ observations ( for n66 in the smc ) . the other h ii regions in the smc are undetected by _ xmm - newton _ and _ chandra _ , and we use these data to set upper limits on hot gas pressure in those targets . in the analyses described below , we assume a filling factor @xmath99 of the hot gas ( i.e. that the hot gas occupies the full volume of our sources ) . for the purposes of measuring the large - scale dynamical role of the hot gas , the appropriate quantity is the volume - averaged pressure . we explain in detail why this approach is critical when assessing global dynamics in appendix [ app : hot gas ] . for the _ rosat _ analyses of the lmc h ii regions , we used ftools , a software package for processing general and mission - specific fits data @xcite , and xselect , a command - line interface of ftools for analysis of x - ray astrophysical data . we produced x - ray images of the sources ( shown in blue in figure [ fig : lmcthreecolor ] ) , and we extracted spectra from within the radii given in table [ tab : sample ] as well as from background regions to subtract from the source spectra . appropriate response matrices ( files with probabilities that a photon of a given energy will produce an event in a given channel ) and ancillary response files ( which has information like effective area ) were downloaded for each observation s date and detector . resulting background - subtracted source spectra ( shown in figure [ fig : rosat15 ] ) were fit using xspec version 12.4.0 @xcite . spectra were modeled as an absorbed hot diffuse gas in collisional ionization equilibrium ( cie ) using the xspec components _ phabs _ and _ apec_. in these fits , we assumed a metallicity @xmath100 , the value measured in h ii regions in the lmc @xcite , and we adopted the solar abundances of @xcite . in some sources ( n11 , 30 dor , and n160 ) , we found the addition of a power - law component was necessary in order to account for excess flux at energies @xmath12 kev , a feature that is likely to be from non - thermal emission from supernova remnants or from point sources in the regions . llccll n4 & 1.6 & 0.15@xmath1010.04 & 0.28@xmath1010.27 & 34.1 & 13/9 + n11 & 1.9 & 0.20@xmath1010.01 & 0.04@xmath1010.01 & 36.3 & 100/99 + n30 & 1.9 & 0.67@xmath1010.30 & 0.27@xmath1010.09 & 34.6 & 20/52 + n44 & 6.0 & 0.22@xmath1010.07 & 0.12@xmath1010.07 & 37.0 & 156/107 + n48 & 4.7 & 0.54@xmath1010.41 & 0.03@xmath1010.02 & 35.6 & 135/123 + n55 & 1.2 & 0.62@xmath1010.16 & 0.01@xmath1010.01 & 34.4 & 34/53 + n59 & 1.6 & 0.63@xmath1010.13 & 0.04@xmath1010.02 & 35.6 & 19/54 + n79 & 1.6 & 0.45@xmath1010.12 & 0.02@xmath1010.01 & 35.1 & 47/47 + n105 & 2.1 & 0.25@xmath1010.03 & 0.09@xmath1010.04 & 35.6 & 68/74 + n119 & 2.1 & 0.23@xmath1010.01 & 0.06@xmath1010.02 & 35.9 & 181/109 + n144 & 2.0 & 0.25@xmath1010.01 & 0.07@xmath1010.02 & 36.0 & 166/115 + 30 dor & 3.0 & 0.39@xmath1010.04 & 0.08@xmath1010.03 & 36.8 & 204/165 + n160 & 8.1 & 0.54@xmath1010.17 & 0.04@xmath1010.03 & 34.8 & 62/40 + n180 & 2.5 & 0.30@xmath1010.06 & 0.06@xmath1010.03 & 35.2 & 11/31 + n191 & & & & & + n206 & 3.0 & 0.28@xmath1010.14 & 0.05@xmath1010.04 & 36.3 & 141/96 + n66 & 3.3 & 0.38@xmath1010.01 & 0.06@xmath1010.03 & 35.7 & 128/86 + [ table : pxresults ] lcccclc dem s74 & 5.06 & 0.0293 & 1.8@xmath102 & 4.6@xmath103 & 36.3 & 0.37 + n13 & 3.58 & 0.0013 & 8.7@xmath104 & 1.0@xmath105 & 33.6 & 0.69 + n17 & 3.33 & 0.0078 & 5.3@xmath105 & 5.2@xmath102 & 35.4 & 0.31 + n19 & 4.76 & 0.0026 & 1.6@xmath105 & 3.6@xmath102 & 35.2 & 0.76 + n22 & 4.44 & 0.0025 & 1.6@xmath105 & 3.0@xmath102 & 35.1 & 0.52 + n36 & 5.02 & 0.0241 & 1.5@xmath102 & 3.7@xmath103 & 36.2 & 0.41 + n50 & 4.86 & 0.0532 & 3.3@xmath102 & 7.7@xmath103 & 36.5 & 0.24 + n51 & 4.41 & 0.0137 & 8.7@xmath105 & 1.6@xmath103 & 35.9 & 0.39 + n63 & 4.60 & 0.0065 & 4.1@xmath105 & 8.3@xmath102 & 35.6 & 0.55 + n71 & 2.49 & 0.0002 & 1.1@xmath104 & 6.5@xmath104 & 33.5 & 0.70 + n76 & 3.45 & 0.1821 & 2.9@xmath103 & 3.1@xmath106 & 37.1 & 0.46 + n78 & 3.49 & 0.0853 & 1.3@xmath103 & 1.5@xmath106 & 36.8 & 0.41 + n80 & 3.48 & 0.0173 & 1.2@xmath102 & 1.3@xmath103 & 35.8 & 0.25 + n84 & 3.52 & 0.2549 & 1.7@xmath103 & 1.9@xmath106 & 36.9 & 0.23 + n90 & 2.10 & 0.0194 & 1.4@xmath102 & 6.4@xmath102 & 35.5 & 0.26 + [ table : pxupperlimits ] for the _ chandra _ analysis of n66 , we extracted a source spectrum using the ciao command _ specextract _ ; a background spectrum was obtained from a circular region of radius @xmath250 offset @xmath21 northeast of n66 . the resulting background - subtracted spectrum ( grouped to 25 counts per bin ) is shown in figure [ fig : n66spectrum ] . we first attempted to fit the spectrum with an absorbed hot diffuse gas in cie as above ( with xspec components _ phabs _ and _ apec _ ) assuming a @xmath107 metallicity plasma . the fit was statistically poor ( with reduced chi - squared values of @xmath108/d.o.f . @xmath109 317/90 ) , with the greatest residuals around emission line features . consequently , we considered an absorbed cie plasma with varying abundances ( with xspec components _ phabs _ and _ vapec _ ) . in this model , we let the abundances of elements in the spectrum ( o , ne , mg , si , and fe ) vary freely . the fit was dramatically improved ( with @xmath108/d.o.f . @xmath109 128/86 ) in this case . we found that the mg and fe abundances were consistent with those of the smc , while o , ne , and si had enhanced abundances of @xmath20.7 @xmath110 . the elevated metallicity of the hot plasma is suggestive that the x - ray emission is from a relatively young ( a few thousand years old ) supernova remnant ( snr ) , and the enhanced abundances are signatures of reverse shock - heated ejecta . a young snr in n66 has been identified previously as snr b0057@xmath8724 based on its non - thermal radio emission @xcite , its high - velocity h@xmath7 emission @xcite , and its far - ultraviolet absorption lines @xcite . the _ rosat _ and _ chandra _ x - ray spectral fit results are given in table [ table : pxresults ] , including the absorbing column density @xmath111 , the hot gas temperature @xmath112 , the hot gas electron density @xmath97 , their associated 90% confidence limits , and the reduced chi - squared for the fits , @xmath108/d.o.f . hot gas temperatures were generally low , with @xmath113 0.150.6 kev . comparing _ results for 30 dor to those from _ chandra _ in @xcite , we find that the integrated _ chandra _ spectral fits gave temperatures a factor of @xmath260% above those given by _ rosat_. this difference can be attributed to the fact that the _ rosat _ spectra were extracted from a much larger aperture than those from _ chandra_. broadly , the x - ray luminosity @xmath114 derived from our fits are consistent with previous x - ray studies of h ii regions in the lmc @xcite . for the smc h ii regions ( except n66 ) , we calculate upper limits on @xmath115 based on the non - detections of these sources in _ chandra _ ( for n76 and n78 ) and _ xmm - newton _ data . in particular , we measured the full - band count rates ( 0.58.0 kev ) within the aperture of our sources and converted these values to absorbed x - ray flux @xmath116 upper limits using webpimms , assuming the emission is from a @xmath107 metallicity plasma with @xmath117 kev . we then corrected for absorption to derive unabsorbed ( emitted ) x - ray fluxes @xmath118 , assuming an absorbing column equal to the weighted average @xmath111 in the source direction , given by the @xcite survey of galactic neutral hydrogen . finally , we simulated spectra of the @xmath107 , @xmath117 kev plasma to determine the emission measure @xmath119 ( and consequently , the electron density @xmath120 ) . the results of these analyses for the 15 smc h ii regions are listed in table [ table : pxupperlimits ] . each pressure term calculated using the methods described above will have an associated error , and there are many uncertainties which will contribute given the variety of data and analyses required . nonetheless , we attempt to assess these errors in the following ways . for the direct radiation pressure @xmath32 , the dominant uncertainty is the relation of @xmath37 to @xmath29 , as described in section [ sec : pdir ] . thus , for our error bars on @xmath32 have incorporated the factor of 2 uncertainty in the conversion of @xmath37 to @xmath29 . our calculation of @xmath60 is fairly robust , and the largest error comes from the 2% uncertainty in the _ spitzer _ photometry , which corresponds to a 2.8% error in the flux ratios of figure [ fig : lmc_models ] . therefore , we interpolated the @xmath69@xmath75 grid for @xmath1012.8% of our flux ratios to obtain a corresponding error in @xmath69 . these uncertainties lead to errors of the order 510% in @xmath60 . in the case of @xmath121 , we have uncertainty in the flux density @xmath63 over the radii of our h ii regions due to the low resolution of the radio data . therefore , we have measured @xmath63 for @xmath101one resolution element in our radio image and obtained the corresponding uncertainty in @xmath91 . this error is relatively small , @xmath21015% in @xmath91 and @xmath121 . finally , the range of @xmath115 is given by the uncertainty in the x - ray spectral fits of emission measure ( and correspondingly , the hot gas density @xmath97 ) and of the temperature @xmath112 . we employ these 90% confidence limits derived in our spectral fits , as listed in table [ table : pxresults ] . generally , the density @xmath97 was poorly constrained in lower signal sources ( e.g. , n4 , n30 , and n59 ) , as further evidenced by the poor reduced chi - squared values in those fits . therefore , in some cases , the error bars on @xmath115 can be relatively large , although the typical uncertainties were around @xmath23050% in @xmath97 . following the multi - wavelength analyses performed above , we calculate the pressure associated with the direct stellar radiation pressure @xmath32 , the dust - processed radiation pressure @xmath60 , the warm ionized gas pressure @xmath121 , and the hot x - ray gas pressure @xmath115 . table [ tab : presults ] gives the pressure components and associated errors measured for all the h ii regions , and figure [ fig : pdirvsp ] plots the pressure terms versus their sum , @xmath122 , to facilitate visual comparison of the parameters . as shown in figure [ fig : pvsr ] , we do not find any trends in the pressure terms versus size @xmath35 of the h ii regions . in all the targets except one , @xmath121 dominates over @xmath60 and @xmath115 . the exception is n191 , which has a @xmath60 roughly equal to its @xmath121 , although the errors on @xmath60 are quite large . for all sources detected in the x - rays except n30 , @xmath121 is a factor 27 above @xmath115 and @xmath123 in all sources . broadly , the relation between the terms is @xmath124 . in the entire sample , @xmath32 is 12 orders of magnitude smaller than the other pressure components . we note that while @xmath125 at distances @xmath075 pc from r136 in the giant h ii region 30 doradus @xcite , the warm ionized gas is what is driving the expansion currently and dominates the energetics when averaged over the entire source . lcccc n4 & 18.2@xmath126 & 2.13@xmath127 & 13.8@xmath128 & 2.31@xmath1012.29 + n11 & 5.08@xmath129 & 0.66@xmath130 & 1.38@xmath131 & 0.22@xmath1010.08 + n30 & 3.31@xmath132 & 0.72@xmath133 & 1.51@xmath134 & 5.64@xmath1013.17 + n44 & 4.21@xmath135 & 0.65@xmath136 & 1.69@xmath137 & 0.83@xmath1010.52 + n48 & 1.57@xmath138 & 0.40@xmath139 & 1.33@xmath137 & 0.43@xmath1010.43 + n55 & 4.41@xmath140 & 0.58@xmath141 & 1.28@xmath137 & 0.22@xmath1010.11 + n59 & 11.4@xmath142 & 1.15@xmath143 & 3.35@xmath144 & 0.78@xmath1010.35 + n79 & 4.96@xmath145 & 0.94@xmath146 & 2.25@xmath130 & 0.29@xmath1010.16 + n105 & 9.34@xmath147 & 0.99@xmath139 & 3.63@xmath148 & 0.66@xmath1010.33 + n119 & 5.24@xmath149 & 0.57@xmath137 & 1.62@xmath150 & 0.44@xmath1010.13 + n144 & 6.18@xmath151 & 0.78@xmath152 & 1.97@xmath153 & 0.51@xmath1010.14 + 30 dor & 55.7@xmath154 & 2.47@xmath155 & 6.99@xmath144 & 0.98@xmath1010.39 + n160 & 21.1@xmath156 & 1.10@xmath157 & 3.32@xmath158 & 0.70@xmath1010.57 + n180 & 9.03@xmath159 & 0.67@xmath152 & 3.21@xmath160 & 0.51@xmath1010.32 + n191 & 1.34@xmath161 & 1.43@xmath162 & 1.43@xmath131 & + n206 & 3.26@xmath163 & 0.41@xmath164 & 1.28@xmath137 & 0.39@xmath1010.39 + dem s74 & 0.67@xmath165 & 0.11@xmath131 & 0.69@xmath166 & @xmath830.88 + n13 & 16.9@xmath167 & 0.81@xmath134 & 7.28@xmath168 & @xmath831.65 + n17 & 2.37@xmath169 & 0.33@xmath150 & 2.00@xmath170 & @xmath830.75 + n19 & 4.67@xmath171 & 0.40@xmath130 & 4.40@xmath172 & @xmath831.82 + n22 & 5.78@xmath173 & 2.12@xmath174 & 4.31@xmath175 & @xmath831.25 + n36 & 4.34@xmath176 & 0.22@xmath150 & 1.63@xmath134 & @xmath830.99 + n50 & 1.25@xmath177 & 0.15@xmath131 & 0.63@xmath131 & @xmath830.58 + n51 & 0.71@xmath178 & 0.39@xmath179 & 0.87@xmath131 & @xmath830.94 + n63 & 2.20@xmath180 & 0.26@xmath137 & 1.57@xmath181 & @xmath831.31 + n66 & 12.1@xmath182 & 1.10@xmath183 & 2.92@xmath139 & 0.65@xmath1010.39 + n71 & 16.6@xmath184 & 0.68@xmath152 & 9.16@xmath185 & @xmath831.69 + n76 & 4.10@xmath186 & 0.38@xmath179 & 2.01@xmath143 & @xmath831.10 + n78 & 3.02@xmath187 & 1.66@xmath188 & 1.96@xmath152 & @xmath830.98 + n80 & 2.21@xmath189 & 0.26@xmath150 & 1.27@xmath179 & @xmath830.60 + n84 & 2.01@xmath190 & 0.47@xmath179 & 0.91@xmath131 & @xmath830.55 + n90 & 4.25@xmath191 & 0.33@xmath179 & 1.47@xmath192 & @xmath830.62 + [ tab : presults ] for the 32 h ii regions . dashed lines are meant to show how much each term contributes to the total pressure . the light blue arrows represent the @xmath115 upper limits of the 15 smc h ii regions that are not detected in archival _ xmm - newton _ and _ chandra _ data ; for our calculation of @xmath122 , we assume the smc @xmath115 upper limits are the pressures of the hot gas . section [ sec : uncertainty ] describes how error bars were calculated for each term . ] of the 32 h ii regions . the light blue arrows represent the @xmath115 upper limits of the 15 smc h ii regions that are not detected in archival _ xmm - newton _ and _ chandra _ data . see section [ sec : uncertainty ] for how error bars were assessed for each term . ] from section [ sec : results ] , it is evident that direct radiation pressure does not play a significant role in the dynamics of the regions . however , given the age and size of our sources , they are too large / evolved for the radiation pressure to be significant . the reason is that the pressure terms have a different radial dependence : @xmath193 , while @xmath194 , where @xmath195 is the shell radius . one can obtain a rough estimate of the characteristic radius @xmath196 where a given source transitions from radiation - pressure driven to gas - pressure driven by setting the total radiation pressure ( i.e. , the direct radiation as well as the dust - processed radiation ) equal to the warm gas pressure and solving for @xmath196 . in this case , we find @xmath197 where @xmath198 ev , the photon energy necessary to ionize hydrogen , @xmath199 is the case - b recombination coefficient , and @xmath200 is a dimensionless quantity which accounts for dust absorption of ionizing photons and for free electrons from elements besides hydrogen . in a gas - pressure dominated h ii region , @xmath200 = 0.73 if he is singly ionized and 27% of photons are absorbed by dust @xcite . the @xmath201 represents the factor by which radiation pressure is enhanced by trapping energy in the shell through several mechanisms , including trapping of stellar winds , infrared photons , and ly@xmath7 photons . here , we adopt @xmath202 , as in @xcite , although we note this factor is uncertain and debated , as discussed in section [ sec : dusty ] . lastly , @xmath203 is the ratio of bolometric power to the ionizing power in a cluster ; we set @xmath204 using the @xmath205 and the @xmath206 relations of @xcite . using these values , the above equation reduces to @xmath207 where @xmath9 is the ionizing photon rate , and @xmath208 s@xmath16 . we note that the derivation of equations [ eq : rch ] and [ eq : rch_s49 ] required several simplifying assumptions ( e.g. , regarding the coupling of the radiation to dust ) , and thus the estimate of @xmath196 should be viewed as a rough approximation of the true radius when an h ii region transitions from radiation- to gas - pressure dominated . we can estimate @xmath209 for our h ii regions based on their h@xmath7 luminosity @xcite : @xmath210 we list the resulting ionizing photon rates @xmath9 for our sample in table [ tab : extinction ] . given these values , we find a range @xmath211 0.017 pc for 31 h ii regions and @xmath212 33 pc for 30 dor . as our sample have radii @xmath210150 pc , the 32 h ii regions are much too large to be radiation - pressure dominated at this stage . this result demonstrates the need to investigate young , small h ii regions to probe radiation pressure dominated sources . the best candidates would be hypercompact ( hc ) h ii regions , which are characterized by their very small radii @xmath00.05 pc and high electron densities @xmath213 @xmath74 @xcite . hc h ii regions may represent the earliest evolutionary phase of massive stars when they first begin to emit lyman continuum radiation , and thus they offer the means to explore the dynamics before the thermal pressure of the ionized gas dominates . giant h ii regions which are powered by more massive star clusters may also be radiation pressure dominated . for example , @xcite showed that the super star clusters ( with masses @xmath214 ) in the starburst galaxy m82 are likely radiation pressure dominated . in section [ sec : results ] , we have demonstrated that the average x - ray gas pressure @xmath115 is below the @xmath215 k gas pressure @xmath121 . for the x - ray detected h ii regions , the median @xmath216 is 0.22 , with a range in @xmath217 0.130.50 ( excluding n30 , which has @xmath218 ) . for the 15 non - detected sources , we set upper limits on @xmath115 requiring at least 13 of the 15 h ii regions to have @xmath219 and nine to have @xmath220 . the low @xmath115 values we derive are likely due to the partial / incomplete confinement of the hot gas by the h ii shells ( e.g. , @xcite ) . if completely confined by an h ii shell expanding into a uniform density ism , the hot gas pressure @xmath115 would be large @xcite . conversely , a freely expanding wind would produce a negligible @xmath115 @xcite . in the intermediate case , a wind bubble expands into an inhomogeneous ism , creating holes in the shell where the hot gas can escape and generating a moderate @xmath115 . for example , @xcite argue the carina nebula is experiencing hot gas leakage based partly on its observed x - ray gas pressure of @xmath221 dyn @xmath222 , whereas the complete confinement model predicts @xmath223 dyn @xmath222 and the freely expanding wind model predicts @xmath224 dyn @xmath222 for carina . recent observational and theoretical evidence has emerged that hot gas leakage may be a common phenomenon . simulations have demonstrated that hot gas leakage can be significant through low - density pores in molecular material @xcite . observationally , signatures of hot gas leakage in individual h ii regions has been noted based on their x - ray luminosities and morphologies , such as in m17 and the rosette nebula @xcite , the carina nebula @xcite , and 30 dor @xcite . the results we have presented here on a large sample demonstrate that hot gas leakage may be typical among evolved h ii regions , implying that the mechanical energy injected by winds and sne can be lost easily without doing work on the shells . although we have found that the warm gas pressure @xmath121 dominates at the shells of our sources , a couple h ii regions ( n191 in the lmc and n78 in the smc , although we caution that the uncertainty in @xmath60 in n191 is large ) have nearly comparable @xmath60 and @xmath121 , and all 32 sources have @xmath225 . physically , this scenario can occur if the shell is optically thick to the dust - processed ir photons , amplifying the exerted force of those photons . in all 32 regions of our sample , the amplification factor caused by trapping the photons @xmath226 is quite large , with @xmath227 4100 and a median value of @xmath227 10 . from a theoretical perspective , it has been debated in the literature how much momentum can be deposited in matter by ir photons . @xcite argued that the imparted momentum would be limited to @xmath228 a few because holes in the shell would cause the radiation to leak out of those pores . conversely , if every photon is absorbed many times , then all the energy of the radiation field is converted to kinetic energy of the gas ; this scenario imparts the most momentum to the shell . an intermediate case is in optically thick systems , where photons are absorbed at least once , and the momentum deposition is dependent on the optical depth @xmath229 of the region @xcite . recent simulations by @xcite indicate that @xmath230 can be large as long as the radiation flux is below a critical value that depends on the dust optical depth . this critical value corresponds to the radiation flux being large enough so that the pressure of the dust - trapped radiation field is at the same order of magnitude as the gas pressure . at fluxes above the critical value , a radiation - driven rayleigh - taylor ( rrt ) instability develops and severely limits the value of @xmath230 by creating low - density channels through which radiation can escape . for example , in one case in @xcite where the rrt instability does not develop , they obtain @xmath231 , whereas when the radiation flux is increased so that radiation forces become significant and there is instability , @xmath230 drops to a few . clearly in the case of our sources , we are in the regime where the radiation pressure is not dominant compared to the warm gas pressure , and rrt instability is not expected ( though two of our sources are near the threshold of instability ) . thus , the high values of @xmath230 we obtain are consistent with these models . in this paper , we have performed a systematic , multi wavelength analysis of 32 h ii regions in the magellanic clouds to assess the role of stellar feedback in their dynamics . we have employed optical , ir , radio , and x - ray images to measure the pressures associated with direct stellar radiation , dust - processed radiation , warm ionized gas , and hot x - ray emitting plasma at the shells of these sources . we have found that the warm ionized gas dominates over the other terms in all sources , although two h ii regions have comparable dust - processed components . the hot gas pressures are relatively weaker , and the direct radiation pressures are 12 orders of magnitude below the other terms . we explore three implications to this work . first , we emphasize that younger , smaller h ii regions , such as hypercompact h ii regions , should be studied to probe the role of direct radiation pressure and the hot gas at early times . secondly , the low x - ray luminosities and pressures we derive indicate the hot gas is only partially confined in all of our sources , suggesting that hot gas leakage is a common phenomenon in evolved h ii regions . finally , we have demonstrated that the dust - processed component can be significant and comparable to warm gas pressure , even if the direct radiation pressure is comparatively less . these observational results are consistent with recent numerical work showing that the dust - processed component can be largely amplified as long as it does not drive winds . support for this work was provided by national aeronautics and space administration through chandra award number go213003a and through smithsonian astrophysical observatory contract sv373016 to mit and ucsc issued by the chandra x - ray observatory center , which is operated by the smithsonian astrophysical observatory for and on behalf of nasa under contract nas803060 . support for lal was provided by nasa through the einstein fellowship program , grant pf1120085 , and the mit pappalardo fellowship in physics . mrk acknowledges the alfred p. sloan foundation , nsf career grant ast0955300 , and nasa atp grant nnx13ab84 g . adb acknowledges partial support from a research corporation for science advancement cottrell scholar award and the nsf career grant ast0955836 . err acknowledges support from the david and lucile packard foundation and nsf grant ast0847563 . dc acknowledges support for this work provided by nasa through the smithsonian astrophysical observatory contract sv373016 to mit for support of the chandra x - ray center , which is operated by the smithsonian astrophysical observatory for and on behalf of nasa under contract nas803060 . the conversion of emission measure @xmath119 to hot gas electron density @xmath97 requires an assumption about the volume occupied by the hot gas , parametrized by a filling factor @xmath232 . for a fixed gas temperature @xmath112 ( which is determined from the spectral fitting and is independent of the assumed @xmath232 ) , the inferred density and pressure scale as @xmath233 . one can attempt to deduce @xmath232 from a combination of morphology and spectral modeling ( as in e.g. , @xcite ) . however , for the purposes of understanding the global dynamics , this approach can be misleading , as we demonstrate here . following the reasoning outlined below , we set @xmath234 . we are interested in the global dynamics of the regions , which are described by the virial theorem . neglecting magnetic fields ( which may not be negligible , but we lack an easy means to measure them ) , the eulerian form of the virial theorem is @xcite : here , @xmath34 is the volume , @xmath9 is the surface of this volume , @xmath237 , @xmath238 , and @xmath239 are the gas density , velocity , and pressure , @xmath240 is the fluid pressure tensor , @xmath241 is the frequency - integrated radiation energy density , @xmath242 is the radiation pressure tensor , @xmath200 is the gravitational potential , and @xmath243 is the identity tensor . the terms @xmath244 , @xmath245 , @xmath246 , and @xmath247 may be identified , respectively , as the moment of inertia , the total thermal plus kinetic energy , the total radiation energy , and the gravitational binding energy . the terms subscripted with @xmath248 represent external forces exerted at the surface of the volume , and are likely negligible in comparison with the internal terms for an h ii region with large energy input by massive stars . since manifestly @xmath249 either is very positive now , or was in the recent past ( otherwise the shell would not have expanded ) , the goal of this work is to understand the balance between the various positive terms on the right - hand side of the equation . the terms @xmath60 and @xmath32 are simply two different parts of @xmath246 , corresponding to energy stored in different parts of the electromagnetic spectrum , while @xmath121 and @xmath115 are part of @xmath245 . writing out the virial theorem in this manner makes the importance of the filling factor clear . the term we are interested in evaluating is the kinetic plus thermal energy of the x - ray emitting gas , where we have dropped the @xmath251 term on the assumption that the flow velocity is subsonic with respect to the hot gas sound speed , and in the second step we have defined the volume - averaged pressure @xmath252 , as distinct from the local pressure at a given point . the quantity @xmath252 can be understood as the partial pressure of the hot gas , including proper averaging down for whatever volumes it does not occupy . thus we see that the quantity of interest is _ not _ the local number density or pressure of the hot gas , it is the volume - averaged or partial pressure . now recall that , for fixed @xmath98 and fixed observed emission measure , local pressure scales with filling factor as @xmath253 , so a small volume filling factor increases @xmath115 . however , since the volume occupied by the hot gas scales as @xmath254 , it follows that @xmath255 , i.e. , a small volume filling factor implies that the hot gas is less , not more , important for the large - scale dynamics . this analysis has two important implications . first , the choice that makes the hot gas as dynamically - important as possible is to set @xmath256 , i.e. , to assume that the hot gas fills most of the available volume . in this case we simply have @xmath257 , and this is the choice we make in this work . a detailed assessment of @xmath232 that gives a value @xmath258 , as performed by @xcite , can imply an even smaller dynamical role for the hot gas , but not a larger one ( although understanding of filling factors is important for other considerations , such as the internal dynamics of h ii regions ) . the second implication is that it is inconsistent to treat @xmath115 as the quantity of interest for the global dynamics while simultaneously adopting @xmath259 . once can certainly attempt to measure @xmath254 and thus obtain a more accurate assessment of @xmath115 , but in this case the quantities that should be compared with other pressures is @xmath260 , _ not _ @xmath115 . the volume - averaged pressure is the relevant quantity for global dynamics , not the local pressure . we note that the above discussion of the filling factor applies to the warm gas as well , and we have also assumed a filling factor of order unity for the warm 10@xmath90 k gas . , c. l. , krumholz , m. r. , ballesteros - paredes , j. , et al.et al . 2013 , arxiv : 1312.3223 , protostars and planets vi , ed . h. beuther , r. s. klessen , c. p. dullemond , & t. henning ( university of arizona press ) , in press
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stellar feedback is often cited as the biggest uncertainty in galaxy formation models today .
this uncertainty stems from a dearth of observational constraints as well as the great dynamic range between the small scales ( @xmath01 pc ) where the feedback originates and the large scales of galaxies ( @xmath11 kpc ) that are shaped by this feedback . to bridge this divide , in this paper we aim to assess observationally the role of stellar feedback at the intermediate scales of h ii regions ( @xmath210100 pc ) .
in particular , we employ multiwavelength data to examine several stellar feedback mechanisms in a sample of 32 h ii regions ( with ages @xmath2310 myr ) in the large and small magellanic clouds ( lmc and smc , respectively ) . using optical , infrared , radio , and x - ray images
, we measure the pressures exerted on the shells from the direct stellar radiation , the dust - processed radiation , the warm ionized gas , and the hot x - ray emitting gas .
we find that the warm ionized gas dominates over the other terms in all of the sources , although two have comparable dust - processed radiation pressures to their warm gas pressures .
the hot gas pressures are comparatively weak , while the direct radiation pressures are 12 orders of magnitude below the other terms .
we discuss the implications of these results , particularly highlighting evidence for hot gas leakage from the h ii shells and regarding the momentum deposition from the dust - processed radiation to the warm gas .
furthermore , we emphasize that similar observational work should be done on very young h ii regions to test whether direct radiation pressure and hot gas can drive the dynamics at early times .
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the experimental realization of bose - einstein condensates ( becs ) in ultracold atomic gases has led to tremendous advances from traditional atomic , molecular , and optical ( amo ) physics @xcite to current quantum information science @xcite . recently , an intriguing atom - molecule dark state was observed in coherent two - color photoassociation ( pa ) @xcite , which has been considered as an efficient way to achieve higher production rates of molecules @xcite from ultracold atoms . in view of their internal properties and long - range anisotropic interactions @xcite , the assembly of heteronuclear molecules @xcite have also been actively pursued with various important applications @xcite , such as a polar molecular quantum computer @xcite . in the light of these developments it is timely to investigate the method of encoding and manipulating quantum optical state through the atom - molecule dark state . such processes will provide new insights on current efforts of optical pa or quantum superchemistry with the goal of designing a hybrid atom - molecule device for quantum control of photonic information . in this work we study such a scenario by transferring the quantum state of an associating light to an atom - heternuclear molecule dark state @xcite . this allows us to study the effects of initial populations imbalance on the optical storage process . in particular , our work compares the results for atom - molecule systems with the more familiar light - storage schemes in atomic samples @xcite . for a given number of atoms , the signal light is slowed more in the atom - molecule hybrid system , indicating some advantages over atomic slow - light media . hence our present proposal , together with e.g. a cascaded molecular transition , may indicate a hybrid device for optical storage , processing , and retrieval . as fig . 1 illustrated , the initial ultracold bosonic two - species atomic condensates ( with populations @xmath0 or @xmath1 ) are photoassociated into the excited molecular state @xmath2 by a quantized signal light , which is then dumped into the ground molecular state @xmath3 by another classical coupling light . the signal pulse is described by the dimensionless operator @xmath4 where @xmath5 is the quantization length in the @xmath6 direction , @xmath7 is the pa light frequency and @xmath8 is the slowly varying amplitude . we focus on the role of coherent couplings of photons and matter waves by ignoring the collisions of a dilute or feshbach - resonance - tuned medium @xcite . this is a safe approximation for the short lifetime of associated dimers @xcite . the operators of signal light and matter waves satisfy the commutation relations , @xmath9 = \frac{\nu}{\epsilon_0}\delta ( z - z')$ ] , @xmath10=\delta_{ij } \delta ( z - z'),$ ] respectively . the dynamics of this system is described in the simplest level by the interaction hamiltonian ( @xmath11 ) @xmath12,\end{aligned}\ ] ] where @xmath13 or @xmath14 is the one- or two - photon detuning , @xmath15 is the rabi frequency of the coupling field , and @xmath16 is the photon - matter waves coupling coefficient with @xmath17 being the transition - dipole moment of @xmath18 transition by @xmath19 @xcite . without loss of generality , we assume that the signal field amplitude @xmath20 and control field amplitude @xmath15 are real whose phase factor can be absorbed by a global gauge transformation of the field operators @xcite . here we first drop off the usual kinetic and the trapping terms by considering a uniform system and the effects due to these terms will be discussed later . with the slowly varying amplitude approximation @xcite , the propagation equation of the signal light can be written as @xmath21 meanwhile , the evolutions of atomic field operators are described by the following heisenberg equations @xmath22 where @xmath23 , @xmath24 , @xmath25 and @xmath26 denote the decay rates of corresponding matter - wave states . in order to obtain a closed - form signal - light propagation equation , it is a key step to study the evolutions of the following hybrid operators , @xmath27 @xmath28 with the transversal decay rates @xmath29 and @xmath30 . these equations can be rewritten as @xmath31 @xmath32 it should be noted that eq . ( [ eqn : field2 ] ) and eq . ( [ eqn : field3 ] ) can be greatly simplified under the weak excitation approximation ( wea ) : the control field is much stronger than the signal light at all times and thus the density of signal photons can be taken as much less than that of atoms . this means that only a small ratio of atoms are converted into molecules , which is the case in the recent two - color pa experiment @xcite . with the wea at hand , after some algebra we find in the lowest non - vanishing order @xmath33 hence eq . ( [ eqn : field2 ] ) can be rewritten as @xmath34 where @xmath35 is the population of atoms a or b , which can be assumed as constant in the wea . substituting eq . ( [ eqn : weak3 ] ) into eq . ( [ eqn : light1 ] ) yields @xmath36 clearly , for a time - independent coupling field , we have a steady group velocity of the signal , and the temporal profile or the spectrum of the signal pulse remains unchanged during its slowing down process , just as in a three - level atomic ensemble @xcite . for a time - dependent coupling field , however , the rand - hand side of eq . ( [ eqn : light2 ] ) leads to an adiabatic raman enhancement of the signal pulse @xmath37 where @xmath38 is the group velocity of the signal light and @xmath39 is the mixing angle between light and matter - wave components , i.e. , @xmath40 with @xmath41 . obviously , if the classical field is adiabatically turned off by rotating the mixing angle @xmath39 for @xmath42 , the signal light will be fully stopped within the medium or in the created atom - molecule dark state [ 4 ] . for the atomic slow - light medium @xcite , the group velocity of signal light is : @xmath43 , i.e. , being proportional to @xmath44 , where @xmath45 can be regarded as the number of initial trapped atoms in the wea ; however , in our situation , this velocity is proportional to @xmath46 ( for an initial balanced sample @xmath47 ) . hence the technique of atom - molecule dark state may have some advantage over the scheme of purely atomic spin waves since the signal light can be much slowed down by starting from the same total number of atoms . the quantum state transfer process is also observed through the form of the closed - channel molecular field . from eqs . ( 7 - 9 ) , it is straightforward to find @xmath48 obviously , for the initial stage ( @xmath49 ) , we have a purely photonic state , i.e. , @xmath50 or @xmath51 but when the coupling light is shut down adiabatically , the quantum state of the associating light is fully encoded into the created molecules via the mapping @xmath52 which thus indicates a possible quantum memory device based on coherent two - color pa technique . as fig . 2 shows , the initial populations imbalance of the atoms a , b ( @xmath53 ) also plays a role in the optical storage process , and the optimal conversion is reached for the balanced case ( @xmath54 ) . in our calculations , the initial total atomic number is taken as @xmath55 . in current experimental conditions @xcite , the strength of coupling field can be chosen as ( in the unit of mhz ) @xmath56 \nonumber\\ & + 0.5\tanh[0.15(t-125)]),\end{aligned}\ ] ] which guarantees that the group velocity can be adiabatically reduced to zero in the photon - storage stage and then the signal light can be re - accelerated in the retrieval stage . the control field being similar to eq . ( 16 ) is also used in the atomic slow - light medium @xcite . for the other cases of initial population imbalance , e.g. , @xmath57 as in the very recent experiment of creating polar molecules krb @xcite , some deviation from the optimized case can appear ( see fig . the quantized length @xmath58 , the coupling coefficient @xmath59 , the velocity is scaled by @xmath60 and the time is in the unit of @xmath61s . ] fig . 3 shows the different features of slow light propagation in four different kinds of matter - wave mediums : ( i ) the atomic ensemble , ( ii)-(iii ) the assembly of homonuclear or heteronuclear diatomic molecules [ 5 , 21 ] , and ( iv ) the assembly of heteronuclear triatomic molecules abc [ 5 ] ( for the trimer case , the calculation of the group velocity is completely similar to the above dimer cases ) . in addition , we see from fig . 3 that , by choosing a higher initial atomic populations , the optical storage process can be significantly improved . it is worth mentioning that in the above discussions we have ignored the decay of molecular states . however , it is readily to show that , after including these decay terms , the group velocity of the signal light is still in the form of eq . ( [ eqn : group1 ] ) but with the following substitution @xmath62 clearly , due to the decay terms , one may reach a @xmath63 group velocity even when the classical field is turned off . for the typical parameters of the molecules krb @xcite , we can take @xmath64 , @xmath65 , @xmath66 s@xmath67 , and @xmath68 the velocity limit can be estimated as @xmath69 km@xmath70s@xmath67 . in particular , for a sufficient state transfer , the pa time duration should satisfy @xmath71 ms @xcite , which can be fulfilled in current experiments @xcite . we also note that some optimized methods exist for a maximum efficiency of optical storage and retrieval , such as the recent works of novikova _ et al . _ by using an optimized signal pulse shape in an atomic medium @xcite . for the present atom - molecule system , in order to avoid incoherent absorptive loss , the frequency components of the signal pulse must fit well with the slow - light spectral window , i.e. , @xmath72 , where @xmath73 is the temporal length of the signal pulse , @xmath74 is optical depth of the medium . thereby , to avoid leakage " of the pulse outside the medium , the following condition should be fulfilled @xmath75which means that @xmath76 must be small enough for the entire signal pulse to be spatially compressed into the medium , with also a large optical depth d. for contrast , by using a purely atomic medium , we have @xmath77 @xcite and thus @xmath78 @xcite . obviously , for the same initial total atomic number , the quantum light storage using the atom - molecule dark state may have some advantage over the familiar atomic spin - wave scheme . now we show that , by taking into account of the particle collisions in the quantum state transfer process , it is also possible to realize a molecular matter - wave soliton laser . to this end , we consider coherent atom - molecule conversion process which is described by the following total hamiltonian @xmath79 where @xmath80 , as in eq . ( 2 ) , denotes the coherent free - quasi - bound - bound transition , @xmath81 and @xmath82 describe the free motions and the particle collisions , respectively , @xmath83 were @xmath84 denotes the longitudinal external effective potential and one can choose @xmath85=@xmath86 in the following derivation @xcite , @xmath87 is the particle mass , @xmath88 denotes the @xmath89-wave scattering collisions between the particles @xcite . then the evolution of molecular matter - wave field is written as @xmath90\hat\phi_g -\omega\hat\phi_e.\end{aligned}\ ] ] . the units of time and trap size are in @xmath61s and @xmath61 m , respectively . ] for simplicity , we consider an initially trapped atomic ensemble and also introduce the mean - field approach by replacing the operators by the @xmath60-numbers , i.e. , @xmath91 . thereby , for the closed - channel molecules , i.e. , with the mixing angle @xmath92 , we obtain the nonlinear mean - field gross - pitaevskii equation @xmath93 where the effective potential @xmath94 can also be moved by suitably tuning the value of @xmath95 @xcite . for @xmath96 , due to some balance between the repulsive molecular collisions and the molecular kinetic energy , the above well - known nonlinear equation can support a gray - soliton solution @xcite @xmath97\bigr\},\end{aligned}\ ] ] with @xmath98 . @xmath99 is chosen such that @xmath100 , the slowly - varying background function is @xmath101 and @xmath6 is the inside - trap position . in addition , the grayness " parameter is @xmath102 with the bogoliubov sound speed @xcite @xmath103^{1/2},\ ] ] and the dark - soliton speed @xmath104@xmath105 , @xmath106 being the central position of the molecular matter - wave soliton . clearly , @xmath107 corresponds to a dark soliton with @xmath108 density depletion . we see from fig . 4 that , by applying two counter - propagating control fields , a second - order molecular grey soliton ( @xmath109 ) starting from the same position ( @xmath110 ) can split into two solitons propagating at opposite directions . in conclusion , we have studied the slow light and quantum optical storage process in coherent two - color pa process by considering a quantized associating light . by taking into account of the particle collisions , one may also create the molecular matter - wave solitons . this may indicate a hybrid atom - molecule quantum device for storage and retrieve of optical information . it is straightforward to study other interesting configurations , such as the light - molecule entanglement by applying a non - classical pa light @xcite , or the quantum switch by considering a multi - level atom - molecule system . as far as we know , our work sets up the first link between the research fields of quantum memory and coherent atom - molecule conversion . due to the rapid experimental advances in both two fields , this atom - molecule system may be potentially useful for designing a hybrid atom - molecule quantum device of optical storage , processing , and retrieval . in the future , we plan to study the quantum memory in a boson - fermion mixture [ 5 , 8 ] , the slow light in the bec - bcs crossover of a fermionic atomic sample @xcite , and the polarization rotation of slow light @xcite with the laguerre - gaussian signal modes . are supported by the national science foundation of china ( grant no . 10874041 ) , the ncet , and the ecnu key lab open fund . weiping zhang is supported by the national science foundation of china ( grant no . 10588402 and no . 10474055 ) , the national basic research program of china ( grant no . 2006cb921104 ) , the science and technology commission of shanghai municipality ( grant no . 06jc14026 and no . 05pj14038 ) , the program of shanghai subject chief scientist ( grant no . 08xd14017 ) , the program for changjiang scholars and innovative research team , shanghai leading academic discipline project ( grant no . b480 ) , and the research fund for the doctoral program of higher education ( grant no . 20040003101 ) . h. y. ling , h. pu , and b. seaman , phys . 93 * , 250403 ( 2004 ) ; a.nunnenkamp , d. meiser , and p. meystre , new j. phys . * 8 * , 88 ( 2006 ) ; h. jing , y. j. jiang , weiping zhang , and p. meystre , _ ibid . _ * 10 * , 123005 ( 2008 ) . r. zhao , y. o. dudin , s. d. jenkins , c. j. campbell , d. n. matsukevich , t. a. b. kennedy , and a.kuzmich , nature phys . * 5 * , 100 ( 2009 ) ; b. zhao , y. a. chen , x. h. bao , t. strassel , c. s. chuu , x. m. jin , j. schmiedmayer , z. s. yuan , s. chen , and j. w. pan , nature phys . * 5 * , 95 ( 2009 ) . j. denschlag , j. e. simsarian , d. l. feder , c. w. clark , l. a. collins , j. cubizolles , l. deng , e. w. hagley , k. helmerson , w. p. reinhardt , s. l. rolston , b. i. schneider , and w. d. phillips , science , * 287 * , 97 ( 2000 ) .
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we propose to use a quantized version of coherent two - color photoassociation to realize a hybrid device for quantum control of light .
the dynamical features of this system are exhibited , including the slowing down or storage of light and the molecular matter - wave solitons .
this may indicate a hybrid atom - molecule quantum device for storage and retrieve of optical information .
| 4,862 | 90 |
* quantum dot structures . * inas / gaas qds are grown by molecular - beam epitaxy on ( 001 ) gaas substrates , and then thermally annealed at 940 @xmath30c @xcite . annealing interdiffuses indium and gallium , resulting in ( in , ga)as / gaas qds with effective localization volumes of order 2000 nm@xmath31 , giving @xmath010@xmath2 nuclei within the spatial extent of the resident hole s wavefunction . the sample contains 20 layers of qds , separated by 60 nm gaas barriers . each layer contains @xmath0@xmath32 qds/@xmath33 . although not intentionally doped , these qds are weakly _ p_-type due to background carbon doping ; we estimate that @xmath010% of the qds contain a single resident hole . the inhomogeneously - broadened photoluminescence ( pl ) spectrum of these qd ensembles is typically peaked at @xmath01.385 ev ; see fig . * spin noise spectroscopy . * the qd samples are mounted on the cold finger of a small optical cryostat . a linearly - polarized continuous - wave probe laser is tuned in wavelength to within the pl spectrum of the qd ensemble and is weakly focused through the sample ( @xmath34 , where * n * is the sample normal ) . stochastic fluctuations of the ensemble hole spin projection along the * z * axis , @xmath10 , impart faraday rotation fluctuations @xmath11 on the transmitted probe laser via the usual optical selection rules for positively - charged trions . balanced photodiodes detect @xmath7 , and the amplified output voltage @xmath35 is continuously digitized and fourier - transformed to obtain the frequency spectrum of the measured noise power @xcite . external coils provide longitudinal ( @xmath23 ) and transverse ( @xmath20 ) applied magnetic fields . background noise densities due to photon shot noise and amplifier noise are eliminated by interleaving and subtracting spectra acquired at large @xmath20 ( @xmath362000 g ) , which shifts any spin noise to high frequencies outside the measured range . this procedure leaves behind only the noise signals arising from fluctuating hole spins at low fields . typically the cw probe laser power is a few hundred @xmath1w , and it is focused to a rather large ( 100 @xmath1 m ) spot on the sample to minimize heating and self - pumping of the qds ( see fig . s2 ) . crucially , and in comparison with previous work @xcite , the present setup uses low - noise , stabilized probe lasers that now permit accurate and quantitative recovery of the small low - frequency spin noise signals that exist in the zero - field limit . a consistent measure of the characteristic timescale @xmath8 of the hole spin correlations is obtained from the measured half - width @xmath15 of the spin noise peak that is centered on zero frequency . specifically , we use @xmath37 , which is precise for lorentzian noise lineshapes that indicate single - exponential relaxation dynamics . this definition of @xmath8 is also used when the noise lineshapes deviate from lorentzian , even though , strictly speaking , power - law lineshapes can not be characterized by a specific timescale . in this case , @xmath15 is determined relative to the peak spin noise power spectral density that is measured in the lowest frequency bin . figure s1 shows the inhomogeneously - broadened photoluminescence ( pl ) spectra of the ( in , ga)as / gaas qd ensemble ( solid black lines ) under very low excitation conditions by a 1.58 ev ( 785 nm ) laser . this pl arises from ground - state recombination of both positively - charged trions @xmath38 ( from qds containing a single resident hole ) , as well as from neutral excitons @xmath39 ( from qds that are empty ) . note that these transition energies are typically very close in ( in , ga)as qds ( @xmath38 being higher in energy by @xmath40 mev @xcite ) and therefore they overlap in this ensemble pl spectrum and can not be separately resolved . the pl spectrum therefore directly reflects the inhomogeneously - broadened distribution of fundamental qd transition energies in the ensemble . the good correspondence between the pl and total spin noise indicates that the spin noise signals arise from resident holes in these qds , and not from holes or electrons residing in , _ e.g. _ , the wetting or buffer layers of the structure . ] when performing spin noise spectroscopy of these qds , the narrow - band , continuous - wave probe laser is tuned in energy to within this pl spectrum . figure s1 also shows the frequency - integrated ( _ i.e. _ , total ) measured spin noise power as a function of the photon energy of the probe laser . the integrated spin noise power provides a relative measure of the number of fluctuating spins being measured . its dependence largely follows the pl spectrum with a small blueshift , commensurate with the expected energy difference between @xmath38 and @xmath39 transition energies . this correspondence indicates that the measured spin noise arises from the resident holes that are trapped in the singly - charged subset of the qds ( rather than from spins in , _ e.g. _ , the buffer or wetting layers or in the bulk of the semiconductor wafer ) . further , at all probe laser energies where spin noise is detected , the spin noise exhibits the same narrow spectral width at zero applied magnetic field ( as shown in figure 1 of the main text ) , and the measured spin noise has the same behavior in transverse and longitudinal fields as shown in the main text ( verified for a number of different probe energies ) . the particular implementation of spin noise spectroscopy employed in these experiments to detect fluctuations of @xmath41 ( the net spin polarization of the resident holes in the qd ensemble ) is based on optical faraday rotation . the faraday rotation angle @xmath42 depends on the difference between the indices of refraction for right- and left - circularly polarized light , @xmath43 and @xmath44 . in particular , @xmath45 $ ] , where @xmath46 is the sample thickness , @xmath47 denotes energy , and @xmath48 is the speed of light . before discussing the case of an inhomogeneously - broadened qd ensemble , first consider a spin noise measurement of a single homogeneously - broadened optical absorption resonance @xmath49 having a lorentzian line - shape centered at @xmath50 and half - width @xmath51 ; namely , @xmath52 , as shown in figure s2a . let us say that this absorption resonance is spin - dependent as for the case of the optical transition between a resident hole and a positively - charged trion @xmath38 . following the usual optical selection rules , if the resident hole is spin - up " then it can only absorb @xmath53 circularly - polarized light , but if it is in the spin - down " state then it can only absorb @xmath54 circularly - polarized light . and @xmath53 circularly polarized absorption resonances @xmath55 and the associated indices of refraction @xmath56 of an idealized homogeneously - broadened ( lorentzian ) optical transition ( _ e.g. _ , from a @xmath38 transition ) . @xmath56 decay much more slowly than @xmath55 as a function of large detuning @xmath57 . faraday rotation , @xmath58 , can therefore remain sensitive to the spin state @xmath41 of the resident hole , even for large @xmath57 where @xmath59 . importantly , note that @xmath60 when @xmath61 . ( b ) an illustration of the inhomogeneously - broadened distribution of @xmath38 transitions within the qd ensemble . the probe laser is most sensitive to hole spin fluctuations in those qds having @xmath38 transition energies detuned by about @xmath51 and is _ not _ sensitive to those qds that are on resonance ( _ i.e. _ , to those qds that are unavoidably pumped by the linearly - polarized probe laser ) . ( c ) the half - width @xmath15 of the zero - field hole spin noise spectrum measured using different probe laser power : there is no perceptible influence of the probe laser on the hole spin correlation time . ( d ) the total ( frequency - integrated ) spin noise , in volts , as a function of probe laser power . the total spin noise scales nearly linearly with the probe laser power , indicating no contribution to the spin noise from excited carriers . the dots are measured values , and the solid lines are power - law fittings . in ( c ) and ( d ) the red and black dots denote two measurements with different spot sizes . ] the dispersive part of this optical transition that is , the part complementary to the absorption resonances @xmath62 are the indices of refraction @xmath63 . as a function of detuning @xmath64 from resonance , the indices @xmath65 decay much more slowly than the absorption @xmath62 : as @xmath66 versus @xmath67 , respectively . therefore , a measurement of faraday rotation can remain sensitive to the spin state of the hole even for large detuning @xmath68 where @xmath69 and the number of photons absorbed by the system becomes vanishingly small . in this regard , the measurement of @xmath41 can be considered non - perturbative , and spin noise measurements of this type were performed on alkali vapors at large probe laser detunings from the @xmath70 ( d1 and d2 ) lines of potassium and rubidium @xcite , and were also performed on conduction - band electrons in bulk _ n_-gaas with the probe laser detuned well below the gaas band - edge @xcite . note that @xmath42 and therefore the spin noise measurement is most sensitive to spin fluctuations of this idealized system when the probe laser and the absorption resonance are separated by an energy of @xmath71 . the sensitivity falls off slowly as @xmath66 at larger detunings . even more importantly , note that @xmath72 at small detunings @xmath73 , and that @xmath74 exactly on resonance ( @xmath61 ) because there is no difference between @xmath75 and @xmath76 . _ the spin noise measurement is not sensitive to transitions lying at the same energy as the probe laser . _ this brings us to the case of measuring spin noise in an inhomogeneously - broadened qd ensemble . here we tune the laser to directly within the inhomogeneously - broadened distribution of qd transition energies . as shown above in figure s1 , this maximizes the measured spin noise signal . undoubtedly , those qds in the ensemble that happen to be resonant with the probe laser are optically pumped by the probe laser . however as noted in the previous paragraph , it is precisely these resonant qds that do not , to leading order , give any spin noise ( moreover , any photogenerated electrons , holes and/or trions that are pumped by the linearly - polarized probe laser are not spin - polarized ) . rather , the probe laser will be primarily sensitive to those resident holes in qds that have detuned @xmath38 transitions , and these qds are not pumped ( see fig . s2b ) . in this regard , the faraday rotation measurement still functions as a non - perturbative probe of @xmath41 , the net spin polarization of resident holes in the qd ensemble . the fact that @xmath77 is not sensitive to spins in qd ensembles that are weakly pumped at @xmath78 was demonstrated in several recent ultrafast studies using independently - tunable pump and probe lasers @xcite . nonetheless , great care is taken to ensure that the probe laser functions only as a passive detector of the resident hole spin fluctuations , and does not inadvertently perturb the measurement of @xmath41 . primarily , we wish to ensure that the probe laser i ) does not heat the qds which could lead to incorrect spin lifetime measurements , and ii ) does not inadvertently detect any particles ( electrons , holes , excitons or trions ) that are photoexcited by the probe laser itself . thus we use low probe laser intensities by using low laser power ( @xmath0200 @xmath1w ) and large spot sizes ( @xmath0100 @xmath1 m ) . although this significantly reduces the measured spin noise signals , it ensures that we operate in a regime where the measured spin correlation times @xmath8 ( _ i.e. _ , the inverse width of the spin noise spectra , @xmath79 ) are independent of probe laser power . in this regime , figure s2(c ) shows that the measured half - width @xmath15 of the low temperature , zero - field hole spin noise spectrum is independent of the probe laser power . significantly higher probe laser intensity leads to a broadening of the spin noise peak , indicating shorter hole spin lifetimes . for the same series of measurements , figure s2(d ) shows the total ( frequency - integrated ) spin noise signal in units of volts of detected signal . as expected , the total spin noise signal detects no contribution from photoexcited carriers , since it increases nearly linearly with probe laser power ( all else being equal , doubling the laser power simply doubles the voltages at the photodetectors ) . if the probe laser were inadvertently measuring spin noise from photogenerated ( rather than resident ) particles in the resonant qds , the total spin noise would be expected to increase super - linearly ( doubling the laser power would not only double the voltages at the photodetectors , it would also double or at least increase the number of particles being measured , leading to a superlinear dependence ) . figure s3 shows that magnetic fields in the longitudinal ( * z * ) direction , either real or effective , necessarily lead to some spin noise centered at zero frequency . to demonstrate this , we show spin noise measurements of electron - doped bulk gaas . free conduction band electrons in _ n_-type bulk gaas are delocalized and sample a huge number of lattice nuclei . therefore the influence of the fluctuating nuclear spin bath on these free electrons is extremely small . to leading order , the only magnetic fields felt by the electrons are those that are externally applied . in this case , a purely transverse applied magnetic field @xmath20 uniformly shifts the spin noise of these electrons out to the larmor frequency @xmath80 , and leaves no remnant of spin noise at zero frequency ( figure s3a ) . this is in marked contrast to the spin noise spectra of qd holes shown in figure 2 of the paper , in which some spin noise clearly remains at zero frequency despite application of a purely transverse @xmath20 this remaining spin noise is due to the longitudinal components of the effective nuclear field @xmath3 that is felt by the holes . = 0 , 2.5 , 5.0 , 7.5 , and 10 g ( black to green ) . note the absence of residual noise at 0 hz , as expected from particles that feel little influence of the nuclear spin bath . the probe laser s wavelength is 845.6 nm ( well below the gaas band - edge ) , its power is 2 mw , and the spot size is large ( 100 @xmath1 m ) . this @xmath81-gaas wafer is 350 @xmath1 m thick and is doped at @xmath82 @xmath83 ( it is sample a " in ref . ( b ) spin noise in a 60 g canted applied magnetic field ( @xmath84 ) , showing that longitudinal ( * z * ) field components lead to spin noise at zero frequency . ] that longitudinal fields generate spin noise at zero frequency is explicitly demonstrated in figure s3(b ) , which shows electron spin noise spectra in an intentionally tilted applied magnetic field . a 60 g applied field * b * is rotated from the transverse to the longitudinal direction ( @xmath85 ) . as * b * acquires a longitudinal component , zero frequency spin noise grows as sin@xmath86 , with a commensurate cos@xmath86 suppression of noise signal at the electron larmor frequency . it is useful to construct a toy model of hole spin noise in the considerably oversimplified limit of _ static _ nuclear overhauser fields @xmath87 . the purpose of this exercise is three - fold : \1 ) it demonstrates that hole spin precession about @xmath3 generates a broad hole spin noise spectrum at high frequencies between 5 - 100 mhz . the broadness of this high frequency noise is due to the statistical distribution of @xmath3 over the qd ensemble , and also to the inhomogeneous distribution of hole g - factors within the qd ensemble . \2 ) it demonstrates that the longitudinal overhauser fields @xmath88 give a delta - function ( or at least very narrow ) spin noise peak at zero frequency , that is _ not _ expected to narrow or broaden with applied fields @xmath20 or @xmath23 ( in contrast to actual experimental observation ) . this noise peak is not statistically broadened by the ensemble , since each qd gives some noise at exactly zero frequency . \3 ) it shows that , in zero applied field , the high - frequency precessional noise is strongly suppressed as compared to the zero - frequency noise due to the large anisotropy of the hole g - factor . in marked contrast with electrons , holes couple very anisotropically to in - plane versus out - of - plane magnetic fields . for _ pure _ heavy holes , the longitudinal ( out - of - plane ) g - factor @xmath89 is finite while the transverse ( in - plane ) g - factor @xmath90 is zero . however , hole eigenstates in typical _ p_-type self - assembled iii - v quantum dots invariably contain some admixture of light hole states in addition to their predominantly heavy - hole character . this leads to a small in - plane g - factor @xmath90 that is of order 0.15 in our qds , which is about an order of magnitude less than @xmath89 ( @xmath91 ) . note also that there exists a large inhomogeneous dispersion of these g - factors within the qd ensemble , likely due to differences in qd shape and strain . here we assume that the overhauser field @xmath3 in each quantum dot has components @xmath92 , and @xmath88 that are each gaussian - distributed with typical dispersion of @xmath025 gauss ( taken from experimental data of figure 3c ) . in any given qd , hole spin precession about the transverse component of @xmath3 generates noise in @xmath41 ( the measured quantity ) at the hole s larmor precession frequency , @xmath93 ( where here we have generalized slightly to allow for real applied magnetic fields @xmath20 and/or @xmath23 , and we assume no in - plane anisotropy of @xmath90 for simplicity ) . in each qd , @xmath94 occurs at a different frequency depending on the magnitude and direction of @xmath3 in that qd , and also depending on @xmath90 and @xmath89 in that qd . using the typical values stated above , @xmath94 can range from a few mhz out to @xmath0100 mhz . averaging over many qds , _ this precessional noise generates a very broad spectrum _ , weakly peaked at about 10 mhz and spanning the range from 5 - 100 mhz . it is this precessional noise that represents the trivial ensemble dephasing of an ensemble of hole spins that are all initially oriented at @xmath95 , as in a pump - probe measurement . . in each qd , hole spin precession about @xmath3 induces a high - frequency noise peak at the hole larmor frequency @xmath94 . in addition , longitudinal ( * z * ) components of @xmath3 generate a noise peak at zero frequency ( for the analogous case of electrons in arbitrary applied fields , see fig . s3 above ) . for the purposes of this toy model , these peaks are represented by delta functions . the total noise power contained within the high - frequency peak is generally much smaller than that within the zero - frequency peak , due to the large anisotropy of the hole g - factor . statistical averaging over the qd ensemble smears the high - frequency precessional noise over a large frequency range . on the other hand , the zero - frequency noise peak is contributed to by every qd . _ in our noise experiments , it is precisely the width and lineshape of this zero - frequency peak that we seek to measure _ , as these parameters reveal the intrinsic spin correlation timescales and decay mechanisms of the hole spins . ] however , in each and every qd , the longitudinal component of @xmath3 ( _ i.e. _ , @xmath88 ) , generates a finite time - averaged value of the hole s spin projection , @xmath13 , that does _ not _ decay in time . in a noise measurement , and within the limitations of this toy model , this simply gives a delta function at zero frequency , and applied magnetic fields do not alter the lineshape of this peak . of course in reality , @xmath13 _ does _ decay because @xmath3 is _ not _ static , and the associated noise peak is not a delta function _ it is precisely the width and lineshape of this zero - frequency peak that reveals the long - time decay mechanisms of the central hole spins _ that we are interested in . within this simple model , it can be shown that the total ( frequency - integrated ) power of the noise peak at @xmath94 is @xmath96 while the integrated power of the noise peak at zero frequency is @xmath97 therefore in the absence of any applied fields ( @xmath98 ) , the noise power at @xmath94 is typically suppressed as compared to the zero - frequency noise by @xmath99 ( except when @xmath88 is very small ) . this , together with the above - noted fact that ensemble averaging smears out the high - frequency precessional noise over a very broad bandwidth , may explain why we do not observe any clear sign of precessional noise occurring at high frequencies in the absence of applied fields . only the much larger zero - frequency noise is apparent . however , as we apply large magnetic fields in the transverse direction ( @xmath20 ) , all the hole noise power is expected to shift to the precessional noise component at @xmath94 , in agreement with our experimental data ( figure 2 of the main paper ) . further , as we apply large fields in the longitudinal direction ( @xmath23 ) , the total noise power contained within the zero - frequency component will increase only very slightly ( all the noise power is essentially already contained in this peak ) , again in agreement with experimental observation . finally , note that the above expressions also hold for electrons , in which case @xmath100 and the high - frequency and zero - frequency noise peaks are expected to have comparable integrated power . previous theoretical studies of nuclear spin co - flips " ( a flip - flop of two distant nuclei mediated by virtual hole - nuclear spin flips ) revealed non - exponential decay of the central spin due to non - markovian evolution and strongly - correlated dynamics @xcite . the goal of this section is to show that the time scale of @xmath01 @xmath1s , which roughly corresponds to the inverse width of the spin noise power in zero magnetic field , is in agreement with a simple theory of how nuclear co - flips lead to central spin relaxation . we will also highlight differences between this theory and the observed behavior of noise power spectrum at longer time scales . consider the effective hamiltonian of a single hole spin interacting with @xmath101 nuclear spins , in the presence of an applied out - of - plane ( longitudinal ) magnetic field @xmath23 . we have @xmath102 @xmath103 where @xmath104 stands for the hole spin operator , @xmath105 is the @xmath106-th nuclear spin operator , @xmath107 and @xmath108 are the coupling strengths , out - of - plane and in - plane respectively , between the central spin and nuclear spins . we assume , for simplicity , that all spins are @xmath109 , and @xmath107 and @xmath108 are in the same ratio as the ratio of longitudinal and transverse hole g - factors , @xmath89 and @xmath90 . thus , @xmath107 is about ten times greater than @xmath108 . note that in general @xmath110 is complex , reflecting the fact that transverse coupling involves both @xmath111 and @xmath112 spin components , but as this does not influence the following discussion we treat these couplings as real positive parameters . the large value of @xmath113 and the coupling anisotropy enable a perturbative approach @xcite , in which the zeroth order wave function of a typical state can be approximated to be the eigenstate of the total spin operator along @xmath114-axis , for example , where @xmath116 or @xmath117 indicates the state of the hole central spin ( up or down along @xmath114-axis ) and other arrows show states of the nuclear spins . we will call the expectation value of operator @xmath118 in the state @xmath119 the _ bias _ , and we will denote it by @xmath120 . we will show that this part of the hyperfine coupling resists the flip - flop transitions . diffusive dynamics of the overhauser field @xmath3 are possible because the couplings of central spin to different nuclear spins are not equal so that exchanging direction of a pair of nuclear spins changes the bias value by the amount @xmath121 . according to the hamiltonian ( [ ham ] ) , such pair - wise co - flips of nuclear spins happen due to transitions through the virtual states with a flipped central spin . the accumulation of pair - wise nuclear spin - flips leads to diffusion in the space of overhauser field values . the expectation value of the central spin follows the direction of the overhauser field . when the bias changes across the region with @xmath122 , the variation of this direction becomes substantial , corresponding to the relaxation of the central spin expectation value . there is no direct coupling between the state ( [ zero - order ] ) and the state @xmath123 where @xmath124 means that nuclear spins @xmath106 and @xmath125 change their directions in comparison to the state @xmath126 . therefore , to estimate the rate of nuclear spin co - flips we use first order perturbation theory to incorporate transitions to the virtual states in our wave functions : @xmath127 where @xmath128 and where @xmath129 means that @xmath106-th nuclear spin is flipped from up to down state , and @xmath130 ensures a normalization . this definition of a typical state @xmath119 includes fast transition processes to the virtual states @xmath131 . one can see that @xmath132 already directly couples states @xmath119 and @xmath133 . further perturbation expansion breaks down because there are many states @xmath133 with almost the same energy as the initial state @xmath134 . the presence of such quasi - degenerate states leads to incoherent transitions from @xmath126 into one of the states @xmath135 . the rate @xmath136 of such transitions can be estimated by fermi s golden rule as follows : there are @xmath137 pairs of distinct states @xmath138 ( again , we are considering spin 1/2 nuclei for convenience ) , distributed around the mean value with interval of energies @xmath139 , where @xmath140 is the width of the distribution of @xmath141 around their mean value . thus the rate , @xmath142 , of an incoherent transition from the initial state @xmath134 to one of the states @xmath133 that differ from @xmath134 by co - flips of two nuclear spins is given by where @xmath144 is the density of states represented by all possible pairs @xmath145 , and @xmath146 is the strength of a typical coupling between states @xmath145 and the state @xmath119 , _ i.e. _ @xmath147 . we use @xmath148 , where according to various estimates , @xmath149 3 - 13 @xmath1ev @xcite is the hyperfine coupling per nuclear spin per unit probability of the hole being in the unit cell that contains this nuclear spin . the factor @xmath150 is the ratio of volumes of the unit cell @xmath151 ( 0.57 nm)@xmath31 to the volume of the quantum dot , @xmath152 2000 nm@xmath31 . hence @xmath153 and @xmath154 s@xmath155 . the fastest spin relaxation happens in states that have initial bias @xmath156 because such states experience comparable couplings along longitudinal ( along @xmath114-axis ) and transverse axes . in such states , changes in total field along @xmath114-axis ( also the bias @xmath120 ) leads to a substantial change of the orientation of the total field that acts on the hole s spin . if we consider a state with initial @xmath120 close to @xmath157 we find from ( [ gam ] ) that the typical time of an incoherent transition between states that differ by a co - flip energy is @xmath158 s. this time @xmath159 should not be confused with a hole s spin relaxation time @xmath8 because a single co - flip does not lead to a substantial change of the total hyperfine field . moreover , such random transitions can lead both to an increase or a decrease of @xmath120 . however , accumulation of co - flip transitions leads to a diffusion in the space of bias values . hence the latter changes with time according to the diffusion law : @xmath160 where the coefficient @xmath107 reflects the fact that a single co - flip corresponds to a change of the bias of order @xmath107 . the hole s spin rotation angle becomes of order unity when the change of the bias becomes comparable to the initial bias , @xmath161 . this gives us an estimate of the hole s spin relaxation time : @xmath162 the experimentally obtained value , @xmath163 ns , is in very reasonable agreement with this estimate . here we note , however , that the region of bias values with @xmath156 , accounts for only about 10% of statistically possible states of the nuclear spin bath . when @xmath120 exceeds the size of this region , incoherent flip - flop transition rate is suppressed , as it can be seen from the fact that @xmath120 enters as a second power in the denominator in equation ( [ gam ] ) . thus , for values of @xmath164 , the flip - flop transition rate is suppressed by a factor 100 , in disagreement with the assumption of a single relaxation time for all quantum dots , which is expected from the lorentzian shape of the power spectrum . dipole interactions between nuclear and central spins allow additional coupling terms in the interaction hamiltonian , such as @xmath165 that could contribute to the nuclear spin dynamics , but such terms were estimated to be negligibly small for heavy - hole states in quantum dots @xcite . it is also possible that strains and nonuniform doping introduce high gradients of local electric fields that couple to quadrupole moment of nuclear spins . quadrupole nuclear interactions with nonuniform electric field often broaden the solid state nmr - lineshapes of nuclei to the mhz range @xcite . such interactions do not directly involve the central spin but , if sufficiently strong , they can induce fast intrinsic dynamics of nuclear spins and stir fluctuations of the overhauser field across its typical values . it is possible that , supplemented by fast incoherent co - flip processes near the zero bias values , such fluctuations lead to the relaxation of the central spin at a fraction of a microsecond . finally , we discuss consequences for the case of nonzero applied fields @xmath23 . according to our theory , the window of overhauser field values that allow fast spin relaxation due to co - flips corresponds to @xmath166 . the average number of quantum dots that happen to have the hyperfine field inside this narrow window at same moment of time follows the gaussian distribution of the hyperfine field @xmath167 . consequently , this number is suppressed when @xmath168 , while the relative contribution to power spectrum from the states having the bias far from the fast relaxation window increases with @xmath23 . hence , the width of the noise power spectrum should quickly decrease with @xmath23 when @xmath168 , in agreement with the experimental data .
|
the problem of how single
central " spins interact with a nuclear spin bath is essential for understanding decoherence and relaxation in many quantum systems , yet is highly nontrivial owing to the many - body couplings involved .
different models yield widely varying timescales and dynamical responses ( exponential , power - law , gaussian , etc ) . here
we detect the small random fluctuations of central spins in thermal equilibrium ( holes in singly - charged ( in , ga)as quantum dots ) to reveal the timescales and functional form of bath - induced spin relaxation .
this spin noise indicates long ( 400 ns ) spin correlation times at zero magnetic field , that increase to @xmath05 @xmath1s as hole - nuclear coupling is suppressed with small ( 100 g ) applied fields .
concomitantly , the noise lineshape evolves from lorentzian to power - law , indicating a crossover from exponential to inverse - log dynamics .
single electron or hole
central " spins confined in iii - v semiconductor quantum dots ( qds ) are promising candidates for solid - state qubits @xcite .
although confinement suppresses momentum - dependent spin relaxation pathways , it enhances the hyperfine coupling between the central spin and the dense spin bath of @xmath010@xmath2 lattice nuclei comprising the qd .
these hyperfine interactions dominate decoherence and spin relaxation at low temperatures . within a qd ensemble
, each central spin feels a different effective ( overhauser ) magnetic field from nuclei , @xmath3 .
trivially , this leads to rapid nanosecond - timescale ensemble dephasing of an initially - oriented ensemble of central spins @xcite . on longer timescales , however , decoherence and relaxation of the central spin within _
each _ qd occurs because @xmath3 evolves slowly in time @xcite . in large applied magnetic fields
@xmath4 , the huge difference between electronic and nuclear zeeman energies suppresses flip - flop interactions between the two species , and @xmath3 evolves primarily via weak dipolar coupling between nuclei . as @xmath4@xmath50 ,
however , central spins can facilitate mutual co - flips " between distant nuclei ( a process involving virtual flips of the central spin ) , which changes @xmath3 more rapidly and accelerates spin relaxation @xcite .
it is precisely in this low - field , intimately - coupled regime where the decoherence of the central spins becomes exceedingly difficult to model theoretically .
distinctly different timescales and a wide range of dynamical response functions ( exponential , gaussian , power - law ) have been postulated , with exact solutions derived only under certain limiting assumptions , such as polarized nuclei .
low - field numerical models with unpolarized nuclei suggest interesting non - exponential dynamics with slow 1/log(@xmath6 ) decays @xcite , highlighting the non - markovian and strongly - correlated evolution of this many - body quantum system .
while groundbreaking experimental qd studies focused on electron central spins , considerable attention has recently shifted to holes @xcite , whose _ p_-type wavefunctions avoid strong fermi - contact hyperfine coupling to the lattice nuclei .
instead , hole - nuclear coupling occurs primarily via weaker dipolar ( anisotropic hyperfine ) interactions , reducing @xmath3 by one order of magnitude @xcite .
optical studies of qd holes based on repeated initialization @xcite or continuous pumping @xcite have revealed long spin relaxation and coherence times in large @xmath4 .
however , studies in the @xmath4@xmath50 limit , where hyperfine interactions are manifest most strongly , have received comparatively little attention @xcite . moreover
, the underlying functional form of the dynamical response of holes in a spin bath has not been explored . as an alternative to conventional pump - probe techniques , the fluctuation - dissipation theorem suggests another route to reveal the dynamical response function of holes , that is based on passively detecting the spectrum of intrinsic random spin fluctuations of holes in thermal equilibrium ( _ i.e. _ , without optical pumping or initialization ) .
this
spin noise spectroscopy " has origins in atomic physics @xcite and nuclear magnetic resonance @xcite , and was subsequently demonstrated for electrons in bulk semiconductors @xcite .
as recently applied to qds @xcite , spin noise revealed the precession and rapid ensemble dephasing of holes in large @xmath4 , due to an inhomogeneous distribution of hole g - factors .
however , these noise - based methods have never been used to detect the intrinsic dynamics of central spins interacting with a nuclear spin bath . of resident holes in ( in , ga)as /
gaas qds impart faraday rotation fluctuations @xmath7 on a linearly - polarized probe laser .
the power spectral density of this spin noise " is measured with a balanced photodiode bridge and digital spectrum analyzer ( see also @xcite ) .
( b ) typical spin noise power spectrum of the resident holes at low temperature ( 5 k ) and zero applied magnetic field ( @xmath4=0 ) .
the 400 khz half - width of the spin noise indicates a long @xmath0400 ns correlation time @xmath8 of the hole spins .
( c ) the same spectrum on a log - log scale .
the noise lineshape closely follows a lorentzian , indicating exponentially - decaying hole spin correlations at @xmath4=0 , in contrast with recent theories @xcite . ]
we therefore use a passive optical technique based on faraday rotation to detect intrinsic hole spin fluctuations in ( in , ga)as qds as @xmath4@xmath50 , where coupling to the nuclear spin bath is most important .
crucially , because spin correlations are revealed in the spectral domain , this approach is well - suited to determine slow dynamical response functions with an accuracy sufficient to achieve a novel understanding of coupled spin - bath systems .
in contrast with theoretical predictions @xcite , exponential dynamics with long ( 400 ns ) correlation timescales are found at @xmath4=0 . using small ( 100 g ) applied fields to suppress a dominant hole - nuclear interaction channel , even longer timescales of order 5 @xmath1s are revealed .
concomitantly , the fluctuation spectrum evolves from lorentzian to power - law , indicating a crossover from exponential to inverse - log spin relaxation .
figure 1(a ) summarizes the spin noise experiment .
a linearly - polarized probe laser is focused through an ensemble of weakly _ p_-type ( in , ga)as / gaas qds , where @xmath010@xmath9 of the qds contain a single hole .
stochastic fluctuations of the ensemble hole spin projection along the sample normal * z * , @xmath10 , impart faraday rotation fluctuations @xmath11 on the probe laser via the usual optical selection rules for positively - charged trions .
the power spectral density of @xmath11 is measured , revealing hole spin fluctuations in the frequency domain . via the wiener - khinchin theorem , this is equivalent to the fourier transform of the hole spin correlation function @xmath12 .
we emphasize that this passive technique probes the _ intrinsic _ fluctuations of the resident hole spins _ while in thermal equilibrium _ with the nuclear spin bath to leading order the detected spins are not optically pumped or excited @xcite , in marked contrast to conventional pump - probe methods .
moreover , since @xmath13 is always zero , only the intrinsic long - timescale dynamics of hole spins are detected in the zero - field limit .
in contrast with prior work @xcite , two crucial differences are implemented here : i ) low - noise stabilized probe lasers that permit accurate and quantitative recovery of the small low - frequency spin noise signals that exist in the zero - field limit , and ii ) very low probe powers ( @xmath14 mw ) and very large spot sizes ( @xmath0100 @xmath1 m ) to ensure that we operate in a regime where the probe laser itself does not influence the measured spin noise signals and lineshapes ( for details , see @xcite ) . using this approach
, we find that the spectral density of this hole spin noise in zero applied field consists of a single , well - defined peak centered at zero frequency ( fig .
1(b ) ) .
considerable information is encoded within this noise peak : its half - width @xmath15 reveals the characteristic timescale @xmath8 of hole spin correlations @xmath12 , and most importantly its detailed lineshape directly reveals _ the functional form _ of the central ( hole ) spin decay a parameter of considerable theoretical interest @xcite . on a log - log scale ( fig .
1(c ) ) , we find this hole spin noise closely follows a lorentzian lineshape over three orders of magnitude in frequency and signal , indicating that temporal correlations of @xmath16 decay exponentially at @xmath4=0 .
note this contrasts directly with recent models predicting non - exponential dynamics in this regime @xcite .
interestingly , the narrow 400 khz width of the spin noise indicates very long hole spin correlation times of @xmath8@xmath0400 ns at @xmath4=0 ( @xmath17 for exponential dynamics ) .
@xmath8 greatly exceeds the @xmath4=0 hole relaxation time measured in ingaas qds @xcite and is consistent with the hole relaxation recently inferred at @xmath4=0 from pump - probe studies of inas qd ensembles @xcite .
@xmath8 also exceeds the dephasing time of holes localized in narrow gaas quantum wells studied by resonant spin amplification @xcite . ) .
in addition to the expected shift of the hole spin noise to the hole larmor frequency ( @xmath18 , with @xmath19@xmath00.15 ) , there remains a finite noise component at zero frequency .
this reveals the presence of the longitudinal ( * z * ) components of the nuclear overhauser magnetic field , @xmath3 .
longitudinal fields , real or effective , necessarily result in spin noise at zero frequency @xcite .
the integrated noise power remains constant .
] the presence of hole - nuclear coupling becomes plainly evident upon applying small _ transverse _ fields @xmath20 ( see fig .
2 ) .
as observed previously @xcite , the noise spectrum largely shifts to higher frequencies as fluctuations @xmath21 are forced to precess about @xmath20 at the hole larmor frequency .
more importantly , however , fig .
2 also reveals that a portion of the zero - frequency spin noise peak remains _ despite _ application of purely transverse fields .
this indicates that the holes do feel effective nuclear magnetic fields in the * z * direction , because longitudinal fields ( real or effective ) necessarily generate noise at zero frequency . in general
, spins in an arbitrary magnetic field generate _ two _ noise peaks : one at high frequency due to trivial spin precession , and one centered at zero frequency due to longitudinal field components .
the former is weak at @xmath4=0 ( for holes ) and is strongly broadened due ensemble averaging , while the latter is not ( for details , see @xcite ) .
it is precisely this zero - frequency noise peak that we study , as it reveals the intrinsic timescales of @xmath12 and the dynamical response function of the hole spin decay .
= 0 , 10 , 20 , 40 , 80 g. the spin noise narrows dramatically , indicating over an order - of - magnitude increase in hole spin correlation time @xmath8 from 400 ns to @xmath05 @xmath1s .
( b ) the same spectra on a log - log scale clearly evolve away from a lorentzian lineshape and more closely approach a 1/frequency power - law , indicating slow inverse - log spin decays in the time domain .
the integrated noise power remains constant ( as expected ) .
( c ) @xmath8 as a function of @xmath22 . ] to explore the extent to which hyperfine interactions limit @xmath8 at @xmath4=0 , we apply small _ longitudinal _ magnetic fields @xmath23 to overwhelm @xmath3 and suppress hole - nuclear coupling .
figures 3(a , b ) shows the spectral density of hole spin noise as @xmath23 increases to 80 g. the width of the noise peak narrows dramatically from 400 khz to less than 40 khz , indicating that @xmath8 increases over ten - fold to nearly 5 @xmath1s .
more importantly , the detailed lineshape of the spin noise evolves away from lorentzian and more closely approaches a @xmath24 power - law decay over the measured frequency range , thereby revealing an apparent crossover from exponential dynamics to much slower 1/log(@xmath6 ) spin decays in the time domain .
higher fields to 300 g do not alter the noise lineshape further .
these data highlight an essential aspect of spin noise measurements : the ability to directly reveal detailed spectral lineshapes to explore slow and non - trivial decay mechanisms .
the data appear to contradict recent theories @xcite predicting slow , inverse - log decays of central _ electron _ spins coupled to nuclear spin baths at @xmath4=0 ( we see exponential decays at @xmath4=0 ) .
whether these theories are fully applicable to holes remains an open question .
we _ do _
observe 1/@xmath25 noise spectra and inverse - log dynamics emerging in _ finite _ ( but small ) @xmath23 however , suggesting at least partial validity of these models .
hole spin decays of order 1 @xmath1s can arise within models of hole - mediated nuclear co - flips , but robust lorentzian noise lineshapes at @xmath4=0 are not expected ( for details , see @xcite ) .
one possibility is that quadrupolar nuclear interactions and the local electric fields in ( in , ga)as qds could rapidly ` stir ' fluctuations of @xmath3 at @xmath4=0 , accelerating hole relaxation via the co - flip mechanism and leading to exponential decays .
two - phonon spin relaxation processes @xcite or hybridization of hole states @xcite have also been proposed , but their influence is not explicitly studied here .
note that these ensemble studies do not distinguish whether , in this regime , every hole in the ensemble exhibits 1/log(@xmath6 ) dynamics , or whether there exists a broad distribution of , _
e.g. _ , exponential timescales in the hole ensemble ( due to varying qd strain and anisotropy ) , whose sum could mimic 1/log(@xmath6 ) dynamics .
ongoing efforts are aimed at elucidating this difference . versus temperature at various applied fields @xmath23 , and as ( b ) @xmath8 versus @xmath23 at various temperatures .
three general regimes can be identified ( see text ) . ]
although not strictly applicable to power - law dynamics , if we continue to infer a characteristic timescale @xmath8 via the noise half - width over the measured frequency range , then figure 3(c ) shows that @xmath8 increases rapidly with increasing @xmath22 , but saturates at 5 @xmath1s when @xmath26 g. these data directly reveal the typical field scale of hole - nuclear coupling , @xmath025 g , or about one - tenth of the electron - nuclear coupling in similar qds @xcite , consistent with previous studies @xcite .
finally , figure 4 shows a comprehensive study of how temperature and @xmath23 determine @xmath8 and the dynamical response function .
the plots identify three general regimes : ( i - yellow ) at high temperatures @xmath27 15 k , @xmath8 falls rapidly , independent of @xmath23 , and noise lineshapes are lorentzian .
this suggests straightforward phonon - assisted hole spin relaxation mediated by spin - orbit coupling , and exponential relaxation .
( ii - green ) low temperatures below 10 k and in @xmath28 , single phonon effects are suppressed , @xmath8 is limited to 400 ns by hole - nuclear coupling , and noise spectra remain unexpectedly lorentzian .
( iii - blue ) finally , at low temperature using small @xmath29 80 g to overwhelm @xmath3 , much longer 5 @xmath1s hole spin correlation times are exposed and noise spectra approach @xmath24 power - laws , indicative of a crossover to very slow 1/log(@xmath6 ) decays .
in summary we have demonstrated that spin noise spectroscopy allows unusually detailed studies , at all relevant timescales , of dynamic response functions in strongly - coupled hole - nuclear spin systems an inherently many - body problem that has eluded concise theoretical treatment .
systematic dependencies on temperature and magnetic field are revealed , serving as a test - bed for theoretical models .
the measurement scheme of passively detecting the intrinsic spin fluctuations represents a kind of ` quantum simulator ' , which is of great relevance to other interacting many - body systems of current interest including microcavity polariton condensates and fractional quantum hall phenomena .
we thank m. m. glazov , .
cywiski , i. uti , j. fabian , and f. anders for helpful discussions .
this work was supported by the los alamos ldrd program .
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| 8,608 | 6,868 |
the implementation of a universal two - qubit gate involving an entanglement operation on two quantum bits represents a necessary step toward the construction of a scalable quantum computer @xcite . intense research on solid state nano - devices during the last decade has established the possibility to combine quantum coherent behavior with the existing integrated - circuit fabrication technology . in particular , based on superconducting technologies , a variety of high - fidelity single qubit gates are nowadays available @xcite , two - qubit logic gates @xcite and violations of bell s inequalities @xcite have been demonstrated , high - fidelity bell states generated @xcite . the recent demonstrations of simple quantum algorithms @xcite and three - qubit entanglement @xcite are further important steps toward a practical quantum computation with superconducting circuits . the requirements for building an elementary quantum processor are however quite demanding on the efficiency of the protocols . this includes both a severe constraint on readout and a sufficient isolation from fluctuations to reduce decoherence effects . solid - state noise sources are often characterized by broad - band and non - monotonic power spectrum . similar noise characteristics have been reported in implementations based on cooper - pair - boxes ( cpb ) @xcite , in persistent current @xcite and phase qubits @xcite . usually , the spectrum of at least one of the noise sources is @xmath0 at low - frequencies @xcite . at the system s eigen - frequencies instead ( @xmath4 ghz ) indirect measurements indicate white or ohmic spectrum @xcite . sometimes spurious resonances of various physical origin have been observed @xcite . at the single - qubit level , the effects of the environmental degrees of freedom responsible for the various parts of the spectrum have been clearly identified leading to a convenient classification in terms of _ quantum noise _ and _ adiabatic noise _ effects @xcite . understanding how these mechanisms affect an entanglement - generating two - qubit gate is a relevant issue not yet investigated and it is the subject of the present article . the picture for a single qubit can be summarised as follows . noise at frequencies of the order of the system s splittings may induce incoherent energy exchanges between qubit and environment ( _ quantum noise _ ) . relaxation processes occur only if the qubit - environment interaction induces spin flips in the qubit eigenbasis , i.e. for transverse noise . weakly - coupled markovian noise can be treated by a born - markov master equation @xcite . it leads to relaxation and decoherence times denoted respectively @xmath1 and @xmath2 in nuclear magnetic resonance ( nmr ) @xcite . for transverse noise they are related by @xmath5 . longitudinal noise does not induce spin flips , but it is responsible for pure dephasing with a decay - time denoted @xmath6 @xcite . in general , both relaxation and pure dephasing processes occur and the resulting decoherence time is @xmath7^{-1}$ ] . since quantum measurements require averages of measurements runs , the main effect of fluctuations with @xmath0 spectrum is defocusing , similarly to inhomogeneous broadening in nmr @xcite . fluctuations with large spectral components at low frequencies can be treated as stochastic processes in the adiabatic approximation ( _ adiabatic noise _ ) . the short - times decay of qubit coherences depends on the symmetry of the qubit - environment coupling hamiltonian . for transverse noise , the time dependence is algebraic @xmath8^{-1/4}$ ] , for longitudinal noise it is exponential quadratic @xmath9 ( `` static - path '' @xcite or `` static - noise '' @xcite approximation ) . the simultaneous presence of adiabatic and quantum noise can be treated in a multi - stage approach @xcite . in simplest cases , the effects of the two noise components add up independently in the coherences time - dependence . defocusing is minimized when noise is transverse with respect to the qubit hamiltonian @xcite . the qubit is said to operate at an `` optimal point '' characterised by algebraic short - times behavior followed by exponential decay on a scale @xmath10 . in the present article we perform a systematic analysis of the effects and interplay of adiabatic and quantum noise on a universal two - qubit gate , extending the multi - stage elimination approach introduced in ref . @xcite . understanding these effects is crucial in the perspective of implementing solid - state complex architectures . our system consists of two coupled qubits each affected by transverse and longitudinal noise with broad - band and non - monotonic spectrum . such a general situation has not being studied in the literature . previous studies concentrated on harmonic baths with monotonic spectrum relying on master equation and/or perturbative redfield approach @xcite , or on numerical methods @xcite , or on formal solutions for selected system observables @xcite . we quantify entanglement via the concurrence @xcite . to compare with bit - wise measurements , single qubit switching probabilities are also evaluated . our analysis is based on approximate analytic results and exact numerical simulations . our main results are : ( i ) the identification of characteristic time scales of entanglement decay due to adiabatic noise , quantum noise and their interplay ; ( ii ) the characterization of relaxation and dephasing for an entanglement operation via the time scales @xmath11 , @xmath12 , @xmath13 and @xmath14 . we point out the dependence of these scales on the symmetry of the hamiltonian describing the interaction between each qubit and the various noise sources ; ( iii ) the demonstration that a universal two - qubit gate can be protected against noise by operating at an `` optimal coupling '' , extending the concept of single - qubit `` optimal point '' . the article is organized as follows . in section 2 we introduce the hamiltonian model for two - qubit entanglement generation in the presence of independent noise sources affecting each unit . in section 3 the general features of the power spectra of these fluctuations , as observed in single qubit experiments , are summarized . the relevant dynamical quantities are introduced and the multi - stage approach to eliminate noise variables and obtain a reduced description of the two - qubit system is illustrated . in sections 4 and 5 we derive separately the effect of quantum noise within a master equation approach and the leading order effect ( quasi - static approximation ) of adiabatic noise . finally , in section 6 we discuss their interplay and introduce the relevant time scales characterizing loss of coherence and entanglement of a universal two - qubit gate in a solid - state environment . results are summarized in and in . in [ appendix - cpb ] the entanglement generating model is derived for capacitive coupled cooper pair boxes ( cpbs ) including fluctuations of all control parameters . in [ appendix : sc ] the effect of selected impurities strongly coupled to the device is pointed out and we speculate on the possibility to extend the `` optimal coupling '' scheme under these conditions . entanglement - generating two qubit gates have been implemented based on different coupling strategies . in the standard idea of gate - based quantum computation , the coupling between the qubits is switched on for a quantum gate operation and switched off after it . the easiest way to realize this scheme is to tune the qubits in resonance with each other for efficient coupling and move them out of resonance for decoupling . employing a fixed coupling scheme two - qubit logic gates have been implemented @xcite and high - fidelity bell states have been generated in capacitive coupled phase qubits @xcite . a different idea is to introduce an extra element between the qubits : an adjustable coupler , which can turn the coupling on and off @xcite . alternatively , two - qubit gates are generated by applying microwave signals of appropriate frequency , amplitude and phase @xcite . the core of the entangling operation of most of the above coupling schemes consists of two resonant qubits with a coupling term transverse with respect to the qubits quantization axis , as modeled by @xmath15 here @xmath16 are pauli matrices and @xmath17 is the identity , in qubit-@xmath18 hilbert space ( @xmath19 ) . in our notation , @xmath16 is the qubit-@xmath18 quantization axis and we put @xmath20 . this model applies in particular to the fixed , capacitive or inductive , coupling of superconducting qubits @xcite , where individual - qubit control allows an effective switch on / off of the interaction . eigenvalues and eigenvectors of eq . are reported in . in order to maintain the single qubit identities , the coupling strength , @xmath21 , must be one - to - two orders of magnitudes smaller than the single qubit level spacing , @xmath22 . thus eigenvalues form a doublets structure , as schematically illustrated in . the hilbert space factorizes in two subspaces spanned by @xmath23 and @xmath24 . ccc @xmath25 & @xmath26 & @xmath27 + 0 & @xmath28 & @xmath29 + 1 & @xmath30 & @xmath31 + 2 & @xmath32 & @xmath33 + 3 & @xmath34 & @xmath35 + a couple of qubits described by ( [ h0 ] ) is suitable to demonstrate entanglement generation . the system prepared in the factorized state @xmath36 freely evolves to the entangled state @xmath37/\sqrt 2 $ ] in a time @xmath38 , realizing a @xmath3 operation . the dynamics takes place inside the @xmath23 subspace , which we name `` swap - subspace '' . the orthogonal subspace will be instead named `` z - subspace '' . and @xmath39 mix and an energy splitting @xmath40 develops between the eigenstates @xmath41 , spanning the swap subspace . product states @xmath42 and @xmath43 weakly mix and split , with @xmath44 . the eigenstates @xmath45 span the z subspace . longitudinal noise in the computational basis is responsible for inter - doublet relaxation processes ( effective transverse inter - doublet ) indicated by red wavy lines ( eq . ) . transverse noise in the computational basis originates incoherent energy exchanges inside each subspace ( effective transverse intra - doublet ) indicated by blue wavy lines ( eqs . , ) . , scaledwidth=80.0% ] fluctuations of the control parameters used for the manipulation of individual qubits couple the circuit to environmental degrees of freedom . we consider the general situation where each qubit is affected both by longitudinal noise ( coupled to @xmath16 ) and by transverse noise ( coupled to @xmath46 ) , as described by the interaction hamiltonian @xmath47 \otimes \mathbb{i}_{2 } - \frac{1}{2 } \mathbb{i}_{1 } \otimes \left [ \hat{x}_{2 } \ , \sigma_{2x}+ \hat{z}_{2 } \ , \sigma_{2z } \right ] \,.\ ] ] here @xmath48 and @xmath49 are collective environmental quantum variables coupled to different qubits degrees of freedom . for instance , in the case of two cpb - based qubits at the charge optimal point @xcite , the charge operator is @xmath46 , and the josephson operator is @xmath16 . fluctuations of the gate charge are described by a transverse coupling term , @xmath50 , and noise in the superconducting phase by the longitudinal term , @xmath51 @xcite ( see [ appendix - cpb ] for the derivation ) , irrespective of the working point . thus , @xmath48 and @xmath49 describe physically different processes only at selected operating points . usually , this is the case at the single qubit s optimal points , see [ appendix - cpb ] . ] . the complete device hamiltonian reads @xmath52 , where @xmath53 denotes the free hamiltonian of all environmental variables . in order to identify relaxation and pure dephasing processes for the coupled qubit setup we project @xmath54 in the 4-dim hilbert space generated by the eigenstates of @xmath55 , @xmath56 , @xmath57 , where we may rewrite @xmath58 \ , + \ , \sqrt{\omega^2 + ( \omega_c/2)^2 } \ , \big [ |3\rangle \langle 3| - | 0\rangle \langle 0| \big ] \\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! \mathcal{h}_\mathrm{i } & = & \frac{1}{2 } ( \hat{x}_{1 } + \hat{x}_{2 } ) \ , \big [ a_- |2\rangle \langle 0| + a_+ |2\rangle \langle 3| + { \rm h.c . } \big ] \ , + \nonumber \\ & + & \frac{1}{2 } ( \hat{x}_{1 } - \hat{x}_{2 } ) \ , \big [ - a_+ |1\rangle \langle 0| + a_- | 1\rangle \langle 3| + { \rm h.c . } \big ] \nonumber\\ & - & \frac{1}{2 } ( \hat{z}_{1 } - \hat{z}_{2 } ) \ , \big [ |1\rangle \langle 2| + |2\rangle \langle 1|\big ] \label{hi - proj}\\ & - & \frac{1}{2 } ( \hat{z}_{1 } + \hat{z}_{2 } ) \ , \big [ \ , \cos \varphi \ , ( |0\rangle \langle 0| - | 3\rangle \langle 3| ) + \sin \varphi \ , ( |0\rangle \langle 3| + | 3\rangle \langle 0|)\ , \big ] \nonumber\end{aligned}\ ] ] where @xmath59/\sqrt 2 $ ] , with @xmath60 . in ( [ hi - proj ] ) we distinguish `` effective longitudinal '' and `` effective transverse '' terms . the first ones are diagonal in the eigenbasis @xmath56 and are responsible for pure dephasing processes . `` effective transverse '' terms instead are off - diagonal and originate both intra- and inter - doublet relaxation processes . specifically we have : + * swap subspace @xmath61 * : longitudinal noise in the computational basis @xmath62 , originates `` effective transverse '' noise in the swap - subspace , i.e. the restriction of @xmath54 to @xmath61 reads @xmath63 - \frac{1}{2}(\hat{z}_{1 } - \hat{z}_{2 } ) \ , \big [ |1\rangle \langle 2| + |2\rangle \langle 1|\big ] \label{h - trunc - swap}\ ] ] note that if both qubits were affected by the same longitudinal noise no effective transverse noise in this subspace would be present . this situation may occur in the presence of totally correlated noise affecting both qubits @xcite . + * z subspace @xmath64 * : longitudinal noise in the computational basis originates both `` effective transverse '' and `` effective longitudinal '' noise in the z - subspace , i.e. the projection of @xmath54 on @xmath65 reads @xmath66 \nonumber \\ & + & ( \hat{z}_{1 } + \hat{z}_{2 } ) \ , \big [ \ , \cos \varphi \ , ( |0\rangle \langle 0| - | 3\rangle \langle 3| ) + \sin \varphi \ , ( |0\rangle \langle 3| + | 3\rangle \langle 0|)\ , \big ] \ , . \label{h - trunc - z } \end{aligned}\ ] ] effective longitudinal and transverse components are modulated via the mixing angle , @xmath67 , similarly to a single qubit with operating point @xmath67 . + * inter - doublet processes * : the only effect of transverse noise in the computational basis is to mix the two subspaces via `` effective transverse '' inter - doublet terms ( fig.[splittings ] ( c ) ) @xmath68 \nonumber \\ \ , & + & \ , \frac{1}{2}(\hat{x}_{1 } - \hat{x}_{2 } ) \ , \big [ - a_+ |1\rangle \langle 0| + a_- | 1\rangle \langle 3| + { \rm h.c . } \big ] \label{h - inter}\end{aligned}\ ] ] these inter - doublet terms are responsible , in particular , for relaxation processes from the swap - subspace to the ground state . we will demonstrate that the resulting `` global '' relaxation time sets the upper limit to all other gate operation times , including other decoherence time scales . the considered entanglement generating operation takes place in the presence of broad - band and non - monotonic noise . in this section we review the multi - stage elimination approach to deal with this problem . the method has been introduced for a single qubit in ref . @xcite where the various approximations have been checked by comparing with the exact numerical solution of the system evolution . this approach allowed to accurately explain the observed dynamics in different experiments @xcite , confirming its appropriateness to deal with the more complex system studied in the present article . the method has been extended to a multi - qubit gate in ref . @xcite , here we summarize the main steps . the multi - stage elimination approach is based on a classification of the noise sources according to their effects and circumvents the problem of a microscopic description of noise sources , which are often non gaussian and non markovian @xcite . in this perspective , the required statistical information on the environment depends on the specific quantum operation performed and on the measurement protocol . even if the statistical characterization of the environment requires going beyond the second order cumulant , often knowledge of the power spectrum of the bath variables , here @xmath69 and @xmath70 denoted generically as @xmath71 , @xmath72 \ , , \label{sealpha}\ ] ] is sufficient . where @xmath73 denotes the equilibrium average with respect to @xmath53 and we assumed stationary processes with @xmath74 . the typical power spectrum reported in various single qubit experiments is sketched in . to make explicit reference to some practical situations , in we summarize the characteristics of transverse and longitudinal noise spectra at low- and high - frequencies reported in a cpb - based circuit @xcite and in the recent experiment on a flux qubit @xcite . the low - frequency part of the spectra of each variable is @xmath0 , @xmath75 . the amplitude @xmath76 can be estimated from spectral measurements . if @xmath77 and @xmath78 denote respectively the low- and high - frequency cut - offs of the @xmath0 region , then @xmath79^{-1}$ ] , where @xmath80 is the variance , @xmath81 . it can be approximated as @xmath82 where @xmath83 is the overall acquisition time for a single data point which results from averaging over several measurement trials . spectrum depends on the specific microscopic source and it is usually not detectable in experiments . in the present article we discuss measurements protocols where the details of the behavior of the power spectrum below @xmath77 and around @xmath78 are not relevant and results depend logarithmically on the ratio @xmath84 . ] the high - frequency power spectrum is usually inferred indirectly from measurements of the qubit relaxation times under various protocols @xcite . from the resulting figures the expected spectra for the corresponding quantum variables @xmath85 at the relevant frequencies , @xmath86 and @xmath87 are derived . experiments tuning the single qubit level spacing @xmath22 reveal either ohmic @xcite or white @xcite power spectrum in the ghz range . evidence of spurious resonances in the spectrum have often been reported , they show up as beatings in time resolved measurements @xcite . ( logarithmic scale ) . measurements of @xmath0 noise usually extend between @xmath88 hz and @xmath89 mhz , whereas the ohmic or white spectrum region typically ranges around @xmath90 ghz @xcite . the region classified as adiabatic noise and quantum noise are indicated . , scaledwidth=50.0% ] ccc & charge - phase qubit & flux qubit + @xmath91 & @xmath92 & @xmath93 + @xmath94 & @xmath95 & @xmath96 + @xmath97 & @xmath98s@xmath99 & @xmath100s@xmath99 + @xmath101 & @xmath102 s@xmath99 & - + in our analysis noise sources belong to three classes . low frequency noise with @xmath0 spectrum is adiabatic since it does not induce transitions , but mainly defocuses the signal , we classify it as _ adiabatic noise_. noise at frequencies of the order of the qubits splittings is responsible for dissipation and ultimately for spontaneous decay , thus it is classified as _ quantum noise_. possible resonances in the spectrum pertain to the class named _ strongly coupled noise_. noise sources belonging to different classes act on different frequency scales and are treated via specific approximation schemes . the distiction can be illustrated as follows . we are interested to a reduced description of the @xmath103-qubit system , expressed by the reduced density matrix ( rdm ) , @xmath104 obtained by tracing out environmental degrees of freedom from the total density matrix @xmath105 , which depends on quantum ( q ) , adiabatic ( a ) and strongly coupled ( sc ) bath variables . since bath s degrees of freedom belonging to different classes of noise act of different time scales we separate in each part of the interaction hamiltonian , @xmath106 ( @xmath107 ) , the contribution from various noise classes as follows @xmath108 adiabatic noise , @xmath109 , is typically correlated on a time scale much longer than the inverse of the qubit s frequencies , @xmath110 , then in the spirit of the born - oppenheimer approximation it can be seen as a classical stochastic field @xmath111 . this approach is valid when the contribution of adiabatic noise to spontaneous decay is negligible , a necessary condition being @xmath112 . this condition is usually satisfied at short enough times , since @xmath113 is substantially different from zero only at frequencies @xmath114 . this fact suggests how to trace - out different noise classes in the appropriate order . the total density matrix parametrically depends on the specific realization of the slow random drives @xmath115 and may be written as @xmath116 . the first step is to trace out quantum noise . in the simplest cases this requires solving a master equation . in a second stage , the average over all the realizations of the stochastic processes , @xmath115 , is performed . this leads to a reduced density matrix for the @xmath103-qubit system plus the strongly coupled degrees of freedom . these latter have to be traced out in a final stage by solving the heisenberg equations of motion , or by approaches suitable to the specific microscopic hamiltonian or interaction . for instance , the dynamics may be solved exactly for some special quantum impurity models at pure dephasing , when impurities are longitudinally coupled to each qubit @xcite . the multi - stage elimination can be formally written as @xmath117 \ , p[\vec e(t ) ] \ ; tr_{\rm q } \big [ \ , \rho^{\rm q , sc}\big(t | \vec e(t)\big ) \ , \big]\right\ } \ , , \ ] ] where @xmath118 and @xmath119 indicate respectively the trace over the q and sc degrees of freedom . in the following sections we apply the multi - stage approach to the two - qubit gate by tracing out first quantum noise and secondly the adiabatic noise . in [ appendix : sc ] the effect of a sc impurity will be analyzed . we focus on the @xmath3 operation @xmath120/\sqrt{2}$ ] which generates by free evolution an entangled state at @xmath121 . as a unambiguous test of entanglement generation and its degradation due to noise , we calculate the evolution of concurrence during the gate operation . introduced in ref . @xcite , the concurrence quantifies the entanglement of a pair of qubits , being @xmath122 for separable states and @xmath123 for maximally entangled states . for the situations discussed in the present article and specified in the following sections , the two - qubits rdm takes the `` x - form '' , i.e. the rdm expressed in the eigenstates basis is non - vanishing only along the diagonal and anti - diagonal at any time . under these conditions , the concurrence can be evaluated in analytic form @xcite . in general , it depends both on diagonal and off - diagonal elements of the rdm . in order to directly compare with experiments where bit - wise readout is performed , we also evaluate the qubit 1 switching probability @xmath124 , i.e. the probability that it will pass to the state @xmath125 starting from the state @xmath126 ; and the probability @xmath127 of finding the qubit 2 in the initial state @xmath128 . in terms of the two qubit rdm in the eigenstate basis they read ( @xmath129 denotes partial trace over qubit @xmath25 of the two - qubit density matrix ) @xmath130 + \rho_{00}(t ) \nonumber \\ & + & \left [ \ , \rho_{33}(t ) - \rho_{00}(t ) \right ] \sin^2 \frac{\varphi}{2 } + { \rm re}[\rho_{12}(t ) ] + { \rm re } [ \rho_{03}(t ) ] \sin \varphi \label{eq : psw1 } \\ p_2(t ) & = & _ 2 \ ! \langle - | { \rm tr}_1 \rho(t ) | - \rangle_2 = \frac{1}{2 } \left [ \ , \rho_{11}(t ) + \rho_{22}(t ) \right ] + \rho_{00}(t ) \nonumber \\ & + & \left [ \ , \rho_{33}(t ) - \rho_{00}(t ) \right ] \sin^2 \frac{\varphi}{2 } - { \rm re}[\rho_{12}(t ) ] + { \rm re}[\rho_{03}(t ) ] \sin \varphi . \label{eq : psw2}\end{aligned}\ ] ] for preparation in the state @xmath131 , in the absence of external fluctuations the above probabilities read @xmath132 the cyclic anti - correlation of the probabilities signals the formation of the entangled state , as reported in various recent experiments @xcite . both the concurrence and the switching probabilities depend on combinations of populations and coherences in the eigenbasis . therefore the relevant time scale to quantify the `` quality factor '' or the efficiency of the universal two - qubit gate is not simply related to a specific rdm element , as a difference with a single qubit gate , where the decay time of the qubit coherence quantifies the quality factor of the operation ( with @xmath2 due partly to relaxation processes ( @xmath133 ) , partly to markovian pure dephasing processes @xmath6 or originated from inhomogeneous broadening ) . in the following we will study the time dependence of coherences and populations , and identify the environmental processes ( transverse , longitudinal , low frequency , high frequency , etc . ) which originate various decay times . based on this analysis we will discuss the resulting effect on the decay of the switching probabilities and of the concurrence . to begin with , we consider the effect of quantum noise replacing in @xmath134 . the system dynamics is obtained by solving the born - markov master equation for the rdm . in the system eigenstate basis and performing the secular approximation ( to be self - consistently checked ) it takes the standard form @xcite : @xmath135 the rates @xmath136 , @xmath137 and the frequency shifts @xmath138 , where @xmath139 , depend respectively on the real and imaginary parts of the lesser and greater green s functions which describe emission ( absorption ) rates to ( from ) the reservoirs @xmath140 in terms of the corresponding power spectra they read @xmath141 where @xmath142 denotes the principal value of the integral . due to the symmetry of @xmath143 , eqs . - , the only independent emission rates , @xmath144 , are @xmath145 , @xmath146 , @xmath147 , @xmath148 , see . symmetric relations hold between the corresponding absorption rates @xmath149 . these processes originate from `` effective transverse '' noise , in particular transverse fluctuations ( @xmath150 ) enter the rates connecting the swap and the z subspaces , whereas longitudinal fluctuations ( @xmath151 ) enter the intra - subspace rates , @xmath147 , @xmath148 , cfr eqs . - . they read @xmath152 \qquad \ , { \rm inter - subspace } \\ \label{gamma20 } \gamma_{20 } = \frac{1}{8 } \ , ( 1-\sin \varphi ) \ , [ c_{x_1}(\omega_{20 } ) + c_{x_2}(\omega_{20 } ) ] \qquad \ , { \rm inter - subspace } \\ \label{gamma30 } \gamma_{30 } = \frac{1}{4 } \ , \sin^2 \varphi \ , [ c_{z_1}(\omega_{30 } ) + c_{z_2}(\omega_{30 } ) ] \qquad \qquad \ , { \rm intra - subspace } \\ \label{gamma21 } \gamma_{21 } = \frac{1}{4 } \ , [ c_{z_1}(\omega_{21 } ) + c_{z_2}(\omega_{21 } ) ] \qquad \qquad \qquad \ ; \ ; \;{\rm intra - subspace } \end{array } \label{eq : ratesgeneral}\end{aligned}\ ] ] absorption rates have the same form with @xmath153 replaced by @xmath154 . the imaginary parts of the corresponding terms take similar forms . they enter the frequency shifts as reported in [ appendix : shifts ] . in the secular approximation , the swap and z coherences decay exponentially with rates @xmath155 \label{gamma12tilde}\\ \widetilde \gamma_{30 } & = & \frac{1}{2 } \ , [ \gamma_{10 } + \gamma_{01 } + \gamma_{20 } + \gamma_{02 } + \gamma_{30 } + \gamma_{03 } ] + \gamma^*_z\ , , \end{aligned}\ ] ] both inter - subspace and intra - subspace rates enter the decay of the swap and z coherences . note that the decay rate of the coherence @xmath156 is only originated from dissipative processes ( intra- or inter- effective transverse ) since no pure dephasing processes ( effective longitudinal noise ) inside the swap subspace exist , cfr eq . . on the contrary , the coherences in the z subspace also decay because of the effective longitudinal terms @xmath157 , which originate @xmath158 $ ] . this pure dephasing factor adds up to a decoherence rate due to intra - subspace effective transverse noise having the characteristic form @xmath159^{-1 } = ( \gamma_{30}+ \gamma_{03})/2 $ ] , as implied by , and to inter - doublet relaxation rates . equations for the populations do not decouple even in the secular limit . general solutions are quite cumbersome , so here we report expressions in the small temperature limit with respect to the uncoupled qubits splittings , @xmath160 . in this regime , if the system is initially prepared in the state @xmath161 , level @xmath162 is not populated , @xmath163 , and the z - coherences vanish , @xmath164 . the remaining populations are conveniently expressed in terms of the escape rates from levels @xmath88 and @xmath103 @xmath165 which enter the evolution of the populations in the following combinations @xmath166 for the chosen initial conditions the populations read @xmath167 \ , e^{-\gamma_k t } \label{population1 } \\ \rho_{22}(t ) & = & \frac{1}{2(\gamma_- - \gamma_+ ) } \sum_{k=\pm } k \left [ \gamma_{12 } - \gamma_k + \gamma_1^{\rm e } \right ] \ , e^{-\gamma_k t } \label{population2 } \\ \label{population0 } \rho_{00}(t ) & = & 1 -(\rho_{11}(t ) + \rho_{22}(t ) ) \ , .\end{aligned}\ ] ] note that , since @xmath168 , thermal excitation processes internal to the swap subspace , expressed via the absorption rate @xmath169 , can not be neglected and @xmath170 . on the contrary , inter - doublet thermal excitation processes are exponentially suppressed with respect to the corresponding decay rates , @xmath171 , with @xmath172 . thus the swap coherences decay rate is approximately half the sum of the escape rates ( [ eq : escape ] ) @xmath173 in order to observe generation of entanglement inside the swap subspace it is necessary that relaxation processes to the ground state take place on a sufficiently long time scale . this is guaranteed when inter- and intra - subspace rates satisfy the condition @xmath174 , which requires that the spectra of the originally transverse and longitudinal fluctuations are @xmath175 ( from ) . in this regime , the swap decoherence rate is due to effective transverse processes internal to the subspace @xmath176 \ , , \ ] ] and the scales entering the populations take the approximate forms @xmath177 \qquad \qquad { \rm relaxation \ , to \ , the \ , ground \,state } \\ \label{gammaminus } \gamma_- & \approx & \gamma_{12 } + \gamma_{21 } \qquad \qquad \quad \ , { \rm relaxation \ , inside \ , the \ , swap \ , subspace}\end{aligned}\ ] ] with @xmath178 . therefore , the time scales resulting from quantum noise , considering that @xmath168 , when @xmath175 , are * `` global '' relaxation time to the ground state , analogous to the single qubit @xmath1 : its order of magnitude is the spectrum of _ transverse fluctuations _ in the computational basis at frequency @xmath22 @xmath179 where the approximate form comes from eq . ( [ eq : ratesgeneral ] ) , where @xmath180 . * relaxation / decoherence times inside the swap subspace : they are due to `` effective transverse '' fluctuations inside this subspace , physically originated from _ longitudinal noise _ on each qubit at frequency @xmath21 . since there is no effective longitudinal noise in the swap subspace , relaxation and dephasing times are related by the typical relation @xmath181 where @xmath182 we now briefly comment on the validity of the secular approximation . it consists in separating the evolutions of elements @xmath183 and @xmath184 provided that @xmath185 , where @xmath186 denotes the typical evolution time scale of the system @xcite . in the present case this condition is fulfilled if @xmath187 . this constraint , on the other side , _ needs _ to be satisfied in order to observe generation of entanglement in the presence of quantum noise , as expressed for instance from anti - correlation of the probabilities ( [ eq : psw1 ] ) and ( [ eq : psw2 ] ) . thus it can be regarded as a _ necessary condition _ , whose validity has to be checked case by case and requires ( from ) @xmath188 in conclusion , entanglement generation in the presence of transverse and longitudinal quantum noise is guaranteed when @xmath189 under these conditions , the `` global '' relaxation time and the swap relaxation and decoherence times are given respectively by eqs . and , . we note that , as a limiting case , the @xmath3 operation can be realized also releasing the condition @xmath178 , provided both rates are much smaller than the coupling strength @xmath21 , and @xmath190 . for instance , when @xmath191 in eq . @xmath192 and the rates @xmath193 are still given by eqs . , leading to @xmath194 and @xmath195 . the swap dephasing time in this case also depends on trasverse fluctuations , @xmath196 $ ] . in the secular approximation , for preparation at @xmath197 in @xmath198 and for @xmath160 , the probabilities and take the simpler form @xmath199 + { \rm re}[\rho_{12}(t ) ] + \cos^2\left ( \frac{\varphi}{2}\right ) \\ \label{switch2 } p_2(t ) = & - & \frac{1}{2 } \ , \cos \varphi \ , \left [ \rho_{11}(t ) + \rho_{22}(t ) \right ] - { \rm re}[\rho_{12}(t ) ] + \cos^2\left ( \frac{\varphi}{2}\right ) \ , .\end{aligned}\ ] ] where @xmath200 and @xmath201 are given by eqs . , and @xmath202 anti - correlation of the above probabilities directly follows from the coherence @xmath156 entering with different signs in @xmath124 and in @xmath127 . in order to check the efficiency of the gate we will therefore consider only the qubit 1 switching probability . neglecting the frequency shift of @xmath203 ( see [ appendix : shifts ] ) , @xmath124 can be approximated as @xmath204 \ , + \ , \cos^2\left ( \frac{\varphi}{2}\right ) \ , .\ ] ] here we explicitly see that , in order to perform the @xmath3 operation , it is necessary that @xmath205 . efficient entanglement generation is guaranteed when @xmath206 , i.e. when transverse and longitudinal quantum noise are such that @xmath175 . under this condition , the efficiency of the gate is limited by @xmath13 , i. e. by longitudinal noise in the computational basis , @xmath207 , which is responsible for the short - times behavior . decay towards the equilibrium value @xmath208 occurs in a time of the order of @xmath11 , due to transverse noise in the computational basis . ccc & qubit 1 & qubit 2 + @xmath91 & @xmath209 & @xmath210 + @xmath94 & @xmath211 & @xmath212 + @xmath213 & @xmath214s@xmath99 & @xmath214s@xmath99 + @xmath215 & @xmath216 s@xmath99 & @xmath217s@xmath99 + : the relevant rates can be estimated in this specific case and are reported in [ appendix - cpb ] . for the charge - phase two - ports architecture , control is via the gate voltage , @xmath218 and magnetic flux dependent phase , @xmath219 , entering the josepshon energy . the single qubit optimal point is at @xmath220 , @xmath221 . the resonant condition between the two qubits with a capacitive coupling is achieved by displacing one of the two qubits from the `` double '' optimal point . in order to limit the sensitivity to charge noise , resonance is achieved tuning @xmath222 . the noise characteristics are reported in , where we note that @xmath175 . because of the operating conditions , absorption ( and emission ) rates due to phase noise on qubit 2 dominate over rates due both to charge noise and to phase noise on qubit 1 . relaxation of the populations takes place on a scale @xmath223 due to transverse ( charge ) noise on both qubits , whereas the swap coherence decay rate is dominated by longitudinal ( phase ) noise on qubit 2 , @xmath224 . the efficiency of the @xmath3 gate is mainly limited by phase noise on qubit 2 , which gives @xmath225 ns . we note that , under these conditions , @xmath13 is comparable with the decoherence time of qubit 2 , @xmath226 \approx 2/s_{z2}(0 ) \approx 2/s_{z2}(\omega_c)$ ] . the phase - dominated behavior is followed by a slower charge - dominated decay towards the equilibrium value @xmath227 . these features are illustrated in ( left ) where for comparison the real part of the swap coherence is shown in the inset . we note that the contribution of high - frequency charge noise to the efficiency of the @xmath3 gate is relatively small . indeed if only polarization fluctuations were present , we could approximate @xmath228 \ , e^{- t / t_2^{swap}}\ , + \ , \cos^2\left ( \frac{\varphi}{2}\right ) \ ] ] and @xmath229 @xmath230s . this situation is illustrated in ( right ) . note that for the estimated noise figures , the condition for the secular approximation , @xmath231 , is satisfied . and @xmath215 reported in . the long - time decay is due to charge noise entering the populations of levels @xmath88 and @xmath103 . inset : the real part of the swap coherence which is responsible for the short - time behavior of the switching probability due to phase noise on qubit @xmath103 . right panel : switching probability of qubit 1 in the presence of charge noise with white spectrum , @xmath232s@xmath99 . , title="fig:",scaledwidth=40.0% ] and @xmath215 reported in . the long - time decay is due to charge noise entering the populations of levels @xmath88 and @xmath103 . inset : the real part of the swap coherence which is responsible for the short - time behavior of the switching probability due to phase noise on qubit @xmath103 . right panel : switching probability of qubit 1 in the presence of charge noise with white spectrum , @xmath232s@xmath99 . , title="fig:",scaledwidth=40.0% ] for the considered initial condition , the rdm takes the `` x - form '' @xcite and the concurrence is given by @xmath233)^2 } -|\sin \varphi| \rho_{00}(t ) , \nonumber \\ & & |\sin \varphi| \rho_{00}(t ) - \sqrt { ( \rho_{11}(t ) + \rho_{22}(t ) ) ^2 - ( 2{\rm re } [ \rho_{12}(t)])^2 } \big ] \ , . \label{concurrence - general}\end{aligned}\ ] ] at times shorter than the global relaxation time , @xmath11 , the ground state is almost unpopulated , @xmath234 , and @xmath235 is given by the second term in @xmath236^{1/2 } - |\sin \varphi | \ , ( 1- e^{-t / t_r } ) \ , , \ ] ] for @xmath237 , the concurrence is approximately given by the swap coherence @xmath238 . like the switching probabilities , the concurrence evolves with the swap coherence decay time , @xmath13 , due to originally longitudinal noise . : the concurrence for the charge - phase 2-qubit gate is illustrated in . we note that the long - time behavior is instead due to populations relaxation to the ground state and @xmath235 is given by the third term in @xmath239^{1/2 } \nonumber \\ & \approx & - |\sin \varphi| \rho_{00}(t ) \to |\sin \varphi |\end{aligned}\ ] ] the finite asymptotic value @xmath240 , reflects the entangled thermalized state . because of the interaction between the two qubits the phenomenon of entanglement sudden death does not take place @xcite . and for quantum noise values @xmath213 and @xmath215 reported in . inset : at short times @xmath241 ( diamonds ) . , scaledwidth=50.0% ] let s consider now the effect of low frequency fluctuations , replacing in @xmath242 . in the adiabatic and longitudinal approximation @xcite populations do not evolve and the system dynamics is related to instantaneous eigenvalues , @xmath243 , which depend on the noise realization , @xmath115 . they enter the coherences in the eigenbasis of @xmath244 in the form @xmath245 \ , p[\vec e(s ) ] \ , e^{-i \int_0^t ds \ , \omega_{ij}(\vec e(s ) ) } \ , \label{path - int}\ ] ] where the probability of the realization @xmath246 , @xmath247 $ ] , also depends on the measurement protocol . a standard approximation of the path - integral consists in replacing @xmath248 with statistically distributed values @xmath249 at each repetition of the measurement protocol . the `` static - path approximation '' ( spa ) @xcite or `` static - noise '' approximation @xcite gives the leading order effect of low - frequency fluctuations in repeated measurements . in the spa , the level splittings @xmath250 are random variables , with standard deviation @xmath251 , where @xmath252 . the coherences reduce to ordinary integrals @xmath253 where the probability density , in relevant cases , can be taken of gaussian form , @xmath254 with @xmath255/\sqrt{2 \pi } \sigma_{e_\alpha}$ ] @xcite . the splittings @xmath250 come both from `` effective longitudinal '' and from `` effective transverse '' terms in ( [ hi - proj ] ) . this is analogous to a single qubit , where longitudinal noise gives the leading order linear terms and transverse noise is responsible for second order terms which dominate at the optimal point , where the first order longitudinal contributions vanish @xcite . here , the z - splitting @xmath256 has a linear contribution due to the effective longitudinal noise @xmath257 in . the swap splitting @xmath258 instead , in the absence of leading - order effective - longitudinal intra - doublet terms in , comes from higher order contributions due to `` effective transverse '' noise . evaluating them requires considering the complete hilbert space of the coupled qubit system ( inter - doublet processes included ) . the systematic approach to obtain these contributions consists in treating in perturbation theory effective transverse terms in @xmath259 , where , in the adiabatic approximation , @xmath260 and @xmath261 are replaced by classical stochastic fields @xmath262 and @xmath263 . we obtain @xmath264 where @xmath60 . the swap - splitting has been evaluated up to @xmath265 order @xcite , whereas the z splitting expansion is considered up to second order since the fourth - order terms are much smaller , scaling with @xmath266 . the z - splitting consists of a linear term due to the effective longitudinal noise , which dominates with respect to the quadratic term due to effective transverse intra - subspace noise ( scaling with @xmath267 ) . transverse inter - doublet noise gives an additional second order contribution . performing the average ( including in the @xmath268 and @xmath265 order terms in only the contributions @xmath269 ) we obtain the swap coherence in the spa @xmath270 \label{eq : spa - swap}\ ] ] where @xmath271 and @xmath272 $ ] is the k - bessel function of order zero @xcite . in the absence of mixed terms in the expansion , different contributions to the z - coherence factorize and lead to @xmath273 here we assumed the same variance for the transverse noise components , @xmath274 . instead we maintained @xmath275 and @xmath276 distinct , considering that the two qubits may operate at different working points ( [ appendix - cpb ] ) . in the exponential factor comes from linear terms in due to effective longitudinal noise in the z - subspace . the algebraic decay instead comes from quadratic terms in the expansion of @xmath277 , due to effective transverse inter - doublet processes . + we remark that the applicability of the gaussian approximation to the fields @xmath278 depends on the relation between each @xmath278 and the fluctuations of the system s physical parameters . for instance , for a charge - phase @xmath3 gate the resonant condition occurs for @xmath279 , @xmath280 ( [ appendix - cpb ] ) . thus it is @xmath281 , where @xmath282 ( not @xmath283 ) is reasonably assumed gaussian distributed . this would result in a modification of the terms @xmath284 in eqs . and . for instance , the integral over @xmath283 in @xmath285 would reduce to @xmath286 , instead of @xmath287 . the quantitative effect in @xmath288 ( and similarly in @xmath289 ) is however negligible , because of the smallness of longitudinal noise at the optimal point , @xmath290 ( see ) . validity regime of the spa : the adiabatic approximation is tenable for times shorter than the relaxation times , for this problem this condition requires that @xmath291 . the static approximation is exact for times smaller that @xmath292 ( in case of a sharp high frequency cut - off ) . we verified that it is a good approximation also for times @xmath293 if @xmath294 and the @xmath0 spectrum is originated from an ensemble of bistable fluctuators with switching rates @xmath295 $ ] , leading to a @xmath296 decay above @xmath78 ( numerical simulations and analytic first correction to the spa @xcite ) . exploiting the band structure of coupled nano - devices it is possible to reduce the influence of @xmath0 fluctuations @xcite . the basic idea of `` optimal tuning '' is to fix control parameters to values which minimize the variance of the splittings @xmath250 , @xmath297 . this naturally results in a enhancement of the decay time of the corresponding coherence due to inhomogeneous broadening , i. e. of @xmath298 . this is simply understood considering the short times expansion of @xmath299 @xmath300 the short - times decay of the coherence in the spa is therefore given by @xmath301 resulting in reduced defocusing for minimal variance @xmath302 . for a single - qubit gate , the `` optimal tuning '' idea immediately leads to the well - known `` magic point '' . in fact , if @xmath303 is monotonic in a region @xmath304 , we can approximate @xmath305 ^ 2 \sigma_{e_\alpha}^2 $ ] , thus the variance attains a minimum for vanishing differential dispersion . for the charge - phase two - port architecture , control is via gate voltage , @xmath306 , and magnetic - flux dependent phase @xmath219 , thus @xmath278 corresponds to the fluctuations @xmath307 and the optimal point , @xmath308 , is at the a saddle point of the energy bands @xcite . when bands are non - monotonic in the control parameters , minimization of defocusing necessarily requires their tuning to values depending on the noise variances . for a multi - qubit gate , the optimal choice has to be done considering the most relevant coherence for the considered operation . here we show how this program applies to the coherence in the swap subspace and partly to @xmath288 . . panels ( a ) and ( b ) : @xmath309 from numerical diagonalization of @xmath143 . in panel ( b ) a zoom around the origin highlights the interplay of @xmath310 and @xmath265 order terms of the expansion , the barrier height is @xmath311 . panel ( c ) : swap exact splitting ( blue ) , expansion ( [ eq : splittingswap ] ) for @xmath312 , @xmath313 ( dashed ) , @xmath310 order expansion ( dash - dotted green ) , @xmath314 splitting ( red ) from and single qubit dispersion ( diamonds ) . panel ( d ) : longitudinal dispersions in the swap ( blue ) and z ( red ) subspaces . , title="fig:",width=264 ] . panels ( a ) and ( b ) : @xmath309 from numerical diagonalization of @xmath143 . in panel ( b ) a zoom around the origin highlights the interplay of @xmath310 and @xmath265 order terms of the expansion , the barrier height is @xmath311 . panel ( c ) : swap exact splitting ( blue ) , expansion ( [ eq : splittingswap ] ) for @xmath312 , @xmath313 ( dashed ) , @xmath310 order expansion ( dash - dotted green ) , @xmath314 splitting ( red ) from and single qubit dispersion ( diamonds ) . panel ( d ) : longitudinal dispersions in the swap ( blue ) and z ( red ) subspaces . , title="fig:",width=283 ] . panels ( a ) and ( b ) : @xmath309 from numerical diagonalization of @xmath143 . in panel ( b ) a zoom around the origin highlights the interplay of @xmath310 and @xmath265 order terms of the expansion , the barrier height is @xmath311 . panel ( c ) : swap exact splitting ( blue ) , expansion ( [ eq : splittingswap ] ) for @xmath312 , @xmath313 ( dashed ) , @xmath310 order expansion ( dash - dotted green ) , @xmath314 splitting ( red ) from and single qubit dispersion ( diamonds ) . panel ( d ) : longitudinal dispersions in the swap ( blue ) and z ( red ) subspaces . , title="fig:",width=264 ] . panels ( a ) and ( b ) : @xmath309 from numerical diagonalization of @xmath143 . in panel ( b ) a zoom around the origin highlights the interplay of @xmath310 and @xmath265 order terms of the expansion , the barrier height is @xmath311 . panel ( c ) : swap exact splitting ( blue ) , expansion ( [ eq : splittingswap ] ) for @xmath312 , @xmath313 ( dashed ) , @xmath310 order expansion ( dash - dotted green ) , @xmath314 splitting ( red ) from and single qubit dispersion ( diamonds ) . panel ( d ) : longitudinal dispersions in the swap ( blue ) and z ( red ) subspaces . , title="fig:",width=264 ] a key feature is that the swap splitting , @xmath203 in eq . is non - monotonic in the small coupling @xmath40 . this is due to the fact that effective transverse fluctuations originate both from longitudinal and transverse noise , and give rise respectively to intra - swap and inter - doublet transitions , see , . for instance , second order corrections to @xmath315 from effective transverse intra - doublet processes are @xmath316 , from effective transverse inter - doublet transitions are @xmath317 . non - monotonicity in @xmath21 results in a competition between @xmath310 and @xmath265 order @xmath262-terms in and in non - monotonic band structure , ( panels ( a ) and ( b ) ) . because of this subtle feature , identification of the best operating condition necessarily requires consideration of the noise characteristics . indeed the _ optimal coupling _ which minimizes the swap variance @xmath318 + \frac{\sigma_{z_2}^4}{2 } \right \ } \label{eq : variance}\ ] ] is given by @xmath319 where we assumed @xmath290 . the effectiveness of the optimal coupling choice has been discussed in details in ref . the advantage of operating at optimal coupling with respect to a generic @xmath21 can be parametrized by the error of the gate at time @xmath320 when system should be in the entangled state @xmath321/\sqrt 2 $ ] , @xmath322 . as shown in for two cpb - qubits , in the presence of moderate amplitude transverse ( charge ) noise , at the optimal coupling the error can be reduced even one order of magnitude with respect to a generic coupling . ccc @xmath323 & @xmath324 & @xmath325 + @xmath326 & @xmath327 & @xmath328 + @xmath329 & @xmath330 & @xmath331 + @xmath332 & @xmath333 & @xmath334 + @xmath335 & @xmath336 & @xmath337 + @xmath338 & @xmath339 & @xmath340 + @xmath341 & @xmath342 & @xmath343 + the splitting in the z - subspace , on the contrary , is monotonic both in @xmath262 and in @xmath263 , similarly to a single qubit operating at @xmath344 and at the two different @xmath345 , , panels ( c ) and ( d ) . the most relevant contribution to z - variance is given by the effective longitudinal noise @xmath346 which can be reduced only by increasing the qubit s coupling strength @xmath21 , which is however limited by single qubit splittings @xmath22 . therefore , no special optimal point exists if the two - qubit operation involves the dynamics inside the z - subspace . a comparison of the evolution of the coherences in the two subspaces and of the single qubit - coherence is shown in for two cpb - based qubits . in order to keep the advantage of optimal tuning in the swap - subspace , the two - qubit operation should not involve the z - subspace . . the dashed red line is the coherence in the swap - subspace for optimal coupling , @xmath347 . the z coherence does not appreciably change by changing @xmath21 and decays similarly to a single qubit at the double optimal point . noise characteristics are reported in . , scaledwidth=50.0% ] finally , we remark that the above results only follow considering the 4-level spectrum of the coupled devices . in fact , because of mixing between subspaces , limiting the analysis to noise projections inside the subspaces would miss a relevant part of the defocusing processes coming from originally transverse fluctuations @xmath348 , see eqs . , . this is clear if we consider the evolution of the swap coherences truncating the system hilbert space to the swap subspace . since the reduced hamiltonian is , effective transverse intra - doublet noise would lead to quadratic corrections to the swap - splitting . in fact , treating in perturbation theory @xmath263 up to second order , we would get @xmath349 and the swap coherence would decay algebraically , as it is typically originated from `` effective transverse '' low frequency noise @xmath350 under this approximation , reduction of defocusing could only be achieved by increasing the coupling strength . similarly , reducing the analysis to the z - subspace , cfr eq . , would miss the quadratic dependence of the z - splitting due to transverse noise . in the multi - stage elimination approach , where quantum noise is traced out first and adiabatic noise is retained as a classical stochastic drive , we have to replace in eqs . - and in @xmath351 , @xmath352 with @xmath353 and perform the path - integral over the realizations of @xmath115 . note that the dependence on adiabatic noise enters also the rates , which should be averaged . if the dependence of the power spectra on the splittings is sufficiently smooth , it can be neglected and the effect of low frequency noise reduces to averaging phase factors in the coherences and @xmath263 enter parametrically also in the frequency shifts resulting from the solution of the master equation . here we neglect this dependence , see [ appendix : shifts ] . ] . thus , provided that the dependence on @xmath354 of @xmath355 and @xmath356 can be neglected , the effects of quantum and adiabatic noise can be treated independently and lead to @xmath357 this result is confirmed by numerical solution of the stochastic schrdinger equation for classical fluctuations leading to dynamical @xmath0 noise ( @xmath358 hz and @xmath359 hz ) . the evolution of populations is only due to quantum noise ( in the adiabatic approximation they do not evolve ) , thus they are given by eqs . - and the switching probabilities are given by eqs . , ( [ switch2 ] ) where @xmath183 are replaced by and . these matrix elements enter also the concurrence which reads , for times @xmath237 , @xmath360)^2 } - |\sin \varphi| \rho_{00}(t)$ ] . the relevant time scales characterizing the efficiency of the @xmath3 operation depend both on the swap coherence and on the populations of the first three levels . due to the interplay of adiabatic and quantum noise , the time dependence is not a superposition of exponentials . we can distinguish two time regions : a asymptotic long - time regime and a intermediate - to - short time regime . cc time scale & physical origin + relaxation to the ground state & transverse noise + @xmath361 & at frequency @xmath22 + swap relaxation time & longitudinal noise + @xmath362 & at frequency @xmath203 + swap decoherence time & longitudinal noise + @xmath363 & at frequency @xmath203 + swap dephasing time @xmath14 & low frequency noise + @xmath364 & ( longitudinal and transverse ) + swap total decoherence time & if @xmath365 i. e. + @xmath366^{-1}$ ] & @xmath367 + the asymptotic behavior is entirely due to populations relaxation to the ground state , it is exponential and takes place with the `` global '' relaxation time @xmath368 , resulting from `` effective transverse '' inter - doublet processes , whose order of magnitude is the spectrum of transverse fluctuations at frequencies of order @xmath22 . in order to avoid leakage from the swap - subspace , any two - qubit operation has to take place on a time much shorter than @xmath11 . this constraint also applies to the swap decoherence times . + the intermediate - to - short time behavior , @xmath237 , gives more relevant information on the gate performance . in this time regime , relevant quantities are populations and coherences in the swap subspace . we distinguish a intermediate time regime , characterized by @xmath369 and @xmath370 due to _ effective transverse _ quantum noise inside this subspace , physically originated from longitudinal noise on each qubit . adiabatic noise leads to additional decay of the swap coherence , @xmath371 . the resulting defocusing is analogous to a `` pure dephasing '' process , we may name the typical time scale @xmath372 , defined as the time at which @xmath373 . the time @xmath372 is found by numerical inversion of . the _ intermediate - time _ behavior of the two - qubit gate is characterized by @xmath374 and @xmath375 , analogously to a two - state system . these time scales and the responsible processes are summarized in . switching probability @xmath124 ( red ) and probability @xmath127 ( blue ) to find qubit @xmath103 in the initial state @xmath376 in the presence of @xmath0 and white noise for @xmath377 and for optimal coupling @xmath378 ( gray ) . to emphasize the robustness of the optimal coupling choice here we consider @xmath0 charge noise of large amplitude , @xmath379 . the remaining noise figures are those reported in . left panel : plot of @xmath124 showing the exponential short - time behavior at @xmath380 and the algebraic decay for generic coupling . right panel : @xmath124 and @xmath381 anti - phase oscillations for @xmath380 ( main ) , @xmath382 ( inset ) . , title="fig:",scaledwidth=45.0% ] switching probability @xmath124 ( red ) and probability @xmath127 ( blue ) to find qubit @xmath103 in the initial state @xmath376 in the presence of @xmath0 and white noise for @xmath377 and for optimal coupling @xmath378 ( gray ) . to emphasize the robustness of the optimal coupling choice here we consider @xmath0 charge noise of large amplitude , @xmath379 . the remaining noise figures are those reported in . left panel : plot of @xmath124 showing the exponential short - time behavior at @xmath380 and the algebraic decay for generic coupling . right panel : @xmath124 and @xmath381 anti - phase oscillations for @xmath380 ( main ) , @xmath382 ( inset ) . , title="fig:",scaledwidth=41.0% ] on the other side , in a quantum information perspective , it is important to estimate the time behavior ( of concurrence or of switching probabilities ) at times of the order of @xmath121 , the first moment in which the entangled state is reached by free evolution . in order to generate entanglement within the swap subspace it is necessary that @xmath383 . it is therefore relevant to estimate the system behavior at short - times @xmath384 . both for the concurrence and the switching probability , the swap - coherence rules the short - time limit which is approximately latexmath:[\[\label{short - limit } the leading contribution , linear or quadratic , does not only depend on the noise amplitude but also on the operating point . in fact @xmath21 enters the swap - splitting variance @xmath386 and , in principle , also the swap decoherence time due to quantum noise @xmath387 . this is expected to be a smooth dependence . for the present considerations we assume a white spectrum in the relevant frequency range . for optimal coupling the short - time behavior is linear ( since the effect of low frequency noise is considerably reduced ) whereas for a different coupling strength the behavior is typically algebraic ( see ) . the short - time expansion is valid at @xmath388 if @xmath383 . in we consider two illustrative cases . for the expected noise characteristics as reported in , for couplings @xmath389 we have @xmath390 . thus the short - time expansion is valid up to @xmath388 . note that for the optimal coupling @xmath391 is about one order of magnitude larger than @xmath386 ( left panel ) , thus @xmath392 . for a larger amplitude of @xmath0 transverse noise , @xmath393 , we have @xmath394 ( right panel ) . also in this case the short - time expansion holds up to @xmath388 . in this case however even for optimal coupling the linear and quadratic terms are comparable . the various possible short - times behaviors and validity regimes are summarized in . ( red ) and of @xmath395 ( blue ) as a function of @xmath323 ( logarithmic scale ) for @xmath396 ( left panel ) and @xmath397 ( right panel ) . the dashed line marks the optimal coupling eq . . , title="fig:",scaledwidth=45.0% ] ( red ) and of @xmath395 ( blue ) as a function of @xmath323 ( logarithmic scale ) for @xmath396 ( left panel ) and @xmath397 ( right panel ) . the dashed line marks the optimal coupling eq . . , title="fig:",scaledwidth=45.0% ] ccc short times & validity & expansion + expansion & conditions & up to @xmath388 if + @xmath398 & optimal coupling and @xmath399 & @xmath400 + @xmath401 & generic coupling or @xmath402 & @xmath403 + in the present article we have identified the relevant decoherence times of an entangling operation due to the different decoherence channels originated from broad - band and non - monotonic noise affecting independently two qubits . results depend on the interplay of noise at low and at high frequencies ( with respect to both the single qubit splittings , @xmath22 , and the qubits coupling strength , @xmath21 ) and on the symmetries of the qubit - environment coupling hamiltonian . in addition , `` optimal '' operating conditions for the universal two - qubit gate have been identified . even if the relevant dynamics is within the swap subspace , the important time scales ( both due to quantum and to adiabatic noise ) can only be predicted considering the whole multilevel nature of the coupled systems . in particular , relaxation processes from the swap subspace to the ground state only depend on _ transverse _ noise at frequency @xmath22 . decoherence processes internal to the swap subspace are instead originated from _ longitudinal _ noise at frequency @xmath21 . this apparently counter - intuitive result simply follows from the symmetries of the hamiltonian expressed in the eigenbasis of coupled qubits , cfr eqs . - . as a consequence , the conditions to generate entangled states within the swap - subspace involve the spectra of transverse and longitudinal fluctuations at two different frequencies . an efficient @xmath3 operation can be realized when @xmath404 . the long - to - intermediate time behavior and the relevant decoherence times are summarized in . defocusing processes instead originate both from transverse and longitudinal adiabatic noise . remarkably , consideration of the coupled systems band structure allows identification of operating conditions of reduced sensitivity to low frequency fluctuations . an `` optimal coupling '' can be identified where defocusing is minimized . the error at the first @xmath3 operation at optimal coupling can be reduced of a factor of @xmath405-to-@xmath406 with respect to operating at generic coupling strength . the possibility of optimal tuning is ultimately due to the absence of effective longitudinal noise in the swap - subspace and to the interplay of effective transverse intra- and inter - doublet fluctuations . the orthogonal z - subspace instead , because of the presence of effective longitudinal noise , turns out to be much more sensitive to low - frequency noise which can not be limited by properly choosing system parameters . therefore , populating this subspace may severely limit the gate efficiency . the short - times behavior of the relevant dynamical quantities crucially depends on adiabatic noise and its interplay with quantum noise . the dependence turns from quadratic to linear depending on the largest component , parametrized by the swap - splitting variance resulting from adiabatic noise , @xmath386 , and the swap - decoherence rate due to quantum noise @xmath407^{-1}$ ] , as summarized in . the considerable protection from 1/f noise achievable in the swap - subspace for optimal coupling may suggest to use this subspace to encode a single quantum bit , similarly to a decoherence - free - subspace @xcite . such an encoding would be convenient if the swap decoherence time was about two orders of magnitude larger than the qubit decoherence time @xmath2 . in fact , because of the smaller swap oscillation frequency , @xmath408 , the quality factor of a single qubit rotation within this subspace would be @xmath409 , therefore @xmath410 if @xmath411 . realizing this condition requires improving the relaxation times with respect to present day experiments . a more realistic possibility is instead to employ the swap subspace as a single qubit quantum memory . the requirement in this case would be satisfying the weaker condition , @xmath412 . finally , we would like to comment on the effects of selected impurities strongly coupled to the two - qubit gate . the detrimental effect of charged bistable fluctuators on single qubit charge or charge - phase gates has been observed in experiments @xcite and explained in theory @xcite . effects on a two - qubit gate may even be worse , as reported in the recent experiment on two coupled quantronium @xcite . explanation of the rich physics which comes out when selected impurities couple to the nano - device is beyond the scope of the present article . in the following [ appendix : sc ] , we illustrate the possible scenario when an impurity considerably changes the 4-level spectrum of the coupled qubits and we speculate on the possibility to limit its effects by proper tuning the system parameters , somehow extending the optimal tuning recipe . the usefulness of such a choice however critically depends on the interplay with quantum noise at intermediate frequencies , an information unavailable in present day experiments . the analysis in [ appendix : sc ] is the first step towards the identification of parameter regimes where a multilevel nano - device may be protected also from strongly coupled degrees of freedom of a structured bath . stimulating discussions with g. schn and u. weiss are gratefully acknowledged . the hamiltonian @xmath413 in eq . is the central model of any non trivial two - qubit gate based on a fixed coupling scheme . in particular , it describes coupled superconducting qubits in the various implementations @xcite . with @xmath414 , given in , it includes independent fluctuations responsible for incoherent processes of different physical origin . in this section we derive @xmath415 for capacitive coupled cooper - pair - box - based ( cpb ) nano - devices . the cpb is the main building block of many superconducting qubits @xcite . here we consider two charge - phase qubits ( quantronia ) @xcite electrostatically coupled via a fixed capacitor , as schematically illustrated in . each qubit is operated via two control parameters , the gate voltage @xmath416 and the magnetic flux across the junction loop , @xmath417 . they enter the cpb hamiltonian via the dimensionless parameters @xmath418 and @xmath419 : @xmath420 here @xmath421 and @xmath422 is the josephson energy , modulated by the phase @xmath219 around the zero phase value @xmath423 . charge @xmath424 and phase @xmath425 are conjugate operators , @xmath426=\rmi$ ] . the loop capacitance @xmath427 is the sum of the gate @xmath428 and the junctions @xmath429 capacitances . the two cpbs are coupled by inserting a fixed capacitance @xmath430 between the two islands . the presence of the coupling capacitance leads to a renormalization of the cpb charging energies @xmath431 , being @xmath432 the cpb charging energy of the uncoupled qubit @xmath18 , @xmath433 the total inverse capacitance of the device . the coupling energy is @xmath434 , and the full device hamiltonian reads @xmath435 ^ 2 - e_{\alpha,\mathrm{j}}(\delta_\alpha ) \cos \hat{\varphi}_\alpha , \nonumber \\ \mathcal{h}_\mathrm{c } = e_{\mathrm{cc } } [ \hat{q}_1 - q_{1,\mathrm{x } } \mathbb{i}_{1 } ] \otimes [ \hat{q}_2 - q_{2,\mathrm{x } } \mathbb{i}_{2 } ] , \label{eq:2qbham - coupl}\end{aligned}\ ] ] where the control parameters @xmath436 and @xmath345 , and the charge @xmath437 and phase operators @xmath438 play the same role as in the single qubit case . . each qubit has the characteristic two - port design , with control parameters the gate voltage @xmath439 and the magnetic flux @xmath440 threading the superconducting loop @xcite.,scaledwidth=60.0% ] at sufficiently low temperatures , the evolution of each cpb approximately takes place within the bi - dimensional subspace spanned by the lowest energy eigenstates of @xmath441 , @xmath442 with a splitting depending on the control parameters @xcite , @xmath443 , where the projection operator reads @xmath444 and @xmath445 . the restricted dynamics of the coupled cpbs can be described in a pseudo - spin formalism by projection in the eigenstates basis @xmath446 . in this subspace the charge and josephson operators are expressed in terms both of @xmath447 and of the transverse component @xmath448 as follows @xmath449 where @xmath450 the restriction of the device hamiltonian to @xmath451 is to indicate @xmath452 , @xmath453 . ] @xmath454 ( q_{\beta,++ } + q_{\beta,--}- 2 q_{\beta , x } ) \nonumber \\ \!\!\!\!\ ! \!\!\!\!\ ! + e_{\mathrm{cc } } \prod_\alpha \left [ - \frac{1}{2 } ( q_{\alpha,++ } - q_{\alpha,-- } ) \sigma_{\alpha z } + q_{\alpha,+- } \sigma_{\alpha x } \right ] \label{eq : htrunc}\end{aligned}\ ] ] note that because of the coupling , in addition to interaction terms between the two qubits , each qubit is effectively displaced from its own operating point ( second term in ) . because of fluctuations of control parameters the device couples to the external environment . the form of interaction terms can be easily deduced considering classical fluctuations of @xmath436 and @xmath345 , i.e. by replacing in and , @xmath455 and @xmath456 . we therefore obtain @xmath457 , where @xmath458 fluctuations of each cpb hamiltonian and of the coupling term read @xmath459 where @xmath460 is related to gate charge fluctuations , @xmath461 and @xmath462 to fluctuations of the phase or equivalently of the josephson energy , @xmath463 , where @xmath464 , and we put @xmath465 . in the computational subspace the additional terms reduce to @xmath466 and @xmath467 \otimes \mathbb{i}_{2 } \nonumber \\ \!\!\!\!\!+ g_1 x_1 \mathbb{i}_{1 } \otimes \left [ - \frac{1}{2 } ( q_{2,++ } - q_{2,-- } ) \sigma_{2 z } + q_{2,+- } \sigma_{2 x}+ \frac{1}{2 } ( q_{2,++ } + q_{2,-- } ) \mathbb{i}_{2 } \right ] .\end{aligned}\ ] ] in conclusion , the coupled cpbs hamiltonian at a general working point , in the four dimensional subspace @xmath451 , including fluctuations of the control parameters and neglecting constant terms , takes the form @xmath468 with @xmath469 given in eq . and @xmath470 \nonumber \\ \!\!\!\!\ ! \!\!\!\!\ ! \!\!\!\!\ ! \!\!\!\!\ ! \!\!\!\!\ ! \!\!\!\!\ ! \!\!\!\!\ ! + g_2 x_2 \left [ \frac{1}{2 } ( q_{1,-- } - q_{1,++ } ) \sigma_{1 z } + q_{1,+- } \sigma_{1 x } \right ] + g_1 x_1 \left [ \frac{1}{2 } ( q_{2,-- } - q_{2,++ } ) \sigma_{2 z } + q_{2,+- } \sigma_{2 x } \right ] \end{aligned}\ ] ] note that due to the capacitive coupling a cross - talk effect takes place , gate charge fluctuations of qubit @xmath18 being responsible for fluctuations of the polarization of qubit @xmath471 . effects are scaled with the coupling energy @xmath472 , thus they are expected to be less relevant compared to fluctuations acting directly on each qubit . a detailed analysis of cross - talk and correlations between gate charge fluctuations has been reported in @xcite . in our analysis we disregarded cross - talk and correlations between noise sources acting on each qubit . lll parameter & qubit 1 & qubit 2 + @xmath473 ( ghz ) & 9.997 & 10.41 + @xmath423 ( ghz ) & 14.16 & 15.62 + @xmath427 ( ff ) & 7.73 & 7.42 + @xmath474 ( ghz ) & 12.666 & 13.81 + @xmath475 ( ghz ) & 9.92 & 10.33 + @xmath22 ( ghz ) & 12.645 & 13.79 + the implementation of a two qubit gate in a fixed coupling scheme requires the qubits to be at resonance during gate operation . because of experimental tolerances on bare parameters ( about @xmath406% ) , the resonant condition can only be achieved by a proper choice of the single - qubit working points via tuning @xmath476 and @xmath345 . this is a critical choice since at least one qubit has to be moved away from the working point of minimal sensitivity to parameters variations , the `` optimal point '' , @xmath477 , @xmath478 @xcite . since cpb - based devices are severely affected by low - frequency charge noise it is desirable to maintain the charge optimal point , @xmath479 and modulate resonance / detuning by tuning the phases @xmath345 . the most reasonable choice consists in maintaining one qubit at its own phase optimal point , @xmath279 and tune the phase of the other qubit , @xmath480 . for set of parameters close to those planned in experiments @xcite ( see ) resonance occurs for @xmath481 . since at @xmath479 it results @xmath482 and @xmath483 , the projected charge is transverse and the josephson operator is longitudinal with respect to each hamiltonian @xmath484 . therefore , the truncated hamiltonian is of the form @xmath485 given by eqs . and respectively @xmath486 \otimes \mathbb{i}_{\beta } \label{eq : hreson - noise } \end{aligned}\ ] ] where @xmath487 , @xmath488 etc . denote the specific values taken by these matrix elements at the resonance point obtained for a specific choice of @xmath480 . values of the charge and phase matrix elements at this working point are reported in table [ tab : meanvalues ] and lead to @xmath489 ghz . the interaction hamiltonian @xmath490 results from @xmath491 considering the quantized version of the classical fluctuations @xmath492 and @xmath493 . where , since @xmath279 we have @xmath494 whereas for @xmath280 it results @xmath495 . ccc & qubit 1 & qubit 2 + @xmath496 & 0 & 0 + @xmath497 & -0.5962161 & -0.5891346 + @xmath498 & 0.7049074 & 0.7396952 + @xmath499 & 0 & 0 + the accuracy of the truncation of the multistate device hamiltonian to the lowest four eigenstates has been checked numerically by exact diagonalization of using the numerical values of the physical parameters reported in and considering 14 charge states for each qubit , see . fluctuations of the control parameters @xmath500 , @xmath345 have different physical origin . they partly stem from microscopic noise sources , partly from the circuitry itself . the resulting stochastic processes are sometimes non - gaussian , therefore complete characterization of the processes requires in principle higher - order cumulants . such a complete description is often unavailable in experiments , whereas it is usually possible to provide a characterization of the power spectrum of the various stochastic processes . noise characteristics for the two quantronia are summarized in table [ tab : noise ] ( ) . experiments on various josephson implementations have revealed the presence of spurious resonances in the spectrum which manifest themselves as beatings in time resolved measurements @xcite . in charge and charge - phase qubits they are due to strongly coupled ( sc ) background charges and may severely limit the reliability of these devices @xcite . in the following [ appendix : sc ] we will discuss the detrimental effect of a single strongly - coupled impurity on the visibility of a @xmath3 operation . finally , we mention that in the present analysis we have not explicitly considered fluctuations of the josephson energy due to critical current fluctuations . the effect of @xmath423 noise may be straightforwardly included in our analysis within the multistage approach . experiments on various josephson implementations have revealed the presence of spurious resonances in the spectrum which manifest themselves as beatings in time resolved measurements @xcite . in charge and charge - phase qubits they are due to strongly coupled ( sc ) background charges and may severely limit the reliability of these devices @xcite . several experiments have shown that these impurities can be modeled as bistable fluctuators ( bf ) , switching between states @xmath501 and @xmath88 with a rate @xmath502 . often the qubit - bf coupling is transverse , i.e. of the form @xmath503 with @xmath504 @xcite . each bistable state corresponds to a different qubit splitting , @xmath22 for @xmath505 and @xmath506 $ ] for @xmath507 ( @xmath508 ) . the parameter quantifying the strength of the qubit - bf coupling is @xmath509 @xcite . when @xmath510 the impurity is strongly coupled and it is visible both in spectroscopy and in the time - resolved dynamics . similarly , a single bf acting on one of the two qubits forming a universal gate induces a bi - stability in the swap - splitting : @xmath511 for @xmath505 and @xmath512 $ ] for @xmath507 ( in this case the stronger condition on the qubit - bf coupling has been assumed @xmath513 ) . the strong coupling condition for the @xmath3 gate is therefore @xmath514 . considering the above expansions and the constraint @xmath515 the two ratios satisfy the condition @xmath516 . therefore the visibility of a specific bf is expected to be reduced in a two - qubit operation with respect to a single qubit gate . the effect of a single bf fluctuator depends on the way experimental data are collected . for spectroscopic measurements the single - shot time is about @xmath517 ns and the acquisition time of a single point in the spectrum requires about @xmath518 repetitions , resulting in a recording time for a single data point of about @xmath328 s @xcite . therefore a single bf will be visible in spectroscopy if its switching time is @xmath519 . for instance a bf switching at @xmath520 khz can be averaged during the spectroscopic measurements and it is visible if it is sufficiently strongly coupled to one qubit , for instance this is the case for @xmath521 . the expected effect of this impurity in spectroscopy is shown in where it results in the simultaneous presence of two avoided crossings depending on the impurity state . the occurrence of a similar bf represents a major problem for charge and charge - phase implementations @xcite . effects are also visible in time resolved measurements . if the considered bf is one out of the several ones responsible for @xmath0 noise , the global effect of the structured bath is a considerable reduction of the oscillation amplitude , as shown in . under these conditions , in principle , an `` optimal coupling '' can still be identified . since the main problem is the beating pattern , optimal tuning is defined by the condition that the average @xmath522 vanishes , rather than minimizing the swap - splitting variance . in the presence of only charge noise , this condition leads to a modified `` optimal coupling '' , @xmath523 , at which effectively the swap visibility is improved ( ) . the efficiency of such a choice however critically depends on the details ( presently not available ) of transverse noise at frequencies in the khz - mhz range . relaxation processes due to quantum noise at these frequencies may in fact represent an additional liming factor for the gate fidelity . for qubit - qubit coupling @xmath524 ( blue ) and for uncoupled qubits ( black ) . in the presence of a bf coupled to one qubit with strength @xmath525 the coupled qubits ( @xmath524 ) eigenenergies are shifted ( red ) . right : numerical simulations of the switching probabilities in the presence of the above bf ( orange / cyan ) , and of the bf plus @xmath0 noise with @xmath526 ( blue and red ) . for optimal coupling , @xmath523 , a considerable recover of the signal can be obtained ( gray ) . , title="fig:",scaledwidth=47.0% ] for qubit - qubit coupling @xmath524 ( blue ) and for uncoupled qubits ( black ) . in the presence of a bf coupled to one qubit with strength @xmath525 the coupled qubits ( @xmath524 ) eigenenergies are shifted ( red ) . right : numerical simulations of the switching probabilities in the presence of the above bf ( orange / cyan ) , and of the bf plus @xmath0 noise with @xmath526 ( blue and red ) . for optimal coupling , @xmath523 , a considerable recover of the signal can be obtained ( gray ) . , title="fig:",scaledwidth=45.0% ] in this appendix we report the frequency shifts entering the off - diagonal elements of the rdm as resulting from the solution of the master equation in the secular approximation reported in . the frequency shifts take the simple form @xmath527 \\ \tilde \omega_{03 } - \omega_{03 } \,= \,\frac{1}{2 } \ , \left [ \sum_{k \neq 0 } { \mathcal e}_{0k } - \sum_{k \neq 3 } { \mathcal e}_{3k } \right ] \ , . \end{array } \label{shifts}\end{aligned}\ ] ] for the swap coherence the different contributions read @xmath528 \nonumber \\ { \mathcal e}_{13 } & = & \frac{1}{8 } \ , ( 1-\sin \varphi ) \,[{\mathcal e}_{x_1}(\omega_{13 } ) + { \mathcal e}_{x_2}(\omega_{13 } ) ] \nonumber \\ { \mathcal e}_{20 } & = & \frac{1}{8 } \ , ( 1-\sin \varphi ) \,[{\mathcal e}_{x_1}(\omega_{20 } ) + { \mathcal e}_{x_2}(\omega_{20 } ) ] \nonumber \\ { \mathcal e}_{23 } & = & \frac{1}{8 } \ , ( 1+\sin \varphi ) \,[{\mathcal e}_{x_1}(\omega_{23 } ) + { \mathcal e}_{x_2}(\omega_{23 } ) ] \nonumber \\ { \mathcal e}_{21 } & = & \frac{1}{8 } \ { [ \mathcal{e}_{z_1 } ( \omega_{21 } ) + \mathcal{e}_{z_2 } ( \omega_{21 } ) ] \nonumber \\ { \mathcal e}_{12 } & = & \frac{1}{8 } \ { [ \mathcal{e}_{z_1 } ( \omega_{12 } ) + \mathcal{e}_{z_2 } ( \omega_{12 } ) ] \nonumber \end{aligned}\ ] ] from single qubit measurements , as reported in , charge noise at frequencies of order of @xmath22 is white . assuming that also the phase variables @xmath529 have white spectrum at frequencies of order @xmath21 , results in @xmath530s@xmath99 . the spectrum of @xmath531 takes instead the ohmic form @xmath532 , where @xmath533s@xmath99 . under these conditions , phase shifts due to polarization noise and phase fluctuations on qubit 2 identically vanish ( from ) . a frequency shift contribution results from ohmic phase noise on qubit 1 , similarly to a qubit affected by transverse noise @xcite , as it can be argued from @xmath534 = { \mathcal p } \int_{-\infty}^{\infty } \frac{d \omega}{4\pi } \ , \frac{s_{z_1}(\omega)}{1+e^{-\omega / k_b t } } \ , \frac{\omega_{21}}{\omega^2 -\omega_{21}^2 } \ , . \label{shift - 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controlled generation of entangled states of two quantum bits is a fundamental step toward the implementation of a quantum information processor . in nano - devices this operation is counteracted by the solid - state environment , characterized by broadband and non - monotonic power spectrum often @xmath0 at low frequencies
. for single qubit gates , incoherent processes due to fluctuations acting on different time scales result in peculiar short- and long - time behaviors .
markovian noise originates exponential decay with relaxation and decoherence times , @xmath1 and @xmath2 , simply related to the symmetry of the qubit - environment coupling hamiltonian .
noise with @xmath0 power spectrum at low frequencies is instead responsible for defocusing processes and algebraic short - times behavior . in this article
we identify the relevant decoherence times of an entangling operation due to the different decoherence channels originated from solid state noise .
entanglement is quantified by the concurrence , which we evaluate in analytic form employing a multi - stage approach .
`` optimal '' operating conditions of reduced sensitivity to noise sources are identified .
we apply this analysis to a superconducting @xmath3 gate for experimental noise spectra .
| 30,164 | 296 |
the recent complete dna sequences of many organisms are available to systematically search of genome structure . for the large amount of dna sequences , developing methods for extracting meaningful information is a major challenge for bioinformatics . to understand the one - dimensional symbolic sequences composed of the four letters ` a ' , ` c ' , ` g ' and ` t ' ( or ` u ' ) , some statistical and geometrical methods were developed@xcite . in special , chaos game representation ( cgr)@xcite , which generates a two - dimensional square from a one - dimensional sequence , provides a technique to visualize the composition of dna sequences . the characteristics of cgr images was described as genomic signature , and classification of species in the whole bacteria genome was analyzed by making an euclidean metric between two cgr images@xcite . based on the genomic signature , the distance between two dna sequences depending on the length of nucleotide strings was presented@xcite and the horizontal transfers in prokaryotes and eukaryotes were detected and charaterized@xcite . recently , a one - to - one metric representation of the dna sequences@xcite , which was borrowed from the symbolic dynamics , makes an ordering of subsequences in a plane . suppression of certain nucleotide strings in the dna sequences leads to a self - similarity of pattern seen in the metric representation of dna sequences . self - similarity limits of genomic signatures were determined as an optimal string length for generating the genomic signatures@xcite . moreover , by using the metric representation method , the recurrence plot technique of dna sequences was established and employed to analyze correlation structure of nucleotide strings@xcite . as a eukaryotic organism , yeast is one of the premier industrial microorganisms , because of its essential role in brewing , baking , and fuel alcohol production . in addition , yeast has proven to be an excellent model organism for the study of a variety of biological problems involving the fields of genetics , molecular biology , cell biology and other disciplines within the biomedical and life sciences . in april 1996 , the complete dna sequence of the yeast ( saccharomyces cevevisiae ) genome , consisting of 16 chromosomes with 12 million basepairs , had been released to provide a resource of genome information of a single organism . however , only 43.3% of all 6000 predicted genes in the saccharomyces cerevisiae yeast were functionally characterized when the complete sequence of the yeast genome became available@xcite . moreover , it was found that dna transposable elements have ability to move from place to place and make many copies within the genome via the transposition@xcite . therefore , the yeast complete dna sequence remain a topic to be studied respect to its genome architecture structure in the whole sequence . in this paper , using the metric representation and recurrence plot methods , we analyze global transposable characteristics in the yeast complete dna sequence , i.e. , 16 chromosome sequences . for a given dna sequence @xmath0 ( @xmath1 ) , a plane metric representation is generated by making the correspondence of symbol @xmath2 to number @xmath3 or @xmath4 and calculating values ( @xmath5 , @xmath6 ) of all subsequences @xmath7 ( @xmath8 ) defined as follows @xmath9 where @xmath3 is 0 if @xmath10 or 1 if @xmath11 and @xmath12 is 0 if @xmath13 or 1 if @xmath14 . thus , the one - dimensional symbolic sequence is partitioned into @xmath15 subsequences @xmath16 and mapped in the two - dimensional plane ( @xmath17 ) . subsequences with the same ending @xmath18-nucleotide string , which are labeled by @xmath19 , correspond to points in the zone encoded by the @xmath18-nucleotide string . taking a subsequence @xmath20 , we calculate @xmath21 where @xmath22 is the heaviside function [ @xmath23 , if @xmath24 ; @xmath25 , if @xmath26 and @xmath27 is a subsequence ( @xmath28 ) . when @xmath29 , i.e. , @xmath30 , a point @xmath31 is plotted in a plane . thus , repeating the above process from the beginning of one - dimensional symbolic sequence and shifting forward , we obtain a recurrence plot of the dna sequence . for presenting correlation structure in the recurrence plot plane , a correlation intensity is defined at a given correlation distance @xmath32 @xmath33 the quantity displays the transference of @xmath18-nucleotide strings in the dna sequence . to further determine positions and lengths of the transposable elements , we analyze the recurrent plot plane . since @xmath34 and @xmath27 @xmath35 , the transposable element has the length @xmath18 at least . from the recurrence plot plane , we calculate the maximal value of @xmath36 to satisfy @xmath37 i.e. , @xmath38 and @xmath39 . thus , the transposable element with the correction distance @xmath40 has the length @xmath41 . the transposable element is placed at the position @xmath42 and @xmath43 . the saccharomyces cevevisiae yeast has 16 chromosomes , which are denoted as yeast i to xvi . using the metric representation and recurrence plot methods , we analyze correlation structures of the 16 dna sequences . according to the characteristics of the correlation structures , we summarize the results as follows : \(1 ) the correlation distance has a short period increasing . the yeast i , ix and xi have such characteristics . let me take the yeast i as an example to analyze . fig.1 displays the correlation intensity at different correlation distance @xmath44 with @xmath45 . a local region is magnified in the figure . it is clearly evident that there exist some equidistance parallel lines with a basic correlation distance @xmath46 . ( 4 ) , we determine positions and lengths of the transposable elements in table i , where their lengths are limited in @xmath47 . many nucleotide strings have correlation distance , which is the integral multiple of @xmath48 . they mainly distribute in two local regions of the dna sequence ( 25715 - 26845 ) and ( 204518 - 206554 ) or ( 11.2 - 11.7% ) and ( 88.8 - 89.7% ) expressed as percentages . the yeast ix and xi have similar behaviors . the yeast ix has the basic correlation distance @xmath49 . many nucleotide strings ( @xmath50 ) with the integral multiple of @xmath48 mainly distribute in a local region of the dna sequence ( 391337 - 393583 ) or ( 89.0 - 89.5% ) expressed as percentages . the yeast xi has the basic correlation distance @xmath51 . many nucleotide strings ( @xmath50 ) with the integral multiple of @xmath48 mainly distribute in a local region of the dna sequence ( 647101 - 647783 ) or ( 97.1 - 97.2% ) expressed as percentages . \(2 ) the correlation distance has a long major value and a short period increasing . the yeast ii , v , vii , viii , x , xii , xiii , xiv , xv and xvi have such characteristics . let me take the yeast ii as an example to analyze . fig.2 displays the correlation intensity at different correlation distance @xmath44 with @xmath45 . the maximal correlation intensity appears at the correlation distance @xmath52 . a local region is magnified in the figure . it is clearly evident that there exist some equidistance parallel lines with a basic correlation distance @xmath53 . in table ii , positions and lengths ( @xmath50 ) of the transposable elements are given . the maximal transposable elements mainly distribute in two local regions of the dna sequence ( 221249 - 224565 , 259783 - 263097 ) or ( 27.2 - 27.6% , 31.9- 32.4% ) expressed as percentage . near the positions , there also exist some transposable elements with approximate values for @xmath48 . moreover , many nucleotide strings have correlation distance , which is the integral multiple of @xmath48 . they mainly distribute in a local region of the dna sequence ( 391337 - 393583 ) or ( 89.0 - 89.5% ) expressed as percentages . in the other 9 dna sequences , the yeast v , x , xii , xiii , xiv , xv and xvi have the same basic correlation distance @xmath53 and similar behaviors with different major correlation distance @xmath5449099 , 5584 , 9137 , 12167 , 5566 , 447110 and 45988 , respectively . the yeast vii and viii have different basic correlation distance @xmath55 and 135 , and similar behaviors with the major correlation distance @xmath56 and 1998 , respectively . \(3 ) the correlation distance has a long quasi - period increasing . the yeast iii has such characteristics . 3 displays the coherence intensity at different correlation distance @xmath57 with @xmath45 . the correlation intensity has the maximal value at the correlation distance @xmath58 and two vice - maximal values at the correlation distance @xmath59 and @xmath60 . since @xmath61 , the coherence distance has a quasi - period increasing . a local region is magnified in the figure . these does not exist any clear short period increasing of the correlation distance . using eq . ( 4 ) , we determine positions and lengths ( @xmath50 ) of the transposable elements in table iii . the maximal and vice - maximal transposable elements mainly distribute in local regions of the dna sequence ( 11499 - 13810 , 197402 - 199713 ) , ( 198171 - 199796 , 291794 - 293316 ) and ( 12268 - 12932 , 291794 - 292460 ) or ( 3.6 - 4.4% , 62.6 - 63.6% ) , ( 62.8 - 63.4% , 92.5 - 93.0% ) and ( 3.9 - 4.1% , 92.5 - 92.7% ) expressed as percentage . \(4 ) the correlation distance has a long major value and a long quasi - period and two short period increasing . the yeast iv has such characteristics . 4 displays the coherence intensity at different correlation distance @xmath44 with @xmath45 . the maximal coherence intensity appears at the correlation distance @xmath62 . there also exist three vice - maximal values at the correlation distance @xmath63 , @xmath64 and @xmath65 , which forms a long quasi - period increasing of the correlation distance , i.e. , @xmath66 . a local region is magnified in the figure . it is clearly evident that there exist two short period increasing with @xmath67 and @xmath68 in the correlation distance . in table iv , positions and lengths ( @xmath47 ) of the transposable elements are determined by using eq . ( 4 ) . all correlation distance with the long major value and the long quasi - period and two short period increasing are denoted . the transposable elements with @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 and @xmath74 mainly distribute in local regions of the dna sequence ( 527570 - 538236 ) , ( 871858 - 876927 , 981207 - 986276 ) , ( 645646 - 651457 , 878346 - 884257 ) , ( 646379 - 651032 , 987600 - 992253 ) , ( 1307733 - 1308591 ) and ( 758135 - 759495 ) or ( 34.4 - 35.1% ) , ( 56.9 - 57.2% , 64.0 - 64.4% ) , ( 42.1 - 42.5% , 57.3 - 57.7% ) , ( 42.2 - 42.5% , 64.4 - 64.8% ) , ( 85.36 - 85.41% ) and ( 49.5 - 49.6% ) expressed as percentages . \(5 ) the dna sequence is hardly relevant . the yeast vi has such characteristics . 5 displays the coherence intensity at different correlation distance @xmath44 with @xmath45 . the maximal coherence intensity appears at the correlation distance @xmath75 . a local region is magnified in the figure . the sequence has not a short period increasing of the coherence distance . in table v , positions and lengths ( @xmath50 ) of the transposable elements are given . only one nucleotide string with the length 337 has the correlation distance @xmath69 . the yeast vi is almost never relevant , so the yeast vi approaches a random sequence . global transposable characteristics in the yeast complete dna sequence is determined by using the metric representation and recurrence plot methods . positions and lengths of all transposable nucleotide strings in the 16 chromosome dna sequences of the yeast are determined . in the form of the correlation distance of nucleotide strings , the fundamental transposable characteristics displays a short period increasing , a long quasi - period increasing , a long major value and hardly relevant . the 16 chromosome sequences are divided into 5 groups , which have one or several of the 4 kinds of the fundamental transposable characteristics . p. j. deschavanne , a. giron , j. vilain , g. fagot , and b. fertil , genomic signature : characterization and classification of species assessed by chaos game representation of sequences . * 16 * ( 1999 ) 1391 .
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global transposable characteristics in the complete dna sequence of the saccharomyces cevevisiae yeast is determined by using the metric representation and recurrence plot methods . in the form of the correlation distance of nucleotide strings , 16 chromosome sequences of the yeast , which are divided into 5 groups , display 4 kinds of the fundamental transposable characteristics : a short period increasing , a long quasi - period increasing , a long major value and hardly relevant . * keywords * yeast , dna sequences , coherence structure , metric representation , recurrence plot +
| 3,776 | 154 |
dans le modle cosmologique , dit de concordance car il est en conformit avec tout un ensemble de donnes observationnelles , la matire ordinaire do nt sont constitus les toiles , le gaz , les galaxies , etc . ( essentiellement sous forme baryonique ) ne forme que 4% de la masse - nergie totale , ce qui est dduit de la nuclosynthse primordiale des lments lgers , ainsi que des mesures de fluctuations du rayonnement du fond diffus cosmologique ( cmb ) le rayonnement fossile qui date de la formation des premiers atomes neutres dans lunivers . nous savons aussi quil y a 23% de matire noire sous forme _ non baryonique _ et do nt nous ne connaissons pas la nature . et les 73% qui restent ? et bien , ils sont sous la forme dune mystrieuse nergie noire , mise en vidence par le diagramme de hubble des supernovae de type ia , et do nt on ignore lorigine part quelle pourrait tre sous la forme dune constante cosmologique . le contenu de lunivers grandes chelles est donc donn par le `` camembert '' de la figure [ fig1 ] do nt 96% nous est inconnu ! la matire noire permet dexpliquer la diffrence entre la masse dynamique des amas de galaxies ( cest la masse dduite du mouvement des galaxies ) et la masse de la matire lumineuse qui comprend les galaxies et le gaz chaud intergalactique . mais cette matire noire ne fait pas que cela ! nous pensons quelle joue un rle crucial dans la formation des grandes structures , en entranant la matire ordinaire dans un effondrement gravitationnel , ce qui permet dexpliquer la distribution de matire visible depuis lchelle des amas de galaxies jusqu lchelle cosmologique . des simulations numriques trs prcises permettent de confirmer cette hypothse . pour que cela soit possible il faut que la matire noire soit non relativiste au moment de la formation des galaxies . on lappelera matire noire _ froide _ ou cdm selon lacronyme anglais , et il y a aussi un nom pour la particule associe : un wimp pour `` weakly interacting massive particle '' . il ny a pas dexplication pour la matire noire ( ni pour lnergie noire ) dans le cadre du modle standard de la physique des particules . mais des extensions au - del du modle standard permettent de trouver des bons candidats pour la particule ventuelle de matire noire . par exemple dans un modle de super - symtrie ( qui associe tout fermion un partenaire super - symtrique qui est un boson et rciproquement ) lun des meilleurs candidats est le _ neutralino _ , qui est un partenaire fermionique super - symtrique dune certaine combinaison de bosons du modle standard . , qui fut introduit dans une tentative pour rsoudre le problme de la violation cp en physique des particules , est une autre possibilit . il y a aussi les tats de kaluza - klein prdits dans certains modles avec dimensions suplmentaires . quant lnergie noire , elle apparat comme un milieu de densit dnergie _ constante _ au cours de lexpansion , ce qui implique une violation des `` conditions dnergie '' habituelles avec une pression ngative . lnergie noire pourrait tre la fameuse constante cosmologique @xmath0 queinstein avait introduite dans les quations de la relativit gnrale afin dobtenir un modle dunivers statique , puis quil avait abandonne lorsque lexpansion fut dcouverte . depuis zeldovich on interprte @xmath0 comme lnergie du vide associe lespace - temps lui - mme . le problme est que lestimation de cette nergie en thorie des champs donne une valeur @xmath1 fois plus grande que la valeur observe ! on ne comprend donc pas pourquoi la constante cosmologique est si petite . malgr lnigme de lorigine de ses constituents , le modle @xmath0-cdm est plein de succs , tant dans lajustement prcis des fluctuations du cmb que dans la reproduction fidle des grandes structures observes . une leon est que la matire noire apparat forme de particules ( les wimps ) grande chelle . la matire noire se manifeste de manire clatante dans les galaxies , par lexcs de vitesse de rotation des toiles autour de ces galaxies en fonction de la distance au centre cest la clbre courbe de rotation ( voir la figure [ fig2 ] ) . les mesures montrent qu partir dune certaine distance au centre la courbe de rotation devient pratiquement plate , cest - - dire que la vitesse devient constante . daprs la loi de newton la vitesse dune toile sur une orbite circulaire ( keplerienne ) de rayon @xmath2 est donne par @xmath3 o @xmath4 est la masse contenue dans la sphre de rayon @xmath2 . pour obtenir une courbe de rotation plate il faut donc supposer que la masse crot proportionnellement @xmath2 ( et donc que la densit dcrot comme @xmath5 ) , ce qui nest certainement pas le cas de la matire visible . on est oblig dinvoquer lexistence dun gigantesque halo de matire noire invisible ( qui ne rayonne pas ) autour de la galaxie et do nt la masse dominerait celle des toiles et du gaz . cette matire noire peut - elle tre faite de la mme particule que celle suggre par la cosmologie ( un wimp ) ? des lments de rponse sont fournis par les simulations numriques de cdm en cosmologie qui sont aussi valables lchelle des galaxies , et qui donnent un profil de densit universel pour le halo de matire noire . a grande distance ce profil dcroit en @xmath6 soit plus rapidement que ce quil faudrait pour avoir une courbe plate , mais ce nest pas trs grave car on peut supposer que la courbe de rotation est observe dans un rgime intermdiaire avant de dcrotre . plus grave est la prdiction dun pic central de densit au centre des galaxies , o les particules de matire noire tendent sagglomrer cause de la gravitation , avec une loi en @xmath7 pour @xmath2 petit . or les courbes de rotation favorisent plutt un profil de densit sans divergence , avec un coeur de densit constante . dautres problmes rencontrs par les halos simuls de cdm sont la formation dune multitude de satellites autour des grosses galaxies , et la loi empirique de tully et fisher qui nest pas explique de faon naturelle . cette loi montre dans la figure [ fig3 ] relie la luminosit des galaxies leur vitesse asymptotique de rotation ( qui est la valeur du plateau dans la figure [ fig2 ] ) par @xmath8 . noter que cette loi ne fait pas rfrence la matire noire ! la vitesse et la luminosit sont bien sr celles de la matire ordinaire , et la matire noire semble faire ce que lui dicte la matire visible . mais le dfi le plus important de cdm est de pouvoir rendre compte dune observation tonnante appele _ loi de milgrom _ @xcite , selon laquelle la matire noire intervient uniquement dans les rgions o le champ de gravitation ( ou , ce qui revient au mme , le champ dacclration ) est plus _ faible _ quune certaine acclration critique mesure la valeur `` universelle '' @xmath9 . tout se passe comme si dans le rgime des champs faibles @xmath10 , la matire ordinaire tait acclre non par le champ newtonien @xmath11 mais par un champ @xmath12 donn simplement par @xmath13 . la loi du mouvement sur une orbite circulaire donne alors une vitesse _ constante _ et gale @xmath14 . ce rsultat nous rserve un bonus important : puisque le rapport masse - sur - luminosit @xmath15 est approximativement le mme dune galaxie lautre , la vitesse de rotation doit varier comme la puissance @xmath16 de la luminosit @xmath17 , en accord avec la loi de tully - fisher ! pour avoir une rgle qui nous permette dajuster les courbes de rotation des galaxies il nous faut aussi prendre en compte le rgime de champ fort dans lequel on doit retrouver la loi newtonienne . on introduit une fonction dinterpolation @xmath18 dpendant du rapport @xmath19 et qui se ramne @xmath20 lorsque @xmath10 , et qui tend vers 1 quand @xmath21 . notre rgle sera donc @xmath22 ici @xmath12 dsigne la norme du champ de gravitation @xmath23 ressenti par les particules dpreuves . une formule encore plus oprationnelle est obtenue en prenant la divergence des deux membres de ce qui mne lquation de poisson modifie , o @xmath24 est le laplacien et @xmath25 le potentiel newtonien local . loprateur @xmath26 appliqu une fonction scalaire est le gradient , appliqu un vecteur cest la divergence : @xmath27 . par convention , on note les vecteurs en caractres gras . ] : @xmath28 = -4 \pi \ , g\,\rho_\text{b } \,,\ ] ] do nt la source est la densit de matire baryonique @xmath29 ( le champ gravitationnel est irrotationnel : @xmath30 ) . on appellera lquation la formule mond pour `` modified newtonian dynamics '' . le succs de cette formule ( on devrait plus exactement dire cette _ recette _ ) dans lobtention des courbes de rotation de nombreuses galaxies est impressionnant ; voir la courbe en trait plein dans la figure [ fig2 ] . cest en fait un ajustement un paramtre libre , le rapport @xmath15 de la galaxie qui est donc _ mesur _ par notre recette . on trouve que non seulement la valeur de @xmath15 est de lordre de 1 - 5 comme il se doit , mais quelle est remarquablement en accord avec la couleur observe de la galaxie . beaucoup considrent la formule mond comme `` exotique '' et reprsentant un aspect mineur du problme de la matire noire . on entend mme parfois dire que ce nest pas de la physique . bien sr ce nest pas de la physique _ fondamentale _ cette formule ne peut pas tre considre comme une thorie fondamentale , mais elle constitue de lexcellente physique ! elle capture de faon simple et puissante tout un ensemble de faits observationnels . au physicien thoricien dexpliquer pourquoi . la valeur numrique de @xmath31 se trouve tre trs proche de la constante cosmologique : @xmath32 . cette concidence cosmique pourrait nous fournir un indice ! elle a aliment de nombreuses spculations sur une possible influence de la cosmologie dans la dynamique locale des galaxies . face la `` draisonnable efficacit '' de la formule mond , trois solutions sont possibles . 1 . la formule pourrait sexpliquer dans le cadre cdm . mais pour rsoudre les problmes de cdm il faut invoquer des mcanismes astrophysiques compliqus et effectuer un ajustement fin des donnes galaxie par galaxie . 2 . on est en prsence dune modification de la loi de la gravitation dans un rgime de champ faible @xmath10 . cest lapproche traditionnelle de mond et de ses extensions relativistes . la gravitation nest pas modifie mais la matire noire possde des caractristiques particulires la rendant apte expliquer la phnomnologie de mond . cest une approche nouvelle qui se prte aussi trs bien la cosmologie . la plupart des astrophysiciens des particules et des cosmologues des grandes structures sont partisans de la premire solution . malheureusement aucun mcanisme convainquant na t trouv pour incorporer de faon naturelle la constante dacclration @xmath31 dans les halos de cdm . dans la suite nous considrerons que la solution 1 . est dores et dj exclue par les observations . les approches 2 . de gravitation modifie et 3 . que lon peut qualifier de _ matire noire modifie _ croient toutes deux dans la pertinence de mond , mais comme on va le voir sont en fait trs diffrentes . notez que dans ces deux approches il faudra expliquer pourquoi la matire noire semble tre constitue de wimps lchelle cosmologique . cette route , trs dveloppe dans la littrature , consiste supposer quil ny a pas de matire noire , et que reflte une violation fondamentale de la loi de la gravitation . cest la proposition initiale de milgrom @xcite un changement radical de paradigme par rapport lapproche cdm . pour esprer dfinir une thorie il nous faut partir dun lagrangien . or il est facile de voir que dcoule dun lagrangien , celui - ci ayant la particularit de comporter un terme cintique non standard pour le potentiel gravitationnel , du type @xmath33 $ ] au lieu du terme habituel @xmath34 , o @xmath35 est une certaine fonction que lon relie la fonction @xmath18 . ce lagrangien a servi de point de dpart pour la construction des thories de la gravitation modifie . on veut modifier la relativit gnrale de faon retrouver mond dans la limite non - relativiste , cest - - dire quand la vitesse des corps est trs faible par rapport la vitesse de la lumire @xmath36 . en relativit gnrale la gravitation est dcrite par un champ tensoriel deux indices appel la mtrique de lespace - temps @xmath37 . cette thorie est extrmement bien vrifie dans le systme solaire et dans les pulsars binaires , mais peu teste dans le rgime de champs faibles qui nous intresse ( en fait la relativit gnrale est le royaume des champs gravitationnels forts ) . la premire ide qui vient lesprit est de promouvoir le potentiel newtonien @xmath25 en un champ scalaire @xmath38 ( sans indices ) et donc de considrer une thorie _ tenseur - scalaire _ dans laquelle la gravitation est dcrite par le couple de champs @xmath39 . on postule , de manire _ ad - hoc _ , un terme cintique non standard pour le champ scalaire : @xmath40 o @xmath41 est reli @xmath18 , et on choisit le lagrangien deinstein - hilbert de la relativit gnrale pour la partie concernant la mtrique @xmath37 . tout va bien pour ce qui concerne le mouvement des toiles dans une galaxie , qui reproduit mond . mais notre thorie tenseur - scalaire est une catastrophe pour le mouvement des photons ! en effet ceux - ci ne ressentent pas la prsence du champ scalaire @xmath38 cens reprsenter la matire noire . dans une thorie tenseur - scalaire toutes les formes de matire se propagent dans un espace - temps de mtrique _ physique _ @xmath42 qui diffre de la mtrique deinstein @xmath37 par un facteur de proportionalit dpendant du champ scalaire , soit @xmath43 . une telle relation entre les mtriques est dite conforme et laisse invariants les cnes de lumire de lespace - temps . les trajectoires de photons seront donc les mmes dans lespace - temps physique que dans lespace - temps deinstein ( cela se dduit aussi de linvariance conforme des quations de maxwell ) . comme on observe dnormes quantits de matire noire grce au mouvement des photons , par effet de lentille gravitationnelle , la thorie tenseur - scalaire est limine . pour corriger cet effet dsastreux du mouvement de la lumire on rajoute un nouvel lment notre thorie . puisque cest cela qui cause problme on va transformer la relation entre les mtriques @xmath42 et @xmath37 . une faon de le faire est dy insrer ( encore de faon _ ad - hoc _ ) un nouveau champ qui sera cette fois un vecteur @xmath44 avec un indice . on aboutit donc une thorie dans laquelle la gravitation est dcrite par le triplet de champs @xmath45 . cest ce quon appelle une thorie _ tenseur - vecteur - scalaire _ ( teves ) . la thorie teves a t mise au point par bekenstein et sanders @xcite . comme dans la thorie tenseur - scalaire on aura la partie deinstein - hilbert pour la mtrique , plus un terme cintique non standard @xmath40 pour le champ scalaire . quant au champ vectoriel on le munit dun terme cintique analogue celui de llectromagntisme , mais dans lequel le rle du potentiel lectromagntique @xmath46 est tenu par notre champ @xmath44 . la thorie teves rsultante est trs complique et pour linstant non relie de la physique microscopique . il a t montr que cest un cas particulier dune classe de thories appeles thories einstein-_ther _ dans lesquelles le vecteur @xmath44 joue le rle principal , en dfinissant un rfrentiel priviligi un peu analogue lther postul au xix@xmath47 sicle pour interprter la non - invariance des quations de maxwell par transformation de galile . si elle est capable de retrouver mond dans les galaxies , la thorie teves a malheureusement un problme dans les amas de galaxies car elle ne rend pas compte de toute la matire noire observe . cest en fait un problme gnrique de toute extension relativiste de mond . cependant ce problme peut tre rsolu en supposant lexistence dune composante de matire noire _ chaude _ sous la forme de neutrinos massifs , ayant la masse maximale permise par les expriences actuelles soit environ @xmath48 . rappelons que toute la matire noire ne peut pas tre sous forme de neutrinos : dune part il ny aurait pas assez de masse , et dautre part les neutrinos tant relativistes auraient tendance lisser lapparence des grandes structures , ce qui nest pas observ . nanmoins une pince de neutrinos massifs pourrait permettre de rendre viables les thories de gravitation modifie . de ce point de vue les expriences prvues qui vont dterminer trs prcisment la masse du neutrino ( en vrifiant la conservation de lnergie au cours de la dsintgration dune particule produisant un neutrino dans ltat final ) vont jouer un rle important en cosmologie . teves a aussi des difficults lchelle cosmologique pour reproduire les fluctuations observes du cmb . l aussi une composante de neutrinos massifs peut aider , mais la hauteur du troisime pic de fluctuation , qui est caractristique de la prsence de matire noire sans pression , reste difficile ajuster . une alternative logique la gravit modifie est de supposer quon est en prsence dune forme particulire de matire noire ayant des caractristiques diffrentes de cdm . dans cette approche on a lambition dexpliquer la phnomnologie de mond , mais avec une philosophie nouvelle puisquon ne modifie pas la loi de la gravitation : on garde la relativit gnrale classique , avec sa limite newtonienne habituelle . cette possibilit merge grce lanalogue gravitationnel du mcanisme physique de polarisation par un champ extrieur et quon va appeler `` polarisation gravitationnelle '' @xcite . la motivation physique est une analogie frappante ( et peut - tre trs profonde ) entre mond , sous la forme de lquation de poisson modifie , et la physique des milieux dilectriques en lectrostatique . en effet nous apprenons dans nos cours de physique lmentaire que lquation de gauss pour le champ lectrique ( cest lune des quations fondamentales de maxwell ) , est modifie en prsence dun milieu dilectrique par la contribution de la polarisation lectrique ( voir lappendice ) . de mme , mond peut - tre vu comme la modification de lquation de poisson par un milieu `` digravitationnel '' . explicitons cette analogie . on introduit lanalogue gravitationnel de la susceptibilit , soit @xmath49 qui est reli la fonction mond par @xmath50 . la `` polarisation gravitationnelle '' est dfinie par @xmath51 la densit des `` masses de polarisation '' est donne par la divergence de la polarisation soit @xmath52 . avec ces notations lquation devient @xmath53 qui apparat maintenant comme une quation de poisson ordinaire , mais do nt la source est constitue non seulement par la densit de matire baryonique , mais aussi par la contribution des masses de polarisation @xmath54 . il est clair que cette criture de mond suggre que lon est en prsence non pas dune modification de la loi gravitationnelle , mais dune forme nouvelle de matire noire de densit @xmath54 , cest - - dire faite de moments dipolaires aligns dans le champ de gravitation . ltape suivante serait de construire un modle microscopique pour des diples gravitationnels @xmath55 ( tels que @xmath56 ) . lanalogue gravitationnel du diple lectrique serait un vecteur @xmath57 sparant deux masses @xmath58 . on se heurte donc un problme svre : le milieu dipolaire gravitationnel devrait contenir des masses ngatives ! ici on entend par masse lanalogue gravitationnel de la charge , qui est ce quon appelle parfois la masse grave . ce problme des masses ngatives rend _ a priori _ le modle hautement non viable . nanmoins , ce modle est intressant car il est facile de montrer que le coefficient de susceptibilit gravitationnelle doit tre ngatif , @xmath59 , soit loppos du cas lectrostatique . or cest prcisment ce que nous dit mond : comme la fonction @xmath18 interpole entre le rgime mond o @xmath60 et le rgime newtonien o @xmath61 , on a @xmath62 et donc bien @xmath59 . il est donc tentant dinterprter le champ gravitationnel plus intense dans mond que chez newton par la prsence de `` masses de polarisation '' qui _ anti - crantent _ le champ des masses gravitationnel ordinaires , et ainsi augmentent lintensit effective du champ gravitationnel ! dans le cadre de ce modle on peut aussi se convaincre quun milieu form de diples gravitationnels est intrinsquement instable , car les constituants microscopiques du diple devraient se repousser gravitationnellement . il faut donc introduire une force interne dorigine _ non - gravitationnelle _ , qui va supplanter la force gravitationnelle pour lier les constituants dipolaires entre eux . on pourrait qualifier cette nouvelle interaction de `` cinquime force '' . pour retrouver mond , on trouve de faon satisfaisante que ladite force doit dpendre du champ de polarisation , et avoir en premire approximation la forme dun oscillateur harmonique . par leffet de cette force , lquilibre , le milieu dipolaire ressemble une sorte d``ther statique '' , un peu limage du dilectrique do nt les sites atomiques sont fixes . les arguments prcdents nous laissent penser que mond a quelque chose voir avec un effet de polarisation gravitationnelle . mais il nous faut maintenant construire un modle cohrent , reproduisant lessentiel de cette physique , et _ sans _ masses graves ngatives , donc respectant le principe dquivalence . il faut aussi bien sr que le modle soit _ relativiste _ ( en relativit gnrale ) pour pouvoir rpondre des questions concernant la cosmologie ou le mouvement de photons . on va dcrire le milieu comme un fluide relativiste de quadri - courant @xmath63 ( o @xmath64 est la densit de masse ) , et muni dun quadri - vecteur @xmath65 jouant le rle du moment dipolaire . le vecteur de polarisation est alors @xmath66 . on dfinit un principe daction pour cette matire dipolaire , que lon rajoute laction deinstein - hilbert , et la somme des actions de tous les champs de matire habituels ( baryons , photons , etc ) . on inclue dans laction une fonction potentielle dpendant de la polarisation et cense dcrire une force interne au milieu dipolaire . par variation de laction on obtient lquation du mouvement du fluide dipolaire , ainsi que lquation dvolution de son moment dipolaire . on trouve que le mouvement du fluide est affect par la force interne , et diffre du mouvement godsique dun fluide ordinaire . ce modle ( propos dans @xcite ) reproduit bien la phnomnologie de mond au niveau des galaxies . il a t construit pour ! mais il a t aussi dmontr quil donne satisfaction en cosmologie o lon considre une perturbation dun univers homogne et isotrope . en effet cette matire noire dipolaire se conduit comme un fluide parfait sans pression au premier ordre de perturbation cosmologique et est donc indistinguable du modle cdm . en particulier le modle est en accord avec les fluctuations du fond diffus cosmologique ( cmb ) . en ce sens il permet de rconcilier laspect particulaire de la matire noire telle quelle est dtecte en cosmologie avec son aspect `` modification des lois '' lchelle des galaxies . de plus le modle contient lnergie noire sous forme dune constante cosmologique @xmath0 . il offre une sorte dunification entre lnergie noire et la matire noire _ la _ mond . en consquence de cette unification on trouve que lordre de grandeur naturel de @xmath0 doit tre compatible avec celui de lacclration @xmath31 , cest - - dire que @xmath67 , ce qui est en trs bon accord avec les observations . le modle de matire noire dipolaire contient donc la physique souhaite . son dfaut actuel est de ne pas tre connect de la physique microscopique fondamentale ( _ via _ une thorie quantique des champs ) . il est donc moins fondamental que cdm qui serait motiv par exemple par la super - symtrie . ce modle est une description effective , valable dans un rgime de champs gravitationnels faibles , comme la lisire dune galaxie ou dans un univers presque homogne et isotrope . lextrapolation du modle au champ gravitationnel rgnant dans le systme solaire nest pas entirement rsolue . dun autre ct le problme de comment tester ( et ventuellement falsifier ) ce modle en cosmologie reste ouvert . m. milgrom , astrophys . j. * 270 * , 365 ( 1983 ) . bekenstein , phys . rev . d * 70 * , 083509 ( 2004 ) . sanders , mon . not . 363 * , 459 ( 2005 ) . l. blanchet , class . * 24 * , 3529 ( 2007 ) . l. blanchet and a. le tiec , phys . d * 78 * , 024031 ( 2008 ) ; and submitted , arxiv:0901.3114 ( 2009 ) . un dilectrique est un matriau isolant , qui ne laisse pas passer les courants , car tous les lectrons sont rattachs des sites atomiques . nanmoins , les atomes du dilectrique ragissent la prsence dun champ lectrique extrieur : le noyau de latome charg positivement se dplace en direction du champ lectrique , tandis que le barycentre des charges ngatives cest - - dire le nuage lectronique se dplace dans la direction oppose . on peut modliser la rponse de latome au champ lectrique par un diple lectrique @xmath68 qui est une charge @xmath69 spare dune charge @xmath70 par le vecteur @xmath71 , et align avec le champ lectrique . la densit des diples nous donne la polarisation @xmath72 . le champ cre par les diples se rajoute au champ extrieur ( engendr par des charges extrieures @xmath73 ) et a pour source la densit de charge de polarisation qui est donne par la divergence de la polarisation : @xmath74 . ainsi lquation de gauss ( qui scrit normalement @xmath75 ) devient en prsence du dilectrique @xmath76 en utilisant les conventions habituelles , avec @xmath77 . on introduit un coefficient de susceptibilit lectrique @xmath78 qui intervient dans la relation de proportionalit entre la polarisation et le champ lectrique : @xmath79 , ainsi : @xmath80 . la susceptibilit est positive , @xmath81 , ce qui implique que le champ dans un dilectrique est plus faible que dans le vide . cest leffet d_crantage _ de la charge par les charges de polarisation . ainsi garnir lespace intrieur aux plaques dun condensateur avec un matriau dilectrique diminue lintensit du champ lectrique , et donc augmente la capacit du condensateur pour une tension donne .
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pour lastrophysicien qui aborde le puzzle de la matire noire , celle - ci apparat sous deux aspects diffrents : dune part en cosmologie , cest - - dire trs grandes chelles , o elle semble tre forme dun bain de particules , et dautre part lchelle des galaxies , o elle est dcrite par un ensemble de phnomnes trs particuliers , qui paraissent incompatibles avec sa description en termes de particules , et qui font dire certains que lon est en prsence dune modification de la loi de la gravitation .
rconcilier ces deux aspects distincts de la matire noire dans un mme formalisme thorique reprsente un dfi important qui pourrait peut - tre conduire une physique nouvelle en action aux chelles astronomiques .
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hard x - ray surveys are the most direct probe of supermassive black hole ( smbh ) accretion activity , which is recorded in the cosmic x - ray background ( cxb ) , in wide ranges of smbh masses , down to @xmath3 , and bolometric luminosities , down to @xmath4 erg / s . x - ray surveys can therefore be used to : study the evolution of the accreting sources ; measure the smbh mass density ; constrain models for the cxb @xcite , and models for the formation and evolution of the structure in the universe @xcite . these studies have so far confirmed , at least qualitatively , the predictions of standard agn synthesis models for the cxb : the 2 - 10 kev cxb is mostly made by the superposition of obscured and unobscured agns ( @xcite and references therein ) . quantitatively , though , rather surprising results are emerging : a rather narrow peak in the range z=0.7 - 1 is present in the redshift distributions from ultra - deep chandra and xmm - newton pencil - beam surveys , in contrast to the broader maximum observed in previous shallower soft x - ray surveys made by rosat , and predicted by the above mentioned synthesis models . however , the optical identification of the faint sources in these ultra - deep surveys is rather incomplete , especially for the sources with very faint optical counterparts , i.e. sources with high x - ray to optical flux ratio ( x / o ) . indeed , the optical magnitude of @xmath5 of the sources , those having the higher x / o , is r@xmath6 , not amenable at present to optical spectroscopy . this limitation leads to a strong bias in ultra - deep chandra and xmm - newton surveys against agn highly obscured in the optical , i.e. against type 2 qsos , and in fact , only 10 type 2 qsos have been identified in the cdfn and cdfs samples @xcite . to help overcoming this problem , we are pursuing a large area , medium - deep surveys , the hellas2xmm serendipitous survey , which , using xmm - newton archival observations @xcite has the goal to cover @xmath7 deg@xmath8 at a 2 - 10 kev flux limit of a few@xmath9 . at this flux limit several sources with x / o@xmath1 have optical magnitudes r=24 - 25 , bright enough for reliable spectroscopic redshifts to be obtained with 10 m class telescopes . we have obtained , so far , optical photometric and spectroscopic follow - up of 122 sources in five xmm - newton fields , covering a total of @xmath10 deg@xmath8 ( the hellas2xmm ` 1df ' sample ) , down to a flux limit of f@xmath11 erg @xmath12 s@xmath13 . we found optical counterparts brighter than r@xmath14 within @xmath15 from the x - ray position in 116 cases and obtained optical spectroscopic redshifts and classification for 94 of these sources @xcite . the source breakdown includes : 61 broad line qso and seyfert 1 galaxies , and 33 _ optically obscured agn _ , i.e. agn whose nuclear optical emission , is totally or strongly reduced by dust and gas in the nuclear region and/or in the host galaxy ( thus including objects with optical spectra typical of type 2 agns , emission line galaxies and early type galaxies , but with x - ray luminosity @xmath16 erg s@xmath13 ) . we have combined the hellas2xmm 1df sample with other deeper hard x - ray samples including the cdfn @xcite , lockman hole @xcite , and ssa13 @xcite samples , to collect a `` combined '' sample of 317 hard x - ray selected sources , 221 ( 70% ) of them identified with an optical counterpart whose redshift is available . the flux of the sources in the combined sample spans in the range @xmath17 and the source breakdown includes 113 broad line agn and 108 optically obscured agn . -5.7truecm [ xos ] -0.5truecm fig . [ xos ] shows the x - ray ( 2 - 10 kev ) to optical ( r band ) flux ratio ( x / o ) as a function of the hard x - ray flux for the combined sample . about 20% of the sources have x / o@xmath1 , i.e ten times or more higher than the x / o typical of optically selected agn . at the flux limit of the hellas2xmm 1df sample several sources with x / o@xmath1 have optical magnitudes r=24 - 25 , bright enough to obtain reliable spectroscopic redshifts . indeed , we were able to obtain spectroscopic redshifts and classification of 13 out of the 28 hellas2xmm 1df sources with x / o@xmath18 ; _ 8 of them are type 2 qso at z=0.7 - 1.8 _ , to be compared with the total of 10 type 2 qsos identified in the cdfn @xcite and cdfs @xcite . [ xolx ] show the x - ray to optical flux ratio as a function of the x - ray luminosity for broad line agn ( left panel ) and non broad line agn and galaxies ( central panel ) . while the x / o of the broad line agns is not correlated with the luminosity , a striking correlation between log(x / o ) and log(l@xmath19 ) is present for the obscured agn : higher x - ray luminosity , optically obscured agn tend to have higher x / o . a similar correlation is obtained computing the ratio between the x - ray and optical luminosities , instead of fluxes ( because the differences in the k corrections for the x - ray and optical fluxes are small in comparison to the large spread in x / o ) . all objects plotted in the right panel of fig . [ xolx ] do not show broad emission lines , i.e. the nuclear optical - uv light is completely blocked , or strongly reduced in these objects , unlike the x - ray light . indeed , the optical r band light of these objects is dominated by the host galaxy and , therefore , _ x / o is roughly a ratio between the nuclear x - ray flux and the host galaxy starlight flux_. the right panel of figure [ xolx ] helps to understand the origin of the correlation between x / o and l@xmath19 . while the x - ray luminosity of the optically obscured agns spans about 4 decades , the host galaxy r band luminosity is distributed over less than one decade . the ratio between the two luminosities ( and hence the ratio between the two fluxes , see above ) results , therefore , strongly correlated with the x - ray luminosity . -0.5truecm we have obtained spectroscopic redshifts and classification of 13 out of the 28 hellas2xmm 1df sources with x / o@xmath1 : the majority of these sources ( 8) are type 2 qsos at z=0.7 - 1.8 , a fraction of type 2 qsos much higher than at lower x / o values . we find a strong correlation between x / o and the x - ray luminosity of optically obscured agn , x / o=10 corresponding to an ( average ) 2 - 10 kev luminosity of @xmath20 erg s@xmath13 . sources of this luminosity and flux @xmath21 , reachable in chandra and xmm - newton ultra - deep surveys , would be at z@xmath22 . although only 20% of the x - ray sources have such high x / o , they may carry the largest fraction of accretion power from that shell of universe . intriguingly , mignoli et al . ( 2003 in preparation ) find a strong correlation between the r - k color and the x / o ratio for a selected sample of 10 high x / o hellas2xmm 1df sources , all of them having r - k@xmath23 , i.e. they are all extremely red objects . 1 setti , g. , & woltjer , l. 1989 , , 224 , l21 comastri , a. , setti , g. , zamorani , g. , & hasinger , g. 1995 , , 296 , 1 haehnelt , m. carnegie observatories astrophysics series , vol . 1 : coevolution of black holes and galaxies , ed . l. c. ho ( cambridge univ . press ) , 2003 , astro - ph/0307378 menci , n. et al . 2003 , , 587 , l63 hasinger , g. 2003 , proceedings of the conference : the emergence of cosmic structure , maryland , stephen s. holt and chris reynolds ( eds ) , astro - ph/0302574 fiore , f. 2003 , proceedings of the symposium `` the restless high - energy universe '' , e.p.j . van den heuvel , j.j.m . in t zand , and r.a.m.j . wijers eds , astro - ph/0309355 cowie l. , barger a. , bautz , m.w . , brandt , w.n . , & garnire , g.p . 2003 , , 584 , l57 fiore , f. brusa , m , cocchia , f. et al . 2003 , a&a in press , astro - ph/0306556 baldi , a. , molendi , s. , comastri , a. , fiore , f. , matt , g. , & vignali , c. 2002 , , 564 , 190 barger a. , et al . 2002 , , 124 , 1839 barger , a. , cowie , l. , mushotzky , r.f . , & richards , e.a . 2001 , , 121 , 662 mainieri , v. et al . 2002 , , 393 , 425
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we present results from the photometric and spectroscopic identification of 122 x - ray sources recently discovered by xmm - newton in the 2 - 10 kev band ( the hellas2xmm 1df sample ) .
one of the most interesting results ( which is found also in deeper sourveys ) is that @xmath0 of the sources have an x - ray to optical flux ratio ( x / o ) ten times or more higher than that of optically selected agn . unlike the faint sources found in the ultra - deep chandra and xmm - newton surveys , which reach x - ray ( and optical )
fluxes @xmath1 times lower than in the hellas2xmm sample , many of the extreme x / o sources in our sample have r@xmath2 and are therefore accessible to optical spectroscopy .
we report the identification of 13 sources with extreme x / o values .
while four of these sources are broad line qso , eight of them are narrow line qso , seemingly the extension to very high luminosity of the type 2 seyfert galaxies .
x - ray : background , x - ray : surveys , qso : evolution
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the discovery of the dna double - helical structure , some 50 years ago , motivated the elaboration of the helix - coil model to account for the separation of the two strands , on physical bases @xcite . the importance of this model from the biological point of view is obvious , since processing of the genetic information involves precisely the separation of the strands . of course , under physiological conditions , the opening of the double - helix is not under the effect of temperature , but the differential stabilities in dna sequences , as revealed by helix - coil analysis , could be sensed by biological effectors , such as proteins , under various types of constraints . the successful development of the helix - coil denaturation model required appropriate elaborations for the physics and the algorithmics , allowing accurate tests through comparisons with experimental data ( melting curves ) . this field , very active in the sixties and seventies , has benefited recently from a renewed interest both from the biological side , for example in the context of genomic analysis , and from the physics side , notably in relation with questions relevant to the order of the transition in the homogeneous case and the effect of sequence heterogeneity . in the light of these still debated issues , both from the theoretical and the numerical points of view , the main focus of the present work is the numerical investigation of the relevance of disorder in a realistic dna denaturation model _ la _ poland - scheraga , in which self - avoidance between loops and the rest of the chain is also taken into account . in what follows , before further detailing the particular system considered and the open questions , we first recall briefly the general background in terms of biological models , numerical methods and previous results . _ basics for dna denaturation : _ dna denaturation is an entropy driven transition , in which at some critical temperature @xmath6 the energy loss @xmath7 with the opening of base pairs is compensated by the entropic gain @xmath8 associated with the increased number of configurations accessible to the separated single strands . experimentally , it is found that @xmath6 depends on different factors , in particular the @xmath9 of the solution and the gc composition of the sequence , related to the ratio of the guanine - cytosine , gc , pairs to the adenine - thymine , at , pairs . for homogeneous sequences , for @xmath10 , typical values for @xmath6 are @xmath11 and @xmath12 , respectively for gc and at cases . such differences reflect of course the fact that the pairing of guanine to cytosine involves three hydrogen bonds whereas that of adenine to thymine involves only two . for a given biological sequence of length @xmath2 , here identified , following at and gc pairs , by the coupling energies @xmath13 , the denaturation transition can be followed with uv absorption . correspondingly , the fraction @xmath14 of closed base pairs , which is the order parameter of the transition in the thermodynamic limit @xmath15 , can be measured in such experiments based on differential absorptions for closed and open base pairs . the resulting curves display usually multi - stepped structures , with abrupt variations on small ( sequence - depending ) temperature ranges around @xmath6 . therefore , for a biological sequence of fixed length , the finite size order parameter @xmath14 varies from zero to one ( associated with complete denaturation ) , with a sequence - dependent behavior . accordingly , the derivative with respect to temperature , @xmath16 , displays typically a series of sharp peaks . from the theoretical point of view , modeling dna denaturation was essentially following two main directions : 1 ) for biological applications , in relation with melting experiments ( sixties , seventies ) , sequence - dependent algorithmic elaborations for the handling of realistic physical models @xcite , concerning notably the representation of denaturation loops , and , 2 ) for the study of the underlying physics , detailed characterizations of the properties for pure systems , neglecting sequence - specificity @xcite . _ physics of dna denaturation for homogeneous sequences : _ dna denaturation is understandable in the framework of _ almost unidimensional _ systems @xcite , and it is therefore associated with a peculiar kind of transition . in fact , the first models displayed no thermodynamic singularity @xcite , as they corresponded to @xmath17 ising models with only short - range ( nearest - neighbor ) interactions , with open and closed base pair states represented by an ising spin . it was subsequently shown , notably by poland and scheraga @xcite ( ps , in what follows ) , that the observed denaturation behavior can indeed be described in terms of a simple @xmath17 model , the helix - coil model , that consists of alternating regions of contiguous open base pairs ( coiled regions or _ loops _ ) and double - stranded ones ( helical _ segments _ ) . in this model the transition in the thermodynamic limit is made possible through the adoption of appropriate long - range entropic weights for the single - stranded loops . more recently , several other models have been considered and studied , using in particular more realistic potential forms between base pairs @xcite . since sharp transitions are observed experimentally , with abrupt changes in @xmath14 on small temperature ranges , it is expected that a model , accounting correctly for such results , should undergo a first order transition in the pure case . indeed , this point has been studied rather extensively recently @xcite . in particular , it was demonstrated @xcite that the transition is of first order in pure ps models in which excluded volume effects for loops are not only with themselves , but also with the rest of the chain . notably , with the probability distributions for loops with lengths @xmath18 at the critical point following a power law , @xmath19 , the transition is of first order for @xmath20 exponents larger than 2 @xcite ( see also @xcite ) . it was shown that in three dimensions , with the two strands described as self - avoiding walks ( saws ) , the value for the exponent is @xmath21 @xcite . in comparison , @xmath22 for random walk ( rw ) loops @xcite and @xmath23 for saw loops , with excluded volume interactions with the rest of the chain neglected @xcite . _ biological and algorithmic backgrounds for sequence - specific dna denaturations : _ the algorithmic problem was initially encountered for the implementation of sequence - specific calculations allowing notably experimental / theoretical comparisons in the study of melting curves . it seemed natural , in the beginning , to resort to transfer matrix formalisms as developed in physics because of the ising - type formulation of the problem @xcite . indeed , neglecting loop - entropy long - range effects , the calculation of the partition function for a sequence of size @xmath2 can be expressed simply as the product of @xmath2 @xmath24x@xmath24 matrices . the extension to realistic models was at first handled through extended transfer matrices , of sizes growing up to @xmath2x@xmath2 , for the proper description of interactions throughout the lengths of the sequences @xcite . because of calculation burdens associated with such matrix sizes , alternative formulations were sought for the implementation of realistic models with affordable computation times . representing the culmination of a series of developments , through some twenty years , an appropriate algorithmic solution was proposed in 1977 by fixman and freire @xcite ( ff method ) , in which calculation efficiency was not at the price of oversimplifications in the physics , but relied instead on the numerical representation of the long - range effect as a multiexponential function . in this formulation , the time complexity for the evaluation of a complete denaturation map for a sequence of length @xmath2 is essentially proportional to @xmath2 . this reduced complexity is to be compared with the intrinsic complexity of the model scaling as @xmath25 , if we were to consider exact one - way calculations along the sequence . in this background , no generalizations were proposed for the ideas in the ff for a long period of time , possibly because of the formulation of this method in the prolongation of an algorithm by poland @xcite , expressed in the rather specialized context of conditional probabilities recursions specific to the linear dna helix - coil model . as a matter of fact , the only applications of the ff concerned the implementation of the algorithms in computer programs , for dna melting calculations ( such as in the poland @xcite or in the meltsim programs @xcite ) . however , upon revisiting the original derivations , it appears that the idea associated with the multiexponential representation , relying on the fundamental property of exponential function , corresponds to a powerful concept amenable to many generalizations for realistic models with long - range effects . accordingly , based on explicit partition function calculations , the simex ( simulations with exponentials ) method was first derived as a reformulation of the ff for the linear helix - coil model @xcite , with further generalizations to higher - order models involving several , mutually coupled , long - range effects @xcite . for such systems , with two or more long - ranges , the reductions of complexities by several orders of magnitudes can be associated with calculation times reduced by million folds . the basic concepts for higher - order models were originally illustrated with a circular dna helix - coil model - problem involving two long - range contributions @xcite . the corresponding principles were further transposed to a linear helix - coil model with non - symmetrical loops @xcite . on the experimental side , linear helix - coil models were successfully compared with experimental denaturation curves @xcite . in the beginning , one of the motivations in the elaboration of dna denaturation models was to ask the question of possible relations between genetics ( coding / non - coding ) and physical ( helix / coil ) segmentations , because of the importance of the separation of the two strands in the processing of the genetic information . with little genomic sequences available at the end of the seventies and beginning of eighties , no clearcut conclusions were reached for such relations . more recently , with the availability of complete genomes , it was possible to resume such investigations on much larger bases , demonstrating variable correspondences between the two types of segmentations , depending on the genomes @xcite . for genomes with very sharp correlations it was even possible to propose _ ab initio _ gene identification methods purely on physical bases @xcite . _ physics of dna denaturation with disorder : _ on the physical side , dna denaturation models are associated with important open questions such as , notably , the relevance of sequence - heterogeneity for the thermodynamic limit behavior . at the beginning of the seventies , it was noted by poland and scheraga @xcite that `` sequence heterogeneity dramatically broadens the transition '' , but it is only recently that such problem has been addressed on more rigorous bases and the transition in disordered ps models with @xmath26 was also investigated @xcite . indeed , in the homogeneous case , such systems exhibit peculiar first order transitions characterized by a diverging correlation length , and it is therefore not clear to which extent general theoretical results on the effect of disorder @xcite can be applied . the question has been addressed both with analytical @xcite and numerical approaches , with either off - lattice @xcite or on - lattice @xcite implementations in this latter case . it appears however hard to reconcile these various results . the off - lattice studies of @xcite , involving very large chain lengths , suggest a peculiar transition , of first order as in the pure case but not obeying usual finite size scaling and exhibiting two different correlation length exponents , associated respectively with _ typical _ and _ average _ quantities . on the other hand , the monte carlo like numerical simulations in @xcite , limited to small chain lengths , agree with a second order transition in the presence of disorder , though it was not possible to rule out completely a transition still of first order . finally , from the analytical standpoint , under quite general hypotheses encompassing ps models with @xmath26 , it was shown that the transition is expected to be at least of second order and possibly smoother @xcite . in the background above , in addition to their relevance to experimental dna denaturation , ps models with sequence heterogeneity represent interesting toy - models for addressing general open questions relative to the proprieties of random fixed points . the detailed study of such systems could also help elaborating the correct approaches to be used in the interpretation of data on disordered models . in this direction , we perform here a numerical analysis of a disordered ps model with @xmath0 . relying on appropriate algorithmic formulations ( simex ) we consider long sequences . with the definition adopted for the model , and the choices for the parameter values , the calculations are made directly comparable with previous on - lattice results @xcite . in addition , the observed behavior should be also related to that found in the previous off - lattice studies of a different disordered ps model with @xmath0 @xcite , in which the same multiexponential representation for the long - range loop entropy law was adopted . our findings show the existence of very strong corrections to scaling . moreover it appears that , for a given size , the effect of disorder is qualitatively described by an appropriately defined _ intrinsic _ length scale @xmath1 depending on model parameters . these observations provide a possible explanation for the discrepancies between previous results @xcite , as well as for an apparent dependence of the evaluated critical exponents on model parameters noted in @xcite . in fact , in the frame of the picture proposed here , the size at which the effect of disorder becomes evident could diverge exponentially with @xmath1 . more precisely , for the value @xmath3 chosen for the present detailed study , it is possible to observe a crossover between a nearly _ pure system like _ behavior , consistent with the one observed in simulations @xcite , and the apparent asymptotic one . with corrections to scaling taken into account , the model clearly displays a smooth transition , corresponding to a value for the correlation length exponent @xmath5 . nevertheless , since our results refer to average quantities , they do not rule out the possibility suggested in @xcite of a transition governed by two different correlation lengths . the analysis for the clarification of this point is left for a forthcoming work @xcite . pure ps models for dna denaturation are described in a rather extensive literature , and in particular the ingredients which make the transition of first order are discussed in several recent works @xcite . here we will only recall results for linear ps models with symmetric loops , in which one fully takes into account self - avoidance through an appropriate choice of the loop length distribution probability exponent @xmath20 . the position of a base pair along the sequence is labeled by @xmath27 ( @xmath28 ) and its configurational state is represented by @xmath29 , with @xmath30 for a closed pair and @xmath31 for an open pair . in the corresponding on - lattice representation , the two strands in the model can be visualized as two interacting rws with the same origin in @xmath32 dimensions . a pair is in the closed state if and only if the two bases are in the same position @xmath27 along the two strands and occupy the same lattice point @xcite ( rw - dna ) . one can write the canonical partition function for the system in the form : @xmath33 where @xmath34 is the number of configurations with the same total energy @xmath35 and @xmath36 is the inverse temperature , taking boltzmann constant @xmath37 for simplicity . the contribution of a single base pair in the closed state to the total energy @xmath35 is @xmath38 , independent from the position @xmath27 in the pure case . the number of configurations of a closed segment of length @xmath39 increases as @xmath40 and therefore its contribution to the total entropy @xmath41 is @xmath42 . here @xmath43 is a parameter of the model , interpretable as the on - lattice connectivity constant ( @xmath44 for @xmath32-dimensional rws on a cubic lattice ) . a denatured region of length @xmath18 is associated with a single - stranded loop of length @xmath45 , and the corresponding number of configurations is given by @xmath46 . the factor @xmath47 takes into account the fact that the two separated chains have to meet again at some distant point , and the relation @xmath48 holds for @xmath32-dimensional rws . assuming that the first and the last base pairs are always coupled , a given configuration of the system is described by the lengths of the closed segments @xmath49 and by those of the denatured loops @xmath50 , with @xmath51 . its energy is @xmath52 , only depending on @xmath53 . therefore , in the partition function , the degeneracy factor @xmath34 includes both the entropic contribution of segments @xmath54 and that of loops @xmath55 , the latter being to be summed over all the possible sets @xmath56 associated with a given total length @xmath57 . correspondingly , the partition function can be also written as : @xmath58 it can be noted that the factor @xmath59 contributes an additive constant to the entropy and it is therefore not relevant for the description of the thermodynamics of the system . in what follows , we study properties of @xmath60 . accordingly , the number of loops of length @xmath61 is taken equal to @xmath62 and a negative contribution to the entropy @xmath63 is associated to each base pair in the closed state . it is seen qualitatively that the possible change in the thermodynamic limit behavior occurs at the temperature for which @xmath64 . from the last expression in ( [ zn ] ) it is moreover clear that , in the computation of the grand canonical partition function @xmath65 , the contributions of helical segments and loops are decoupled and one obtains a geometric series @xcite : @xmath66 \left [ \sum_{l=2}^{\infty } { \frac{z^l}{(2l)^{c_p}}}\right ] \right \}^\rho = \frac{{\cal z}_{s}{\cal z}_{l } } { 1-{\cal z}_{s } { \cal z}_{l}},\ ] ] where we introduced the fugacity @xmath67 and @xmath68 and @xmath69 refer to the _ segment _ ( helical ) and _ loop _ ( coil ) grand partition functions respectively : @xmath70 since the behavior for @xmath15 is dictated by the fugacity value @xmath71 corresponding to the pole nearest to the origin , the system undergoes a phase transition when a critical temperature @xmath6 is found below which the zero of the denominator becomes smaller than the smallest pole of the numerator . the possibility of the transition and its order both depend on the value of the exponent @xmath20 @xcite . in detail , for @xmath72 there is no thermodynamic singularity , whereas for @xmath73 the following situations must be distinguished : a smooth transition for @xmath74 with a specific heat exponent @xmath75 , a second order transition for @xmath76 and finally a first order transition for @xmath77 . in fact , these distinctions can be understood considering the properties of @xmath69 , _ i.e. _ those of the distribution probability of the loop length at the ( possible ) critical point @xmath78 : @xmath79 in the case @xmath80 , the mean length @xmath81 for loops at the critical point diverges and the system exhibits large coiled regions , in which most of the bases are involved . on the contrary , for @xmath26 , the mean loop length at @xmath6 is finite and correspondingly it is possible to show that the density of closed base pairs @xmath82 , and therefore the energy density , varies abruptly in the thermodynamic limit , from the value zero at high temperatures to a finite value at @xmath6 . in detail , when approaching the critical point from the low temperature ( helical ) phase , one finds @xcite : @xmath83 for example , the rw - dna model in three dimensions exhibits a second order transition with @xmath84 , since @xmath22 , whereas in five dimensions it undergoes a first order transition , since @xmath85 , as confirmed both by exact computations of thermodynamic quantities and by on - lattice numerical simulations @xcite . all interactions between different loops and helical segments are neglected in classical calculations of the grand canonical partition functions for helix - coil models . it is possible to account for self - avoidance of each loop with itself through the appropriate choice of the exponent @xmath23 @xcite , corresponding roughly to the value adopted usually for comparisons with experimental data @xcite . more recently , it was demonstrated that self - avoidance of the loops with the rest of the chain can be also taken into account , and that intriguingly the pure ps models exhibit first order transitions in this case @xcite . the exponent @xmath20 , corresponding to a self - avoiding loop embedded in a self - avoiding chain , can be predicted from conformal theory results @xcite , and in particular it was found that @xmath21 in three dimensions , the transition being of first order also in @xmath86 . it is notable that such determination provides the appropriate value of @xmath20 to be used as an _ input _ in off - lattice calculations . in the monte carlo like simulations , one studies an on - lattice model ( saw - dna ) in which self - avoidance is completely taken into account , by considering two interacting saws , with two monomers allowed to occupy the same lattice point if and only if their positions along the two chains are identical , thus representing complementary base pairs @xcite . in the pure @xmath87 case , it was found that this system exhibits a first order transition , with the maximum of the specific heat diverging linearly with the chain length . it was subsequently shown @xcite that the value of the exponent describing the probability distribution for the loop lengths at the critical point is in perfect agreement with the theoretical prediction , @xmath88 . an off - lattice pure ps model with @xmath0 was also studied numerically @xcite , finding the same scaling behavior than in @xmath87 saw - dna , apart from the strong finite size corrections which appear to be more important in the on - lattice situation . even though of first order , the transition is characterized by a diverging correlation length , which can be identified from the behavior of @xmath89 @xcite : @xmath90 with @xmath91 where @xmath92 for @xmath26 and @xmath93 for second order ( or smoother ) transitions . it can also be predicted that the free energy density @xmath94 takes the value zero in the high temperature phase and that it behaves proportionally to @xmath95 for @xmath96 , leading again to the behavior of the energy density and of the order parameter for different @xmath20 values given in ( [ orpa ] ) . therefore , the hyperscaling relation @xmath97 is clearly fulfilled , both for @xmath80 and for the first order ( @xmath98 ) case . disorder is introduced to account for sequence - heterogeneity , with parameter values depending on the chemical nature of base pairs ( at or gc ) at a given position along a sequence . there are a few studies on the effect of disorder on general properties of dna denaturation models in which self - avoidance is neglected @xcite . previous numerical works on disordered models _ la _ ps was mainly for comparison of the predictions with experimental data and genetic signals @xcite and for the study of the effect of base pair mismatches @xcite , where one usually takes also into account the stacking contributions , with the coupling energies depending on the chemical nature of base pairs at positions @xmath27 and @xmath99 . for comparisons with experimental melting curves , it is moreover necessary to take into account the possibility for complete dissociation of the two strands in the molecule @xcite . we also notice that biological sequences exhibit long - range correlations and strong variability in gc compositions , both according to genomes and within chromosomes . letting aside for the present such sophistication , we sum up some recent results for simple disordered models _ la _ ps with self - avoidance , such as those considered in off - lattice @xcite and on - lattice @xcite studies , which allow nevertheless to capture essential features of the effect of disorder . in these various works , disorder enters only through the position dependent contribution of a closed base pair to the total energy . in detail , the @xmath100 are quenched random variables distributed following a binomial probability , corresponding to gc composition equal to 1/2 : @xmath101 , \label{peps}\ ] ] with @xmath102 . one is interested in the thermodynamic properties of the quenched free energy density : @xmath103 where , as usual , @xmath104 denotes the average over disorder and @xmath105 is a self - averaging quantity in these models @xcite . generally speaking it is known , from the theoretical point of view , that disorder can modify very significantly the fixed point of a system , and therefore its critical exponents . in what follows , the notations with subscripts @xmath106 and @xmath107 refer to the ( possibly different ) _ pure _ and _ random _ system fixed points , respectively . a series of results @xcite , and notably the well known harris criterion @xcite , demonstrate that disorder is relevant as soon as the specific heat exponent fulfills the condition @xmath108 . correspondingly , in the presence of disorder , the transition becomes smoother and it is in particular expected that @xmath109 , _ i.e. _ , from the hyperscaling relation , a correlation length exponent @xmath110 @xcite . it is however important to stress that these results are obtained essentially for magnetic systems and it is not clear to which extent they are relevant to the case here considered , concerning interacting polymers undergoing a first order transition characterized by a diverging correlation length . an analysis in terms of pseudo - critical temperatures @xcite was applied recently to disordered ps models with different @xmath20 values @xcite . in these studies , an appropriate sample - dependent @xmath111 was defined and measured , looking in particular at the associated probability distribution . the results point towards irrelevance of disorder for @xmath22 , corresponding to the marginal case @xmath84 . on the contrary , relevance of disorder was found for @xmath112 . importantly , peculiar behavior was observed for the value @xmath0 , whereas the situation appears to be clear for @xmath113 . indeed , for @xmath113 , compatible estimates for @xmath114 were obtained from the scaling of the average pseudo - critical temperature @xmath115 and from that of ( the square root of ) its fluctuations @xmath116 . in the case @xmath0 , instead , a scaling of the average critical temperature @xmath117 was reported , suggesting a still first order transition with exponent @xmath118 . however it was also found a scaling for the fluctuations following @xmath119 , which was associated to a different exponent @xmath120 . it was then suggested @xcite that these observations are compatible with a system still exhibiting a first order transition but in which scaling laws are no more fulfilled , characterized by two different correlation lengths : a _ typical _ one , @xmath121 , describing the behavior of a typical sample in the thermodynamic limit , with @xmath118 , and an _ average _ one , @xmath122 , describing the behavior of average quantities , dominated by rare fluctuations , with @xmath123 . the resulting two - sided scenario is therefore that disorder is irrelevant to the typical sample and , in the same time , the obtainment of the @xmath124 value is in agreement with the theoretical expectations @xcite . by contrast , usual finite size scaling analysis of monte carlo like simulations results on a @xmath87 disordered saw - dna model ( dsaw - dna ) @xcite , suggested a transition governed by an ( average ) correlation length exponent @xmath125 . however , in this study , average energy curves were observed to cross at the same point within the errors in the estimations , and accordingly the possibility @xmath126 could not be completely ruled out . the findings were further confirmed by the analysis of the behavior of @xmath127 at the critical point , which led to the compatible value @xmath128 when considering the largest sizes and taking into account the presence of a finite correlation length . but again , particularly with the smallest chain lengths , estimations @xmath129 , still compatible with a first order transition , were obtained . it is noticeable that the affordable sizes in monte carlo like studies are significantly smaller ( factors of order @xmath130 ) than the sequence lengths accessible to off - lattice recursive canonical partition function calculations for ps models . finally , in recent theoretical works based on a probabilistic approach @xcite , it was shown for a general class of interacting polymer models that the transition becomes at least of second order in the presence of disorder . the frame of this approach covers the ps models with @xmath112 , including the case @xmath26 with corresponding first order transition in the pure system . accordingly , these conclusions are expected to also cover the @xmath87 dsaw - dna case . following such studies , it is expected that @xmath131 both for average and typical quantities , though the possibility of different correlation lengths is not ruled out @xcite . on the other hand , according to other theoretical results in which self avoidance is neglected @xcite , disordered models _ la _ ps could undergo a definitely smoother transition , corresponding to an essential singularity in the free energy . before presenting our numerical findings , it is worth discussing qualitative features expected , at fixed chain lengths @xmath2 , for the behavior of disordered ps models . as previously recalled , disorder should be relevant to these models as soon as @xmath112 . moreover , on general grounds , one can argue that the behavior of a system near the transition point is governed by given critical exponents which do not depend on model details . correspondingly , one would expect that both the form of @xmath132 and the precise choices for the parameters ( here @xmath133 , @xmath43 and gc composition ) should not correspond to different thermodynamic limit singularities . on the other hand , such choices could have strong influence on finite size effects . the disordered ps model in @xcite and the @xmath87 dsaw - dna in @xcite involve the same @xmath0 , either as a direct input for the recursive calculations or as consequence of the implementation of self - avoidance in the simulated model . nevertheless , there is a first noticeable difference between the two systems studied , as the off - lattice calculations were inspired from a wetting transition model @xcite , in which it is forbidden for two consecutive elements to be in the closed state simultaneously ( _ i.e. _ , in our notation , if @xmath30 then @xmath134 ) . also the connectivity constant @xmath43 is not the same , since it was fixed to the value @xmath135 in @xcite , whereas it is an output of the model in the on - lattice simulations , and one finds @xmath136 for saws on a @xmath87 cubic lattice . in addition , the two studies involved significantly different @xmath137 values . in @xcite the choice @xmath138 was adopted , for obtaining a critical temperature ratio @xmath139 close to the experimental value . on the other hand , in @xcite , the value @xmath140 was studied in detail and the values @xmath141 and @xmath142 ( corresponding to the choice @xmath143 and @xmath144 ) were also considered . in the latter study , preliminary results suggested that the ( average ) correlation length exponent @xmath145 could increase with @xmath137 , ranging from @xmath146 for @xmath140 to @xmath147 for @xmath142 . for proper understanding of these findings , and for a qualitative analysis of the expected finite size behavior , it can be important to consider in some detail the potential key role of _ rare regions _ in the generated sequences . indeed , the possible presence of such regions , of large enough size , can explain the presence of strong corrections to scaling , which could therefore depend both on model details and on the precise choice for the parameters . it can be noted that temperature and disorder appear only in the @xmath148 $ ] terms in the partition function , which are clearly invariant under the transformation @xmath149 . moreover , in the pure system , the transition occurs around the temperature @xmath150 , at which the energetic contribution for the two bound chains is of the same order than the entropic loss . in the presence of disorder , for a given sequence , it is expected to observe the multi - step behavior in @xmath14 displayed by experimental dna denaturation curves . this results from the presence of regions with different local contents in terms of gc to at ratios , associated accordingly with different local melting temperatures . in the simplest extreme case , one imagines two regions @xmath151 and @xmath152 , of about the same length @xmath153 , completely dominated by at and gc compositions respectively . in such situation , the local transition in region @xmath151 is driven by @xmath154 energies , with local critical temperature @xmath155 , whereas the local transition in region @xmath152 is associated with the higher local critical temperature @xmath156 . in this illustrative example , for a given temperature , the contributions in the partition function of total @xmath157 factors corresponding to the configurations with base pairs in the closed state , will be significantly different for @xmath151 and @xmath152 regions . one obtains , for @xmath158 and @xmath159 : @xmath160 \sim o(1 ) , \nonumber \\ \pi^{tot}_{b } & = & \prod_{i \in b } \pi_i = \exp \left [ { l ( \beta_{c , loc}(b ) \epsilon_{gc}-\log \mu ) } \right ] \sim \exp(-l / x).\end{aligned}\ ] ] since , whereas @xmath161 , one has @xmath162 defining the parameter @xmath1 . from these expressions it is possible to argue that the larger the value of @xmath153 with respect to @xmath1 the more the effect of disorder will be _ felt _ by the finite size system , _ i.e. _ the difference between the weights of configurations corresponding to closed @xmath151 and @xmath152 regions in the partition function will be higher . on the other hand , the probability for such an extreme case in a particular sequence of size @xmath2 is quite small . with the choice for @xmath163 in ( [ peps ] ) , large @xmath2 and @xmath153 values and @xmath164 , the probability of @xmath153 contiguous elements of the same type is simply @xmath165 . therefore this probability , though approaching 1 in the thermodynamic limit for any finite @xmath153 , becomes rapidly negligible with increasing @xmath153 for fixed chain length @xmath2 . following these considerations , for the finite size system to feel the effect of disorder , the chain lengths necessary for observing _ rare regions _ with @xmath166 could be not reachable for large @xmath1 values . for more quantitative analysis , at least in the extreme case considered , let us suppose that at the length scale @xmath167 a region of size @xmath168 is observed for the parameter value @xmath169 , such that @xmath170 , with non negligible probability @xmath171 . then , in order to make the same observation for @xmath172 , it will be necessary to consider length scales of order @xmath173 $ ] , thus involving exponential increases with the ratio @xmath174 for the sequence lengths @xmath2 . for attempting to understand the role of the long - range loop entropic effect , one can impose that , at the temperature @xmath175 , the weight of the configuration associated with region @xmath151 in the closed state is significantly smaller than that of the configuration associated with an open region corresponding to a single loop ( of size @xmath153 ) , getting the condition @xmath176 . correspondingly , one can argue that , with increasingly larger @xmath20 values , increasingly larger _ rare region _ lengths will be necessary for observing cooperative melting behavior at different temperatures . moreover , the considered extreme case seems particularly appropriate for a qualitative description when @xmath77 . here , the first order character of the transition in the pure system and the corresponding favored formation of small loops suggests that larger differences of local at to gc content ratio are necessary for obtaining different local melting temperatures . in conclusion , important finite size corrections to scaling are expected qualitatively , which could in particular depend strongly on the parameter @xmath1 introduced above , involving both the energy ratio @xmath137 and the connectivity constant @xmath43 . specifically , the effect of disorder on the behavior of the system could become evident only for chain lengths diverging exponentially with @xmath1 . this parameter seems therefore to play the role of an _ intrinsic _ length scale for the _ rare regions _ , corresponding to the logarithm of an intrinsic length scale for the system itself . setting aside other differences , the parameter choices in @xcite correspond to @xmath177 , whereas in @xcite they give @xmath178 . it is accordingly possible that results in the two studies can be explained following the described picture , on the basis of the underlying finite size effects . it is nevertheless to be noted that in @xcite very large sequences , up to @xmath179 , were considered . the present qualitative picture could anyway explain the observations in @xcite , for an apparent dependence of @xmath145 on @xmath137 , as an increase in @xmath137 at fixed @xmath180 amounts to a decrease in @xmath1 . it is possible that the chain lengths @xmath181 affordable in the simulations were not large enough and that also the value @xmath147 obtained with @xmath142 was affected by finite size corrections to scaling , being therefore to be interpreted as a lower bound for the ( asymptotic ) @xmath145 . we consider the disordered ps model , with @xmath0 , described by the canonical partition function : @xmath182 where @xmath183 runs over all the loop lengths @xmath184 , associated with a given configuration of the @xmath185 . importantly , here and in the following we only impose the condition @xmath186 , thus allowing for free ends , with separated strands not involved in a loop at one end - extremity of the sequence . such free - ends are not expected to modify the thermodynamic limit behavior @xcite , but the high temperature phase of the model will consist accordingly of two strands linked only at @xmath187 ( instead of a large loop , for a system with both extremities required to be in the coupled state ) . this so - called _ bound - unbound _ ( @xmath188 ) model corresponds more closely to the one studied usually in on - lattice simulations @xcite , and it was also adopted in @xcite . we notice that the factor @xmath189 cancels out when looking at @xmath190 . in the present study , we are moreover implicitly adopting the value @xmath191 for the cooperativity factor , where the parameter @xmath192 gives a measure for the barrier to overcome for the initiation of a loop opening . in realistic sequence - specific calculations @xcite , one uses typically @xmath193 . it is however not clear what choice for @xmath192 is appropriate when @xmath0 since in experimental / theoretical comparisons an exponent @xmath194 is generally taken and in a recent study @xcite it was suggested that the values for @xmath192 and @xmath20 should vary in parallel in order to reproduce correctly experimental melting curves . we note that small @xmath192 values could increase corrections to scaling , whereas this parameter is not expected to influence the thermodynamic properties . disordered ps models can be solved numerically by writing down recursive equations for the partition function with a simex scheme @xcite , taking into account efficiently the long - range entropic loop weights . a basic idea in the recursive scheme is that the _ forward _ partition function @xmath195 , which accounts for all configurations up to position @xmath196 along the sequence with both base pairs at positions @xmath187 and @xmath197 in the closed state ( @xmath198 ) , can be obtained from @xmath199 : @xmath200^{c_p } } \right ] . \label{zf}\ ] ] we can similarly write down the equation for the _ backward _ partition function : @xmath201^{c_p } } + 1 \right ] , \label{zb}\ ] ] with the last term in the equation above corresponding to the free - end configuration ( @xmath202 for @xmath203 ) . the canonical partition function for the complete chain of length @xmath2 is given by : @xmath204 where the forward sum takes into account the free - ends . with these calculations one obtains in particular the probability for a base pair at position @xmath27 ( along the sequence of length @xmath2 ) to be in the closed state as : @xmath205 where the division with @xmath206 is to rectify the double counting of the corresponding factor ( involved both in @xmath207 and in @xmath208 ) . for a given disorder sequence @xmath209 , at fixed chain length @xmath2 and temperature @xmath210 , we can derive from the @xmath211 ( [ ps ] ) quantities of interest such as the density of closed at base pairs @xmath212 ( respectively gc , @xmath213 ) , the total density of closed base pairs @xmath214 and the energy density @xmath215 : @xmath216 . \end{aligned}\ ] ] we can also consider the specific heat @xmath217 as well as the derivative of the density of opened base pairs @xmath218 , which is relevant to experimental determinations : @xmath219 since @xmath220 , the energy density @xmath215 and the order parameter @xmath221 can exhibit different behaviors , and accordingly such can be also the case for @xmath222 and @xmath223 . in the same direction we consider also the susceptibility , obtained as : @xmath224 = \nonumber \\ & = & \frac{1}{\beta } \left [ \frac { d\theta_{at,\epsilon}(t , n)}{d\epsilon_{at}}+ \frac { d\theta_{gc,\epsilon}(t , n)}{d\epsilon_{gc } } + \frac { d\theta_{at,\epsilon}(t , n)}{d\epsilon_{gc}}+ \frac { d\theta_{gc,\epsilon}(t , n)}{d\epsilon_{at } } \right ] , \label{chi}\end{aligned}\ ] ] providing interestingly a possibility for checking numerical accuracy in the computations , from the fulfillment of the equality @xmath225 . we study moreover the behavior of the ( non - normalized ) loop length probability distribution : @xmath226 with @xmath227 , independent from the disorder sequence . therefore we introduce the quantity : @xmath228 noting that in the pure model @xmath229 ( see ( [ plt ] ) ) and correspondingly @xmath230 for @xmath231 . as a first step in the analysis of such data , we will check the validity of standard finite size scaling for quantities averaged over disorder . with the usual definition of the critical exponents and @xmath232 , one expects : @xmath233 where it is possible to find @xmath234 and @xmath235 . for the average loop length probability distribution , we still look for a behavior at the critical point described by a power law as in the pure case : @xmath236 and therefore @xmath237 interestingly , based on this relation , it should be particularly simple to seek numerical evidence for @xmath238 . we resort to the simex scheme @xcite , based explicitly on partition function evaluations , instead of recursions for specific conditional probabilities as in @xcite . besides this conceptual difference ( important for generalizations , notably to higher - order models ) , for the simple helix - coil model in linear molecules , as considered here , the reduction of the computational complexity by one order of magnitude in the simex method relies on the numerical representation of the long - range effects in the model as a sum of @xmath239 exponentials , as already formulated in the ff method . the other important ingredient in the ff , also implemented straightforwardly in the simex , corresponds to a forward - backward scheme as described in section 3.1 , classical in dynamic programming and associated with an additional order of magnitude reduction in complexities . for the linear case , the complexities for a complete probability map calculation reduce overall from @xmath25 ( for a one - way progressive treatment ) to @xmath240 . in order to make the scheme operational in practice , it is necessary to obtain appropriate numerical representations for long - range effects as sums of exponentials . the general numerical problem associated with the analysis of multiexponential functions is notoriously a delicate one . it covers two distinct -in principle- situations , concerning either identifications or approximations , the relevant case for the present study . in the identification situation , it is necessary to recover the correct number of exponentials , and of course the correct associated parameters , from curves ( usually experimental ) supposed to be of multiexponential type for theoretical reasons . a general solution to this problem is provided by the pad - laplace method @xcite , requiring no _ a priori _ hypotheses for the identification of components in sums of general exponentials ( real and/or complex ) . this formulation encompasses , and generalizes , in a unified frame , a series of solutions since prony s method and the so - called method of moments @xcite . even though originally formulated as an identification approach , the numerical application of the pad - laplace method to power - law functions ( such as for loop - entropies here ) revealed an identification - like behavior in this approximation problem , in the sense that , for given maximal long - range lengths , a fixed number of significant exponential components are obtained . for example , for a series of biologically - oriented studies , an approximation of @xmath241 with @xmath242 exponentials was shown to be appropriate ( with further refinements of the parameters with least - squares procedures ) @xcite . in the present study , in order to be in strictly comparable conditions with this respect , we adopt the numerical representation of @xmath243 with @xmath244 exponentials in @xcite . for the numerical computation of the recursive equations for the forward and backward partition functions , it is moreover necessary to avoid underflow / overflow problems . for this purpose different schemes can be implemented @xcite , in order to normalize the numerators and denominators the ratios of which are involved in the evaluation of probabilities ( [ ps ] ) . we consider here the normalization described in @xcite , based on the introduction of _ free energy like quantities _ for the handling of the logarithms of @xmath207 and @xmath208 . the details of the implementations are provided in the appendix , along with the description of boundary conditions . we study extensively the case @xmath245=1.3 $ ] , using the same energies and connectivity constant as in @xcite : @xmath140 and @xmath246 , in three dimensions . we consider sequence lengths ranging between @xmath247 and @xmath248 . for @xmath3 , such @xmath2-values appear to be large enough both for the clarification of the thermodynamic limit behavior and for the study of corrections to scaling . on the other hand , the numerical computations are reasonably fast up to this length , which makes it possible to consider closely spaced temperatures and to obtain correspondingly , with negligible numerical errors , quantities related to derivatives , such as in particular the values of the maxima of the specific heat . in detail , for given chain lengths , we consider @xmath249 different @xmath210-values , equally spaced in intervals @xmath250 $ ] around the corresponding @xmath115 , evaluated roughly from the position of the maximum of the average specific heat in some preliminary results . for the different chain lengths , the number of samples @xmath251 as well as the @xmath252 and @xmath253 temperatures are detailed in [ tab . 1 ] . .tabulation of the chain lengths @xmath2 , the number of samples @xmath251 and the range of temperatures @xmath250 $ ] in the numerical computations . for each disordered sequence @xmath249 equally spaced temperatures , in the corresponding intervals , are considered . [ cols="^,^,^,^",options="header " , ] it was checked in particular that with the various choices above the errors on the maximum of specific heat and of susceptibility were both consistently smaller than fluctuations between samples . without any loss of generality we set in all calculations @xmath254 , i.e. temperature is in @xmath154 unities . the evaluation of the susceptibility is obtained by numerical derivations with respect to @xmath154 and @xmath255 ( see ( [ chi ] ) ) , with @xmath256 and @xmath257 , which ensures the desired numerical accuracy at all temperatures . finally , in all calculations the errors on average quantities are computed from sample - to - sample fluctuations . we consider first the qualitative behavior of the model for a given sample - sequence of length @xmath258 , with different @xmath245 $ ] values . results shown in this section are obtained with a particular disorder configuration . it was however checked that the corresponding qualitative observations are also valid with various arbitrarily chosen sequences . in order to cover different significant situations , the following values for the parameter @xmath1 were considered : @xmath259 , respectively . notably , the choice @xmath260 is for comparisons with the results in @xcite . in detail , we used the same @xmath261 , and we set in addition @xmath262 , for obtaining close critical temperatures in the pure case . the choice @xmath3 is for compatibility with the conditions in @xcite , and accordingly we set in this case @xmath140 and @xmath263 . on the other hand , the choice of the close value @xmath264 is following parameters setting usual in comparisons with experimental results @xcite . in this latter case , the value for @xmath1 is not related to a large @xmath137 value , but rather to a large average @xmath265 ( in @xmath266 unities ) . it can be noticed that typically used coupling energies lead to @xmath267 , as obtained by averaging over the different stacking energies for neighbor base pairs @xmath268 . finally , for clarification of potential differences resulting from choices of large @xmath137 or alternatively large @xmath43 values , @xmath269 was retained as corresponding to the two choices for ( @xmath137 , @xmath180 ) couples : @xmath270 , @xmath271 ( case @xmath272 ) ; and @xmath273 , @xmath263 ( case @xmath274 ) . we plot in [ fig . [ fig1 ] ] the susceptibilities @xmath275 for the sample - sequence for the different @xmath1 values . we observe that the results depend strongly on @xmath1 . however , the shapes of the curves obtained with @xmath3 and @xmath276 appear to be strikingly similar . moreover , also the two curves corresponding to the two different cases associated with @xmath277 ( case @xmath272 and case @xmath274 ) are qualitatively similar . and different @xmath1 values ( see text).,title="fig:",width=302 ] + and different @xmath1 values ( see text).,title="fig:",width=302 ] and different @xmath1 values ( see text).,title="fig:",width=302 ] and different @xmath1 values ( see text).,title="fig:",width=302 ] and different @xmath1 values ( see text).,title="fig:",width=302 ] and different @xmath1 values . @xmath278 $ ] data are plotted as function of @xmath279 \log \mu/(r-1)$ ] , with @xmath280 the temperature associated with the absolute maximum of susceptibility . in the case @xmath272 with the value @xmath281 , specific choice was needed for defining the pseudo-@xmath6 , and we looked in this case to correspondence of positions of the peaks with the other curves , by taking it as the temperature of the second maximum.,width=377 ] the results here are in agreement with the overall picture given in section 2.4 , with indications for an @xmath1-dependent finite size behavior . the extreme case considered there , with pure at and gc regions , is clearly a very rough approximation of a typical sequence . it is nevertheless clear that the parameter @xmath1 appears to capture some essential ingredients of the model . for more quantitative analysis , we note that for @xmath15 one expects a transition temperature close to @xmath282 . for a given finite sequence , a pseudo - critical temperature @xmath280 must be adopted , as the critical temperature is not well defined . here we take as @xmath280 the temperature associated with the highest maximum value for susceptibility @xcite . in particular , with such a choice , we find that the behavior of @xmath275 displays some scaling when plotted as function of @xmath279 t_c(\epsilon , x , n)/(r-1)$ ] , for different @xmath137 and @xmath1 values . we present in [ fig . [ fig2 ] ] correspondingly scaled @xmath283 data ( also multiplied by the factor @xmath284 $ ] , for obtaining close behaviors in the high temperature phase ) . this figure further makes evident the dependence on @xmath1 for finite size systems . the results suggest that , at fixed length scale @xmath2 , for large @xmath1 , and in particular here already for the value @xmath260 , the system exhibits typically only one very sharp peak . this observation should be related to the fact that the probability of large enough _ rare regions _ is negligible ( though one could still encounter such a case when considering a large number of sequences ) , and the system behaves essentially as a pure model with @xmath285 . for smaller @xmath1 values , we observe on the contrary an increasing number of peaks , with decreased sharpness . this finding is coherent with the qualitative picture following which , with increasingly smaller @xmath1 values , _ rare regions _ with increasingly smaller lengths @xmath153 are sufficient in order to observe multistep behaviors , since the relevant quantity should be the ratio @xmath286 . nevertheless , for the smallest considered value @xmath287 , obtained with two different parameters choices , we observe the same number ( four ) of peaks , but the position with respect to the other peaks for the absolute maximum of the curve is shifted for @xmath288 as compared to the corresponding one for @xmath289 and the larger @xmath1 values . this confirms that the introduced @xmath1 describes the finite size behavior only approximately . the changes in the behavior of a typical sample , as described above , are also expected at larger sequence sizes for a given @xmath1-value , since this parameter appears to behave as ( the logarithm of ) an intrinsic length scale of the system . some numerical evidence in this direction was already given in @xcite for the on - lattice dsaw - dna , where the qualitative analysis of the specific heat suggested the appearance of multi - peaked curves only for large enough @xmath2-values , and an increasing number of peaks in typical sequences as function of @xmath2 . here we obtain the same qualitative results for the value @xmath3 studied in detail . nevertheless , detailed quantitative bases for such conclusions are left for future work , with an extensive study of the finite size effects for different @xmath1 values . it is to be emphasized that in the suggested picture it is the behavior of the _ typical _ finite size sample which is expected to change when varying @xmath1 , and therefore one would not expect different _ typical _ and _ average _ correlation lengths in the thermodynamic limit , though the sizes necessary for confirming this hypothesis could be out of reach for large @xmath1 values . an analysis in terms of pseudo - critical temperatures is clearly necessary to distinguish between this situation and the alternative one proposed in @xcite , which is left to a forthcoming work @xcite . in what follows , we investigate the behavior of average quantities for @xmath3 ( obtained by setting @xmath140 and @xmath290 ) . the average energy density @xmath291 and the average closed base pair density @xmath292 , for different chain lengths , are represented in [ fig . [ fig3 ] ] . it is clear that the system undergoes a phase transition in the thermodynamic limit , with the energy density and the order parameter going from zero at high temperature to finite values below @xmath6 . we can moreover observe from [ fig . [ fig3 ] ] very similar behaviors for the two quantities ( apart from the sign difference ) and we expect to find , as a consequence , @xmath293 and @xmath294 . we have also checked that the average densities of closed at @xmath295 and gc @xmath296 base pairs exhibit qualitatively similar behaviors . and averaged total density of closed base pairs @xmath292 , for various chain lengths , plotted as functions of temperature.,title="fig:",width=302 ] and averaged total density of closed base pairs @xmath292 , for various chain lengths , plotted as functions of temperature.,title="fig:",width=302 ] nevertheless , the data do not agree with the corresponding expected scaling laws ( see ( [ scle ] ) and ( [ sclop ] ) ) , making clear the presence of strong corrections to the asymptotic behavior . even though it is therefore difficult to evaluate the critical exponents , the fact that the curves do not cross at the same point ( as particularly evident for the largest sizes ) suggests a transition with ( an average ) @xmath297 . more in detail , the energy density and the order parameter appear to converge both towards functions which are continuously vanishing at the critical point and possibly also differentiable , which would imply a transition at least of second order . still better evidence for a smooth transition comes from the average specific heat @xmath298 data , plotted in [ fig . [ fig4 ] ] . one can notice from this figure that the maximum appears to diverge with the chain length for the smallest sizes , but saturates for larger @xmath2-values , as expected for a critical point characterized by @xmath299 , _ i.e. _ @xmath5 . interestingly , the qualitative behavior for chain lengths smaller than @xmath300 appears to be similar to the one found in @xcite . this observation suggests that the on - lattice dsaw - dna and the off - lattice disordered ps model considered could exhibit the same kind of finite size effects , also supporting the hypothesis that the value @xmath301 obtained from the monte carlo like numerical simulations represents a lower bound for the average correlation length critical exponent . , for the chain lengths considered , plotted as functions of temperature.,width=377 ] , for the chain lengths considered , plotted as functions of temperature.,width=377 ] , for the chain lengths considered , plotted as functions of the loop lengths @xmath18 at temperatures @xmath115 , corresponding to maxima of average specific heat . normalization factors are arbitrarily chosen in order to make values comparable at small @xmath18.,width=377 ] we find a very similar behavior for the susceptibility @xmath302 , represented in [ fig . [ fig5 ] ] , as well as for the derivative with respect to the temperature of the total density of closed base pairs @xmath303 ( not shown ) . these findings further suggest that these quantities , as well as the energy and the order parameter , are described also in the disordered case by the same critical exponents . again , it is difficult to evaluate the exponents by applying standard finite size scaling analysis to the data , because of the obvious presence of strong corrections to the expected laws ( here ( [ maxlaw ] ) and ( [ sclsu ] ) ) . finally , [ fig . [ fig6 ] ] displays data for @xmath304 ( see ( [ pstar ] ) ) at the size - dependent critical temperatures @xmath115 , identified here with the temperatures associated with the maximum of the average specific heat . we note also in this case a strongly @xmath2-dependent behavior . in particular , the considered quantity is nearly constant in the range @xmath305 for the smallest sizes , whereas for the largest ones , expected to be the most meaningful , it is increasing with @xmath18 rather linearly ( on logarithmic scale ) on a wide ranged interval , which should mean that the expected power law representation ( [ plpl ] ) is valid , but with @xmath306 . we notice that the observed strong @xmath2-dependence of averaged quantities and the fact that data do not obey usual scaling laws on the whole @xmath2-range studied is in agreement with the qualitative picture given in section 2.3 , clearly suggesting that , at fixed @xmath1 , the effect of disorder becomes evident , and the system reaches the asymptotic behavior , only for large enough size values . from this point of view , in particular the saturation of the specific heat and susceptibility maxima can be related to the appearance of a larger number of less sharp peaks ( apparently in the typical sequences ) for increasing @xmath2 , in analogy with the behavior discussed in the previous section when decreasing @xmath1 at fixed chain length . we consider in terms of quantitative analysis data for the maximum of the average specific heat [ fig . [ fig7]a ] and the maximum of the average susceptibility [ fig . [ fig7]b ] , as functions of chain lengths @xmath2 . using the law @xmath307 , which is a particular case of ( [ maxlaw ] ) , we obtain from the analysis of these data @xmath308 and correspondingly @xmath309 when considering only the smallest chain lengths . in detail , the exponent would be still compatible with the value @xmath310 characterizing a first order transition , for @xmath311 . for both @xmath312 and @xmath313 the asymptotic saturation becomes obvious only for sizes larger than @xmath314 . following @xcite , we consider a fit of the data to the form @xmath315 , with @xmath316 and where the exponent is @xmath317 for the specific heat and @xmath318 for the susceptibility . using the whole data sets in the fits , negative exponents , compatible with zero within the errors , are obtained in both cases . on the other hand , with corrections to scaling , strictly negative exponents are obtained . letting aside the two smallest sizes ( @xmath247 and @xmath319 ) , the best fit in the case of @xmath312 corresponds to @xmath320 , and we get a compatible value for @xmath318 from @xmath321 . this result confirms that @xmath322 and , from the hyperscaling relation , it implies @xmath323 . intriguingly , the correlation length exponent value is close to the one obtained for the disordered ps model with @xmath113 considered in @xcite . we notice that limiting the analysis to @xmath324 we obtain a still larger @xmath145 , but the statistics in our study do not allow more accurate evaluations . nevertheless , the important point is that , when looking at average quantities , it clearly appears that the transition is at least of second order and , at the same time , the crossover between a _ pure system like _ behavior for small sizes and the ( apparent ) asymptotic one is quantitatively confirmed . and on the maximum of the average susceptibility @xmath321 , plotted as functions of chain lengths , together with the best fits to the law @xmath315 with @xmath316 . , title="fig:",width=302 ] and on the maximum of the average susceptibility @xmath321 , plotted as functions of chain lengths , together with the best fits to the law @xmath315 with @xmath316 . , title="fig:",width=302 ] , plotted as function of @xmath325 . the results are from the fits of the probability distribution of the loop length to the expected law ( [ pstartc ] ) @xmath326 ( with @xmath0 and @xmath115 taken as the temperature for which the average specific heat is maximum ) . for each chain length , data are fitted in the @xmath18-range ( with @xmath327 ) in which @xmath328 is increasing with @xmath18 . here the errors are only indicative , as they depend strongly on the number of points in the range of the fit.,width=415 ] , at temperatures @xmath115 for which average specific heats reach their maxima . the expected asymptotic behavior @xmath329 with @xmath330 ( i.e. @xmath331 with @xmath145 obtained from the fit of the maximum of the specific heat ) is also plotted.,width=377 ] for further validation of this result , and for a better understanding of finite size corrections to scaling , we consider the fits of @xmath304 to the expected behavior @xmath332 ( [ pstartc ] ) . we get correspondingly the _ size - dependent _ estimations of the critical exponent @xmath333 , which are presented in [ fig . [ fig8 ] ] . here the ( finite size ) critical temperatures @xmath115 are taken as the temperatures for which the average specific heat reaches its maximum and we disregard the possible presence of a finite correlation length . we fit data by considering only the @xmath18-range ( with @xmath327 ) in which @xmath304 is an increasing function of @xmath18 ( see [ fig . [ fig6 ] ] ) . the obtained values of @xmath333 , and therefore of @xmath334 , for different chain lengths are definitely not compatible within the ( though indicative ) errors and one observes a clear trend towards decreasing @xmath333 values for larger chain lengths . it is in particular interesting to notice that , for @xmath319 , we still have @xmath335 and correspondingly @xmath336 , whereas for @xmath337 , we obtain @xmath338 , again in perfect agreement with the results of @xcite concerning the @xmath87 dsaw - dna on - lattice model . on the other hand , with the study of larger chain lengths it becomes clear that the transition is at least of second order with @xmath131 . in fact , for the largest size considered @xmath248 , the exponent obtained is @xmath339 . this implies that the correct evaluation of the ( asymptotic ) critical exponents , apart from finite size corrections , as @xmath340 leads to @xmath341 . this value can be compatible with @xmath342 obtained from the maximum of the specific heat . it is nevertheless important to stress that the above results concern average quantities . the thermodynamic limit behavior of the typical sample could be therefore blurred by the fluctuations of the sequence - dependent pseudo - critical temperature , if they are governed by an exponent @xmath343 as suggested in @xcite . it is possible that we are only looking at the _ average _ correlation length @xmath344 , and in particular the value @xmath345 would be in agreement with the result @xmath346 in @xcite . accordingly , we consider also data on @xmath347 , evaluated by averaging over disorder after taking the logarithm . to be precise , following @xcite , the quantity expected to be described by the typical correlation length @xmath348 is @xmath349/z^*_{\epsilon , n } \}$ ] . here we are instead considering a kind of mixed average , but the behavior of @xmath347 should anyway display differences with that of @xmath350 in the presence of two distinguishable correlation lengths . on the contrary , the comparison between [ fig . [ fig9 ] ] and [ fig . [ fig6 ] ] shows that , at least for @xmath351 , @xmath352 and @xmath353 behave similarly . in particular , @xmath354 is also an increasing function of @xmath18 for the largest considered sizes , and on quantitative grounds the evaluated @xmath333 values are essentially compatible within errors . in order to emphasize this point , the expected asymptotic behavior @xmath355 with @xmath356 ( obtained from the average specific heat ) is also plotted in [ fig . [ fig9 ] ] . more quantitative analysis of these results are left for a forthcoming work @xcite . we studied numerically a disordered ps model for dna denaturation with @xmath0 , which displays a first order transition in the homogeneous case , by solving recursively the equations for the canonical partition function with the simex scheme . the model is made as similar as possible to the @xmath87 dsaw - dna previously studied by monte carlo like simulations @xcite and it is expected that the results of the study could also be relevant to different disordered ps models with @xmath26 . we introduced the parameter @xmath245 $ ] , where @xmath133 is the ratio of the guanine - cytosine to the adenine - thymine coupling energies and @xmath43 is the connectivity constant of the corresponding on - lattice model . we showed that this parameter , at fixed @xmath20 value and gc composition ( taken to be 1/2 ) , appears to play the role of ( the logarithm of ) an _ intrinsic _ length scale , and to describe , in a first approximation , the finite size behavior . in particular , for a given @xmath2-value , the manifestation of the effect of disorder appears to be the most evident for the smallest @xmath1 values , in agreement with a qualitative explanation based on the possible occurrence of large enough _ rare regions_. it is interesting to notice that , within this picture , the system size necessary for observing the asymptotic behavior and making evident the effect of disorder diverges exponentially with @xmath1 . we studied in detail the value @xmath3 , obtained with @xmath140 and @xmath290 ( as in @xcite ) , for sequence sizes up to @xmath248 ( larger than the sizes accessible to monte carlo like simulations by a factor 20 ) . we found that the model exhibits strong corrections to scaling , displaying a crossing between a still nearly _ pure system like _ behavior for small chain lengths @xmath4 and the observed ( apparently asymptotic ) large one for @xmath357 . in particular , the maximum of the average specific heat , which behaves as the susceptibility , increases with @xmath2 for small chain lengths . considering the whole size range , it appears instead to be clearly saturating . this result shows that , at least from the point of view of average quantities , the thermodynamic limit is described by a random fixed point with @xmath299 and correspondingly @xmath358 . by fitting data with a scaling law of the form @xmath359 , and by taking into account corrections to scaling , we find in particular @xmath360 , which by using the hyperscaling relation gives @xmath323 . the average loop length probability distribution at the critical temperature appears still described by a power law at least on an interval @xmath305 of the range . upon fitting data according to @xmath361 one finds @xmath2-dependent values for the exponent which is compatible with @xmath20 for the smallest sizes whereas when looking at the whole @xmath2 range it appear to converge towards an asymptotic limit @xmath341 ( possibly compatible with @xmath342 ) . moreover , @xmath362 exhibits also a similar behavior , suggesting that there is no difference between _ typical _ and _ average _ correlation lengths . our best - fit estimation , @xmath323 , is close to the estimation in @xcite for the case @xmath113 . this observation would support the hypothesis that disorder is relevant as soon as @xmath112 and that the various disordered ps models considered could be described by the same random fixed point corresponding to a transition which is at least of second order ( and probably smoother ) , in agreement with recent analytical findings @xcite . our statistics do not allow nevertheless to completely rule out the possibility that @xmath363 , particularly from @xmath364 data , and in any event an analysis in terms of pseudo - critical temperatures @xcite is in order for clarifying the situation . we leave this development to a forthcoming work @xcite . it is also interesting to notice that there are very recent theoretical studies @xcite on the loop dynamics in ps models , which in particular relates the equilibrium loop length distribution probability to the correlation function , therefore suggesting a new intriguing method for measuring experimentally the @xmath365 value . from this point of view , the expected behavior of @xmath366 in presence of disorder and the possibility of observing differences between the average and the typical sequence cases seems to us important questions to be clarified . in conclusion , our results provide numerical evidence for strong finite size corrections to the asymptotic behavior of the disordered ps model considered . the data show moreover that disorder is relevant , at least from the analysis of average quantities . the findings here appear also to confirm that the evaluation @xmath125 in previous numerical study concerning on - lattice @xmath87 dsaw - dna model @xcite is to be considered as lower bound for the correct ( average ) correlation length exponent . the observed behavior is in agreement with a proposed qualitative picture for finite size effects , which could also explain the difference with the results of previous studies on a different disordered ps model with the same @xmath20 @xcite . a preliminary presentation for part of the findings and hypotheses here can be found in @xcite . b.c . would like to acknowledge an enlightening discussion that she had with david mukamel some time ago . we are moreover grateful to thomas garel , cecile monthus , andrea pagnani and giorgio parisi for comments . we use in the numerical computations the simex implementation of the ff scheme @xcite , which relies on the numerical approximation of the powers @xmath367 by sums of exponentials : @xmath368 in the present study we consider @xmath0 and the values for the coefficients @xmath369 and @xmath370 , with @xmath371 , provided in @xcite . the computation of the recursive equations for the forward and backward partition functions were implemented with the introduction of _ free energy like quantities _ , in order to handle logarithms of @xmath207 and @xmath208 ( following @xcite ) . in detail , the equation ( [ zf ] ) for the forward partition function becomes : @xmath372 \right \}.\ ] ] by defining @xmath373,\ ] ] one obtains @xmath374-\exp[\mu_k(\rho-1)-2 b_k]\ ] ] and a recursion relation for @xmath375 : @xmath376 \right \ } \nonumber \\ c_f&=&\exp(\beta\epsilon_{\rho+1}-\log \mu ) \left \ { \sum_{j=1}^{n_{ff } } a_j \exp[-4b_j+\mu_j(\rho-1)-\mu_k(\rho ) ] \right \ } \nonumber \end{aligned}\ ] ] analogously , one writes the equation for the backward partition function : @xmath377 + 1\right \},\ ] ] and defines @xmath378,\ ] ] obtaining @xmath379 \right \ } \nonumber \\ c_b&=&\exp(\beta\epsilon_{\rho-1}-\log \mu ) \left \ { \sum_{j=1}^{n_{ff } } a_j \exp[-4b_j+\nu_j(\rho+1)-\nu_k(\rho ) ] \right \ } \nonumber \\ d_b&=&\exp(\beta\epsilon_{\rho-1}-\log \mu)\exp[-\nu_k(\rho ) ] . \nonumber \end{aligned}\ ] ] we used the boundary conditions ( with the implicit assumption @xmath380 ) : @xmath381 \nonumber \\ z^b_\epsilon(n ) & = & \exp(\beta \epsilon_n-\log \mu ) , \end{aligned}\ ] ] and correspondingly : @xmath382 \nonumber \\ \nu_k(n-1)&=&\beta \epsilon_n-\log \mu + \log \left \ { \exp(-2b_k)+\exp(\beta \epsilon_{n-1}-\log \mu ) [ 1+\exp(-\beta \epsilon_{n}+\log \mu ) ] \right \}\nonumber \\ \nu_k(n)&=&\beta \epsilon_n-\log \mu . \end{aligned}\ ] ] b. coluzzi and e. yeramian , _ numerical evidence for the relevance of disorder in a poland - scheraga model for dna denaturation with self - avoidance : an analysis in terms of pseudo - critical temperatures _ , work in progress . b. coluzzi and e. yeramian , _ on the disordered saw model for dna denaturation _ , proceedings of the x international workshop on disordered systems , andalo ( italy ) , 18 - 21 march 2006 . philosophical magazine ( in press , published on - line ) .
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we study numerically the effect of sequence heterogeneity on the thermodynamic properties of a poland - scheraga model for dna denaturation taking into account self - avoidance , _
i.e. _ with exponent @xmath0 for the loop length probability distribution . in complement to previous on - lattice monte carlo like studies , we consider here off - lattice numerical calculations for large sequence lengths , relying on efficient algorithmic methods .
we investigate finite size effects with the definition of an appropriate _ intrinsic _ length scale @xmath1 , depending on the parameters of the model . based on the occurrence of large enough _ rare regions _
, for a given sequence length @xmath2 , this study provides a qualitative picture for the finite size behavior , suggesting that the effect of disorder could be sensed only with sequence lengths diverging exponentially with @xmath1 .
we further look in detail at average quantities for the particular case @xmath3 , ensuring through this parameter choice the correspondence between the off - lattice and the on - lattice studies . taken together
, the various results can be cast in a coherent picture with a crossover between a nearly _ pure system like _ behavior for small sizes @xmath4 , as observed in the on - lattice simulations , and the apparent asymptotic behavior indicative of disorder relevance , with an ( average ) correlation length exponent @xmath5 .
\a ) _ ceres - erti ( plateforme environnement ) _ , + ecole normale suprieure , 24 rue lhomond , 75005 paris b)_unit de bio - informatique structurale _ , + institut pasteur , 25 - 28 rue du docteur roux , 75724 paris cedex 15 + pacs : 64.60.fr equilibrium properties near critical points , critical exponents + pacs : 82.39.pj nucleic acids , dna and rna bases + pacs : 02.60.cb numerical simulation , solution of equations
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there is presently a growing body of circumstantial evidence linking some long - duration gamma ray bursts ( grbs ) with afterglows to the explosions of massive stars . supernovae ( sn ) or supernova - like features have been identified in seven afterglows although some of these could be light echoes from dust clouds @xcite . the afterglows of six other grbs have been interpreted in terms of a wind - like ambient medium as expected around a massive star at the end of its life . however these are often equally well explained by a collimated flow in a uniform medium . x - ray lines have been detected with moderate confidence in about half the afterglows for which they were investigated @xcite ; however , the data analysis has been questioned ( see * ? ? ? if real , these are most easily explained by dense material surrounding the burst engine , suggesting a stellar origin ( e.g. , * ? ? ? @xcite have recently derived beaming angles for a number of grbs from observations of their afterglows . these authors derive gamma - ray energies reminiscent of supernovae : roughly @xmath0 erg for the observed lobes of grbs . similar results were reported by @xcite and @xcite . clear evidence that grbs occur very close to massive star formation would be almost as conclusive as a sn signature in an individual grb . several grb afterglows show evidence for high column densities ( 980703 and 980329 ; @xcite ) or high local gas densities ( 000926 and 980519 ; @xcite , @xcite ) , both of which connote star - forming regions . likewise , the intrinsic extinction of grb 000926 is characteristic of a galaxy disk @xcite . @xcite have shown that the observed locations within hosts imply a tight correlation between grbs and stellar populations , considered too tight @xcite to be matched by merging neutron stars . note however that the locations of merging neutron star pairs depends on their uncertain distribution of natal kicks . if grbs are a rare byproduct of star formation , rapidly star - forming galaxies should be over - represented as grb hosts . in optical light host galaxies tend to look ordinary compared to contemporaries in the hubble deep field , but [ ne iii ] and [ o ii ] and infrared observations often indicate elevated star formation rates @xcite . at least eight afterglows have been associated with starburst or interacting galaxies @xcite . although the association between long - duration grbs and sne is tentative ( and applies only to the long - duration bursts for which afterglows are observed ) , the above evidence warrants a careful evaluation . there are two ways a sn can create a grb . @xcite predicted that gamma rays might be produced in the very fastest , outermost ejecta of an ordinary supernova explosion . this proposal was recently revived by @xcite and @xcite . these authors showed that the grb ( 980425 ) most compellingly associated with a sn ( 1998bw ) is likely to be the result of trans - relativistic sn ejecta colliding with a stellar wind ( see also * ? ? ? * ) . in their model , as conjectured by @xcite , sn 1998bw was spherically symmetric or mildly asymmetric , and produced the grb in an external shock . scaled - up versions of this model could produce external shock grbs , at the expense of vast amounts ( @xmath1 erg ) of energy in nonrelativistic ejecta . in contrast , @xcite have argued that most grbs require internal emission ( e.g. , by internal shocks ) within unsteady ultrarelativistic winds or jets , as originally suggested by @xcite . the arguments for internal emission are strongest for rapidly - fluctuating cosmological bursts with hard spectra , those least resembling grb 980425 ; also , see @xcite for arguments in support of external shocks . i shall assume for the purposes of this investigation that cosmological grbs involve internal emission within optically thin , ultrarelativistic outflows . for this to result from a sn , a jet must emanate from a star s core and pierce its envelope shedding the baryons in its path prior to producing the gamma rays observed at earth . such a jetlike explosion is the conventional model ( e.g. , * ? ? ? * ) for a supernova origin of cosmological grbs . the goal of this paper will be to develop analytical models for the phase of this latter model in which a jet , already created by the stellar core , must traverse the envelope and shove aside material in its path . these models , which are complementary to numerical simulations @xcite , are meant to elucidate under what conditions the hypothesis of a stellar origin is viable for the observed grbs . in [ inside star ] and [ envelopeconstraints ] i assume that jets travel ballistically within their stars ; this allows one to place strict constraints on stellar envelopes . in [ widening ] this assumption is reconsidered . it is shown that a phase in which the jet is hot , pressure confined , and mixing with its environs would have interesting consequences . figure [ fig1 ] sketches the typical masses and radii of the stellar grb progenitor candidates considered in this paper . in general , those that retain an outer envelope ( e.g. , supergiants ) have quite large radii ( @xmath2 ) at the time of core collapse , whereas those depleted in hydrogen due to winds or binary interaction ( e.g. , those that have been through a wolf - rayet phase ) are quite compact ( @xmath3 ) . among post - wolf - rayet stars , those containing helium ( `` he wolf - rayets '' on the plot ) are less compact than their he - depleted peers ( `` c / o wolf - rayets '' ) . very massive objects ( vmos ) might have formed at high redshift due to the difficulty of cooling in the absence of metals and thus the large jeans mass in primordial gas . vmos may also form today in rare conditions . those initially more massive than @xmath4 die when their cores collapse to black holes , and are candidates for producing grbs ; @xcite discuss their evolution . again , the pre - collapse radii of these stars depend on their mass loss . if present , their h envelope is quite diffuse ( see * ? ? ? otherwise the remaining convective core is very compact at the point of collapse . in addition to winds and binary mass transfer , vmos may shed their envelopes in a super - eddington phase during helium core contraction . lone vmos are thought to retain h envelopes if they are formed from sufficiently low - metallicity gas ( although this is uncertain ; * ? ? ? the `` supranova '' model of @xcite posits that a sn explosion produces a rapidly spinning neutron star massive enough to collapse after shedding its angular momentum @xcite along with @xmath5 erg of rotational energy . the object collapses to a black hole with an accretion torus , firing a jet through the pulsar s wind nebula and its sheath of stellar ejecta . because of the short viscous times of such tori , this model is most appropriate for short grbs rather than the long bursts for which there is evidence of a sn connection . also , we will see in [ envelopeconstraints ] that the expanding stellar ejecta would prevent the production of a grb entirely , unless it has either become thompson optically thin or has been cleared aside by the breakout of the pulsar wind nebula . the latter is possible in this model , because the pulsar spindown energy exceeds the typical kinetic energy of sn ejecta . @xcite and @xcite discuss a scenario in which the stellar core collapse leading to a grb results from the coalescence of a helium wolf - rayet star with a compact companion . similarly , @xcite argue that binary mergers are likely to dominate the production stars whose cores collapse to black holes accreting through a disk . progenitors created in this fashion will typically be stripped of their outer envelopes , hence compact . however the stripped envelope poses a potential barrier to jet propagation , as in the supranova model , unless it is strictly confined to the equator of the system or has expanded to the point of being optically thin . in sections [ s : freefalltimes ] and [ s : pressureconf ] i examine the properties of stellar cores at the point of collapse . for this i shall assume that collapse sets in at the oxygen ignition temperature , @xmath6 ( e.g. , * ? ? ? consider the progress of a relativistic jet outward from a star s core through the stellar envelope . schematically , three distinct regions develop : the propagating jet ; the _ head _ of the jet , where jet material impacts the stellar envelope , and a _ cocoon _ consisting of shocked jet and shocked ambient material . these are familiar components from the theory of radio galaxies @xcite ; see figure [ fig : diagram ] . stellar envelopes pose a problem for the propagation of grb jets only if they have not collapsed prior to the launching of the jet . the stellar envelope s collapse timescale is greater ( probably by a factor of at least a few ) than its free - fall time @xmath7 . in figure [ fig2 ] i compare stellar free - fall times with the intrinsic durations of grbs for several possible redshifts . only the very densest progenitors , the cores of very massive objects ( @xmath8)^{1/4}$ ] s at oxygen ignition ) and helium depleted post - wolf - rayet stars ( @xmath9 s ) could plausibly collapse entirely in the durations of the longest grbs , and then only at low redshift . in all other cases a stellar envelope remains to impede grb jets . for stellar core collapse to successfully produce a cosmological grb , it must emit a jet that clears the stellar envelope from the observer s line of sight to the core . this is required if the jet is to achieve high lorentz factors and if its internal shocks are to be unobscured by overlying material . this can not occur if the lorentz factor of the jet head , @xmath10 ( where @xmath11 is the head s velocity ) , exceeds the inverse beaming angle @xmath12 . if it did , then the jet head would be causally disconnected from its edges and would behave like a spherical blastwave @xcite . in this case essentially no material would escape sideways to form the cocoon . if instead @xmath13 then there is ample opportunity for shocked jet and envelope material to flow sideways and inflate the cocoon , so a successful grb requires @xmath14 observers of grb afterglows often identify an achromatic break in the light curve with the deceleration of the swept up shell from @xmath15 to @xmath13 @xcite . in order for this to be possible , the head must have satisfied @xmath16 while it was still being driven forward by the jet ( prior to its deceleration ) . condition ( [ gamthetawind ] ) only applies to grbs whose afterglows exhibit such a break in their afterglows . the expansion velocity of the jet head is given to a very good approximation by the balance of jet and ambient ram pressures ( momentum fluxes ) in the frame of the jet head . this approximation is most accurate when ambient material is cast aside into the cocoon , as is the case if equation ( [ gamthetaenv ] ) is satisfied . ram pressure balance means @xmath17 where @xmath18 is the relative four - velocity between the jet and its head , @xmath19 is the four - velocity of the head into the ambient medium , @xmath20 , @xmath21 , and @xmath22 are density , pressure , and enthalpy ( @xmath23 is total comoving energy density ) , and the subscripts @xmath24 and @xmath25 refer to jet and ambient material . as @xmath26 in a stellar envelope @xmath27 to a good approximation and @xmath28 can be ignored on the right - hand side . to leading order in @xmath29 , @xmath30 can be ignored on the left . for a stationary ambient medium , these approximations give @xmath31 where @xmath32 the second equality , which holds to leading order in @xmath29 , is derived by noting that the kinetic plus internal energy density of the jet , evaluated in the lab frame , is @xmath33 ; its isotropic luminosity @xmath34 is this quantity times @xmath35 ; and the mass per unit length of the ambient medium is @xmath36 . here @xmath37 is the radius and the true jet luminosity is @xmath38 . the relation between @xmath39 and @xmath40 implied by equation ( [ vhead ] ) takes simple limits in two regimes . if @xmath41 , then the reverse shock into the jet is nonrelativistic @xcite and @xmath42 . this case is physically unattainable within a star and is achieved outside only if the ambient density is extremely low . in the opposite limit of a relativistic reverse shock , @xmath43 ( see also * ? ? ? * ) . therefore , @xmath44 is useful in determining the observer s time @xmath45 for the jet head to expand to radius @xmath37 if viewed at redshift @xmath46 . since @xmath47 , @xmath48 for both non - relativistic and relativistic jet heads , so long as @xmath49 . in presupernova envelopes and stellar winds @xmath50 is relatively constant , and since @xmath34 is likely to vary slowly @xmath44 can be approximated with its average value . more generally , one might know how @xmath34 varies as a function of @xmath51 ( which is also @xmath52 at the origin , @xmath53 ) . if one also knows @xmath54 ( e.g , from a model of the star and its collapse ) , then equation ( [ tobs ] ) integrates to @xmath55 giving @xmath56 implicitly . the jet cocoon is the region containing spent jet material and shocked ambient material . its extent is equal to that of the jet , but its width is determined either by pressure balance with the surrounding gas , if there is time for this to be achieved , or else by the expansion of a lateral shock into the envelope . the latter case holds so long as it predicts a cocoon pressure @xmath57 in excess of the hydrostatic pressure or collapse ram pressure @xmath28 in its environment . let us first consider the case of an adiabatic cocoon whose pressure exceeds that of its surroundings , and check this assumption in [ s : pressureconf ] . let us also restrict attention to the case where the head velocity is subrelativistic ( @xmath58 , eq . [ [ limitsofgammah ] ] ) , in which case the cocoon pressure @xmath57 is roughly constant away from the jet head . the cocoon created by a jet of constant opening angle expands self - similarly so long as its width @xmath59 expands in proportion to its extent @xmath37 , and so long as no other size scales affect its structure . under these conditions , numerical simulations can determine cocoon structures exactly . however , analytical estimates ( e.g. , * ? ? ? * ) , while not as accurate , elucidate how cocoon properties scale with @xmath44 and @xmath60 . the cocoon expands nonrelativistically in a direction normal to its surface at the velocity @xmath61 given by @xmath62 if @xmath63 is evaluated at the point where the cocoon is widest , the normal direction is sideways and thus @xmath64 . the pressure @xmath57 is related to the energy @xmath65 deposited in the cocoon and the cocoon volume @xmath66 through @xmath67 ( an overestimate , as part of @xmath65 is kinetic ) . if a cocoon of length @xmath37 and width @xmath59 is idealized as a cone , @xmath68 and @xmath69 in equation ( [ betacprelim ] ) . but , since @xmath70 and @xmath71 , @xmath72 . now , @xmath65 is the energy emitted in time to catch up with the jet head at radius @xmath37 . ( after breakout , @xmath65 is available to drive an observable outflow and possibly a precursor : see @xcite and [ breakout ] . ) as long as @xmath73 the flight time of the jet can be neglected compared to @xmath74 , so @xmath75 is essentially all of the energy emitted up to then : @xmath76^{1/2 } \end{aligned}\ ] ] using equation ( [ tobs ] ) . along with the expressions for @xmath57 , @xmath66 and @xmath61 , this implies @xmath77 this implies that the cocoon is nonrelativistic ( and slower than the jet head ) so long as the jet head is also nonrelativistic , for @xmath78 in this case ( eq . [ [ limitsofgammah ] ] ) , and @xmath79 for a collimated jet . equation ( [ betacsoln ] ) assumes that the length of the cocoon is set by the advance of the jet head ; this requires @xmath80 . if this condition is violated , the cocoon expands around the jet and develops into a roughly spherical blastwave ( see also * ? ? ? for a nonrelativistic jet head , equations ( [ limitsofgammah ] ) and ( [ betacsoln ] ) indicate that @xmath80 so long as @xmath81 . for a more precise criterion , consider the velocity @xmath82 of a spherical blastwave powered by two jets : @xmath83^{1/2 } = 0.74 \theta^{2/3 } { { \tilde{l}}}^{1/3},\ ] ] where the coefficient is given by the pga/@xmath84 approximation of @xcite , for a wind bubble with a ratio of specific heats @xmath85 in a medium with @xmath86 . the head outruns this blastwave ( @xmath87 ) if @xmath88 , i.e. , @xmath89 this condition can only be violated if @xmath90 , so that only jets driving nonrelativistic heads can violate it . i assumed a nonrelativistic head in deriving equation ( [ l : no - bw : l<1 ] ) ; however there is nothing to suggest that relativistic jet heads can be swallowed by their cocoons . a successful grb requires that stellar envelope material be cleared from the path of the jet so it can sustain @xmath91 at @xmath92 1 au where internal shocks are thought to form . a jet - cocoon structure is required for this : a spherical blastwave would not accomplish it . for this reason grbs must satisfy @xmath93 within the stellar envelope ( eqs . [ [ gamthetaenv ] ] , [ [ limitsofgammah ] ] , and [ [ l : no - bw : l<1 ] ] ) so that a cocoon forms but does not overcome the jet . of these limits on @xmath44 , only the lower need be considered because @xmath94 for any reasonable combination of opening angle and stellar model . one can use equation ( [ l : no - bw : l<1 ] ) in equation ( [ ein - from - l - tobs ] ) to eliminate either @xmath95 , which gives a lower limit on @xmath75 , or to eliminate @xmath96 , which gives an upper limit on @xmath75 . in terms of the total energy per lobe @xmath97 deposited in the stellar envelope , the formation of a cocoon rather than a spherical blastwave implies @xmath98 a spherical blastwave is not sensitive to the site of energy injection , whether at a jet head or near the collapsing core , and for this reason the upper bound applies to energy from _ any _ source that is entrained in the stellar envelope before the jet breaks out of the star . note that the lower bound implies that @xmath97 must be at least a fraction @xmath99 of the rest energy of the envelope in the jet s path , @xmath100 . one further and very important constraint derives from the durations of grbs . in the internal - shock model , fluctuating gamma - ray emission reflects variability in the central engine and persists only while this engine is running @xcite . define @xmath101 as the observed duration of a grb , including any precursor but excluding its afterglow . in the collapsar model , this must be preceded by the cocoon phase during which the jet crosses the stellar envelope . the central engine must therefore be active for at least @xmath102 in the observer s frame . it is unlikely though not impossible for @xmath101 to be much shorter than @xmath103 , for this would require the central engine to shut off just as its effects become observable . so @xmath104 where @xmath105 is the efficiency with which the jet s kinetic luminosity is converted into gamma rays in the observed band . the second inequality derives from the first as long as the jet s luminosity is relatively constant . i have assumed a constant luminosity jet up to this point , but grbs are often observed to fluctuate significantly in intensity on very short timescales . how should the above results be adjusted for jet variability ? first , note that the process producing gamma rays ( e.g. , internal shocks ) is likely to accentuate the intrinsic variability of the source . second , the observed propagation speed of the jet head , @xmath106 , is equal to @xmath107 ( eq . [ [ tobs ] ] ) . a fluctuating jet thus progresses more slowly than a steady jet of the same mean luminosity . if one uses the average value of @xmath44 to constrain a star s mass and radius by requiring that @xmath103 is not long compared to the observed burst ( eqs . [ [ tobs - tgamma - rstar ] ] above and [ [ r - m - l - tobs ] ] below ) , then this constraint is only tightened if one accounts for variability . similarly , a variable jet deposits more energy per radius than a steady jet of the same mean luminosity ; this only tightens the constraints derived by requiring a jet - cocoon structure ( eq . [ [ l : no - bw : l<1 ] ] ) . to apply the above constraints to stellar progenitors for grbs , observed quantities must be related to the the parameters of equations ( [ l : no - bw : l<1 ] ) , ( [ einconstraint ] ) , and ( [ tobs - tgamma - rstar ] ) . this is possible for cases where , in addition to @xmath101 , the redshift @xmath46 and jet opening angle @xmath60 have been derived from afterglow observations . to construct @xmath44 one requires the isotropic ( equivalent ) kinetic luminosity @xmath34 and the mass per unit radius @xmath50 in the environment . i adopt for grbs energy , luminosity , and duration the following definitions : @xmath108 ; and @xmath109 . here the isotropic gamma ray energy is @xmath110 , and @xmath105 is the average efficiency factor relating the gamma - ray energy to the total ( kinetic , poynting , and photon ) luminosity of the jet . the net energy @xmath111 and luminosity @xmath112 represent only the approaching jet , which presumably has a counterjet . a key assumption employed throughout this section is that the @xmath113-ray half opening angle @xmath60 is equal to the jet s half opening angle while it crosses the outer stellar envelope . this is essentially the same as the assumption that the jet is ballistic ( rather than hot and pressure - confined ) at that point ; this assumption is revisited in [ widening ] . i assume the value of @xmath60 derived from afterglow observations @xcite can be used to characterize the jet as it crosses the outer stellar envelope and emerges from the star . the above definitions identify @xmath114 as the mean value ( energy @xmath115 in source - frame duration @xmath116 ) . this is somewhat arbitrary , since grbs are highly variable ; see [ s : variability ] for justification . for numerical evaluations i use either @xmath117 erg / s , or @xmath118 erg . the former is a characteristic value for grbs and can be observed without determining @xmath60 . the latter is motivated by @xcite s result that @xmath119 for ten grbs whose @xmath60 could be determined . in a stellar envelope the radial average value of @xmath50 is @xmath120 , the ratio of envelope mass to stellar radius . indeed , many presupernova envelopes have density profiles close to @xmath121 ( @xcite ) , for which @xmath50 is constant at its average value . the average value of @xmath44 is @xmath122 in order for the jet head to be relativistic ( @xmath123 ) , the star must have a mass per length much lower than @xmath124 , or the gamma - ray efficiency @xmath105 must be small . under the assumption of a ballistic jet , the observed time of jet breakout is @xmath125 this , along with the constraint @xmath126 ( eq . [ [ tobs - tgamma - rstar ] ] ) illustrates that typical long - duration grbs are most easily produced in compact stars @xcite . this constraint is best expressed @xmath127 ^ 2;\ ] ] the left - hand side pertains to a hypothetical stellar model and to the efficiency of gamma radiation and is constrained by observables on the right . note that the combination @xmath128 ^ 2 $ ] is related to an observed burst s fluence and duration through its comoving distance ( i.e. , physical distance at redshift zero ) , rather than its luminosity distance . note also that the above constraint is independent of @xmath60 . in figure [ figtiming ] i illustrate the derivation of @xmath129 and constraint ( [ r - m - l - tobs ] ) from the observed fluence and duration and an observed or estimated redshift ( for @xmath130 , @xmath131 , @xmath132 ) . bursts with well - observed redshifts are plotted as solid diamonds ; those for which @xcite have estimated redshifts from a luminosity - variability correlation are plotted as open circles . a progenitor model passes the timing constraint if it lies to the left of its burst . most of the bursts plotted are consistent with @xmath133 ; for none of them does this limit exceed @xmath134 . this constraint is nearly independent of redshift and derives primarily from the distribution of @xmath135 among grbs . vmo cores and wolf - rayet stars are compatible with many bursts . blue supergiants are compatible only with the very brightest and longest . red supergiants and vmos that retain their envelopes are ruled out , as are optically - thick shrouds of expanding ejecta which might persist in the `` supranova '' or stellar - merger models of central engines . equation ( [ l : no - bw : l<1 ] ) gives a complimentary constraint on the basis that a jet - cocoon structure exists : @xmath136 this constraint is derived from observations of bursts in figure [ figcocoon ] . similar to figure [ figtiming ] , filled diamonds here represent bursts for which afterglow observations have allowed an estimate of @xmath60 in addition to @xmath34 ( as reported by * ? ? ? * and references therein ) ; open circles represent redshifts estimated by @xcite . for these i have estimated @xmath60 by requiring that @xmath137 erg as suggested by @xcite . these points fall in a narrow band on the plot because @xmath138 . with @xmath139 held fixed , the dispersion in this quantity is dominated by @xmath140 ( 2 dex rms , for the points plotted ) rather than @xmath141 ( 0.5 dex rms ) . grbs from supernovae are likely to occur within a dense stellar wind , an environment that potentially affects both the grb itself and its afterglow @xcite . the internal - shock model for grb emission posits that significant variability on a time scale @xmath142 ( @xmath143 ) arises from the collision of shells within the jet at radii of about @xmath144 as discussed by @xcite . @xcite argue that an external shock with the ambient medium to which the afterglow is attributed can not create a fluctuating gamma ray burst . for internal shocks to occur , the external shock ( jet head ) must move beyond @xmath145 on a timescale not long compared to the duration of the burst : @xmath146 . @xcite define @xmath147 as the number of pulses that fit within the burst duration given a characteristic separation @xmath148 . with equations ( [ limitsofgammah ] ) , ( [ ris ] ) , and this definition , the criterion @xmath146 becomes @xmath149 or @xmath150 @xcite argue that @xmath151 is typical , although a wide variety of time scales is observed within bursts @xcite . for a wind @xmath50 is the ratio of mass loss rate to wind velocity , @xmath152 , so @xmath153 where @xmath154 and @xmath155 are normalized to characteristic values for wolf - rayet stars , although the presupernova mass loss rates of such stars is not well known @xcite . expressing condition ( [ lt - head - past - ris ] ) in terms of @xmath40 , @xmath156 the analogous criterion for a uniform ambient medium was presented by @xcite . this upper limit on @xmath40 must exceed the lower limit required for the escape of the observed gamma rays , described most recently by @xcite . these authors inferred @xmath157 for five of the ten ordinary bursts in their table 2 ; equation ( [ gjmax ] ) suggests that these could not have occurred in a stellar wind representative of wolf - rayet stars unless , for instance , @xmath158 ( which does not appear to characterize the bursts in their table 2 ) . however , condition ( [ gjmax ] ) might be circumvented by the efficient sweeping - forward of the ambient medium by runaway pair production , as discussed by @xcite , @xcite and @xcite . if the relativistic flow has generated a fraction of the observed photons at a radius smaller than @xmath159 , then these can clear optically - thin ambient gas from the region . a spreading break ( albeit one difficult to detect : * ? ? ? * ) will occur in the afterglow if equation ( [ gamthetawind ] ) is satisfied . the remaining isotropic kinetic luminosity after the gamma - rays have been emitted is @xmath160 . along with ( [ ltildewind ] ) , this becomes @xmath161^{1/4}.\ ] ] this criterion could only be violated by the narrowest of grb jets in especially dense stellar winds . the energy deposited by the jet as it crosses a surrounding stellar envelope ( @xmath65 in [ inside star ] ) is available to power ( concievably ) observable phenomena distinct from the grb and its afterglow . these include transients associated with the emergence of the jet head from the stellar surface ( _ breakout _ ) , and the escape of hot cocoon material along the path of the jet ( _ blowout _ ) . note that a type ic light curve is not a product of jet energy , as sn ic are powered by radioactive nickel , which is not produced at the jet head . the energy @xmath65 is typically less than the total grb jet energy ( for successful grbs ) according to condition ( [ tobs - tgamma - rstar ] ) . nevertheless the effects of deposited energy may be of observational interest if they cover a wider opening angle , dominate a different observed band , are emitted with higher efficiency , or occur where the jet stalls before breakout . in ordinary supernovae the fastest ejecta are produced at the surface of the star . there , the rapid decrease in stellar density leads to a whip - like acceleration of the shock front followed by additional postshock acceleration @xcite . sufficiently energetic explosions can produce relativistic ejecta this way , so long as the stellar progenitor is sufficiently compact @xcite . the impact of these ejecta with a circumstellar wind may produce a transient of hard photons ; this is a plausible explanation for the association between supernova 1998bw and grb 980425 @xcite . erg emerging from a sn of comparable energy . ] @xcite , @xcite , and @xcite have suggested that shock breakout might lead to the x - ray precursors seen in some bursts @xcite . @xcite and @xcite concentrate on the prompt flash from the shock - heated stellar photosphere ; however , the fast ejecta have a greater store of kinetic energy to be tapped . @xcite discuss several cases of precursor activity in gamma rays , including one burst for which iron - line emission has been suggested ( grb 991216 : * ? ? ? if precursors are due to shock breakout then their energy is a small fraction of @xmath97 and is therefore limited by conditions ( [ einconstraint ] ) and ( [ tobs - tgamma - rstar ] ) ; however it may yet be observable . @xcite argue that the x - ray precursor of grb 980519 resembles its x - ray afterglow to the extent that the afterglow effectively preceded the grb ; as argued by @xcite , this is consistent with the emission from material with @xmath162 ejected prior to the grb emission . it is therefore worthwhile to estimate the distribution of ejecta kinetic energies from shock breakout in a jetlike explosion . to do this one must identify at what point the jet head makes a transition from the state of ram pressure balance described in [ vhead ] to the state of whip - like shock acceleration discussed by @xcite . the jet head obeys ram pressure balance only if shocked ambient material exits the jet head more rapidly than the head accelerates . while this holds , the forward ambient shock can not travel far ahead of the jet reverse shock . at some point near the surface , however , ambient material can not exit the jet head prior to breakout . the forward shock will then accelerate away down the density gradient , and the flow becomes progressively more normal to the surface @xcite . if @xmath163 is a typical value for the perpendicular component of velocity in the jet head , and @xmath164 is the fractional depth within the stellar envelope , then material is trapped within the head if @xmath165 , i.e. , if the time to exit the head exceeds the time to reach the surface . a reasonable guess for @xmath163 is that shocked ambient gas exits the jet head at the postshock sound speed , in the frame of the head . for a nonrelativistic head , then , @xmath166 . in the relativistic case , the transverse velocity saturates at @xmath167 in the head s frame ; in the star s frame , @xmath168 @xcite . so the transition occurs when @xmath169 where @xmath170 are uncertain parameters . in the relativistic case , @xmath171 is determined implictly once @xmath172 is known . for this , apply equation ( [ vhead ] ) to the outer density distribution of the stellar progenitor @xmath173 where @xmath174 is the effective polytropic index and the coefficient @xmath175 is an extrapolation to @xmath176 . in a radiative outer layer @xmath175 can be derived from the mass , radius , and luminosity of the progenitor star @xcite ; @xcite present formulae ( their equations [ 25 ] , [ 45 ] , and [ 48 ] ) for @xmath175 in kramers or thomson atmospheres . once @xmath171 and @xmath177 or @xmath172 have been identified , the production of fast ejecta follows from @xcite s theory ( their eq . [ 54 ] ) if one matches the shock four - velocity @xmath178 with the head four - velocity @xmath179 at the depth @xmath171 . with equations ( [ vhead ] ) and ( [ xwhip ] ) setting a reference depth , external mass , and shock velocity , their theory predicts an isotropic - equivalent ejecta kinetic energy @xmath180 when the transition occurs in the nonrelativistic regime , and @xmath181 } \nonumber \\ & ~&\times \left(\frac{{l_{\rm iso}}}{r^2 \rho_h c^3}\right)^{\frac{4.32\gamma_p-3}{n(4\gamma_p-3 ) } } { l_{\rm iso}}\frac{r}{c } , \end{aligned}\ ] ] when the head is relativistic at the transition . here , @xmath182 is the envelope s polytropic exponent . thesen equations have been simplified by the restriction @xmath183 . in general , the appropriate value of @xmath184 is the minimum of the values given by equations ( [ ekisonr ] ) and ( [ ekisoxr ] ) . note that @xmath184 is an isotropic equivalent ; the _ total _ kinetic energy per lobe in ejecta above @xmath185 is smaller by @xmath186 . if one varies @xmath60 holding the other quantities fixed , this total energy above @xmath185 is maximized when the two expressions are equal , whereas the isotropic value continues to rise slowly as @xmath60 is decreased in the relativistic regime . in the relativistic regime , the above formulae only appy to @xmath187 i.e. , only to those ejecta involved in quasi - spherical shock acceleration . the uncertainty of @xmath188 leads to a much greater uncertainty in @xmath184 for the nonrelativistic than for the relativistic case : for instance , when @xmath189 , @xmath184 varies as @xmath190 in the nonrelativistic and as @xmath191 in the relativistic regime . physically , this difference arises because ram pressure and shock acceleration give very different velocity laws in the nonrelativistic regime ( @xmath192 and @xmath193 , respectively ) whereas they give very similar velocity laws in the relativistic regime ( @xmath194 and @xmath195 , respectively ) . another uncertainty concerns whether the head velocity should be matched to the shock or postshock velocity ; the latter choice increases the ejecta energies by only @xmath196 for both relativistic and nonrelativistic transitions . to illustrate these estimates of jet - driven shock breakout , let us consider the progenitor model for sn 1998bw adopted by @xcite and studied by @xcite . the parameters @xmath174 , @xmath197 , and @xmath175 can be derived from @xcite s tables 2 and 3 . adopting their fit for the outermost regions ( whose rest energy is @xmath198 erg ) , @xmath199 , @xmath200 , and @xmath201 cm . the isotropic kinetic energy in breakout ejecta is therefore : @xmath202 \times 10^{48 } \gamma_f^{-0.975 } ~{\ifmmode { \rm erg}\else erg\fi}. \end{aligned}\ ] ] if this formula were to give a result @xmath203 erg , a different envelope fit would be appropriate ; however , the qualitative result would be unchanged . even in isotropic equivalent , the energy predicted by equation ( [ eq98bwbreakout ] ) is small compared to a cosmological grb . it is comparable in magnitude to the values derived by @xcite for the spherical explosion of sn 1998bw . while sufficient to produce grb 980425 at a redshift of 0.0085 , it would not contribute to the appearance of a grb at @xmath204 . it should not be surprising that the energy of motion in shock breakout is intrinsically much smaller than that available in the jet , as the accelerating shock is powered by the jet for a brief period ( a small range of radii ) prior to breakout . breakout does produce a spray of ejecta with a variety of lorentz factors , which may produce weak transients if observed off the jet axis . the cocoon inflated by a jet prior to breakout is filled with hot gas that is free to expand away from the star after breakout , constituting a `` dirty '' fireball @xcite which may be visible either through its thermal emission , through its circumstellar interaction , or through line absorption and fluorescence . the cocoon energy @xmath65 ( eq . [ ein - from - l - tobs ] ) comprises @xmath205^{1/2 } \left(\frac{\theta}{3^\circ}\right)^2 { \rm erg}.\ ] ] two assumptions made in sections [ inside star ] and [ envelopeconstraints ] remain to be checked , both concerning the effects of an external pressure on the cocoon or on the jet . [ s : pressureconf ] addresses the assumption that the cocoon drives a strong shock into the stellar envelope . [ s : selfconf ] concerns the possibility that the jet is self - confined by its own cocoon pressure , then spreads sideways outside the star before producing gamma rays . if the ambient pressure is greater than the pressure within the cocoon , then the assumption of a strong shock in equation ( [ betacprelim ] ) is incorrect and the cocoon does not inflate as described in [ cocoon ] . the cocoon is overpressured relative to the envelope by @xmath218 , where @xmath219 is the envelope s isothermal sound speed : @xmath220 with @xmath221 so defined , the virial theorem stipulates @xmath222 when the mean is weighted by binding energy and @xmath223 when weighted by thermal energy within the star . in general , @xmath224 wherever the scale height is of order the radius . ( in a polytrope of index @xmath174 , @xmath225 is related to @xcite s variable @xmath226 by @xmath227 . ) using equations ( [ betacsoln ] ) and ( [ defalpha ] ) , @xmath228 when @xmath229^{7/3 } \frac{c^5}{g}. \end{aligned}\ ] ] this constraint is expressed in terms of the total jet luminosity @xmath230 ; i shall evaluate it at the fiducial collapse temperature @xmath231 ( o ignition ; * ? ? ? the appropriate value of @xmath232 , though different from @xmath233 , can not exceed unity . in ordinary core - collapse supernovae ( below the pair instability limit ) the core is degenerate but the envelope above the collapsing core is not ; also , gas pressure dominates over radiation pressure . at the oxygen ignition temperature , condition ( [ pc > p ] ) becomes @xmath234^{7/3 } ~{\ifmmode { \rm erg}\else erg\fi}~{\ifmmode { \rm s}\else s\fi}^{-1}\ ] ] where @xmath235 is the mean mass per particle in a.m.u . in contrast to ordinary supernova cores , vmo cores are dominated by radiation and are approximately @xmath189 polytropes in structure @xcite . they obey @xmath236^{1/2 } ( 3.2/t_9 ) ~{\ifmmode { r_\odot}\else r$_\odot$ \fi}$ ] ; hence @xmath237 when @xmath238 the critical luminosities in equations ( [ pc > p::ordinary ] ) and ( [ pc > p::vmo ] ) should be compared to the value @xmath239 implied by @xcite s standard value @xmath137 erg . in both ordinary supernovae and collapsing vmos , the jet cocoon could possibly be pressure - confined in the collapsing region ( @xmath240 ) , but not in the hydrostatic region where @xmath241 . the constraints on stellar envelopes derived above assume that observational determinations of the intensity ( @xmath34 ) and opening angle of gamma - ray emission can be applied to the earlier phase in which the grb crossed the stellar envelope . this requires that the jet is dynamically cold in the stellar interior , so that it does not widen once outside the star . it is important to consider the alternative : that the jet may contain internal energy sufficient to spread in angle after breakout . in this case the emission angle @xmath233 will exceed the jet angles @xmath232 attained within the stellar envelope . the jet would then be more intense within the star than outside , vitiating ( or at least relaxing ) the constraints on stellar envelopes derived in [ inside star ] and [ envelopeconstraints ] . for the flow to expand after breakout it must possess relativistic internal energy , @xmath242 , and it must be slow enough to spread , @xmath243 . ( note that cold jets will be heated and decelerate locally to @xmath243 , if they are crossed by an stationary oblique shock . ) in this case the final opening angle will be set by the relativistic beaming of the jet at breakout : @xmath244 ; a suggestion made by r. blandford . in general , @xmath245 \end{aligned}\ ] ] where the two possibilities refer to hot , confined jets ( with the confining pressure is released at @xmath246 ) and cold , ballistic jets , respectively . ) does not describe their numerical results ; however , it is not clear what @xmath246 or @xmath247 is applicable to the wide - angle ejecta they see . although the choice of a single @xmath246 is perhaps oversimplified , it remains a useful parameterization . ] one must have @xmath248 for pressure confinement to be significant ; if @xmath249 , the ballistic - jet constraints of the previous section would still hold for the outer envelope . for confined jets , @xmath250 is determined by an observational determination of the final opening angle . at the same time , when @xmath251 , and so long as @xmath40 is still significantly above unity , the jet luminosity is @xmath252 ( see the discussion after equation [ [ ltildedef ] ] ) , where @xmath253 is the jet radius . if evaluated at @xmath254 , @xmath40 may be replaced with @xmath255 . what determines @xmath30 ? the confining pressure is clearly time - dependent in this scenario , which complicates matters , but one can set @xmath256 at @xmath246 . the cocoon pressure is @xmath257 where @xmath61 is given by equation ( [ betacsoln ] ) : @xmath258 the primary assumption being that the jet head expands nonrelativistically ( @xmath58 ) . setting @xmath259 and using @xmath260 , @xmath261 which provides a means to evaluate pressure confinement if one has a constraint on @xmath262 . one such constraint arises if the jet expands adiabatically from its launching region , since this provides a formula for @xmath263 . for the steady flow of a relativistically hot gas ( @xmath264 ) at relativistic speeds ( @xmath265 ) through a channel , the simultaneous conservation of luminosity @xmath266 and mass flux @xmath267 , along with the relation @xmath268 , imply @xmath269 where @xmath270 is the radius from which the flow accelerates through @xmath271 and should reflect the dimensions of the central engine , tens or hundreds of kilometers . in equation ( [ confinement ] ) , this implies @xmath272 the @xmath113-ray opening angle produced by pressure confinement of an adiabatic jet is thus lower than any of the derived opening angles , the discrepancy being much greater in terms of pressure ( since @xmath273 at fixed @xmath270 ) . a jet expanding adiabatically from @xmath274 10 km can therefore be confined in the core of the star , but would become cold and ballistic in the outer envelope . we found in [ widening ] that the ambient core pressure may be high enough crush the cocoon , suggesting that the jet can also be confined by ambient pressure in this region . this scenario was described by @xcite ; as they note , it holds only in the stellar core and fails ( producing a ballistic jet of fixed @xmath275 ) in the envelope . however , we must question whether @xmath270 might also change . the above argument shows that a jet expanding from a launching region @xmath270 of order 10 km will no longer be pressure - confined in the outer stellar envelope . but , what if the propagation is not adiabatic ? this can result from shocks that cross the jet , or from mixing between the jet and its environment ( cocoon or stellar envelope ) . heuristically one expects the jet s memory of @xmath270 to be erased in either process ; a larger effective @xmath270 would ameliorate the discrepancy highlighted in [ s : selfconf ] , perhaps making pressure confinement more realistic . first , consider a stationary oblique shock that crosses the jet . the postshock jet pressure ( @xmath276 ) is brought into equilibrium with @xmath57 , the external pressure that launches the shock . the shock front must be causal with respect to the postshock flow , i.e. , sound travels across the ( postshock ) jet faster than the shock itself does . if the postshock lorentz factor is @xmath277 , sound fronts can propagate at an angle @xmath278 to the jet axis . this must exceed the angle @xmath279 of the crossing shock relative to the jet axis ( although usually not by much : * ? ? ? in order for the shock to cross the jet , one must have @xmath280 ; so , @xmath281 . the jet therefore makes a transition to a hot , pressure - confined state at its current radius ; the new effective value of @xmath270 for the postshock jet is @xmath282 . one can , in fact , estimate @xmath277 and the shock obliquity @xmath279 from the expression @xmath283 ( valid for hot jets ) , from the condition @xmath284 , and from the requirement that @xmath285 . if @xmath286 is estimated via equation ( [ betacsoln ] ) , one finds @xmath287^{1/8},\ ] ] which can be read as an estimate of the number of crossing shocks along the jet axis . note that eq . ( [ thetapossibilities ] ) predicts @xmath288 , roughly speaking , if the jet is effectively released after one of its crossing shocks . these results appear broadly compatible with the work of @xcite , and indicate the potential importance of this route to pressure confinement . second , consider the mixing of ambient material into the jet during a pressure - confined phase . nonadiabatic mixing can be described by imagining relativistically hot jet material flowing though a uniform channel of fixed cross - sectional area @xmath289 . at some initial time the channel is allowed to mix with some area @xmath290 of ambient material ( density @xmath291 , pressure @xmath57 ) at rest . assuming perfect mixing and nonrelativistic surroundings ( @xmath292 ) , the change in jet parameters @xmath30 , @xmath293 , and @xmath40 are determined by the conservation of energy , momentum , and rest mass through the mixing event . that is , @xmath294 algebra then yields differential equations for @xmath30 , @xmath293 , and @xmath40 as functions of @xmath295 . a couple interesting conclusions can be drawn from this exercise . first , the heuristic argument that @xmath270 would be forgotten is correct : the effective value of @xmath296 increases monotonically in non - adiabatic jet expansion , because @xmath262 increases whereas @xmath40 decreases ( or if @xmath297 , stays constant ) . this in turn implies that pressure confinement is extended somewhat by mixing . secondly , mixing forces the nondimensional jet parameters to trace a characteristic trajectory that whose implications would be observable . recall that the final opening angle @xmath233 is the inverse of @xmath40 if the jet is confined . also , final lorentz factor @xmath185 is equal to the ratio of the jet s luminosity to its mass flux , @xmath298 in terms of these dimensionless parameters , equation ( [ mixing1 ] ) can be restated @xmath299 where @xmath300 represents the fractional mass per unit length mixed in . the ratio of these equations ( in which the denominators cancel ) describes set mixing trajectories for @xmath301 , as shown in fig . [ figmix ] . there exists an attractor solution , which is @xmath302 ( i.e. , @xmath303 ) when @xmath304 and approaches @xmath305 . mixing trajectories rapidly join onto it by decreasing either @xmath40 or @xmath306 as mass is mixed in . as shown in figure [ figmix ] , mixing trajectories do cross through reasonable values of @xmath185 and @xmath233 . does this mean that these jet parameters arise from mixing ? this seems plausible if there ever exists phase of pressure - confined evolution in the stellar core or mantle . if @xmath307 , adiabatic expansion and mixing both act to bring @xmath308 . if @xmath309 , the adiabatic tendency for @xmath40 to increase is counterbalanced by the nonadiabatic decrease in @xmath40 . this makes the mixing attractor solution a probable one for the end state of jets that are pressure - confined at some stage . ( a more detailed analysis would require a dynamical analysis of jet expansion . ) if so , then the attractor solution would imprint the correlation @xmath310 which may have observational implications in that @xmath311 for a fixed @xmath312 . such a trend has been implicated in the correlations between spectral lag and luminosity , variability and luminosity , and afterglow break time and luminosity among grbs ( e.g. , * ? ? ? * ; * ? ? ? the mixing attractor provides a motivation for a rather tight correlation among bursts with similar jet luminosities . the variability - luminosity correlation would be affected also by the filtering of jet variations by pressure gradients , which occurs only when the jet is pressure - confined . now , the interal dynamical time of a relativistic jet is @xmath313 , or @xmath314 in the lab frame . if a strobe approached the observer at @xmath40 , pulsating once per dynamical time , the observed period would be foreshortened to @xmath315 the same result can be derived by considering a standing - wave pattern of wavelength @xmath262 in the jet s frame , or @xmath316 in the lab frame , that is swept past a fixed pressure - release radius @xmath254 and leads to jet pulsations as the peaks and troughs pass by . as we saw above , crossing shocks and mixing increase the effective value of @xmath270 and thus increase the characteristic variability timescale . shocks and mixing also decrease @xmath114 by decreasing @xmath40 and thereby increasing @xmath233 ; this implies a correlation between jet variability and @xmath113-ray brightness . what should we make of the possibility that pressure - confined jets will evade the constraints on stellar envelopes derived in [ inside star ] and [ envelopeconstraints ] for ballistic jets ? equation ( [ adconfinement ] ) shows that jet confinement works if @xmath317 a value that does not appear to violate timing constraints ( eq . [ [ deltatobspressure ] ] ) . this indicates pressure - confined jets are plausible if shocks or jet mixing increase @xmath270 sufficiently . why would a jet change from pressure - confined to ballistic ? this could occur in a couple ways . first , the jet could accelerate in such a way that sound no longer crosses it despite it being hot , i.e. , @xmath318 for some range of @xmath37 . however this appears to require the ambient density to drop to a near vacuum , implying it would not occur spontaneously within a star . alternatively , the jet could become internally nonrelativistic ( @xmath319 ) . as we have seen , adiabatic evolution works in this direction whereas crossing shocks and non - adiabatic mixing oppose it . the transition would then depend on the specifics of jet propagation and mixing . finally , the jet could be confined by internal magnetic stresses , a possibility highlighted by the @xmath113-ray polarization discovered by @xcite , but beyond the scope of this paper . jet pressure confinement would naturally lead to time - dependent jet properties , since the confining pressure evolves during the cocoon phase and after breakout . crossing shocks and mixing with ambient material are also unlikely to be steady processes . one would expect evolution during grbs as a result ; however , no trends are discernible except for changes in pulse asymmetry @xcite . apart from this contraindication , it is difficult to judge whether pressure confinement persists across entire stellar radii . if so , then the constraints on stellar envelopes in [ inside star ] and [ envelopeconstraints ] are relaxed . numerical simulations can potentially solve detailed questions such as this one . however , caution should be used in interpreting them , because of the many decades separating the launching radius from the stellar radius . this paper constrains possible stellar progenitors for grbs by requiring that jets of grb - like luminosities and durations can clear a path for themselves in the star s envelope prior to producing gamma rays . of all the possible progenitors , only the compact carbon - oxygen post - wolf - rayet stars ( sn ic progenitors ) and the bare cores of very massive objects can plausibly collapse in the durations of long grbs at low redshift ( [ s : freefalltimes ] , figures [ fig1 ] and [ fig2 ] ) . in more extended stars , the outer stellar envelope remains to impede the progress of the jet , and it is likely to violate observational constraint . therefore , i conclude that grbs with sn progentiors come primarily from type ic or vmo - core events . type ib is not thoroughly ruled out by this work , but type ii ( supergiant stars ) are . the most stringent constraint on a stellar envelope arises from the requirement that the jet can traverse it in an observed time not much longer than the duration of the grb . under the assumption of a ballistic rather than a pressure - confined jet , this constraint is independent of the inferred opening angle of the burst , and ( given an observed fluence and duration ) depends on its inferred comoving distance rather than its luminosity distance . this makes it insensitive to an uncertainty in redshift . given the luminosities and durations of grbs ( regardless of their redshift ; fig . [ figtiming ] ) , only post - wolf - rayet stars and vmo cores are compact enough to satisfy this criterion . the variability - luminosity correlation discussed by @xcite and @xcite allows this constraint to be applied to a large number of bursts in the batse catalog ( figure [ figtiming ] ) . only a few are compatible with blue supergiant progenitors ; red supergiants and vmos with envelopes are ruled out . also ruled out is the `` supranova '' model of @xcite ( a supernova followed by a grb ) , unless the sn ejecta are optically thin by the time of the grb , or unless the pulsar nebula is energetic enough to clear them aside . post - wolf - rayet stars have been favored among stellar grb progenitors since the work of @xcite , on the basis that their compact envelopes delay the grb jet breakout the least . the above constraint quantifies and strengthens this conclusion , and relates it to the observed properties of grbs rather than those of a specific model for the central engine . for ballistic jets , an additional constraint arises from the requirement that the grb jet cocoon should not overtake its driving jet and produce a spherical explosion . this is generally not as restrictive as the constraint from burst durations discussed above , but it does become important for jets with opening angles exceeding @xmath320 ( figure [ figcocoon ] ) . vmo cores , being the most compact , are the progenitors most likely to be ruled out this way if they have not collapsed prior to the grb . the requirement of a jet - cocoon structure also gives interesting upper and lower bounds on the energy entrained in the stellar envelope during the phase of jet propagation ( eq . [ [ einconstraint ] ] ) . this energy is stored in the jet cocoon and is available to drive a `` dirty '' fireball @xcite of expanding cocoon material after the jet breaks out . the breakout phenomenon is itself a candidate for producing a transient , as may have happened in sn 1998bw to produce grb 980425 . in section [ breakout ] it is calculated how much kinetic energy is channeled into relativistic envelope ejecta during a jet s breakout , by matching the propagation law for the jet s terminal shock onto the relativistic shock and post - shock acceleration behavior described by @xcite . i find the energy of this ejecta to be small compared to that of the burst . @xcite have recently discussed how the upscattering of photons from the shocked envelope by the jet may produce a precursor of hard gamma rays ; however , note that this is reduced in importance for the compact stellar progenitors favored by the timing constraints . @xcite have suggested breakout ejecta as the origin of short , hard bursts ; however their estimate of the energy is at odds with that calculated here . if the gamma - ray photons are not able to clear away a stellar wind in the region around the star in the manner described by @xcite , @xcite , and @xcite , then the presence of this wind places an upper limit on the jet lorentz factor . this limit arises in the internal shock model for grb emission because the presence of the external shock limits the distance within which internal shocks can form . the equivalent limit has been presented previously for uniform ambient media @xcite ; however , for a sufficiently dense stellar wind it can conflict with the lower limits on jet lorentz factor ( e.g. , * ? ? ? figure [ fig418 ] illustrates the above criteria for the specific case of grb 000418 , assuming a ballistic jet with @xmath321 . its seven - second duration is briefer even than the free - fall times of vmo cores . because of its large opening angle ( @xmath322 ; * ? ? ? * ) , it would not succesfully form a jet - cocoon structure in any uncollapsed portion of the vmo core . in fact , a jet of its inferred luminosity could cross nothing more extended than the most compact of wolf - rayet stars in the grb duration . i conclude from this that it came either from a compact carbon - oxygen wolf - rayet star , or from a vmo core that managed to produce a grb of briefer duration than its free - fall time , or that it did not have a supernova origin . these restrictions change quantitatively , but not qualitatively , if @xmath323 . pressure confinement of jets provides a way for grb hosts to evade the constraints on stellar envelopes listed above for ballistic jets . this occurs because , for a given @xmath113-ray opening angle , confined jets are narrower and more intense than their ballistic counterparts . the computational results of @xcite and @xcite correspond to pressure - confined jets . in [ adiabatics ] it was shown that jets that do not mix with their environs can not remain pressure - confined ( see also * ? ? ? simulations that do not resolve the jets launching scale may not observe this effect , however . mixing of jet and envelope ( [ ss : mixing ] ) is capable of sustaining a pressure - confined state . intriguingly , mixing of jet and envelope imprints on the jet a tight correlation ( the `` mixing attractor '' of eq . [ [ mixingattractor ] ] and figure [ figmix ] ) between its lorentz factor and energy per unit mass . this leads to a correlation between final opening angle and lorentz factor , which in turn may be related to the lag - luminosity and variability - luminosity correlations observed in burst catalogs . excessive mixing , however , has the effect of filtering rapid fluctuations from jets . this work was stimulated by a visit to uc santa cruz and by interactions with stan woosley , andrew macfadyen , alex heger , and weiqun zhang while i was there . it was further motivated by conversations with and encouragement from roger blandford , reem sari , sterl phinney and josh bloom during a visit to caltech . it is a pleasure to thank woosley , blandford , and phinney for their hospitality during those visits . i am especially grateful to nicole lloyd - ronning for explaining the grb luminosity - variability relation and suggesting clarifications . i also thank andrei beloborodov and chris thompson for discussing jet - envelope interactions and chris fryer for discussing progenitor scenarios . comments from beloborodov , charles dermer , jonathan tan , chris mckee , john monnier , and the referee , ralph wijers , are also appreciated . this work was supported by nserc and by the canada research chairs program . e. , diercks a. , frail d. a. , kulkarni s. r. , bloom j. s. , sari r. , halpern j. , mirabal n. , taylor g. b. , hurley k. , pooley g. , becker k. m. , wagner r. m. , terndrup d. m. , statler t. , wik d. r. , mazets e. , cline t. , 2001 , , 556 , 556 d. a. , kulkarni s. r. , sari r. , djorgovski s. g. , bloom j. s. , galama t. j. , reichart d. e. , berger e. , harrison f. a. , price p. a. , yost s. a. , diercks a. , goodrich r. w. , chaffee f. , 2001 , , 562 , l55
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i constrain a possible supernova origin for gamma - ray bursts by modeling the dynamical interaction between a relativistic jet and a stellar envelope surrounding it . the delay in observer s time introduced by the jet traversing the envelope should not be long compared to the duration of gamma - ray emission ; also , the jet should not be swallowed by a spherical explosion it powers .
the only stellar progenitors that comfortably satisfy these constraints , if one assumes that jets move ballistically within their host stars , are compact carbon - oxygen or helium post - wolf - rayet stars ( type ic or ib supernovae ) ; type ii supernovae are ruled out .
notably , very massive stars do not appear capable of producing the observed bursts at any redshift unless the stellar envelope is stripped prior to collapse .
the presence of a dense stellar wind places an upper limit on the lorentz factor of the jet in the internal shock model ; however , this constraint may be evaded if the wind is swept forward by a photon precursor .
shock breakout and cocoon blowout are considered individually ; neither presents a likely source of precursors for cosmological grbs .
these envelope constraints could conceivably be circumvented if jets are laterally pressure - confined while traversing the outer stellar envelope .
if so , jets responsible for observed grbs must either have been launched from a region several hundred kilometers wide , or have mixed with envelope material as they travel
. a phase of pressure confinement and mixing would imprint correlations among jets that may explain observed grb variability - luminosity and lag - luminosity correlations .
[ firstpage ] gamma rays : bursts , supernovae : general , shock waves , relativity
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the large hadron collider ( lhc ) probes collisions of protons at very high energies , resulting in a multitude of final - state particles . with increasing energy , the probability that one hadron - hadron collision leads to more than one scattering process also increases . these additional scattering processes beside the primary hard scattering belong to the group of multi - parton interactions ( mpi ) . their estimation is important for the correct determination of background from standard model processes , for instance when the signal process consists of new physics particles . in particular , double parton scattering ( dps ) , where two distinct parton interactions arise from the same proton - proton collision , can become likely enough to compete with single parton scattering ( sps ) processes , see fig . [ fig : dpsfeyn ] . therefore , a thorough understanding of these additional contributions is needed for a precise theoretical description of the background at the lhc and will also help to explore the inner structure of protons and nucleons , not being accessible by perturbative calculations . double parton scattering has been searched for both in pre - lhc experiments like afs , ua2 , cdf , and d0 as well as by the lhcb and atlas collaborations , in 4-jet @xcite , @xmath6-jet @xcite , di-@xmath7-jets @xcite , @xmath8-jets @xcite , @xmath9 @xcite , @xmath10 @xcite , open charm @xcite , @xmath0+charm @xcite , @xmath11+charm @xcite , @xmath12 @xcite and @xmath13 @xcite final states . on the theoretical side the efforts have concentrated on improving the understanding of the underlying scattering mechanism as well as providing phenomenological predictions . in particular related issues such as correlations and interferences between the two hard scatterings , the role of the perturbative splitting contributions ( so - called `` 2v1 '' ) and the definition of double parton scattering cross section as well as double parton distributions have been addressed , see e.g. @xcite for a comprehensive review . a @xmath0 pair is a very good candidate to study double parton scattering at the lhc due to relatively high production rates and subsequent decays into muons giving a clear and easily distinguishable signal . results for the production of @xmath0 pairs have been published by lhcb in @xcite , by d0 in @xcite , and by cms in @xcite . correspondingly , since then there has been a considerable interest to improve theoretical predictions for double @xmath0 production both for the sps and dps production modes @xcite . the calculation of conventional single parton scattering contributions to @xmath0 pair - production is non - trivial and requires specific methods to account for the non - perturbative mechanisms involved in meson production as well as the short - distance effects . two widely applied approaches are the colour - singlet model ( csm ) @xcite and non - relativistic quantum chromodynamics ( nrqcd ) @xcite . in the framework of nrqcd , until not long ago , only the lo predictions for hadronic production in the colour singlet production mode @xcite , supplemented by the octet corrections @xcite , were known . recently , the effects of relativistic corrections @xcite , nlo corrections and selected nnlo qcd contributions @xcite as well as an application of the @xmath14 factorisation approach @xcite have been investigated . additionally , the importance of including contributions from all possible @xmath15 fock state configurations relevant for prompt double @xmath0 production has been pointed out in @xcite . this paper documents the predictions of sps and dps production of a pair of @xmath0 , delivered to the lhcb and atlas experiments for their ongoing studies of double parton scattering with run i data . the work presented here updates the study on @xmath0 pair - production reported in @xcite , which in turn was inspired by the first measurement of a double @xmath0 signal @xcite . furthermore , predictions for the current lhc run at a centre - of - mass energy of @xmath16 tev are provided . we also perform a comparison with cms data @xcite and more thoroughly with theoretical predictions for double @xmath0 production obtained by another group @xcite . the outline is as follows . in section [ sec : theo_setup ] , the theoretical setup of @xcite used for both the sps and dps cross section calculations is reviewed , followed by a listing of monte carlo parameters for event simulation in section [ sec : monte_sim ] . we present numerical results for total cross sections and kinematic distributions for a choice of experimentally accessible variables in section [ sec : kin_dis ] . at last , we conclude in section [ sec : conclusions ] . in this work , the sps contributions will be considered utilising a leading - order ( lo ) colour - singlet result presented in @xcite and including radiative corrections from parton showering . the details of the implementation are described in section [ sec : monte_sim ] and the sps results obtained in this way are compared to the nlo calculations of @xcite in section [ sec : complansberg ] . as it was pointed out in @xcite , the prompt production of @xmath0 mesons comprises feed - down from the decay of @xmath17 and @xmath18 at a non - negligible amount of roughly 85% . the sps calculation of @xcite is for direct production of @xmath0 pairs only , so in the following , all sps cross sections will be considered for prompt production , @xmath19 . the dps results implicitely include feed - down contributions due to the fit to experimental data . to include some higher - order effects in our sps predictions , in addition to using nlo pdfs , we enable initial - state radiation or parton showering within the ` herwig ` @xcite framework . furthermore , if denoted , we also add effects of intrinsic transverse momentum of the initial - state partons using a gaussian model in ` herwig ` with a root mean square @xmath20 of 2 gev . we have checked that the predictions do not depend strongly on the actual numerical value , and it will be seen in the following sections that the effect of the intrinsic transverse momentum is rather mild on the distributions . dps production of a @xmath0-pair is described using an approximation in which the dps cross section factorises into a product of two hard - scattering cross sections describing single-@xmath0 production which are independent from each other : @xmath21 this customary approximation assumes factorization of the transverse and longitudinal components in the generalized parton distribution function . we refer the reader to @xcite for a discussion of the validity of the approximation and the status of understanding factorization in dps . the sps cross section for single-@xmath0 production is given as @xmath22 with a sum over the initial - state flavours @xmath23 and the parton distribution functions @xmath24 and @xmath25 the partonic cross section for single @xmath0 production with the corresponding matrix elements @xmath26 and the phase space @xmath27 . @xmath28 denotes any additional final state which is not a @xmath0 , and therefore not of interest for @xmath0 production . the factor @xmath29 is assumed to only depend on the transverse structure of the proton , and should therefore be process and energy independent if the factorisation of eq . holds . it is the main quantity to be extracted by a dps experiment . theoretical description of single quarkonium production @xcite is challenging even within the nrqcd framework @xcite . given that lhcb can trigger over low @xmath30 muons it is important to describe the low @xmath30 production accurately . therefore in this work we choose to model the low @xmath30 region and use the same setup as in @xcite . it relies on the matrix element as in eq . given by : @xmath31^{-n } & \text{for } p_t > \langle p_t\rangle , \end{cases}\label{crystalball}\end{aligned}\ ] ] which describes a fit of data from the lhcb @xcite , atlas @xcite , cms @xcite , and cdf @xcite experiments to a crystal ball function . in eq . , @xmath32 , and the fit parameters are determined to be @xmath33 and @xmath34 for @xmath35 and @xmath36 gev . while the work of @xcite updated the fit parameters to include more recent measurements of single @xmath0 production , we have checked that the change in predictions for dps production is only moderate and well within the uncertainty on @xmath29 . we have also checked that for the available measurement of single @xmath0 production at 13 tev from the lhcb experiment @xcite with @xmath37 @xmath38b , the fit parameters still produce results at 13 tev consistent with the lhcb measurement , @xmath39 @xmath38b . the public monte carlo event generator ` herwig-7.0.3 ` @xcite has been used to simulate double @xmath0 production at the lhc via sps . the central values for the renormalisation and factorisation scales are chosen as the transverse mass of a single @xmath0 , @xmath40 with the physical @xmath0 mass @xmath41 gev @xcite . one parameter appearing in the calculation of the sps cross section of @xcite is the charm quark mass which we set to @xmath42 which corresponds to the lo choice of the hadron mass in a nrqcd calculation @xcite . another input parameter entering the sps calculation is the non - perturbative wave function of the @xmath0 meson at the origin . in the following computations , it is set to @xmath43 gev@xmath44 @xcite . it should be noted that a variation of this parameter can be achieved by multiplying the sps cross section by a factor of @xmath45 , where @xmath46 is the new value of the wave function . we use mstw2008 nlo parton distribution functions @xcite for the sps predictions including initial - state radiation and for dps . the parton distribution functions are accessed via the lhapdf 6 library @xcite . the @xmath0 mesons are assumed to decay isotropically into a pair of opposite - sign ( os ) muons with a branching ratio of @xmath47 . out of the two possible combinations of choosing os muon pairs , the one with an invariant mass closest to @xmath48 is chosen . from these pairs , properties of the @xmath0 are reconstructed . to optimise the data samples collected by the experiments for a dps analysis , a certain set of cuts on transverse momentum and ( pseudo-)rapidity of the @xmath0 as well as their decay products is applied . unless otherwise specified , we use the cdf value of @xmath49 mb @xcite . a more recent double-@xmath0 study by d0 reports a lower value of @xmath50 mb @xcite , but given that most of other experiments measure higher values , see e.g. @xcite , and the difficulty of theoretical modelling of @xmath29 , we choose the cdf value . with its relatively wide error bars it then accounts to a large extent for the observed span in values of @xmath51 . we also note that this value is in accordance with the phenomenological estimates @xcite taking into account in our framework ( eq . [ dpsfact ] ) the so - called `` 2v2 '' and `` 2v1 '' contributions , i.e. contributions from two separate parton ladders or from one ladder and another ladder created by a perturbative splitting of a single parton , respectively . as found out in @xcite , the shapes of the transverse momentum and rapidity distributions for the two types of production mechanisms remain very similar , justifying our effective approach of considering only the conventional 2v2 scattering . in a similar manner as for the non - perturbative wave function at the origin , the dps results for a different value of @xmath29 can be obtained by rescaling our dps cross section with a factor of @xmath52 . the dps predictions have been cross checked with two independent numerical in - house implementations . the lhcb experiment , being a forward spectrometer , mainly selects events in the forward - scattering region with low transverse momentum . the cuts are : * @xmath53 gev , * @xmath54 , with @xmath55 being the transverse momentum and @xmath56 the rapidity of a single @xmath0 . any further cuts on the muons as the decay products of the di-@xmath0 are not relevant for the theoretical predictions presented in this work , as they are already taken into account in the efficiency correction of the data . the atlas experiment probes the @xmath0 in the central region imposing a minimum transverse momentum and a more central rapidity region . additionally , several cuts are also applied to the muons : * @xmath57 gev , * @xmath58 , * @xmath59 gev , * @xmath60 , * at least 1 @xmath0 with both @xmath61 gev , with @xmath62 being the transverse momentum and @xmath63 the pseudorapidity of one muon . .total cross sections for sps and dps production of a @xmath0 pair for different centre - of - mass energies and cuts . ps+@xmath64(2 gev ) denotes the addition of initial - state radiation and intrinsic transverse momentum to the lo calculation in ` herwig ` . all numbers include the branching ratio factor of @xmath65 . the uncertainties on the ps+@xmath64(2 gev ) numbers correspond to a simultaneous variation of the renormalisation and factorisation scales up and down by a factor of two , while the uncertainty on the dps numbers corresponds to the uncertainty of our value of @xmath29 used , see section [ sec : monte_sim ] . [ cols="^,^,^",options="header " , ] also for the atlas predictions , shown in fig . [ fig : atlas_plots13 ] , the dominance of the dps contributions at 13 tev leads to an easier distinction between sps and dps . despite the transverse momentum distribution of the di-@xmath0 system in fig . [ fig : atlas_plots13 ] ( a ) now offering a clearer possibility to separate sps and dps contributions for low @xmath66 due to the dps contributions being larger than sps by almost a factor of 2 , we see that the increase in the centre - of - mass energy complicates the distinction of the shapes of sps and dps , and we again remark that higher - order corrections which are not included here can change the shape of the distribution significantly . cc tev for the invariant mass ( a ) , the rapidity separation ( b ) , and the transverse momentum of the di-@xmath0 system ( c ) . shown are bins which are normalised to the corresponding total cross sections of the different sps and dps distributions and the data.,title="fig : " ] & tev for the invariant mass ( a ) , the rapidity separation ( b ) , and the transverse momentum of the di-@xmath0 system ( c ) . shown are bins which are normalised to the corresponding total cross sections of the different sps and dps distributions and the data.,title="fig : " ] + ( a ) & ( b ) + + the cms experiment has recently measured @xmath0-pair production at @xmath1 tev @xcite . they applied the following cuts to their data : * @xmath67 gev for @xmath68 , * @xmath69 gev for @xmath70 , * @xmath71 gev for @xmath72 . the @xmath30 cut in point 2 scales linearly from 6.5 gev to 4.5 gev with the value of @xmath73 from 1.2 to 1.43 . no further cuts on the muons are applied . in fig . [ fig : cms_comp ] , we compare our predictions to the cms data . we show all bins normalised to the corresponding total cross section of a line to only compare the shape of the distributions and approximately remove the dependence on a specific pdf set . furthermore , for the theoretical sps and dps predictions , we only show the central values without the error bands as described in section [ sub : lhcb_predictions ] . shown are the invariant mass distribution , the transverse momentum of the di-@xmath0 system , and the rapidity separation of the two @xmath0 . we see that our predictions catch the bulk behaviour of the cms data , in particular also when further sources of uncertainty like the exact choice of the parameters which appear in the sps and dps calculations would be taken into account . especially for the invariant mass and the rapidity separation distributions , figs . [ fig : cms_comp ] ( a ) and ( b ) , we see that at the high end of the spectrum the dps contributions can not be neglected . at the same time , the existing discepancies between theory and data call for further improvements in the theoretical description of sps and dps distributions . at last , we compare our results to the recently published ones of lansberg and shao @xcite . the authors present predictions for similar scenarios of @xmath0-pair production at the lhcb and atlas experiments , with the difference of using full calculations of real gluon emission at nlo@xmath74 , the asterisk denoting the lack of virtual corrections . this method differs from ours by also taking into account hard gluon emission , while parton showering only considers soft gluons , however to all orders in @xmath75 . in this regard , it is interesting to see how the two approaches compare for the lhcb and atlas cuts . in order to minimise the sources of uncertainty , we choose parameters and pdf sets as close as possible to lansberg et al . these are the wave function of the @xmath0 meson at origin @xmath76 gev@xmath44 , the charm mass in the range @xmath77-@xmath78 gev , and the pdf sets cteq6l1 for lo @xcite , cteq6 m for sps ( ps+@xmath64 ) @xcite , and mstw2008 nlo for dps . the renormalisation and factorisation scales for sps production are set to @xmath79 , where @xmath80 . the error bands are obtained from a simultaneous variation of @xmath81 and @xmath82 as @xmath83;@xmath84;@xmath85 . for dps , we additionally use the effective dps cross section @xmath86 mb and the best fit parameters of @xcite with @xmath87 and @xmath88 . the factorisation scale in this case is set , as before , to the transverse mass of a single @xmath0 . the error bands are now obtained from the uncertainty of @xmath29 . [ fig : comp_lhcb_lansberg ] shows the comparison of distributions for three kinematic variables : the transverse momentum of the di-@xmath0 system @xmath66 , the rapidity separation between the two @xmath0 @xmath89 , and the invariant mass of the di-@xmath0 system @xmath90 , for the lhcb cuts at a collider energy of @xmath1 tev . we note that in @xcite , the sps predictions for the @xmath89 and @xmath90 distributions are only given at lo without taking into account additional gluon radiation . in fig . [ fig : comp_lhcb_lansberg ] ( a ) , it can be seen that the transverse momentum distributions of sps in our calculation and the one of @xcite agree within the error bands for intermediate values of @xmath66 , while they differ at low and high @xmath66 . we see at low @xmath66 the typical suppression from the all - order structure of parton showering , while the nlo@xmath74 prediction is growing towards small @xmath66 . we expect that the inclusion of parton showering describes the shape of the distribution at low @xmath66 better than a fixed - order calculation at nlo@xmath74 , since a major part of the contribution in this region comes from soft - collinear gluon emission , which is approximately taken into account at all orders in @xmath75 in the parton shower formalism . on the other hand , the high-@xmath66 region can not be described properly by parton showering due to the lack of hard gluon emission which dominates this region . we remark that the good agreement for intermediate @xmath66 is also related to the @xmath53 gev cut , effectively cutting off the high-@xmath66 region where hard gluon emission becomes important . the dps predictions for the transverse momentum distribution agree very well between our calculation and @xcite due to the same functional form of the cross section fit of eq . . cc system ( a ) , the rapidity separation ( b ) , and the invariant mass ( c ) . the `` sps nlo ( lansberg et al . ) '' and `` dps ( lansberg et al . ) '' uncertainty bands are read off from the corresponding plots in @xcite . the cross sections here are not multiplied by the squared branching ratio @xmath65.,title="fig : " ] & system ( a ) , the rapidity separation ( b ) , and the invariant mass ( c ) . the `` sps nlo ( lansberg et al . ) '' and `` dps ( lansberg et al . ) '' uncertainty bands are read off from the corresponding plots in @xcite . the cross sections here are not multiplied by the squared branching ratio @xmath65.,title="fig : " ] + ( a ) & ( b ) + + cc for the case of the atlas@xmath74 cuts . these differ from the atlas cuts defined in section [ subsec : atlascuts ] by imposing a lower @xmath55 cut of @xmath91 gev instead of @xmath57 gev . furthermore , the predictions are shown for @xmath92 tev instead of 8 tev.,title="fig : " ] & for the case of the atlas@xmath74 cuts . these differ from the atlas cuts defined in section [ subsec : atlascuts ] by imposing a lower @xmath55 cut of @xmath91 gev instead of @xmath57 gev . furthermore , the predictions are shown for @xmath92 tev instead of 8 tev.,title="fig : " ] + ( a ) & ( b ) + + the rapidity separation in fig . [ fig : comp_lhcb_lansberg ] ( b ) shows , as expected , a very good agreement between the sps and dps predictions from us and @xcite because of the lo calculations and the same parametrisation of eq . . for the invariant mass distributions of fig . [ fig : comp_lhcb_lansberg ] ( c ) , our sps and dps predictions agree well with @xcite for an invariant mass up to approx . 20 gev , while there are differences for sps in the last two bins and for dps in the last bin . these differences might be related to numerical precision , as the differential cross sections become very small for a high invariant mass . [ fig : comp_atlas_lansberg ] for the atlas@xmath74 predictions at a collider energy of @xmath1 tev shows the same set of distributions as for the lhcb predictions . it should be noted that the asterisk denotes a changed atlas cut with @xmath91 gev instead of @xmath57 gev . the transverse momentum distribution of fig . [ fig : comp_atlas_lansberg ] ( a ) displays a larger @xmath66 range than for the lhcb cuts , which shows that for large values of @xmath93 gev , the parton shower and nlo@xmath74 results of sps differ by a notable amount region ; instead the nrqcd framework should be used . ] while there is again an agreement for low @xmath94 - 15 gev within the error bands . we point out that the bulk of the cross section comes from the region for @xmath95 gev , as seen e.g. in fig . [ fig : atlas_plots ] ( b ) for a similar setup at 8 tev ( on a linear axis ) , so that the differences for @xmath93 gev between the parton shower and nlo@xmath74 results affect the description of only a small portion of events . interestingly , while there is a small difference for sps in the bin with small rapidity separation , fig . [ fig : comp_atlas_lansberg ] ( b ) , the two predictions agree well within the errors for @xmath96 . the invariant mass distribution of fig . [ fig : comp_atlas_lansberg ] ( c ) shows that , while there is again a difference for the smallest bin , the predictions for sps with parton shower and nlo@xmath74 corrections almost agree within the error bands ( and they in fact do for some bins ) , although it can be seen more clearly here that the lack of hard gluon emission leads to the parton shower result always being below the nlo@xmath74 result . the dps predictions agree very well for the transverse momentum distribution , while for the rapidity separation and the invariant mass , there are slight deviations in the high-@xmath89 and @xmath90 bins , possibly related to the difference in numerics and codes used to compute these predictions . from these comparisons , we see that , as one would expect , the rapidity separation distribution is most stable with respect to higher - order corrections from hard gluon emission that are not included in our approach , while the transverse momentum distribution of the di-@xmath0 system is most strongly affected by them . we remark that here the sps and dps predictions have been computed with different input pdfs ( cteq6l1 , cteq6 m , and mstw2008 nlo , respectively ) for the purpose of comparing to @xcite , while the comparison between the magnitudes of sps and dps presented in section [ sec : kin_dis ] avoids introducing pdf effects unrelated to the sps and dps calculations . precise predictions for multi - parton interations are a vital ingredient for the high - energy collisions at the lhc , in particular during the current run at a centre - of - mass energy of @xmath16 tev and future runs at higher energies , where the probability for such subleading scattering processes to happen is significantly increased . in this work we have documented sps and dps predictions for the production of @xmath0 pairs with the updated fiducial volume cuts for the lhcb analyses of run i data , and also for a new dps study of @xmath0-pair production at @xmath97 tev by the atlas experiment . the distributions show interesting indications that dps processes could contribute significantly to certain kinematic regions of the invariant mass and rapidity separation of the di-@xmath0 system , while the transverse momentum distribution is very susceptible to higher - order corrections . the predictions for a collider energy of @xmath16 tev show a considerable increase of the dps contributions with respect to sps . finally , the comparison to the results presented in @xcite indicate a good agreement for regions where it is reasonable to compare a parton shower to a nlo@xmath74 calculation , supporting the parton shower approach as a good approximation in these regions . + * note added : * in the final preparation stages of this report , we have become aware of a new atlas study of double @xmath0 production @xcite . in this study , our dps predictions presented in section [ sec : kin_dis ] are compared with the data - driven estimates of dps . a good agreement between dps theory and data is found for all differential distributions reported in @xcite . the full @xmath0 distributions measured by atlas are then compared with the sum of our dps predictions and nlo sps predictions of @xcite with the collision energy and fiducial volume adjusted according to the experimental analysis and normalised to the fraction of dps events found using data - driven model . we have checked that applying the same normalisation procedure to our predictions leads to a rather good agreement with the measured @xmath0 distributions , apart from the large end of the spectra ( and the first , low end bins in some cases ) , in accordance with observations in @xcite and section [ sec : complansberg ] . it needs to be checked if supplementing the theoretical predictions with full nlo corrections can eliminate the need for introducing the normalisation procedure , as results of @xcite would suggest . the authors thank c.h . kom for sharing his expertise during initial stages of the event simulation . part of this work has been performed on the high performance computing cluster palma maintained by the center for information technology ( ziv ) at wwu mnster , and on the high - 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section in pp collisions at @xmath114 = 8 tev with the atlas detector _ , '' atlas - conf-2016 - 047 .
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double parton scattering ( dps ) is studied at the example of @xmath0 pair - production in the lhcb and atlas experiments of the large hadron collider ( lhc ) at centre - of - mass energies of @xmath1 , 8 , and 13 tev .
we report theoretical predictions delivered to the lhcb and atlas collaborations adjusted for the fiducial volumes of the corresponding measurements during run i and provide new predictions at 13 tev collision energy .
it is shown that dps can lead to noticeable contributions in the distributions of longitudinal variables of the di-@xmath0 system , especially at 13 tev .
the increased dps rate in double @xmath0 production at high energies will open up more possibilities for the separation of single parton scattering ( sps ) and dps contributions in future studies .
ms - tp-16 - 21 * double parton scattering in pair - production of @xmath0 mesons at the lhc revisited * + christoph borschensky@xmath2 and anna kulesza@xmath3 _ @xmath4 institute for theoretical physics , university of tbingen , auf der morgenstelle 14 , d-72076 tbingen , germany + @xmath5 institute for theoretical physics , wwu mnster , d-48149 mnster , germany _
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although physics in cold atoms and halo nuclei are driven by interactions at very different physical scales , these systems share common features in their respective low energy regimes . universal behavior occurs when a system satisfies a separation of a large length scale and a small one . the large length scale is characterized by the scattering length @xmath0 , which determines the total cross section of the two - body s - wave scattering at zero energy by @xmath1 . for identical fermions . ] the small length scale is represented by the range of two - body interactions @xmath2 . in the limit @xmath3 , physics at the scale of @xmath0 is disentangled from physics at the scale of @xmath2 , and is therefore insensitive to the details of the short - range interactions . an example of three - body universality is efimov physics . in systems with three identical bosons , vitaly efimov predicted that , in the unitary limit @xmath4 , an infinite number of three - body bound states ( _ trimers _ ) emerge and accumulate at zero energy @xcite . these trimers have a geometric spectrum that satisfies a discrete scaling symmetry . this spectrum behavior , together with many other few - body features satisfying the discrete scaling symmetry in the limit @xmath5 , are often called `` the efimov effect '' . evidence of the efimov effect was found in the recombination processes in ultracold atomic gases , such as @xmath6cs @xcite , @xmath7li @xcite , @xmath8k @xcite , and @xmath9rb @xcite . in these experiments , the atom - atom scattering length @xmath0 is tuned through an external magnetic field to arbitrarily large values near feshbach resonances @xcite , where free atoms form shallow dimers ( two - atom bound states ) or trimers . the atomic recombination rates are measured as a function of @xmath0 . by tuning the magnetic field , critical features such as recombination minima and resonances occur at different values of @xmath0 . the discrete scaling symmetry has been observed in the critical recombination phenomena , which are labeled by the values of @xmath0 . universality also exists in molecular clusters of helium atoms . as observed by luo _ et al . _ @xcite , two @xmath10he atoms form a shallow dimer . the atom - atom scattering length is @xmath11 , about @xmath12 times the range of the van der waals potential @xcite . the @xmath10he trimer has been calculated using various realistic potential models @xcite , which indicated the existence of two consecutive ( ground and excited ) trimer states . the ground state was observed two decades ago @xcite , but it is only until recently that the excited - state trimer has been observed using coulomb explosion imaging techniques @xcite . in nuclear systems , the nucleon - nucleon s - wave scattering length is 3 times the range of nuclear forces ( the inverse pion mass ) in the spin - triplet channel , and is 15 times in the singlet channel @xcite . this separation of scales yields universal properties in few - nucleon systems . for example , the calculated values of the triton binding energy and the spin - doublet neutron - deuteron scattering length obey a linear correlation , which does not depend on the nucleon - nucleon potential models or potential parameterizations . this linear correlation is well known as the phillips line @xcite . another candidate for investigating few - body universal physics is the halo nucleus @xcite , _ i.e. _ a nucleus that contains one or several nucleons loosely attached to a tightly bound nuclear core . the valence nucleons can move far away from the core , thus forming a halo that substantially extends the spatial distribution of the nucleus . the scale separation between the shallow valence - nucleon separation energy and the deep core excitation energy allows connecting the clustering mechanism in halo nuclei with universal features . one successful approach to describe universal physics in few - body systems is an effective field theory ( eft ) . this theory utilizes the separation of scales and integrates out the short - range dynamics beyond the eft description . the short - range effects to low - energy physics are embedded into a series of two- and three - body effective contact interactions , which are constructed based on a systematic expansion of the ratio between two momentum scales , @xmath13 . the low momentum @xmath14 denotes the typical momentum of particles in the system , and the high momentum @xmath15 quantifies when the eft breaks down . the coupling constants of the counterterms are determined from low - energy observables . the resulting eft with contact interactions is known as the pionless eft @xcite in nuclear physics . it has also been applied to cold atomic and halo physics , and is often dubbed respectively as short - range eft ( _ e.g. _ in refs . @xcite ) and halo eft ( _ e.g. _ in refs . i will refer hereafter effective field theories with contact interactions simply as `` eft '' . detailed reviews of efimov signatures in cold atomic physics @xcite and nuclear / particle physics @xcite already exist in the literature . in this review , i will discuss the study of three - body universal physics using eft approaches , focusing on the description of cold atomic and halo nuclear systems . based on the systematic expansion in @xmath13 , we discuss the leading - order eft predictions , the extension to various higher - order effects and other contributions . the system of three identical bosons interacting with short - range potentials has been studied by bedaque _ et al . _ @xcite using eft in the limit @xmath3 . an effective lagrangian is constructed as a series of two- and three - body contact interactions : @xmath16 where @xmath17 and @xmath18 represents respectively the single boson field and the auxiliary dimer field . @xmath19 indicates the bare mass of the dimer and @xmath20 ( @xmath21 ) is the two - body ( three - body ) coupling constant . @xmath22 with the sign determined from the effective range @xmath23 . the ellipsis represents higher - order two- and three - body interactions in the eft expansion , which contains either more derivatives or more fields . using renormalization - group methods , several groups @xcite showed that the eft expansion is equivalent to the effective - range expansion for a scattering momentum @xmath24 . the later expands the two - body low - energy s - wave phase shift in powers of @xmath25 by @xcite @xmath26 where @xmath23 denotes the effective range and is of order @xmath2 . the ellipsis represents the shape - parameter and higher effective - range terms of order @xmath27 and above . the eft description of a system satisfying @xmath3 instead expands the two - body s - wave scattering amplitude , @xmath28 , around the singularity pole . this expansion is equivalent to eq . provided that @xmath24 . the leading order ( lo ) eft corresponds to parameterizing physical quantities in terms of @xmath0 . since terms of order @xmath27 do not enter eft calculations until three orders beyond lo , the inclusion of range effects can describe the large scattering - length physics at least up to a next - to - next - to - leading - order ( n@xmath29lo ) accuracy . corrections from finite - range effects can be evaluated perturbatively in the @xmath30 expansion . therefore , the four - momentum dimer propagator @xmath31 is expanded in powers of @xmath23 as @xmath32 , with the @xmath33th - order piece defined by @xmath34 where @xmath35 is the two - body binding momentum obeying @xmath36 . the three - body problem can be studied by calculating the atom - dimer scattering amplitude . as illustrated in fig . [ pic : t0 ] , contributing diagrams to the lo amplitude @xmath37 , which arise from the exchange of an atom between a dimer and an atom ( proportional to @xmath38 ) and the atom - dimer contact interaction ( proportional to @xmath21 ) , need to be iterated due to the non - perturbative feature of large scattering - length physics . for atom - dimer scattering in the integral equation : the double line and the round dot indicate the lo dimer propagator @xmath39 and three - body coupling . ] @xmath37 is solved in the modified skorniakov - ter - martirosian ( stm ) equation @xcite , which is equivalent to a non - relativistic faddeev equation @xcite with two- and three - body contact interactions . the s - wave projection yields @xmath40 where the kernel function is defined by @xmath41 here the three - body coupling @xmath42 , with @xmath43 denoting the regulation cutoff of the integral equation . @xmath44 is tuned to fit one three - body observable to ensure a cutoff independent result . the bound - state case is solved in eq . without the inhomogeneous term . the binding energies @xmath45 of efimov trimers are manifested as poles in @xmath37 . in fig . [ pic : efimov ] , the binding momentum @xmath46 is plotted as a function of @xmath47 , where several critical phenomena appear along the spectrum curve : ( a ) the binding momentum @xmath48 in the unitary limit @xmath49 , ( b ) the scattering length @xmath50 where free atoms form the efimov trimer at zero energy , ( c ) the scattering length @xmath51 at which the efimov trimer dissociates into an atom and a dimer , and ( d ) @xmath52 where the recombination reaches a local minimum . these efimov features are related by universal numbers predicted from the lo eft : @xmath53 where @xmath54 , @xmath55 @xcite , and @xmath56 @xcite . of efimov trimers is plotted as a function of @xmath47 . the discrete scaling constant is artificially adjusted for ease of illustration . ] [ pic : efimov ] also displays a series of efimov trimers obeying a discrete scaling symmetry . for example , the binding momentum @xmath57 of the @xmath33th efimov trimer in the unitary limit is related to @xmath48 in the @xmath58th branch by @xmath59 with @xmath60 denoting the scaling constant . similarly , other efimov features encoding the same critical phenomena also obey the scaling symmetry : @xmath61 the discrete scale invariance can be explained as a result of a limit cycle under the renormalization - group flow @xcite . several such analyses have been discussed in the eft framework @xcite . the three - body running coupling @xmath62 has an analytic expression in the zero - range limit @xcite : @xmath63 where @xmath64 , and the multiplicative factor is a regulator - dependent number that can be obtained numerically @xcite . @xmath65 is a three - body parameter that is determined by the lo renormalization condition . for example , it relates to @xmath48 by @xmath66 . as shown in both fig . [ pic : hlambda ] and eq . , @xmath44 is invariant under a discrete scale transformation @xmath67 . therefore , three - body observables in the efimov system display the discrete scaling symmetry due to the limit cycle behavior in the three - body hamiltonian . as a function of the cutoff @xmath43 at a fixed renormalization condition . the dots and solid line represent respectively the numerical results and the analytic expression given in eq .. ] another crucial efimov feature related to @xmath37 is the three - body recombination rate measured in ultracold atomic gases . the recombination is a collision process in which three free atoms collide and form a shallow dimer . the released energy is carried away kinetically by the third atom , which escapes the magnetic trap . the rate of the recombination process @xmath68 is connected with the density of trapped atoms @xmath33 by @xmath69 . in the zero - temperature approximation , @xmath68 can be calculated using lo eft by @xcite @xmath70 the existence of deep dimers in experimental setups opens additional recombination channels . a deep dimer , whose binding energy is of order @xmath71 or larger , can occur regardless of the existence ( @xmath72 ) or absence ( @xmath73 ) of shallow dimers . if deep dimers exist , an additional recombination process occurs as the scattering between an atom and a deep dimer with large kinetic energy but small total energy . the deep - dimer effects on @xmath68 , which is beyond the eft description for large - scattering - length physics , has been implemented by introducing a parameter @xmath74 encoding information of inelastic channels @xcite . the resulting @xmath75 is @xmath76 @xmath77 + \sinh^2 \eta _ * } \ ; \frac{\hbar a^4}{m } \qquad ( a<0).\ ] ] as shown in fig . [ pic : cs - rec ] , the eft prediction in the recombination rate was successfully confirmed in the measurement of ultracold @xmath6cs atoms done at innsbruck @xcite . is measured in the ultracold @xmath6cs atomic gas as a function of @xmath0 and is compared with the eft calculation . the solid curve represents the full lo eft calculation including both shallow- and deep - dimer effects . the straight line indicates a lower limit for @xmath73 and an upper limit for @xmath72 from the eft prediction . the figure is adapted from the original publication @xcite under copyright license no . 3714730785537 . ] deep dimers also open inelastic channels in the atom - dimer scattering , named the dimer relaxation . in this process , the deep dimer is formed by an atom scattering with a shallow dimer at low energies . the relaxation rate @xmath78 has also been calculated in the eft framework as @xcite @xmath79 + \sinh^2 \eta_*}\frac{\hbar a}{m}\ , . \label{beta - ad}\end{aligned}\ ] ] experiments of the relaxation rate in @xmath6cs @xcite are in good agreement with the eft predictions . although the ultracold atomic gases can be created at near - zero temperatures ranging from a few nk to a few @xmath80k @xcite , the finite - temperature effects make visible corrections to recombination and relaxation rates . such effects have also been studied in the eft framework @xcite . beyond the universal physics predicted in lo eft , corrections from a finite effective range @xmath23 enter at higher orders in the eft expansion . such effects are visible in the experiments of ultracold atoms @xcite , where the measured efimov features , _ i.e. _ @xmath81 , @xmath82 and @xmath83 , display finite deviations from the universal relations predicted in eq . [ eq : a - kappa * ] . the explanation of such discrepancies relies on including finite effective range into the eft analysis of three - body systems . expanding observables in @xmath30 , one can construct the eft formalism @xcite to analyze the range effects in three - body systems . the next - to - leading - order ( nlo ) correction to the atom - dimer scattering amplitude , @xmath84 , is schematically represented by fig . [ pic : t1 ] , which corresponds to inserting the linear - in-@xmath23 pieces of the dimer propagator and three - body coupling , _ i.e. _ @xmath85 and @xmath86 , in the three - body system . in perturbation theory : the crossed double and the square box denote respectively the nlo dimer propagator and the nlo three - body coupling . ] the nlo correction to the efimov spectrum was calculated by platter _ et al . _ they demonstrated that the first - order - in-@xmath23 corrections to the trimer binding energies vanish in the unitary limit . the range corrections in systems with a fixed @xmath0 were studied in refs . @xcite and applied to three - nucleon systems , where they concluded that renormalization at nlo does not depend on additional three - body observables . however , in cold atomic systems , the scattering length varies dramatically near the feshbach resonance . _ developed a perturbative eft formalism to study the range corrections to three - boson systems @xcite . they found that in systems with a variable scattering length , the nlo three - body coupling @xmath87 contains a ( @xmath30)-dependent piece , which requires an additional three - body input for consistent renormalization at nlo . @xcite calculated the range corrections to the three - body recombination in @xmath88li atoms , which were measured at bar - ilan university @xcite and rice university @xcite . the universal relations between efimov features were reanalyzed in ref . @xcite with the inclusion of linear - in-@xmath23 corrections and benchmarked with the @xmath88li experiments . by analyzing the running of the three - body coupling in the renormalization group , ji _ et al . _ recently revealed a pattern of the first - order range corrections to efimov features @xcite . the @xmath30 dependent piece in the three - body coupling , which contains a second three - body parameter , modifies the analytic expression of @xmath89 up to first - order in @xmath23 by @xmath90 where @xmath91 encapsulates the additional three - body parameter besides @xmath65 , and the ellipsis represents other linear - in-@xmath23 terms of @xmath89 that do not depend on @xmath91 @xcite . the term proportional to @xmath92 in eq . indicates a logarithmic breaking of the discrete scaling symmetry . it can be absorbed into the lo three - body coupling @xmath44 by replacing @xmath65 with a parameter @xmath93 that runs logarithmically with the momentum scale at a rate proportional to @xmath30 : @xmath94 the running three - body parameter yields new universal correlations among the range - corrected efimov features , and modifies the lo universal correlations in eq . ( [ eq : a - kappa*],[eq : buni],[eq : ani ] ) by @xcite @xmath95 where @xmath96 , @xmath97 , @xmath98 , and @xmath99 . @xmath100 is tuned to reproduce the second three - body observable . this modified scaling symmetry showed good agreement with various potential - model - dependent calculations @xcite . the innsbruck group recently discovered that in experiments of different atomic species or at different feshbach resonances the measured @xmath101 is always @xmath102 times the van der waals length @xmath103 within a @xmath104 accuracy @xcite . this correlation , named _ van der waals universality _ , may originate from the underlying short - range interactions in various cold atomic systems . the origin of the van der waals universality was studied in refs . using functional renormalization group , horinouchi and ueda recently described the van der waals universality as the universality for the onset of the limit cycle @xcite , that is independent of short - range potentials . the insensitivity to underlying short - range physics suggests the connection between eft and the van der waals universality , which requires future investigations . when the typical momentum scale in a few - body system is smaller , but within the same order of , the scale of the underlying short - range physics , range effects beyond the first order become important . this applies to few - nucleon systems where @xmath105 . in a system of three @xmath10he atoms , although the excited - state trimer is shallowly bound as a signature of efimov physics , an accurate description of the @xmath10he trimer , especially for its ground state , requires analyzing corrections from higher - order range effects . the second - order range corrections to three - body observables can be determined by calculating the next - to - next - to - leading order ( n@xmath29lo ) atom - dimer scattering amplitude in the eft framework . such n@xmath29lo calculations for three - boson systems with a fixed @xmath0 were carried out by bedaque _ et al . _ @xcite , by griehammer @xcite and by platter and phillips @xcite . although these works used a partial resummation formalism , they reached different conclusions on the running of the three - body coupling at this order . _ showed that an additional , energy - dependent , three - body counterterm enters in n@xmath29lo renormalization @xcite . this finding was later confirmed numerically by griehammer @xcite , who also extended this finding to three - nucleon systems in general settings @xcite . on the contrary , platter and phillips concluded that this additional piece vanishes in the limit @xmath106 @xcite . however , the partial resummation can arbitrarily include @xmath107 ( and beyond ) range effects , which cause complications to renormalization . to solve the controversy of higher - order three - body couplings raised in refs . @xcite , ji and phillips performed a rigorous perturbative three - body calculation up to n@xmath29lo @xcite . the second - order range effects to the atom - dimer scattering amplitude were calculated using perturbation theory . as schematically shown in fig . [ pic : t2 ] , the contributions arise from the insertion of the n@xmath29lo pieces of the dimer propagator and the three - body coupling , and from inserting twice the corresponding nlo terms . through regularizing the n@xmath29lo atom - dimer amplitude , they verified the needs of the additional energy - dependent three - body coupling at n@xmath29lo , which requires a second three - body input for proper renormalization . this indicates that the partial resummation is only consistent with a rigorous perturbative range expansion under the condition @xmath108 . lo atom - dimer scattering amplitude @xmath109 in second - order perturbation theory . the crossed ( double crossed ) double line denotes the nlo ( n@xmath29lo ) dimer propagator , and the square ( round ) box represents the nlo ( n@xmath29lo ) three - body counterterm . ] @xcite applied the rigorously perturbative range expansion to calculate the up - to-@xmath110 corrections to the trimer binding energies and atom - dimer scattering phase shifts in the system of three @xmath10he atoms , and compared with calculations from realistic potentials @xcite and the partially - resummed eft @xcite . the results displayed a correction at nlo ( at n@xmath29lo ) for the excited - state @xmath10he trimer binding energy and atom - dimer scattering length , and a correction at nlo ( at n@xmath29lo ) for the ground - state @xmath10he trimer binding energy . the perturbative @xmath30 expansion formalism has also been developed by vanasse _ et al . _ @xcite , in the pionless eft framework , to investigate range effects in few - nucleon bound - state and scattering problems . besides in systems of ultracold bosonic atoms , the efimov scenario can also be observed in a quantum degenerate gas of fermionic atoms . the efimov trimers have been measured in the @xmath111li atomic mixture of three degenerate hyperfine states . in this system , the @xmath111li dimers have three different scattering lengths which vary respectively near their own feshbach resonances . due to the overlap of these resonance regimes , three scattering lengths can be tuned simultaneously through an external magnetic field , creating a three - hyperfine - component atomic mixture . several experiments have measured the recombination features of the three - component @xmath111li atoms associated with the ground - state efimov trimer @xcite and the excited - state trimer @xcite . shortly after these experiments , et al . _ @xcite observed an enhanced resonance of a dimer relaxation process in @xmath111li due to the existence of efimov trimers . the binding energy of the @xmath111li trimer was determined by the radio - frequency association method @xcite . using contact eft approaches , braaten _ et al . _ calculated various aspects of the efimov physics in the three - spin mixture of @xmath111li atoms @xcite . with three scattering lengths at different but large values in this system , they constructed the coupled - channel stm equations to calculate the spectrum of efimov trimers , the three - body recombination rate , and the dimer relaxation rate near the resonance when the efimov trimer crosses the atom - dimer scattering threshold . the universal properties in @xmath111li atoms , predicted by the eft calculations @xcite in the @xmath112 limit , displayed good agreements with experimental results @xcite . another area to observe efimov scenario is in heteronuclear systems , which are a mixture of two ultracold atomic species near an interspecies feshbach resonance . the heteronuclear system opens new possibilities to study the discrete scaling symmetry , which is a critical feature at the heart of efimov physics . the scaling constant @xmath60 in systems of three identical bosons yields a large energy gap between the ground- and excited - state efimov trimers . this gap makes the experimental observation of an excited efimov state , which is already near the saturation regime due to thermal motion , a very challenging task . this obstacle can be potentially circumvented in heteronuclear systems . the effective field theory calculation by helfrich _ et al . _ indicated the existence of efimov physics in a heteronuclear trimer containing a mixture of two atomic species , which has a large interspecies scattering length and a small intraspecies scattering length @xcite . their studies found that the scaling factor is much larger than @xmath113 for a heteronuclear trimer containing two species with comparable masses , but much smaller in a trimer consisting of one light and two heavy atoms . the later case makes the observation of several efimov states connected by a small geometric scaling possible . several experiments have investigated the recombination and relaxation in a bose - bose mixture of @xmath114k-@xmath115rb atoms @xcite and a fermi - bose mixture of @xmath116k-@xmath115rb @xcite . although the scaling constant is large in these k - rb trimers , _ i.e. _ 131 for @xmath114k-@xmath115rb-@xmath115rb , 348000 for @xmath115rb-@xmath114k-@xmath114k , and 123 for @xmath116k-@xmath115rb-@xmath115rb predicted from eft @xcite , the observed resonance features , associated with ground - state trimers of the k - rb mixture , showed reasonably good agreement with eft calculations in the zero - range limit . the theory - experiment deviation is due to finite range effects and finite temperature effects . recent experiments in the @xmath117li-@xmath6cs mixture , that has an extreme mass imbalance , have observed the discrete scaling symmetry @xcite . by measuring the three - body loss resonance at negative scattering lengths near a @xmath117li-@xmath6cs feshbach resonance , pires _ _ discovered a ratio of @xmath118 between the two @xmath83 s associated with the ground- and excited - state @xmath117li-@xmath6cs-@xmath6cs @xcite . a later work by tung _ et al . _ observed a geometric scaling factor @xmath119 among three consecutive @xmath117li-@xmath6cs-@xmath6cs efimov trimers @xcite . these measured scaling constants are consistent with a universal zero - range theory calculation , finding @xmath120 @xcite . this work by petrov and werner also implemented effects from finite temperatures and cs - cs interactions into the zero - range theory . their results displayed good agreement with the measured three - body recombination in the @xmath117li-@xmath6cs mixture . halo nuclei , either neutron rich or proton rich , are found to have large nuclear charge / matter radii and shallow binding energies . for the scale separation in halo nuclei , the short - range scale refers to the size of the nuclear core , @xmath121 , and the large - distance scale is characterized by the average distance from a valence nucleon to the core , @xmath122 , satisfying @xmath123 . one can study physics in halo nuclei by treating the core as an inert point - like particle and work in cluster degrees of freedom consisting of valence nucleons and a core . in effective field theory for halo nuclei , the contact interactions among valence nucleons and the core are constructed based on the expansion of @xmath124 . the dynamic core - excitation effects and the effects of valence- and core - nucleon anti - symmetrization are embedded in contact interactions . the eft for halo nuclei was first constructed by bertulani _ et al . _ @xcite and bedaque _ et al . _ they studied @xmath125he as a neutron-@xmath68 two - body system interacting in a p - wave resonance , where different power - counting approaches based on the @xmath124 expansion are employed in these two works . by implementing the long - range coulomb potential , in addition to the short - range interaction , into the eft framework , theoretical investigations have been extended to @xmath126be as an @xmath68-@xmath68 scattering resonance @xcite , the proton-@xmath68 scattering @xcite , and @xmath127f as a one - proton - halo @xcite . eft studies of halo nuclei with one valence nucleon have also been applied to calculate electric - dipole strengths in the photo - dissociation of @xmath128be @xcite and @xmath129c @xcite . recently , the eft method for halo nuclei have been applied actively to describe the radiative neutron - captures on @xmath88li @xcite and @xmath130c @xcite , and the radiative proton - captures on @xmath88be @xcite and @xmath131o @xcite . the halo physics also exists in the three - body sector , which typically involves a halo nucleus with two valence neutrons ( @xmath132 halo ) . the neutron - core ( @xmath33-@xmath133 ) interaction is parameterized in the eft framework through the effective range expansion of the partial - wave decomposed t - matrix . if the @xmath33-@xmath133 t - matrix is dominated at low - energies by a resonance in a particular partial wave @xmath134 ( @xmath135 ) , we refer to such nucleus as an @xmath134-wave halo . the eft formalism for a @xmath132 halo is similar to that for a heteronuclear trimer of one heavy bosonic and two light fermionic atoms . however , differently from the heteronuclear system , where only the interspecies scattering length is large , the interaction between the two valence neutrons , dominated by a low - energy s - wave virtual state , also has a large scattering length . therefore , the stm equation for an @xmath134-wave @xmath132 halo requires both the @xmath33-@xmath33 s - wave and @xmath33-@xmath133 @xmath134-wave interactions , represented by their own _ dimeron _ operators . the bound - state problem is to solve the scattering amplitudes of @xmath133-(@xmath136 ) ( @xmath137 ) and @xmath33-(@xmath138 ) ( @xmath139 ) in coupled homogeneous integral equations , as illustrated schematically in fig . [ pic : halo - stm ] . a @xmath33-@xmath33-@xmath133 three - body counterterm is also needed for renormalization . in the following , i focus on the eft achievements in calculating s- and p - wave @xmath132 halo nuclei . and @xmath139 in coupled integral equations . the solid and dashed lines are respectively the @xmath33 and @xmath133 propagators , the double line and thick line denote respectively the @xmath136 and @xmath138 dimeron propagator , and the round box represents the @xmath140 three - body force . ] in an s - wave @xmath132 halo nucleus , the neutron - core interaction is dominated by a low - energy s - wave bound ( virtual ) state with a large positive ( negative ) scattering length . due to its similarity to a heteronuclear trimer , the s - wave @xmath132 halo engenders tremendous theoretical and experimental interest in unveiling efimov effects and the discrete scaling symmetry in halo nuclei . canham and hammer have performed eft calculations of binding energies , root - mean - square ( rms ) matter radii , and three - body configuration distributions of several s - wave @xmath132 halos @xcite , with finite range effects included in a following paper @xcite . working in the zero - range limit , they also explored the possibilities of finding an excited efimov state in different halo nuclei , by taking inputs from the experimental values of the @xmath132 separation energy @xmath141 , neutron - core energy @xmath142 ( @xmath143 for a bound state and @xmath144 for a virtual state ) , and the neutron - neutron energy @xmath145 of a given nucleus . they constructed a boundary region constrained by the contour plot of the ratios @xmath146 versus @xmath147 . as shown in fig . [ pic : halo - efimov ] , the contour curve converges to a universal region by increasing the mass number of the core @xmath148 . if there exists an excited efimov state below the @xmath140 breakup threshold , the measured ground - state properties of a @xmath132 halo yield a point inside the boundary region . this analysis was applied to @xmath128li , @xmath149c and @xmath150be as halo candidates based on experimental data from the `` nuclear data evaluation project '' of tunl @xcite . they showed that the only possible candidate for an excited efimov state is @xmath151c , mainly due to the large uncertainty of the experimental @xmath142 of @xmath151c , _ i.e. _ @xmath152 kev @xcite . similar searches for efimov states were also done by frederico , tomio and collaborators in a zero - range three - body model @xcite . however , newer experimental data , from the coulomb dissociation of @xmath129c ( @xmath153 kev @xcite ) and the 2012 atomic mass evaluation ( ame2012 ) ( @xmath154 kev @xcite ) , excluded an excited efimov state in @xmath151c . versus @xmath147 for the ground - state @xmath132 halos at mass numbers @xmath148 of the cores . the excited efimov state is at threshold along the contour curves . the figure is adapted from the original publication @xcite under copyright license no . 3718851192616 . ] the most neutron - rich carbon isotope @xmath155c is another halo candidate . the measurement of the reaction cross section of @xmath156 on a hydrogen target @xcite determined a @xmath156 rms matter radius @xmath157 fm . this measurement also suggests that the @xmath156 ground state has an s - wave @xmath132 halo configuration , where the @xmath158 subsystem is known to be unbound @xcite . this halo picture was confirmed by a neutron removal reaction measurement on neutron - rich carbon isotopes @xcite . except @xmath159 , other properties of @xmath156 were measured with very large uncertainties . from ame2012 @xcite , the @xmath33-@xmath160 interaction has a continuum energy of @xmath161 kev , while the three - body binding energy of @xmath156 is @xmath162 kev . acharya _ et al . _ @xcite applied the eft analysis and predicted the universal correlations among @xmath159 , @xmath142 and @xmath163 in @xmath155c . as shown in fig . [ pic:22c - contour ] , three sets of @xmath163 vs @xmath142 correlation are plotted in ref . @xcite by fixing the experimental values of @xmath159 within a @xmath164 uncertainty , _ i.e _ , 4.5 fm , 5.4 fm and 6.3 fm , along with the theoretical error bands from estimating the @xmath165 corrections . the correlation plot indicates that the @xmath159 data @xcite sets an upper limit of 100 kev on the @xmath132 separation energy @xmath163 of @xmath156 , regardless of the poorly determined @xmath142 . this upper bound on @xmath163 is consistent with the ame2012 data . it is also @xmath104 lower than the calculation in a zero - range three - body model @xcite , and significantly lower than other model - dependent calculations @xcite . based on the @xmath142 and @xmath163 correlation , acharya _ et al . _ @xcite explored the possibility for an excited efimov state of @xmath155c , whose existence only occurs if the @xmath33-@xmath151c continuum energy is of the order of a few kev . versus @xmath142 in @xmath156 with fixed values of the matter radius at @xmath166 = 5.4 fm ( blue , dashed ) , 6.3 fm ( red , solid ) , and 4.5 fm ( green , dotted ) . the shaded bands indicate the theoretical errors based on estimates of higher - order eft corrections . ] based on the eft approach , hagen _ et al . _ developed a formalism to calculate the charge form factor @xmath167 of s - wave @xmath132 halos @xcite . the low - momentum expansion of the calculated charge form factor is related to the charge radius of a halo nucleus @xmath168 by @xmath169 @xmath168 shows universal correlations with other observables in @xmath132 halos , such as @xmath163 , @xmath142 , and @xmath159 . these correlations were applied in ref . @xcite to investigate @xmath170ca as the heaviest @xmath132 halo nucleus , where the information on the n - core interaction was obtained from an _ ab initio _ coupled - cluster calculation of the @xmath33@xmath171ca s - wave scattering phase shift . this combination of eft and _ ab initio _ methods predicted efimov physics in @xmath170ca . the dipole response functions of @xmath128li and @xmath155c , which are associated with their coulomb dissociation cross sections , have been recently calculated as s - wave @xmath132 halo systems using a leading - order halo eft @xcite . in the case of @xmath128li , the calculation showed an agreement with the experimentally measured coulomb dissociation @xcite at low transition energies , within the error bars arising from corrections beyond the leading order of the halo eft . the p - wave two - body phase shift can be expanded in the low - energy limit by @xmath172 where @xmath173 is the scattering volume , and the p - wave effective `` range '' @xmath174 has the dimension of momentum . recent studies @xcite indicated that in a three - body system with resonant pairwise p - wave interactions , the discrete scaling symmetry can not be realized in the limit @xmath175 and @xmath176 because this p - wave unitary limit is not physical : it inevitably yields a two - body state with a negative probability density and therefore violates the causality condition . despite the absence of discrete scaling symmetry in p - wave - pairwise three - body systems , low - energy observables in such systems can still display interesting universal features . one example is @xmath111he , which is a p - wave @xmath132 halo nucleus with the neutron - core ( @xmath33-@xmath68 ) interaction dominated by a @xmath29p@xmath177 resonance . the existence of a three - body bound state in @xmath111he shows the evidence of universal physics controlled by low - energy dynamics in higher partial waves . ji _ et al . _ described the @xmath111he ground state as a @xmath178 system in the halo eft framework @xcite . the @xmath179 @xmath180p@xmath177 interaction was expanded by employing a `` narrow resonance '' power counting , _ i.e. _ @xmath181 and @xmath182 , developed by bedaque _ et al . _ @xcite . this power counting avoids the expansion near the unphysical unitary limit but preserves the low - energy p - wave resonance . to obtain a renormalized @xmath132 separation energy @xmath163 in the integral equations ( fig . [ pic : halo - stm ] ) of the @xmath178 system , ji _ et al . _ introduced a p - wave @xmath178 counterterm . by reproducing the experimental value @xmath183 mev , they analyzed the running of the three - body coupling @xmath44 as a function of the regulation cutoff @xmath184 . as shown in fig . [ pic:6he-3bf ] , @xmath44 exhibits a log - periodic behavior with decreasing periods when @xmath43 increases . this different behavior from @xmath44 in three - boson systems ( fig . [ pic : hlambda ] ) indicates the breaking of discrete scaling invariance by the presence of the p - wave pairwise interaction . the @xmath111he ground state was also calculated by rotureau and van kolck @xcite by combining eft potentials with the gamow - shell - basis - expansion few - body techniques . -counterterm parameter @xmath44 as a function of the cutoff @xmath43 . @xmath44 is tuned to reproduce @xmath185 mev at different values of @xmath43 . ] if the three - body behavior in @xmath111he is universal , the eft analysis can also be applied to investigate other p - wave @xmath132 halo nuclei . one example is @xmath128li , which is another borromean @xmath132 halo system . the measurement of the @xmath132 transfer reaction , @xmath186h(@xmath128li,@xmath187li)@xmath188h , implied that both the s- and p - wave components of @xmath189li interactions play important roles in forming the @xmath128li ground state @xcite . therefore , the binding and structural properties of @xmath128li may display unique three - body features that depend on both s- and p - wave two - body and three - body interactions . three - body systems display universal features at large distance scales , when the range of the pairwise interaction is much shorter than the two - body scattering length . the universal three - body physics exists in systems spanning over very different energy scales , including few - nucleon , cold atomic , and halo nuclear systems . utilizing the scale separation between the short interaction range @xmath2 and the large scattering length @xmath0 , low - energy behaviors in three - body systems can be investigated in the effective field theory framework by constructing two- and three - body contact interactions based on an @xmath190 expansion . in this review , we have discussed the effective field theory studies of three - body universal physics with an emphasis on cold atomic and halo nuclear systems . universal correlations among the efimov features , related to critical phenomena of the three - body recombination and dimer relaxation in ultracold bosonic atomic gases , were predicted by eft in the zero - range limit . the discrete scale invariance was explained , in renormalization group , through the running of the three - body coupling built in eft . finite - range corrections to efimov physics , which breaks the discrete scaling symmetry , are also explained in the eft framework as higher - order effects in the @xmath190 expansion . the investigation of range effects is applied to recombination features in ultracold atoms and the binding and scattering properties of @xmath10he atomic trimers . studies of efimov physics are also extended to the inclusion of deep - dimer effects in cold atoms , fermionic atoms with spin mixtures , and heteronuclear atomic mixtures . in halo nuclei , the size of the nuclear core is much smaller than the size of the halo . this scale separation provides three - body universal physics in halo nuclei with two valence neutrons . a number of two - neutron halo nuclei with either s- or p - wave pairwise interactions are investigated using halo eft . studies of universal correlations among the halo spectrum at low - energies , structural distributions , charge and matter - radii , and electro- and photo - induced reactions are discussed in this review . these universal correlations are also combined with experimental constraints to explore the possibility of finding an excited efimov state in halo nuclei . the author is grateful to lucas platter and wolfram weise for valuable comments on the manuscript . this research was supported in part by the natural sciences and engineering research council ( nserc ) and the national research council canada . s. r. beane , p. f. bedaque , w. c. haxton , d. r. phillips and m. j. savage , in m. shifman ( ed . ) : _ at the frontier of particle physics _ , vol . 1 . , 133 - 269 ( 2001 ) http://arxiv.org/abs/nucl-th/0008064 [ arxiv : nucl - th/0008064 ] .
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few - body systems , such as cold atoms and halo nuclei , share universal features at low energies , which are insensitive to the underlying inter - particle interactions at short ranges .
these low - energy properties can be investigated in the framework of effective field theory with two - body and three - body contact interactions .
i review the effective - field - theory studies of universal physics in three - body systems , focusing on the application in cold atoms and halo nuclei .
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entanglement is a uniquely quantum mechanical phenomenon in which quantum systems exhibit correlations not possible for classical systems . as such , entanglement is a vital resource for many aspect of quantum information processing including quantum computation , quantum metrology , and quantum communication @xcite . but despite its fundamental and practical importance and much work in the subject , there are many aspects of entanglement , especially multi - partite entanglement , that are in need of further study @xcite . a major challenge facing experimental implementations of quantum computation , sensing , and communication is decoherence , unwanted interactions between the system and environment . decoherence may be especially detrimental to highly non - classical , and hence most potentially useful , entangled states @xcite . a manifestation of this is entanglement suddent death ( esd ) in which entanglement is completely lost in a finite time @xcite despite the fact that the coherence loss of the system is asymptotic . this aspect of entanglement has been well explored in the case of bi - partite systems and there are a number of studies looking at esd in multi - partite systems @xcite . in addition , there have been several initial experimental studies of this phenomenon @xcite . however , even when analyzing a multi - partite system , previous works demonstrate esd only for bi - partite entanglement , either via concurrence or negativity rather than using measures for purely multi - partite entanglement . it is important to note that the characterization and quantification of true multi - partite entanglement is still very much an unsettled area for pure states and even more so for mixed states . in this paper i explore the loss of detectable entanglement in a three qubit system by invoking entanglement witnesses , observables that can detect the presence of entanglement . entanglement may be present in a system but still not be practically useful @xcite . for entanglement to be useful its presence should be efficiently detectable experimentally . multi - partite entanglement can be detected inefficiently via quantum state tomography or a violation of bell inequalities . it can be detected efficiently by utilizing properly constructed entanglement witnesses @xcite . i compare the entanglement detection abilites of tri - partite entanglement witnesses to esd of tri- and bi - partite entanglement in a given system . in this exploration i find a state which has no concurrence and no tri - partite entanglement as measured by the tri - partite negativity but is entangled as measured by the negativity . i then apply these results to a three qubit quantum error correction ( qec ) code and explore how esd affects the working of this code . three qubit pure states can assume a ` standard ' form with respect to local unitary operations @xcite , @xmath0 where @xmath1 , @xmath2 and @xmath3 . these states can be separated into four broad categories : separable ( in all three qubits ) , biseparable , and there exist two types of locally inequivalent tri - partite entanglement ( ghz and w - type ) @xcite . similar classification schemes exist for mixed states . reference @xcite defines four classes of three qubit mixed states each of which includes the preceeding classes as special cases . they are separable ( s ) states , bi - separable ( b ) states , w states , and ghz states , which encompasses the complete set of three qubit states . note that additional subtlety exists in characterizing the entanglement within each of these classes @xcite . to determine in which class a given state belongs one can use entanglement witnesses , observables which give a positive or zero expectation value for all states of a given class and negative expectation values for at least one state in a higher ( i.e. more inclusive ) class . specifically , i will make use of entanglement wintesses @xcite to identify whether a state is in the ghz@xmath4w class ( i. e. a state in the ghz class but not in the w class ) , in which case it certainly has ghz type tri - partite entanglement , the w@xmath4b class , in which case the state certainly has true tri - partite entanglement either of the ghz - type or w - type , or the b class in which case it is not certain that the state has any tri - partite entanglement . while the witnesses we explore may not be of the sort that can be implemented efficiently for experimentally determining the presence of tri - partite entanglement , they are among the most sensitive , or finest , known witnesses . thus , if these witnesses do not detect the presence of entanglement neither will any of the effeciently implementable witnesses i will compare the detection ability of the witnesses to the evolution of the concurrence @xcite , @xmath5 , for measuring the bi - partite entanglement between qubits @xmath6 and @xmath7 after partial trace over one qubit . the concurrence of a two qubit state with density matrix @xmath8 is defined as the maximum of zero and @xmath9 , where @xmath10 and the @xmath11 are the eigenvalues of @xmath12 in decreasing order , where @xmath13 is the @xmath14 pauli matrix of qubit @xmath15 . other entanglement measures that i will look at are the negativity , @xmath16 , for which i will use the sum of the absolute values of the negative eigenvalues of the partial transpose of the density matrix @xcite with respect to one qubit , and the tri - partite negativity , @xmath17 , a tri - partite entanglement measure for mixed states which is simply the third root of the product of the negativities with respect to each of three qubits @xcite . if the negativity is the same when taking the partial transpose with respect to any of the three qubits , @xmath18 . we look at a three qubit system , with no interaction between the qubits , placed in a dephasing environment fully described by the kraus operators @xmath19 where the dephasing parameter @xmath20 can also be written in a time - dependent fashion @xmath21 . when all three qubits undergo dephasing we have eight kraus operators each of the form @xmath22 where @xmath23 and @xmath24 . the next three sections are dedicated to exploring the entanglement dynamics and detectability of three types of three qubit states : ghz - type states in section [ sghz ] , w - type states in section [ sw ] , and a state with ghz - type tri - partite entanglement and bi - partite entanglement in section [ sgb ] . in section [ sqec ] we apply our results to the three qubit phase flip quantum error correction code and study the affect of esd on the code . section [ sconc ] contains some further observations and conclusions . in this section i explore the affects of dephasing on the following three qubit ghz - type states , @xmath25 where @xmath26 . these states have no bi - partite entanglement and non - zero negativity and tri - partite negativity equal to @xmath27 $ ] . the state @xmath28 is certainly in the ghz@xmath4w class for @xmath29 since it can be detected via the witness @xcite @xmath30 . @xmath28 is in at least the class w@xmath4b for @xmath31 since it can be detected via the witness @xcite @xmath32 . the effect of dephasing all of the qubits of @xmath28 is simply to reduce the magnitude of the off diagonal elements @xmath33 . for @xmath34 the entanglement witness @xmath35 gives @xmath36 which decays exponentially with time . in other words , the state can always be detected by a w - state witness and does not exhibit tri - partite esd @xcite . however , the dephased state can not always be detected as belonging to the ghz@xmath4w class . this can be seen from the expectation value @xmath37 which is negative ( thus detected ) only as long as @xmath38 . thus , we have a ` sudden death ' type phenomenon with respect to the ghz entanglement witness . experimentally we can not be sure the state exhibits ghz - type entanglement . the inability to detect any tri - partite entanglement in the state @xmath28 can occur for @xmath39 . in such a case we find @xmath40 and @xmath41 . the behavior , as a function of time , for both entanglement witnesses is similar , as shown in fig . [ ghzstate ] , and clearly demonstrates the inability of the witnesses to detect any entanglement at sufficiently long but finite time . comparing this to the entanglement measures , we note that there is only one eigenvalue of the partially transposed density matrix that can be negative and so @xmath42 $ ] . for @xmath34 , i.e. initially pure states , this is equal to @xmath43 and thus the witness always detects the tri - partite entanglement . for @xmath39 the initial state is mixed and @xmath43 is no longer equal to @xmath16 . instead , the entanglement witness goes to zero at a rate approaching three times that of @xmath16 demonstrating that there is remaining tri - partite entanglement in the system beyond the detection of the entanglement witness . nevertheless , even @xmath16 decays to zero in finite time exhibiting the esd phenomenon , fig . [ ghzstate ] . as expected , @xmath44 between any two qubits of @xmath28 after partial trace over the third qubit is always zero thus the system never exhibits bi - partite entanglement . as a function of the initial state paramaterized by @xmath45 , and the dephasing constant @xmath46 . at sufficiently low values of @xmath45 or high values of @xmath46 the state can no longer be detected by a ghz - type entanglement witness . top - right : @xmath43 as a function of the same parameters . this entanglement witness shows when the state can no longer be detected as one with tri - partite entanglement . bottom : the negativity with respect to any qubit partition ( and thus in this case the negativity is equivalent to the tri - partite negativity ) . when this is zero the state has no distillable entanglement . for @xmath34 ( initial pure state ) the negativity is equal to @xmath43 meaning that the @xmath35 entanglement witness always detects the presence of the tri - partite entanglement . when @xmath39 the state is mixed and the witness does not always detect the tri - partite entanglement which is known to be present via the tri - partite negativity measure . , title="fig:",width=160 ] as a function of the initial state paramaterized by @xmath45 , and the dephasing constant @xmath46 . at sufficiently low values of @xmath45 or high values of @xmath46 the state can no longer be detected by a ghz - type entanglement witness . top - right : @xmath43 as a function of the same parameters . this entanglement witness shows when the state can no longer be detected as one with tri - partite entanglement . bottom : the negativity with respect to any qubit partition ( and thus in this case the negativity is equivalent to the tri - partite negativity ) . when this is zero the state has no distillable entanglement . for @xmath34 ( initial pure state ) the negativity is equal to @xmath43 meaning that the @xmath35 entanglement witness always detects the presence of the tri - partite entanglement . when @xmath39 the state is mixed and the witness does not always detect the tri - partite entanglement which is known to be present via the tri - partite negativity measure . , title="fig:",width=160 ] as a function of the initial state paramaterized by @xmath45 , and the dephasing constant @xmath46 . at sufficiently low values of @xmath45 or high values of @xmath46 the state can no longer be detected by a ghz - type entanglement witness . top - right : @xmath43 as a function of the same parameters . this entanglement witness shows when the state can no longer be detected as one with tri - partite entanglement . bottom : the negativity with respect to any qubit partition ( and thus in this case the negativity is equivalent to the tri - partite negativity ) . when this is zero the state has no distillable entanglement . for @xmath34 ( initial pure state ) the negativity is equal to @xmath43 meaning that the @xmath35 entanglement witness always detects the presence of the tri - partite entanglement . when @xmath39 the state is mixed and the witness does not always detect the tri - partite entanglement which is known to be present via the tri - partite negativity measure . , title="fig:",width=160 ] unlike ghz states the w state @xmath47 , retains a high degree of ( bi - partite ) entanglement upon partial trace over one qubit . thus , w - type states allow some comparison between tri - partite and bi - partite entanglement evolution . we start with the state @xmath48 where @xmath49 . states of this sort in the w@xmath4b class may be detected by the entanglement witness @xcite @xmath50 . for @xmath34 the expectation value of the entanglement witness is @xmath51 and the concurrence of any two qubits is @xmath52 . the entanglement witness detects the state , for all @xmath53 . @xmath44 , after partial trace of a qubit , remains positive for all @xmath54 and @xmath16 , which is equal to @xmath17 since the results of the partial trace over any of the three qubits are equivalent , is non - zero for @xmath55 . once again the witness is successful in detecting tri - partite entanglement only up to a certain point despite the continued presence of tri - partite entanglement . in addition , for @xmath56 there is surviving tri - partite @xmath57-type entanglement , as measured by @xmath17 , even when there is no longer any residual bi - partite entanglement . the effect of dephasing on the state @xmath28 is to degrade the off - diagonal terms by @xmath58 . the state is detected by @xmath59 only for @xmath60 while @xmath44 , after tracing out any qubit , is @xmath61 $ ] . we compare these values to @xmath16 where again there is only one negative eigenvalue upon partial transpose of the density matrix : @xmath62 $ ] . for @xmath34 and sufficiently large @xmath46 we do not find esd for @xmath44 or @xmath16 but nevertheless find that the tri - partite entanglement can not be detected by @xmath59 . for @xmath39 esd occurs for @xmath44 and @xmath16 as shown in fig . [ wstate ] with the tri - partite entanglement witness going to zero first followed by @xmath44 and finally the @xmath18 . in other words , after a certain time there is no longer bi - partite entanglement in the system despite the remaining w - type tri - partite entanglement . the above expressions also show that each entanglement measure decreases at the exponential rate @xmath63 and there is no difference in the exponential rate of decay between the bi - partite and tri - partite entanglement . the difference in entanglement measure evolution comes from different coefficients mulitplying the exponential and additional constants . ( solid line ) , the bi - partite concurrence ( medium dashed line ) , and the negativity ( large dashed line ) as a function of the dephasing constant @xmath46 . top left : @xmath34 , the initial state is pure . in this case @xmath64 goes to zero in finite time , but the other entanglement measures do not . top right : @xmath65 , for any @xmath39 the concurrence undergoes esd before the negativity ( which is equal to the tri - partite negativity ) goes to zero . bottom : @xmath66.,title="fig:",width=151 ] ( solid line ) , the bi - partite concurrence ( medium dashed line ) , and the negativity ( large dashed line ) as a function of the dephasing constant @xmath46 . top left : @xmath34 , the initial state is pure . in this case @xmath64 goes to zero in finite time , but the other entanglement measures do not . top right : @xmath65 , for any @xmath39 the concurrence undergoes esd before the negativity ( which is equal to the tri - partite negativity ) goes to zero . bottom : @xmath66.,title="fig:",width=151 ] ( solid line ) , the bi - partite concurrence ( medium dashed line ) , and the negativity ( large dashed line ) as a function of the dephasing constant @xmath46 . top left : @xmath34 , the initial state is pure . in this case @xmath64 goes to zero in finite time , but the other entanglement measures do not . top right : @xmath65 , for any @xmath39 the concurrence undergoes esd before the negativity ( which is equal to the tri - partite negativity ) goes to zero . bottom : @xmath66.,title="fig:",width=151 ] we would like to explore a system that contains both bi - partite and tri - partite ghz - type entanglement to compare and contrast the two types of entanglement evolution . we look at the following initial state @xmath67 where @xmath68 and @xmath69 . in order to detect the tri - partite entanglement of such states we must first find the proper ghz and w - type entanglement witnesses . the closest state to @xmath70 with no w - type vectors is likely to be the ghz state with an @xmath71-rotation about the third qubit . with this in mind we construct ghz and w state witnesses @xmath72 and @xmath73 , for third qubit rotation of state @xmath74 by angle @xmath75 . the @xmath75 that minimizes the witnesses is @xmath76 . with these tools we can explore the entanglement of this system . with no dephasing the state @xmath28 is detected by @xmath77 for @xmath78 and detected by @xmath79 for @xmath80 . taking a partial trace over one of the qubits leads to non - zero concurrence only when tracing over the third qubit . this concurrence , @xmath81 , is non - zero only when @xmath82 . the lowest lying eigenvalue of the partial transpose of @xmath83 is when the partial transpose is taken with respect to the first or second qubit and gives non - zero negativity for @xmath84 . the tri - partite negativity in this case is not equal to the negativity and is nonzero only for @xmath85 . the effect of dephasing on the entanglement of this state is shown in fig . [ ghzbstate ] . for @xmath34 the tri - partite entanglement witnesses can not detect the entanglement below a certain threshold @xmath46 , but @xmath81 , @xmath16,and @xmath17 do not exhibit esd . for @xmath39 all of the entanglement measures exhibit esd . it is interesting to compare and contrast the behavior of the various entanglement measures especially with respect to @xmath81 . when @xmath39 the entanglement that takes the longest amount of time to exhibit esd is @xmath16 . for sufficiently high values of @xmath45 the tri - partite negativity goes to zero before @xmath81 . for the time between the subsequent sudden death of @xmath81 and the esd of @xmath16 the entanglement present in the system is not of the bi - partite type since it is not destroyed by the partial trace , nor is it tri - partite in the sense that it can be measured by @xmath17 . the ability of the tri - partite entanglement witnesses to detect entanglement is weaker than any of the entanglement measures . as @xmath45 decreases @xmath81 decreases at a faster rate than any of the other entanglement measures . for sufficiently low values of @xmath45 the concurrence , @xmath81 , exhibits esd before @xmath17 such that the bi - partite entanglement is the first to experience esd , followed by the tri - partite entanglement measured by @xmath17 . after this time there is entanglement that can be measured only by @xmath16 . the behavior of @xmath17 is different than that of the other entanglement measures . rather than an exponential decay minus some constant there is initial exponential decay and then an inflection point before @xmath17 goes to zero . decreasing @xmath45 futher we see that @xmath81 undergoes esd even before the w - state entanglement witness can no longer detect entanglement . this is in sharp contrast to the w - state case in which @xmath86 always went to zero before the bi - partite @xmath44 . this again shows the possibility of states with tri - partite w - type entanglment without the presence of @xmath44 when tracing over one of the qubits . -type states . each plot shows @xmath81 ( solid line ) , @xmath16 ( large dashed line ) , @xmath17 ( medium dashed line ) , @xmath87 ( small dashed line ) , and @xmath88 ( dotted line ) as a function of the dephasing constant @xmath46 . top left : @xmath34 , the initial state is pure . in this case the tri - partite entanglement witnesses go to zero in finite time but the concurrence , negativity , and tri - partite negativity do not undergo esd . top right : @xmath89 , for any sufficiently high value of @xmath45 where @xmath39 the concurrence undergoes esd before the negativity but after the tri - partite negativity . bottom : @xmath90 , the concurrence undergoes esd before the tri - partite negativity and even before the tri - partite entanglement witness . thus , there is detectable tri - partite entanglement in the system though there is no concurrence remaining when one of the qubits is traced over.,title="fig:",width=151 ] -type states . each plot shows @xmath81 ( solid line ) , @xmath16 ( large dashed line ) , @xmath17 ( medium dashed line ) , @xmath87 ( small dashed line ) , and @xmath88 ( dotted line ) as a function of the dephasing constant @xmath46 . top left : @xmath34 , the initial state is pure . in this case the tri - partite entanglement witnesses go to zero in finite time but the concurrence , negativity , and tri - partite negativity do not undergo esd . top right : @xmath89 , for any sufficiently high value of @xmath45 where @xmath39 the concurrence undergoes esd before the negativity but after the tri - partite negativity . bottom : @xmath90 , the concurrence undergoes esd before the tri - partite negativity and even before the tri - partite entanglement witness . thus , there is detectable tri - partite entanglement in the system though there is no concurrence remaining when one of the qubits is traced over.,title="fig:",width=151 ] -type states . each plot shows @xmath81 ( solid line ) , @xmath16 ( large dashed line ) , @xmath17 ( medium dashed line ) , @xmath87 ( small dashed line ) , and @xmath88 ( dotted line ) as a function of the dephasing constant @xmath46 . top left : @xmath34 , the initial state is pure . in this case the tri - partite entanglement witnesses go to zero in finite time but the concurrence , negativity , and tri - partite negativity do not undergo esd . top right : @xmath89 , for any sufficiently high value of @xmath45 where @xmath39 the concurrence undergoes esd before the negativity but after the tri - partite negativity . bottom : @xmath90 , the concurrence undergoes esd before the tri - partite negativity and even before the tri - partite entanglement witness . thus , there is detectable tri - partite entanglement in the system though there is no concurrence remaining when one of the qubits is traced over.,title="fig:",width=151 ] [ cols="^,^,^,^",options="header " , ] [ tab1 ] the results of the above explorations are summarized in the table [ tab1 ] . there are a few points worth noting . first , we see the shortcomings of the entanglement witnesses in detecting mixed state entanglement . this is especially important for experimental realizations where entanglement witnesses may suggest there is no entanglement present between systems when in fact there is . second , for the ghz and w - type states the different entanglement measures decrease at the same rate , _ i. e. _ , the exponential term is the same . however , this is not so for the state that has both ghz and bi - partite entanglement . finally , we have found states where the negativity is non - zero despite the states having no tri - partite negativity and no concurrence between any set of two qubits . with what we have learned concerning the entanglement behavior of tri - partite systems and the advent of esd , we address the question of whether esd affects the reliability of quantum error correction ( qec ) or can esd be used as a signature that error correction has only been successful up to a certain accuracy . initial explorations of a possible connection were reported in ref . @xcite . due to the fragility of certain quantum states qec will be a necessary component of any working quantum computer . the multi - qubit logical states in which quantum information is encoded by a qec code are typically highly entangled at the level of the physical qubits and thus may be subject to esd . i explore the reliability of a qubit of quantum information encoded into a qec code capable of correcting phase flips on one ( physical ) qubit . the goal is to determine whether esd is a reliable signature to the success or failure of the code . to encode the state @xmath91 into the phase flip code one simply uses the unencoded qubit as the control of two controlled - not gates each gate implemented between itself and an additional qubit in state @xmath92 this is followed by hadamard gates on all qubits . for simplicity we have chosen @xmath93 . unless @xmath94 or @xmath95 there is always some tri - partite entanglement present in this state which can be detected by a proper entanglement witness and measured by @xmath16 ( which in this case is equivalent to @xmath17 ) . we use the follwing w - state witness to detect all types of tri - partite entangement @xmath96 where @xmath97 is a hadamard on all three qubits . detection of a possible error is done via syndrome measurement . when only one of the three physical qubits undergoes dephasing there are parameters of the intial state @xmath98 and the dephasing parameter , @xmath20 , which lead to finite time non - detection of entanglement in the error state @xmath99 by the entanglement witness @xmath100 . however , there is no esd of @xmath16 or @xmath17 ( though @xmath101 ) . the three qubit qec code always corrects the error . thus , there is no clear correlation between the detection of tri - partite entanglement by the entanglement witness or any of the entanglement measures and the success of the qec . when all three qubits are placed in a dephasing environment ( where , for simplicity we have assumed that the dephasing parameter @xmath20 is the same for all three qubits and thus ensure that @xmath18 ) the density matrix after the dephasing , @xmath102 , exhibits tri - partite esd , as shown in fig . moreover , the qec code will not , in general , completely protect the encoded quantum information . we can measure the effectiveness of the qec code by looking at the purity , @xmath103 of the final single qubit state , @xmath104 after error correction and decoding or by looking at the fidelity of @xmath104 to the initial state . for the case at hand these two measures exhibit almost the exact same behavior . we can then ask whether the parameters for which this indicator is minimized ( maximized ) are those which exhibit ( do not exhibit ) esd . we note that in general @xmath104 , and thus the purity , @xmath105 , will depend on whether or not the syndrome measurement detects an error or not . the purity of the final decoded state for both cases is shown in fig . [ qec ] . when @xmath106 the qec code perfectly corrects even complete dephasing on all three qubits depite the onset of tri - partite esd . as @xmath107 moves away from this point the purity decreases as a function of @xmath20 , due to the inability of the qec to correct the dephasing . not surprisingly , some regions of low purity correspond to regions that exhibit esd . however , even error regions in which the entanglement is detected by the entanglement witness can exhibit purity below @xmath108 . this suggests that , for the particular qec code studied here , any correlation between esd and the success of the code is not fundamental . a comparison of the onset of esd due to the dephasing error with the purity is shown in fig . in conclusion , i have studied the effect of dephasing on entanglement in three qubit systems . i have demonstrated esd for both bi - partite and tri - partite entanglement using concurrence , the negativity , and the tri - partite negativity . these were compared to the detection ability of tri - partite entanglement witnesses . in general we find that for mixed states the entanglement witnesses fail to detect the presence of entanglement well before the entanglement goes to zero . in addition , i have found a state in which there is no concurrence nor tri - partite negativity but entanglement as measured by the negativity still exists . based on these exploration i considered whether esd affects the workings of the three - qubit qec phase flip code and concluded that there is no fundamental relationship between esd and the failure of the code . as the number of system qubits grow studies of entanglement become more complex as the number of different types of entanglement continually increase . however , a study of four qubit systems would allow for explorations of cluster state entanglement by using the proper entanglement witnesses @xcite . this would be of central importance to issues of cluster state quantum computation @xcite . m nielsen , i. chuang , _ quantum information and computation _ ( cambridge university press , cambridge , 2000 ) . for a recent review see r. horodecki , p. horodecki , m. horodecki , k. horodecki , arxiv : quant - ph/0702225 . c. simon and j. kempe , phys . a * 65 * , 052327 ( 2002 ) ; w. dur and h .- j . briegel , phys . . lett . * 92 * 180403 ( 2004 ) ; m. hein , w. dur , and h .- j . briegel , phys . a * 71 * , 032350 ( 2005 ) ; s. bandyopadhyay and d.a . lidar , phys . a * 72 * , 042339 ( 2005 ) ; o. guhne , f. bodosky , and m. blaauboer , arxiv:0805.2873 . p.j . dodd and j.j . halliwell , phys . a * 69 * , 052105 ( 2004 ) . t. yu and j.h . eberly , phys . lett . * 93 * , 140404 ( 2004 ) ; _ ibid . _ * 97 * , 140403 ( 2006 ) . i. sainz and g. bjork , phys . a * 76 * , 042313 ( 2007 ) . l. aolita , r. chaves , d. cavalcanti , a. acin , and l. davidovich , phys . * 100 * , 080501 ( 2008 ) . lopez , g. romero , f. lastra , e. solano , and j.c . retamal , arxiv:0802.1825 . m. yonac , t. yu , j.h . eberly , j. phys . b * 39 * , 5621 ( 2006 ) ; _ ibid . _ * 40 * , 545 ( 2007 ) . almeida , _ et al . _ , science * 316 * , 579 ( 2007 ) ; j. laurat , k.s . choi , h. deng , c.w . chou , and h.j . kimble , phys . 99 * , 180504 ( 2007 ) ; a. salles , f. de melo , m.p . almeida , m. hor - meyll , s.p . walborn , p.h . souto ribeiro , and l. davidovich , phys . a * 78 * , 022322 ( 2008 ) . a. acin , d. bru , m. lewenstein , a. sanpera , phys . rev . lett . * 87 * , 040401 , ( 2001 ) . c. sabin and g. garcia - alcaine , eur . j. d * 48 * , 435 ( 2008 ) . w. dur , g. vidal , and j.i . cirac , phys . rev . a * 62 * , 062314 ( 2000 ) . g. toth and o. guhne , phys . rev . lett . * 94 * , 060501 ( 2005 ) . a. acin , a. andrianov , l. costa , e. jane , j.i . latorre , and r. tarrach , phys . . lett . * 85 * , 1560 ( 2000 ) . s. hill and w.k . wootters , phys . lett * 78 * , 5022 ( 1997 ) . g. vidal and r.f . werner , phys . a * 65 * 032314 ( 2002 ) . i. sainz and g. bjork , phys . rev a * 77 * , 052307 ( 2008 ) . r. raussendorf and h.j . briegel , phys . * 86 * , 5188 ( 2001 ) .
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i explore entanglement dynamics in a three qubit system comparing the ability of entanglement witnesses to detect tri - partite entanglement to the phenomenon of entanglement sudden death ( esd ) . using a system subject to dephasing
i invoke entanglement witnesses to detect tri - partite ghz and w - type entanglement and compare the evolution of their detection capabilites with the evolution of the negativity , bi - partite concurrence , and tri - partite negativity .
interestingly , i find a state in which there is no concurrence or tri - partite negativity but there is entanglement .
finally , i utilize a three qubit quantum error correction ( qec ) code to address how esd affects the abilities of quantum error correction .
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the interplay between superconducting and magnetic order in ruthenocuprates raised considerable interest for these materials . they exist in three modifications , rusr@xmath0recu@xmath0o@xmath1 ( ru1212 ) , rusr@xmath0re@xmath2ce@xmath3cu@xmath0o@xmath4 ( ru1222 ) and , recently synthesized @xcite , rusr@xmath0rece@xmath0cu@xmath0o@xmath5 ( ru1232 ) ( re = gd , eu for ru1212 and ru1222 ; re = y , dy for ru1232 ) . neutron scattering measurements on ru1212 @xcite have revealed the existence of the antiferromagnetic order in ru - sublattice below 130 k , although ferromagnetic - like features have been observed in magnetization measurements @xcite . measurements of microwave absorption @xcite , ac susceptibility @xcite , nmr @xcite have shown an evidence of the spontaneous vortex phase as an explanation for the coexistence of the superconductivity and magnetism below sc transition around 30 k. on the other hand , xue et al . @xcite have suggested the nanoscale separation between ferromagnetic and antiferromagnetic islands . as for the ru1222 composition , the magnetic structure is still unknown . esr measurements @xcite have displayed a ferromagnetic resonance below t@xmath6 = 180 k with an enhancement in magnetism around t@xmath7 = 100 k. xue et al . @xcite have found evidence of the clusters above the main peak at t@xmath7 ( around 100 k , depends on the eu / ce ratio ) . detailed magnetization and mssbauer study @xcite have shown two component magnetism which has been also supported with the muon spin rotation results @xcite . spin - glass - like behavior , as suggested by cardoso et al . @xcite , occurs at t@xmath7 . various dynamical features have been reported by us @xcite , including pronounced time relaxation of ac susceptibility and the inverted butterfly hysteresis loops in ac susceptibility . in this paper we present further investigation of the unusual behavior of the ru1222 material . we have concentrated on the peculiarities of the butterfly hysteresis and have found supporting evidence in favor of the above mentioned two component magnetism picture . samples used in this study are the same as the ones used in our previous article @xcite . ac susceptibility measurements were taken by the use of commercial cryobind system with the frequency of the alternating field equal to 431 hz and the field amplitude of 0.1 oe . the stabilization of the temperature for the hysteresis measurement was better than 10 mk . when a standard ferromagnet , characterized with the domain structure , is swept in a dc field up and down , its magnetization m shows a well - known m h hysteresis . the characteristic elements of the m h hysteresis loop are the coercive field and the remanent magnetization . another type of hysteretic behavior of a ferromagnet can be studied in the mutual inductance arrangement of ac susceptibility technique , imposing an ac field superimposed over the sweeping and cycling dc field . in the latter case one measures a butterfly hysteresis @xcite . in increasing dc field it is characterized by a monotonously decreasing virgin curve , followed by the repetitive branches of further reduced susceptibility values . the repetitive branches are characterized by the characteristic maxima which are related to the coercive field . the values of the coercive field , as obtained from m h and butterfly hysteresis are not necessarily the same due to the inherent difference between dc and ac techniques . in fig . [ srruo3hys ] butterfly hysteresis is shown for srruo@xmath8 , an itinerant ferromagnet . . ( @xmath9 ) - virgin branch ; ( @xmath10 ) - descending field branch ; ( @xmath11 ) - ascending field branch . ] at variance with a standard butterfly hysteresis , the one discovered to characterize ru1222 material @xcite shows very different behavior when subjected to the sweeping dc field . in fig . [ ru1222hys]a we present the butterfly hysteresis for the ru1222euce ( x = 1.0 ) material . the virgin branch has a maximum ( denoted with h@xmath12 ) , followed by the descending field branch which has systematically _ higher _ susceptibility than the virgin branch . instead of one there are two maxima types , at h@xmath13 before and h@xmath14 after h@xmath15 = 0 . in the ascending field branch , h@xmath13 maximum appears again , followed by the h@xmath14 maximum for h@xmath16 . such an unusual magnetism hasnt been reported so far and it reveals that the ru1222 material is a rather unique magnetic system , very different from an ordinary ferromagnet . [ ru1222hys]b displays the m h curve measured with a vibrating sample magnetometer ( see also @xcite ) . apart from a somewhat unusual virgin curve ( overlapping with the right - hand side hysteresis branch ) , the m h hysteresis appears quite regular and standard , implying that the unusual butterfly - hysteresis features rely on magneto - dynamics of the ru1222 compound . ) - virgin branch ; ( @xmath10 ) - descending field branch ; ( @xmath11 ) - ascending field branch . ( b ) normal hysteresis obtained with a vibrating sample magnetometer taken at the same temperature and the same maximum dc field as in ( a ) . ] temperature dependence of ac susceptibility of the ru1222 materials is shown in fig . [ tempdep ] . for x = 1.0 the main peak ( t@xmath7 ) is at 120 k , while for x = 0.7 it is at 90 k. the x = 0.7 sample is superconducting below 30 k. the anomaly at 130 k can also be seen , especially for the x = 1.0 sample . it has been shown by felner et al . @xcite that this anomaly has a different magnetic origin than the main peak at t@xmath7 . also , @xmath17sr experiments @xcite revealed no bulk character of the latter anomaly . for the x = 1.0 composition the main peak at t@xmath7 and the anomaly at 130 k are very close and overlapping , so we will use the x = 0.7 material to exclude the possibility that the butterfly hysteresis are connected to this anomaly . a qualitative change in the behavior of the butterfly hysteresis can be seen in fig . [ hystemp ] . three characteristic temperatures have been chosen in order to illustrate the range of peculiar behaviors of butterfly hysteresis covered by varying the temperature . all the curves have been scaled by the value @xmath18(h@xmath19 = 0 oe ) . the most important feature to notice is the disappearance of the inverted part , characterized by the maximum at h@xmath13 in fig . [ ru1222hys ] . for temperatures above 30 k the butterfly hysteresis is inverted but below it starts to accommodate the single - peak shape associated with the ferromagnets ( fig . [ srruo3hys ] ) , except for the virgin branch . the h@xmath12 maximum in the virgin curve has been linked with the possible spin - flop mechanism @xcite where the initial afm state changes into a fm state under the influence of the external dc field . the exact temperature where the h@xmath12 peak and the butterfly hysteresis set in is indicated with the arrows in the fig [ tempdep ] . for the x = 1.0 composition there is just a small difference in the temperature between the main peak at t@xmath7 and the anomaly at 130 k and one can not be sure with which peak to associate this event . for that reason we have measured the x = 0.7 material for which the main peak is at 90 k and it clearly indicates that the observed dynamics is not connected with the small anomaly around 130 k but rather with the main peak at t@xmath7 which , in turn , depends on the eu / ce ratio . ce@xmath20 ( x = 0.7 ) at 95 k , just above the temperature at which the inverted peak emerges . this featureless curve resembles the shape of the butterfly hysteresis characterizing ru1212 material below its magnetic ordering transition @xcite . at 78 k ( bottom panel ) the inverted peak is well - defined and dominates in the hysteretic behavior . ( @xmath9 ) - virgin branch ; ( @xmath10 ) - descending field branch ; ( @xmath11 ) - ascending field branch . ] in addition to the temperature dependence of the butterfly hysteresis for the ru1222 material , we have investigated the influence of the maximum dc field one reaches in cycling the dc field . [ maxdc ] shows how the inverted peak ( h@xmath13 ) emerges for small dc fields ( above 25 oe ) and eventually completely overwhelms the underlying ferromagnetic behavior for larger dc fields where it tends to saturate ( here _ larger _ denotes a few hundred oe ) . after the careful analysis has been done and the ferromagnetic background has been subtracted , one gets the dependence of the inverted peak h@xmath13 vs. maximum dc field imposed on the system h@xmath21 , shown in the inset of fig . [ maxdc ] . the line represents a fit to the formula : @xmath22 with the parameters h@xmath23 = 42(1 ) oe and h@xmath24 = 72(4 ) oe ( correlation coefficient : r = 0.987 ) . at the same time , there is no observable shift in the h@xmath14 peak , as is expected for a coercive maximum . one should note that there is no sign of the h@xmath13 peak in the descending field branch ( squares ) for h@xmath25 0 nor in the ascending field branch ( triangles ) for h@xmath26 0 . also , it seems that the h@xmath12 peak is linked only to the virgin curve resembling the s - shaped vsm virgin curve ( see fig . [ srruo3hys]b ) and has no influence on the behavior of the inverted part . for h@xmath21 = 350 oe . inset : the maximum of the inverted peak h@xmath27 vs. the maximum dc field h@xmath21 . the line is the fit to the exponential saturation ( see text ) . ( @xmath9 ) - virgin branch ; ( @xmath10 ) - descending field branch ; ( @xmath11 ) - ascending field branch . ] as has been recently proposed @xcite , ru1222 materials consist of two phases , the minority one that orders around 180 k and the majority one that orders around 100 k ( depending on the eu / ce ratio ) . exact nature of both orderings is still unclear . although ferromagnetic - like features have been observed , detailed investigation of butterfly hysteresis in ru1222 reveals a more complex magnetism characterizing this material . temperature dependence of the inverted part ( fig . [ hystemp ] ) shows that it emerges at the main peak t@xmath7 where the majority phase orders , progressively freezing out as the temperature is lowered . we have investigated samples with x from 0.5 to 1.0 and they all show the same behavior , suggesting similar magnetic ordering to take place in all compositions . the h@xmath14 peak , which marks the coercive field , gradually increases as the temperature is decreased and reaches the value of 100 oe for 4.2 k. it can be compared with the values of 250 oe obtained in @xcite for much larger maximum fields ( 50 koe ) . from the figs . [ hystemp ] and [ maxdc ] it is obvious that h@xmath14 is not affected by the presence of the inverted peak , suggesting two separate contributions to the ac susceptibility . presently , we are unable to provide a full interpretation of the exponential dependence shown in the inset of the fig . [ maxdc ] . we note however that it indicates the presence of some fundamental interaction which is susceptible to small magnetic fields . also , a remarkable fact is that no feature can be seen in the magnetization measurements ( either vsm or squid ) that would correspond to the inverted part of the butterfly hysteresis . we suggest that the inverted behavior might reflect the interaction of the two magnetic phases , namely the ferromagnetic clusters and the background matrix , ordering at t@xmath6 and at t@xmath7 , respectively . as suggested by cardoso et al . @xcite the phenomenology of spin glasses could be relevant for ru1222 as well . indeed , we have observed the frequency dependence of the main peak at t@xmath7(not shown ) . the magnitude of the shift follows the spin - glass - like behavior @xcite but the magnitude of the ac susceptibility signal is orders of magnitude larger than for the usual spin - glass material . we speculate that the ferromagnetic clusters , randomly distributed and oriented in matrix , could impose a frustration on the surrounding matrix , giving rise to the spin - glass behavior and the observed frequency dependence . we have measured the butterfly hysteresis for the ru1222 material and have found that it consists of two components . the first one comes from the already observed ferromagnetic - like behavior of the compound and is represented by the coercive field peak . the second component is responsible for the onset of the inverted peak in butterfly hysteresis , formed after the dc field reaches its maximum value and starts reducing . the inverted part disappears as the temperature is lowered while the coercive maximum just shifts to larger values . exponential dependence of the inverted peak vs. maximum dc field has been observed but it s microscopic background is not well understood as yet . interaction between ferromagnetic clusters and the ordered matrix embedding the clusters has been proposed to account for the observed behavior . we thank prof . i. felner for providing us with samples and giving valuable comments . awana et al . , j. appl . ( 2005 ) 10b111 j.w . lynn et al . , phys . b 61 ( 2000 ) r14964 g.w.m . williams and s. krmer , phys . b 61 ( 2000 ) 6401 m. poek et al , phys . rev . b 61 ( 2000 ) r14964 i. ivkovi et al . , europhys . lett 60 ( 2002 ) 917 h. sakai et al . , b 67 ( 2003 ) 184409 y.y . xue et al . , phys . b 67 ( 2003 ) 224511 y.y . xue et al . , phys . b 67 ( 2003 ) 184507 k. yoshida , h. kojima , h. shimizu , j. phys . soc . jpn . 72 ( 2003 ) 3254 i. felner et al . , phys.rev . b 70 ( 2004 ) 094504 ; i. felner , e. galstyan , i. nowik , phys . rev . b 71 , ( 2005 ) 064510 a. shengelaya et al . b 69 ( 2004 ) 024517 c.a . cardoso et al . , phys . b 67 ( 2003 ) 020407(r ) i. ivkovi et al . , phys . rev . b 65 ( 2002 ) 144420 f.h . salas and d. weller , jmmm 128 , 209 ( 1993 ) , and references therein . j.a . mydosh , spin glasses : an experimental introduction , taylor & francis , london , 1993
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we report detailed studies of the ac susceptibility butterfly hysteresis on the ru1222 ruthenocuprate compounds .
two separate contributions to these hysteresis have been identified and studied .
one contribution is ferromagnetic - like and is characterized by the coercive field maximum .
another contribution , represented by the so called inverted maximum , is related to the unusual inverted loops , unique feature of ru1222 butterfly hysteresis .
the different nature of the two identified magnetic contributions is proved by the different temperature dependences involved . by lowering the temperature the inverted peak gradually disappears while the coercive field slowly raises .
if the maximum dc field for the hysteresis is increased , the size of the inverted part of the butterfly hysteresis monotonously grows while the position of the peak saturates . in reaching saturation exponential field dependence has been demonstrated to take place . at t
= 78 k the saturation field is 42 oe .
| 4,499 | 253 |
given @xmath4 and a bounded open set @xmath5 with @xmath6-boundary , the @xmath0-perimeter of a ( measurable ) set @xmath7 in @xmath2 is defined as @xmath8 where @xmath9 denotes the complement of @xmath1 , and @xmath10 denotes the following nonlocal interaction term @xmath11 here we are using the standard convention for which @xmath12 if either @xmath13 or @xmath14 . this notion of @xmath0-perimeter and the corresponding minimization problem were introduced in @xcite ( see also the pioneering work @xcite , where some functionals related to the one in have been analyzed in connection with fractal dimensions ) . recently , the @xmath0-perimeter has inspired a variety of literature in different directions , both in the pure mathematical settings ( for instance , as regards the regularity of surfaces with minimal @xmath0-perimeter , see @xcite ) and in view of concrete applications ( such as phase transition problems with long range interactions , see @xcite ) . in general , the nonlocal behavior of the functional is the source of major difficulties , conceptual differences , and challenging technical complications . we refer to @xcite for an introductory review on this subject . the limits as @xmath3 and @xmath15 are somehow the critical cases for the @xmath0-perimeter , since the functional in diverges as it is . nevertheless , when appropriately rescaled , these limits seem to give meaningful information on the problem . in particular , it was shown in @xcite that @xmath16 approaches the classical perimeter functional as @xmath15 ( up to normalizing multiplicative constants ) , and this implies that surfaces of minimal @xmath0-perimeter inherit the regularity properties of the classical minimal surfaces for @xmath0 sufficiently close to @xmath17 ( see @xcite ) . as far as we know , the asymptotic as @xmath3 of @xmath18 was not studied yet ( see however @xcite for some results in this direction ) , and this is the question that we would like to address in this paper . that is , we are interested in the quantity @xmath19 whenever the limit exists . of course , if it exists then @xmath20 since @xmath21 we will show that , though @xmath22 is subadditive ( see proposition [ t1 ] below ) , in general it is not a measure ( see proposition [ t2 ] , and this is a major difference with respect to the setting in @xcite ) . on the other hand , @xmath22 is additive on bounded , separated sets , and it agrees with the lebesgue measure of @xmath23 ( up to normalization ) when @xmath1 is bounded ( see corollary [ cor ] ) . as we will show below , a precise characterization of @xmath24 will be given in terms of the behavior of the set @xmath1 towards infinity , which is encoded in the quantity @xmath25 whenever it exists ( see theorem [ tf ] and corollary [ cor ] ) . in fact , the existence of the limit defining @xmath26 is in general equivalent to the one defining @xmath22 ( see theorem [ tf1](ii ) ) . as a counterpart of these results , we will construct an explicit example of set @xmath1 for which both the limits @xmath24 and @xmath27 do not exist ( see example [ exx ] ) : this says that the assumptions we take can not , in general , be removed . also , notice that , in order to make sense of the limit in , it is necessary to assume that for any @xmath28 . moreover , if @xmath29 is smooth , then is always satisfied . ] @xmath30 to stress that can not be dropped , we will construct a simple example in which such a condition is violated ( see example [ ex2 ] ) . the paper is organized as follows . in the following section , we collect the precise statements of all the results we mentioned above . section [ sec_proofs ] is devoted to the proofs . we define @xmath31 to be the family of sets @xmath7 for which the limit defining @xmath24 in exists . we prove the following result : [ t1 ] @xmath22 is subadditive on @xmath31 , i.e. @xmath32 for any @xmath1 , @xmath33 . first , it is convenient to consider the normalized lebesgue measure @xmath34 , that is the standard lebesgue measure scaled by the factor @xmath35 , namely @xmath36 where , as usual , we denote by @xmath37 the @xmath38-dimensional sphere . now , we recall the main result in @xcite ; that is , [ thm_ms](see ( * ? ? ? * theorem 3 ) ) . let @xmath4 . then , for all @xmath39 , @xmath40 an easy consequence of the result above is that when @xmath41 and @xmath42 then @xmath24 agrees with @xmath43 ( in fact , we will generalize this statement in theorem [ tf ] and corollary [ cor ] ) . based on this property valid for subsets of @xmath2 , one may be tempted to infer that @xmath22 is always related to the lebesgue measure , up to normalization , or at least to some more general type of measures . the next result points out that this can not be true : [ t2 ] @xmath22 is not necessarily additive on separated sets in @xmath44 , i.e. there exist @xmath45 such that @xmath46 , but @xmath47 . also , @xmath22 is not necessarily monotone on @xmath31 , i.e. it is not true that @xmath48 implies @xmath49 . in particular , we deduce from proposition [ t2 ] that @xmath22 is not a measure . on the other hand , in some circumstances the additivity property holds true : [ t3 ] @xmath22 is additive on bounded , separated sets in @xmath31 , i.e. if @xmath1 , @xmath33 , @xmath1 and @xmath50 are bounded , disjoint and @xmath46 , then @xmath51 and @xmath52 . there is a natural condition under which @xmath24 does exist , based on the weighted volume of @xmath1 towards infinity , as next result points out : [ tf ] suppose that @xmath53 for some @xmath54 , and that the following limit exists @xmath55 then @xmath56 and @xmath57where @xmath58 as a consequence of theorem [ tf ] , one obtains the existence and the exact expression of @xmath24 for a bounded set @xmath1 , as described by the following result : [ cor ] let @xmath1 be a bounded set , and @xmath53 for some @xmath54 . then @xmath56 and @xmath59 in particular , if @xmath42 and @xmath53 for some @xmath54 , then @xmath60 . condition is also in general necessary for the existence of the limit in . indeed , next result shows that the existence of the limit in is equivalent to the existence of the limit in , except in the special case in which the set @xmath1 occupies exactly half of the measure of @xmath2 ( in this case the limit in always exists , independently on the existence of the limit in ) . [ tf1 ] suppose that @xmath53 , for some @xmath54 . then : 1 . if @xmath61 , then @xmath56 and @xmath62 . 2 . if @xmath63 and @xmath56 , then the limit in exists and @xmath64 there exists a set @xmath1 with @xmath65-boundary for which the limits in and do not exist . [ exx special ] there exists a set @xmath1 with @xmath65-boundary for which the limit in exists and the limit in does not exist . notice that examples [ exx ] and [ exx special ] are provided by smooth sets , and therefore they have finite @xmath0-perimeter for any @xmath4 ( see , e.g. , lemma 11 in @xcite ) . on the other hand , as regards condition , we point out that it can not be dropped in general , since there are sets that do not satisfy it ( and for them the limit in does not make sense ) : [ ex2 ] there exists a set @xmath1 for which @xmath66 for any @xmath4 . we observe that @xmath67 to check this , let @xmath68 , @xmath69 be open sets of @xmath70 . we remark that @xmath71 & & \qquad \quad= l((e\cap\omega_1)\cup(f\cap\omega_1),({{\mathscr}{c}}e)\cap({{\mathscr}{c}}f)\cap\omega_2)\\[1ex ] & & \qquad\quad{\leqslant}l ( e\cap\omega_1,({{\mathscr}{c}}e)\cap({{\mathscr}{c}}f)\cap \omega_2 ) + l(f\cap\omega_1,({{\mathscr}{c}}e)\cap({{\mathscr}{c}}f)\cap \omega_2 ) \\[1ex ] & & \qquad\quad{\leqslant}l ( e\cap\omega_1,({{\mathscr}{c}}e)\cap \omega_2 ) + l(f\cap\omega_1,({{\mathscr}{c}}f)\cap \omega_2).\end{aligned}\ ] ] by taking @xmath72 and @xmath73 we obtain @xmath74 while , by taking @xmath75 and @xmath76 , we conclude that @xmath77 by summing up , we get @xmath78 & & \quad{\leqslant}l ( e\cap\omega,{{\mathscr}{c}}e ) + l(f\cap\omega,{{\mathscr}{c}}f)\\ & & \quad \quad + \ , l ( e\cap({{\mathscr}{c}}\omega),({{\mathscr}{c}}e)\cap \omega ) + l(f\cap({{\mathscr}{c}}\omega),({{\mathscr}{c}}f)\cap \omega ) \\[1ex ] & & \quad = \text{\rm per}_s(e;\omega)+ \text{\rm per}_s(f;\omega ) .\end{aligned}\ ] ] this establishes and then proposition [ t1 ] follows by taking the limit as @xmath3 . @xmath79 here and in the sequel , we denote by @xmath80 the open ball centered at @xmath81 of radius @xmath82 . we observe that if @xmath83 and @xmath84 then @xmath85 , therefore @xmath86 for some positive constants @xmath87 , @xmath88 and @xmath89 . now we take @xmath90 , @xmath91 . then @xmath92 therefore @xmath93 by sending @xmath3 , we conclude that @xmath94 , so @xmath22 is not additive . now we show that @xmath22 is not monotone either . for this we take @xmath1 such that @xmath95 ( for instance , one can take @xmath1 a small ball inside @xmath2 ; see corollary [ cor ] ) , and @xmath96 : with this choice , @xmath97 and @xmath98 , so @xmath99 . @xmath79 * observation 2 . * now we would like to remark that the quantity @xmath104 is independent of @xmath105 , if the limit exists . more precisely , we show that for any @xmath106 @xmath107 to prove this , we notice that @xmath108 and so , by taking limit in @xmath0 , @xmath109 which establishes . * observation 4 . * for any @xmath4 , we define @xmath111 and we prove that , for any bounded set @xmath112 , and any set @xmath7 , @xmath113 to prove this , we take @xmath114 such that @xmath115 and @xmath116 ( later on @xmath105 will be taken as large as we wish ) . we observe that , for any @xmath117 and @xmath118 , @xmath119 therefore , if , for any fixed @xmath118 we consider the map @xmath120 we have that @xmath121 for any @xmath117 , which implies @xmath122 therefore @xmath123 for some @xmath124 independent of @xmath0 . as a consequence @xmath125 this and ( applied here with @xmath126 ) imply . * observation 6 . * now we point out that , if @xmath128 for some @xmath82 , and @xmath50 has finite @xmath129-perimeter in @xmath2 for some @xmath54 , then @xmath130 indeed , for any @xmath28 , @xmath131 which implies . in particular , thanks to ( * ? ? ? * proposition 16 ) , the argument above also shows that if @xmath132 and @xmath133 , then @xmath50 has finite @xmath0-perimeter in @xmath2 for any @xmath134 . we prove proposition [ t3 ] by suitably modifying the proof of proposition [ t1 ] . given two open sets @xmath68 and @xmath69 , and two disjoint sets @xmath1 and @xmath50 , we have that @xmath138 & & \qquad= l((e\cap\omega_1)\cup(f\cap\omega_1),({{\mathscr}{c}}e)\cap({{\mathscr}{c}}f)\cap\omega_2)\\[1ex ] & & \qquad = l(e\cap\omega_1,({{\mathscr}{c}}e)\cap({{\mathscr}{c}}f)\cap\omega_2 ) + l(f\cap\omega_1,({{\mathscr}{c}}e)\cap({{\mathscr}{c}}f)\cap\omega_2).\end{aligned}\ ] ] by taking @xmath72 and @xmath73 we obtain @xmath139 while , by taking @xmath75 and @xmath76 , we conclude that @xmath140 as a consequence , @xmath141 & & \qquad = \ , l(e\cap\omega,({{\mathscr}{c}}e)\cap({{\mathscr}{c}}f ) ) + l(f\cap\omega,({{\mathscr}{c}}e)\cap({{\mathscr}{c}}f ) ) \\ & & \qquad \quad + \ , l(e\cap({{\mathscr}{c}}\omega),({{\mathscr}{c}}e)\cap({{\mathscr}{c}}f)\cap\omega ) + l(f\cap({{\mathscr}{c}}\omega),({{\mathscr}{c}}e)\cap({{\mathscr}{c}}f)\cap\omega ) \\[1ex ] & & \qquad = \,{\text{\rm per}}_s(e;\omega)+ { \text{\rm per}}_s(f;\omega ) \\ & & \qquad \quad -\ , l(e\cap\omega , ( { { \mathscr}{c}}e)\cap f ) -l(f\cap\omega , e\cap({{\mathscr}{c}}f ) ) \\ & & \qquad \quad -\,l(e\cap({{\mathscr}{c}}\omega ) , ( { { \mathscr}{c}}e)\cap f\cap\omega ) -l(f\cap({{\mathscr}{c}}\omega ) , e\cap({{\mathscr}{c}}f)\cap\omega).\end{aligned}\ ] ] we remark that the last interactions involve only bounded , separated sets , since so are @xmath1 and @xmath50 , therefore , by , @xmath142 which completes the proof of proposition [ t3 ] . @xmath79 we suppose that @xmath143 , for some @xmath114 , and we take @xmath116 . let @xmath144 and @xmath136 . notice that , for any @xmath145 , which has finite @xmath129-perimeter in @xmath2 for some @xmath146 , @xmath147 and so gives that @xmath148 provided that @xmath50 has finite @xmath129-perimeter in @xmath2 . using this and , we conclude that , for any @xmath145 of finite @xmath129-perimeter in @xmath2 , @xmath149 & \qquad\qquad\qquad\qquad=\,\alpha(e)\,|f|.\end{split}\ ] ] in particular and @xmath150 have finite @xmath129-perimeter in @xmath2 if so has @xmath1 , thanks [ foot ] to our smoothness assumption on @xmath151 . we check this claim for @xmath152 , the other being analogous . first of all , fixed @xmath153 , we have that @xmath154 also @xmath155 ( see , e.g. , lemma 11 in @xcite ) , therefore @xmath156 that is finite . ] , by taking @xmath157 and @xmath158 , and recalling and , @xmath159 we now claim @xmath160 indeed , since @xmath161 , this is a plain consequence of theorem [ thm_ms ] ( see also remark 4.3 in @xcite for another elementary proof ) by simply choosing @xmath162 there : @xmath163 & = \,{{\mathscr}{h}}^{n-1}(s^{n-1})\,\| \chi_{e_1}\|_{l^2({{\mathds r}}^n)}^2 \ = \ { { \mathscr}{h}}^{n-1}(s^{n-1})\,|e_1| , \end{split}\ ] ] as desired . thus , using , , and , we obtain @xmath164 which is the desired result . @xmath79 we fix @xmath105 large enough so that @xmath165 , hence @xmath166 . by the expression of @xmath27 in @xmath167 , we have that the limit in @xmath168 exists and @xmath169 . then the result follows by theorem [ tf ] . @xmath79 we suppose that @xmath143 , for some @xmath114 , and we take @xmath116 . let @xmath170 and @xmath136 . by , @xmath171 & \qquad= s l(e_2,\omega\setminus e_1 ) - s l(e_1,e_2 ) \\[1ex ] & \qquad = s \int_{\omega\setminus e_1 } \int_{e_2 \cap b_r } \frac{1}{|x - y|^{n+s}}\ , dx \ , dy + s \int_{\omega\setminus e_1 } \int_{e_2 \cap ( { { \mathscr}{c}}b_r ) } \frac{1}{|x - y|^{n+s}}\ , dx \ , dy \\ & \qquad \quad- s \int _ { e_1 } \int_{e_2 \cap b_r } \frac{1}{|x - y|^{n+s}}\ , dx \ , dy - s \int _ { e_1 } \int_{e_2 \cap ( { { \mathscr}{c}}b_r ) } \frac{1}{|x - y|^{n+s}}\ , dx \ , dy . \end{split}\ ] ] by rearranging the terms , we obtain @xmath172 & \qquad \quad \ = s { \text{\rm per}}_s(e;\omega ) -s { \text{\rm per}}_s(e_1;\omega ) - s \int_{\omega\setminus e_1 } \int_{e_2 \cap b_r } \frac{1}{|x - y|^{n+s}}\ , dx \ , dy \\ & \qquad \qquad \ + s \int _ { e_1 } \int_{e_2 \cap b_r } \frac{1}{|x - y|^{n+s}}\ , dx \ , dy . \end{split}\ ] ] by using with @xmath158 and @xmath173 ( which have finite @xmath129-perimeter in @xmath2 , recall the footnote on page ) , we have that the last two terms in converge to zero as @xmath174 , thus @xmath175 we now recall the notation in and we write @xmath176 and @xmath177 by subtracting term by term , we obtain that @xmath178 as a consequence , by using ( applied here both with @xmath158 and @xmath157 ) , @xmath179=0.\ ] ] now , if @xmath180 then @xmath181 , and from , , and corollary [ cor ] we get @xmath182 which proves that @xmath183 and @xmath62 . this establishes theorem [ tf1](i ) . on the other hand , if @xmath184 , then by , , and corollary [ cor ] we obtain the existence of the limit @xmath185 & & \qquad \qquad \qquad = \lim_{r\rightarrow+\infty}\lim_{s\searrow0}\bigg\ { \left [ \alpha_s(e)\,\big ( |\omega\setminus e_1|-|e_1|\big ) -i(s , r)\right ] + \ , i(s , r)\bigg\ } \\[2ex ] & & \qquad \qquad \qquad = \mu(e)-\mu(e_1)= \mu(e)-{\mathscr}{m}(e\cap\omega),\end{aligned}\ ] ] which completes the proof of theorem [ tf1](ii ) . @xmath79 we start with some preliminary computations . let @xmath186 , for any @xmath187 , and let @xmath188 notice that @xmath189 may be written as the disjoint union of the @xmath190 s . let @xmath191\big)$ ] be such that @xmath192 in @xmath193\cup i_0 $ ] , @xmath194 in @xmath195 , and then @xmath196 smoothly interpolates between @xmath197 and @xmath17 in @xmath198 . we claim that there exist two sequences @xmath199 and @xmath200 such that @xmath201 to check , we take @xmath202 and @xmath203 . we observe that , by construction , @xmath192 in @xmath204 and @xmath194 in @xmath205 , so @xmath206 for any @xmath207 and @xmath208 in @xmath209 , where @xmath210 we deduce that @xmath211 this implies by noticing that @xmath212 first of all , since @xmath216 and @xmath217 , it is easy to see that @xmath218 for any @xmath134 ( notice that , since @xmath1 has smooth boundary , the fact that @xmath1 has finite @xmath0-perimeter is also a consequence of lemma 11 in @xcite ) . then , recalling we have @xmath219 therefore , by the change of variable @xmath220 , we have @xmath221 and , by the further change @xmath222 , we have @xmath223 if we set @xmath224 , the limit in becomes the following : @xmath225 and , by , we get that such a limit does not exist . this shows that the limit in does not exist . since @xmath226 , by theorem [ tf1](ii ) , the limit in does not exist either . @xmath79 it is sufficient to modify example [ exx ] inside @xmath216 in such a way that @xmath227 . notice that , since the set @xmath1 has smooth boundary , then it has finite @xmath0-perimeter for any @xmath134 ( see lemma 11 in @xcite ) . then is not affected by this modification and so the limit in does not exist in this case too . on the other hand , the limit in exists , thanks to theorem [ tf1](i ) . @xmath79 now , we define @xmath236 notice that @xmath237 and @xmath238 an integral computation shows that if @xmath239 then @xmath240.\ ] ] by plugging this into , we obtain @xmath241 \\ & \qquad= \sum_{j=1}^{+\infty } \beta_{2j+2}^{1-s}+\beta_{2j+1}^{1-s}- ( \beta_{2j+2}+\beta_{2j+1})^{1-s}. \end{split}\ ] ] now we observe that the map @xmath242 is concave , therefore @xmath243 for any @xmath244 , that is @xmath245 by taking @xmath246 and then multiplying by @xmath247 , we obtain @xmath248 by plugging this into and using , we conclude that @xmath249 as desired.@xmath79 : geometric analysis of fractional phase transition interfaces . in `` geometric properties for parabolic and elliptic pde s '' , a. alvino , r. magnanini and s. sakaguchi eds . , springer indam series , springer
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we deal with the asymptotic behavior of the @xmath0-perimeter of a set @xmath1 inside a domain @xmath2 as @xmath3 .
we prove necessary and sufficient conditions for the existence of such limit , by also providing an explicit formulation in terms of the lebesgue measure of @xmath1 and @xmath2 .
moreover , we construct examples of sets for which the limit does not exist . serena dipierro alessio figalli giampiero palatucci enrico valdinoci ( communicated by the associate editor name )
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the rayleigh - taylor instability occurs when a heavy fluid is being pushed by a light fluid . two plane - parallel layers of fluid , colder on top , are in equilibrium while the slightest perturbation leads to the denser fluid moving down and the lighter material being displaced upwards . the early , linear stage of the instability was described by rayleigh @xcite and taylor @xcite , and summarized in @xcite . further development of the instability leads to enhancement of the mixing and to a gradual increase of the mixing zone , which is the domain where proportions of heavy in light and light in heavy are comparable . dimensional arguments , supported by large - scale modeling @xcite , suggest that the half - width of the mixing zone , @xmath0 , grows quadratically at late time , @xmath1 , where @xmath2 is the atwood number characterizing the initial density contrast , @xmath3 is the gravitational acceleration , and @xmath4 is a dimensionless coefficient . the coefficient @xmath4 was the focus of almost every paper written on the subject of rayleigh - taylor turbulence ( rtt ) during the last fifty years . the first attempts to look inside the mixing zone were initiated only in late 1990s @xcite , due to advances in experimental and numerical techniques . the results of many studies and the controversies surrounding the @xmath4-coefficient were recently summarized in the review combining and analyzing the majority of existing @xmath4-testing simulations and experiments @xcite . in this article we also discuss the developed regime of rt turbulence . our main focus is on the analysis of the internal structure of the mixing zone , and we trace the @xmath4-coefficient only for validation purposes . our analysis of the mixing zone develops and extends previous experimental @xcite and numerical @xcite observations on the subject , and it is also guided by phenomenological considerations discussed in @xcite . the essence of the phenomenology , which utilizes the classical kolmogorov-41 approach @xcite , can be summarized in the following statements : ( i ) the mixing zone width , @xmath0 , and the energy containing scale , @xmath5 , are well separated from the viscous , @xmath6 , and diffusive , @xmath7 , scales . in the inertial range , realized within the asymptotically large range bounded by @xmath8 from above / below , turbulence is adjusted adiabatically to the large - scale buoyancy - controlled dynamics . ( ii ) in three dimensions , the velocity fluctuations at smaller scales are asymptotically decoupled from weaker buoyancy effects phenomenological prediction of @xcite was numerically confirmed in @xcite . ] . ( iii ) typical values of velocity and density fluctuations scale the same way as in the stationary , homogeneous kolmogorov turbulence , @xmath9 , and @xmath10 , where the energy kolmogorov flux , @xmath11 , increases with time while the density fluctuations flux , @xmath12 , remains constant , according to the buoyancy prescribed large scale dynamics . all of these three theses of the phenomenology are consistent with available experimental @xcite and numerical @xcite observations of the velocity and density spectra . one particularly important consequence of the phenomenology , the decrease of the viscous and dissipative scales , was also predicted in @xcite and numerically confirmed in @xcite . in spite of its relative success in explaining rtt , the phenomenology @xcite is , obviously , not free from deficiencies . first , the asymptotic , large time character of the theory turns into a handicap in explaining numerical and experimental data , taken at finite , and actually modest , times . second , the phenomenology treats all @xmath13-slices within the mixing zone equally . third , the phenomenology does not differentiate between the mixing zone width , @xmath0 , and the energy containing scale , @xmath5 , for the turbulent fluctuations . improving the phenomenology from within itself , or by some complementary theoretical means , does not seem feasible , and one needs to rely on resolving these questions / uncertainties through experiments and simulations . this article reports a step in this direction . here we raise and give partial answers , based on simulations , to the following subset of key questions concerning the internal structure of the rtt mixing zone : * analyzing the evolution of @xmath0 , @xmath5 , @xmath7 and @xmath6 with time one often observes a non - universal , simulation / experiment specific behavior , especially at transient , so - called early self - similarity , times @xcite . will the relative dependence of scales be a more reliable indicator of a universal behavior than the time - dependence of the scales ? * how does the energy containing scale , @xmath5 , compare with the width of the mixing layer , @xmath0 ? this question was already addressed in @xcite . here , we will elaborate on this point . * how different are the turbulent spectra at different vertical positions in the mixing zone within a given time snapshot ? * how different are the scales and spectra corresponding to qualitatively different initial perturbations ? the material in this article is organized as follows . we start by describing our simulations , we then proceed to the definitions and subsequently the observations of the various spatial scales characterizing snapshots of the mixing zone . finally we discuss self - similarity and universality of the emerging spatio - temporal picture of the rtt . we conclude by answering the questions posed above . we consider 3d incompressible , miscible rayleigh - taylor flow in the boussinesq regime , @xmath14 where @xmath15 and @xmath16 are the diffusion and viscosity coefficients , while @xmath17 is the normalized density . the boussinesq approximation for gravity corresponds to fluids with small density contrast , @xmath18 , where @xmath19 is the atwood number . here , we restrict ourself to the case of @xmath20 . we solve equations ( [ ns]-[bm_v ] ) using the spectral element code of fischer et al @xcite designed specifically for dns of boussinesq fluids . the equations are solved in the nondimensional units , @xmath21 = ( 2ag)^{-\frac{1}{3 } } \nu^{\frac{2}{3 } } , \hspace{15 mm } [ t ] = ( 2ag)^{-\frac{2}{3 } } \nu^{\frac{1}{3}};\ ] ] the results are presented in the same units . this choice of units is based solely on the dimensional parameters in eqs . ( [ ns]-[bm_v ] ) thus reflecting free boundary conditions ( absence of any wall constraints ) . the critical wavelength and the wavelength of the linearly most unstable mode are constants in these units . the boundary conditions are periodic in the horizontal directions and no - slip in vertical direction . the initial conditions include a quiescent velocity and a slightly perturbed interface between the layers , @xmath22 , where @xmath23 $ ] is the function describing the density profile across the interface and @xmath24 is the perturbation . we use spectral elements of size @xmath25 with 12 collocation points in each direction . this is equivalent to the spectral resolution with the spacing between points @xmath26 . the size of our largest computational domain is @xmath27 physical units , or @xmath28 collocation points . we stop our simulation at , @xmath29 , when the width of the mixing layer reaches the domain size . at the end of simulation , the reynolds number reaches @xmath30 to @xmath31 depending on the initial conditions , where @xmath32 . for comparison , the largest rayleigh - taylor simulation to date @xcite was performed in a @xmath33 domain at resolution @xmath34 and reaching time @xmath35 and @xmath36 . our relatively coarse resolution might raise concerns , especially in diagnostics of small structures . nevertheless , all our results , including the spectra and the estimates for microscale @xmath6 , are in a very good agreement with @xcite , as well as with our finer - resolved ( but smaller ) simulation with @xmath34 . in all cases studied , the ( initial ) fastest growing mode is located at @xmath37 . most of the presented results were obtained in the simulations in the domain of @xmath38 physical units with initial perturbation in the form of a narrow initial spectrum , with modes @xmath39 and spectral index @xmath40 . ( here the spectral index refers to the exponent of the wavenumber , as in @xcite , and describes the shapes of the spectra . ) to investigate the influence of initial condition we also performed additional simulations in smaller domain of size @xmath41 ( in physical units ) with : ( 1 ) a narrower initial spectrum , with modes @xmath42 and spectral index 0 ( figure [ fig_cubes ] ) ; and ( 2 ) a broader initial spectrum , with modes @xmath43 and spectral index @xmath44 . the two regimes were identified in previous studies as giving distinctly different @xmath45 at transient times @xcite . according to @xcite , the first system develops a mode - merging regime and exhibits scalings with universal @xmath4 , while the second system develops in the regime of mode - competition , with @xmath4 depending on the amplitude of the initial perturbation . we observed that , in spite of the early stage differences , the additional simulations gave the same results in the turbulent ( advanced time ) regime as the main set . one important focus of our simulations / analysis is on resolving the vertical inhomogeneity of the mixing zone ( figure [ fig_color ] ) . to achieve this goal we differentiate vertical slices within a given snapshot , thus calculating various characteristics of the mixing zone such as the energy containing scale , the energy spectra and the viscous scale . we collect statistics within a given slice @xmath13 , e.g. contrasting results for the mixing zone center and its periphery . the mixing zone width is the standard characteristic used in the @xmath4-studies @xcite . according to the most recent analysis @xcite , the mixing zone width obeys the scaling , @xmath46 , where @xmath47 is an initial - conditions - dependent constant . in the simulation with narrow initial spectrum , we reproduce this scaling relatively well ; in the faster - developing simulation with a broad initial spectrum the scaling is affected by the finite domain size ( figure [ fig_mixing_width ] ) . in simulations with narrow ( solid line ) and wide ( dashed line ) initial spectra . , scaledwidth=47.0% ] the value of @xmath4 depends on the definition of the mixing zone width . following @xcite we consider the definition based on the mixing function , @xmath48 , @xmath49 where the overbar denotes averaging over the horizontal plane . we prefer integral definitions of the mixing zone width over the common definitions based on the values of @xmath50 ( for example the half - distance @xmath51 between two heights where @xmath52 and @xmath53 ) simply due to the fact that integral quantities are less sensitive to the profile of @xmath54 at the edges of the mixing layer , and consequently the quantities are less sensitive to the size of the computational domain . in the established self - similar regime unrestricted by domain boundaries , two definitions of the mixing zone width are actually within an @xmath55 systematic factor of each other . the value of the coefficient @xmath4 , determined from the slopes of the curves shown in figure [ fig_mixing_width ] at @xmath56 are @xmath57 for the narrow initial spectrum and @xmath58 for the broad initial spectrum . the obtained values of @xmath4 are in a good agreement with other simulations ( see reviews in @xcite ) . experimental values are higher , @xmath59-@xmath60 , which is usually attributed to the presence of longer wavelengths in the initial spectra , and our simulation with broader initial spectrum follows the same tendency . as we show below , in spite of the difference in @xmath4 , the two systems are very similar . further broadening of the initial spectrum would require larger a and more expensive simulation , while we do not expect the results to be significantly different . an important thesis of the phenomenology @xcite is that the internal structure of the mixing zone senses the overall time scale only adiabatically through slowly evolving large scale characteristics , of which the mixing zone width , @xmath0 , is the benchmark one . therefore , our intention is to separate the `` large scale '' question of the overall time dependence of the mixing zone width from the set of focused `` small scale '' questions about internal structure of the mixing zone . to achieve this goal , we track the dependence of the various internal characteristics of the mixing zone ( see below ) on the mixing zone width . the energy - containing scale represents the size of a typical turbulent eddy which , intuitively , corresponds to the size of the large scale vortices seen in the mixing zone snapshot , e.g. shown in figure [ fig_color ] . formally , it is convenient to define this scale , @xmath5 , as the correlation length of the normalized two - point pair correlation function of velocity , @xmath61 , where @xmath62 is one spatial component of the velocity vector , @xmath63 . we estimate @xmath5 as a half width of the correlation function , @xmath64 . defined this way , @xmath5 is consistent ( up to some @xmath65-dependent constant ) with the wavelength ( inverse of the wave vector ) where the turbulent energy spectra achieves its maximum . see e.g. figure [ fig_spectra ] . energy containing scale based on the two - point correlation function , @xmath66 energy containing scale based on the single point measurements ( see discussion in the text for details ) , and @xmath6 the viscous scale , all plotted at time @xmath67 in simulations with narrow initial spectrum as functions of the distance from the middle of the mixing layer . dashed and dotted lines correspond to scale @xmath5 computed using vertical and horizontal components of velocity respectively.,scaledwidth=47.0% ] in the developed regime , the correlation length taken at the center of the mixing zone grows linearly with the mixing layer width , @xmath68 and @xmath69 for correlations between horizontal and vertical velocities respectively for both types of initial perturbation ( figure [ fig_scales ] ) . in the limit of large @xmath5 , this suggests significant separation between the two scales , @xmath70 or more . ristorcelli and clark @xcite introduced their version of the energy containing scale as @xmath71 , and found this scale to be of the order of the width of the mixing layer , @xmath72 . note that defining @xmath66 requires a single point measurement , while @xmath5 characterizes the two - point correlations . this single - point nature of @xmath66 makes it the preferred large scale characteristic in the engineering closure modeling . our simulations show that @xmath66 is significantly larger than @xmath5 , where both @xmath66 and @xmath5 scale linearly with @xmath0 : @xmath73 . the scale separation of @xmath66 and @xmath5 may be viewed as a very favorable fact for the engineering modeling of the rtt , thus suggesting a numerical justification for the closure schemes , e.g. of the type discussed in @xcite . in our simulations @xmath74 , thus the viscous scale and the dissipation scale are equal to each other . ( this simplification reflects our desire to focus on the larger scale physics while keeping the resolution domain sufficiently large . ) we estimate the viscous scale in the middle of the mixing layer ( @xmath75 ) directly as @xmath76 . ( the `` @xmath77''-factor here is an artifact of an old tradition , see e.g. @xcite . ) in magnitude , the viscous scale agrees with @xcite and with the respective phenomenology estimate @xcite , @xmath78 . the viscous scale decreases slowly with time ( see figure [ fig_scales ] ) . however , our data are too noisy to claim anything more than rough consistency with the @xmath79 predicted in @xcite and observed in @xcite . in view of our focus on the internal structure of the mixing zone , we choose to study the relative dependence of the relevant scales . thus , figure [ fig_scales_vsz ] shows dependence of the energy - containing and viscous scales on the mixing zone width . analyzing simulations of rtt corresponding to different initial perturbations , we confirm the earlier observed @xcite sensitivity of time - evolution of the mixing zone width , scales @xmath6 and @xmath5 , and the integral quantities on initial conditions . ( see left panels in figures [ fig_scales ] and [ fig_energies ] . ) however , we also find that the same quantities re - plotted as functions of @xmath0 look very much alike ( figures [ fig_scales ] and [ fig_energies ] , right panels ) . therefore , one conclusion we draw here is that the relative scale representation is actually a better universal indicator of turbulence within the mixing zone . figure [ fig_velotemp_vsz ] shows dependencies of the mixing function and of the rms - averaged velocities across the mixing layer on the height , @xmath13 . in magnitude , the velocities are @xmath80 , as suggested by the @xmath81-dependence of the total kinetic energy : @xmath82 ( see figure [ fig_energies ] ) . the curves taken at three different times are almost indistinguishable from each other . this suggests that the mixing zone , viewed from the large - scale perspectives , is self - similar . self - similarity of the averaged density and the averaged velocities was observed in @xcite . the self - similarity itself does not define the specific form of the @xmath13-averaged profiles , only suggesting that these profiles are smooth functions of @xmath83 . in particular , self - similarity is in principle consistent with specific parabolic predictions for the size of the dominant eddy and total kinetic energy contained in the mixing layer @xcite , @xmath84 and @xmath85 , where @xmath86 and @xmath87 are respective characteristics measured in the middle of the mixing layer . notice , however , that our simulation results , shown in fig . [ fig_scales_vsz ] , suggest a much flatter profile for kinetic energy than the parabolic one . we notice that when illustrating self - similarity ( see for instance @xcite ) , it is common to rescale the mean profiles of different quantities using the values at the center of the mixing layer and plot these profiles as function of @xmath83 . here , we propose to use @xmath0-based scalings not only for the @xmath13-coordinate but also for the discussed quantity . thus , in figure [ fig_velotemp_vsz ] we rescale velocities with @xmath88 . in addition to the mixing function dependence on @xmath13 , figure [ fig_velotemp_vsz ] shows @xmath89 to be almost constant inside of the mixing layer , with a sharp drop - off near the edges . profiles of @xmath90 , sometimes called the molecular mixing fraction , were obtained experimentally and numerically in @xcite , and most of the observations agree on the fact that at the later stages of the rt instability @xmath90 remains constant across the mixing layer at approximately 0.75 - 0.8 . this suggests that a @xmath91-based definition of the mixing zone width can be advantageous because it generates a more robust scaling . the @xmath0-scaling and self - similarity of the mean profiles are also observed in the probability distribution functions ( pdfs ) as well as in the correlation functions for density and velocities computed at @xmath75 ( see figures [ fig_pdf_z0 ] and [ fig_pcf_z0 ] ) . the self - similarity is observed at sufficiently large times , @xmath92 , but is lacking at the earlier times ( not shown in the figures ) . the dynamics one sees at the earlier times for the pdf points to transition from a single peak curve to two peaks and to a single peak again . the explanation for this phenomenon is as follows . the initial , single peak distribution is dependent on the initial perturbation of the originally sharp interface . the transformation from one to two peaks corresponds to transition to the non - linear regime of the rt instability , associated with the secondary kelvin - helmholtz type shear instability and the formation of rt mushrooms . the transition from two peaks to one corresponds to the destruction of rt mushrooms and formation of the turbulent mixed zone . notice that the emergence of the earlier time transitions is consistent with results reported in @xcite for the pdf of density , where , probably , the asymptotic self - similar regime was not reached yet . a discussion of scales is rare in the existing literature , not to mention a discussion of how the scales vary across the mixing zone . this is partly due to difficulties with experimental diagnostics . notable exceptions are @xcite and @xcite , which discuss the energy containing scale within simulations and one - dimensional modeling , respectively . the model in @xcite predicts a parabolic profile for the energy containing scale , and this appears consistent with earlier time observations in @xcite and in our setting . at later stages the profile in @xcite changes to a much flatter one , which also agrees with our estimates of the energy containing scale based ( as in @xcite ) on single point measurements . moreover , we show that later in time these profiles stabilize in a nontrivial self - similar solution ( figure [ fig_scales_vsz_2 ] ) . our measurements , based on the two - point correlation functions of density and velocities as well as estimates of the dissipation scale @xmath6 , also exhibit monotonic dependence on @xmath83 . as shown in figure [ fig_scales_vsz ] , both @xmath5 and @xmath6 vary insignificantly across the mixing layer with a slight increase towards the edges . among the scales we measured , @xmath5 is the only scale which the self - similarity in time is still questionable ( figure [ fig_scales_vsz_2 ] ) . it might be related to a resolution - related systematic error in obtaining the correlation functions near zero ( figure [ fig_pcf_z0 ] ) . comparing the respective curves for different initial spectra , we find that the dependence of @xmath93 and @xmath94 on initial perturbation is weak . as expected the level of fluctuations grows with time , resulting in a monotonic shift of the turbulence spectra maxima towards larger wavelengths . the energy - containing wavelength , @xmath95 , ( corresponding to the maximum ) is in agreement with the correlation radius , @xmath5 . this applies to various spectra . for example , the maximum of the @xmath96-spectrum obtained at @xmath75 and @xmath67 , and shown in figure [ fig_spectra ] ( right ) , is located at @xmath97 , i.e. @xmath98 . the correlation radius at this time is @xmath99 . at time @xmath100 the spectrum maximum shifts to @xmath101 ( @xmath102 ) while @xmath103 . extracting the kolmogorov scaling for velocity fluctuations at different wavelengths as well as the largest wavelength cutoff ( corresponding to the inverse of the viscous scale ) from the spectral data ( e.g. figure [ fig_spectra ] ) is problematic due to the lack of spatial resolution . in this regard our simulations , as well as many others , e.g. @xcite , lack the power of the simulations performed by cook and cabot @xcite : the latter provide the only reliable confirmation ( so far ) of the expected kolmogorov features of the spectra . ( these record simulations are an les run @xcite and a dns run with @xmath33 points resolution @xcite . ) based on our simulation results , we can only state that the range of scales compatible with the kolmogorov scaling grows with time and that the viscous scale decreases with time in accordance with predictions of @xcite . we conclude by presenting a question - and - answer style summary for the observations made in the article . * _ does the relative dependence of the characteristic scales constitute a better indicator of self - similarity within the rtt than the dependence of the individual scales on time ? _ we found that the energy - containing scale , @xmath5 , and the viscous scale , @xmath6 , exhibit monotonic evolution with @xmath0 . at transient times both @xmath5 and @xmath6 demonstrate much clearer scaling with @xmath0 than with the observation time @xmath104 . * _ how does the energy containing scale , @xmath5 , compare with @xmath0 ? _ we found that at late time , the ratio of @xmath5 to @xmath0 taken at the center of the mixing zone is @xmath105 and fluctuates little with time . * _ do the turbulent spectra vary with vertical position of the mixing zone ? _ analyzing spatial correlations for a given time snapshot , we did not observe any qualitatively new features of the turbulence spectra with transition from the vertically central slice to an off - centered one within the mixing zone . * _ how different are the scales and the spectra corresponding to qualitatively different initial perturbations ? _ we found that the dependencies of @xmath5 and @xmath6 , as functions of @xmath83 , on the initial perturbations are weak . we wish to thank p. fischer for the permission to use the nekton code , a. obabko and p. fischer for the detailed help in using the code , and j.r . ristorcelli for useful comments . this work was supported by the u.s . department of energy at los alamos national laboratory under contract no . de - ac52 - 06na25396 and under grant no.b341495 to the center for astrophysical thermonuclear flashes at the university of chicago . rayleigh , lord , `` investigation of the character of the equilibrium on an incompressible heavy fluid of variable density , '' proc . of the london math * 14 * , 170 ( 1883 ) . g. taylor , taylor , sir geoffrey ingram , `` the instability of liquid surfaces when accelerated in a direction perpendicular to their planes , '' proc . of the royal soc . of london . series a , mathematical and physical sciences , * a201 * , no . 1065 , 192 - 196 ( 1950 ) . g. dimonte et al , `` a comparative study of the turbulent rayleigh - taylor instability using high - resolution three - dimensional numerical simulations : the alpha - group collaboration , '' phys . fluids * 16 * , 1668 ( 2004 ) .
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we report and discuss case study simulations of the rayleigh - taylor instability in the boussinesq , incompressible regime developed to turbulence .
our main focus is on a statistical analysis of density and velocity fluctuations inside of the already developed and growing in size mixing zone .
novel observations reported in the article concern self - similarity of the velocity and density fluctuations spectra inside of the mixing zone snapshot , independence of the spectra of the horizontal slice level , and universality showing itself in a virtual independence of the internal structure of the mixing zone , measured in the re - scaled spatial units , of the initial interface perturbations . ,
| 7,139 | 157 |
scully and co - workers @xcite have introduced a model of a carnot heat engine based on a bath comprised of partly - coherent three - level atoms ( nicknamed `` phaseonium '' ) that interact with a cavity - mode `` working fluid '' ( wf ) while they cross the cavity . their astounding conclusion was that the efficiency of such an engine may exceed the universal carnot bound @xcite because the phaseonium bath endows the cavity mode with a temperature @xmath0 that , depending on the phase @xmath1 of the atomic coherence , may surpass the corresponding temperature of thermal atoms without coherence . this research has initiated diverse suggestions of quantum resources for boosting heat - machine efficiency above the carnot bound , with a focus on non - thermal baths possessing quantum coherence @xcite . in traditional heat engines , energy exchange between the wf and the ( hot and cold ) thermal baths are entropy - changing heat transfers , whereas parametric changes of the wf hamiltonian are isentropic processes that produce or invest work @xcite . the main questions we raise here are : does the same division between heat and work necessarily hold in engines fuelled by non - thermal ( quantum - coherent ) baths and how does this division affect the engine efficiency ? to what extent is the quantum character of non - thermal baths and ( or ) the wf relevant to the engine performance ? here we address the above questions by means of the fundamental notion of _ non - passivity _ @xcite that defines the ability of a quantum state to deliver work . the maximal work extractable from a non - passive state is known as ergotropy @xcite ( [ app_ergotropy ] ) . the significance of non - passivity as a work resource has been previously demonstrated for a heat machine with a quantised piston @xcite . by resorting to this notion , we point out that there are two kinds of machines fuelled by non - thermal baths . in machines of the first kind ( exemplified by the intriguing @xcite ) the energy imparted by a non - thermal bath to the wf consists of an isentropic part that transfers ergotropy ( work ) to the wf , which has hitherto been unaccounted for , and an entropy - changing part that corresponds to heat transfer , but the total energy received by the wf can not be associated with heat . by contrast , in machines of the second kind ( exemplified by the pioneering @xcite ) the entire energy transfer from the non - thermal bath to the wf can indeed be considered as heat . a correct division of the energy transfer from the bath to the wf into heat and work is crucial for the realisation that the efficiency of machines of the first kind does not have a thermodynamic bound that may be deduced from the second law . this becomes evident when the energy of the non - thermal bath has a vanishingly small thermal component : the engine can then produce work without heat input . our analysis of these two kinds of machines is focused on an otto cycle for an harmonic - oscillator wf under the assumption that the non - thermal bath that powers the machine is unitarily generated from a thermal one . a central result of this analysis is that such non - thermal baths may produce a non - passive steady state of the wf and thereby change its ergotropy . we use this result to identify the two distinct kinds of machines powered by quantum non - thermal baths : ( i ) machines of the first kind are exemplified by setups fuelled by a squeezed thermal bath or a coherently - displaced thermal bath @xcite ) which render the wf state _ non - passive _ ( and therefore non - thermal ) . our central finding is that this kind of machine does not act as a heat engine , but rather as a _ hybrid thermo - mechanical machine energised by work as well as heat _ imparted by this bath . the thermodynamic carnot bound does not apply to the efficiency of such a machine , which is shown to operate not only as an engine , but concurrently as a heat pump / refrigerator that moves heat from the `` cold '' bath to the `` hot '' non - thermal bath , at the expense of mechanical work invested by the latter . ( ii ) machines of the second kind are obtained for wf bath interactions whereby , in contrast to machines of the first kind , the wf is rendered _ thermal _ ( i.e. , passive ) by the non - thermal bath . an engine fuelled by a phaseonium bath @xcite exemplifies this kind of machines . it is shown to act as a genuine heat engine , whose efficiency is limited by the carnot bound corresponding to the _ real _ temperature of the wf . in the case of a phaseonium bath @xcite , this temperature is @xmath0 . we analyse an otto cycle @xcite for both kinds of machines ( sections [ sec_otto ] and [ sec_thermal_otto ] ) . for machines of the first kind we then propose a modification of the otto cycle ( section [ sec_modified_otto ] ) , aimed at attaining an efficiency as high as unity , well above the otto - cycle bound , again at the expense of mechanical work provided by the non - thermal bath . the general criteria allowing us to distinguish between the two kinds of machines are analysed ( section [ sec_conditions ] ) and the role of their quantum features is discussed ( section [ sec_quantum ] ) . our conclusions ( section [ sec_conclusions ] ) are that despite their superior performance bounds compared to standard heat engines , machines powered by non - thermal baths still adhere to the traditional rules of thermodynamics , whether or not they are powered by quantum baths or exhibit quantum features . we first revisit the analysis @xcite of a four - stroke quantum otto cycle @xcite for thermal baths , wherein the wf is taken to be a quantised harmonic oscillator . in the isentropic strokes @xmath2 and @xmath3 , the wf undergoes compression and expansion , respectively , measured by the corresponding frequency ratio @xmath4 . in the isochoric strokes @xmath5 and @xmath6 , the wf is alternately coupled to an energising ( `` hot '' ) bath at temperature @xmath7 and an entropy - dump ( `` cold '' ) bath at temperature @xmath8 , respectively . at the four end points of the otto cycle in figure [ fig_quantum_otto_cycle_1 ] , the respective energies of the harmonic - oscillator wf read @xcite [ eq_wf_energies ] @xmath9 where @xmath10 . here @xmath11 and @xmath12 denote the wf s excitation number at points @xmath13 and @xmath14 , respectively . for thermal baths , @xmath15 and @xmath16 , where @xmath17 ( @xmath18 ) is the excitation number of an oscillator at temperature @xmath19 and frequency @xmath20 . these excitation numbers are not changed in the isentropic strokes @xmath2 and @xmath3 . and @xmath3 , in which the wf frequency is increased and decreased by the piston , respectively . in the isochoric ( constant @xmath21 ) strokes @xmath5 and @xmath6 the wf is in contact with a `` hot '' ( possibly non - thermal ) or `` cold '' bath , respectively . ] at the heart of the analysis is the substitution of the hot thermal bath at temperature @xmath7 ( coupled in the second stroke ) by a non - thermal bath . in order to compare the two cycles , we restrict ourselves to a non - thermal bath generated by a unitary transformation of a thermal bath state @xcite . this restriction allows us to relate the mean energy delivered by the non - thermal bath to its thermal counterpart . as noted above , the ( originally ) thermal bath causes a thermal excitation of the wf to @xmath22 , which we use as reference . namely , we parameterise @xmath23 , where @xmath24 is the additional excitation of the wf when the thermal bath is replaced in the second stroke by the non - thermal one . by contrast , the cold thermal bath in stroke @xmath6 remains unaltered , @xmath15 . from the wf energies at the four end points of the strokes forming the cycle in figure [ fig_quantum_otto_cycle_1 ] we can compute the energy transfer in each stroke @xcite , [ eq_general_energies ] @xmath25 in the first and third strokes only the classical piston drives the wf so that the time evolution of the machine is unitary , hence the energy exchanged between the wf and the piston is unambiguously in the form of work and no heat transfer is involved , @xmath26 . by contrast , for the strokes @xmath5 and @xmath6 , wherein the wf is coupled to the ( possibly non - thermal ) baths , the nature of the energy exchanges @xmath27 and @xmath28 calls for further consideration , as detailed below . under the conditions discussed in section [ sec_conditions ] , the non - thermal bath in stroke @xmath5 promotes the wf to a non - passive state , @xmath29 , expressed by a unitary transformation of a thermal state at temperature @xmath7 ( see [ app_ergotropy ] ) . this unitary transformation necessarily increases the energy of the state , so that @xmath30 , because a thermal state has the lowest energy for a given entropy @xcite . the wf energy at point @xmath14 has two distinct parts : the part proportional to @xmath22 corresponds to the wf s thermal ( passive ) energy and the part proportional to @xmath24 corresponds to its non - passive energy ( ergotropy ) , which increases from zero to @xmath31 on account of the bath . this ergotropy change corresponds to an `` internal work '' stored in the wf , i.e. , useful energy extractable by a piston @xcite . this work evaluates to @xmath32 which is the difference between the wf energy and its passive ( thermal ) counterpart @xcite . the heat transferred in the second stroke , @xmath33 is the same that a thermal bath at temperature @xmath7 would provide . hence , such a machine is powered by both _ work and heat _ from the non - thermal bath . similarly , the energy @xmath28 exchanged in the fourth stroke can be divided into @xmath34 and @xmath35 . this division of the energies @xmath27 and @xmath28 ( exchanged between the wf and the bath ) into work and heat ( for a non - passive wf state ) is one of our central results that bears important operational consequences . the efficiency of this machine , as of any engine , is the ratio of the work output to the input energy @xcite , @xmath36 which holds as long as @xmath37 , i.e. , work is extracted by the piston . substituting the energies yields the same expression as for a thermal bath @xcite , @xmath38 however , the bounds on the efficiency may be strongly modified by the wf s non - passivity . to derive these bounds , we require the work to be non - positive ( as fits an engine ) , implying @xmath39 . for thermal baths ( @xmath40 ) this condition reduces to the carnot bound , @xmath41 the carnot bound has here been obtained from the argument that an engine ( by definition ) has to provide work . this _ thermodynamic _ bound also follows from applying the first and second laws of thermodynamics to a cyclic heat engine @xcite . for @xmath30 we may parameterise the wf by an excitation parameter that has the appearance of a _ fictitious _ `` temperature '' @xmath42 , defined as @xmath43^{-1}.\ ] ] here @xmath44 only parameterises the wf s excitation , but _ not _ its state , which is _ non - passive _ and hence _ non - thermal_. the condition on work extraction then yields @xmath45 which implies an apparent violation of the carnot bound for a non - passive wf , since @xmath46 . for a non - passive wf . in the right sub - carnot ( yellow ) region ( thermo - mechanical engine ) the non - thermal bath provides heat and work . in the central super - carnot ( green ) region the wf dumps heat into the `` hot '' bath as a heat pump , but the machine still provides work . for even smaller frequency ratios ( left white area ) the machine is not an engine . ] let us examine this alleged violation ( figure [ fig_efficiency_nonthermal_1 ] ) . the `` hot '' non - thermal bath only provides heat @xmath47 as long as @xmath48 , corresponding to the `` sub - carnot '' range @xmath49 ( figure [ fig_quantum_otto_cycle_2_new]a ) . in the `` super - carnot '' range @xmath50 this bath does not provide any heat to the wf . on the contrary , in this range @xmath51 , causing the wf to dump heat into the nominally `` hot '' bath , we find that @xmath52 ( figure [ fig_quantum_otto_cycle_2_new]b ) . the hybrid machine then acts as an engine but also pumps heat @xmath53 out of the `` cold '' bath , along with work dumping @xmath54 into this bath . still , the negativity of @xmath28 in the super - carnot range can make this heat pump distinct from a refrigerator ( [ app_refrigerator ] ) . the input energy @xmath27 is then still positive , owing to the work contribution @xmath55 [ equation ] endowed by the non - thermal bath s ergotropy . the first and second laws @xcite hold in both the sub- and super - carnot ranges under the conditions [ eq_first_second_law ] @xmath56 and @xmath57 here we have again relied on our assumption that the non - thermal bath is unitarily generated from a thermal bath , hence equation represents the equilibrium version of the second law ( see [ app_second_law ] ) . we have arrived at a crucial result : as opposed to the genuine carnot bound , the efficiency bound in equation _ can not _ be derived from the laws of equilibrium thermodynamics that are expressed by equations : @xmath44 is _ not _ a temperature and hence does not appear in equation . namely , although the efficiency can only take values up to @xmath2 by the first law of thermodynamics ( energy conservation ) , the _ maximal _ efficiency for given parameters _ does not _ follow from the second - law statement for this equilibrium scenario : this second - law statement only restricts the heat transfer , but not the work imparted by the non - thermal bath , because this work is transferred via a unitary transformation of the wf state , which is an isentropic process . the bound is thus _ not _ a thermodynamic bound . we shall revisit this result below . as follows from the discussion above , since this machine is fed by mechanical work , its efficiency [ equation ] is bounded by equation , rather than by the carnot bound that only applies to heat engines ( figure [ fig_efficiency_nonthermal_1 ] ) . whereas the energy conversion efficiency of a heat engine is limited by @xmath58 , its mechanical - motor counterpart is bounded by the input work , @xmath59 . hence , machines powered by coherent ( or squeezed ) baths realise the regime of hybrid thermo - mechanical operation ( where heat and work supplied by the bath are converted into work ) that lies between the heat- and mechanical - engine regimes . the difference between a standard heat engine ( which , by definition , is energised exclusively by heat ) and this hybrid thermo - mechanical machine of the first kind becomes apparent in the extreme case @xmath60 . a machine of the first kind can then still deliver work , @xmath61 although no heat is imparted to the wf by the ( pure - state ) bath . the machine is then an _ entirely mechanical _ engine , energised by @xmath62 , which is exclusively work transfer from the zero - temperature bath , since the wf state becomes pure and non - passive ( e.g. , squeezed vacuum ) as a result of the wf bath interaction . namely , for a pure state of both the cold and the hot ( non - thermal ) baths , the evolution of the wf is unitary , so that the wf energy increase in the second stroke , wherein the wf is coupled to the non - thermal bath , is _ isentropic _ and has nothing to do with heat transfer from the bath . we may replace the non - thermal otto cycle ( figures [ fig_quantum_otto_cycle_2_new]a and [ fig_quantum_otto_cycle_2_new]b ) by an equivalent cycle ( figure [ fig_quantum_otto_cycle_2_new]c ) involving a hot thermal reservoir at temperature @xmath7 and an external work source ( which is not the piston ) . after the second stroke this external device performs a unitary transformation on the thermal wf state that promotes it to the same non - passive state it would have attained via contact with a non - thermal bath . the amount of work @xmath63 invested by this device is the same as @xmath64 , the work stored in the wf state that a non - thermal bath would have provided . this equivalent cycle demonstrates the _ hybrid thermo - mechanical _ nature of the engine , since the non - thermal bath is generated by a ( work - investing ) unitary transformation of a thermal state ( see [ app_mapping ] and reference @xcite ) . the equivalence of this cycle and the non - thermal otto cycle follows from our analysis at the beginning of this section , where we found that the heat @xmath65 [ equation ] provided by the non - thermal bath is the _ same _ as the heat that a thermal bath at temperature @xmath7 would have provided . the energy surplus imparted by the non - thermal bath was identified to be the work @xmath64 in equation . this equivalence supports our conclusion that the maximum efficiency is not a thermodynamic bound the work @xmath66 imparted by the auxiliary work reservoir is not bounded by the second law of thermodynamics , whereas the heat exchanges are . our analysis can be illustrated for a _ squeezed thermal bath _ @xcite , for which the oscillator wf evolves into a ( non - passive ) squeezed thermal state owing to work and heat imparted by the bath . the deviation of the wf s excitation number from thermal equilibrium is @xmath67 with @xmath68 denoting the squeezing parameter . at high temperature @xmath7 , @xmath69 but @xmath44 should _ not _ be mistaken for a temperature , as stressed above . an alternative is a _ coherent thermal state _ of the bath @xcite , which , in turn , yields a coherent thermal state of the wf , represented in phase space by a gaussian displaced by @xmath70 , for which @xmath71 . at high @xmath7 we then find @xmath72 the energy obtained from such a bath is clearly a combination of heat and work : the master equation for a wf in contact with this bath contains a thermalising liouvillian term and a hamiltonian term ( `` cavity pump '' ) that generates coherent displacement @xcite . . in the subsequent isentropic stroke the wf frequency is reduced from @xmath73 to @xmath74 , extracting the work @xmath75 . the figure ( like figure [ fig_quantum_otto_cycle_2_new]b ) shows the operation for @xmath76 , where the machine acts simultaneously as an engine and a refrigerator . ] potentially extractable work is lost in the otto cycle because the ergotropy stored by a non - passive wf is dumped into the `` cold '' bath in stroke @xmath6 . to avoid this loss of extractable work and make the stored ergotropy useful , we suggest to modify the otto cycle as follows : before the adiabatic stroke @xmath3 the piston will perform on the wf the inverse of the unitary transformation that rendered the wf non - thermal in stroke @xmath5 ( e.g. , the inverse of the squeezing or displacement transformations ) . this ( ideally cost - free ) operation will release the excess ergotropy of the wf and transform it back to a passive state . after this unitary transformation , the wf undergoes the same adiabatic frequency change as in the standard otto cycle . note , however , that the order of the two actions in stroke @xmath3 can be arbitrary . the work extracted by the piston in this modified stroke ( figure [ fig_quantum_otto_cycle_3 ] ) is @xmath77 where @xmath75 and @xmath78 denote work extraction after and before the transformation , respectively . the last term on the r.h.s . is the ergotropy obtained from the non - thermal bath , but with a negative sign . in the parameter range @xmath48 the modified otto cycle represents a thermo - mechanical engine that dumps heat @xmath79 into the `` cold '' bath , and equation yields the efficiency ( figure [ fig_efficiency_nonthermal_2 ] ) @xmath80 in the regime @xmath48 . parameters ( in an arbitrary energy unit ) : @xmath81 , @xmath82 , @xmath83 and @xmath84 . in the range @xmath76 the efficiency of the machine is unity . ] if , however , @xmath76 , the machine acts simultaneously as a thermo - mechanical engine and a refrigerator , i.e. , it extracts heat @xmath85 from the nominally `` cold '' bath ( [ app_refrigerator ] ) . hence , the input energy in equation is then @xmath86 , yielding the maximal efficiency @xmath87 . thus , in this regime the machine not only operates as the most efficient engine possible , but , surprisingly , also refrigerates the `` cold '' bath . the heat extraction @xmath88 from the cold bath is the same as for the _ thermal _ otto refrigerator , i.e. , the refrigerator obtained by inverting the otto cycle in figure [ fig_quantum_otto_cycle_1 ] for thermal baths . note , however , that our dual action machine operates in the regime @xmath76 , whereas a thermal refrigerator requires @xmath89 . the coefficient of performance ( cop ) of this refrigerator , following the standard definition @xcite , reads @xmath90 where @xmath91 denotes the net invested work ( see figure [ fig_quantum_otto_cycle_3 ] ) . this cop has the standard form despite an unusual feature : @xmath64 is counted here in the net invested work @xmath92 , even though @xmath64 is imparted by the bath `` for free '' . however , if the non - thermal bath is decomposed into a thermal bath and a work reservoir ( as in figure [ fig_quantum_otto_cycle_2_new ] ) , the cop regains its traditional meaning : the unitary transformation between the strokes @xmath5 and @xmath3 produces the work @xmath93 due to the ergotropy obtained by the wf in the second stroke . the machine then invests the work @xmath92 for refrigeration . since the net work produced by the engine is @xmath94 [ cf . equation ] , we obtain the work balance equation @xmath95 , which yields the denominator in equation . thus , full exploitation of the wf s non - passivity ( as in a quantum battery @xcite ) can increase the machine efficiency , and allow for simultaneous production of work and refrigeration . we have thus far considered machines of the first kind in which the wf draws both work and heat from the non - thermal reservoir , promoting it to a non - passive state . however , there are machines wherein the wf is thermalised by the non - thermal bath ( see section [ sec_conditions ] ) , so that no work is imparted by the bath , despite it being non - thermal . this means that the excitation @xmath96 in equations corresponds to a _ temperature @xmath97 of the wf @xcite ( instead of the parameter @xmath44 used for mere convenience ) . the first and second laws then read [ in contrast to equations ] [ eq_first_second_law_thermal_wf ] @xmath98 and @xmath99 where @xmath100 and @xmath101 with the energies @xmath102 being defined in equations . the notation @xmath103 for the heat in stroke @xmath104 is meant to distinguish them from @xmath105 considered in section [ sec_otto ] . consequently , in this regime the machine operates as a genuine heat engine ( see figure [ fig_quantum_otto_cycle_4 ] ) whose efficiency is restricted by the carnot bound @xmath106 corresponding to this real temperature . the ( `` original '' ) temperature @xmath7 of the bath , prior to the unitary transformation , plays no role ; the only temperatures of consequence are @xmath107 and @xmath97 . these are the temperatures that appear in the second - law expression , which , together with the first law , gives rise to the carnot bound . for a harmonic oscillator wf , this regime is realised , e.g. , for a cavity being fuelled by a phaseonium bath where @xmath108 @xcite , its @xmath109-level generalisation @xcite , or by a beam of thermal entangled atoms @xcite . note that contrary to machines of the first kind treated in section [ sec_otto ] , in machines of the second kind @xmath24 may , in principle , become negative , thereby decreasing @xmath12 compared to @xmath22 in equations and . this case is exemplified by a phaseonium bath with the wrong choice of phase @xmath1 @xcite . only if @xmath30 is @xmath110 . we have seen in section [ sec_otto ] that in the regime of a non - passive wf state the engine is rendered purely mechanical and capable of providing work for @xmath60 [ equation ] . is this behaviour changed when the wf remains thermal ? close inspection of the baths @xcite ( which are all generated by atomic beams ) that thermalise the wf to @xmath97 reveals that @xmath111 entails @xmath112 , hence no work is generated by a thermal wf in the limit of pure bath states . however , this statement must be examined for each specific bath . for instance , for a cavity - mode wf interacting with an atomic - beam bath , the cavity mode may attain a finite temperature @xmath113 even if the atoms are originally at zero temperature , i.e. , in a pure state @xcite . this is due to the fact that the atoms in the bath are removed and traced out after each interaction , thereby increasing the wf entropy . to sum up , in machines of the second kind the wf remains thermal , namely its bath - induced evolution is governed by a gibbs - preserving map . we then recover the traditional heat - engine operation @xcite . the hybrid regime realised in machines of the first kind ( sections [ sec_otto ] and [ sec_modified_otto ] ) does not arise in machines of the second kind , as no work ( ergotropy ) is exchanged with the bath when the map is gibbs - preserving . whether or not the wf thermalises depends on the possible change of the wf bath interaction hamiltonian in the interaction picture , @xmath114 , under the unitary operation @xmath115 that transforms the bath from a thermal to a non - thermal state . as detailed in [ app_conditions ] , if the master equation @xcite derived for the original @xmath114 and a thermal bath yields a thermal wf state , then also the master equation derived following the transformation @xmath115 will yield a thermal state , provided the transformed hamiltonian @xmath116 where @xmath117 is the unity operator acting on the wf , _ retains its original form _ ( apart from possible renormalisations of the wf bath coupling strengths , as , e.g. , in @xcite ) . the wf will then thermalise to some real temperature , which may differ from the original bath temperature . if , however , the transformed hamiltonian _ changes its form _ , as , e.g. , in the engines discussed in @xcite , the wf may be driven into a non - passive steady state . hence , a change in the form of @xmath114 under transformation is a necessary ( but not sufficient ) condition for a non - passive wf steady state . physically , any interaction of the wf with the bath that causes energy exchange leads to thermalisation , provided that the wf is not initially in a dark state @xcite . by contrast , parametric processes , in which the energy supplied by an undepleted pump compensates for the wf bath energy exchange ( so that no energy flows between them ) , are the key to the formation of a non - passive steady state of the wf . one such process involves bath squeezing , described by operator @xmath118 , for which @xmath119 is transformed into @xmath120 here @xmath121 labels the bath modes , @xmath122 are the coupling constants to these modes and @xmath123 and @xmath124 describe the effect of squeezing @xcite . this hamiltonian , originally written in the rotating - wave approximation , possesses , following its transformation , terms of the form @xmath125 and @xmath126 that are now resonant thanks to the ( undepleted ) pump action . these terms , as opposed to @xmath127 and @xmath128 , do not contribute to thermalisation since they do not involve net energy exchange of the wf and the bath . this feature allows for non - passive steady states of the wf . we note that although squeezed baths in general give rise to a non - passive wf state , there exists one known exception , namely the two - level wf @xcite . another parametric process involves a coherent - pumping transformation , described by the displacement operator @xmath129 @xcite . the hamiltonian is then transformed into @xmath130 where @xmath131 is the displacement amplitude of the mode @xmath132 that acts as a resonant pump term on the wf , rendering its steady state non - passive . a necessary , but not sufficient , condition for a state @xmath133 of the wf to be passive with respect to its free hamiltonian @xmath134 is that @xmath133 and @xmath134 commute @xcite , which is , e.g. , not the case for squeezed thermal states : these non - equilibrium states are non - passive , although they are centred , i.e. , fulfill @xmath135 . the ability of squeezed states to provide work has also been noted in @xcite . we have shown that the operation mode of machines powered by non - thermal baths crucially depends on the quantum state of the wf , namely , on whether it is thermal ( and thus passive ) or non - passive ( and thus non - thermal ) . however , the concept of non - passivity also exists in a purely classical context @xcite which motivates the question : to what extent is the performance of these machines truly affected by quantum features of the bath and ( or ) the wf?this question calls for the elucidation of two points : ( i ) what are the relevant criteria for the bath or wf quantumness ? ( ii ) is there a compelling link between such quantumness and the machine performance ? for the machines discussed here ( sections [ sec_otto][sec_thermal_otto ] ) the first point hinges on the quantumness of a harmonic - oscillator wf , assuming that the bath is quantum ( e.g. , it exhibits intermode entanglement ) . there is a well - established criterion whereby the harmonic - oscillator state is non - classical if its glauber sudarshan @xmath136-function is either negative or does not exist at all ( in the sense of tempered distributions ) , namely , it is even more singular than the dirac @xmath137-distribution @xcite . such non - classical states can not be described by ( semi-)classical stochastic processes . in all machines powered by non - thermal baths proposed thus far @xcite , the dynamics of the harmonic - oscillator wf is described by linear or quadratic operators . consequently , since the initial state of the cycle ( point @xmath13 in figure [ fig_quantum_otto_cycle_1 ] ) is a thermal state and therefore gaussian @xcite , its evolution , whether hamiltonian or liouvillian , conserves the gaussian character of the wf state @xcite . hence , the only types of wf states that can emerge under such interactions are ( i ) thermal states , ( ii ) coherent states , ( iii ) squeezed states and combinations thereof , e.g. , squeezed thermal states . according to the above - mentioned criterion , a displaced and ( or ) squeezed thermal state with thermal photon number @xmath138 and squeezing parameter @xmath68 is non - classical if its fluctuations are smaller than the minimum uncertainty limit ) , which amounts to @xcite @xmath139 realistic squeezing parameters of @xmath140 @xcite require @xmath141 , i.e. , a very low temperature of the `` hot '' bath . in a recent experiment , squeezing of @xmath142 has been achieved @xcite , which amounts to @xmath143 and @xmath144 . all remaining gaussian states are deemed to be `` classical '' , meaning that the evolution of the wf may be mapped onto a ( semi-)classical stochastic process @xcite , even though the bath may possess quantum properties . the possible non - classicality of the wf ( according to the above definition ) is , however , not reflected in the machine s operational principles , i.e. , the analysis of the _ standard cycle _ in section [ sec_otto ] does not discriminate between classical and non - classical states_only the wf s energy matters_. by contrast , in the _ modified cycle _ of section [ sec_modified_otto ] the _ wf state is crucial _ for the machine s operation : the additional unitary transformation must be chosen according to this state and the extracted work in the modified cycle depends explicitly on the wf s ergotropy via @xmath24 instead of the wf s excitation via @xmath96 as in the standard cycle . consequently , for work optimisation a fixed wf energy is best divided into a small thermal component @xmath138 and a large mechanical component @xmath24 that stems from ergotropy transfer and is parameterised by either @xmath145 or @xmath146 . for a squeezed thermal bath , such a division favours a non - classical wf [ equation ] , whereas for the displaced bath the wf remains classical for any choice of @xmath138 and @xmath146 . this most favourable ( optimal ) regime of the modified cycle corresponds to an almost purely mechanical operation of the machine , enabled by the ergotropy imparted by the bath to the wf . the passive ( heat ) contribution should best be chosen as small as possible . hence , only the wf state s non - passivity plays a role in the modified cycle , as it determines the hybridisation of the machine s operation mode . it is in general impossible to relate non - classicality and non - passivity : coherent thermal states are classical but non - passive , whereas squeezed thermal states are non - passive but may be either classical or non - classical . these conclusions also hold if we lift the restriction on gaussian states : for a given wf energy , the best performance of the modified machine is realised for a wf with the highest possible ergotropy allowed for this energy . this , however , clearly shows that such machines have little in common with `` heat '' engines . they defy the need for a thermodynamic cycle : at @xmath60 the modified cycle simply realises a quantum battery that is charged by the bath and discharged by the piston . the preceding discussion has dealt with the rapport between quantumness and non - passivity of the wf . however , the bath may be truly quantum - mechanical . yet , its quantumness only affects the parameters of the wf evolution . to decide whether or not the quantumness of the bath is useful , one should ask : given a certain energy to modify a thermal bath , what would be its optimal unitary transformation for obtaining either the largest possible wf energy ( for the standard cycle ) , or the largest possible ergotropy of the wf ( for the modified cycle ) after the second stroke ? only if this optimal transformation has either no classical counterpart ( for electromagnetic - field baths ) or renders the bath state coherent ( for atomic - beam baths ) , may quantumness be considered beneficial this question must be individually answered for each type of bath . to sum up , although the bath may be designed to exhibit quantum features , such as intermode entanglement , the _ operational principles _ of the thermo - mechanical machine , whether being operated in the standard cycle ( section [ sec_otto ] ) or the modified cycle ( section [ sec_modified_otto ] ) , do not rely on the non - classicality of the harmonic - oscillator wf state . extracted work and efficiency of the modified cycle , on the other hand , are optimised by maximising the non - passivity ( ergotropy ) and minimising the passive ( thermal ) energy of the harmonic - oscillator wf , but are not directly determined by its non - classicality . to conclude , we have related the unconventional performance bounds of otto machines powered by non - thermal baths to the quantum state of their harmonic - oscillator working fluid ( wf ) . in order to allow for a comparison with traditional otto engines ( which are energised by a thermal bath ) , we have assumed that the non - thermal bath is unitarily generated from a thermal one . we have dubbed as `` machines of the first kind '' those where the wf state becomes non - passive as a result of the wf bath interaction , and the machines act concurrently as engines and heat pumps or refrigerators . if we fully exploit the ergotropy they have received from the non - thermal bath , these machines may attain maximal efficiency @xmath87 . this maximal efficiency can not be solely determined by thermodynamic arguments , as the second law only limits the heat , but not the work exchanged with the non - thermal baths that has been hitherto unaccounted for . by contrast , in what we dubbed as `` machines of the second kind '' the wf is thermalised by the bath . the carnot bound applies to such machines , which are true heat engines ( as opposed to the first kind ) . their wf temperature and hence their efficiency may depend on the coherence or entanglement of the bath . a key insight is that the state of the bath is unimportant if all that one can measure are the extracted work and the efficiency : a thermal bath at a _ real _ temperature @xmath110 would yield the same extracted work as a non - thermal one at the fictitious `` temperature '' @xmath147 . however , the operation mode of the respective machines is completely different . this may be revealed by observing concurrent refrigeration and work production or by diagnosing the non - passivity of the wf state , e.g. , by homodyning or tomography . under prevalent ( quadratic ) wf bath interactions the steady state of the harmonic - oscillator wf is non - classical only if it interacts with a low - temperature squeezed bath . otherwise , the wf dynamics is described by a classical stochastic process , whose parameters , however , are determined by possible quantum features of the bath ( e.g. , intra - atomic coherence or entanglement between atoms ) . yet , non - classicality does not directly affect the machine s performance , _ only non - passivity matters_. different choices of wfs may entail different criteria for their quantumness but such choices will not change this conclusion . for example , a multipartite wf state can be non - passive , regardless of whether it is separable or entangled and hence the performance criteria in sections [ sec_otto ] and [ sec_modified_otto ] will be unaffected . this issue is an example of the subtle rapport of thermodynamic and quantum features @xcite which have prompted proposed reformulations of thermodynamics @xcite . these effects are realisable in optical or optomechanical setups @xcite . bath squeezing can be implemented in multimode cavities by highly - nonadiabatic periodic modulations of the cavity @xcite , by rapid modulation of atomic resonance frequencies @xcite , as well as by laser interactions with trapped ions @xcite . a coherently displaced bath is obtainable in a laser - driven cavity @xcite . these cavity states can also be generated by beams of entangled atoms @xcite . an interesting application of the present analysis may concern a spin bath that interacts with a harmonic - oscillator wf ( e.g. , a cavity mode ) @xcite since collective ( angular - momentum ) states of a spin bath are also amenable to squeezing @xcite . we note that the present considerations do not account for wf bath correlation and information - thermodynamic effects @xcite . nor are we concerned with bath - preparation costs @xcite . we are only concerned with the question : how to best exploit non - thermal baths as resources for machine operation ? we thank robert alicki for helpful discussions and the isf and bsf for support . d. g .- k . acknowledges the support of the center for excitonics , an energy frontier research center funded by the u.s . department of energy under award de - sc0001088 ( energy conversion process ) , and the conacyt ( non - equilibrium thermodynamics ) . ergotropy is a function of a quantum state @xmath133 and a hamiltonian @xmath134 that quantifies the maximal work extractable from this state @xcite , @xmath148 where the minimisation is over the set of all possible unitary transformations of the hilbert space . a state @xmath149 is called passive if no work can be extracted from it , i.e. , @xmath150 for all @xmath151 and therefore @xmath152 . ergotropy is thus the difference of the state s energy and the corresponding passive energy , @xmath153 hence , the internal energy of a non - passive quantum state @xmath133 can be divided into passive energy and ergotropy . when two systems are in contact , work transfer between them is associated with ergotropy change . note that while ergotropy is non - negative , work extracted from a system by a piston is negative . in equations ( section [ sec_otto ] ) we have used the feature that if the state @xmath133 is obtained by a ( entropy - conserving ) unitary transformation of a thermal state , the corresponding passive state remains this thermal state . this follows from the fact that a thermal state , which is necessarily passive @xcite , is the minimum - energy state for a given entropy @xcite . the generalisation of equations to an _ arbitrary _ non - thermal bath would amount to the replacement of @xmath22 by the excitation number @xmath154 corresponding to the passive energy of the wf at point @xmath14 of the cycle ( figure [ fig_quantum_otto_cycle_1 ] ) . yet , @xmath24 would still correspond to the non - passive energy ( ergotropy ) obtained from the mechanical work imparted by the bath via a unitary transformation . the parameterisation of the wf energy by @xmath44 [ equation ] still holds . by contrast , there is no notion of @xmath7 when the bath coupled to the wf ( in the second stroke ) is not generated by a unitary operation from a thermal bath . in such cases , the strokes @xmath5 and @xmath6 can no longer be regarded as coupling to `` hot '' and `` cold '' baths , but rather as an energising stroke and a state - resetting stroke ( that closes the cycle ) . in the quantum otto cycle powered by a non - thermal bath that renders the wf non - passive , stroke @xmath6 in the super - carnot range ( figure [ fig_quantum_otto_cycle_2_new]b ) corresponds to work dumping into the `` cold '' bath , @xmath155 , along with heat pumping from this bath , @xmath85 . the effect of such work dumping is subtle : typically , this work corresponds to a coherent excitation ( displacement ) of a bath mode @xcite that may be extracted by an appropriate piston . if this work is not extracted , it may ( in the case of a finite heat capacity ) heat up this bath and thereby counter the heat pumping out of it . by contrast , the modified otto cycle ( figure [ fig_quantum_otto_cycle_3 ] ) , where the work dumping in stroke @xmath6 is absent , gives rise to genuine `` cold''-bath refrigeration in the super - carnot range . consider what we refer to as machines of the first kind , wherein the second stroke ( which describes the interaction with a non - thermal bath ) transforms the wf into some non - passive state . according to [ app_ergotropy ] , the wf state at point @xmath14 ( figure [ fig_quantum_otto_cycle_1 ] ) can be written ( for an arbitrary non - thermal bath ) as @xmath156 , i.e. , as a unitarily - transformed passive state . consequently , the second stroke can be thought of as consisting of a stroke that drives the wf into the passive state @xmath149 ( via heat exchange and an increase of the wf s entropy , @xmath157 ) and a subsequent isentropic ergotropy - transfer stroke that is associated to work performed by the bath via the unitary transformation @xmath151 , leading to the state @xmath158 . in our study , the passive state is an _ equilibrium _ gibbs state . hence , the entropy exchange in this stroke is given by the standard ( equilibrium ) form of the second law , @xmath159 . the ( lindblad ) master equation ( me ) in the interaction picture for a bosonic wf in contact with a phase - sensitive unitarily - transformed bath ( from a thermal to a non - thermal state ) with liouvillian @xmath160 and initial condition @xmath161 , @xmath162 is unitarily equivalent @xcite to a me with liouvillian @xmath163 , i.e. , assuming a thermal state of the bath , @xmath164 here the wf state @xmath165 signifies the solution of the me with a thermal bath . when in contact with a non - thermal bath , the wf state can then be obtained by unitarily transforming ( e.g. , squeezing ) the wf state determined by equation , @xmath166 for the otto cycle depicted in figure [ fig_quantum_otto_cycle_1 ] only the steady - state solution of equation is required . hence , figures [ fig_quantum_otto_cycle_2_new]a and [ fig_quantum_otto_cycle_2_new]b correspond to obtaining the steady - state solution of equation , while figure [ fig_quantum_otto_cycle_2_new]c describes the equivalent procedure of finding the thermal steady - state solution of the me ( which is independent of the initial condition ) and unitarily transforming it subsequently according to equation . upon tracing out the bath , @xmath167 , the solution of the von neumann equation in the interaction picture involving a thermal bath yields the wf state @xmath168.\ ] ] here @xmath169 , @xmath170 being the time - ordering operator , denotes the time evolution operator induced by @xmath114 . the thermal bath state is now replaced by the non - thermal state @xmath171 so that equation now reads @xmath172.\end{gathered}\ ] ] owing to the cyclic property of the partial trace over the bath , i.e. , @xmath173 for any @xmath13 , and since @xmath174 , we can rewrite equation as @xmath175.\end{gathered}\ ] ] both equations and involve the _ same _ thermal bath state , the only possible difference being due to the transformation of the time evolution operator and thus of @xmath114 . 74ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1126/science.1078955 [ * * , ( ) ] link:\doibase 10.1063/1.1523786 [ * * , ( ) ] http://www.maik.ru/cgi-perl/search.pl?type=abstract&name=lasphys&number=3&year=3&page=375 [ * * , ( ) ] @noop _ _ , ed . ( , , ) http://stacks.iop.org/0295-5075/88/i=5/a=50003 [ * * , ( ) ] link:\doibase 10.1103/physreve.84.051122 [ * * , ( ) ] link:\doibase 10.1103/physreve.86.051105 [ * * , ( ) ] http://stacks.iop.org/0295-5075/106/i=2/a=20001 [ * * , ( ) ] link:\doibase 10.1103/physreve.89.052132 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.112.030602 [ * * , ( ) ] link:\doibase 10.1038/srep12953 [ * * , ( ) ] link:\doibase 10.1103/physreve.93.012145 [ * * , ( ) ] http://stacks.iop.org/0305-4470/12/i=5/a=007 [ * * , ( ) ] link:\doibase 10.1063/1.461951 [ * * , ( ) ] link:\doibase 10.1007/bf01614224 [ * * , ( ) ] link:\doibase 10.1007/bf01011769 [ * * , ( ) ] http://stacks.iop.org/0295-5075/67/i=4/a=565 [ * * , ( ) ] link:\doibase 10.1103/physreve.91.032119 [ * * , ( ) ] http://stacks.iop.org/0295-5075/103/i=6/a=60005 [ * * , ( ) ] link:\doibase 10.1103/physreve.90.022102 [ * * , ( ) ] link:\doibase 10.1103/physreva.40.2494 [ * * , ( ) ] link:\doibase 10.1080/09500349114552501 [ * * , ( ) ] link:\doibase 10.1103/physreva.47.4474 [ * * , ( ) ] link:\doibase 10.1103/physreve.70.046110 [ * * , ( ) ] link:\doibase 10.1103/physrevx.5.031044 [ * * , ( ) ] link:\doibase 10.1103/physreva.42.487 [ * * , ( ) ] @noop _ _ ( , , ) @noop _ _ , ed . ( , , ) link:\doibase 10.1103/revmodphys.85.553 [ * * , ( ) ] link:\doibase 10.1103/physreve.87.042123 [ * * , ( ) ] http://stacks.iop.org/1367-2630/17/i=7/a=075015 [ * * , ( ) ] link:\doibase 10.1088/1367 - 2630/17/11/115012 [ * * , ( ) ] link:\doibase 10.3390/e18070244 [ * * , ( ) ] link:\doibase 10.1063/1.1523787 [ * * , ( ) ] @noop _ _ ( , , ) @noop _ _ ( , ) @noop _ _ , ed . ( , , ) link:\doibase 10.1007/bf00943428 [ * * , ( ) ] link:\doibase 10.1063/1.524949 [ * * , ( ) ] link:\doibase 10.1088/0305 - 4470/20/12/034 [ * * , ( ) ] link:\doibase 10.1103/physrev.131.2766 [ * * , ( ) ] link:\doibase 10.1088/1464 - 4266/4/1/201 [ * * , ( ) ] @noop _ _ ( , , ) @noop _ _ , ed . ( , , ) link:\doibase 10.1088/1751 - 8113/40/28/s01 [ * * , ( ) ] link:\doibase 10.1103/physreva.81.023816 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.92.153601 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.98.070502 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.104.251102 [ * * , ( ) ] link:\doibase 10.1103/physreve.85.031102 [ * * , ( ) ] http://www.mdpi.com/1099-4300/15/6/2100 [ * * , ( ) ] http://arxiv.org/abs/1508.04128 [ ( ) ] http://arxiv.org/abs/1508.06099 [ ( ) ] link:\doibase 10.1063/1.883034 [ * * , ( ) ] http://www.nature.com/ncomms/2013/130626/ncomms3059/full/ncomms3059.html [ * * ( ) ] http://www.nature.com/ncomms/2014/140627/ncomms5185/full/ncomms5185.html [ * * ( ) ] http://www.nature.com/ncomms/2014/140627/ncomms5185/full/ncomms5185.html [ * * ( ) ] link:\doibase 10.1103/physrevx.3.031012 [ * * , ( ) ] in link:\doibase 10.1007/978 - 3 - 642 - 55312 - 7_3 [ _ _ ] , ( , ) pp . link:\doibase 10.1103/physreva.89.023849 [ * * , ( ) ] link:\doibase 10.1103/physreva.50.5301 [ * * , ( ) ] link:\doibase 10.1103/physreva.87.013841 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.93.260404 [ * * , ( ) ] link:\doibase 10.1038/nphys1821 [ * * , ( ) ] link:\doibase 10.1103/physrevx.3.041003 [ * * , ( ) ] link:\doibase 10.1103/physreva.88.022112 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.114.060602 [ * * , ( ) ] link:\doibase 10.1103/physreve.89.042134 [ * * , ( ) ] http://stacks.iop.org/1367-2630/17/i=6/a=065006 [ * * , ( ) ] link:\doibase 10.1088/1367 - 2630/17/5/055018 [ * * , ( ) ] link:\doibase 10.1103/physreve.92.042126 [ * * , ( ) ] link:\doibase 10.1088/1751 - 8113/49/14/143001 [ * * , ( ) ] link:\doibase 10.1063/pt.3.2912 [ * * , ( ) ] link:\doibase 10.1038/nphys3230 [ * * , ( ) ]
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diverse models of engines energised by quantum - coherent , hence non - thermal , baths allow the engine efficiency to transgress the standard thermodynamic carnot bound .
these transgressions call for an elucidation of the underlying mechanisms . here
we show that non - thermal baths may impart not only heat , but also mechanical work to a machine .
the carnot bound is inapplicable to such a hybrid machine .
intriguingly , it may exhibit dual action , concurrently as engine and refrigerator , with up to 100% efficiency .
we conclude that even though a machine powered by a quantum bath may exhibit an unconventional performance , it still abides by the traditional principles of thermodynamics .
| 15,257 | 167 |
given their ubiquity in nature , long chain macromolecules have been the subject of considerable study . whereas there is now a reasonably firm basis for understanding the physical properties of homopolymers@xcite , considerably less is known about the heteropolymers of biological significance . from a biologist s perspective , it is the specific properties of a particular molecule that are of interest . after all the genetic information is coded by very specific sequences of nucleic acids , which are in turn translated to the chain of amino acids forming a protein@xcite . the energy of the polymer is determined by the van der waals , hydrogen bonding , hydrophobic / hydrophilic , and coulomb interactions between its constituent amino acids . in accord to these interactions , the protein folds into a specific shape that is responsible for its activity . given the large number of monomers making up such chains , and the complexity of their interactions , finding the configuration of a particular molecule is a formidable task . by contrast , a physicist s approach is to sacrifice the specificity , in the hope of gleaning some more general information from simplified models@xcite . there are in fact a number of statistical descriptions of _ ensembles _ of molecules composed of a random linear sequence of elements with a variety of interactions that determine their final shapes@xcite . these simple models of heteropolymers are of additional interest as examples of disordered systems with connections to spin glasses @xcite , with the advantage of faster relaxation @xcite . there are a number of recent experimental studies of solutions@xcite and gels@xcite of polymers that incorporate randomly charged groups . as statistical approaches only provide general descriptions of such heteropolymers , we focus on simple models which include the essential ingredients . the overall size and shape of a polymer with charged groups is most likely controlled by the coulomb interactions that are the strongest and with the longest range . we shall consider the typical properties of a model _ polyampholyte _ ( pa)@xcite : a flexible chain in which each of the @xmath5 monomers has a fixed charge @xmath0 selected from a well defined ensemble of quenches . the polymer has a characteristic microscopic length @xmath6 ( such as range of the excluded volume interaction , or nearest neighbor distance along the chain ) . in the numerical studies we further simplify the model by considering only self avoiding walk ( saw ) configurations on a cubic lattice with lattice constant @xmath6 . the long range nature of the coulomb interactions , combined with the randomness of the charge sequence , produces effects quite distinct from systems with short range interactions . in section [ secgend ] we use the knowledge accumulated in previous studies@xcite to explore the phase diagrams of quenched pas in @xmath7 dimensions . in particular , we show that for @xmath8 , the behavior of pas is similar to that of random chains with short range interactions , while for @xmath9 the spatial conformations of a pa strongly depend on its excess charge @xmath10 . in every space dimension @xmath9 , there is a critical charge @xmath11 such that pas with @xmath12 can not form a compact state . the probability of a randomly charged pa to have such an excess charge depends on both @xmath7 and its length . in the @xmath13 limit the excess charge will always ( i.e. with probability 1 ) be `` small '' for @xmath14 and `` big '' for @xmath15 . thus investigation of the `` borderline '' three dimensional case provides valuable insight into the behavior of the system in general space dimensions . in section [ secgen ] we summarize previous results for pas in @xmath16 : analytical arguments and monte carlo ( mc ) studies indicate that the pa undergoes a transition from a dense ( `` globular '' ) to a strongly stretched configuration as @xmath1 exceeds @xmath17 . the mc simulations@xcite were performed for polymer sizes up to @xmath18 and in a wide range of temperatures . they , however , could not provide information on the energy spectrum of pas , and on very low temperature properties . in this work we undertake a complete enumeration study of pas for all possible quenches up to @xmath19 , and are thus able to present very detailed results regarding energetics and spatial conformations of short pas . the details of the enumeration procedure are explained in section [ secenum ] , while the results are described in sections [ secenspec ] and [ secshape ] . the majority of these results add further support to the predictions of mc studies , and provide some details which could not be measured by mc ( e.g. , density of states , condensation energy , and surface tension in the globular phase ) . we also find some indication that pas with small @xmath1 may undergo a phase transition between two dense states . no signs of this transition could be detected in the mc studies , because it occurs at temperatures too low for that procedure to equilibrate . it is helpful to view the problem in the more general context of a variable space dimension @xmath7 . let us consider a continuum limit in which configurations of the pa are described by a function @xmath20 . the continuous index @xmath21 is used to label the monomers along the chain , while @xmath22 is the position of the monomer in @xmath7dimensional embedding space . the corresponding probabilities of these configurations are governed by the boltzmann weights of an effective hamiltonian , @xmath23\over t } & = & { k\over2}\int dx\left({d\vec{r}\over dx}\right)^2 + { v\over2}\int dxdx'\delta^d(\vec{r}(x)-\vec{r}(x ' ) ) \nonumber\\ & & + { 1\over 2t}\int dxdx'{q(x)q(x')\over & \equiv & h_0+h_v+h_q\ .\end{aligned}\ ] ] in this equation @xmath24 represents the entropic properties of the connected chain ( ideal polymer ) , @xmath25 is the continuum description of the excluded volume interactions , while @xmath26 represents the @xmath7dimensional electrostatic energy . for each pa , there is a specific ( quenched ) function @xmath27 representing the charges along the chain . ( in this work we set @xmath28 and measure @xmath2 in energy units . ) in the simplest ensemble of quenches , each monomer takes a charge @xmath0 independent of all the others ; i.e. @xmath29 , where the overline indicates averaging over quenches . while the average charge of such pas is zero , a `` typical '' sequence has an excess charge of about @xmath30 , with @xmath31 . this statement , as well as the definition of @xmath32 , are unrelated to the embedding dimension @xmath7 . however , the importance of charge fluctuations ( both for the overall polymer , or large segments of it ) does depend on the space dimension . the electrostatic energy of the excess charge , spread over the characteristic size of an ideal polymer ( @xmath33 ) , grows as @xmath34 . this simple dimensional argument shows that for @xmath8 weak electrostatic interactions are irrelevant . ( the excluded volume effects are also irrelevant in @xmath8 . ) thus , at high temperatures the pa behaves as an ideal polymer with an entropy dominated free energy of the order of @xmath35 . however , on lowering temperature it collapses into a dense state , taking advantage of a condensation energy of the order of @xmath36 . this collapse is similar to the well known @xmath37transition of polymers with short range interactions and will be discussed later in this section . for @xmath9 , electrostatic interactions are relevant and the high temperature phase is no longer a regular self avoiding walk . at high temperatures the behavior of the polymer can be studied perturbatively . for the above ensemble of uncorrelated charges , the lowest order ( @xmath38 ) correction to the quench averaged @xmath39 vanishes@xcite . however , if we restrict the ensemble of quenches to sequences with a fixed overall excess charge of @xmath1 , there is a lowest order correction term proportional to @xmath40 . thus pas with @xmath1 less than @xmath32 contract while those with larger charges expand . this trend appears in any space dimension @xmath7 , and is indicated by the vertical line at the top of fig . it should be noted that restricting the ensemble to yield fixed @xmath1 , slightly modifies the quench averaged charge charge correlations . in particular , the two point correlation function becomes @xmath41 for @xmath42 . this small ( order of @xmath43 ) correction to the correlation function may cause a significant change in @xmath39 due to the long range nature of the coulomb interaction . the above discussion can be extended to pas with short range correlations along the sequence : if neighboring charges satisfy @xmath44@xcite where @xmath45 , with no further restrictions , then @xmath46 . the resulting ensembles continuously interpolate between the deterministic extremes of an alternating sequence ( @xmath47 ) and a uniformly charged polyelectrolyte ( @xmath48 ) . as in the case of uncorrelated charges , we can impose an additional constraint on the overall charge , resulting in correlations @xmath49 where @xmath50 . we note that the variance of @xmath1 in such a correlated sequence also becomes @xmath51 . thus the proportion of quenches with @xmath1 above or below @xmath52 is independent of @xmath53 . all the results for uncorrelated sequences remain valid if we substitute @xmath52 for @xmath32 . as @xmath54 , the behavior of the pa crosses over from that of a random sequence to the deterministic ( alternating or homogeneous ) one . however , the crossover occurs only for @xmath55 . as typical of the qualitative behavior outside this narrow interval , we concentrate on the uncorrelated case of @xmath56 . a short distance cutoff @xmath6 , such as the range of the excluded volume interaction , introduces a temperature scale @xmath57 . for @xmath8 the electrostatic interactions in random pas are effectively short ranged . previous results on a random short range interaction model@xcite ( rsrim ) in @xmath16 indicate that , as long as the positive and negative charges are approximately balanced , the polymer assumes spatial conformations where the interactions are predominantly attractive . to maximize this attraction , the chain undergoes a transition from an expanded to a collapsed ( dense ) state at a @xmath37transition . for truly short range interactions , the @xmath37transition disappears only for a rather strong charge imbalance of @xmath58 . even if as a result of the relevant coulomb interactions in @xmath9 , the high temperature phase of uncorrelated pas turns out to be compact , we can not exclude the possibility of a transition into another dense ( possibly glassy ) state when @xmath2 decreases below a critical @xmath59 . such a potential `` @xmath59transition '' , indicated by the horizontal dashed line in fig . [ figa ] , must be different from a regular collapse since the lower density phase is not a self avoiding walk . the compact phase can also be destroyed by increasing the net charge as described in the following paragraph . a dense globular pa droplet of radius @xmath60 has a surface energy of @xmath61 , where @xmath62 is the surface tension . for small @xmath1 , the surface tension keeps the pa in an approximately spherical shape . however , as shown in appendix [ secrayleigh ] , at sufficiently large @xmath1 electrostatic forces destabilize the droplet . comparing the electrostatic ( @xmath63 ) and surface energies indicates that the droplet shape is controlled by the parameter @xmath64 , where @xmath65 is the _ rayleigh charge_. for a large enough @xmath66 a spherical shape is unstable ( a charged liquid droplet disintegrates ) . the rayleigh charge in @xmath16 is proportional to @xmath67 , while for @xmath14 ( @xmath15 ) it increases faster ( slower ) than @xmath32 . the solid vertical line at the bottom of fig . [ figa ] shows the position of this instability in @xmath16 . clearly , any @xmath59transition ( if at all present ) must also terminate at @xmath11 . only a negligible fraction of random quenches in @xmath14 have @xmath1 exceeding @xmath11 , and thus a typical pa is a spherical droplet at low @xmath2 . conversely , in @xmath15 almost all pas have charges larger than @xmath11 and the dense phase does not exist . the borderline case of @xmath16 , where a finite fraction of pas have @xmath1 exceeding @xmath11 , is the most controversial : an analogy with uniformly charged polyelectrolytes@xcite suggests@xcite that the pa is fully stretched ( @xmath68 ) in this case@xcite . by contrast , a debye hckel inspired theory@xcite predicts that low@xmath2 configurations are compact . partial resolution of this contradiction comes from the observation@xcite that pas in @xmath16 are extremely sensitive to the excess charge @xmath1 . in the following section we shall briefly review the main features of three dimensional pas obtained by mc simulations@xcite . numerical simulations are performed on a discretized version of eq . ( [ continuumh ] ) . configurations of a polymer are specified by listing the position vectors @xmath69 ( @xmath70 ) of its monomers . the shape and spatial extent of the polymer are then characterized by the tensor , @xmath71 with the greek indices labeling the various components . thermal averages of the eigenvalues @xmath72 of this tensor ( sometimes referred to as moments of inertia ) are used to describe the mean size and shape ; their sum is the squared radius of gyration , @xmath73 . since we are dealing with sequences of quenched disorder , these quantities must also be averaged over different realizations of @xmath74 . in three dimensions , uniform uncharged polymers in good solvents are swollen ; their @xmath75 scaling as @xmath76 with @xmath77 as in self avoiding walks . polymers in poor solvents are `` compact '' , i.e. described by @xmath78 . in previous work@xcite we used monte carlo ( mc ) simulations ( along with analytical arguments ) to establish the following properties for pas immersed in a good solvent : \(a ) the radius of gyration strongly depends on the total excess charge @xmath1 , and is weakly influenced by other details of the random sequence . \(b ) a @xmath38expansion indicates that the size of a pa tends to decrease upon lowering temperature if @xmath1 is less than a critical charge @xmath31 , and increases otherwise . this behavior is confirmed by mc simulations . \(c ) at low temperatures , neutral polymers ( @xmath79 ) are compact in the sense that their spatial extent in any direction grows as @xmath80 , where @xmath6 is a microscopic length scale . \(d ) the low@xmath2 size of the pa exhibits a sharp dependence on its charge : @xmath75 is almost independent of @xmath1 for @xmath81 , and grows rapidly beyond this point . this increase becomes sharper as the temperature is lowered , or as the length of the chain is made longer . we interpret the low temperature results by an analogy to the behavior of a charged liquid drop . the energy ( or rather the quench averaged free energy ) of the pa is phenomenologically related to its shape by @xmath82 the first term is a condensation energy proportional to the volume ( assumed compact ) , the second term is proportional to the surface area @xmath83 ( with a surface tension @xmath84 ) , while the third term represents the long range part of the electrostatic energy due to an excess charge @xmath1 ( @xmath85 is a dimensionless constant of order unity ) . the optimal shape is obtained by minimizing the overall energy . the first term is the same for all compact shapes , while the competition between the surface and electrostatic energies is controlled by the dimensionless parameter @xmath86 here @xmath87 and @xmath88 are the radius and volume of a spherical drop of @xmath5 particles , and we have defined the _ rayleigh charge _ @xmath11 . ( see eq . ( [ eqr ] ) of the appendix for the definition of @xmath11 in a general dimension @xmath7 . ) the dimensionless parameter @xmath66 controls the shape of a charged drop : a spherical drop becomes unstable and splits into two equal droplets for @xmath89 . we argued in ref.@xcite that the quenched pa has a similar instability at the vertical line in the bottom of fig . [ figa ] : charged beyond a critical @xmath66 , the pa splits to form a _ necklace _ of blobs connected by strands . from the definition of @xmath11 it is clear that it is proportional to @xmath32 ; the dimensionless prefactor relating the two depends on @xmath84 and is estimated in this work . \(e ) while confirming the general features of fig . [ figa ] , the mc simulations provide no indication of the suggested @xmath59 transition . however , these simulations are not reliable at very low temperatures@xcite due to the slowing down of the equilibration process . all the above results were obtained by mc simulations for pas of between 16 and 128 monomers . since the mc procedure does not provide good equilibration at low @xmath2 , we could not determine the properties of the ground states ( although some conclusions were drawn from the low@xmath2 data ) . systems with coulomb interactions are particularly poorly equilibrated , even at densities of only 10% the maximal value . to remedy these difficulties , in this work we resorted to complete enumeration of all possible spatial conformations , and all possible quenched charge sequences . while such an approach enables us to obtain exact results , and detailed information not available in mc studies , the immensity of configuration space restricts the calculation to chains of at most 12 steps ( 13 monomers ) . we considered self avoiding walks ( saws ) on a simple cubic lattice of spacing @xmath6 . an @xmath90step saw has @xmath91 sites ( atoms ) , and the randomly charged polymer is defined by assigning a fixed sequence of charges ( @xmath92 ) to its monomers . the charge sequence is considered to be quenched , i.e. it remains unchanged when the spatial conformation of the walk changes . the energy of any particular configuration is given by @xmath93 , where @xmath94 is the position of the @xmath95th atom on the three dimensional lattice . thermal averages of various quantities are calculated by summing over all conformations with the boltzmann weights of this energy . the resulting average is quench specific . we then obtain quenched averages by summing over all possible realizations of the sequence , possibly with certain restrictions such as on the total excess charge @xmath96 . our calculation consists of three steps : ( a ) generating lists of all spatial conformations and quench sequences ; ( b ) using the lists to calculate various thermodynamic quantities for each quenched configuration ; ( c ) averaging of results over restricted ensembles of quenches , and analyzing the data . since the calculational procedure is extremely time consuming , we used precalculated lists of all saws , and of all possible sequences of charges along the chains . other programs then used these two lists as input . since the first list of all @xmath90step saws with @xmath97 is extremely large , we tried to reduce it by including only `` truly different '' configurations and listing their degeneracies . as the actual position of a walk in space is not important , we disregard it and only give the _ directions _ of the @xmath90 steps . as the energy of a configuration is independent of its overall orientation , we assume that the first step is taken in the @xmath98 direction . the above trivial symmetries are not included in our counting ; e.g. we assign a completely straight line the degeneracy of @xmath99 . all saws , except for the straight line , have a four fold degeneracy related to rotation around the direction of the initial step . we shall therefore assume that the first step which is not in @xmath98 direction is taken along the @xmath100 axis , and attribute a degeneracy @xmath101 to all walks which are not straight lines . every non planar walk has an additional degeneracy due to reflection in the @xmath102 plane , leading to a total degeneracy of @xmath103 . thus the list of all 12step saws consists of the directions of 11 successive steps , along with a degeneracy factor . the total number of 4,162,866 chains consists of one straight line ( @xmath99 ) , 40,616 planar saws ( @xmath101 ) , and 4,122,249 non planar saws ( @xmath103 ) . accounting for degeneracies reduces the length of the list and the time needed to calculate various quantities by almost a factor of eight . some chains possess additional symmetries ; e.g. , by inverting the sequence of steps we may get some other saw in the list . we did not take advantage of this symmetry because the distribution of quenched charges on the chain is not necessarily symmetric under interchange of its two ends . in any case , we use the end to end exchange symmetry in the listing of all possible quenches . a second input list contains all possible charge sequences ; each quench " for an @xmath90step walk has @xmath91 charges . since the energy is unchanged by reversing the signs of all charges , we considered only configurations in which the total charge is @xmath104 . this shortens the list by almost a factor of 2 . the majority of quenches are not symmetric under order reversal , i.e. the sequence does not coincide with itself when listed backwards . since we are exploring all spatial configurations , without accounting for the end reversal symmetry mentioned in the previous paragraph , the list can be reduced by almost another factor of 2 by considering only one of each pair of such quenches ( keeping track of the degeneracy ) . after accounting for both charge and sequence reversal symmetries , the list for @xmath105 ( @xmath19 ) has only 2080 entries . for any sequence , the number of computer operations required to calculate the energy of a single configuration grows as @xmath106 . the total number of saws grows as @xmath107 , where @xmath108 is the effective coordination number for saws on a cubic lattice , and @xmath109 . the number of quenches " grows as @xmath110 . these factors limit the size of chains which can be investigated to @xmath105 steps . at this @xmath90 we needed 4 weeks of cpu time on a silicon graphics r4000 workstation . an increase of @xmath90 by a single unit multiplies the calculation time by an order of magnitude . thus it is impractical to employ our procedure for chains that are much longer than 12 steps . if , instead of completely enumerating all possible charge configurations we confine ourselves to sampling a few hundred quenches , the calculations can be extended to @xmath111 , but not much further . the order in which the calculations were performed is as follows : for each saw configuration we calculated the radius of gyration , squared end to end distance , _ and _ the energies of all possible quenched charge configurations along the backbone . these energies were then used to update histogram tables ( a separate histogram of possible energies for each quench ) . due to the long range nature of the coulomb interaction , the allowed energies form almost a continuous spectrum which we discretized in units of @xmath112 . this discretization was sufficient to accurately reproduce properties of the system on the temperature scales of interest . ( for @xmath113 , we used a finer division of the histograms to verify that the discretization process does not distort calculation of such properties as the specific heat , except at extremely low temperatures . ) in addition to histograms , we also collected data about the energy , the radius of gyration , and the end to end distance at the ground state of each quench . the issue of the multiplicity of the ground state is of much interest , and hotly debated in the context of models of proteins@xcite . in the presence of coulomb interactions , due to the quasi continuous nature of the energy spectrum , the ground state is almost never degenerate ( except for the trivial degeneracies mentioned above ) . it is quite likely that , for sufficiently long polymers and specific quenches , there may be exactly degenerate ground states which are not related by symmetry operations . however , for @xmath114 such cases are extremely rare . moreover , the distance between the ground state and the second lowest energy state remains of the order of @xmath112 for all @xmath90s in our calculation , even though at higher energies the densities of the states increase very rapidly with @xmath90 . we used this data to obtain averages over quenches ( with or without a constraint on the net charge ) . it should be mentioned that creation of histograms , as well as calculation of the thermal averages , required the correct accounting of degeneracies of spatial conformations , while averages over quenches needed proper care of the sequence degeneracies . for few selected quenches we also performed a calculation of the density of states as a function of two variables , the energy and the squared radius of gyration . due to large amount of data , we could not do such detailed studies for all possible quenches . fig . [ figb ] depicts ground states of four cases of quenched charges with different excess charges @xmath1 . it provides qualitative support for the conclusions previously obtained for mc simulations : fig . [ figb]a depicts the ground state configuration of an almost neutral pa which is quite compact . the pa in fig . [ figb]b has @xmath1 slightly smaller than @xmath32 ; while the configuration is still compact we see the beginning of a stretching . figs . [ figb]c and [ figb]d show strongly stretched configurations in cases where @xmath1 exceeds @xmath32 . in the following sections we will quantify this qualitative observations . as a by product of the above procedure we also obtained similar data for the model with short range interactions : in the random short range interaction model ( rsrim ) , a quenched sequence of dimensionless charges @xmath115 is defined along the chain . the interaction energy is @xmath116 , where @xmath117 if @xmath118 , and @xmath119 , otherwise . while we shall compare and contrast several properties of short and long range models in this paper , detailed results for rsrim can be found in ref.@xcite . the additional data were gathered without a substantial increase in the total execution time of the programs . there are a few minor differences in the data collection process in the two models : ( a ) as the energies of the rsrim are naturally discretized , the resulting histograms are exact . ( b ) the ground state of most quenches is highly degenerate . this required keeping track of the degeneracy , and obtaining the @xmath39 in the ground state as an average lowest energy configurations . we begin our analysis by testing the validity of eq . ( [ edefenerg ] ) for the ground states of the polymers . obviously , the exact value of each ground state energy depends on the details of the charge sequence . however , eq . ( [ edefenerg ] ) implies that the effect of the overall charge can be ( approximately ) separated ; the remaining parts of the energy depending only weakly on the details of the sequence . the basic energy unit of our model is @xmath120 , and a useful system for comparison is the regular crystal formed by alternating charges ( the `` sodium chloride '' structure ) . the condensation energy per atom of such a crystal ( @xmath121 ) is much smaller than the interaction energy per atom between the nearest neighbors ( @xmath122 ) . this demonstrates the importance of the long range coulomb interaction : although the system is locally neutral , the ground state energy depends on an extended neighborhood . similarly , the surface tension @xmath123 of the crystal is quite small . our first observation is that , for a fixed @xmath1 , the ground state energy is quite insensitive to the details of the sequence : fig . [ figc ] depicts the ground state energies of _ all _ 2080 possible quenches for 12step ( 13atom ) chains . ( the horizontal axis represents an arbitrary numbering of the quenches . ) the energies are clearly separated into 7 bands , corresponding to excess charges of @xmath124 , 3 , 5 , @xmath125 , 13 . ( there is only one quench with @xmath126 . ) while each band has a finite width , we see that the energy of a pa can be determined rather accurately by only specifying its net charge @xmath1 ! _ this is not the case for short range interactions : _ a comparison of histograms of ground state energies between ( a ) pas and ( b ) rsrim of length @xmath127 in fig . [ figd ] clearly shows the importance of long range interactions . there is a rather clear separation of energies into ` bands ' with fixed values of @xmath1 for the pas , which is almost absent in the rsrim . of course , the finite width of each ` band ' shows that the details of the sequence can not be completely neglected , although their influence on the ground state energy is rather small . using a debye hckel approximation@xcite , wittmer _ et al_@xcite have performed a systematic study of the dependence of the free energy of neutral pas ( @xmath79 ) at high @xmath2 on the correlations between neighboring charges along the chain ( see eq . ( [ qqcorr ] ) ) . they obtain an expression which smoothly interpolates between the free energy densities of a completely random sequence ( @xmath128 , where @xmath129 is the debye screening length ) , and non - random alternating sequence ( @xmath130 ) . ( the latter model was also studied in ref.@xcite . ) these results exclude the electrostatic self interaction energy , which is infinite in the continuum model used in ref.@xcite . thus the free energy of the alternating chain is roughly 20 times smaller than the random one . while these results can not be directly extended to the ground states , we may attempt to obtain crude estimates by setting @xmath131 and @xmath132 . however , our results indicate that the ground state energy of alternating polymers ( @xmath133 ) is only smaller by about 16% than the mean condensation energy of unrestricted sequences . such inconsistency is partially explained by the fact that the alternating pa has negative mean electrostatic energy ( approximately @xmath134 per atom ) even at @xmath135 , while such an energy for a completely random pa ( averaged over all quenches ) vanishes . thus , only about 1/4 of the ground state energy of an alternating pa is its condensation energy . ( this part of the energy explicitly depends on the discreteness of the chain and is not accounted for in ref.@xcite . ) this argument brings the approximate conclusions based on ref.@xcite in better qualitative agreement with our exact enumeration results . the dependence of the quench averaged ground state energies on the length of the chain is depicted in fig . a restricted average is performed at each value of @xmath1 ( indicated next to each line ) . the scaling of the axes is motivated by the re - casting of eq . ( [ edefenerg ] ) in the form @xmath136 where @xmath137 , with a prefactor @xmath138 depending on the average shape . the small number of data points makes an accurate determination of @xmath139 ( and hence the surface tension ) rather difficult . the value used in fig . [ fige ] is @xmath140 , for which the curves with different @xmath1 extrapolate to approximately the same value , giving a condensation energy of @xmath141 . for this choice of @xmath139 the slopes of the curves with @xmath142 approximately scale as @xmath143 . the condensation energy @xmath144 is surprisingly close to that of a regular crystal ( @xmath121 ) , despite the fact that in a random chain on average one neighbor ( along the chain ) has the `` wrong '' sign ( compared to the alternating arrangement ) , costing an energy of the order of @xmath145 . this again confirms our contention that the ground state energy is determined by very extended neighborhoods of each particle . if the ground state configuration has approximately cubic or spherical shape , then @xmath146 , while for the slightly elongated objects that we obtain , @xmath138 can be somewhat larger ( @xmath147 ) . therefore , we estimate @xmath148 . the error bars indicate our uncertainty in the values of @xmath138 and @xmath139 , and disregard possible systematic errors in attempts to evaluate surface tension from such small clusters . using these numbers we estimate that the rayleigh charge of the model pa is approximately the same as @xmath32 , since @xmath149 ( the relation @xmath150 assumes pas of maximum possible density . ) from fig . [ fige ] it is not clear that the ( charge unconstrained ) average energies ( indicated by the @xmath151 symbols ) of all quenches , also extrapolate to the same condensation energy of @xmath144 . this apparent inconsistency can be understood by noting that since the quench averaged @xmath143 is equal to @xmath152 , the last term in eq . ( [ eqnscale ] ) scales as @xmath153 . thus , the linear approach ( in the variables used in fig . [ fige ] ) to asymptotic value ( as @xmath154 ) is replaced by a very small power law . such a slow decay can not be detected for the small values of @xmath5 used in our enumeration study . since our model is defined on a discrete lattice , the allowed energies are discrete . however , as the length of the chain increases the separations between the states are reduced . the density of states becomes quasi continuous and can be described by a function @xmath155 . [ figf ] depicts @xmath156 , where the overline denotes averaging over all quenches with a fixed @xmath1 . ( note that this quantity is _ not _ the quench averaged free energy as the average is performed on @xmath157 rather than on @xmath158 . ) not surprisingly , the densities of states for different @xmath1s are shifted with respect to each other . for every quench the density of states is very high near the middle of the band and decreases towards the edges . we find that almost all pas have a unique ground state ( up to trivial symmetry transformations ) . this is not the case for short range interactions@xcite and may be an important clue to the problem of protein folding . ( for ease of calculation , most studies of similar random copolymers have focused on short - range interactions , and typically find highly degenerate ground states . ) furthermore , the gaps to the second lowest energy states typically remain of order of @xmath159 ( up to the studied size of @xmath105 ) , while most interstate separations decrease with @xmath90 . in the @xmath160 limit , the density of lowest energy excitations of our model pas appears to decay faster than a power law . ( of course our lattice model does not include any vibrational modes . ) this decay manifests itself in a vanishing heat capacity in the @xmath161 limit , as depicted in fig . the solid lines represent the quench averaged heat capacities per degree of freedom @xmath162 , of pas with @xmath79 at low temperatures . ( since the energy fluctuations of a polymer depend only on changes of its shape , and are independent of its overall position and orientation , we assumed that an @xmath5atom pa has @xmath163 degrees of freedom , where @xmath164 represents subtraction of translational and rotational degrees of freedom . such a choice decreases the bias in the @xmath5dependence of @xmath162 which would appear for very small values of @xmath5 . ) the vanishing heat capacity was _ not _ observed in mc studies@xcite , where poor equilibration at low @xmath2 hinders measurement of @xmath162 . it is instructive to compare and contrast the behavior of random pas with vanishing excess charge to that of an ordered alternating sequence ; the latter is a highly atypical member of the ensemble with @xmath79 . numerical investigations of alternating charge sequences by victor and imbert@xcite show that such polymers undergo a collapse transition , similar to saws with _ short range _ attractive interactions . this is because the exact compensation in the charges of any pair of neighboring monomers leads to large scale properties determined by dipole dipole ( and faster decaying ) interactions . thus coulomb interactions are irrelevant in the high temperature phase of the alternating chain that consequently behaves as a saw . by contrast , even though we consider a sub ensemble of quenches with @xmath79 , the charge fluctuations can not be neglected in random pas and control the long distance behavior of the chain . such pas are compact at _ any _ temperature . the attractive dipole dipole interactions eventually cause the collapse of the alternating charge sequence to a compact state at temperatures below a @xmath37point . of course , the ground state of such a chain is the ordered nacl crystal discussed earlier . however , it is not clear if the state of the chain immediately below the @xmath37 temperature is the ordered crystal . another possibility is that the initial collapse is into a molten globular " ( liquid like ) state @xcite , which then crystallizes at a lower temperature . we singled out the alternating pas in our complete ensemble of quenches ; the dashed lines in fig . [ figg ] depict the heat capacity of this sequence . the presence of a phase transition manifests itself in the peak in @xmath162 at @xmath165 ( for @xmath166 ) which grows ( and slightly shifts towards higher temperatures ) as @xmath90 increases . [ figg ] shows that the average heat capacity of random pas with @xmath79 also has a peak at @xmath167 ( for @xmath166 ) . as the high temperature phase is no longer swollen ( for @xmath79 ) , there are again two possible interpretations of this heat capacity peak . one is that it represents a crossover remnant of the @xmath37 transition , with an increase in the density of the compact polymer . indeed , the peak is lower and broader than that of alternating chains . another possibility is that there is a glass " transition in which the molten globule " freezes into its ` ground state ' . the proximity of the peak temperature to the energy gap for the first excited state supports the latter conclusion . no corresponding anomaly was observed in the mc simulations@xcite . since finite size effects are extremely important in such small systems , the heat capacity peak should be regarded only as a suggestion for the presence of a @xmath59transition " . as indicated by the dashed line in fig . [ figa ] , the location of such a transition may depend on @xmath1 , disappearing at @xmath168 , consistent with other features of the phase diagram . this behavior is analogous to that of the @xmath37point in the rsrim@xcite , although in that case the limiting charge scales linearly with @xmath5 . additional , studies are needed to establish the @xmath59transition . the contour plots in fig . [ figh ] depict the number of states as a function of both @xmath39 and @xmath169 , for three sequences of @xmath127 with charges @xmath124 ( a ) , 5 ( b ) , and 11 ( c ) . at high temperatures the typical configurations correspond to the highest densities . in all three cases these configurations are located in the middle of the diagram , and behave essentially as saws . on lowering temperature the polymer seeks out states of lowest energy which are very different in the three cases . the approximately neutral chain of fig . [ figh]a assumes a very compact shape represented by the lower left corner of the contours . the presence of a specific heat peak is consistent with the shape of this contour plot . while @xmath39 increases monotonically with @xmath2 , the chains are too short to permit a quantitative test for the presence of a @xmath59point from the scaling of @xmath39 . the lowest energy contour of the chain with @xmath170 ( fig . [ figh]b ) is almost horizontal . hence , upon lowering temperature the chain will not collapse , maintaining an extended shape . thus a putative transition must disappear for larger @xmath1 . finally , the fully charged polymer in fig . [ figh]c expands from a saw to the completely stretched configurations represented by the lower right corner of the contour plot . the low temperature results from mc simulations suggest that @xmath39 of a pa strongly depends on its charge , crossing over from compact configurations at small @xmath1 to extended states for larger @xmath1 . this is qualitatively supported by the ground state shapes in fig . [ figb ] , and will be more quantitatively examined here . [ figi]a depicts the @xmath90dependence of @xmath39 for several choices of @xmath1 . the vertical axis is scaled so that compact , i.e. fixed density , structures are represented by horizontal lines . since @xmath1 is fixed , the influence of the excess charge diminishes as the length of the polymer is increased , and thus all curves must asymptotically converge to the same horizontal line . there is some indication of this in fig . [ figi]a , although the crossover is rather delayed for larger values of @xmath1 . since the unrestricted ensemble ( solid circles ) includes a large range of @xmath1s , it is not surprising that the corresponding averages are not compact . the chains are too short to extract a meaningful value for the exponent @xmath4 . nevertheless , the effective slope of @xmath171 , strongly suggests that the average over an unrestricted ensemble is not compact . by comparison , the corresponding results for the rsrim in fig . [ figi]b clearly indicate that the averages both at fixed and varying @xmath1 have similar fixed density ground states . since the quench averaged @xmath39 of the unrestricted ensemble scales differently from the sub ensembles of fixed @xmath1 , the former set must contain a non negligible portion of non compact configurations for every @xmath90 . it is natural to assume that the borderline between compact and stretched states is controlled by @xmath67 . in previous work@xcite we argued that pas undergo a transition to an expanded state when @xmath1 exceeds @xmath11 ( @xmath172 ) : the transition is more pronounced for larger @xmath90 and lower @xmath2 . in the mc simulations@xcite we were able to use long pas , but were restricted to finite , albeit small , temperatures which slightly smeared the transition in @xmath75 with increasing @xmath1 . in this study we know the exact ground states but are limited to small @xmath90s where the difference between @xmath75 of compact and stretched states is less visible . the sum of all eigenvalues of the shape tensor , @xmath39 is somewhat insensitive to an expansion since the increase in the largest eigenvalue is partially compensated by the decrease of the other two eigenvalues . a clearer view is provided by the ratios of the mean eigenvalues of the shape tensor as depicted in fig . these ratios for different @xmath90s can be collapsed after scaling the charges by @xmath32 , consistent with the mc simulations . [ figk ] depicts ( on a logarithmic scale ) the distribution of values of @xmath39 in the ground states of all quenches for @xmath111 . the distribution is peaked near the smallest possible value of @xmath39 , but has a broad ( possibly power law ) tail . if the tail falls off sufficiently slowly , it will determine the asymptotic value of the exponent @xmath4 : as @xmath90 increases the very large values of @xmath39 of the ( minority ) stretched configurations will eventually dominate the total average . we thus expect @xmath173 to increase with @xmath90 , and the value of @xmath173 extracted from the slope of the solid line on fig . [ figi]a , probably underestimates the true asymptotic value . to get further insight into the behavior for larger @xmath90 , we performed separate averages for the 80% of configurations which have the smallest @xmath39 , and for the remaining top 20% . these averages are depicted in fig . the vertical axis is again scaled so that compact structures are represented by horizontal lines . the bottom 80% indeed scale as compact chains while the top 20% , which stand for the tail of the distribution , have radii that grow with @xmath90 with an effective exponent of @xmath174 . we thus conclude that the @xmath39 of the unrestricted ensemble increases with @xmath90 at least as fast as a saw . as a byproduct of our study , since we have access to the complete set of quenches , we can find which particular sequence , restricted only by its net charge , has the lowest energy . as this is the sequence that is selected in a model in which the charges are free to change positions along the chain , we shall refer to the results as describing the ground states of _ annealed pas_. for long chains , neither the sequence , nor its spatial conformation , need to be unique . however , for the sizes considered here , we always found a single ground state , several of which are shown in fig . [ figm ] for @xmath111 and different values of @xmath1 . it appears that the optimal configurations correspond to a uniform distribution of excess charge along the backbone . in particular , for small @xmath1 the preferred arrangement is the alternating sequence which then folds into a nacl structure . previously@xcite we suggested that annealed pas expel their excess charge ( provided @xmath175 ) into highly charged `` fingers '' . as a result of such `` charge expulsion '' the spanning length of annealed pas should increase dramatically ( @xmath176 ) . however , since most of the mass remains in a compact globule , @xmath39 is not substantially modified ( as long as @xmath177 ) . the chains used in our study are too short to exhibit an increased spanning length with no change in @xmath75 . moreover , the effects of lattice discreteness are much more pronounced for annealed pas where ground states correspond to a single sequence . in the quenched case , averaging over all sequences partially smoothens out lattice effects . as partial evidence we note that plots for the charge dependence of ratios of eigenvalues of the shape tensor ( analogous to fig . [ figj ] ) exhibit better collapse with the variable @xmath178 than with @xmath179 . however , given the scatter of the few data points , the evidence for the appearance of `` fingers '' is not really any more convincing than any conclusions drawn from inspection of the ground states in fig . [ figm ] . as noted earlier , we expect the ground state of a sufficiently long annealed pa with fixed @xmath1 to be the nacl structure . to test the approach to this limit , in fig . [ fign ] we plot the energies per atom of the ground states . as in the case of quenched pas ( fig . [ fige ] ) , we check for finite size corrections proportional to the surface area . ( unlike the case of quenched pas , each point in this figure represents a _ single _ configuration . ) here we used a value of @xmath180 although the results are rather insensitive to this choice , and we estimate the accuracy of this quantity as @xmath181 . the point of intersection with the @xmath182 axis is close to the known value of the @xmath145 . furthermore , @xmath183 corresponds to a surface tension of @xmath184 which is also consistent with the known value of @xmath185 . these consistency checks add further confidence to the values of @xmath144 and @xmath84 deduced for quenched pas . this work was supported by the us israel bsf grant no . 9200026 , by the nsf through grants no . dmr9400334 ( at mit s cmse ) , dmr 9115491 ( at harvard ) , and the pyi program ( mk ) . in this appendix we discuss instabilities of charged @xmath7dimensional drops . a detailed discussion of the three dimensional case can be found in appendices b and c of ref.@xcite , which also provides other references to the subject . the energy of a charged _ conducting _ ( hyper)sphere of radius @xmath87 with charge @xmath1 is given by @xmath186 where the first term is the surface energy ( @xmath84 is the surface tension , and @xmath187 denotes the @xmath7dimensional solid angle ) , while the second term is the electrostatic energy . ( we have used units such that , in @xmath7 dimensions , the electrostatic potential at a distance @xmath188 from a charge @xmath189 is @xmath190 ; and @xmath191 in @xmath192 . ) for small @xmath1 , the sphere is stable with respect to infinitesimal shape perturbations . however , when the electrostatic and surface energies are comparable , the drop becomes unstable . to explore this instability we differentiate eq . ( [ eeq ] ) with respect to @xmath87 to find the pressure difference between the interior and the exterior of the drop as @xmath193 the pressure difference vanishes when @xmath1 equals the _ rayleigh charge _ @xmath11 , where @xmath194 for @xmath195 a ( hyper)spherical shape is unstable to small perturbations ; initially the drop becomes distorted and subsequently it disintegrates . note that @xmath196 , where @xmath88 is the volume of the drop . when applied to pas , up to a dimensionless prefactor , @xmath197 we can regard the first term in eq . ( [ eeq ] ) as setting the overall energy scale , while the shape of the drop is determined by the dimensionless ratio @xmath198 while from the above argument we conclude that the spherical shape is ( locally ) unstable for @xmath199 , even for @xmath200 , the energy of the drop can be lowered by splitting into smaller droplets . in particular , we may split away from the original drop a large number @xmath157 , of small droplets of radius @xmath201 and charge @xmath202 , and remove them to infinity . it can be directly verified that for @xmath203 , the total electrostatic energy , total surface area of the small droplets , as well as their total volume , vanishes in the @xmath204 limit . thus the energy of any charged conducting drop can be lowered to that of an uncharged drop by expelling a large number of `` dust particles '' which carry away the entire charge . ( of course this argument neglects the finite size of any particles making up the drop ! ) the globular phase of a quenched random pa is better represented by a drop of immobile charges . therefore , we next consider a drop in which the charges are _ uniformly _ distributed over the volume . the sum of the surface and electrostatic energies is now given by @xmath205 for sufficiently large @xmath1 , the drop can lower its energy by splitting into two droplets of equal size . this will occur when @xmath66 exceeds a critical value of @xmath206 which is equal to 0 , 0.293 , 0.323 , 0.322 , and @xmath207 , for @xmath192 , 3 , 4 , 5 , and @xmath208 , respectively . as the value of @xmath66 increases further , the drop splits into a larger number of droplets . by examining the energy of a system of @xmath157 equal spherical droplets , we find that the optimal number is proportional to @xmath209 . if the typical @xmath143 is proportional to @xmath5 ( as happens in unrestricted pas ) , while @xmath210 is given by eq . ( [ edefqrind ] ) , the number of droplets scales as @xmath211 . thus @xmath16 is a special dimension , above which a typical pa prefers to stay in a single globule . m. mzard , g. parisi , and m. a. virasoro , _ spin glass theory and beyond _ , world scientific , singapore ( 1987 ) . yu , a. tanaka , k. tanaka , and t. tanaka , j. chem . phys . * 97 * , 7805 ( 1992 ) ; yu x .- h . , ph . d. thesis , mit ( 1993 ) . m. annaka and t. tanaka , nature * 355 * , 430 ( 1992 ) . m. scouri , j.p . munch , s.f . candau , s. neyret , and f. candau , macromol . * 27 * , 69 ( 1994 ) . y. kantor and m. kardar , europhys . * 27 * , 643 ( 1994 ) . y. kantor and m. kardar , phys . rev . * e51 * , in press ( 1995 ) . j. wittmer , a. johner and j.f . joanny , europhys . lett . * 24 * , 263 ( 1993 ) . y. kantor and m. kardar , europhys . * 28 * , 169 ( 1994 ) . p. pfeuty , r.m . velasco , and p.g . de gennes , j. phys . ( paris ) lett . * 38 * , l5 ( 1977 ) . the coulomb potential used in refs.@xcite was slightly distorted at short distances , with different short range cutoff and intermonomer distances . therefore , there is no exact equivalence between @xmath2 of this work and that in refs.@xcite . an approximate correspondence is obtained by multiplying the @xmath2s in refs.@xcite by a factor of 2 to 3 . see , e.g. , h. frauenfelder and p.g . wolynes , physics today * 47*(2 ) , 58 ( 1994 ) . landau and e.m . lifshitz , _ statistical physics _ , part 1 , pergamon , ny ( 1981 ) . j.m . victor and j.b.imbert , europhys . lett . * 24 * , 189 ( 1993 ) . in all figures in this paper lengths are measured in units of @xmath6 , charges in units of @xmath212 , temperatures and energies in units of @xmath120 . for the rsrim , charges are dimensionless , and energies are instead measured in units of @xmath213 .
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we perform an exact enumeration study of polymers formed from a ( quenched ) random sequence of charged monomers @xmath0 .
such polymers , known as polyampholytes , are compact when completely neutral and expanded when highly charged .
our exhaustive search included all spatial conformations and quenched sequences for up to 12step ( 13site ) walks .
we investigate the behavior of the polymer as a function of its overall excess charge @xmath1 , and temperature @xmath2 . at low temperatures
there is a phase transition from compact to extended configurations when the charge exceeds @xmath3 .
there are also indications of a transition for small @xmath1 between two compact states on varying temperature .
numerical estimates are provided for the condensation energy , surface tension , and the critical exponent @xmath4 .
| 14,691 | 202 |
the fermi accelerator is a dynamical system , proposed by enrico fermi @xcite to describe cosmic ray acceleration , in which a charged particle collides with a time dependent magnetic field . with base in different applications , the original model was later modified and studied by other authors . one of them is the well known fermi - ulam model ( fum ) @xcite , in which a ball is confined between a rigid fixed wall and an oscillatory moving one . in this model the ball collides elastically with the walls and the system is described by an area preserving map . in high energies regime the phase space presents kolmogorov - arnold - moser ( kam ) islands surrounded by locally chaotic regions , which are limited by spanning curves . below the invariant spanning curve of lower energy a globally chaotic sea involves kam islands . the presence of invariant spanning curves limits the orbits in phase space impeding unlimited energy growth ( fermi acceleration ) @xcite . another model , very similar to fum , in which the ball is in a gravitational field , so - called bouncer @xcite , presents , differently from fum , the property of fermi acceleration depending on values of control parameter and initial conditions . this difference between the two models was later explained by lichtenberg et al @xcite . hybrid versions @xcite involving both fum and bouncer and versions of fum with energy dissipation @xcite were also explored . quantum models based on fum and bouncer have also been studied @xcite . the study of such systems is interesting because it allows to compare theoretical predictions with experimental results @xcite ; moreover the knowledge about how time - dependent perturbations affect the dynamics of hamiltonian systems is something that needs to be more explored . therefore it is useful to study such perturbations in simple models because they furnish insights about more complex systems ; even more , the formalism used in its characterization can be extended to the billiard class of problems @xcite . the simplified fermi - ulam model ( sfum ) @xcite is an approximation in which the position of the _ moving wall _ is considered as _ fixed _ , but it transfers momentum and energy to the particle . this geometrical change of the complete model can be neglected when the amplitude of oscillation is much smaller than the distance between the two walls and the velocity of the particle is larger than the wall maximal velocity . it is clear that the simplified versions can speed up the simulations . however the main interest in these simplified models is that they can be studied by analytical methods whose results are often compared with the numerical results of the simplified model . moreover , these analytical results are , sometimes , also useful in study of the full models . the moving wall in the fermi - ulam model represents an external force that acts as a perturbation in the system . if the oscillation amplitude of the moving wall is zero the system is integrable but as soon as this amplitude is different from zero the fermi - ulam model behaves chaotically @xcite . near the integrable to non - integrable transition , average quantities can be described by scaling functions . this kind of analysis was originally proposed in a study of the sfum , where the collision number with the moving wall is employed as independent variable @xcite . in a recent work @xcite , also based on the sfum , we proposed a similar analysis in which time is considered as independent variable and we showed that the average energy can also be described by scaling functions but with a different exponents set . moreover , we showed that the average energy _ decays _ at long times . in the present work we investigate the chaotic sea below the spanning curve of lowest energy for the full fermi - ulam model , where scaling properties of average energies are studied on variables _ time _ and _ iteration _ ( or _ collision _ ) _ number_. we show that if the time is employed as independent variable , then the exponents related to the scaling properties of fum are not the same that the ones of sfum . however , if the iteration number is employed as independent variable , then both fum and sfum present the same set of exponents . we show that the energy decay found for long time in the sfum @xcite does not exist for the fum and we justify physically the origins of the similarities and differences between the fum and sfum . we provide also some analytical results for the scaling exponents and , moreover , we show that although the exponents related to the variables _ time _ and _ collision number _ are not the same a relation between them can be established . this paper is organized as follows : in the next section the fermi - ulam model and the average quantities of interest are defined . we present also the procedures to obtain these averages when time is the independent variable . in section iii we present the results of the numerical simulations , the scaling properties of average energies as function of both time and collision number , and we determine the exponents related to these averages . in section iv analytical arguments to determine some exponents and to derive a relation between the exponents of time and collision number analyses are discussed . finally , we draw the conclusions in section v , where we also present a summary of this present work . the fermi - ulam model describes the motion of a classical particle bouncing between two rigid walls , one of which is fixed at position @xmath7 and other that is moving periodically in time whose position is given by @xmath8 . here @xmath9 is the equilibrium position , @xmath10 is the oscillation amplitude , @xmath11 is a frequency , @xmath12 is time and @xmath13 is the initial phase . in order to work with dimensionless variables we perform scale changes in length @xmath14 and in time @xmath15 . with these new variables the system has just one parameter , namely @xmath16 , and we write the position of the moving wall as @xmath17 . the particle moves freely between the walls and collides elastically with them . in this manner the fum can be described by a map @xmath18 which gives the velocity of the particle and the phase of the moving wall immediately after each collision @xcite @xmath19 here the term @xmath20 gives the fraction of velocity gained or lost in collision and @xmath21 is the time between two collisions with the moving wall , which is given by the smallest solution of @xmath22 . \label{eq2}\ ] ] the plus sign in the above equations corresponds to the situation in which the particle collides with the fixed wall before hitting the moving one ( _ indirect collisions _ ) ; the minus sign corresponds to the situation in which the particle hits successively with the moving wall ( _ direct collisions _ or _ successive collisions _ ) . let us define @xmath23 as the square velocity at time @xmath24 . we are interested in the scaling properties of the dimensionless energy @xmath25 averaged over an ensemble of @xmath26 samples that belong to the chaotic sea . such samples are characterized by initial phases @xmath13 of the moving wall randomly chosen in an interval @xmath27 . if the initial velocity @xmath28 is small enough then we can use @xmath29 . moreover we consider that the particle starts at the fixed wall position with velocity @xmath30 . thus , the time of the first collision of the particle with the moving wall is given by @xmath31 and we define a variable @xmath1 as @xmath32 . in this way the time @xmath1 starts at the first collision instant and we write the average energy as @xmath33 where @xmath34 refers to a sample . we consider also another kind of average where the square velocity is firstly averaged over the orbit of a sample as @xmath35 and then we perform the average of the energy in an ensemble of @xmath26 samples as defined below @xmath36 as the particle moves freely between the walls , the square of its velocity is constant between two impacts with the moving wall and the integral in eq . ( [ eq4 ] ) can be numerically evaluated without difficulties . we will now describe some numerical procedures used to evaluate the averages quantities on time . the dynamics of fum as described in eq . ( [ eq1 ] ) evolves in a discrete variable , namely the collision number @xmath2 . as we are interested in the evolution of the average energies as defined in eqs . ( [ eq3 ] ) and ( [ eq5 ] ) , where time @xmath1 is a continuous variable , we can use the map given by eq . ( [ eq1 ] ) to speed up the calculation process . we evaluate the averages in eqs . ( [ eq3 ] ) and ( [ eq5 ] ) at discrete , logarithmic spaced , values of time @xmath37 . we known that at @xmath38 , the first collision instant , the energy of the particle is @xmath39 and , as the particle is at the moving wall position , the next collision takes place at @xmath40 , where @xmath41 is obtained by solving eq . ( [ eq2 ] ) . to evaluate the energy @xmath42 of the first sample of the ensemble at time @xmath43 we follow the procedure : ( 1 ) if @xmath44 , the energy of the particle at @xmath45 will be @xmath46 . ( 2 ) otherwise @xmath47 and a collision occurs at time @xmath48 . then we employ eq . ( [ eq1 ] ) updating the velocity of the particle to @xmath49 and the next collision time is given by @xmath50 . it means that if case 1 is satisfied then @xmath51 was determined . otherwise we update the next collision instant and the velocity of the particle . if now @xmath44 ( case 1 ) then @xmath52 but if @xmath45 is still larger than @xmath53 , we repeat case 2 updating both @xmath54 and @xmath53 . the reasoning is basically to repeat the above procedure until @xmath44 and then to update the energy @xmath51 . we follow a similar proceeding , with appropriate time intervals , to evaluate @xmath55 . the procedure is the same for all samples of the ensemble giving @xmath56 from @xmath45 until @xmath57 . then the average energy in eq . ( [ eq3 ] ) is performed doing @xmath58/m,~\ldots,~ e(t_n)=[e_1(t_n)+e_2(t_n)+\ldots+e_m(t_n)]/m$ ] . the average energy @xmath59 in eq . ( [ eq5 ] ) can be evaluated by a similar procedure . since the map in eq . ( [ eq1 ] ) gives the velocity and phase after each collision with the moving wall , then the averages on variable @xmath2 can be defined in a more direct manner . we first consider the average velocity over the orbit generated from an initial phase @xmath13 , defined as @xmath60 . considering an ensemble with @xmath26 samples , characterized by initial conditions that belong to the chaotic sea , we define the average energy as @xmath61 in fig . [ fig1](a ) and [ fig1](b ) we show , respectively , the numerical results for the average energy @xmath62 when the initial velocity is small , or @xmath63 , and the situation in which @xmath64 . the averages defined in eqs . ( [ eq3 ] ) and ( [ eq5 ] ) were performed in an ensemble with @xmath65 samples . although @xmath64 , the energy curves shown in [ fig1](b ) are obtained from orbits that belong to the chaotic sea . in fig . [ fig1](b ) we used the same values of @xmath0 as those shown in fig . [ fig1](a ) . defined in eq . ( [ eq3 ] ) , averaged over @xmath65 samples , as function of time for ( a ) three values of @xmath0 and @xmath66 ( @xmath63 ) and ( b ) three values of @xmath0 , the same ones as those shown in ( a ) , and three values of initial velocity @xmath64.,width=264,height=170 ] defined in eq . ( [ eq3 ] ) , averaged over @xmath65 samples , as function of time for ( a ) three values of @xmath0 and @xmath66 ( @xmath63 ) and ( b ) three values of @xmath0 , the same ones as those shown in ( a ) , and three values of initial velocity @xmath64.,width=264,height=170 ] as we can see more clearly in fig . [ fig1](a ) the energy is constant until a time @xmath45 , grows up to a time @xmath67 and then reaches a stationary value for large values of time . as show in @xcite the energy in the simplified fum presents a slow decay in time . since in sfum the oscillating wall is considered fixed at position @xmath68 , the time between two collisions with the oscillating wall is @xmath69 , and successive collisions do not occur . therefore , if after a collision the particle has very low velocity , then it remains for a long time , @xmath69 , with low energy . in this way , at time @xmath1 many realizations are in such condition originating the slow decay in the average energy for @xmath70 . in fum this situation does not occur . if the particle , after a collision , losses almost all its energy , a successive collision occurs increasing the energy of the particle . therefore , in fum the average energy does not decay , but presents a saturation regime for @xmath70 . moreover , the energy for @xmath70 can be described as @xmath71 the value of exponent @xmath72 can be determined by searching for the best collapse of the energy curves in the asymptotic regime . we performed simulations for values of @xmath0 between @xmath73 and @xmath74 and we obtained the average value @xmath75 . the crossover time @xmath67 obeys the relation @xmath76 and the better fit in a plot of @xmath67 as a function of @xmath0 gives @xmath77 , as shown in fig . [ te ] . as function of parameter @xmath0 . , width=264,height=170 ] the time @xmath45 is an average time in which the second indirect collisions happen after that new collisions occur and the energy @xmath62 begins to increase . moreover , @xmath45 is related to an initial transient and it affects the behavior of @xmath62 for @xmath78 . therefore , between @xmath45 and @xmath67 we have a crossover region in which the growth exponent can not be directly evaluated . for large values of time ( @xmath79 ) , the energy @xmath62 can be written as a scaling function , namely @xmath80 here @xmath81 , @xmath82 and @xmath83 are scaling exponents and @xmath84 is the scaling factor . in the limit @xmath63 ( fig . [ fig1](a ) ) we can choose @xmath85 and write the above equation as @xmath86 . for @xmath70 this relation can be written as @xmath87 . then , from eq . ( [ eq6 ] ) we have that @xmath88 . using the simulation value @xmath75 we obtain @xmath89 . from eq . ( [ eq7 ] ) we derive the relation @xmath90 . since the simulations furnish @xmath77 , it follows that @xmath91 . [ fig2](a ) shows the rescaled energy @xmath92 as function of rescaled time @xmath93 . we can see that with these new coordinates the energy curves , originally depicted in fig . [ fig1](a ) , collapse onto a universal curve in the limit of long time , after an initial transient . we emphasize that the scaling behavior is valid only for small values of @xmath0 . as function of rescaled time @xmath93 for three values of @xmath0 in limit of long time . in ( a ) it is shown the collapse for @xmath63 ( @xmath66 ) and in ( b ) the collapse for three values of initial velocity @xmath64 . we chose @xmath85 and the exponents are @xmath91 , @xmath89 and @xmath94 . , width=264,height=170 ] as function of rescaled time @xmath93 for three values of @xmath0 in limit of long time . in ( a ) it is shown the collapse for @xmath63 ( @xmath66 ) and in ( b ) the collapse for three values of initial velocity @xmath64 . we chose @xmath85 and the exponents are @xmath91 , @xmath89 and @xmath94 . , width=264,height=170 ] the argument to obtain the @xmath83 exponent , namely @xmath95 , is presented in the next section . using this relation we obtain that @xmath94 . [ fig2](b ) shows the collapse of the curves depicted in fig . [ fig1](b ) for the appropriate chosen initial velocities . depending on initial velocity and phase , the velocity after a collision with the moving wall can be very small in such way that a direct collision occurs and gives an immediate increase in energy . this scenario is more common for small values of @xmath28 and @xmath1 and does not allow us to find a scaling description to the initial transient at small values of @xmath1 . however , this transient is important because it affects the power - law growth of the energy between @xmath45 and @xmath67 . we can use eqs . ( [ eq6 ] ) and ( [ eq7 ] ) to describe the average energy @xmath59 , defined in eq . ( [ eq5 ] ) , as a scaling function just changing @xmath96 by @xmath97 . figs . [ fig3](a ) and [ fig3](b ) present the energy @xmath59 as function of time @xmath1 for @xmath63 ( small initial velocity ) and @xmath64 , respectively . the values of @xmath0 are the same as those in fig . [ fig1](a ) . in figs . [ fig4](a ) and [ fig4](b ) it is shown the rescaled energy as function of rescaled time in limit of large @xmath1 ( @xmath79 ) . following the same reasoning employed in analysis of the average energy @xmath62 , we obtain the exponents @xmath98 , @xmath99 and @xmath100 . considering the uncertainties we observe that both averages @xmath62 and @xmath59 are described by scaling functions with basically the same exponents set . therefore , we will now use the average exponents @xmath98 , @xmath101 and @xmath102 . note that these values are different than the ones of the sfum @xcite , namely , @xmath103 , @xmath104 and @xmath105 . , defined in eq . ( [ eq5 ] ) , as function of time @xmath1 for three values of @xmath0 and ( a ) three values of initial velocity @xmath63 and ( b ) three values of velocity @xmath64 . the averages were performed with @xmath65 samples . , width=264,height=170 ] , defined in eq . ( [ eq5 ] ) , as function of time @xmath1 for three values of @xmath0 and ( a ) three values of initial velocity @xmath63 and ( b ) three values of velocity @xmath64 . the averages were performed with @xmath65 samples . , width=264,height=170 ] as function of rescaled time @xmath93 in limit of large time for three values of @xmath0 . we have that ( a ) @xmath63 and ( b ) @xmath64 . we chose @xmath85 and the exponents are @xmath98 , @xmath99 and @xmath100 . , width=264,height=170 ] as function of rescaled time @xmath93 in limit of large time for three values of @xmath0 . we have that ( a ) @xmath63 and ( b ) @xmath64 . we chose @xmath85 and the exponents are @xmath98 , @xmath99 and @xmath100 . , width=264,height=170 ] fig . [ fig5](a ) shows the average energy @xmath106 for two values of @xmath0 and different initial velocities , including the situations ( i ) @xmath63 and ( ii ) @xmath64 . and three values of initial velocity , @xmath28 , as function of collision number , @xmath2 . ( b ) the rescaled average energy as function of rescaled collision number shows that , after an initial transient , the energy curves collapse onto a universal curve.,width=264,height=170 ] and three values of initial velocity , @xmath28 , as function of collision number , @xmath2 . ( b ) the rescaled average energy as function of rescaled collision number shows that , after an initial transient , the energy curves collapse onto a universal curve.,width=264,height=170 ] now we have that @xmath107 we can determine the exponents @xmath108 and @xmath4 from the energy curves for @xmath63 . when @xmath66 , we observe in fig . [ fig5](a ) that the average energy @xmath106 presents two regimes . for @xmath109 the energy has a power - law growth and for @xmath110 the energy is constant . moreover , we obtain that @xmath111 with @xmath112 . since in the limit of large @xmath2 the energy depends only on @xmath0 , we have that @xmath113 with @xmath114 . we can choose @xmath115 and rewrite the above equation as @xmath116 . from this relation we obtain that @xmath117 . in limit @xmath110 we can write @xmath118 . therefore , we have that @xmath119 . this implies that @xmath120 and @xmath121 . we use a connection between the simplified fum and the standard map , described in the next section , to obtain the value of the exponent @xmath3 , namely , @xmath122 . when @xmath64 we observe in fig . [ fig5](a ) that the energy curves present two characteristic iteration numbers , @xmath123 and @xmath124 . we can also observe that @xmath125 for @xmath63 . therefore , we must consider two situations : @xmath126 , for small initial velocities ( @xmath63 ) , and @xmath127 , for @xmath64 . the energy curves with @xmath66 ( @xmath125 ) present only two regimes : ( 1 ) a power - law growth for @xmath128 and ( 2 ) a saturation regime for @xmath129 . on the other hand , for the energy curves with @xmath130 and @xmath131 ( @xmath132 ) shown in fig . [ fig5](a ) , we have three regimes : ( 1 ) the energy is basically constant for @xmath133 , ( 2 ) for @xmath134 the energy grows and begins to follow the curve of @xmath66 and ( 3 ) the energy curves reach a saturation regime for @xmath129 . in fig . [ fig5](b ) it is shown the rescaled energy @xmath135 as function of the rescaled interactions number @xmath136 . as we can observe , the energy curves , after a small initial transient , collapse onto a universal curve , even for @xmath64 , with the exponents @xmath121 , @xmath120 and @xmath137 . it is important to note that this set of exponents is the same , within the uncertainties , to that of the sfum @xcite . the exponents of the average energies as function of both time @xmath1 and collision number @xmath2 for the fum and its simplification are shown in table [ tab1 ] . .numerical estimations of the scaling exponents for both fum and sfum . the exponents @xmath81 , @xmath82 and @xmath83 describe the scaling properties of the average energy as function of time @xmath1 while the scaling relations of the average energy as function of collision number @xmath2 are characterized by the exponents @xmath108 , @xmath4 and @xmath3 . the uncertainties are given between parenthesis . [ tab1 ] [ cols="<,^,^,^,^ " , ] let us first derive a relation between the exponents @xmath83 ( @xmath3 ) and @xmath82 ( @xmath4 ) . we follow the same lines of leonel et al . @xcite . performing the variable change @xmath138 , where @xmath139 is a typical velocity near to the lowest spanning curve , and a linearization around @xmath139 , the fum transforms into a standard map which is described by @xmath140 and @xmath141 . here @xmath142 is an effective control parameter given by @xmath143 . note that the standard model presents a transition from local to globally stochastic behavior at @xmath144 . the values of @xmath139 in the lowest spanning curve of the fum furnish a @xmath142 with value approximately the same as @xmath145 , independent of @xmath0 . then , we use the scaled variables @xmath146 and @xmath147 to obtain @xmath148 . this implies that @xmath95 . now we present an heuristic argument to support that @xmath5 . from eq . ( [ eq1 ] ) we can write , for @xmath149 , that @xmath150 a similar equation can be obtained for @xmath151 by changing @xmath28 by @xmath152 and @xmath153 by @xmath154 . then , we replace @xmath39 by eq . ( [ jaff1 ] ) . this iteration procedure can be done for @xmath155 with arbitrary @xmath2 . if now we take the average in the ensemble of initial phases @xmath156 , we always find a sum of three kinds of terms : ( i ) @xmath157 , ( ii ) a set of terms @xmath158 and ( iii ) a set of terms @xmath159 . observe that @xmath160 and @xmath161 take values in the interval @xmath162 $ ] . since we are in the region below the first spanning curve , the maximal initial value @xmath163 must be of order of @xmath164 . therefore , for small @xmath0 , @xmath165 and @xmath2 small enough we have that @xmath166 , implying that @xmath167 . assuming that the scaling relation is valid in this limit , we obtain the relation @xmath168 , which furnishes @xmath5 . it is worth mentioning that the numerical results for @xmath64 shows that the scaling occurs in the beginning of the simulation without any transient ( see fig . ( [ fig5 ] ) . moreover , using the result @xmath169 , we also obtain that @xmath6 . finally we present an argument that relates the exponents @xmath108 and @xmath81 , the scaling dimensions of variables @xmath2 and @xmath1 , respectively . as the average energy presents a saturation regime at large values of time , we can define the average velocity @xmath170 and @xmath171 as the average time between two arbitrary collisions . we can also define @xmath172 as the average distance that the particle travels between these collisions in such way that we can write @xmath173 , where @xmath174 is the average collisions number with the moving wall that take place in the time interval @xmath171 . in terms of the rescaled variables we have that @xmath175 , or @xmath176 . as @xmath177 we have , therefore , that the exponent of collision number , @xmath108 , is related to the exponent of time , @xmath81 , by @xmath178 . this result is in good agreement with the simulations . we studied the scaling properties of the chaotic sea below the lowest energy spanning curve of fum considering average energies as function of time @xmath1 and collision number @xmath2 . in limit of large @xmath1 ( @xmath79 ) , the average energies @xmath96 and @xmath97 of fum can be described by scaling functions with exponents @xmath179 , @xmath180 and @xmath181 ( see table [ tab1 ] ) . this values are different than the ones of the sfum @xcite , namely , @xmath103 , @xmath104 and @xmath105 . the scaling descriptions of the average energies are also hold on variable @xmath2 , with exponents @xmath182 , @xmath183 and @xmath184 . these values are basically the same as those of the sfum @xcite . we employed some analytical arguments to determine the exponents @xmath5 and @xmath6 . we observe also that , in the full model , the scaling exponents related to variables @xmath185 and @xmath28 are , within the uncertainties , the same for both @xmath2 and @xmath1 analyses , or @xmath186 and @xmath187 . we can also note that the exponents related to the variables @xmath1 and @xmath2 ( @xmath81 and @xmath108 , respectively ) are note the same . however , we provide an heuristic argument that establishes a connection between these exponents by the relation @xmath178 . it is important to stress that the scaling analysis only holds for small values of the rescaled amplitude @xmath0 , close enough to the integrable ( @xmath188 ) to non - integrable ( @xmath189 ) transition . moreover , we observe that the energy of the fum is constant up to a time @xmath45 , grows to a time @xmath67 and , then , reaches a stationary value . in the sfum the average energy , differently of fum , presents a slow decay for large @xmath1 ( @xmath70 ) @xcite . this striking result is a consequence of the approximation used to define the sfum . as the oscillating wall is considered fixed in space , the time between collisions is @xmath190 and successive collisions do not occur . therefore , eventually after a collision with the moving wall , the particle has very low velocity @xmath54 and remains for a long time ( @xmath190 ) with low energy , originating a slow decay in energy for @xmath70 . in the full model it is different : if after a collision with the moving wall the particle losses almost all its energy , then a successive collision occurs increasing the energy of the particle and the decay in energy at @xmath70 is not observed . the analyses on variable @xmath2 furnish basically the same results for both fum and sfum because between one arbitrary collision and the next one we always have that @xmath191 , independently if the collisions are direct ( successive ) or indirect . summarizing , we employed scaling analyses to describe the properties of average energies as function of @xmath1 and as function of @xmath2 in regime of small amplitudes of oscillation of the moving wall , @xmath192 , for the full fermi - ulam model . we observe that i ) the scaling exponents related to the variables @xmath0 and @xmath28 are basically the same for both @xmath1 and @xmath2 analyses , namely , @xmath193 and @xmath194 . ii ) we also observe that the scaling exponents related to the variables @xmath1 and @xmath2 are not the same , @xmath179 and @xmath182 , respectively , and , by a simple analytical reasoning , we show that these exponents are connected by the relation @xmath178 . iii ) performing some analytical analyses on variable @xmath2 we derive that @xmath6 and @xmath5 . iv ) considering also the results of the simplified fermi - ulam model , @xcite and @xcite , we observe that the scaling analyses of fum and sfum on variable @xmath1 are characterized by different exponents sets , while the analyses on variable @xmath2 furnish , basically , the same exponents set for both fum and sfum . v ) we observe also that the successive collisions play an important rule when the fermi - ulam model is studied on variable @xmath1 by , mainly , preventing the energy decay for long times . vi ) in the analysis on variable @xmath2 direct ( successive ) and indirect collisions play a similar rule and , therefore , the energy curves as function of @xmath2 present , for both fum and sfum , the same behavior and they are described by a single set of scaling exponents . we thank to j.a . plascak for the careful reading of the manuscript . d.g.l . and j.k.l.s . were partially supported by conselho nacional de pesquisa ( cnpq ) . also thanks to fundao de amparo pesquisa do estado de minas gerais ( fapemig ) . + @xmath195 electronic address : dgl@fisica.ufmg.br + @xmath196 electronic address : jaff@fisica.ufmg.br + l. d. pustilnikov , trudy moskov . mat . * 34 * ( 2 ) , 1 ( 1977 ) , l. d. pustilnikov , theor . * 57 * , 1035 ( 1983 ) ; l. d. pustilnikov , sov . math . dokl . * 35(1 ) * , 88 ( 1987 ) ; l. d. pustilnikov , russ . sb . math . * 82(1 ) * , 231 ( 1995 ) .
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the chaotic sea below the lowest energy spanning curve of the complete fermi - ulam model ( fum ) is numerically investigated when the amplitude of oscillation @xmath0 of the moving wall is small .
we use scaling analysis near the integrable to non - integrable transition to describe the average energy as function of time @xmath1 and as function of iteration ( or collision ) number @xmath2 . if @xmath1 is employed as independent variable , the exponents related to the energy scaling properties of the fum are different from the ones of a well known simplification of this model ( sfum )
. however , if @xmath2 is employed as independent variable , the exponents are the same for both fum and sfum . in the collision number analysis , we present analytical arguments supporting that the exponents @xmath3 and @xmath4 related to the initial velocity and to @xmath0 are given by @xmath5 and @xmath6 .
we derive also a relation connecting the scaling exponents related to the variables time and collision number .
moreover , we show that , differently from the sfum , the average energy in the fum _ saturates _ for long times and we justify the physical origins for some differences and similarities observed between the fum and its simplification .
| 9,119 | 323 |
neutron stars are believed to form from the core collapse of massive stars and the accretion induced collapse of massive white dwarfs . if the stellar core or white dwarf is rotating , conservation of angular momentum implies that the resulting neutron star must rotate very rapidly . it has been suggested @xcite that such a rapidly rotating star may develop a non - axisymmetric dynamical instability , emitting a substantial amount of gravitational radiation which might be detectable by gravitational wave observatories such as ligo , virgo , geo and tama . rotational instabilities arise from non - axisymmetric perturbations having angular dependence @xmath4 , where @xmath5 is the azimuthal angle . the @xmath0 mode is called the bar mode , which is usually the strongest mode for stars undergoing instabilities . there are two types of instabilities . dynamical _ instability is driven by hydrodynamics and gravity , and it develops on a dynamical timescale , i.e. the timescale for a sound wave to travel across the star . a _ secular _ instability , on the other hand , is driven by viscosity or gravitational radiation reaction , and its growth time is determined by the relevant dissipative timescale . these secular timescales are usually much longer than the dynamical timescale of the system . in this paper , we focus on the dynamical instabilities resulting from the new - born neutron stars formed from accretion induced collapse ( aic ) of white dwarfs . these instabilities occur only for rapidly rotating stars . a useful parameter to characterize the rotation of a star is @xmath6 , where @xmath7 and @xmath8 are the rotational kinetic energy and gravitational potential energy respectively . it is well - known that there is a critical value @xmath9 so that a star will be dynamically unstable if its @xmath10 . for a uniform density and rigidly rotating star , the maclaurin spheroid , the critical value is determined to be @xmath11 @xcite . numerous numerical simulations using newtonian gravity show that @xmath9 remains roughly the same for differentially rotating polytropes having the same specific angular momentum distribution as the maclaurin spheroids @xcite . however , @xmath9 can take values between 0.14 to 0.27 for other angular momentum distributions @xcite ( the lower limit @xmath12 is observed only for a star having a toroidal density distribution , i.e. the maximum density occurs off the center @xcite ) . numerical simulations using full general relativity and post - newtonian approximations suggest that relativistic corrections to newtonian gravity cause @xmath9 to decrease slightly @xcite . most of the stability analyses to date have been carried out by assuming that the star rotates with an _ ad hoc _ rotation law or using simplified equations of state . the results of these analyses might not be applicable to the new - born neutron stars resulting from aic . recently , fryer , holz and hughes @xcite carried out an aic simulation using a realistic rotation law and a realistic equation of state . their pre - collapse white dwarf has an angular momentum @xmath13 . after the collapse , the neutron star has @xmath1 less than 0.06 , which is too small for the star to be dynamically unstable . however , they point out that if the pre - collapse white dwarf spins faster , the resulting neutron star could have high enough @xmath1 to trigger a dynamical instability . they also point out that a pre - collapse white dwarf could easily be spun up to rapid rotation by accretion . the spin of an accreting white dwarf before collapse depends on its initial mass , its magnetic field strength and the accretion rate , etc . @xcite . liu and lindblom @xcite ( hereafter paper i ) in a recent paper construct equilibrium models of new - born neutron stars resulting from aic based on conservation of specific angular momentum . their results show that if the pre - collapse white dwarfs are rapidly rotating , the resulting neutron stars could have @xmath1 as large as 0.26 , which is slightly smaller than the critical value @xmath9 for maclaurin spheroids . however , the specific angular momentum distributions of those neutron stars are very different from that of maclaurin spheroids . so there is no reason to believe that the traditional value @xmath14 can be applied to those models . the purpose of this paper is first to determine the critical value @xmath9 for the new - born neutron stars resulting from aic , and then estimate the signal to noise ratio and detectability of the gravitational waves emitted as a result of the instability . we do not intend to provide an accurate number for the signal to noise ratio , which requires a detailed non - linear evolution of the dynamical instability . instead , we use newtonian gravitation theory to compute the structure of new - born neutron stars . then we evolve the linearized newtonian hydrodynamical equations to study the star s stability and determine the critical value @xmath9 . relativistic effects are expected to give a correction of order @xmath15 , which is about 8% for the rapidly rotating neutron stars studied in this paper . here @xmath16 is a typical sound speed inside the star and @xmath17 is the speed of light . this paper is organized as follows . in sec . [ sec : eqm ] , we apply the method described in paper i to construct a number of equilibrium neutron star models with different values of @xmath1 . in sec . [ sec : stab ] , we study the stability of these models by adding small density and velocity perturbations to the equilibrium models . then we evolve the perturbations by solving linearized hydrodynamical equations proposed by toman et al @xcite . from the simulations , we can find out whether the star is stable , and determine the critical value @xmath9 . in sec . [ sec : gw ] , we estimate the strength and signal to noise ratio of the gravitational waves emitted by this instability . in sec . [ sec : mag ] , we estimate the effects of a magnetic field on the stability results . finally , we summarize and discuss our results in sec . [ sec : dis ] . in this section , we describe briefly how we construct new - born neutron star models from the pre - collapse white dwarfs . a more detailed description is given in paper i. we consider two types of pre - collapse white dwarfs : those made of carbon - oxygen ( c - o ) and those made of oxygen - neon - magnesium ( o - ne - mg ) . the collapse of a massive c - o white dwarf is triggered by the explosive carbon burning near the center of the star @xcite . the central density of the pre - collapse c - o white dwarf must be in the range @xmath18 in order for the collapse to result in a neutron star , rather than exploding as a type ia supernova @xcite . the collapse of a massive o - ne - mg white dwarf , on the other hand , is triggered by electron captures by @xmath19 and @xmath20 when the central density reaches @xmath21 @xcite . we construct three sequences of pre - collapse white dwarfs , with models in each sequence having different amounts of rotation . sequences i and ii correspond to c - o white dwarfs with central densities @xmath22 and @xmath23 respectively . sequence iii is for o - ne - mg white dwarfs with @xmath24 . all white dwarfs are assumed to rotate rigidly , because the timescale for a magnetic field to suppress differential rotation is much shorter than the accretion timescale ( see sec . [ sec : dis ] ) . the pre - collapse white dwarfs constructed in this section are described by the equation of state ( eos ) of a zero - temperature ideal degenerate electron gas with electrostatic corrections derived by salpeter @xcite . at high density , the pressure is dominated by the ideal degenerate fermi gas with electron fraction @xmath25 that is suitable for both c - o and o - ne - mg white dwarfs . electrostatic corrections , which depend on the white dwarf composition through the atomic number @xmath26 , contribute only a few percent to the eos for the high density white dwarfs considered here . equilibrium models are computed by hachisu s self - consistent field method @xcite , which is an iteration scheme based on the integrated euler equation for hydrostatic equilibrium : @xmath27 where @xmath28 is the rotational angular frequency of the star ; @xmath29 is a constant ; @xmath30 is the radius from the rotation axis ; @xmath31 is the specific enthalpy , which is related to the density @xmath32 and pressure @xmath33 by @xmath34 the gravitational potential @xmath35 satisfies the poisson equation @xmath36 where @xmath37 is the gravitational constant . the self - consistent field method determines the structure of the star for fixed values of two adjustable parameters . in ref . @xcite , the maximum density and axis ratio ( the ratio of polar to equatorial radii ) are the chosen parameters . however , it is more convenient to choose the central density @xmath38 and equatorial radius @xmath39 as the two parameters for the models studied here . sequence i : c - o white dwarfs with @xmath22 .properties of pre - collapse white dwarfs . here @xmath28 is the rotational angular frequency ; @xmath40 is the maximum rotational angular frequency of the white dwarf in the sequence without mass - shedding ; @xmath39 , @xmath41 , @xmath42 , @xmath43 and @xmath1 are respectively the equatorial radius , polar radius , mass , angular momentum and the ratio of rotational kinetic to gravitational potential energies . [ cols="^,^,^,^,^,^ " , ] the eigenfunctions of the most unstable bar mode for the other unstable equilibrium neutron stars are similar to those displayed above . table [ tab : omega ] summarizes the oscillation frequencies [ @xmath44 and e - folding time @xmath45 of the unstable models we have studied . the table also shows the ratio of the rotational frequency of the pre - collapse white dwarfs to the maximum frequency @xmath40 of the white dwarf in the sequence . we find that the oscillation frequencies are almost the same ( @xmath46 ) for all the cases . we do not observe any instability in our simulations for stars with @xmath47 . hence we conclude that @xmath9 is somewhere between 0.241 and 0.251 , and the pre - collapse white dwarf has to have @xmath48 in order for the collapsed star to develop a dynamical instability . in this section , we estimate the strength of the gravitational radiation emitted by neutron stars undergoing a dynamical instability . we also estimate the signal to noise ratio and discuss the detectability of these sources . the rms amplitude of a gravitational wave strain , @xmath49 , depends on the orientation of the source and its location on the detector s sky . when averaged over these angles , its value is given by @xcite @xmath50 where @xmath51 and @xmath52 are the rms amplitudes of the plus and cross polarizations of the wave respectively , and @xmath53 denotes an average over the orientation of the source and its location on the detector s sky . in the presence of perturbations , the density and velocity of fluid inside the star become @xmath54 where the perturbation functions @xmath55 and @xmath56 have angular dependence @xmath4 . the amplitude of the gravitational waves produced by time varying mass and current multipole moments can be derived from ref . the result is @xmath57 \label{eq : h0}\ ] ] where @xmath58 is the distance between the source and detector ; @xmath17 is the speed of light , and @xmath59 ^ 2 } \ ; \\ d_{lm}^{(l ) } & = & \frac{d^l}{dt^l } d_{lm } \ ; \\ s_{lm}^{(l ) } & = & \frac{d^l}{dt^l } s_{lm } \ .\end{aligned}\ ] ] for a newtonian source , the mass moments @xmath60 and current moments @xmath61 are given by @xmath62 where @xmath63 are the magnetic type vector spherical harmonics . the functions @xmath60 and @xmath61 have the property that @xmath64 and @xmath65 . hence it is sufficient to consider only positive values of @xmath66 and eq . ( [ eq : h0 ] ) becomes @xmath67 \ . \label{eq : h}\ ] ] the energy and angular momentum carried by the gravitational waves can also be derived from @xcite . the result is @xmath68 } \ ; \label{eq : edot } \\ \dot{j } & = & \sum_{l=2}^{\infty } \sum_{m=0}^l \frac{g}{c^{2l+1}}\ , 2imn_l \overline { \left [ d_{lm}^{(l)*}d_{lm}^{(l+1)}+s_{lm}^{(l)*}s_{lm}^{(l+1)}\right ] } \ , \label{eq : jdot0}\end{aligned}\ ] ] where the overline denotes time average over several periods . when a neutron star develops a dynamical instability and the bar mode ( @xmath0 ) is the only unstable mode , the values of @xmath31 , @xmath69 and @xmath70 will be dominated by the term involving @xmath71 . since the unstable bar mode has even parity under reflection about the equatorial plane , @xmath72 and the next leading term will involve @xmath73 and @xmath74 . these terms are expected to be smaller than the @xmath71 term by a factor of @xmath75 for @xmath69 and @xmath70 , and a factor of @xmath15 for @xmath31 . in our models , @xmath76 , so the contribution of higher order mass and current multipole moments are small and will be neglected . strictly speaking , the above analysis only applies when the amplitudes of the perturbations are small . when the amplitudes are large , however , the fluid motion does not separate neatly into decoupled fourier components , so all @xmath60 and @xmath61 will contribute . however , it is expected that the @xmath71 term will still be the most important term . since the detailed non - linear evolution of the dynamical instability is not known , the aim of this section is to provide an order of magnitude estimate of the gravitational radiation from these sources . hence we shall only consider the effect of the mass quadrupole moment and assume @xmath71 can be approximated by the bar - mode eigenfunctions computed in sec . [ sec : lsaresult ] . in this approximation , ( [ eq : h])([eq : jdot0 ] ) become @xmath77 where @xmath78 is the oscillation frequency of the bar mode . substituting the bar - mode eigenfunctions ( from sec . [ sec : lsaresult ] ) into eq . ( [ eq : dlm ] ) , we find that @xmath79 for all the unstable models we have studied . here @xmath80 is the amplitude of the bar mode defined in eq . ( [ def : alpha ] ) . the mass quadrupole moment @xmath71 has a time dependence @xmath81 , where @xmath82 is the angular frequency of the mode . hence the time derivative @xmath83 and we obtain @xmath84 the signal to noise ratio of these sources depends on the detailed evolution of the bar mode when the density perturbation reaches a large amplitude and non - linear effects take over . recently , new , centrella and tohline @xcite and brown @xcite perform long - duration simulations of the bar - mode instability . they find that the mode saturates when the density perturbation is comparable to the equilibrium density , and the mode pattern persists , giving a long - lived gravitational wave signal . here we assume that this is the case , and that the mode dies out only after a substantial amount of angular momentum is removed from the system by gravitational radiation . we then follow the method described in refs . @xcite to estimate the signal to noise ratio . in the stationary phrase approximation , the gravitational wave in the frequency domain @xmath85 is related to @xmath49 by @xmath86 combining eqs . ( [ eq : h2 ] ) , ( [ eq : jdot ] ) and ( [ eq : stapp ] ) , we obtain @xmath87 the signal to noise ratio is given by @xmath88 where @xmath89 is the spectral density of the detector s noise . if we assume that the oscillation frequency remains constant in the entire evolution , we obtain @xcite @xmath90 where @xmath91 is the total amount of angular momentum emitted by gravitational waves . to estimate @xmath91 , we assume that the mode dies out when the angular momentum of the star decreases to @xmath92 , which is the angular momentum of the marginally bar - unstable star . then we have @xmath93 for all the unstable stars , and the signal to noise ratio for ligo - ii broad - band interferometers @xcite is @xmath94 the timescale of the gravitational wave emission can be estimated by the equation @xmath95 where @xmath96 is the amplitude @xmath80 of the density perturbation at which the mode saturates . we have used eqs . ( [ eq : jdot ] ) and ( [ res : d22 ] ) to calculate the numerical value in the last equation . the detectability of this type of sources also depends on the event rate . the event rate for the aic in a galaxy is estimated to be between @xmath97 and @xmath98 per year @xcite . of all the aic events , only those corresponding to the collapse of rapidly rotating o - ne - mg white dwarfs can end up in the bar - mode instability , and the fraction of which is unknown . if a signal to noise ratio of 5 is required to detect the source , an event rate of at least @xmath3/galaxy / year is required for such a source to occur at a detectable distance per year . hence these sources will not be promising for ligo ii if the event rate is much less than @xmath3 per year per galaxy . the event rate of the core collapse of massive stars is much higher than that of the aic . the structure of a pre - supernova core is very similar to that of a pre - collapse white dwarf , so our results might be applicable to the neutron stars produced by the core collapse . if the core is rapidly rotating , the resulting neutron star might be able to develop a bar - mode instability . if a significant fraction of the pre - supernova cores are rapidly rotating , the chance of detecting the gravitational radiation from the bar - mode instability might be much higher than expected . as mentioned in sec . [ sec : ns ] , a new - born hot proto - neutron star is dynamically stable because its @xmath1 is too small . it takes about 20 s for the proto - neutron star to cool down and evolve into a cold neutron star , which may have high enough @xmath1 to trigger a dynamical instability . the proto - neutron stars , as well as the cold neutron stars computed in sec . [ sec : ns ] , show strong differential rotation ( paper i ) . this differential rotation will cause a frozen - in magnetic field to wind up , creating strong toroidal fields . this process will result in a re - distribution of angular momentum and destroy the differential rotation . if the timescale of this magnetic braking is shorter than the cooling timescale , the star may not be able to develop the dynamical instability discussed in secs . [ sec : stab ] and [ sec : gw ] . in this section , we estimate the timescale of this magnetic braking . in the ideal magnetohydrodynamics limit , the magnetic field lines are frozen into the moving fluid . the evolution of magnetic field @xmath99 is governed by the induction equation @xmath100 in our equilibrium models , @xmath101 . hence @xmath102 and eq . ( [ eq : bind ] ) becomes @xmath103 where @xmath104 is the time derivative in the fluid s co - moving frame . ( [ eq : bind2 ] ) can be integrated analytically ( see e.g. appendix b of @xcite ) . the magnetic field @xmath105 at the position @xmath106 of a fluid element at time @xmath107 is related to the magnetic field @xmath108 at the position @xmath109 of the same fluid at time @xmath110 by @xmath111 where @xmath112 is the coordinate strain between @xmath110 and @xmath107 . with @xmath113 , it is easy to show that the induced magnetic field has components only in the @xmath114 direction . its magnitude @xmath115 , after a time @xmath107 , is easy to compute from eq . ( [ eq : bevol ] ) . the result is @xmath116 where @xmath117 is the component of magnetic field in the @xmath118 direction . the induced magnetic field will significantly change the equilibrium velocity field when the energy density of magnetic field @xmath119 is comparable to the rotational kinetic energy density @xmath120 . this will occur in a timescale @xmath121 set by @xmath122 . using eq . ( [ eq : bi ] ) , we obtain @xmath123 where @xmath124 is the length scale of differential rotation , and @xmath125 is the speed of alfvn waves . observational data suggest that the magnetic fields of most white dwarfs are smaller than @xmath126 , although a small fraction of `` magnetic white dwarfs '' can have fields in the range @xmath127 . assuming flux conservation , the magnetic fields of the hot proto - neutron stars just after collapse would be @xmath128 for those @xmath126 white dwarfs . using the angular velocity distribution in paper i for the hot proto - neutron star , we find that the magnetic timescale in the dynamically important region ( @xmath129 ) is @xmath130 which is much longer than the neutrino cooling timescale ( @xmath131 ) . hence the angular momentum transport caused by the magnetic field is negligible during the cooling period . the magnetic timescale for the cold neutron stars can be calculated from the angular frequency distribution computed in sec . [ sec : ns ] . we find that @xmath121 for the cold models is about half of that given by eq . ( [ res : taub ] ) , which is still much longer than the timescale of gravitational waves @xmath132 calculated in the previous section . the instability results presented in the previous two sections remain unchanged unless the neutron star s initial magnetic field @xmath117 is greater than @xmath133 . in that case , a detailed magnetohydrodynamical simulation has to be carried out to compute the angular momentum transport . the magnetic timescale for these nascent neutron stars is significantly different from that estimated by baumgarte , shapiro and shibata @xcite and shapiro @xcite . they consider differentially rotating `` hypermassive '' neutron stars , which could be the remnants of the coalescence of binary neutron stars . those neutron stars are very massive ( @xmath134 ) and have much higher densities than the new - born neutron stars studied in this paper . they also use a seed magnetic field of strength @xmath135 , which is much larger than our estimate . these two differences combined make our magnetic braking timescale two orders of magnitude larger than theirs . it should be noted that it is the magnetic field just after the collapse that is relevant to our analysis here . the strong differential rotation of the neutron star will eventually generate a very strong toroidal field ( @xmath136 ) and destroy the differential rotation . the final state of the neutron star will be in rigid rotation , and its magnetic field will be completely different from the initial field . for this reason , the field strength @xmath137 observed in a typical pulsar is probably not relevant here . we have applied linear stability analysis to study the dynamical stability of new - born neutron stars formed by aic . we find that a neutron star has a dynamically unstable bar mode if its @xmath1 is greater than the critical value @xmath2 . in order for the neutron star to have @xmath10 , the pre - collapse white dwarf must be composed of oxygen , neon , magnesium and have a rotational angular frequency @xmath138 , corresponding to 93% of the maximum rotational frequency the white dwarf can have without mass shedding . the eigenfunction of the most unstable bar mode is concentrated within a radius @xmath139 . the oscillation frequency of the mode is @xmath140 . when the amplitude of the mode is small , it grows exponentially with an e - folding time @xmath141 for the most rapidly rotating star ( @xmath142 ) , which is about 5.5 rotation periods at the center of the star . the signal to noise ratio of the gravitational waves emitted by this instability is estimated to be 15 for ligo - ii broad - band interferometers if the source is located in the virgo cluster of galaxies ( @xmath143 ) . the detectability of these sources also depends on the event rate . the event rate of aic is between @xmath97 and @xmath144 . only those aic events corresponding to the collapse of rapidly rotating o - ne - mg white dwarfs can end up in the bar - mode instability . while it is likely that the white dwarfs would be spun up to rapidly rotation by the accretion gas prior to collapse @xcite , it is not clear how many of the aic events are related to the o - ne - mg white dwarfs . if the event rate is less than @xmath145 , it is not likely that ligo ii will detect these sources . however , the event rate of the core collapse of massive stars is much higher than that of the aic . a bar - mode instability could develop for neutron stars formed from the collapse of rapidly rotating pre - supernova cores . if a significant fraction of the cores are rapidly rotating , the chance of detecting the gravitational radiation from bar - mode instability would be much higher . if the pre - collapse white dwarf is differentially rotating , the resulting neutron star can have a higher value of @xmath1 . the bar - mode instability is then expected to last for a longer time . however , any differential rotation will be destroyed by magnetic fields in a timescale @xmath146 , where @xmath147 is the size of the white dwarf and @xmath148 . for a massive white dwarf with @xmath149 , @xmath150 which is much shorter than the accretion timescale . hence rigid rotation is a good approximation for pre - collapse white dwarfs . the magnetic field of a neutron star is much stronger than that of a white dwarf . the timescale for a magnetic field to suppress differential rotation depends on the initial magnetic field @xmath117 of the proto - neutron star . if the magnetic field of the pre - collapse white dwarf is of order @xmath126 , the initial field will be @xmath151 according to conservation of magnetic flux . in this case , the magnetic timescale is @xmath152 . this timescale is much longer than the time required for a hot proto - neutron star to cool down and turn into a cold neutron star , and go through the whole dynamical instability phase . if @xmath153 , a significant amount of angular momentum transport will take place during the cooling phase . a detailed magnetohydrodynamical simulation has to be carried out to study the transport process in this case . however , such a strong initial magnetic field is possible only if the pre - collapse white dwarf has a magnetic field @xmath154 . finally , we want to point out that the collapse of white dwarfs will certainly produce asymmetric shocks and may eject a small portion of the mass . we expect that our neutron star models describe fairly well the inner cores of the stars but not the tenuous outer layers . our stability results are sensitive to the region with @xmath155 . the results could change considerably if the structure in this region is very different from that of our models . this issue will hopefully be resolved by the future full 3d aic simulations . i thank lee lindblom for his guidance on all aspects of this work . i also thank kip s. thorne and stuart l. shapiro for useful discussions . this research was supported by nsf grants phy-9796079 and phy-0099568 , and nasa grant nag5 - 4093 . we see from figs . [ fig : cdeneq]-[fig : veleq ] that the bar - mode eigenfunction has peciliar structures at the corotation radius ( @xmath156 ) at which @xmath157 . the density perturbation has a small peak and the velocity perturbation is almost parallel to the @xmath5 direction . in this appendix , we shall show that these are caused by the resonance of the fluid driven by the mode . for simplicity , we only consider the fluid s motion on the equatorial plane . assume that the perturbations are dominated by a mode that goes as @xmath158 . we also assume that this mode is even under the reflection @xmath159 . hence we have @xmath160 and @xmath161 . in cylindrical coordinates , the linearized euler equation takes the form @xmath162 the density perturbation @xmath163 is related to the pressure perturbation @xmath164 by @xmath165 the @xmath30-component of the lagrangian displacement is given by @xmath166 our numerical simulations show that @xmath164 is well - behaved and smooth near the corotation radius at which @xmath167 . the perturbed gravitational potential @xmath168 is expected ( and is confirmed by our numerical simulations ) to be smooth since it depends on the overall distribution of the density perturbation . we can then use eqs . ( [ eq : euler1])-([eq : ximode ] ) to express all the other perturbed quantities in terms of @xmath164 and @xmath168 . near the corotation radius , the expressions are : @xmath169 \label{eq : vphi } \ , \\ \kappa^2 & = & \varpi \partial_{\varpi}\omega^2 + 4\omega^2 \ , \\ b & = & \frac{\partial_{\varpi } p \partial_{\varpi}\rho}{\rho^2}\left(1- \frac{\gamma_{\rm eq}}{\gamma_p } \right ) \ .\end{aligned}\ ] ] it follows from eqs . ( [ eq : ximode ] ) and ( [ eq : drhomode ] ) that if @xmath170 is not of order @xmath171 near the corotation radius , both @xmath172 and @xmath173 will be large . the large magnitude of the lagrangian displacement is caused by the fluid being driven in resonance by the mode . the large displacement of the fluid causes @xmath173 to be large due to the second term of eq . ( [ eq : drhomode ] ) . this term arises becuase of the different compressibilities of stationary and oscillating fluid ( i.e. @xmath174 ) . in the case of the bar mode ( @xmath0 ) , the corotation radius is located at @xmath175 . the equilibrium density on the equator @xmath176 and the stationary fluid is very compressible ( @xmath177 ) . the high compressibility of the stationary fluid make the background equilibrium density @xmath32 drop rapidly as @xmath30 increases , i.e. @xmath178 is large . the oscillating fluid is far less compressible ( @xmath179 ) . as a result , when the oscillating fluid moves to a new location , it does not expand or compress to an extent that can compensate for the difference between the background densities at the old and new locations . since both @xmath172 and @xmath178 are large , @xmath163 is dominated by the second term of eq . ( [ eq : drhomode ] ) near the corotation radius . this explains the narrow secondary peak of @xmath163 seen in fig . [ fig : cdeneq ] . we see from eq . ( [ eq : vphi ] ) that @xmath180 .
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the dynamical instability of new - born neutron stars is studied by evolving the linearized hydrodynamical equations .
the neutron stars considered in this paper are those produced by the accretion induced collapse of rigidly rotating white dwarfs .
a dynamical bar - mode ( @xmath0 ) instability is observed when the ratio of rotational kinetic energy to gravitational potential energy @xmath1 of the neutron star is greater than the critical value @xmath2 . this bar - mode instability leads to the emission of gravitational radiation that could be detected by gravitational wave detectors . however , these sources are unlikely to be detected by ligo ii interferometers if the event rate is less than @xmath3 per year per galaxy
. nevertheless , if a significant fraction of the pre - supernova cores are rapidly rotating , there would be a substantial number of neutron stars produced by the core collapse undergoing bar - mode instability .
this would greatly increase the chance of detecting the gravitational radiation .
0.5 cm pacs : 04.30.db , 95.30.sf , 97.60.jd 2
| 8,753 | 277 |
electronic - structure calculations on periodic systems are conventionally done using the so - called bloch orbital based approach which consists of assuming an itinerant form for the single - electron wave functions . this approach has the merit of incorporating the translational invariance of the system under consideration , as well as its infinite character , in an elegant and transparent manner . an alternative approach to electronic - structure calculations on periodic systems was proposed by wannier @xcite . in this approach , instead of describing the electrons in terms of itinerant bloch orbitals , one describes them in terms of mutually orthogonal orbitals localized on individual atoms or bonds constituting the infinite solid . since then such orbitals have come to be known as wannier functions . it can be shown that the two approaches of description of an infinite solid are completely equivalent and that the two types of orbitals are related by a unitary transformation @xcite . therefore , the two approaches differ only in terms of their practical implementation . however , the description of metallic systems in terms of wannier functions frequently runs into problems as it is found that for such systems the decay of the orbitals away from the individual atomic sites is of power law type and not of exponential type . in other words , the wannier functions for such systems are not well localized @xcite . this behavior is to be expected on intuitive grounds as electrons in metals are indeed quite delocalized . on the other hand , for the situations involving surfaces , impurity states , semiconductors and insulators , where the atomic character of electrons is of importance , wannier functions offer a natural description . recent years have seen an increased amount of activity in the area of solid - state calculations based on localized orbitals @xcite , of which wannier functions are a subclass . most of these approaches have been proposed with the aim of developing efficient order - n methods for electronic structure calculations on solids within the framework of density functional theory . with a different focus , nunes and vanderbilt @xcite have developed an entirely wannier - function based approach to electronic - structure calculations on solids in the presence of electric fields , a case for which the eigenstates of the hamiltonian are no longer bloch states . however , we believe that there is one potential area of application for wannier orbitals which remains largely unexplored , namely in the _ ab initio _ treatment of electron - correlation effects in solids using the conventional quantum - chemical methods @xcite . it is intuitively obvious that an _ ab initio _ treatment of electron correlations on large systems will converge much faster with localized orbitals as compared to delocalized orbitals because the coulomb repulsion between two electrons will decay rapidly with the increasing distance between the electrons . in the quantum - chemistry community the importance of localized orbitals in treating the correlation effects in large systems was recognized early on and various procedures aimed at obtaining localized orbitals were developed @xcite . some of the localized - orbital approaches were also carried over to solids chiefly by kunz and collaborators @xcite at the hartree - fock level . this approach has been applied to a variety of systems @xcite . kunz , meng and vail @xcite have gone beyond the hartree - fock level and also included the influence of electron correlations for solids using many - body perturbation theory . the scheme of kunz et al . is based upon nonorthogonal orbitals which , in general , are better localized than their orthogonal counterparts . however , the subsequent treatment of electron correlations with nonorthogonal orbitals is generally much more complicated than the one based upon true wannier functions . in our group electron correlation effects on solids have been studied using the incremental scheme of stoll @xcite which works with localized orbitals . in such studies the infinite solid is modeled as a large enough cluster and then correlation effects are calculated by incrementally correlating the hartree - fock reference state of the cluster expressed in terms of localized orbitals @xcite . however , a possible drawback of this procedure is that there will always be finite size effects and no _ a priori _ knowledge is available as to the difference in results when compared with the infinite - solid limit . in order to be able to study electron - correlation effects in the infinite - solid limit using conventional quantum - chemical approaches , one first has to obtain a hartree - fock representation of the system in terms of wannier functions . this task is rather complicated because , in addition to the localization requirement , one also imposes the constraint upon the wannier functions that they be obtained by the hartree - fock minimization of the total energy of the infinite solid . in an earlier paper @xcite henceforth referred to as i we had outlined precisely such a procedure which obtained the wannier functions of an infinite insulator within a hartree - fock approach and reported its preliminary applications to the lithium hydride crystal . in the present paper we describe all theoretical and computational details of the approach and report applications to larger systems namely lithium fluoride and lithium chloride . unlike i , where we only reported results on the total energy per unit cell of the system , here we also use the hartree - fock wannier functions to compute the x - ray structure factors and compton profiles . additionally , we also discuss the localization characteristics of the wannier functions in detail . all the physical quantities computed with our procedure are found to be in excellent agreement with those computed using the crystal program @xcite which employs a bloch orbital based _ ab initio _ hartree - fock approach . in a future publication we will apply the present formalism to perform _ ab initio _ correlation calculations on an infinite insulator . the rest of this paper is organized as follows . in section [ theory ] we develop the theoretical formalism at the hartree - fock level by minimizing the corresponding energy functional , coupled with the requirement of translational symmetry , and demonstrate that the resulting hf equations correspond to the hf equations for a unit cell of the solid embedded in the field of identical unit cells constituting the rest of the infinite solid . thus an embedded - cluster picture for the infinite solid emerges rigorously from this derivation . subsequently a localizing potential is introduced in the hf equations by means of projection operators leading to our working equations for the hartree - fock wannier orbitals for an infinite solid . finally , these equations are cast in the matrix form using a linear combination of atomic orbitals approach which is used in the actual calculations . in section [ results ] we present the results of our calculations performed using the aforementioned formalism on lif and licl crystals . finally , in section [ conclusion ] we present our conclusions . various aspects related to the computer implementation of the present approach are discussed in the appendix . we consider the case of a perfect solid without the presence of any impurities or lattice deformations such as phonons . we also ignore the effects of relativity completely so that the spin - orbit coupling is also excluded . in such a case , in atomic units @xcite , the nonrelativistic hamiltonian of the system consisting of the kinetic energy of electrons , electron - nucleus interaction , electron - electron repulsion and nucleus - nucleus interaction is given by @xmath0 where in the equation above @xmath1 denotes the position coordinates of the @xmath2-th electron while @xmath3 and @xmath4 respectively denote the position and the charge of the @xmath5-th nucleus of the lattice . for a given geometry of the solid the last term representing the nucleus - nucleus interaction will make a constant contribution to the energy and will not affect the dynamics of the electrons . to develop the theory further we make the assumptions that the solid under consideration is a closed - shell system and that a single slater determinant represents a reasonable approximation to its ground state . moreover , we assume that the same spatial orbitals represent both the spin projections of a given shell , i.e. , we confine ourselves to restricted hartree - fock ( rhf ) theory . with the preceding assumptions , the total energy of the solid can be written as @xmath6 where @xmath7 and @xmath8 denote the occupied spatial orbitals assumed to form an orthonormal set , @xmath9 denotes the kinetic energy operator , @xmath10 denotes the electron - nucleus potential energy , @xmath11 denotes the nucleus - nucleus interaction energy and @xmath12 etc . represent the two - electron integrals involving the electron repulsion . the equation above is completely independent of the spin degree of freedom which , in the absence of spin - orbit coupling , can be summed away leading to familiar factors of two in front of different terms . clearly the terms involving @xmath13 , @xmath12 , and @xmath11 contain infinite lattice sums and are convergent only when combined together . so far the energy expression of eq.([eq - esolid ] ) does not incorporate any assumptions regarding the translational symmetry of a perfect solid . in keeping with our desire to introduce translational symmetry in the real space , without having to invoke the * k*-space as is usually done in the bloch orbital based theories , we make the following observation . a crystalline solid , in its ground state , is composed of identical unit cells and the orbitals belonging to a given unit cell are identical to the corresponding orbitals belonging to any other unit cell and are related to one another by a simple translation operation . assuming that the number of orbitals in a unit cell is @xmath14 and if we denote the @xmath15-th orbital of a unit cell located at the position given by the vector @xmath16 of the lattice by @xmath17 then clearly the set @xmath18 denotes all the orbitals of the solid . in the previous expression @xmath19 is the total number of unit cells in the solid which , of course , is infinite . henceforth , greek labels @xmath20 will always denote the orbitals of a unit cell . the translational symmetry condition expressed in the real space can be stated simply as @xmath21 where @xmath22 is an operator which represents a translation by vector @xmath23 . using this , one can rewrite the energy expression of eq.([eq - esolid ] ) as @xmath24 where @xmath25 denotes an orbital centered in the reference unit cell , @xmath26 involves the interaction energy of the nuclei of the reference cell with those of the rest of the solid ( @xmath27 ) , and we have removed the subscript @xmath28 from the energy . the preceding equation also assumes the important fact that the orbitals obtained by translation operation of eq.([eq - trsym ] ) are orthogonal to each other . we shall elaborate this point later in this section . an important simplification to be noted here is that by assuming the translational invariance in real space as embodied in eq.([eq - trsym ] ) , we have managed to express the total hartree - fock energy of the infinite solid in terms of a finite number of orbitals , namely the orbitals of a unit cell @xmath14 . if we require that the energy of eq.([eq - esolidf ] ) be stationary with respect to the first - order variations in the orbitals , subject to the orthogonality constraint , we are led to the hartree - fock operator @xmath29 where j and k the conventional coulomb and exchange operators , respectively are defined as @xmath30 any summation over greek indices @xmath20 will imply summation over all the @xmath14 orbitals of a unit cell unless otherwise specified . as mentioned earlier , the terms @xmath10 , @xmath31 and @xmath32 involve infinite lattice sums and their practical evaluation will be discussed in the next section . the eigenvectors of the hartree - fock operator of eq.([eq - hff ] ) will be orthogonal to each other , of course . however , in general , these solutions would neither be localized , nor would they be orthogonal to the orbitals of any other unit cell . this is because the orbitals centered in any other unit cell are obtained from those of the reference cell using a simple translation operation as defined in eq.([eq - trsym ] ) , which does not impose any orthogonality or localization constraint upon them . since our aim is to obtain the wannier functions of the infinite solid , i.e. , all the orbitals of the solid must be localized and orthogonal to each other , we will have to impose these requirements explicitly upon the eigenspace of ( [ eq - hff ] ) . this can most simply be accomplished by including in ( [ eq - hff ] ) the projection operators corresponding to the orbitals centered in the unit cells in a ( sufficiently large ) neighborhood of the reference cell @xmath33 where @xmath34 stands for @xmath25 , an orbital centered in the reference unit cell , @xmath35 , @xmath36 , and @xmath37 collectively denotes the unit cells in the aforementioned neighborhood . clearly the choice of @xmath37 will be dictated by the system under consideration the more delocalized electrons of the system are , the larger will @xmath37 need to be . in our calculations we have typically chosen @xmath37 to include up to third nearest - neighbor unit cells of the reference cell . in the equation above @xmath38 s are the shift parameters associated with the correponding orbitals of @xmath37 . for perfect orthogonality and localization , their values should be infinitely high . by setting the shift parameters @xmath38 s to infinity we in effect raise the orbitals localized in the environment unit cells ( region @xmath37 ) to very high energies compared to those localized in the reference cell . thus the lowest energy solutions of eq.([eq - hff1 ] ) will be the ones which are localized in the reference unit cell and are orthogonal to the orbitals of the environment cells . of course , in practice , it suffices to choose a rather large value for these parameters , and the issue pertaining to this numerical choice is discussed further in section [ results ] . eq.([eq - hff1 ] ) will generally be solved iteratively as described in the next section . if the initial guesses for the orbitals of the unit cell @xmath39 are localized , subsequent orthogonalization by means of projection operators will not destroy that property @xcite and the final solutions of the problem will be localized orthogonal orbitals . therefore , projection operators along with the shift parameters , simply play the role of a localizing potential @xcite as it is clear that upon convergence their contribution to the hartree - fock equation vanishes . the orbitals contained in unit cells located farther than those in @xmath37 should be automatically orthogonal to the reference cell orbitals by virtue of the large distance between them . it is clear that the orthogonalization of the orbitals to each other will introduce oscillations in these orbitals which are also referred to as the orthogonalization tails . combining the orthogonality of the neighboring orbitals to the reference cell orbitals with the translation symmetry of the infinite solid , it is easy to see that the orbitals of any unit cell are orthogonal to all the orbitals of the rest of the unit cells . therefore , orbitals thus obtained are essentially wannier functions . after solving for the hf equations presented above one can obtain the electronic part of the energy per unit cell simply by dividing the total energy of eq.([eq - esolidf ] ) by @xmath19 , which , unlike the total energy , is a finite quantity . in paper i we arrived at exactly the same hf equations as above , although we had followed a more intuitive path utilizing the so - called `` embedded - cluster '' philosophy , whereby we minimized only that portion of the total energy of eq.([eq - esolid ] ) which corresponds to the `` cluster - environment '' interaction . the fact that the derivations reported in paper i , and here , both lead to the same final equations has to do with the translation invariance which allows the total energy to be expressed in the form of eq.([eq - esolidf ] ) . therefore , we emphasize that the equations derived above are exact and do not involve any approximation other than the hartree - fock approximation itself . thus results of all the computations utilizing this approach should be in complete agreement with the equivalent computations performed using the traditional bloch orbital based approach as is implemented , e.g , in the program crystal @xcite . by inspection of eq.([eq - hff1 ] ) it is clear that it is of the embedded - cluster form in the sense that if one calls the reference unit cell the `` central cluster '' , it describes the dynamics of the electrons of this central cluster embedded in the field of identical unit cells of its environment ( rest of the infinite solid ) . we have performed a computer implementation of the formalism presented in the previous section within a linear combination of atomic orbital ( lcao ) approach , whereby we transform the differential equations of eq.([eq - hff1 ] ) into a set of linear equations solvable by matrix methods . atomic units were used throughout the numerical work . we proceed by expanding the orbitals localized in the reference cell as @xmath40 where @xmath41 has been used to denote the reference cell , @xmath16 represents the location of the @xmath42th unit cell ( located in @xmath41 or @xmath37 ) and @xmath43 represents a basis function centred in the @xmath42th unit cell . in order to account for the orthogonalization tails of the reference cell orbitals , it is necessary to include the basis functions centred in @xmath37 as well . clearly , the translational symmetry of the crystal as expressed in eq.([eq - trsym ] ) demands that the orbitals localized in two different unit cells have the same expansion coefficients @xmath44 , and differ only in the location of the centers of the basis functions . the lcao formalism implemented in most of the quantum - chemistry molecular programs , as also in the crystal code@xcite , expresses the basis functions @xmath43 of eq.([eq - lcao ] ) as linear combinations of cartesian gaussian type basis functions ( cgtfs ) of the form @xmath45 where @xmath46 . in the previous equation , @xmath47 denotes the exponent and the vector @xmath48 represents the center of the basis function . the centers of the basis functions @xmath48 are normally taken to be at the locations of the appropriate atoms of the system . cgtfs with @xmath49 are called respectively @xmath50 type basis functions@xcite . the individual basis functions of the form of eq.([eq - cgto ] ) are called _ primitive _ functions while the linear combinations of them are called the _ contracted _ functions . the formalism is totally independent of the type of basis functions , but for the sake of computational simplicity , we have programmed our approach using gaussian lobe - type functions@xcite . in this approach one approximates the @xmath51 and higher angular momentum cgtfs as linear combinations of @xmath52-type basis functions displaced by a small amount from the location of the atom concerned . for example , in the present study a primitive @xmath51 type cgtf centered at the origin was approximated as @xmath53 where , @xmath54 is the normalization constant and @xmath55 . in the present study the value of 0.1 atomic units ( a.u . ) was employed for @xmath56 . for approximating the @xmath57 , @xmath58 and @xmath59 types of basis functions , the displacement vectors @xmath60 are chosen to be along the positive @xmath61 , @xmath62 and @xmath63 directions , respectively . by substituting eq.([eq - lcao ] ) in eq.([eq - hff1 ] ) we obtain the hf equations in the lcao matrix form @xmath64 the fock matrix @xmath65 occuring in the equation above is defined as @xmath66 where the contribution of all the operators appearing in eq.([eq - hff1 ] ) has been replaced by the corresponding matrices in the representation of the chosen basis set . above , unprimed functions @xmath67 and @xmath68 represent the basis functions corresponding to the orbitals of the reference unit cell while the primed functions @xmath69 and @xmath70 denote the basis functions corresponding to the orbitals of @xmath37 . in particular , the overlap matrix is given by @xmath71 and the coulomb and the exchange matrix elements are defined as @xmath72 and @xmath73 where @xmath74 denotes the elements of the density matrix @xmath75 of the orbitals of a unit cell evaluated as @xcite @xmath76 the matrix form of the hf equations ( [ eq - scf ] ) is a pseudo eigenvalue problem which can be solved iteratively to obtain the hf orbitals . the energy per unit cell can be computed by means of a simple matrix - trace operation @xmath77 where above @xmath9 , @xmath10 , @xmath31 and @xmath32 and @xmath75 denote the matrices of the corresponding operators in the representation of the chosen basis set , and @xmath26 was defined after eq.([eq - esolidf ] ) . in practice one proceeds according to the following algorithm : 1 . start with some localized initial guess for the orbitals of the reference cell . for ionic systems considered here we chose these to be the orbitals of the individual ions centered on the corresponding atomic sites . for covalent systems , it would be reasonable to use suitable bonding combinations of atomic orbitals . 2 . use these orbitals to construct the fock matrix as defined in eq.([eq - fock ] ) . 3 . diagonalize the fock matrix to obtain a new set of orbitals of the reference cell . 4 . compute the energy per unit cell by using eq.([eq - ecell ] ) . go to step 2 . iterate until the energy per unit cell has converged . various mathematical formulas and computational aspects related to the evaluation of different contributions to the fock matrix are discussed in the appendix . in this section we describe the evaluation of the x - ray structure factors and compton profiles from the hartree - fock wannier functions obtained from the formalism of the previous section . both these properties can be obtained from the first - order density matrix of the system defined for the present case as @xmath78 where @xmath79 is the @xmath15-th hf orbital of the unit cell located at position @xmath23 . the factor of two above is a consequence of spin summation . by measuring the x - ray structure factors experimentally one can obtain useful information on the charge density of the constituent electrons . theoretically , the x - ray structure factor @xmath80 can be obtained by taking the fourier transform of the diagonal part of the first - order density matrix @xmath81 by means of compton scattering based experiments , one can extract the information on the momentum distribution of the electrons of the solid . in the present study we compute the compton profile in the impulse approximation as developed by eisenberger and platzman @xcite . under the impulse approximation the compton profile for the momentum transfer @xmath82 is defined as @xcite @xmath83 where @xmath84 and @xmath85 are , respectively , the changes in momentum and the frequency of the incoming x- or @xmath86-ray due to scattering , @xmath87 is the compton electron momentum , @xmath88 is the projection of @xmath87 in the direction of @xmath84 , the delta function imposes the energy conservation and @xmath89 denotes the electron momentum distribution obtained from the diagonal part of the full fourier transform of the first - order density matrix @xmath90 by choosing the @xmath63-axis of the coordinate system defining @xmath87 along the direction of @xmath84 , one can perform the @xmath59 integral in eq.([eq - cp1 ] ) to yield @xmath91 integrals contained in the expressions for the x - ray structure factor and the compton profile ( eqs . ( [ eq - xfac ] ) and ( [ eq - cp ] ) , respectively ) can be performed analytically when the density matrix is represented in terms of gaussian lobe - type basis functions . these analytic expressions are used to evaluate the quantities of interest in our computer code , once the hartree - fock density matrix has been determined . in this section we present the results of the calculations performed on crystalline lif and licl . prencipe et al . @xcite studied these compounds , along with several other alkali halides , using the crystal program @xcite . crystal , as mentioned earlier , is a bloch orbital based _ ab initio _ hartree - fock program set up within an lcao scheme , utilizing cgtfs as basis functions . in their study , prencipe et al . employed a very large basis set and , therefore , their results are believed to be very close to the hartree - fock limit . in the present work our intention is not to repeat the extensive calculations of prencipe et al . @xcite , but rather to demonstrate that at the hartree - fock level one can obtain the same physical insights by applying the wannier function based approach as one would by utilizing the bloch orbital based approach . moreover , because of the use of lobe functions as basis functions , we run into problems related to numerical instability when very diffuse @xmath92type ( and beyond ) basis functions are employed . in future we intend to incorporate true cgtfs as basis functions in our program , which should make the code numerically much more stable . therefore , we have performed these calculations with modest sized basis sets . we reserve the use of large basis sets for the future calculations , when we intend to go beyond the hartree - fock level to utilize these wannier functions to do correlated calculations . the reason we have chosen to compare our results to those obtained using the crystal program is because not only is crystal based upon an lcao formalism employing gaussian type of basis functions similar to our case , but also it is a well - tested program and widely believed to be the state of the art in crystalline hartree - fock calculations @xcite . all the calculations to be presented below assume the observed face - centered cubic ( fcc ) structure for the compounds . the reference unit cell @xmath41 was taken to be the primitive cell containing an anion at the @xmath93 position and the cation at @xmath94 , where @xmath95 is the lattice constant . the calculations were performed with different values of the lattice constants to be indicated later . the basis sets used for lithium , fluorine and chlorine are shown in tables [ tab - basli ] , [ tab - basf ] and [ tab - bascl ] , respectively . for lithium we adopted the basis set of dovesi et al . used in their lithium hydride study @xcite , while for fluorine and chlorine basis sets originally published by huzinaga and collaborators @xcite were used . the values of the level - shift parameters @xmath38 s of eq.([eq - fock ] ) should be high enough to guarantee sufficient orthogonality while still allowing for numerical stability . thus this choice leaves sufficient room for experimentation . we found the values in the range @xmath96 suitable for our work . we verified by explicit calculations that our results had indeed converged with respect to the values of the shift parameters . in the course of the evaluation of integrals needed to construct the fock matrix , all the integrals whose magnitudes were smaller than @xmath97 were discarded both in our calculations as well as in the crystal calculations . the comparison of our ground - state energies per unit cell with those obtained using the identical basis sets by the crystal program @xcite is illustrated in tables [ tab - enlif ] and [ tab - enlicl ] for different values of lattice constants . the biggest disagreement between the two types of calculations is 0.7 millihartree . a possible source of this disagreement is our use of lobe functions to approximate the @xmath51-type cgtfs . however , since the typical accuracy of a crystal calculation is also 1 millihartree @xcite , we consider this disagreement to be insignificant . such excellent agreement between the total energies obtained using two different approaches gives us confidence as to the essential correctness of our approach . from the results it is also obvious that the basis set used in these calculations is inadequate to predict the lattice constant and the bulk modulus correctly . to be able to do so accurately , one will have to employ a much larger basis set such as the one used by prencipe et al . @xcite . since hartree - fock lattice constants generally are much larger than the experimental value , we reserve the large - scale hartree - fock calculations for future studies in which we will also go beyond the hartree - fock level to include the influence of electron correlations . valence wannier functions for lif and licl are plotted along different crystal directions in figs . [ fig - lif2p-001 ] , [ fig - lif2p-110 ] , [ fig - licl3p-001 ] and [ fig - licl3p-110 ] . lattice constants for these calculations were assigned their experimental values @xcite of 3.99 @xmath98 and 5.07 @xmath98 , for lif and licl , respectively . although core orbitals were also obtained from the same set of calculations , we have not plotted them here because they are trivially localized . the @xmath51-character of the wannier functions is evident from the antisymmetric nature of the plots under reflection . the additional nodes introduced in the orbitals due to their orthogonalization to orbitals centered on the atoms of region @xmath37 are also evident . the localized nature of these orbitals is obvious from the fact that the orbitals decay rapidly as one moves away from the atom under consideration . the orthogonality of the orbitals of the reference cell to those of the neighborhood ( region @xmath37 ) was always better than @xmath99 . now we discuss the data for x - ray structure factors . these quantities were also evaluated at experimental lattice constants mentioned above . the x - ray structure factors obtained by our method are compared to values calculated with the crystal program , and experimental data , in tables [ tab - lifxfac ] and [ tab - liclxfac ] for lif and licl , respectively . for the case of lif we directly compare the theoretical values with the experimental data of merisalo et al . @xcite , extrapolated to zero temperature by euwema et al . @xcite . for licl it was not possible for us to extrapolate the experimental data of inkinen et al . @xcite , measured at @xmath100 , to the corresponding zero temperature values . therefore , to compare our licl calculations to the experiment , we correct our theoretical values for thermal motion using the debye - waller factors of @xmath101 and @xmath102 , measured also by inkinen et al . @xcite the debye - waller corrections were applied to the individual form factors of li@xmath103 and cl@xmath104 ions . from both the tables it is obvious that our results are in almost exact agreement with those of crystal . this implies that our wannier function hf approach based description of the charge density of systems considered here , is identical to a bloch orbital based hf description as formulated in crystal @xcite . for the case of lif the agreement between our results and the experiment is also quite good , maximum errors being @xmath105 5% . for the case of licl , our corrected values of x - ray structure factors deviate from the experimental values at most by approximately 3% . perhaps by using a larger basis set one can obtain even better agreement with experiments . finally we turn to the discussion of compton profiles . directional and isotropic compton profiles , computed using our approach and the crystal program , are compared to the isotropic compton profiles measured by paakkari et al . @xcite , in tables [ tab - lifcp ] and [ tab - liclcp ] . we obtain the isotropic compton profiles from our directional profiles by performing a directional average of the profiles along the three crystal directions according to the formula @xmath106 valid for an fcc lattice @xcite . while experimental data for directional compton profiles exist in the case of lif @xcite , no such measurements have been performed for licl , to the best of our knowledge . for lif there is close agreement between our results and the ones calculated using the crystal program . for licl our results disagree with the crystal results somewhat for small values of momentum transfer , although relatively speaking the disagreement is quite small the maximum deviation being @xmath105 0.3% for @xmath107 and the @xmath108 $ ] direction . the possible source of the disagreement may be that to get the values of compton profiles for all the desired values of momentum transfer , we had to use the option of crystal @xcite where the compton profiles are obtained by using the real - space density matrix rather than its more accurate * k*-space counterpart . however , as is clear from the tables , even for those worst cases , there is no significant difference between the averaged out isotropic compton profiles obtained in our computations and those obtained from crystal . at the larger values of momentum transfer , our results are virtually identical to the crystal results . the close agreement with crystal clearly implies that our wannier function based description of the momentum distribution of the electrons in the solid is identical to the one based upon bloch orbitals . considering the fact that we have used a rather modest basis set , it is quite surprising that the values of isotropic compton profiles obtained by us are in close agreement with the corresponding experimental values @xcite . an inspection of tables [ tab - lifcp ] and [ tab - liclcp ] reveals that the calculated values always agree with the experimental ones to within 6% . however , ours as well as the crystal calculations presented here are not able to describe the observed anisotropies in the directional compton profiles @xcite for lif which is also the reason that we have not compared the theoretical anisotropies to the experimental ones . for small values of momentum transfer the calculated values are even in qualitative disagreement with the experimental results , although for large momentum transfer the qualititative agreement is restored . this result is not surprising , however , because , as berggren et al . have argued @xcite in their detailed study , the proper description of the compton anisotropy mandates a good description of the long - range tails of the crystal orbitals . to be able to do so with the gaussian - type of basis functions used here , one will unlike the present study have to include basis functions with quite diffuse exponents . in conclusion , an _ ab initio _ hartree - fock approach for an infinite insulating crystal which yields orbitals in a localized representation has been discussed in detail . it was applied to compute the total energies per unit cell , x - ray structure factors and directional compton profiles of two halides of lithium , lif and licl . the close agreement between the results obtained using the present approach , and the ones obtained using the conventional bloch orbital based hf approach , demonstrates that the two approaches are entirely equivalent . the advantage of our approach is that by considering local perturbations to the hartree - fock reference state by conventional quantum - chemical methods , one can go beyond the mean - field level and study the influence of electron correlations on an infinite solid in an entirely _ ab initio _ manner . presently projects along this direction are at progress in our group , and in a future publication we will study the influence of electron correlations on the ground state of a solid . one of us ( a.s . ) gratefully acknowledges useful discussions with prof . roberto dovesi , and his help regarding the use of the crystal program . in this section we discuss the calculation of various terms in the fock matrix . since the kinetic - energy matrix elements @xmath109 and the overlap - matrix elements @xmath110 have simple mathematical expressions and are essentially unchanged from molecular calculations , we will not discuss them in detail . however , we will consider the evaluation of the rest of the contributions to the fock matrix at some length . the electron - nucleus attraction term of the fock matrix contains the infinite lattice sums involving the attractive interaction acting on the electrons of the reference cell due to the infinite number of nuclei in the solid . when treated individually , this term is divergent . however , when combined with the coulombic part of the electron repulsion to be discussed in the next section , convergence is achieved because the divergences inherent in both sums cancel each other owing to the opposite signs . this fact is a consequence of the charge neutrality of the unit cell and is used in the ewald - summation technique @xcite to make the individual contibutions also convergent by subtracting , from the corresponding potential a shadow potential emerging from a ficitious homogeneous charge distribution of opposite sign . in addition , in the ewald method , one splits the lattice potential into a short - range part whose contribution is rapidly convergent in the @xmath111-space and a long - range part which converges fast in @xmath84-space . therefore , in the ewald - summation technique one replaces the electron - nucleus interaction potential due to a lattice composed of nuclei of charge @xmath112 , by the effective potential @xcite @xmath113 where @xmath23 represents the positions of the nuclei on the lattice , @xmath114 are the vectors of the reciprocal lattice , @xmath85 is the volume of the unit cell , @xmath115 is a convergence parameter to be discussed later and erfc represents the complement of the error function . matrix elements of the ewald potential of eq.([eq - ewaldp ] ) with respect to primitive @xmath52-type basis functions were derived by stoll @xcite to be @xmath116 above @xmath51 and @xmath82 label the primitive basis functions , @xmath117 and @xmath118 represent the positions of the unit cells in which they are located and @xmath119 represents the overlap matrix element between the two primitives given by @xmath120 the vectors @xmath121 and @xmath122 above specify the centers of the two basis functions relative to the origin of the unit cell , @xmath47 and @xmath123 represent the exponents of the two gaussians , @xmath124 and @xmath125 with @xmath126 , @xmath127 and where the parameter @xmath129 takes over the role of the convergence parameter @xmath115 of eq.([eq - ewaldp ] ) . the remaining quantities are the the same as those in eq.([eq - ewaldp ] ) . it is clear that the function @xmath130 involves lattice sums both in the direct space and the reciprocal space . although the final value of the function will be independent of the choice of the convergence parameter @xmath129 , both these sums can be made to converge optimally by making a judicious choice of it . large values of @xmath129 lead to faster convergence in the real space but to slower one in the reciprocal space and with smaller values of @xmath129 the situation is just the opposite . therefore , for optimal performance , the choice of @xmath129 is made dependent on the value of @xmath15 . in the present work we make the choice so that if @xmath131 , @xmath132 and if @xmath133 , @xmath134 . in the former case the sum is both , in the real and the reciprocal space while in the latter case the sum is entirely in the reciprocal space . although we have written an efficient computer code to evaluate the function @xmath130 , it remains the most computer intensive part of our program . the computational effort involved in the computation of these integrals can be reduced by utilizing the translational symmetry . one can verify that as a consequence of translation symmetry @xmath135 where @xmath136 is also a vector of the direct lattice , @xmath137 represents the reference unit cell and the last term is a compact notation for the second term . since the number of unique @xmath138 vectors is much smaller than the number of pairs @xmath139 , the use of eq.([eq - usym ] ) reduces the computational effort considerably . to further reduce the computational effort we also use the interchange symmetry @xmath140 additional savings are achieved if one realizes that matrix elements @xmath141 become smaller as larger the distance @xmath142 between the interacting charge distributions becomes . as is clear from eq.([eq - upq ] ) , a good estimate of the magnitude of an integral is the overlap element @xmath143 @xcite . therefore , we compute only those integrals whose overlap elements @xmath143 are larger than some threshold @xmath144 . in the present calculations we chose @xmath145 . to calculate the coulomb contribution to the fock matrix , one needs to evaluate the two - electron integrals with infinite lattice sum @xmath146 where @xmath147 and @xmath52 represent the primitive basis functions and @xmath148 and @xmath149 represent the unit cells in which they are centered . this integral , treated on its own is divergent , as discussed in the previous section . however , using the ewald - summation technique , one can make this series conditionally convergent with the implicit assumption that its divergence will cancel the corresponding divergence of the electron nucleus interaction . since the details of the ewald - summation technique for the coulomb part of electron repulsion are essentially identical to the case of electron - nucleus interaction , we will just state the final results @xcite @xmath150 where @xmath151 and @xmath152 all the notations used in the equations above were defined in the previous section . the expression @xmath153 used in eq.([eq - jmate ] ) , as against @xmath154 of eq.([eq - jmat ] ) , is meant to remind us that the matrix elements stated in eq.([eq - jmate ] ) are those of the two - electron ewald potential and not those of the ordinary coulomb potential . like in the case of electron - nucleus attraction , one can utilize the translational symmetry for the present case to reduce the computational effort significantly . the corresponding relations in the present case are @xmath155 where as before @xmath137 represents the reference unit cell , @xmath136 , @xmath156 and the last term in eq.([eq - jsym ] ) is a compact notation for the second term . since the number of pairs @xmath157 is much smaller than the number of quartets @xmath158 , use of eq.([eq - jsym ] ) results in considerable savings of computer time and memory . in addition , we also use the four interchange relations of the form of eq.([eq - usym2 ] ) to further reduce the number of nonredundant integrals . additionally , these integrals also satisfy the interchange relation @xmath159 to keep the programming simple , however , at present we do not utilize this symmetry . in future , we do intend to incorporate this symmetry in the code . similar to the case of electron - nucleus integrals , here also we use the magnitude of the product @xmath160 to estimate the size of the integral to be computed and proceed with its calculation only if it is greater than a threshold @xmath161 , taken to be @xmath162 in this study . in order to compute the exchange contribution to the fock matrix , one has to compute the following two - electron integrals involving infinite lattice sum @xmath163 where the notation is identical to the previous two cases . by using the translational symmetry arguments one can show even for the exchange case that @xmath164 where the last term in eq.([eq - ksym ] ) above is a compact notation for the second term . as in the previous two cases , the use of translational symmetry results in considerable savings of computer time and storage . explicitly @xmath165 although eq.([eq - kfin ] ) contains an infinite sum over lattice vectors @xmath166 , the contributions of each of the terms decreases rapidly with the increasing distances @xmath167 and @xmath168 between the interacting charge distributions . a good estimate of the contribution of the individual terms is provided by the product of overlap matrix elements between the interacting charge distributions namely , @xmath169 and @xmath170 @xcite . therefore , in the computer implementation , we arrange the vectors @xmath166 so that the corresponding overlaps are in the descending order and the loop involving the sum over @xmath166 in eq.([eq - kfin ] ) is terminated once the individual overlap matrix elements or their product are less than a specified threshold @xmath171 . the computer code for evaluating these integrals is a modified version of the program written originally by ahlrichs @xcite . the value of the threshold @xmath171 used in these calculations was @xmath162 . the exchange integrals also satisfy interchange symmetries similar to those of eqs.([eq - usym2 ] ) and ( [ eq - jsymn ] ) , which are not used in the present version of the code for the ease of programming . in future , however , we plan to use them as well . as described above , to minimize the need of computer time and storage , we have made extensive use of translational symmetry . however , the integral evaluation can be further optimized considerably by making use of point group symmetry as is done in the crystal program @xcite . implementation of point group symmetry , as well as the use of cgtos instead of lobe - type functions , is planned for future improvements of the present code .
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an _ ab initio _ hartree - fock approach aimed at directly obtaining the localized orthogonal orbitals ( wannier functions ) of a crystalline insulator is described in detail .
the method is used to perform all - electron calculations on the ground states of crystalline lithium fluoride and lithium chloride , without the use of any pseudo or model potentials .
quantities such as total energy , x - ray structure factors and compton profiles obtained using the localized hartree - fock orbitals are shown to be in excellent agreement with the corresponding quantities calculated using the conventional bloch - orbital based hartree - fock approach .
localization characteristics of these orbitals are also discussed in detail .
| 11,996 | 172 |
the dynamics of the lyness recurrence @xmath11 specially when @xmath12 has focused the attention of many researchers in the last years and it is now completely understood in its main features after the independent research of bastien & rogalski @xcite and zeeman @xcite , and the later work of beukers & cushman @xcite . see also @xcite . in particular all possible periods of the recurrences generated by are known and for any @xmath13 infinitely many different prime periods appear . however there are still some open problems concerning the dynamics of rational points . with the computer experiments in mind , and following @xcite , it is interesting to know the existence of rational periodic sequences . the lyness map @xmath14 associated to , leaves invariant the elliptic curves and @xmath2 the curve is not elliptic . we study these values separately . ] @xmath15 and the map action can be described in terms of the group law action of them . in consequence several tools for studying the rational periodic orbits on them are available . in particular , from mazur s torsion theorem ( see for example @xcite ) , we know that , under the above hypotheses , the rational periodic points can only have ( prime ) periods @xmath16 and @xmath17 our first result proves that almost all these periods appear for the lyness recurrence for suitable @xmath18 and rational initial conditions . [ teo ] for any @xmath19 there are @xmath20 and rational initial conditions @xmath21 such that the sequence generated by is @xmath10-periodic . moreover these values of @xmath10 are the only possible prime periods for rational initial conditions and @xmath1 . notice that the value @xmath22 is the only one in mazur s list that is not in the list given in the theorem . following @xcite it is possible to interpret that this period corresponds to the case @xmath23 see remark [ infinity ] in section [ totq ] . concerning the rational periodic points in @xmath24 for the lyness map with @xmath12 it is proved in @xcite that it only can have periods @xmath25 and @xmath26 taking @xmath27 and @xmath28 we obtain trivially 1-periodic integer sequences . the existence of positive rational periodic points of period 5 is well known and simple : they only exist when @xmath29 and in this case all rational initial conditions give rise to them because the recurrence is globally 5-periodic . on the other hand , as far as we know , the case of period 9 has resisted all previous analysis . in particular , the conjecture 2 of zeeman , @xcite says that there are no such points and their existence is left in problem . ] 1 bis of @xcite as an open question . we prove that there are some values of @xmath2 for which the lyness recurrences have positive rational periodic sequences of period 9 and , even more , that this happens for infinitely many values of @xmath8 see more details in theorem [ teoa ] . it is known , see again @xcite , that the periodic points of period 9 of @xmath30 are on the elliptic curve @xmath31 and they have positive coordinates only when @xmath32 where @xmath33 is the biggest root of @xmath34 see also subsection [ grouplaw ] . by using magma ( @xcite ) , and after several trials , we have found some positive rational points on the above curve proving that the zeeman s conjecture 2 has a negative answer . the simplest one that we have obtained is @xmath35 . notice that the sequence , taking @xmath36 and the initial condition @xmath37 , @xmath38 gives @xmath39 other positive rational points that we have found are @xmath40 and many others with much bigger entries . our main result proves that there are infinitely many positive rational values of the parameter @xmath2 giving rise to @xmath7periodic positive rational orbits . [ teoa ] there are infinitely many values @xmath41 for which there exist initial conditions @xmath42 such that the sequence given by the lyness recurrence is @xmath7periodic . furthermore , the closure of these values of @xmath2 contains the real interval @xmath43 , where @xmath44 is the biggest root of @xmath45 notice that there is a small gap between the values @xmath46 and @xmath47 where we do not know if there are or not rational values of @xmath2 for which the lyness recurrence has positive rational periodic orbits of period 9 . as we will see in the proof of the theorem the gap is provoked by our approach and it seems to us that it is not intrinsic to the problem , see the comments in section [ periode9 ] , after the proof of the theorem . from the classical results of mordell , see for example ( * ? ? ? * ch.viii ) , it is well known that the set of rational points on an elliptic curve @xmath48 over @xmath49 together with the point at infinity , form an additive group @xmath50 and @xmath51 where @xmath52 is called the rank of @xmath50 and @xmath53 is the torsion of the group . notice that @xmath54 is a measure of the amount of rational points that the curve contains . the torsion part @xmath53 is well understood from the results of mazur already quoted , and it contains at most 16 points . for the elliptic curve it is easy to check that @xmath55 fixed any @xmath56 among all the allowed possibilities , it is not known if the rank of the elliptic curves having @xmath57 for some @xmath52 , has an upper bound , but it is believed that it has not ( see the related conjecture viii.10.1 in @xcite ) . as far as we know , nowadays when @xmath58 the highest known rank is greater or equal than 28 while when @xmath59 the highest known rank is 4 , see @xcite . we have found values of @xmath2 for which the algebraic curve has ranks @xmath60 or @xmath61 . for instance @xmath62 appears for @xmath63 and the value 4 happens for @xmath64 a key point to obtain these results is the following theorem , which extends the results of @xcite , given for the cases of torsion points with order 5 or 6 , and proves the universality of the level curves invariant for the lyness map . [ nft](lyness normal form ) the family of elliptic curves @xmath65 , @xmath66 together with the points @xmath67 $ ] and @xmath68 $ ] , is the universal family of elliptic curves with a point of order @xmath69 @xmath70 ( including @xmath71 ) . this means that for any elliptic curve @xmath48 over any field @xmath72 ( not of characteristic 2 or 3 ) with a point @xmath73 of order @xmath10 , there exist unique values @xmath74 and a unique isomorphism between @xmath48 and @xmath75 sending the zero point of @xmath48 to @xmath76 and the point @xmath77 to @xmath78 . moreover , given a finite @xmath10 , the relation between @xmath2 and @xmath79 can be easily obtained from the computations made in subsection [ grouplaw ] . from a dynamical viewpoint the above result implies that the lyness map is an affine model for most birrational maps on elliptic curves . this result is similar to the one described by jogia , roberts and vivaldy in ( * ? ? ? * theorem 3 ) , where they use the weierstrass normal form . moreover , notice that in our result the sum operation takes the extremely easy form @xmath80)=[xy : az^2+yz : xz]$ ] ( see section [ grouplaw ] ) . once for some value @xmath18 a periodic orbit in @xmath81 of period @xmath7 for @xmath30 is obtained , it is not difficult to obtain infinitely many different 9-periodic orbits by using the group law on the curve . for instance , for a given @xmath8 we know that if a point @xmath82 is on then @xmath83 is also on it . in particular for @xmath36 if we write @xmath84 we get @xmath85 and @xmath86 can be obtained for instance by plugging @xmath87 in and choosing a suitable solution @xmath88 taking @xmath89 we find @xmath90 which also gives rise to a 9-periodic orbit in @xmath81 . taking now @xmath91 as starting point for the procedure we obtain a new initial condition for a 9-periodic orbit : @xmath92 and so on . in general we have the following result , see subsection [ grouplaw ] . [ propob ] if @xmath41 is a value for which there exist an initial condition @xmath93 such that the sequence is @xmath7periodic , then there exist infinite many different rational initial conditions giving rise to @xmath7periodic sequences . moreover the points corresponding to these initial conditions fill densely the elliptic curve ( [ 9p ] ) . observe that such a point @xmath93 always gives rise to a point on the elliptic curve given by which is not a torsion point , because all the @xmath7-torsion points are not in @xmath94 see again subsection [ grouplaw ] . the case of 5-periodic points is studied in section [ periode5 ] , proving that for @xmath29 there are some elliptic curves @xmath95 with rank @xmath62 and , as a consequence , ovals without points with rational coordinates . finally , last section is devoted to give rational values of @xmath2 for which the lyness map @xmath30 has as many periods as possible , of course with rational initial conditions . the structure of this paper is the following . in subsection [ grouplaw ] we recall some known results which describe the action of the lyness map in terms of a linear translation over elliptic curves . subsection [ nf ] is devoted to prove theorem [ nft ] about the lyness normal form . in section [ totq ] we prove theorem [ teo ] , while the proof our main result , theorem [ teoa ] , and all our outcomes on rational 9-periodic points of are presented in section [ periode9 ] . section [ periode5 ] studies the number of rational points on the elliptic curves invariant for @xmath96 and the last section deals with the problem of finding a concrete @xmath30 with as many periods as possible . the results of this subsection are well known , we refer the reader to references @xcite for instance to get more details . as mentioned above , the phase space of the discrete dynamical system defined by the map is foliated by the family curves @xmath97 this family is formed by elliptic curves except for a few values of @xmath79 . when @xmath98 it is a product of straight lines ; when @xmath99 , @xmath100 , it is the formed by a straight line and an hyperbola ; and when @xmath101 with @xmath102 , it is a rational cubic , having an isolated real singularity . in fact the values @xmath103 correspond to the level sets containing the fixed points of @xmath30 , @xmath104 . the first quadrant @xmath105 is invariant under the action of @xmath30 and it is foliated by ovals with energy @xmath106 in summary , on most energy levels , @xmath30 is a birational map on an elliptic curve and therefore it can be expressed as a linear action in terms of the group law of the curve ( * ? ? ? * theorem 3 ) . indeed , taking homogeneous coordinates on the projective plane @xmath107 the curves @xmath65 have the form @xmath108 and @xmath109 can be seen as the map @xmath80)=[xy : az^2+yz : xz]$ ] . taking the point @xmath67 $ ] as the neutral element , each elliptic curve @xmath110 is an abelian group with respect the sum defined by the usual _ secant tangent chord process _ ( i.e. if a line intersects the curve in three points @xmath111,@xmath78 , and @xmath77 then @xmath112 ) . according to these operation the lyness map can be seen as the linear action @xmath113 where @xmath68 $ ] . when @xmath114 it is not difficult to get that @xmath115,$ ] @xmath116,$ ] @xmath117,\\ 5q&=\left[\frac{ah - a+1}{a-1}:\frac{-a^2-ah+2a-1}{a(a-1)}:1\right],\\ 6q&=\left[\frac{-a^2-ah+2a-1}{a(a-1)}:\frac{a^3 - 2a^2-ah+2a-1}{a(ah - a+1)}:1\right],\\ 7q&=\left[\frac{-a^2-ah+2a-1}{a(a-1)}:\frac{-a^4h+a^3h+a^3+a^2h-3a^2-ah+3a-1}{a^3h - a^3+a^2h^2 - 3a^2h+3a^2 + 2ah-3a+1 } : 1\right],\end{aligned}\ ] ] @xmath118,$ ] @xmath119,$ ] @xmath120,$ ] @xmath121,\\ -5q&=\left[\frac{-a^2-ah+2a-1}{a(a-1)}:\frac{ah - a+1}{a-1}:1\right],\end{aligned}\ ] ] see also @xcite . in terms of the recurrence , the linear action ( [ accio ] ) can be seen as follows : taking the initial conditions @xmath122 $ ] then @xmath123=p+(n+1)q$ ] , where @xmath124 is the group operation . notice also that , from this point of view , the condition of existence of rational periodic orbits is equivalent to the condition that @xmath78 is in the torsion of the group given by the rational points of @xmath125 which , as we have already commented , is described by mazur s theorem . hence , from the above expressions of @xmath126 we can obtain the values of @xmath79 corresponding to a given period . for instance , for period 9 we impose that @xmath127 , or equivalently @xmath128 which gives @xmath129 . from this equality we have that @xmath130 which corresponds to the elliptic curve . for these values of @xmath79 the points corresponding to the torsion subgroup are : @xmath131,\,[-1:0:1],\,[0:-a:1],\ , [ -a : a(a-1):1],\\ & [ a(a-1):-a:1],\ , [ -a:0:1],\,[0:-1:1],\ , [ 0:1:0],\ , { \cal o}=[1:-1:0].\end{aligned}\ ] ] it is also important from a dynamical point of view the following well know property of the secant tangent chord process defined on any real non - singular elliptic curve @xmath48 . let @xmath111 be a point of @xmath48 such that @xmath132 for @xmath133 is never the neutral element @xmath134 . recall that when the elliptic curve is defined over @xmath135 there are at most sixteen points @xmath111 with rational entries not satisfying this property due the mazur classification of the torsion subgroup of @xmath50 . then the adherence of the set @xmath136 is : * either all the curve @xmath48 , when @xmath111 belongs to the connected component of @xmath48 which does not contains the neutral element @xmath137 ; or * the connected component containing @xmath137 when @xmath111 belongs to it . this result is due to the fact that there is a continuous isomorphism between @xmath138 with this operation and the group @xmath139 , with the operation @xmath140 see for instance corollary 2.3.1 of ( * ? ? ? v.2 ) . clearly , by using this construction , proposition [ propob ] follows . notice that when an elliptic curve is given by @xmath65 the bounded component never contains @xmath137 . it is known that any elliptic curve having a point @xmath77 that is not a 2 or a 3 torsion point can be written in the so called tate normal form @xmath141 where @xmath77 is sent to @xmath142 , see ( * ? ? ? our proof will follow by showing that the curves @xmath65 can be transformed into the ones of the tate normal form . in projective coordinates these curves write as @xmath143 and the curves @xmath65 as @xmath144 with the change of variables @xmath145 and the relations @xmath146 both families of curves are equivalent and the theorem follows . observe that the case @xmath147 corresponds to a curve with a @xmath61torsion point . as we will see in the proof of theorem [ teoa ] one advantage of this normal form is that it is symmetric with respect to @xmath148 and @xmath88 from theorem [ nft ] we have that all the known results on elliptic curves with a point of order greater than 4 can be applied to the corresponding lyness curves . in particular we find inside @xmath149 the curves with high rank and prescribed torsion given in @xcite or we can use the list of elliptic curve data " for curves in cremona form ( @xcite ) , taking advantage of the magma or sage softwares that allow to identify a given elliptic curve in it . as was explained in subsection [ grouplaw ] the curves @xmath65 are elliptic for all values of @xmath2 and @xmath79 except for @xmath150 , with @xmath151 given in ( [ hchc ] ) . on the curves corresponding to these values there could be , for the rational periodic orbits , some periods that are not in the list given by mazur s theorem . in this section we prove that no new period appears . [ genere0 ] the periods of the rational periodic orbits of @xmath30 lying on the curves @xmath65 for @xmath152 are @xmath153 and 12 . it is well known that the case @xmath154 is globally 6-periodic . so from now on we consider that @xmath155 we start the study of the rational cubic curves @xmath156 , where @xmath151 are given in ( [ hchc ] ) and moreover @xmath157 . by setting @xmath158 , we have @xmath159 note that @xmath154 implies that @xmath160 ; @xmath161 implies that @xmath162 ; and @xmath163 implies that @xmath164 . observe also that since @xmath2 and @xmath165 are in @xmath135 then @xmath166 by using again a computer algebra software we obtain the following joint parametrization of both curves @xmath156 : @xmath167 moreover @xmath168 on each curve @xmath156 the lyness map @xmath30 can be seen as @xmath169 where @xmath170 has to be determined . by using we obtain that @xmath171 is the linear fractional transformation @xmath172 where recall that @xmath166 therefore the dynamics of @xmath30 on each of the curves @xmath156 is completely determined by the dynamics of the maps @xmath170 . it is a well - known fact that if a linear fractional map , @xmath173 , has a periodic orbit of prime period @xmath174 , @xmath175 then it is is globally @xmath174-periodic . moreover this happens if and only if either : * @xmath176 and @xmath177 and in this case @xmath178 is @xmath179-periodic ; or * @xmath180 and @xmath181 is a primitive @xmath174-root of the unity and in this case @xmath178 is @xmath174-periodic . hence , apart of the fixed points , the maps @xmath170 can have @xmath174-periodic solutions , @xmath182 , if and only if @xmath183 the condition that @xmath184 is a primitive @xmath174-root of the unity , implies that @xmath185 it is also a well - known fact that the only rational values of @xmath186 , where @xmath148 is a rational multiple of @xmath187 are @xmath62,@xmath188,@xmath189 ( see ( * ? ? ? * theorem 6.16 ) or @xcite ) . this fact implies that @xmath190 the only allowed valued is @xmath191 , which implies that @xmath192 and so the corresponding map @xmath171 is not periodic . hence , only the fixed points of @xmath30 appear on this family of curves . concerning the case @xmath98 , observe that @xmath193 . when @xmath29 there are no periodic orbits on this level set . when @xmath194 the three straight lines forming this set are mapped one into the other in cyclical order by @xmath195 and so they are invariant under @xmath196 the cyclical order determined by @xmath30 is : @xmath197 the restriction of @xmath198 on @xmath199 is given by @xmath200 hence the dynamics of @xmath198 is determined by the dynamics of the linear fractional map @xmath201 one of its fixed points corresponds to the continua of three periodic points of @xmath30 given in table 1 . by using again the characterization of the periodicity of the linear fractional maps we obtain that @xmath171 is periodic only when @xmath202 it remains to study the case @xmath203 . in this situation @xmath204 and @xmath30 sends the straight line to the hyperbola and vice versa . so both level sets are invariant under @xmath205 the restriction of @xmath206 on @xmath207 is given by @xmath208 hence the dynamics of @xmath206 is determined by the linear fractional map @xmath209 which has fixed points only when @xmath210 they give rise to 2-periodic points of @xmath30 when @xmath211 and to a fixed point when @xmath212 arguing as in the case @xmath213 , the map @xmath171 is @xmath174-periodic , @xmath214 only when @xmath215 this happens if and only if @xmath216 and @xmath217 or equivalently when @xmath218 ( recall that the case @xmath154 is already considered ) . the case @xmath219 gives a @xmath61-periodic map @xmath171 and @xmath220 a @xmath221-periodic map . these cases correspond , respectively , to the existence of continua of @xmath222 and @xmath4 periodic points for @xmath30 on @xmath223 . from lemma [ genere0 ] we know that when @xmath152 the possible periods on @xmath65 are in the list given in the statement . for those points on the elliptic curves @xmath65 for all values of @xmath2 and @xmath224 we can apply mazur s theorem and we obtain that the only possible periods are the ones of the statement together with the period 4 . the points of ( prime ) period 4 can be discarded by observing that in subsection [ grouplaw ] we prove that @xmath225 it is also possible to perform a direct study with resultants of the system @xmath226 . from this study we get that that its only solutions are the ones corresponding to fix or 2-periodic points . these results together with the ones presented on table 1 prove the theorem . [ cols="^,^,^,^,^",options="header " , ] + table 2 . values of @xmath2 and rational initial conditions for the recurrences + with periodic sequences of several periods . [ incomp ] given @xmath227 the lyness map @xmath30 has not simultaneously rational periodic points with prime periods 1 and 12 , or 2 and 12 . from the results of subsection [ nonelli ] we know that these couple of periods do not coexist when the corresponding curves @xmath65 are not elliptic curves . so , from now one we can assume that the 12 periodic points lie on an elliptic curve @xmath65 . first of all , given @xmath228 , it is easy to obtain that there is a rational periodic point with period 1 if and only if @xmath229 is a square in @xmath135 and that there is one with period 2 if and only if @xmath230 is a square in @xmath135 . by using the expressions of @xmath126 given in section 2.1 ( or by using the known results for the tate curve @xcite ) , we get that the explicit parameterization of the values of @xmath2 and @xmath79 for which @xmath78 can have order 12 is @xmath231 for some @xmath232 . hence , to have a rational periodic points with periods 1 and 12 , we need rational numbers @xmath233 such that @xmath234 is a square in @xmath135 . we will show that this only happens for the values @xmath235 and @xmath236 , which do not give prime period 12 . multiplying by @xmath237 , we need to search for rational solutions of the equation @xmath238 . a standard change of variables , similar to the one used used in section [ periode9 ] , shows that this genus one curve is isomorphic to the elliptic curve with corresponding number in the cremona s list equal to 15a8 , which has only four rational points . these points correspond to the points with @xmath235 and @xmath239 . similarly , to have rational periodic points with periods 2 and 12 , we need rational numbers @xmath233 such that @xmath240 is a square in @xmath135 . arguing as in the previous case we arrive to the equation @xmath241 which number in cremona s list is 39a4 and has only the two rational points corresponding to @xmath242 . so the result follows . in order to find 9 or 10 rational periodic points for @xmath243 , one can not just naively search for points , since their coordinates are too big . so we use the following strategy . first , with the same formulas that in the proof of theorem [ nft ] , we transform the equations @xmath244 for their corresponding @xmath245 and @xmath246 to a weierstrass equation . we need to find non - torsion points on these elliptic curves . since the weierstrass equations have too big coefficients , we apply @xmath179-descent procedure with magma in order to get an equivalent quartic equation with smaller coefficients for the corresponding elliptic curves . we get that they are equivalent respectively to @xmath247 and @xmath248 these are still not sufficient simple to be able to find ( non - torsion ) rational points , so we do another transformation . in the first case , we apply @xmath61-descent in order to obtain another form , this case as intersection of two quadrics ( in the projective space ) . we get the equations @xmath249 @xmath250 finally , an easy search finds the point given in projective coordinates by @xmath251 $ ] . using the transformation rules given by magma one gets the corresponding point in @xmath252 , shown in table 2 . for the second equation , corresponding to the 10 rational periods , we directly transform the quartic equation to an intersection of two quadrics , and then we apply an algorism due to elkies ( @xcite ) to search for rational points in this type of curves ( as implemented in magma ) . the authors are partially supported by mcyt through grants mtm2008 - 03437 ( first author ) , dpi2008 - 06699-c02 - 02 ( second author ) and mtm2009 - 10359 ( third author ) . the authors are also supported by the government of catalonia through the sgr program . _ geometric unfolding of a difference equation _ , hertford college , oxford ( 1996 ) . unpublished paper . a video of the distinguished lecture , with the same title , at pims on march 21 , 2000 , is aviable at : ` http://www.pims.math.ca/resources/multimedia/video ` . the slides can be obtained at : ` http://zakuski.utsa.edu/\simgokhman/ecz/gu.html `
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consider the celebrated lyness recurrence @xmath0 with @xmath1 .
first we prove that there exist initial conditions and values of @xmath2 for which it generates periodic sequences of rational numbers with prime periods @xmath3 or @xmath4 and that these are the only periods that rational sequences @xmath5 can have .
it is known that if we restrict our attention to positive rational values of @xmath2 and positive rational initial conditions the only possible periods are @xmath6 and @xmath7 .
moreover 1-periodic and 5-periodic sequences are easily obtained .
we prove that for infinitely many positive values of @xmath8 positive 9-period rational sequences occur .
this last result is our main contribution and answers an open question left in previous works of bastien & rogalski and zeeman .
we also prove that the level sets of the invariant associated to the lyness map is a two - parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order @xmath9 including @xmath10 infinity .
this fact implies that the lyness map is a universal normal form for most birrational maps on elliptic curves . _
2000 mathematics subject classification : _ ` 39a20 , 39a11,14h52 . `
_ keywords : _ lyness difference equations , rational points over elliptic curves , periodic points , universal family of elliptic curves .
| 7,749 | 375 |
hetero - structures based on iii - nitrides @xcite and in particular on the combination al@xmath0ga@xmath1n / gan represent the basis of a variety of state - of - the - art ( opto)electronic devices like blue and white light - emitting diodes @xcite , laser diodes @xcite , blue lasers @xcite , high - power- @xcite , and high - electron - mobility - transistors @xcite . most of the above mentioned devices are commercially available and their performance continuously improved . furthermore , iii - nitrides doped with transition metals ( tm ) have also been the focus of considerable research efforts towards the demonstration of semiconductor spintronic functionalities @xcite . in this respect , while a remarkable number of reports on gan : mn provide an overview on the structural , optical , magnetic and electric properties of this material system @xcite , little is known about al@xmath0ga@xmath1n : mn @xcite and related nanostructures @xcite . recent findings @xcite indicate this alloy as particularly interesting for _ e.g. _ the self - assembling of functional multilayers and for having revealed the decisive role of mn as surfactant during the epitaxial growth of al@xmath0ga@xmath1n : mn , considerably enhancing the critical thickness of al@xmath0ga@xmath1n : mn on gan , and opening new perspectives for the realization of _ e.g. _ improved reflectors in gan - based laser structures . we report here on al@xmath0ga@xmath1n : mn grown by means of metalorganic vapor phase epitaxy ( movpe ) in a broad range of al concentrations and extensively investigated @xmath2 x - ray absorption spectroscopy ( xas ) , x - ray emission spectroscopy ( xes ) , energy - dispersive spectrometry ( eds ) , x - ray diffraction ( xrd ) , and high - resolution ( hr ) transmission electron microscopy ( tem ) , supported by density functional theory ( dft ) calculations . the results provide fundamental information on the microstructure and local environment in the layers and on the valence state of mn incorporated in the lattice over the whole range of al concentrations . the wurtzite ( wz ) al@xmath0ga@xmath1n : mn samples are grown in an aixtron 200rf horizontal - tube movpe reactor . all structures are deposited on @xmath3-plane sapphire substrates with trimethylgallium ( tmga ) , trimethylaluminum ( tmal ) , bis - methylcyclopentadienyl - manganese ( mecp@xmath4mn ) and ammonia ( nh@xmath5 ) as precursors for respectively ga , al , mn , n , and with h@xmath4 as carrier gas . the epitaxial process , developed from a well established procedure @xcite , consists of : ( i ) substrate nitridation ; ( ii ) low temperature ( 540 @xmath6c ) deposition of a gan nucleation layer ( nl ) ; ( iii ) its annealing under nh@xmath5 ; ( iv ) growth of a 1 @xmath7 m device - quality gan buffer deposited at 1020 @xmath6c ; ( v ) al@xmath0ga@xmath1n : mn layers at 850 @xmath6c , with the same tmga and mecp@xmath4mn flow rates and different - over the sample series - tmal flow rates ranging from 1 to 80 standard cubic centimeters per minute ( sccm ) . in order to have real time control over the entire fabrication process , the movpe system is equipped with an _ in situ _ isa jobin yvon ellipsometer that allows for both spectroscopic and kinetic measurements in the energy range 1.5 ev 5.5ev @xcite . the structures are routinely characterized by atomic force microscopy ( afm ) , secondary - ion mass spectroscopy ( sims ) and ( magneto)photoluminescence ( pl ) in order to get information on the surface roughness , chemical composition and magnetooptical response , respectively . measurements of squid magnetometry in the temperature range between 1.5 k and room temperature , confirm the samples to be paramagnetic . here , we focus on the effect of mn incorporation on the structural arrangement of al@xmath0ga@xmath1n : mn and on the local atomic environment of mn , with particular attention to the xrd and hrtem analysis as essential complement to the synchrotron xas and xes measurements . all considered al@xmath0ga@xmath1n : mn samples are listed together with their growth parameters in table [ tab : growth ] . the mn concentration in all doped layers is @xmath81% cations , as established by sims analysis . + .growth parameters for the al@xmath0ga@xmath1n : mn samples presented in this work . al concentration @xmath9 ( from xrd ) ; tmga and tmal flow rates and the pressure @xmath10 in the reactor during the process . the mecp@xmath4mn and nh@xmath5 flow rates are fixed at 490sccm and 1500sccm , respectively ; the substrate temperature during the growth of the gan buffer layer and during the deposition of the al@xmath0ga@xmath1n : mn layer are , respectively , 1020@xmath11c and 850@xmath11c . the nominal thickness is obtained from the kinetic ellipsometry spectra and confirmed by tem cross - sections . [ cols="^,^,^,^,^,^ " , ] a quantitative analysis @xmath2 a least - squares fit of the exafs data is then performed . due to the complexity of the system under investigation and in order to keep the correlation between the fitted variables as low as possible , a model with a minimum set of parameters to describe the whole al concentration range is found . this corresponds to the best fitting model and consists of a mn@xmath12 defect in al@xmath0ga@xmath1n expanded in three sets of single scattering paths : mn - n , mn - al and mn - ga , corresponding to the first three coordination shells . for each sample , the fit is performed in r - space , limited to the [ 13.5 ] range . both vgi and hgi data sets ( weighted by the noise level ) are included in a single fit in order to correctly account for the polarization effects . this permits to report the average bond distances for the out - of - plane ( vgi , parallel to @xmath3 ) and in - plane ( hgi , perpendicular to @xmath3 ) atomic configurations . the results are shown in table [ tab : exafs - fits ] and in supplementary fig . [ figs : exafs - fits ] . the model is built as follows : the passive electron reduction factor @xcite , @xmath13 , is fixed to the calculated value of 0.935 ; the coordination numbers for mn - n and mn - al are fitted , respectively , @xmath2 the variables @xmath14 and @xmath15 , while the coordination number of the second cation shell is constrained to sum to 12 ; a common debye - waller factor , which accounts for both the structural and thermal disorder , is fitted to @xmath16 for all single scattering paths ; three variables are employed for the mn - n , mn - al and mn - ga average distances , @xmath17 , @xmath18 and @xmath19 , respectively , with a common expansion / contraction factor in the two orthogonal directions ( vgi and hgi ) ; a common variable is fitted also for the shift of the edge energy , @xmath20 . this model permits to keep the numerical correlation between the variables below a 50% level . the r - factor of the fits ranges from 0.009 to 0.04 , affecting the propagated error bars , as reported in table [ tab : exafs - fits ] . several additional fitting models have been tested , either increasing the number of fitted variables or introducing additional scattering paths from other defects , as mn interstitials ( mn@xmath21 and mn@xmath22 ) . in all cases those models do not pass a f - test @xcite , meaning that the improvement in the fit quality is not statistically relevant . + , @xmath23 , @xmath18 and @xmath19 variables ( as reported in table [ tab : exafs - fits ] ) . the horizontal lines are the corresponding average bond distances for gan ( dashed ) and aln ( dotted ) . ] the exafs quantitative analysis indicates that the majority of mn atoms is in a mn@xmath12 configuration . on the other hand , the fitted percentage of @xmath24 does not correspond exactly to the percentage of mn@xmath12 in the samples . in fact , the absolute value of this variable , which represents the coordination of the first coordination shell ( mn-4n tetrahedron ) , is affected by the numerical correlation with @xmath16 and by the presence of nitrogen vacancies , as found in similar samples @xcite . for this reason , we rely on the results of the xld analysis , which is much more sensitive to the symmetry of the crystal , for determining the level of mn@xmath25 in the samples . nevertheless , a strong @xmath26-independent amplitude reduction of the exafs signal is obtained for the aln : mn sample ( # h ) . as shown by the tem micrographs , this sample has a columnar structure , thus the amplitude reduction is attributed to an increased local disorder , as it was demonstrated by exafs simulations combined with molecular dynamics calculations for mn nano - columns in ge : mn @xcite . the second percentage parameter,@xmath15 , is extracted from the fitted coordination number of the second coordination shell , keeping the constraint of 12 total neighbors ( ga / al ) dictated by the wurtzite structure . the results follow a linear dependence and match , within the error bars , with the al concentration found by xrd . furthermore , it is found that the average mn - n bond distance is larger than those of ga - n or al - n and is not affected by the al doping , while mn - al and mn - ga show a contraction going from gan : mn to aln : mn , as expected by the reduction of the lattice parameters . this implies that the lattice distortion introduced by the mn incorporation is local and mainly limited to the first coordination shell . + herfd - xanes spectra ( bottom ) for the al@xmath0ga@xmath1n : mn series with the corresponding simulations ( top ) for hgi ( left panel ) and vgi ( right panel ) geometries . the vertical dashed lines are guides to the eye of the main spectral features . ] in order to further confirm the local structural description obtained @xmath2 exafs analysis , the xanes region is investigated through _ ab initio _ simulations . in fig . [ fig : xanes ] the normalized k@xmath27 herfd - xanes spectra are shown together with their relative simulations ( using the fdmnes @xcite code ) for the hgi and vgi geometries . the herfd - xanes spectra correspond to a diagonal cut in the 2d resonant inelastic x - ray scattering ( rixs ) plane @xcite and can be approximated to a standard xanes spectra only in the region above the main absorption edge , where the spectral features arise from electric dipole transitions from 1@xmath28 to 4@xmath29 empty states of the absorbing atoms ( mn ) . on a first order approximation , this energy range can be described by multiple scattering theory employing simple muffin - tin potentials @xcite within a one - electron approach , _ i.e. _ the level of theory employed for the simulated spectra shown in this study . the spectral features present in the pre - edge region of the herfd - xanes spectra can not be fully described by the level of theory employed here and a quantitative analysis requires to account for the full rixs plane , not only for line cuts @xcite . nevertheless , the presence of an intense pre - edge peak in the k - edge xas spectra of 3@xmath30 transition metals is the fingerprint of tetrahedral ( t@xmath31 ) symmetry @xcite , due to allowed electric dipole transitions to the @xmath29-character of the @xmath32 spin - polarized 3@xmath30 states . the spectral features present in the xanes region do not correlate straightforward with a given coordination shell or scattering species , but are the result of full multiple scattering configurations . this induces an enhanced sensitivity to the geometry around the absorber . on the other hand , this also makes challenging to quantitatively model the xanes via _ ab initio _ methods . as shown in fig . [ fig : xanes ] , all the spectral features and the trend with increasing al concentration are reproduced by the simulations using a substitutional model based on the dft - relaxed supercells , rescaled to the experimental lattice parameters . to better evaluate the quality of each simulation , the supplementary figs . [ figs : xanes - subs ] and [ figs : xanes - ints ] show the comparison with experimental spectra for the nominal wyckoff sites and the dft - relaxed positions . the defects investigated are mn@xmath12 , mn@xmath22 and mn@xmath33 in al@xmath0ga@xmath1n . in order to get more quantitative results , a linear combination fit ( lcf ) analysis of the xanes spectra is performed . the constraints imposed are : the presence of the mn@xmath12 phase ; the number of components is limited to two ( one substitutional and one interstitial ) ; an energy shift for the interstitial phase is allowed ( fitted ) . all combinations among the four interstitial cells are performed and the fits are ranked by @xmath34 . in all samples / geometries it is found that the mn@xmath12 phase is @xmath3580% and the complementary phase is the non relaxed mn@xmath22 defect . on the other hand , the @xmath34 values of the best fits do not pass a statistical test ( f - test ) , meaning that the increase in the fit quality is not relevant . this confirms what previously found by exafs and the formation energies results of the dft , that is , that mn@xmath22 defect in al@xmath0ga@xmath1n is not stable and has a high formation energy . + a more quantitative analysis to establish the percentage of mn atoms incorporating as substitutional defects in the host matrix , is obtained by studying the xld spectra . it is established that xld is extremely sensitive to the symmetry of non - cubic sites @xcite and it was shown to be a powerful tool to determine the quality of substutional inclusions in dilute magnetic semiconductors @xcite . the xld spectra for the studied samples are reported in fig . [ fig : xld ] and are obtained from the difference between the herfd - xanes spectra in vgi and hgi geometries . the amplitude of the xld main oscillation at the edge position highlighted in fig . [ fig : xld ] is taken as a figure of merit for mn@xmath12 . in fact , the maximum xld amplitude would be obtained for 100% mn@xmath12 dilute in a perfect al@xmath0ga@xmath1n lattice . the mn@xmath22 interstitial shows a xld signal too , however it is not in phase with the mn@xmath12 xld signal and the resulting xld amplitude in the region of interest is reduced . as a reference for the 100% case , we arbitrarily rescale the experimental xld amplitudes to the xld amplitude at the ga k - edge of a gan : mn layer from ref . the results are reported in the inset to fig . [ fig : xld ] . the increasing values of the mn@xmath12 percentage for al @xmath3682% are due to the accuracy of the normalization procedure employed . in fact , a more accurate procedure would require to rescale the mn k - edge xld amplitudes to the ga k - edge ( or al k - edge ) xld amplitude measured for each sample in the same experimental conditions . on the other hand , the systematic errors are estimated to be within a @xmath3710% bandwidth . the dramatically low mn@xmath12 value for aln : mn can be safely attributed to an actual reduction of mn@xmath12 in this sample . + ga@xmath1n : mn series . the amplitude in the highlighted region is taken as a figure of merit for mn@xmath38 . inset : quantitative analysis of the results . ] as final point we discuss the mn valence state inferred from the integral of the absolute difference of the k@xmath39 xes data ( integrated absolute difference iad analysis @xcite ) . this method is preferred over the one employing the position of the main absorption edge for the possibility it gives to quantitatively follow the evolution of the effective spin moment on mn ( @xmath40 ) as a function of a given parameter and to directly compare the results with dft calculations @xcite . the total magnetic moment per unit cell calculated with dft is in all cases 4@xmath41 and corresponds to @xmath40@xmath82.0 , as found in the frame of a bader partitioning scheme @xcite . this result is confirmed experimentally , as reported in fig . [ fig : xes ] . the mn valence state is constant within the error bar for the whole series , with the exception of the aln : mn sample , as expected and supporting all previous results . xes data over the whole series of samples . inset : relative iad analysis . the vertical dashed line is a guide to the eye . ] we have carried out an extensive study of epitaxial al@xmath0ga@xmath1n : mn on a series of samples with al concentration up to 100% . by xrd we have found that the al content in the layers matches over the sample series the one expected from growth conditions . the lattice parameters as a function of the al concentration are also obtained by xrd . the dft computations on the formation energy for the incorporation of al in a gan matrix let us to conclude that al and ga are randomly distributed into the lattice , and in al@xmath0ga@xmath1n : mn the mn ions have the tendency to preferentially substitute for ga . the formation of mn interstitial defects is not favored . a coherent growth without local aggregation or precipitation is obtained for al@xmath0ga@xmath1n : mn with al concentrations up to 82% , confirming the surfactant role of mn already reported @xcite . synchrotron radiation xas has been employed to probe the local atomic and electronic structure of mn . from exafs , xanes and xld it is found that the majority of the mn ions is dilute , _ i.e. _ homogeneously distributed over the doped layers . an iad analysis of the xes data allows to determine the valence state of mn as constant up to an al concentration of 82% . due to the reduced lattice parameters with respect to _ e.g. _ gan : mn , enhanced hybridization of the orbitals can be expected in al@xmath0ga@xmath1n : mn , making it a material system worth to be investigated in view of spintronic functionalities . moreover , this work paves the way to the understanding and control of the role played by mn in particular and transition metals in general on the structure and properties of the alloys al@xmath0ga@xmath1n : tm . significantly , the incorporation of mn has been found to promote the growth of al@xmath0ga@xmath1n on gan , to defer the relaxation of the layers and to increase the critical thickness also for al concentrations up to 82% , with remarkable potential effects on the fabrication of _ e.g. _ distributed bragg mirrors for iii - nitride - based optoelectronic devices . the authors gratefully acknowledge the european synchrotron radiation facility ( esrf ) for providing synchrotron radiation beam - time ( proposal he-3825 ) . this work was supported by the esrf trainee program , by the austrian science fundation ( fwf projects 24471 and 26830 ) , by the nato science for peace programme ( project 984735 ) , by the eu 7@xmath42 framework programme through the capacities project regpot - ct-2013 - 316014 and by the european research council ( advanced grant 22790 ) . + * supplemental information * in the plots of fig . [ figs : exafs - fits ] , the quality of the fits performed on the exafs data for both vgi and hgi geometries is reported . + + + the quality of the simulated xanes spectra for mn@xmath12 is shown in fig . [ figs : xanes - subs ] , while the quality of the simulated xanes for the mn@xmath22 and mn@xmath21 interstitials in al@xmath0ga@xmath1n is given in fig . [ figs : xanes - ints ] .
|
synchrotron radiation x - ray absorption and emission spectroscopy techniques , complemented by high - resolution transmission electron microscopy methods and density functional theory calculations are employed to investigate the effect of mn in al@xmath0ga@xmath1n : mn samples with an al content up to 100% .
the atomic and electronic structure of mn is established together with its local environment and valence state .
a dilute alloy without precipitation is obtained for al@xmath0ga@xmath1n : mn with al concentrations up to 82% , and the surfactant role of mn in the epitaxial process is confirmed .
+ = 0 = 0
| 6,037 | 171 |
the atacama large millimeter array ( alma ) is one of the largest ground - based astronomy projects of the next decade , which will revolutionize several fields of astronomy . a large community of scientists is expected to use alma to tackle several outstanding questions in astrophysics . however , mm / submm astronomy is often considered a field restricted to experts . in the case of students and young scientists in particular , the limited familiarity with mm / submm facilities and observations may prevent them to fully exploit the alma capabilities in the future . these lecture notes are aimed at providing students and young researches some background on mm / submm extragalactic astronomy , with a focus on the investigation of agns . i will first provide a quick overview of the current results obtained through extragalactic mm / submm observations , by focusing on agns ( [ sec_mm_astronomy ] ) . i will then summarize the currently available ( and forthcoming ) mm - submm facilities ( [ sec_current_facilities ] ) . then i will shortly describe alma and summarize its observing capabilities ( [ sec_alma ] ) . finally , i will discuss some of the alma prospects for extragalactic studies , and in particular for agns , both in the local universe and at cosmological distances ( [ sec_alma_prospects ] ) . these lecture notes are far from being exhaustive ; several scientific cases will not be discussed at all ; the main goal of these notes is only to provide an introduction to mm / submm extragalactic astronomy and to highlight some scientific cases that alma will be able to tackle . this branch of astronomy includes observations at wavelengths between @xmath010 mm and @xmath0300 @xmath1 m . longer wavelengths are traditionally identified as radio - astronomy domain . shorter wavelengths , out to mid - ir wavelengths , are unobservable from ground because of the nearly complete atmospheric absorption ( although some sites , under exceptional conditions , allow observations out to @xmath2 m . ) . even within the mm - submm range not all wavelengths are equally easy to observe , since the sky transparency on average decreases rapidly at shorter wavelengths . at @xmath3 only a few atmospheric windows are accessible , and only under optimal weather conditions . this issue is clearly illustrated in fig . [ fig_atm_transmission ] , which shows the atmospheric transmission at the alma site . the main source of opacity at these wavelengths is the water vapor . this is the reason for locating mm - submm observatories at dry and high altitude sites , where the amount of water vapor is much reduced . however , even at these optimal sites there are strong variations of the the water vapor , which make the atmospheric transmission change strongly ( fig . [ fig_atm_transmission ] ) both on long ( seasonal ) and short ( day / night ) time scales . given the difficulties of observing at these wavelengths one may wonder why international agencies are investing so much effort to develop facilities with enhanced observing capabilities in these bands . the mm - submm band contains a wealth of information that can not be inferred from any other band . most of the @xmath0150 molecules known so far in the _ cold _ interstellar medium ( see http://astrochemistry.net for an updated list ) emit their rotational transitions in the mm - submm bands , with a density of about 70 lines / ghz . all of these transitions are important diagnostics of the chemistry , of the physics and of the dynamics of the inter stellar medium ( ism ) from which stars form . some of these lines are so strong ( e.g. the co transitions ) to be powerful tools to trace the dynamics and the gas physics even in distant galaxies . furthermore , some of the strongest lines emitted by the ism of any galaxy , such as the [ cii]158@xmath1 m and the [ oi]63@xmath1 m fine structure lines ( the two main coolants of the ism ) , are redshifted into the mm - submm bands at z@xmath424 . within the context of the continuum emission , the mm - submm bands encompass the rayleigh - jeans region of the warm dust thermal emission ( which traces star formation and the dust mass ) , the high frequency tail of the synchrotron emission ( dominating the radio emission in most galaxies ) and of the free - free emission ( tracing hii regions ) . at high redshift the prominent ir dust thermal bump ( which dominates the spectral energy distribution sed in starburst galaxies ) is shifted into the submm band , therefore making this one of the best spectral regions to search and characterize high - z star forming galaxies . this was just a very quick glance at the scientific motivations behind the development of mm - submm facilities , and mostly limited to the extragalactic field . young stellar objects , protostars and proto - planetary systems are , for instance , additional fields where the mm - submm range is crucial for a thorough investigation . the importance of the mm - submm band within the extragalactic context will become more obvious in the following sections , where i will provide some ( shallow ) background on what we currently know of external galaxies based on mm - submm observations , and where some extragalactic alma science cases will be discussed . on the technical side , it is important to mention that the ( sub)mm is currently the shortest wavelength where sensitive , many - elements coherent detection interferometers are feasible from the ground . these can simultaneously provide high angular resolution , sensitivity , and image reconstruction fidelity . direct detection interferometers at shorter wavelengths ( e.g. mid / near - ir ) can achieve similar angular resolution , but are more severely constrained in terms of sensitivity and image fidelity . m ( _ right _ ) . note that most of the far - ir emission comes from a region that is heavily obscured at optical wavelengths . credit of the space telescope science institute ( optical hst image ) and of c. dowell ( submm cso image ) . ] the warm dust emitting at far ir wavelengths is mostly heated by the uv radiation field of young massive stars in star forming regions . as a consequence , the far infrared luminosity @xmath5 and its submm rayleigh - jeans part are considered good tracers of star formation in galaxies . in particular , these bands are useful to trace obscured star formation , since they are virtually unaffected by dust extinction . this is evident in fig . [ fig_antennae ] , where the 350@xmath1 m map of the interacting galaxies `` antennae '' ( obtained at the cso telescope , c. dowell , priv . is compared with the optical hst 20@xmath6 ) . both these issues will no longer be a problem with alma , which will have sensitivities orders of magnitude better and an angular resolution similar to hst . as already mentioned , also most of the _ gas _ phase of the cold ism emits in the mm - submm range . more specifically , it is in this band that most of the molecular gas transitions are observed . however , _ cold _ molecular hydrogen h@xmath7 ( by far the most abundant molecule in the cold gas phase ) can not be detected directly , since it has no electric dipole moment ( therefore rotational transitions with @xmath8 are not allowed ) . carbon monoxide co is the second most abundant molecule : its rotational levels are excited by collisions with h@xmath7 , producing the brightest molecular lines in the spectrum of any galaxy . the luminosity of the co rotational transitions , and in particular the fundamental one j(1@xmath90 ) , are widely used as tracers of the molecular gas mass through a linear relation : @xmath10 . however , the conversion factor @xmath11 is found to depend on the gas metallicity as well as on its physical conditions ( temperature and density ) . the critical density of the co transitions is relatively low , at least for the low rotational levels . for instance , the critical density of co ( 10 ) ( at 115 ghz ) is about @xmath12 . as a consequence , co is generally little effective in tracing the high density regions of molecular clouds . moreover , the large optical thickness the co lines ( at least for the low rational levels ) prevents us to penetrate dense molecular regions . dense regions are better traced by other species , such as hcn , hco@xmath13 and cs , which are characterized by higher critical densities for transitions in the same frequency bands as the co ones . for instance hcn ( 10 ) , hco@xmath13 ( 10 ) and cs ( 21 ) ( observable in the 3 mm band ) , have critical densities of @xmath14 , @xmath15 and @xmath16 , respectively . however , these lines are typically one order of magnitude fainter than co. , indicated by the solid line . the right - hand axis translates the far - ir luminosity into star formation rate . _ right : _ @xmath17 as a function of the far ir luminosity . the right - hand axis gives the gas exhaustion timescale , where @xmath11 is the @xmath18to h@xmath7 conversion factor ( @xmath19 ) . both figures are from @xcite.,title="fig : " ] , indicated by the solid line . the right - hand axis translates the far - ir luminosity into star formation rate . _ right : _ @xmath17 as a function of the far ir luminosity . the right - hand axis gives the gas exhaustion timescale , where @xmath11 is the @xmath18to h@xmath7 conversion factor ( @xmath19 ) . both figures are from @xcite.,title="fig : " ] there is a relationship between infrared luminosity and co luminosity , which is by itself not surprising ( as most luminosity - luminosity relations ) . more interesting is the fact that the relation is not linear , being @xmath20 ( fig . [ fig_fir_co ] , from @xcite ) . since @xmath5 is proportional to the star formation rate sfr , while @xmath21 is proportional to the molecular gas content m(h@xmath7 ) ( which is the fuel for star formation ) , the non linear relation implies that the star formation efficiency , defined as @xmath22 increases with the luminosity of the system . the inverse of the star formation efficiency is the gas exhaustion timescale , @xmath23 , i.e. the time required by the starburst to totally consume the available molecular gas , if the star formation proceeds at the rate currently observed . the non linear relation between @xmath5 and @xmath21 implies that the gas exhaustion timescale decreases with luminosity , reaching values as low as @xmath24 , as illustrated in fig . [ fig_fir_co ] . it is clear that the most powerful starbursts are also short lived . however , it should be mentioned that this picture may be biased , especially for what concerns high redshift sources , due to the incomplete census of galaxy populations . indeed , @xcite have recently identified some _ powerful _ starburst galaxies characterized by _ low _ sfe at z@xmath02 . the far - ir luminosity appears to be _ linearly _ correlated with the luminosity of dense gas tracers , such as hcn . the relation is linear over a very broad luminosity range spanning from galactic star forming regions to powerful starburst galaxies @xcite . this result suggests that star formation always occurs in dense molecular clouds , and that star formation in starburst systems may simply be a scaled up version of galactic star formation . however , also the linearity between @xmath5 and @xmath25 has been recently questioned through indications that the ratio between these two quantities increases with the luminosity @xcite . moreover , the presence of an agn may also affect the intensity of the hcn emission , as discussed in [ sec_past_agn_local ] . interferometers allowed astronomers to map the molecular gas in large samples of galaxies , generally by exploiting the co lines , but more recently also through transitions of various other species . the co emission in galaxies show a variety of morphologies : spiral patterns , bars , rings , nuclear concentrations and irregular distributions @xcite . the co maps also reveal important kinematic information . in the case of disk or ring - like morphologies the gas kinematics is generally dominated by ( nearly ) regular rotation . in the case of barred galaxies , the molecular gas kinematics is often characterized by prominent streaming motions along the stellar bar ( e.g. * ? ? ? * ; * ? ? ? this is regarded as direct evidence that the non - axisymmetric potential of a bar is effective in funneling gas into the central region ( which may eventually produce a central starburst , or be further driven into the nuclear region to fuel an agn ) . agns generally heat their circumnuclear dust to temperatures much higher than starburst galaxies . indeed , active nuclei are generally characterized by strong mid - ir and near - ir emission , indicating dust temperatures of several hundred degrees and reaching the dust sublimation temperature ( 15002000 k ) . at far - ir and submillimeter wavelengths the relative contribution of agn and star formation to the dust heating is debated . spectral decomposition techniques , as well as the correlation between far - ir emission and other tracers of star formation , suggest that the far - ir and submm emission is dominated by star formation in the host galaxies , even in powerful qsos @xcite . disentangling _ spatially _ the far - ir / submm emission due to the agn from the host galaxy emission is very difficult , if not impossible , with current facilities due to the lack of angular resolution at these wavelengths . disentangling the two components is important not only to determine the contribution of agns to the far - ir / submm radiation , but also to constrain models of the obscuring dusty torus , as we shall see in [ sec_alma_local ] . identifying the main mechanism responsible for fuelling agns has been one of the hottest topics in the last decade . non - axisymmetric potentials introduced by stellar bars and galaxy interactions were considered as promising mechanisms , but various studies failed in finding any excess of these morphologies in agns . co observations offer the possibility of directly witnessing the gas fuelling towards the nucleus . intensive campaigns have been performed with millimetric interferometers to investigate this issue by mapping the co emission in galaxies with different types of nuclear activity ( e.g. the nuga project , * ? ? ? * ; * ? ? ? * ; * ? ? ? the main result is that there is no evidence for systematic differences , in terms of molecular gas distribution and kinematics , between galaxies hosting agns and quiescent ones . in particular , seyfert galaxies appear characterized by a wide variety of molecular gas distributions : streaming motions along stellar bars , rings , nuclear concentrations , nuclear voids and irregular distributions . the lack of any relationship between the presence of an agn and the co morphology / dynamics , indicates that there is no ubiquitous evidence for current fuelling of the agn . one possibility , is that the large scale fuelling phase and the agn phase may not be simultaneous . another consideration is that local agns have low luminosities and do not require large fuelling rates from the host galaxies . more specifically , most local seyfert nuclei are characterized by black hole accretion rates of about @xmath26 ; at these rates even a single molecular cloud of @xmath27 can keep the nucleus active for about 1 gyr . the fuelling problem is more serious for powerful qsos , where the accretion rates may exceed @xmath28 . in qsos some mechanism capable of funneling molecular gas from the host galaxy into the nuclear region is actually required . however , qsos are much more distant than seyfert galaxies . the limited sensitivity and angular resolution of current mm interferometers hampers our capability of mapping the molecular gas distribution in qso hosts , which will instead be easy with alma . ngc 1068 is one of the best studied agns also at mm - submm wavelengths . at the highest resolution currently achievable the nuclear co distribution shows a broken ring - like structure , about 2@xmath6 in size ( corresponding to 140 pc ) , as illustrated in fig . [ fig_n1068 ] @xcite . at the specific location of the agn ( cross in fig . [ fig_n1068 ] ) little co emission is observed . does this mean that little or no molecular gas is present in the vicinity of the agn ? this is somewhat contrary to expectations , given the need of gas in the vicinity of the agn both to feed it and to provide the heavy obscuration characterizing ngc 1068 @xcite . actually , we know that in the vicinity of the agn , on scales of 110 pc , dense molecular gas and dust are present , based on radio and mid - ir interferometric observations @xcite . possibly most of the molecular gas in the nuclear region is dense and warm , and therefore not properly sampled by the co lines ( at least not by the low rotational transitions ) . maps of other lines tracing dense gas , such as hcn , have lower angular resolution @xcite , but do show a much higher concentration on the nucleus with respect to co. @xcite detected several molecular species in the nucleus of ngc 1068 tracing not only dense gas , but also a very complex chemistry . these findings indicate that the agn has created a giant `` x - ray dominated region '' ( xdr ) . hard x - ray photons emitted by the agn can penetrate deep into the circumnuclear molecular clouds and keep the temperature high over an extended region . the high gas temperature in xdr s favors the formation of various molecular species such as hcn @xcite . in contrast , photo dissociation regions ( pdr ) , which are generated by the uv photons of star forming regions , are characterized by a much narrower region with enhanced temperature , making the production of various molecular species much less efficient than in xdr . these fundamental differences between xdr s and pdr s suggest that xdr - enhanced species can be used to unveil the presence of heavily obscured agns that escaped detection at other wavelengths . within this context @xcite and @xcite developed a diagnostic diagram involving the line ratios hcn / hco@xmath13 versus hcn / co , where pure seyfert nuclei are clearly separated from starburst nuclei , in the sense that the former show enhanced hcn emission ( but see also caveats discussed in * ? ? ? * ) . this and other complementary diagnostic diagrams will be usable with alma to identify obscured agns even in distant galaxies . ( scaled from the sed of m82 ) at different redshifts ( z@xmath292 , 5 , 7 and 10 , as labelled ) . note that the observed flux at @xmath30 changes very little . the thick blue and red lines show the alma and jwst sensitivities for continuum detection at 5@xmath31 with an integration of @xmath32 . ] as the spectrum of a starburst galaxy moves in redshift , the global flux is reduced according to the 1/d@xmath33 cosmological dimming , but the intrinsic luminosity observed at a fixed mm - submm wavelength increases as a consequence of the steeply rising submm continuum . at z@xmath41 such strong negative k correction counteracts completely the cosmological dimming , so that detecting a source at z@xmath2910 is as easy as detecting a source at z@xmath291 ( for a given intrinsic luminosity ) . this effect is illustrated in fig . [ fig_sed_z ] , which shows the observed flux distribution of a star forming galaxy at different redshifts . the flux observed at @xmath34 mm is essentially unchanged within the redshift range @xmath35 . [ fig_kneg ] shows the same effect by plotting , for a given starburst template , the observed flux at various wavelengths ( and specifically in the alma bands ) as a function of redshift . the observed flux remains nearly constant in the redshift range 1@xmath36z@xmath3610 at wavelengths longer than about 800@xmath1 m . at wavelengths shorter than @xmath0500@xmath1 correction is not strong enough to compensate for the cosmological dimming . at wavelengths longer than @xmath02 mm the k - correction is strong , but the observed flux is more than one order of magnitude fainter than observed at 1 mm ; moreover , at @xmath37 there is an increasing `` risk '' of contamination by non - thermal sources . ( by scaling the m82 template ) observed in the different mm - submm alma bands . note the nearly flat trend at 1@xmath36z@xmath3610 at wavelengths close to @xmath01 mm . the predicted flux density in the optical ( @xmath38 ) is also shown for comparison . ] an interesting consequence of the strongly negative k correction at mm - submm wavelengths is that in any deep field one expects to see many more galaxies at z@xmath41.5 than at lower redshifts . this is just the opposite of what observed in the optical , where the distribution of observed fluxes is dominated by the cosmological dimming ( fig . [ fig_kneg ] ) , which makes any optical deep field dominated by galaxies at z@xmath361.5 , and only a small fraction of galaxies at higher redshifts . the negative k correction allowed the discovery of a large number ( a few 100 ) of starburst galaxies at high - z , thanks to extensive surveys exploiting array of bolometers available on single dish telescopes ( see e.g. * ? ? ? * ; * ? ? ? * for a review ) . dusty starburst galaxies at high - z discovered through the detection of their submm continuum are often dubbed as sub - millimeter galaxies ( smgs ) . although , high - z dusty starbursts have been also discovered through observations in the mm band , for sake of simplicity we will refer to the whole population as `` smgs '' . much effort has been invested for several years in the identification of the optical ( or near - ir ) counterparts of smgs , mostly with the goal of determining their redshift through spectroscopic followup . however , as we will discuss in [ sec_current_facilities ] , the angular resolution of single dish telescopes is so low ( 11@xmath618@xmath6 ) that several optical / near - ir candidate counterparts are found within the telescope beam . the optimal way to identify the true counterpart would be to obtain mm - submm observations at higher angular resolution with a mm - submm interferometer . however , as discussed in [ sec_current_facilities ] , the sensitivity to continuum of current mm - submm interferometers is low , making this approach very expensive in terms of observing time , especially if a statistically meaningful sample is needed . @xcite employed the alternative strategy of exploiting deep radio vla observations . given the tight correlation between far - ir thermal emission and radio synchrotron emission observed in galaxies , most smgs should be associated with a detectable radio source . the higher angular resolution achievable with the vla allowed @xcite to locate the position of the smgs with an accuracy high enough to position the slit for the spectroscopic identification . note also that the large vla primary beam , resulting into a large field of view , allowed the simultaneous radio detection of many smgs in each single pointing . the identification of a prominent ly@xmath11 in many of the deep spectra provided the redshift for several tens smgs . the smgs redshift distribution resulting from these surveys is shown in fig . [ fig_smg_z ] ( left ) , and it is characterized by a median redshift @xmath39 . the redshift distribution is very similar to that of qsos , suggesting a link between the two populations of objects . the sample of smgs for which a redshift has been measured is limited to a submm flux of a few mjy . the inferred redshifts imply far - ir luminosities of several times @xmath40 , i.e. extreme ultra - luminous infrared galaxies ( more luminous , on average , than local ulirg samples ) . therefore , currently identified smgs represent only the most luminous population of high - z dusty starbursts ( the tip of the iceberg ) . in fig . [ fig_smg_z ] ( right ) the inferred contribution of smgs to the density of star formation ( sfrd ) is compared with those inferred by uv / optical surveys and radio / ir tracers @xcite . smgs are indicated with large squares , while small squares show an attempt to account for the incompleteness due to the fact that current surveys only sample the most luminous sources , as discussed above . clearly , once incompleteness is accounted for , smg contribute significantly to the history of star formation , especially at redshifts around two . again , their evolution appears very similar to that of qsos , suggesting a close link between smgs and the evolution of massive halos which hosts qsos . the redshift determination of smgs allowed also their millimetric spectroscopic followup with interferometers aimed at the detection of molecular transitions ( which requires an accurate knowledge of the redshift due to the limited bandwidth ) . intensive campaigns , especially with the iram pdb interferometer yielded the detection of co rotational transitions for several smgs @xcite , indicating that these galaxies also host huge amounts of molecular gas . the profile of the co lines is often double - peaked , indicative of a rotating disk or of a merging of two galaxies . followup interferometric observations at higher angular resolution have found that both the continuum and line emission are very compact in most smgs , a few kpc or even less @xcite . the compactness of the smgs provides constraints on the inferred central densities , which result much higher than in any other population of starburst galaxies , but comparable to those of compact passive galaxies found in the same redshift range . these findings , along with the huge inferred star formation rates , suggest that smgs are progenitors of local massive ellipticals , experiencing their main episode of star formation , but which must undergo further structural evolution in order to reach the sizes observed in local ellipticals . these interferometric data also provide constraints on the global dynamical mass , typically yielding masses of a few times @xmath41 ( e.g. * ? ? ? * ; * ? ? ? interestingly , the inferred high density of massive galaxies ( @xmath42 ) at z@xmath02 is difficult to reconcile with classical hierarchical models of galaxy evolution . millimetric and submillimetric observations of distant agns are currently limited mostly to powerful qsos ( although we will later discuss the case of lower luminosity agns discovered in smgs ) . bolometric observations of large qso samples have detected mm - submm continuum emission at the level of a few mjy in about 60 optically selected qsos and about 20 radio galaxies at z@xmath41 , out to z=6.4 @xcite . generally the detection rate achieved with current facilities is about 30% among bright qsos . detected qsos have far - ir luminosities of about @xmath43 , but these clearly represent the tip of the iceberg due to our current sensitivity limits . m luminosity ( which is a tracer of star formation ) and far - ir luminosity ( inferred from the mm / submm emission in high - z qsos ) . high - z mm - bright qsos follow the same relation of local starburst ulirgs , indicating that the far - ir emission of the former is mostly due to star formation ( from * ? ? ? ] for a fraction of high - z qso the inferred far - ir luminosity follows the same correlation with the radio luminosity as local star forming galaxies , suggesting that the far - ir luminosity is powered mostly by star formation @xcite . this result is supported by the spitzer mid - ir detection , in a number of mm - bright qsos , of strong pah emissionm . these species are mostly destroyed in the strong radiation field produced by agns . ] , which is considered a good tracer of star formation @xcite . moreover , the pah luminosity is found to correlate with the far - ir luminosity and following the same relation of starburst galaxies ( fig.[fig_pah ] ) , further supporting the starburst origin of the far - ir emission . the star formation rates inferred from the far - ir luminosity ( as well as from the pah luminosity ) are as high as a few times @xmath44 . this suggests that in mm - bright qso we are witnessing the simultaneous growth of black holes ( as traced by the optical and x - ray agn emission ) and of the stellar mass in their host galaxy ( as traced by the far - ir ) , which will probably evolve into massive ellipticals @xcite . such a co - coeval growth is expected in models of bh - galaxy evolution aimed at explaining the local relation between spheroids and bh mass @xcite . however , the relation between star formation and black hole accretion ( i.e. the relation between @xmath5 and @xmath45 ) seems to saturate at high luminosities @xcite , suggesting that at high - z the bh growth proceeds more rapidly than the host growth . ) already present in this object close to the epoch of re - ionization . adapted from @xcite . ] the high far - ir luminosities inferred from the mm - submm observations also imply large amount of dust in the host of high - z qsos . accurate measurements of the dust mass are however difficult to obtain due to the unknown dust temperature and emissivity . multiple band observations greatly help to remove the degeneracy between these quantities @xcite . fig.[fig_j1148_dust ] shows the rest - frame infrared sed of one of the most distant qsos ( z=6.4 ) known so far , for which the far - ir thermal bump is relatively well sampled thanks to various submm and mm observations . the inferred dust masses are as high as several times @xmath46 . the discovery of such huge masses of dust in the most distant qsos is puzzling . indeed , in the local universe the main source of dust are the atmospheres of evolved stars ( mostly agb ) . however , at z@xmath46 the age of the universe is less than 1 gyr , which is the minimum time required for agb stars to evolve in large numbers and to significantly enrich the ism with dust . this suggests that in the early universe other mechanisms dominate the dust production . core - collapse sne are candidate sources of dust on short timescales . observations have indeed found dust formed in sn ejecta @xcite , as also expected on theoretical grounds @xcite . the extinction curve inferred in high - z qsos and galaxies appears in agreement with that expected for dust produced in sn ejecta @xcite , supporting the idea that sne may indeed be the major source of dust in the early universe . however , even with the highest dust yield observed so far , the huge mass of dust inferred from mm - submm observations of the most distant qsos remains difficult to account for @xcite . extensive reviews on the co emission in high - z objects , and in agns in particular , are given in @xcite and in @xcite . more than half of the @xmath040 co detections obtained at high redshift ( z@xmath41 ) are in agns ( qsos or radio galaxies ) . in particular , all of the detections at z@xmath43.5 are in agns . the higher detection rate in high - z qsos is mostly due to their huge far - ir luminosities and to the co - fir correlation ( [ sec_past_normal_local ] ) . the molecular gas masses inferred from the co detections are of the order of a few times @xmath47 . except for the huge amounts of molecular gas , the general properties of the co emission in high - z qsos and radio - galaxies do not differ strongly from local and lower redshift powerful starbursts . the observation of multiple co transitions suggests that the co excitation temperature is higher in some qsos ( e.g. * ? ? ? * ) , but the statistics are still very low . the co emission follows the same trend with the fir luminosity observed in fig . [ fig_fir_co ] , indicating that qso host galaxies are experiencing a strong starburst event in the process of rapidly exhausting the available molecular gas , on a time scale of only @xmath48 . it is interesting to note that the relation between co luminosity ( tracing the amount of molecular gas ) and optical luminosity ( tracing the black hole accretion rate in qsos ) is not linear , as shown in fig . [ fig_co_opt ] @xcite . this result indicates that , while the black hole can accrete at very high rates ( limited only by its eddington luminosity ) , the host galaxy has only a limited amount of molecular gas available for star formation ( given by the galaxy evolutionary mechanism ) ; hence the two formation processes probably occur on different time scales ( as already discussed above ) . deep observations have allowed the detection of transitions from other molecular and atomic species , such as hcn , hco@xmath13 , hnc , cn an ci , which are tracers of high density gas and of the gas chemistry and excitation ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? however , these detections are limited to very few bright sources , due to the limited sensitivity of current facilities . within the limited statistics available , the intensity of these lines relative to co and to @xmath5 do not differ strongly from lower redshift and local starburst galaxies ( although there are indications that the intensity of hcn decreases at high luminosities ; * ? ? ? * ; * ? ? ? 158@xmath1 m fine structure line in one of the most distant quasars known , j1148 + 5152 , at z=6.4 @xcite . _ lower panel : _ the co(65 ) transition observed in the same objects is shown for comparison @xcite . note the different flux scales on the two panels . ] recently , deep observations achieved the first detection of the [ cii ] fine structure line at 158@xmath1 m in two high - z qsos ( fig . [ fig_cii ] ) @xcite . this line is emitted by photo - dissociation regions ( pdrs ) in star forming galaxies , and it is the strongest line in the spectrum of nearly any galaxy . the ratio @xmath49}/l_{fir}$ ] observed in these two high - z qsos is about @xmath50 , i.e. similar to the value observed in local ulirgs . the detection of [ cii ] with a luminosity relative to @xmath5 similar to local sources suggests that these high - z systems have already been enriched with heavy elements , and with carbon in particular @xcite . generally the angular resolution and sensitivity are not good enough to trace co rotational curves and therefore to obtain accurate constraints on the dynamical mass . however , in some high - z bright agn the kinematics is resolved , especially in lensed cases , where the lensing shear helps to resolve the galaxy structures ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? in other unresolved , or marginally resolved qsos the width of the co line , along with measurements ( or limits ) on the size of the co emitting region provide some constraints on the dynamical mass ( e.g * ? ? ? typical dynamical masses inferred for qso hosts and radio galaxies are of the order of a few times @xmath41 . however , we should recall that current observations sample only the `` tip of the iceberg '' of the agn population at high redshift . recently , @xcite attempted to calibrate the width of co lines in galaxies as a proxy of their dynamical masses . once this calibration is applied to the host of qsos , the resulting galaxy mass can be compared with the black hole mass inferred from their optical - uv emission lines . an interesting result is that high - z qsos appear to deviate from the local bh - galaxy mass relation , in the sense that their hosts are less massive than expected . this result is in agreement with other independent indications that the bh - galaxy mass relation evolves at high redshift @xcite . however , one should be careful when using the co line width as a tracer of the dynamical mass in ( unobscured ) qsos . indeed , selection effects make qsos hosts to be viewed preferentially face - on , since edge - on views tend to be obscured . as a consequence , the observed width of the co line in qsos is likely reduced due to projection effects @xcite , and may not be representative of the intrinsic rotation of the co disk @xcite . corresponds to about 800 pc . ] an interesting particular result is the co(32 ) map of one of the most distant qsos , j1148 + 5152 , at z=6.4 ( fig . [ fig_j1148_map ] ) @xcite . the map shows resolved emission extending over about 5 kpc and two central peaks separated by 1.7 kpc . from the velocity structure @xcite infer a dynamical mass of about @xmath51 . this mass is much lower than the mass of a few times @xmath52 that would be expected from the black hole mass derived for the same object ( a few times @xmath53 ) if applying the local bh - bulge mass relation . this result would provide further support to the scenario where at high - z black holes grow faster then their host galaxies , while the latter reach the local bh - bulge relation only at lower redshifts . however , observations at higher angular resolution and with higher sensitivity are certainly required to better determine the dynamical mass , and remove the degeneracy with the inclination angle of the putative disk . above we have discussed mm - submm observations aimed at specifically investigating objects known to host powerful agns ( qsos or radio - galaxies ) . we can however consider the orthogonal approach : do galaxies discovered at mm - submm wavelengths ( smgs ) host agns ? this issue has been investigated with the aid of deep x - ray observations . @xcite found that most smgs do host a x - ray source whose luminosity and spectral shape can only be ascribed to the presence of an obscured agn . however , even when corrected for absorption , the inferred agn luminosity can not account for far - ir luminosity observed in smgs , hence the main source of far - ir emission in smgs is due to star formation . similar results were obtained more recently thanks to spitzer mid - ir spectroscopy @xcite . the detection of strong pahs in the spectra of most smgs supports the scenario where most of the luminosity is due to star formation . the detection of a weak mid - ir continuum due to hot dust does trace the presence of an obscured agn in some smgs , but with a bolometric power much lower than the starburst component . summarizing , smgs represent a critical phase in the formation of massive galaxies ; black hole accretion is accompanying this major star formation event , but at a rate much lower than observed in qsos . millimetric and submillimetric facilities can be divided in two main classes : single dish telescopes and interferometers . these two classes of facilities are often complementary , especially in extragalactic astronomy , one generally being characterized by wider field of view and higher sensitivity to continuum , the other providing higher angular resolution and higher sensitivity for line detection . this section provides a short overview of the current facilities in both classes , by also discussing their main relative advantages and limitations . a list of the main currently available single dish mm - submm telescopes is given in tab . [ tab_sing_dish ] . current single dish telescopes have a diffraction limited angular resolution of only about 1020@xmath6 . however , one of the main advantages of single dish telescopes relative to interferometers is their wide field of view , which makes single dish telescopes optimally suited for wide area surveys . all single dish telescopes are equipped with various heterodyne receivers for spectroscopy . however , an additional advantage of single dish telescopes is the possibility of using bolometers , which ( thanks to their wider band width ) are more sensitive to the continuum emission than heterodyne receivers . to fully exploit the large field of view for the detection of mm - submm sources , observatories have developed cameras hosting large bolometer arrays that allow the simultaneous observation of wide areas of the sky . [ tab_sing_dish ] lists the main bolometer cameras currently available , with a short summary of their characteristics ( number of detectors , wavelength of operation , field of view ) . most of the high redshift submillimeter galaxies known so far have been discovered thanks to such bolometer arrays . some of these single dish observatories are dedicating an increasing fraction of their total observing time to extensive surveys by exploiting these bolometer cameras . probably this will be one of the main working modes for some single dish telescopes when alma will be in operation : single dish telescopes will provide continuum targets through wide area surveys to be then observed in depth with alma ( to obtain spectroscopic information , multi - band data , and angular resolution information ) . .main mm - submm single dish telescopes currently available [ cols= " < , < , < , < , < , < , < , < " , ] + notes : + @xmath54 angular resolution in arcsec in the compact ( 200 m ) and in the most extended configuration ( 18 km ) . + @xmath55 field of view in arcsec estimated as the full width half maximum of the primary beam . + @xmath56 continuum sensitivity : rms ( in mjy ) in one minute of integration , derived by using the eso etc . the weather conditions were automatically selected by the etc depending on the band . + @xmath57 emission line sensitivity : rms ( in mjy ) in one minute of integration , derived by using the eso etc and by assuming a line width of 300 km / s . + @xmath58 initially only 6 antennas will be equipped with band 5 ( which explains the much lower sensitivity in this band ) . + @xmath59 funding of band 10 is still pending , awaiting results from the technical development . the sensitivity and frequency range quoted for this band are just indicative , and may be subject to significant variations depending on technical constraints . alma consists of at least 54@xmath6012 m antennas and 12@xmath607 m antennas for a total collecting area of at least 6500 m@xmath33 . the antenna configurations will cover baselines ranging from 200 m out to 18 km , yielding a maximum angular resolution better than 0.1@xmath6 at 3 mm , and better than 0.01@xmath6 at submm wavelengths ( tab . [ tab_alma ] ) . the field of view is limited by the primary beam of individual antennas and it ranges from 1@xmath61 at 3 mm to about 10@xmath6 at submm wavelengths ( tab . [ tab_alma ] ) . [ fig_alma_art ] shows an artistic view of the alma array in extended and compact configurations . the 7 m antennas and 4 of the 12 m antennas are clustered into the atacama compact array , aca ( fig . [ fig_alma_art ] ) , which will have a compact fixed configuration aimed at covering the _ uv _ plane on short spacings . aca will be crucial to recover diffuse emission extending on scales larger than a few arcsec ( which is resolved out by the alma array ) . for more than 50% of the time , allowing very sensitive observations at submm wavelengths . courtesy of the european southern observatory . ] alma will be located on chajnantor plateau , in the atacama desert ( chile ) , at an altitude of 5000 m. this is an exceptionally dry site providing a good transparency even in the 350@xmath1 m and 450@xmath1 m windows for a significant fraction of the time . in particular , the precipitable water vapor ( the main source of opacity ) is @xmath62 for more than 50% of the time , at least during the winter months ( fig . [ fig_pwv ] ) . the vast flat topography of chajnantor plateau is optimal to accommodate the various array configurations , providing some constraints on the location of the antennas only in the most extended configuration . the antennas will be moved from one station to the other by means of huge trucks at a rate of 2 antennas / day . as listed in tab . [ tab_alma ] , receivers for 6 frequency bands are currently fully funded ( bands 3 , 4 and 69 ) . band 5 ( 163211 ghz ) is currently planned only for 6 antennas . the funding and the specifications of band 10 are pending , since the receivers development is still in progress . the distribution of the alma bands relative to the atmospheric windows is illustrated in fig . [ fig_atm_transmission ] . the maximum simultaneous frequency coverage allowed by the huge correlator is 8 ghz , which allows alma to reach high sensitivities even for the continuum detection [ tab_alma ] summarizes the alma sensitivities ( both for continuum and line detection ) . for instance , in band 7 ( 870@xmath1 m ) alma reaches the extraordinary sensitivity of @xmath00.1 mjy ( rms ) in one minute of integration , to be compared with the scuba sensitivity of about 1 mjy ( rms ) in 1 hour of integration . constructions at the alma site started already a few years ago and are now nearly completed . at the time of writing the first eight antennas have arrived for integration and testing at the operations support facilities ( osf , located at about 30 km from the alma site ) . in early 2009 the first three antennas will be moved to the alma operations site ( aos ) for commissioning . in 2010 a call for proposal is planned for early science with an initial set of at least 16 antennas . the full array is scheduled to come in full operation in 2012 . an important feature of the alma organization is the plan of making it an observing facility easily accessible also by non - experts in the field . both the software to prepare the observations and the pipeline for the data reduction are designed to be easy to use also by people that are not necessarily experts in interferometry nor necessarily acquainted with mm / submm observing techniques . this is an important characteristic that will make alma totally open to a broad astronomical community . it is obvious that the alma capabilities will allow astronomers to tackle numerous outstanding issues , spanning most of the research fields in astronomy . a summary of some of the main scientific goals can be found in the design reference science plan ( drsp ) , available at http://www.strw.leidenuniv.nl/alma/drsp11.shtml . it is beyond the scope of these lecture notes to provide an overview of the several outstanding scientific cases for alma . in this section i will only focus on some of the alma prospects for agn studies and some related topics , but even in this field the discussion is not meant to be exhaustive . one of the most hotly debated issues on agns is the physics , structure and dynamics of the circumnuclear molecular medium , which is both responsible for the obscuration of agns and presumably related to their feeding . some models assume a uniform gas distribution in a toroidal geometry , whose thickness is supported by ir radiation pressure ( e.g. * ? ? ? other models propose that the circumnuclear obscuring medium is clumpy and originated by the outflow of the accretion disk @xcite . yet other models assume a clumpy distribution due to the effects of supernova explosions or stellar winds , which are also responsible for making the medium `` thick '' @xcite . the high angular resolution , along with its spectroscopic capabilities , will allow alma to clearly distinguish between these models , both through morphological analyses and by investigating the kinematics of the nuclear and circumnuclear medium . [ fig_wada ] shows the expected distribution of circumnuclear co emission according to the torus model in @xcite , and convolved with the angular resolution of alma projected at the distance of ngc 1068 ( panel _ c _ ) : the clumpy nature of the `` torus '' is clearly resolved . actually , the structure shown in fig . [ fig_wada ] describes the extended component of the torus ( radius @xmath032 pc ) , which contains a large amount of gas , but accounts only for a small fraction of the nuclear obscuration . most of the obscuration is due to dense gas on the parsec scale @xcite , as also inferred by mid - ir interferometric observations @xcite . in clumpy models the dust within each individual cloud spans a wide range of temperatures and therefore emits strongly at all wavelengths , from mid - ir to submm . as a consequence , an interesting prediction of clumpy models is that the nuclear `` torus '' , made of out of several clouds , should show the same morphology at all infrared wavelengths . in contrast , uniform `` torus '' models expect a strong radial temperature gradient and , therefore , the innermost warm dust should emit much more at mid - ir wavelengths , while submm radiation from cold dust should be observed mostly in the outer regions . at high frequencies alma has an angular resolution matching vlti ( i.e. sub - pc scale at the distance of famous agns such as ngc 1068 and circinus ) , therefore it will be possible to directly compare the mid - ir morphology with the submm morphology and directly test the predictions of clumpy and uniform torus models . the gas kinematics inferred from alma observations of the nuclear molecular emission lines will provide important information on the dynamical stability and origin of the circumnuclear medium . in the case of a simple uniform structure , supported by radiation pressure , we expect gas motions to be dominated mostly by rotation . in the case of a medium originated by the accretion disk wind @xcite we expect the molecular gas kinematics to also show a clear outflow component . in the case of a strong contribution by sn and stellar winds we also expect a strong turbulent component , hence a large width of the emission lines even within resolved regions . obviously investigating the molecular gas in the innermost regions will require the observation of species typically tracing dense and highly excited gas ( e.g. hcn , hco@xmath13 , ... ) , which are much fainter than co , but easy to detect with alma . a detailed modelling of the excitation and emissivity distribution of these high density tracers in agn torii has been recently performed by @xcite . according to these models the flux ratios between these molecular lines is expected to be spatially highly inhomogeneous , reflecting the inhomogeneous structure of the molecular torus . only the excellent angular resolution of alma will allow us to reveal such complex structures for the first time . the horizontal blue dashed lines indicate the angular resolution at @xmath63 of various current or planned optical / near - ir facilities ( jwst , 8 m telescope , 30 m telescope , vlti ) . the red shaded area gives the range of angular resolution achievable with alma at different frequencies . courtesy of a. marconi . ] by resolving the molecular gas dynamics on such small scales it will be possible to measure the mass of the nuclear black hole . for instance , in ngc 1068 the @xmath64 black hole has a sphere of influence with a radius of about 4 pc , requiring a beam projected on the sky smaller than 0.05@xmath6 to be resolved , which is achievable with alma at most frequencies . in ngc 1068 , and a few additional nearby sy2s , the bh mass has been already measured thanks to vlbi observations of a maser h@xmath7o disk @xcite ; however , the latter technique can only be exploited in a few cases where the maser disk is observed nearly edge - on ( within 15@xmath65 of the line of sight ) . alma observations of the nuclear molecular gas will allow us to extend the measurement of black hole masses in agns to large samples , without stringent constraints on the geometry of the nuclear gas distribution . [ fig_bh_rinf ] shows the radius of influence , as projected on the sky , of black holes of different masses in the nuclei of galaxies , as a function of their distance ( the central stellar gravitational potential is assumed to match the @xmath66 relation ) . the red shaded area indicates the range of angular resolution achievable with alma at various frequencies . clearly alma is in principle able to measure even very small black holes ( @xmath67 ) in nearby systems ( @xmath68 ) ; therefore alma will allow us to extend the investigation of the @xmath66 relation to very low masses . alma will also allow us to measure massive black holes ( @xmath69 ) in distant systems , @xmath70 ; at these distances we can find the closest cases of merging systems ( most of which are powerful ulirgs ) . in these cases alma will measure the black hole mass within each of the two merging systems , which will allow us to place the two individual black holes on the @xmath66 relation and to compare their location with the evolutionary path expected by models of hierarchical black hole growth ( * ? ? ? * ; * ? ? ? * e.g. ) . at mm - submm wavelengths with the angular resolution achievable with a mm - submm vlbi . the left panels illustrate the result of the model , showing a clear `` shadow '' due to the light bending by the gravitational field of the supermassive black hole . the other panels show the image obtained after convolution with the beam expected for the mm - submm vlbi . the panels on the top are for a rotating black hole , while the panels on the bottom are for a non - rotating black hole . from @xcite . ] alma will also be a powerful machine to discover heavily obscured agns . it is now clear that a population of obscured agn are missed by optical spectroscopic surveys aimed at identifying agns through their characteristic narrow emission lines @xcite . some of these obscured agns are missed because the narrow line region is also obscured by dust in the host galaxy @xcite . in other cases the nlr is not formed at all ( as inferred by the lack of nlr lines even in the mid - ir , * ? ? ? * ) , probably because the ionizing radiation is absorbed in all directions ( i.e. 4@xmath71 obscuration rather than a torus - like geometry ) . however , the x - rays emitted by the agns create extended xdrs , which are characterized by enhanced temperatures favoring the formation of several molecular species characteristics of these regions , as discussed in [ sec_past_normal_local ] . the molecular transitions and diagnostics typical of these regions ( e.g. enhanced hcn / hco@xmath13 * ? ? ? * ; * ? ? ? * ) do not suffer any dust absorption and will be easily detected with alma even in distant systems . therefore , alma will be able to provide an unbiased census of the agn population , by including also those heavily obscured agns that are not detectable at other wavelengths . i conclude this section by discussing the potentialities for the investigation of our galactic center . the nuclear radio source sgra@xmath72 is also a mm / submm source . if the emitting region is uniformly distributed around the supermassive black hole , then @xcite showed that photons are expected to be deviated by the strong gravitational field within a few schwarzschild radii , therefore creating a `` shadow '' in a putative mm / submm high resolution image ( fig . [ fig_falcke ] , left ) . there is a plan of combining alma with other mm / submm observatories distributed world wide to create ( sub-)millimetric vlbi network , reaching an angular resolution of a few 10@xmath1arcsec , which would allow us to resolve the black hole `` shadow '' ( fig . [ fig_falcke ] , right ) . this would really be a major result , essentially the first `` picture '' of a black hole . the shape and the contrast of the `` shadow '' would also allow us to determine whether the black hole is rotating ( upper panels in fig . [ fig_falcke ] ) or not ( lower panels ) . we have seen in [ sec_past_starburst_highz ] that current mm / submm observations of high - z objects are limited to very luminous systems ( @xmath43 ) , not representative of the bulk of the galaxy population at high redshift . alma will be able to detect galaxies at least two orders of magnitude fainter , therefore providing an unbiased view of the evolution of galaxies from the mm / submm perspective . [ fig_sed_z ] , compares the continuum sensitivity of alma ( 24 hours of integration ) with the continuum flux expected by a galaxy with @xmath73 ( i.e. @xmath74 ) at various redshifts . it is clear that with the same integration time alma will be able to detect the thermal dust continuum of galaxies that are three times fainter . it is also interesting to note that for a sed typical of starburst galaxies the alma sensitivity nicely matches the mid / near - ir sensitivity of jwst , another major facility which will be launched in 2013 , i.e. when alma will be already fully operating . these two facilities will be fully complementary for the investigation of faint distant galaxies that are out of reach for current observatories . even in its compact configuration alma will have a resolution high enough ( @xmath75 ) to make the confusion noise negligible , which is instead one of the main limitations of current surveys with single dish telescopes . the excellent alma resolution will allow astronomers to identify the optical / near - ir counterpart of mm - submm sources without ambiguities . the optical / near - ir counterpart may provide the redshift of the galaxy / agn through optical / near - ir followup spectroscopy . however , alma will directly provide the redshift of the mm / submm sources through the detection of molecular lines or fine structure atomic lines . [ fig_lines_z ] shows the observed frequency of the co rotational transitions as a function of redshift overlaid on the alma frequency bands . at z@xmath43 at least two co transitions are observable within band 3 ( the most sensitive one ) , therefore unambiguously providing the redshift of the source . at z@xmath363 the redshift confirmation requires the observation an additional co line in a higher frequency band . [ fig_alma_lines ] illustrates the alma sensitivity for the detection of the co(65 ) line ( which in smgs is close to the peak of the co lines intensities , * ? ? ? * ) for a galaxy with @xmath73 . alma will clearly be able to detect this co transition in luminous infrared galaxies ( lirgs ) out to z@xmath05 . note that to scale the diagram in fig . [ fig_alma_lines ] to other fir luminosities one has to keep in mind the non - linear relation between @xmath21 and @xmath5 ( fig . [ fig_fir_co ] ) . 158@xmath1 m line fluxes expected for a galaxy with @xmath73 as a function of redshift ( keep in mind that the luminosity of these lines does not scale linearly with @xmath5 ) . _ dashed lines : _ sensitivity of alma ( 24 hours of integration ) for a 5@xmath31 detection of emission lines ( width of 300 km / s ) in the frequency bands corresponding to the co(65 ) and [ cii ] redshifted lines . here i have only considered the frequencies around the center of the bands and far from deep atmospheric absorption features ( tab . [ tab_alma ] ) , and i have neglected band 5 ( which has currently much lower sensitivity because available only for 6 antennas ) . the [ cii ] luminosity was inferred by fitting to the @xmath49}/l_{fir}$ ] relation reported in @xcite . the co luminosity was inferred by assuming the @xmath76 relation in fig.[fig_fir_co ] and by assuming the relative luminosities of the co transitions observed in smm j16359 + 6612 ( z=2.5 ) by @xcite . ] at z@xmath47 only co transitions higher than ( 65 ) will be observable within the alma bands . however , such high transitions are generally little excited in most galaxies @xcite and therefore more difficult to observe relative to the lower transitions . yet , already at z@xmath41 the strong [ cii]158@xmath1 m line will be observable in the alma bands , and it will be one of the main tools to identify the redshift of distant sources , and in particular at z@xmath47 ( fig . [ fig_lines_z ] ) . fig . [ fig_alma_lines ] shows that in lirgs ( @xmath77 ) alma will be able to detect [ cii ] out to z@xmath010 . note that also in this case the diagram can not be scaled linearly to other fir luminosities , since the relation between @xmath49}$ ] and @xmath5 is not linear @xcite . the second brightest line is generally [ oi]63@xmath1 m ( in some objects this line is even stronger than [ cii ] , but in some exceptional objects it may be self - absorbed ) . [ oi]63@xmath1 m will be observable at z@xmath44 in the alma bands , and will help to obtain the redshift confirmation along with the [ cii ] detection . the [ oiii]88@xmath1 m line is also generally relatively strong , especially if an agn is present ( @xmath78}\sim 0.5 - 0.3 ~f_{[cii]}$ ] ) , and can be detected with alma at high redshifts . however , since [ oi ] and [ oiii ] will be observable at high - z mostly in the high frequency bands , which are the least sensitive , the detection of these lines will probably be limited to ulirgs ( unless chemical evolutionary effects favor the emission of these oxygen lines , as discussed below ) . the far - ir emission inferred from the mm / submm thermal continuum sed , along with the redshift inferred from the co , [ cii ] and [ oi ] lines , will allow alma to provide a self - consistent , unbiased view of the evolution of the star formation rate through the cosmic epochs . for what concerns high - z agns , alma will allow us to directly trace the coevolution of black hole growth ( through the x - ray and optical emission ) and of the formation of stellar mass in the host galaxies ( through the far - ir emission ) . as discussed in [ sec_past_agn_highz ] , current mm / submm studies have found tentative indications of this co - evolution for very luminous systems ( @xmath79 ) ; alma will extend this investigation to much more quiescent systems ( @xmath80 ) , more representative of the bulk of the galaxy population at high redshift . the luminosity of the co emission will provide the molecular gas mass for most of these galaxies and , when compared with the sfr , will provide a direct measure of the star formation efficiency ( i.e. the star formation per unit gas mass ) as a function of redshift . in high redshift galaxies ( z@xmath41 ) the angular resolution of alma will allow us to resolve morphologies on sub - kpc scales . besides providing the extension of the star formation activity and the distribution of the molecular gas , the alma maps will deliver precious information on the gas dynamics . by resolving rotation curves it will be possible to constrain the dynamical mass of individual galaxies , which can be compared with the expectations by hierarchical models of galaxy evolution . in the case of qsos and type 1 agns it will be possible to directly compare the galaxy dynamical mass to the black hole mass ( inferred from the broad optical / uv emission lines ) , therefore tracing any putative evolution of the @xmath81 relation . the observed evolution of the latter relation as a function of redshift will be directly comparable with the predictions of models dealing with the galaxy - bh co - evolution @xcite , and will therefore provide tight constraints on the same models . the dynamics and morphology of the molecular gas will also provide information on the agn fuelling mechanism . in [ sec_past_agn_local ] we saw that local agns do not show obvious systematic signatures of nuclear fuelling from the host galaxy , but also that such low luminosity agns do not actually require large fuelling from the host galaxy to maintain their low accretion rates . the fuelling demand from the host galaxy is however much larger in powerful qsos , whose accretion rates may exceed a few @xmath82 . therefore , it is important to trace the dynamics of the molecular gas in the host of these powerful qsos , and investigate for instance whether in these systems gas funneled by bars or driven by galaxy merging is common . however , such luminous qsos are not found in the local universe , while are abundant at high redshift . current facilities are in general unable to resolve the molecular gas dynamics in such distant systems , with very few exceptions . alma will allow us to trace in detail the gas dynamics in a large number of qso hosts at high redshift and to statistically investigate the occurrence of various possible fuelling mechanisms . detailed simulations are already being performed to assess the detectability and capability of resolving molecular gas morphologies in high - z agns . @xcite showed that not only alma will be able to resolve and detect the molecular gas in the host galaxy , but also the 100 pc gaseous torii expected to surround most of the supermassive black holes at z@xmath02 . 63@xmath1 m to [ cii]158@xmath1 m intensity ratio as a function of galaxy age , by assuming the chemical evolutionary model in @xcite . the model also assumes log(u)=-2.5 , @xmath83 and that the excitation conditions do not change over time . courtesy of m. kaufman . ] the alma detection of multiple molecular and atomic lines in high - z galaxies and qsos will also provide precious information on the chemical evolutionary status of the galaxy . for instance , the ratio of [ oi]63@xmath1 m and [ cii]158@xmath1 m is sensitive to the relative abundance of oxygen and carbon c / o . since carbon is subject to a delayed enrichment with respect to oxygen ( by several 100 myr ) , the c / o abundance ratio is a sensitive tracer of the evolutionary stage of a galaxy . fig . [ fig_kauf ] shows the expected @xmath8463\mu m}/l_{[cii]158\mu m}$ ] ratio as a function of the age of a galaxy , by exploiting the chemical models in @xcite ( the model assumes the physical conditions derived for the qso at z=6.4 , as discussed in @xcite , and that the excitation conditions do not change over time ) . therefore , by measuring both these lines , alma will allow us to constrain the evolutionary stage of distant galaxies . in particular , it will be extremely interesting to investigate the c / o abundance ratio in agns and galaxies at z@xmath07 , close to the re - ionization epoch , where the age of the universe is close to the minimum enrichment timescale for carbon ; at these redshifts the c / o ratio is expected to be very low , and therefore the [ oi]/[cii ] ratio is expected to be very high . in very luminous high - z sources , such as the qsos already detected with current facilities , alma will be able to easily detect several other molecular ( e.g. hcn , hco@xmath13 , ... ) and atomic ( e.g. [ nii]122@xmath1 m ) transitions , which will allow us to perform an accurate modelling of the physics of the ism and to infer accurate chemical abundances ( hence the evolutionary stage of the system ) . as an example , fig . [ fig_wotten ] shows the simulated alma spectrum ( 24 hours of integration ) of a quasar with the same redshift ( z=6.4 ) and luminosity of the qso j1148 + 5251 , which is currently the most distant qso with a co detection . beside the co(65 ) transition , the 8 ghz alma band is expected to reveal several other molecular transitions with excellent signal - to - noise . finally , alma will allow us to investigate in detail the evolution of dust production in the early universe . the dust mass in galaxies can only be inferred by measuring the ( rest - frame ) infrared to submm sed . currently dust masses have been measured only for very powerful systems at high - z ( qsos and smgs ) , providing an incomplete and very biased view , not representative of the global evolution of dust through the cosmic epochs . alma will measure the dust mass in large samples of high - z systems , even in relatively quiescent ones ( sfr of a few @xmath82 ) . therefore , it will be possible to trace the evolution of dust mass as a function of redshift and for different classes of galaxies , which can be directly compared with models of dust evolution @xcite . in particular , it will be important to clarify whether dust produced by sne can actually be the main source of dust in the early universe , or whether other mechanisms of dust production are required @xcite . i am grateful to the organizers of the school for their kind invitation . i thank m. walmsley , p. caselli , k. wada and l. testi for many useful comments on the manuscript . i am grateful to c. dowell , a. beelen , a. marconi , m. kaufman and a. wotten for providing some of the figures in these lecture notes . some of the images were reproduced with kind permission of the space telescope science institute and of the european southern observatory . finally , i acknowledge financial support by the national institute for astrophysics ( inaf ) and by the italian space agency ( asi ) .
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these lecture notes provide an introduction to mm / submm extragalactic astronomy , focused on agn studies , with the final goal of preparing students to their future exploitation of the alma capabilities .
i first provide an overview of the current results obtained through mm / submm observations of galaxies and agns , both local and at high redshift .
then i summarize the main mm / submm facilities that are currently available .
alma is then presented with a general description and by providing some details on its observing capabilities .
finally , i discuss some of the scientific goals that will be achievable with alma in extragalactic astronomy , and for agn studies in particular .
galaxies : active , evolution , formation , high - redshift , nuclei , quasars , seyfert , starburst , millimeter , submillimeter , instrumentation : high angular resolution , interferometers 95.85.fm , 95.85.bh , 98.54.-h , 98.54.cm , 98.54.ep , 98.54.kt
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half - metallic ferromagnets with high ( room temperature and above ) curie temperatures @xmath1 are ideal for spintronics applications , and as such , much experimental and theoretical@xcite effort has been devoted in recent years to the designing of and search for such materials . among these , cr - doped dilute magnetic semiconductors ( dms ) @xcite or cr - based alloys and in particular cras and crsb @xcite in zinc blende(zb ) structure have attracted particular attention , not only because of the possibility of complete spin polarization of the carriers at the fermi level , but also for their possible high @xmath1 . akinaga @xcite were able to grow zb thin films of cras on gaas ( 001 ) substrates by molecular beam epitaxy , which showed ferromagnetic behavior at temperatures in excess of 400 k and magnetic moments of 3@xmath2 per cras unit . theoretical calculations by akinaga @xcite and several other theoretical calculations since then @xcite have verified the half - metallic character of cras . the high value of @xmath1 has also been supported by some of these studies@xcite . thin films of crsb grown by solid - source molecular beam epitaxy on gaas , ( al , ga)sb , and gasb have been found to be of zb structure and ferromagnetic with @xmath1 higher than 400 k@xcite . galanakis and mavropoulos@xcite , motivated by the successful fabrication of zb cras , crsb and mnas@xcite , have examined the possibility of half - metallic behavior in ordered zb compounds of transition metals v , cr and mn with the @xmath3 elements n , p , as , sb , s , se and te . their theoretical study shows that the half - metallic ferromagnetic character of these compounds is preserved over a wide range of lattice parameters . they also found that the half - metallic character is maintained for the transition element terminated ( 001 ) surfaces of these systems . yamana @xcite have studied the effects of tetragonal distortion on zb cras and crsb and found the half - metallicity to survive large tetragonal distortions . of course , the ground states of many of these compounds in the bulk are known to be different from the zb structure , the most common structure being the hexagonal nias - type . zhao and zunger@xcite have argued that zb mnas , cras , crsb , and crte are epitaxially unstable against the nias structure , and zb crse is epitaxially stable only for lattice constants higher than 6.2 , remaining half - metallic at such volumes . they also find that even though the ground state of crs is zb , it is antiferromagnetic at equilibrium lattice parameter , and thus not half - metallic . these results reveal the challenge experimentalists face in synthesizing these compounds in zb structure . however , the possibility remains open that such difficulties will be overcome with progress in techniques of film - growth and materials preparation in general . recently , deng @xcite were successful in increasing the thickness of zb - crsb films to @xmath4 3 nm by molecular beam epitaxy using ( in , ga)as buffer layers , and li @xcite were able to grow @xmath4 4 nm thick zb - crsb films on nacl ( 100 ) substrates . in view of the above situation regarding the state of experimental fabrication of these compounds and available theoretical results , it would be appropriate to study the variation of magnetic properties , particularly exchange interactions and the curie temperature , of cr - based pnictides and chalcogenides as a function of the lattice parameter . towards this goal , we have carried out such calculations for the compounds crx ( x = as , sb , s , se and te ) and the mixed alloys cras@xmath0x@xmath0 with x = sb , s , se and te . essentially we study the effect of anion doping by choosing elements of similar atomic sizes ( neighboring elements in the periodic table ) , one of which , namely sb , is isoelectronic to as , while the others ( s , se , te ) bring one more valence electron to the system . the mixed pnictide - chalcogenide systems offer further opportunity to study the effects of anion doping . the alloying with other 3d transition metals ( both magnetic , e.g. fe or mn , or non - magnetic , e.g. v ) on cation sublattice would also change the carrier concentration and bring about strong d - disorder which can additionally modify the shape of the fermi surface . this , however , is not the subject of the present paper . almost all theoretical studies on these alloys so far address aspects of electronic structure and stability of these alloys only . although a few theoretical estimates of exchange interactions and the curie temperature for cras at equilibrium lattice parameter have appeared in the literature , a detailed study of the volume dependence of these quantities is missing . for the other alloys , crsb , crs , crse and crte , no theoretical results for the exchange interaction , curie temperature and their volume dependence exist at present . the mixed pnictide - chalcogenide systems offer the possibility of not only creating these alloys over a larger range of the lattice parameter , but also with a larger variation in the exchange interactions . this is because at low values of the lattice parameter the dominant cr - cr exchange interactions in the chalcogenides can be antiferromagnetic , while for the pnictides they are ferromagnetic . the pnictide - chalcogenide alloying is important from the experimental viewpoint of stabilizing the zb structure on a given substrate , via the matching of the lattice parameter of the film with that of the latter . although the present study is confined to the zb structure only , we hope that it will provide some guidance to the experimentalists in their search and growth of materials suitable for spintronic devices . electronic and magnetic properties of crx ( x = as , sb , s , se and te ) and cras@xmath0x@xmath0 ( x= sb , s , se and te ) were calculated for lattice parameters varying between 5.45 and 6.6 , appropriate for some typical ii - vi and iii - v semiconducting substrates . calculations were performed using the tb - lmto - cpa method@xcite and the exchange - correlation potential given by vosko , wilk and nusair@xcite . in our lmto calculation we optimize the asa ( atomic sphere approximation ) errors by including empty spheres in the unit cell . we use the fcc unit cell , with cr and x ( as , sb , s , se and te ) atoms located at ( 0,0,0 ) and ( 0.25,0.25,0.25 ) , respectively , and empty spheres at locations ( 0.5,0.5,0.5 ) and ( -0.25,-0.25,-0.25 ) . for several cases , we have checked the accuracy of the lmto - asa electronic structures against the full - potential lmto results@xcite and found them to be satisfactory . for the mixed alloys cras@xmath0x@xmath0 ( x= sb , s , se and te ) , the as - sublattice of the zb cras structure is assumed to be randomly occupied by equal concentration of as and x atoms . the disorder in this sublattice is treated under the coherent potential approximation ( cpa)@xcite . our spin - polarized calculations assume a collinear magnetic model . in the following we will present results referred to as fm and dlm . the fm results follow from the usual spin - polarized calculations , where self - consistency of charge- and spin - density yields a nonzero magnetization per unit cell . although we call this the fm result , our procedure does not guarantee that the true ground state of the system is ferromagnetic , with the magnetic moments of all the unit cells perfectly aligned . this is because we have not explored non - collinear magnetic states , nor all antiferromagnetic ( afm ) states attainable within the collinear model . indeed , our results for the exchange interactions in some cases do suggest the ground states being of afm or complex magnetic nature . for lack of a suitable label , we refer to all spin - polarized calculations giving a nonzero local moment as fm state calculations . within the stoner model , a nonmagnetic state above the curie temperature @xmath1 would be characterized by the vanishing of the local moments in magnitude . it is well - known and universally accepted that the neglect of the transversal spin fluctuations in the stoner model leads to an unphysical picture of the nonmagnetic state and a gross overestimate of @xmath1 . an alternate description of the nonmagnetic state is provided by the disordered local moment ( dlm ) model , where the local moments remain nonzero in magnitude above @xmath1 , but disorder in magnitude as well as their direction above @xmath1 causes the global magnetic moment to vanish . combining aspects of the stoner model and an itinerant heisenberg - like model , heine and co - workers@xcite have developed a suitable criterion for a dlm state to be a more appropriate description of the nonmagnetic state than what is given by the stoner model . within the collinear magnetic model , where all local axes of spin - quantization point in the same direction , dlm can be treated as a binary alloy problem and thus described using the coherent potential approximation ( cpa)@xcite . we have carried out such dlm calculations , assuming the cr - sublattice to be randomly occupied by an equal number of cr atoms with oppositely directed magnetic moments . the object for performing the dlm calculations is two - fold . if the total energy in a dlm calculation is lower than the corresponding fm calculation , we can safely assume that the ground state ( for the given lattice parameter and structure ) is not fm , albeit of unknown magnetic structure . the exchange interactions in the dlm state can also be used to compute estimates of @xmath1 , and such estimates of @xmath1 may be considered as estimates from above the magnetic - nonmagnetic transition . @xmath1 computed from exchange interactions in the fm state are estimates from below the transition . of course , if the ground state is known to be ferromagnetic , then estimates of @xmath1 based on exchange interactions in the fm reference state are the appropriate ones to consider . in some cases where the fm results point to the possibility of the ground state magnetic structure being afm or of complex nature , we have carried out a limited number of afm calculations to provide some insight into this problem ( see section [ subsec : stabilityjq ] ) . we have computed the spin - resolved densities of states ( dos ) for all the alloys for varying lattice parameters , and for both the fm and dlm configurations . the fm calculations show half - metallic character , due to the formation of bonding and antibonding states involving the @xmath5 orbitals of the cr - atoms and the @xmath3 orbitals of the neighboring pnictogen ( as , sb ) or chalcogen ( s , se , te ) . the hybridization gap is different and takes place in different energy regions in the two spin channels . the critical values of the lattice parameters above which the fm calculations show half - metallic character agree well with those reported by galanakis and mavropoulos@xcite . the dos for the alloys of the type crx ( x = sb , s , se , te ) have been presented by several other authors@xcite and thus will not be shown here . in figs.[fig1 ] and [ fig2 ] we show the dos for the mixed alloys cras@xmath0sb@xmath0 and cras@xmath0se@xmath0 , for lattice parameters above and below the critical values for the half - metallic character . according to galanakis and mavropoulos@xcite , half - metallicity in zb cras appears between the lattice parameters of 5.45 and 5.65 . the latter corresponds to the lattice parameter of the gaas substrate . for crsb half - metallicity appears at a lattice parameter between 5.65 and 5.87 . the mixed alloy cras@xmath0sb@xmath0 , as shown in fig.[fig1 ] , is not quite half - metallic at the lattice parameter of 5.65 , and fully half - metallic at the lattice parameter of 5.76 . replacing sb with se in the above alloy , i.e. for cras@xmath0se@xmath0 , brings the critical lattice parameter down slightly . as shown in fig.[fig2 ] , at a lattice parameter of 5.65 , cras@xmath0se@xmath0 is half - metallic , although barely so . in our calculation crs and crse are half - metallic at a lattice parameter of 5.65 , and not so at a lattice parameter of 5.55 . crte is not half - metallic at a lattice parameter of 5.76 , but at a lattice parameter of 5.87 . for both crs and crse the critical value should be close to 5.65 , and for crte it should be close to 5.87 . note that in general the half - metallic gap is larger in the chalcogenides than in the pnictides . this is due to larger cr - moment ( see section [ sec : mag.mom . ] ) for the chalcogenides , which results in larger exchange splitting . this explains the difference in the half - metallic gaps in figs . [ fig1 ] and [ fig2 ] for similar lattice parameters . fig . [ fig3 ] compares the total dos of cras for the fm and dlm calculations for the equilibrium lattice parameter 5.65 . higher dos at the fermi level for the dlm calculation , compared with the fm calculation , is an indication that the band energy is lower in the fm state . indeed , as indicated in table [ table1 ] , compared with the dlm state the total energy for zb cras is lower in the fm state for the lattice parameters from 5.44 to 5.98 . in fact , this holds for lattice parameters up to 6.62 , showing the robustness of ferromagnetism in cras over a wide range of the lattice parameter . this is also true for crsb . sb@xmath0 for lattice parameters ( a ) 5.55 , ( b ) 5.65 , ( c ) 5.76 @xmath6 and ( d ) 5.87 , respectively . , width=360 ] se@xmath0 for lattice parameters ( a ) 5.55 , ( b ) 5.65 , ( c ) 5.76 @xmath6 and ( d ) 5.87 , respectively . , width=360 ] for the dlm and fm states.,width=360 ] in table [ table1 ] we show the variation of total energies per atom in ry with the lattice parameter for crx ( x = as , sb , s , se and te ) in the dlm and fm states . the purpose of tabulating these energies is not to determine the bulk equilibrium lattice parameters in the zb structure , as this has already been done by several authors@xcite . our results for equilibrium zb phase lattice parameters agree with those found by galanakis and mavropoulos@xcite . the important point is that for crs and crse at low values of lattice parameters the dlm energies are lower than the fm energies , showing clearly that the fm configuration is unstable . the result for crs is in line with the observation by zhao and zunger@xcite , who find zb crs to be antiferromagnetic with an equilibrium lattice parameter of 5.37 . as shown later ( section [ sec : exchange ] ) , the exchange coupling constants for the cr atoms in the fm calculations are negative , indicating the instability of the ferromagnetic spin alignment . the tendency to antiferromagnetism in crse at compressed lattice parameters is also revealed in a study by sasaio@xmath7lu @xcite for crte at lower lattice parameters the fm energy is lower than the dlm energy , but the exchange constants for the cr - atoms in the fm calculations are still negative ( see discussion in section [ sec : exchange ] ) , signaling the possibility of the ground states in crte at low values of the lattice parameter being neither dlm nor fm . note that in our discussion ground state implies the lowest energy state in zb structure . for crs , crse and crte the ground states at low lattice parameters can be of an antiferromagnetic ( afm ) or complex magnetic structure . a lower total energy may also mean a lower band energy , and in some cases , the latter may be reflected in a lower density of states at the fermi level . this is shown in fig.[fig4 ] , where for crs at the lowest lattice parameter of 5.44 @xmath6 the dos at the fermi level is lower in the dlm state than in the fm state . the deviation from ferromagnetism at low values of the lattice parameter for crs , crse and crte is also revealed by our study of the lattice fourier transform of the exchange interaction between the cr atoms in the fm state ( section [ sec : exchange ] ) . the search for an antiferromagnetic state with lower energy is possible within our collinear magnetic model by enlarging the unit cell in various ways . we have pursued this issue to a limited extent , by considering 001 , 111 afm configurations for crs , crse and crte at low values of the lattice parameter ( see discussion in section[sec : exchange ] ) . a satisfactory resolution of such issues is possible only by going beyond the collinear model . lcccccc lattice parameter ( ) & 5.44 & 5.55 & 5.65 & 5.76 & 5.87 & 5.98 + * cras * + dlm energy & -1653.4039 & -1653.4021 & -1653.3995 & -1653.3960 & -1653.3920 & -1653.3876 + fm energy & -1653.4068 & -1653.4055 & -1653.4034 & -1653.40026 & -1653.3966 & -1653.3922 + * crsb * + dlm energy & -3762.4738 & -3762.4781 & -3762.4807 & -3762.4821 & -3762.4822 & -3762.4814 + fm energy & -3762.4770 & -3762.4813 & -3762.4841 & -3762.4857 & -3762.4862 & -3762.4856 + + + * crs * + dlm energy & -723.4504 & -723.4468 & -723.4426 & -723.4381 & -723.4332 & -723.4280 + fm energy & -723.4491 & -723.4464 & -723.4434 & -723.4395 & -723.4351 & -723.4302 + * crse * + dlm energy & -1737.9146 & -1737.9139 & -1737.9123 & -1737.9100 & -1737.9071 & -1737.9036 + fm energy & -1737.9139 & -1737.9132 & -1737.9124 & -1737.9110 & -1737.9087 & -1737.9057 + * crte * + dlm energy & -3918.9074 & -3918.9124 & -3918.9158 & -3918.9179 & -3918.9189 & -3918.9188 + fm energy & -3918.9078 & -3918.9128 & -3918.9161 & -3918.9181 & -3918.9194 & -3918.9201 + our spin - polarized calculations for the fm reference states lead to local moments not only on the cr atoms , but also on the other atoms ( as , sb , s , se , and te ) as well as the empty spheres . sandratskii @xcite have discussed the problem associated with such induced moments in case of the heusler alloy nimnsb and the hexagonal phase of mnas . usually such systems can be divided into sublattices with robust magnetic moments and sublattices where moment is induced under the influence of the former . these authors argue that the treatment of the induced moments as independent variables in a heisenberg hamiltonian may lead to artificial features in the spin - wave spectra , but these artificial features do not drastically affect the calculated curie temperatures of the two alloys , nimnsb and hexagonal mnas . clearly , in our case the sublattice with the robust magnetic moment is the cr - sublattice . among the three other sublattices , the magnitudes of the induced moments decrease in the following order for the two robust ferromagnets cras and crsb : x - sublattice ( x = as , sb ) , sublattice es-1 ( the sublattice of empty spheres that is at the same distance with respect to the cr - sublattice as the x - sublattice ) , sublattice es-2 ( sublattice of empty spheres further away from the cr - sublattice ) . this trend is particularly valid for low values of the lattice parameter . the induced moments originate from the tails of the orbitals ( primarily @xmath8 ) on the nearby cr - atoms . this is particularly true for the moments induced on the empty spheres . the magnitudes of the induced moments on the two empty sphere sublattices decrease as the lattice parameter increases , and so do the differences in their magnitudes . the signs of the moments on es-1 and es-2 may be the same for small lattice parameters , but are opposite for large lattice parameters . the sign of the moment on the x - sublattice is opposite to that on the cr - sublattice and the magnitudes of the moments on the two sublattices increase with increasing lattice parameters , due to decreased hybridization between cr-@xmath8 and x-@xmath3 orbitals . above a critical value of the lattice parameter , the moment per formula unit ( f.u . ) saturates at a value of 3.0 @xmath2 , as the half - metallic state is achieved , while the local moments on the cr- and x - sublattices increase in magnitude , remaining opposite in sign . the maximum ratio between the induced moment on x ( x = as , sb ) and the moment on cr is 0.18 for cras and 0.15 for crsb , occurring at the highest lattice parameter of 6.62 @xmath6 studied . the maximum ratio between the induced moment on es-1 and that on cr is 0.06 , occurring at the lowest lattice parameter of 5.44 @xmath6 studied . magnetic moments of cras and crsb per formula unit ( f.u . ) as well as the local moment at the cr site are shown in fig . [ fig5 ] , where we compare the two compounds with each other for their magnetic moments in the fm and dlm states . the same results are presented in fig . [ fig6 ] , comparing the moment per f.u.in the fm state with the cr local moment in the fm and dlm states separately for each compound . it is to be noted that there are no induced moments for the dlm reference states , i.e. the moments on the non - cr sublattices are several orders of magnitude smaller than the robust moment on the cr atoms . the total moment per formula unit in the dlm state is zero by construction . the local moment on the cr atom for the dlm reference state is usually less than the corresponding fm value for smaller lattice parameters , and larger for larger lattice parameters ( fig.[fig6 ] ) . similar trends in the variation of the local moment on cr and the induced moments on the other sublattices for the fm reference states as a function of lattice parameter are revealed for crx ( x = s , se , te ) , except that the moments on es-1 are always an order of magnitude larger than those on es-2 . i addition , the induced moments on es-2 are @xmath9 times larger than those on x sublattice for smaller values of the lattice parameter , with the two becoming comparable in magnitude for larger lattice parameters . the induced moments on es-1 and x - sublattices are never larger than @xmath4 5% of the moment on the cr atoms . the induced moments for the dlm reference states are several orders of magnitude smaller than the cr - moments , and can be safely assumed to be zero . results for zb crs , crse and crte are presented in figs.[fig7 ] and [ fig8 ] . the moment per f.u . reaches the saturation values of 4@xmath2 for crs , crse , and crte in the half - metallic state , as discussed in detail by galanakis and mavropoulos@xcite . the saturation values of the moments for all these alloys ( crx , x = as , sb , s , se , and te ) satisfy the so - called `` rule of 8 '' : @xmath10 , where @xmath11 is the total number of valence electrons in the unit cell . the number 8 accounts for the fact that in the half - metallic state the bonding @xmath12 bands are full , accommodating 6 electrons and so is the low - lying band formed of the @xmath13 electrons from the @xmath3 atom , accommodating 2 electrons . the magnetic moment then comes from the remaining electrons filling the @xmath8 states , first the @xmath14 states and then the @xmath5 . the saturation value of 3@xmath2/f.u . , or the half - metallic state , appears for a larger critical lattice constant in crsb than in cras . similarly , the critical lattice constants for the saturation magnetic moment of 4@xmath2/f.u . are in increasing order for crs , crse and crte . the local moment on the cr atom can be less / more than the saturation value , depending on the moment induced on the non - cr atoms and empty spheres . -[fig8 ] ) , fm calculations produce induced moments on non - cr spheres representing the x - atoms ( x = as , sb , s , se , te ) , and one set of empty spheres . the dlm calculations produce no such induced moments , i.e. , the moments reside on the cr - atoms only . see text for discussion.,width=360 ] fig.[fig9 ] shows the variation of the magnetic moment with the lattice parameter for the random alloys cras@xmath0x@xmath0 ( x = sb , s , se , te ) , where 50% of the as - sublattice is randomly occupied by x - atoms . the saturation moment per f.u . for cras@xmath0sb@xmath0 in the half - metallic state is 3@xmath2 , with the results falling between those for cras and crsb shown in fig.[fig6 ] . for cras@xmath0x@xmath0 ( x = s , se , te ) , the saturation moment per f.u . is 3.5@xmath2 . the local cr - moment deviates from the saturation value in the half - metallic state , being higher than the saturation value for all lattice parameters above 6.1 . from figs.[fig5]-[fig9 ] it is clear that the magnetic moment per formula unit is closer to the magnetic moment of the cr atoms in the fm calculations than in the dlm calculations . local cr - moments in the dlm calculations are suppressed w.r.t . the fm results for low lattice parameters and enhanced for larger lattice parameters . as shown in table [ table1 ] the total energy of the fm state is lower than that of the corresponding dlm state in almost all cases , except for some compressed lattice parameters for crs and crse . however , the consideration of the dlm state does provide an advantage in that there are no associated induced moments , i.e. , the dlm calculations produce moments that reside on the robust magnetic sublattice only . mapping of the total energy on to a heisenberg hamiltonian , therefore , does not result in exchange interactions involving atoms / spheres with induced moments and all associated artificial / non - physical features referred to by sandratskii @xcite currently , most _ first - principles _ studies of the thermodynamic properties of itinerant magnetic systems proceed via mapping @xcite the system energy onto a classical heisenberg model : @xmath15 where @xmath16 are site indices , @xmath17 is the unit vector pointing along the direction of the local magnetic moment at site @xmath18 , and @xmath19 is the exchange interaction between the moments at sites @xmath18 and @xmath20 . the validity of this procedure is justified on the basis of the adiabatic hypothesis- the assumption that the magnetic moment directions are slow variables on all the characteristic electronic time scales relevant to the problem , and thus can be treated as classical parameters . the energy of the system for a given set of magnetic moment directions is usually calculated via methods based on density functional theory ( dft ) . one of the most widely used mapping procedures is due to liechtenstein @xcite it involves writing the change in the energy due to the deviation of a single spin from a reference state in an analytic form using the multiple scattering formalism and by appealing to the magnetic variant of the andersen force theorem@xcite . the force theorem , derived originally for the change of total energy due to a deformation in a solid , dictates that the differences in the energies of various magnetic configurations can be approximated by the differences in the band energies alone@xcite . the energy of a magnetic excitation related to the rotation of a local spin - quantization direction can be calculated from the spinor rotation of the ground state potential . no self - consistent calculation for the excited state is necessary . a second approach is based on the total energy calculations for a set of collinear magnetic structures , and extracting the exchange parameters by mapping the total energies to those coming from the heisenberg model given by eq.([e1 ] ) . such calculations can be done using any of the standard dft methods . however , unlike the magnetic force theorem method , where the exchange interactions can be calculated directly for a given structure and between any two sites , several hypothetical magnetic configurations and sometimes large supercells need to be considered to obtain the values of a modest number of exchange interactions . in addition , some aspects of environment - dependence of exchange interactions are often simply ignored . the difference between these two approaches is , in essence , the same as that between the generalized perturbation method ( gpm)@xcite and the connolly - williams method@xcite in determining the effective pair interactions in ordered and disordered alloys . a third approach is a variant of the second approach , where the energies of the system in various magnetic configurations corresponding to spin - waves of different wave - vectors are calculated by employing the generalized bloch theorem for spin - spirals@xcite . the inter - atomic exchange interactions can be calculated by equating these energies to the fourier transforms of the classical heisenberg - model energies . this approach , known as the frozen magnon approach , is similar to the frozen phonon approach for the study of lattice vibrations in solids . in this work , we have used the method of liechtenstein , which was later implemented for random magnetic systems by turek , using cpa and the tb - lmto method@xcite . the exchange integral in eq.([e1 ] ) is given by @xmath21 dz \ ; , \ ] ] where @xmath22 represents the complex energy variable , @xmath23 , and @xmath24 , representing the difference in the potential functions for the up and down spin electrons at site @xmath18. in the present work @xmath25 represents the matrix elements of the green s function of the medium for the up and down spin electrons . for sublattices with disorder , this is a configurationally averaged green s function , obtained via using the prescription of cpa . the integral in this work is performed in the complex energy plane , where the contour includes the fermi energy @xmath26 . the quantity @xmath19 given by eq . ( [ eq - jij ] ) includes direct- , indirect- , double - exchange and superexchange interactions , which are often treated separately in model calculations . the negative sign in eq.([e1 ] ) implies that positive and negative values of @xmath19 are to be interpreted as representing ferromagnetic and antiferromagnetic interactions , respectively . a problem with the mapping of the total energy to a classical heisenberg hamiltonian following the approach of liechtenstein @xcite is that it generates exchange interactions between sites , where one or both may carry induced moment(s ) . of course this problematic scenario appears only for the fm reference states , as the dlm reference states do not generate induced moments . in the present work the liechtenstein mapping procedure , applied to fm reference sates , generates exchange interactions between the cr atoms , between cr and other atoms x ( x = as , sb , s , se , te ) , and also between cr atoms and the nearest empty spheres es-1 . depending on the lattice parameter , this latter interaction is either stronger than or at least comparable to that for the cr - x pairs . the exchange interactions between cr atoms and the furthest empty spheres es-2 are always about one or two orders of magnitude smaller than the cr - es1 interactions and can be neglected . in cras , the ratio of the nearest neighbor cr - es1 to cr - cr interaction varies from 0.2 - 0.25 at low lattice parameters to 0.06 - 0.07 at high values of the lattice parameter . in crsb , these ratios are smaller , varying between 0.14 and 0.05 . the cr - es1 exchange interactions are also relatively strong in magnitude in crs , crse and crte . one important point is that while these interactions are positive for nearest neighbors for all lattice parameters , cr - cr nearest neighbor interaction is negative for low values of lattice parameters in crs and crse . in crte , this interaction changes sign from positive to negative and then back , as the lattice parameter is varied in the range 5.44 - 6.62 . as mentioned earlier , the calculation for the dlm reference states do not produce induced moments , and thus no exchange interactions other than those between the cr atoms . sandratskii@xcite have discussed the case when , in addition to the interaction between the strong moments , there is one secondary , but much weaker , interaction between the strong and one induced moment . in this case , the curie temperature , calculated under the mean - field approximation ( mfa ) , seems to be enhanced due to this secondary interaction , irrespective of the sign of the secondary interaction . in other words , the curie temperature would be somewhat higher than that calculated by considering only the interaction between the strong moments . the corresponding results under the random phase approximation ( rpa ) have to be obtained by solving two equations simultaneously . one can assume that the rpa results for the curie temperature follow the trends represented by the mfa results , being only somewhat smaller , as observed in the absence of induced moments . in our case , since there are at least two secondary interactions ( cr - x and cr - es1 ) to consider in addition to the main cr - cr interaction , the influence of these secondary interactions is definitely more complex . in view of the above - described situation involving secondary interactions between cr- and the induced moments for the fm reference states , we have adopted the following strategy . since no induced moments appear in calculations for the dlm reference states , the curie temperature @xmath1 for these can be calculated as usual from the exchange interaction between the cr - atoms , i.e. the strong moments . for these cases the calculation of @xmath1 can proceed in a straightforward manner by making use of the mean - field approximation ( mfa ) or the more accurate random - phase approximation ( rpa)@xcite . one can obtain the mfa estimate of the curie temperature from @xmath27 where the sum extends over all the neighboring shells . an improved description of finite - temperature magnetism is provided by the rpa , with @xmath28 given by @xmath29^{-1 } \ , .\ ] ] here @xmath30 denotes the order of the translational group applied and @xmath31 is the lattice fourier transform of the real - space exchange integrals @xmath32 . it can be shown that @xmath33 is always smaller than @xmath34 @xcite . it has been shown that the rpa curie temperatures are usually close to those obtained from monte - carlo simulations @xcite . as shown by sandratskii @xcite the calculation of @xmath1 using rpa is considerably more involved even for the case where only one secondary interaction needs to be considered , in addition to the principal interaction between the strong moments . the complexity of the problem increases even for mfa , if more than one secondary interaction is to be considered . the same comment applies to stability analysis using the lattice fourier transform of the exchange interactions . the deviation of the nature of the ground state from a collinear and parallel alignment of the cr moments in the fm reference states could be studied by examining the lattice fourier transform of the exchange interaction between the cr atoms : @xmath35 , if all the secondary interactions could be ignored . this is definitely not possible for many of our fm results , where several pairs of interaction need to be considered , and @xmath36 is a matrix bearing a complicated relationship to the energy as a function of the wave - vector * q*. thus , in the following the results for @xmath1 will be presented mostly for the dlm reference states . for comparison , in a small number of cases we will present @xmath1 calculated for the fm reference states using only the cr - cr exchange interactions as the input . of course , this will be done with caution only for cases where we have reason to believe that the results are at least qualitatively correct . some fm results will also be included towards the stability analysis based on @xmath36 derived from cr - cr interactions only . again , this will be done with caution , only if the corresponding results can be shown to be meaningful via additional calculations . the cr - cr exchange interactions for all the alloys studied and for both fm and dlm reference states become negligible as the inter - atomic distance reaches about three lattice parameters or , equivalently , thirty neighbor shells . the same applies to the cr - x and cr - es interactions for the fm cases , these interactions in general being somewhat smaller . the cr - cr interactions for the dlm reference states are more damped compared with the corresponding fm results , showing less fluctuations in both sign and magnitude . the distance dependence of the exchange interactions between the cr atoms in cras is shown in fig.[fig10 ] for several lattice parameters . although the nearest neighbor interaction is always positive ( i.e , of ferromagnetic nature ) , the interactions with more distant neighbors are sometimes antiferromagnetic . such antiferromagnetic interactions are more common in cras for lower lattice parameters . with increasing lattice parameter , interactions become predominantly ferromagnetic , and by the time the equilibrium lattice parameter of 5.52 @xmath6 is reached , antiferromagnetic interactions mostly disappear . we have calculated such interactions up to the 405th neighbor shell , which amounts to a distance of roughly 8 lattice parameters . although the interactions themselves are negligible around and after the 30@xmath37 neighbor shell , their influence on the lattice sums continues up to about 100 neighbor shells . by about the neighbor 110@xmath37 shell ( a distance of @xmath4 5 lattice parameters ) the interactions fall to values small enough so as not to have any significant effect on the calculated lattice fourier transform of the exchange interaction and the curie temperature ( see below ) . it is clear from fig.[fig10 ] that ferromagnetism in cras is robust and exists over a wide range of lattice parameters . the distance dependence of the cr - cr exchange interactions in crsb is very similar to that in cras for both fm and dlm reference states . for crs , crse , and crte the situation is somewhat different . for crs and crse , the fm reference states for some low lattice parameters yield cr - cr interactions that are antiferromagnetic even at the nearest neighbor separation . for crte , at the lowest lattice parameter studied ( 5.44 ) the nearest neighbor interaction for the fm reference state is ferromagnetic , but becomes antiferromagnetic with increasing lattice parameter , changing back to ferromagnetic at higher lattice parameters . for all three compounds , the interactions are predominantly ferromagnetic at higher lattice parameters . figs.[fig11 ] and [ fig12 ] show the distance dependence of the exchange interactions calculated for the fm reference states in crse and crte , respectively , for several lattice parameters . predominant nearest neighbor antiferromagnetic interactions between the cr atoms result in negative values of the curie temperature , when calculated via eqs . ( [ e2 ] ) or ( [ e3 ] ) . these results for the curie temperature for the fm reference states can be discarded as being unphysical on two grounds : because of the neglect of the interactions involving the induced moments and also because they point to the possibility that the ground state is most probably antiferromagnetic or of complex magnetic structure . the antiferromagnetic cr - cr interactions mostly disappear , when calculated for the dlm reference states . this could be interpreted as being an indication that the actual magnetic structure of the ground states for these low lattice parameters in case of crs , crse and crte is closer to a dlm state than to an fm state . in fig.[fig13 ] we show the cr - cr exchange interactions for the dlm reference states in case of crse for the same lattice parameters as those considered for fig.[fig11 ] . a comparison of the two figures shows that all interactions have moved towards becoming more ferromagnetic for the dlm reference states , the nearest neighbor interaction for the lowest lattice parameter staying marginally antiferromagnetic . between the cr atoms in cras for various lattice parameters @xmath38 , calculated for the fm and dlm reference states . the distance between the cr atoms is given in units of the lattice parameter @xmath38 ( the same applies to figs.[fig11]-[fig13 ] ) . the main plot in fig.[fig10 ] shows the distance dependence up to 2.25@xmath38 , while the inset shows the values between 2.25@xmath38 and 5@xmath38 . although the individual values of @xmath19 are small beyond about 2.25@xmath38 , their cumulative effects on the total exchange constant and the curie temerature can not be neglected ( see text for details ) . comparison of the insets for the fm and dlm cases shows that the interactions are more damped for the dlm case , being at least an order of magnitude smaller for distances beyond @xmath42.25 - 2.5@xmath38 or 15 - 20 neighbor shells . similar comments apply to the interactions presented in figs.[fig11]-[fig13].,width=379 ] . , width=336 ] . , width=336 ] . , width=336 ] the deviation of the nature of the ground state from the reference state can be studied by examining the lattice fourier transform of the corresponding exchange interactions between the cr atoms : @xmath35 . as pointed out earlier , for the fm reference states this procedure suffers from the drawback of neglecting the effects of all other interactions involving the induced moments . for the dlm reference states there are no induced moments , so the relationship between the energy and @xmath36 is simpler , but a physical picture of the spin arrangement corresponding to a particular wave - vector @xmath39 is harder to visualize . for the fm reference states , if there were no moments other than those on the cr atoms , a maximum in @xmath36 at @xmath40 would imply that the ground state is ferromagnetic with collinear and parallel cr magnetic moments in all the unit cells . a maximum at symmetry points other than the @xmath41-point would imply the ground state being antiferromagnetic or a spin - spiral state . a maximum at a wave - vector @xmath39 that is not a symmetry point of the bz would imply the ground state being an incommensurate spin spiral . the presence of induced moments and the consequent interactions involving non - cr atoms and empty spheres spoil such interpretations based on @xmath36 derived from cr - cr interactions alone . however , the tendencies they reveal might still be useful . it is for this reason that we study the fourier transform @xmath36 , defined above , for both fm and dlm reference states . in fig.[fig14 ] we have plotted this quantity for cras . the results for crsb are quite similar . the maximum in @xmath36 at the @xmath41-point for all lattice parameters and for both fm and dlm reference states can be taken as an indication that the ground state magnetic structure is ferromagnetic for cras for all the lattice parameters studied . the same comment applies to crsb . the apparent lack of smoothness in @xmath36 shown for the fm reference states is a consequence of the fact that there are other additional bands ( involving induced moments ) , which are supposed to cross the band shown , but have not been computed . for crs , crse , and crte ( see figs.[fig15]-[fig16 ] ) , the deviation of the ground state for low lattice parameters from the parallel arrangement of cr moments is reflected in the result that the maximum moves away from the @xmath41-point for the fm reference states . at high values of the lattice parameter the maximum returns to the @xmath41-point . the curves for crse are similar to those for crs and have therefore not been shown . the fact that the maximum for the dlm reference sates lies at the @xmath41-point in most cases is again an indication that the ground state magnetic structure is closer to the dlm state than to the fm state . the conclusions based on the fm reference state results in figs.[fig15 ] and[fig16 ] may be suspect on ground of neglecting the interactions involving the induced moments . however , to explore whether they do carry any relevant information we have carried out additional calculations for the three compounds crs , crse and crte for two commonly occurring antiferromagnetic configurations : afm[001 ] , afm[111 ] . note that another commonly occurring afm configuration afm[110 ] is not unique , i.e. there are several configurations that could be seen as an afm[110 ] arrangement ( see fig3 . of ref . [ ] , table 2 of ref . the simplest among these is actually equivalent to afm[100 ] . the results for the total energy for the two afm calculations are shown in table [ table2 ] and compared with the corresponding fm and dlm total energies . for crs , the lowest energy state for lattice parameters 5.44 and 5.55 @xmath6 is afm[111 ] , exactly as suggested by the maximum in @xmath36 appearing at the l - point in fig.[fig15 ] for the fm reference state and for these two lattice parameters . as the lattice parameter increases beyond 5.55 , antiferromagnetic interactions diminish . for the next higher lattice parameter 5.66 @xmath6 in table [ table2 ] , the lowest energy state is dlm . this may suggest that the ground state has a complex magnetic structure , which remains to be explored . for higher lattice parameters the fm state has the lowest energy . for crse , afm[111 ] state has the lowest energy up to the lattice parameter 5.66 , as is also supported by the maximum of @xmath36 at l - point . the @xmath36 curves for crse are similar to those of crs , and have not been shown . for crte , at the lowest lattice parameter of 5.44 @xmath6 the lowest energy state is fm , as is also indicated by the maximum of @xmath36 at the @xmath41-point . for higher lattice parameters 5.65 and 5.76 , even though the @xmath36 curves point to the possibility of an afm[111 ] ground state , the fm state energy turns out to be the lowest among the configurations studied . it could be concluded that in this case a proper relationship between the energy and @xmath36 , obtained without the neglect of the induced moments , would point to the ground state being fm . for these three chalcogenides , for lattice parameters above 5.65 - 5.7 @xmath6the ground state should be fm . [ cols="<,^,^,^,^,^,^ " , ] we determine the curie temperature using eqs . ( [ e2 ] ) and ( [ e3 ] ) . for the dlm reference states , these produce estimates of @xmath1 from above the ferromagnetic@xmath42paramagnetic transition , and are free from errors due to induced moments . however , these estimates are high compared with properly derived values of @xmath1 from below the transition . the latter estimates would require the use of fm reference states ( where the ground states are known to be fm ) and thus a proper treatment of the induced moments . for cras and crsb , the magnetic state is ferromagnetic for all the lattice parameters considered . hence , for the sake of comparison we have calculated the @xmath1 for the fm reference states using eqs . ( [ e2 ] ) and ( [ e3 ] ) as well . according to the results of sandratskii @xcite the correctly calculated @xmath1 values , in the presence of interactions involving all the induced moments , would be higher . thus , the correct estimates of @xmath1 should lie somewhere between the dlm results and the fm results obtained with the neglect of the induced moments . in fig . [ fig17 ] we show these results for cras , crsb . we have used up to 111 shells in the evaluation of eq . ( [ e2 ] ) and for the lattice fourier transform of @xmath43 in eq . ( [ e3 ] ) , after having tested the convergence with respect to the number of shells included . the estimated computational error corresponding to the chosen number of shells used in these calculations is below @xmath44 . for comparison we also include the results for the mixed alloy cras@xmath0sb@xmath0 , for which the calculated @xmath1 values fall , as expected , in between those of cras and crsb . since rpa values are more accurate than mfa values , our best estimates of @xmath1 for cras range from somewhat higher than 500 k at low values of the lattice parameter , increasing to 1000 - 1100 k around the mid lattice parameter range ( 5.75 - 5.9 ) and then decreasing to around 600 k for higher lattice parameters ( 6.5 @xmath6 and above ) . for crsb these estimates are consistently higher than those for cras : 1100 k , 1500 k and 1200 k , respectively . the estimates for cras are similar to those provided by sasaiolglu @xcite for crs , crse , the results obtained with the fm reference states would clearly be wrong , in particular , for the low values of the lattice parameters , for which we have shown the ground state to be antiferromagnetic within our limited search . there is a possibility that the ground state for certain lattice parameters might have a complex magnetic structure . for crte , even though the ground state appears to be ferromagnetic , there are considerable antiferromagnetic spin fluctuations , making the fm estimates unreliable . in fig . [ fig18 ] we show the @xmath1 values for crs , crse and crte for the dlm reference states . the values for lattice parameters for which the ground state has been shown to be antiferromagnetic in the preceding section should be discarded as being inapplicable . similar results for the alloys cras@xmath0x@xmath0 ( x = s , se and te ) are shown in fig . [ fig19 ] for dlm reference states . for these , the ground state is ferromagnetic for all lattice parameters . however , because of the neglect of the induced moments related effects , our results for the curie temperatures for the fm reference states are lower than the properly calculated values . thus in fig.[fig19 ] we show the dlm results only , which are devoid of the induced moment effects and provide us with estimates of @xmath1 from above the transition . these are expected to be somewhat higher than the properly computed values for fm reference states . thus , for these alloys the trend revealed in fig . [ fig19 ] for the variation of @xmath1 with lattice parameter is correct . the estimates themselves are qualitatively correct , albeit somewhat higher than the correct values . only the rpa values are plotted in fig . [ fig19 ] , which are more reliable than the mfa values . for comparison , we also show the results for the pnictides cras , crsb , and cras@xmath0sb@xmath0 , which are isoelectronic among themselves , but have half an electron per unit cell less than the mixed alloys cras@xmath0x@xmath0 ( x = s , se and te ) sb@xmath0 are also shown . , width=360 ] x@xmath0 alloys with x = s , se and te . for comparison , the results for cras , crsb and cras@xmath0sb@xmath0 are also shown . all results shown are for dlm reference states , and as such , should be considered as upper limits for @xmath1 . , width=336 ] the differences between the results for the pnictides , chalcogenides and the mixed pnictide - chalcogenides can be summarized as follows . the pnictides , cras , crsb , and cras@xmath0sb@xmath0 , are strong ferromagnets at all the lattice parameters studied ( 5.44 - 6.62 ) . in the dlm description , their @xmath1 stays more or less constant ( apart from a minor increase ) as the lattice parameter increases from 5.4 /aa to 6.1 , and then decreases beyond ( figs . [ fig17 ] and [ fig19 ] . the chalcogenides are antiferromagnetic or have complex magnetic structure for low lattice parameters . in the dlm description , their @xmath1 in the ferromagnetic state increases and then becomes more or less constant as the lattice parameter increases ( fig . [ fig18 ] ) . the mixed alloys cras@xmath0x@xmath0 ( x = s , se , te ) are ferromagnetic at all the lattice parameters studied . in the dlm description , their @xmath1 rises and then falls as the lattice parameter is increased from 5.44 to 6.62 . a comparison of the results presented in figs . [ fig17]-[fig19 ] shows that large changes in @xmath1 take place by changing the number of carriers . changes due to isoelectronic doping are small compared with changes brought about by changing carrier concentration . our _ ab initio _ studies of the electronic structure , magnetic moments , exchange interactions and curie temperatures in zb crx ( x = as , sb , s , se and te ) and cras@xmath0x@xmath0 ( x = sb , s , se and te ) reveal that half - metallicity in these alloys is maintained over a wide range of lattice parameters . the results for the exchange interaction and the curie temperature show that these alloys have relatively high curie temperatures , i.e. room temperature and above . the exceptions occur for the alloys involving s , se and te at some low values of lattice parameters , where significant inter - atomic antiferromagnetic exchange interactions indicate ground states to be either antiferromagnetic or of complex magnetic nature . a comparison of total energies for the fm , dlm , and two zb antiferromagnetic configurations ( afm[001 ] and afm[111 ] ) show the lowest energy configuration to be afm[111 ] for crs and crse for compressed lattice parameters ( table[table2 ] ) . the possibility of afm ground states for compressed lattice parameters for crs was noted by zhao and zunger@xcite and for crse by sasioglu @xcite . our search for the antiferromagnetic ground states is more thorough than what was reported in these two studies . an extensive study of several antiferromagnetic configurations as well as ferrimagnetic and more complex magnetic structures for crs , crse and crte is currently underway . the mixed pnictide - chalcogenide alloys cras@xmath0x@xmath0 ( x= s , se , te ) do not show any tendency to antiferromagnetic spin fluctuations for the entire range of the lattice parameter studied . presumably the pnictogens suppress antiferromagnetic tendencies . such alloys may play an important role in fabricating stable zb half - metallic materials , as the concentration of the pnictogens and the chalcogens may be varied to achieve lattice - matching with a given substrate . as long as the concentration of as or sb is higher than the chalcogen concentration , half - metallic ferromagnetic state can be achieved . there is a large variation in the curie temperature of these alloys ( fig . [ fig19 ] ) as the lattice parameter varies from the low ( @xmath4 5.4 ) to the mid ( @xmath4 6.1 ) range of the lattice parameters studied . this variation is much smaller for the isoelectronic alloys cras , crsb and cras@xmath0sb@xmath0 ( fig . [ fig17 ] ) over this range of lattice parameters . note that most ii - vi and iii - v zb semiconductors have lattice parameters in this range . large changes in @xmath1 can be brought about by changing the carrier concentrations . the pnictides in general have a higher @xmath1 than the chalcogenides . our results for the curie temperature , the lattice fourier transform of the exchange interactions , and the resulting stability analysis are based on the exchange interactions between the cr atoms only . for the fm reference states this causes some errors due to the neglect of the effects of the induced moments . the dlm results are free from such errors . it is expected that the present study will provide both qualitative and quantitative guidance to experimentalists in the field . 99 k. sato and h. katayama - yoshida , semicond . * 17 * , 367 ( 2002 ) . see k. sato , t. fukushima and h. katayama - yoshida , j. phys . : matter * 19 * , 365212 ( 2007 ) , and references therein . see b. belhadji , l. bergqvist , r. zeller , p.h . dederichs , k. sato and h. katayama - yoshida , j. phys . : condens . matter * 19 * , 436227 ( 2007 ) , and references therein . h. saito , v. zayets , s. yamagata , and k. ando , phys . lett * 90 * , 207202 - 1 ( 2003 ) . k. sato and h. katayama - yoshida , jpn . . phys . * 40 * , l651 ( 2001 ) . h. akinaga , t. manago , and m. shirai , jpn . j. appl . phys.*39 * , l1118 ( 2000 ) . s. li , j - g duh , f. bao , k - x liu , c - l kuo , x. wu , liya l , z. huang , and y du , j. phys . phys . * 41 * 175004 ( 2008 ) . m. shirai , j. appl . phys . * 93 * , 6844 ( 2003 ) . h. akinaga , m. mizuguchi , k. nagao , y. miura , and m. shirai in _ springer lecture notes in physics _ * 676 * , 293 - 311 ( springer - verlag , berlin 2005 ) . k. yamana , m. geshi , h. tsukamoto , i. uchida , m. shirai , k. kusakabe , and n. suzuki , j. phys . : condens . matter * 16 * , s5815 ( 2004 ) . l. kahal , a. zaoul , m. ferhat , j. appl . phys . * 101 * , 093912 ( 2007 ) . i. galanakis and p. mavropoulos , , 104417 ( 2003 ) ; 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we present calculations of the exchange interactions and curie temperatures in cr - based pnictides and chalcogenides of the form crx with x = as , sb , s , se and te , and the mixed alloys cras@xmath0x@xmath0 with x = sb , s , se , and te .
the calculations are performed for zinc blende ( zb ) structure for 12 values of the lattice parameter between 5.44 and 6.62 , appropriate for some typical ii - vi and iii - v semiconducting substrates .
electronic structure is calculated via the linear muffin - tin - orbitals ( lmto ) method in the atomic sphere approximation ( asa ) , using empty spheres to optimize asa - related errors . whenever necessary
, the results have been verified using the full - potential version of the method , fp - lmto .
the disorder effect in the as - sublattice for cras@xmath0x@xmath0 ( x = sb , s , se , te ) alloys is taken into account via the coherent potential approximation ( cpa ) .
exchange interactions are calculated using the linear response method for the ferromagnetic ( fm ) reference states of the alloys , as well as the disordered local moments ( dlm ) states .
these results are then used to estimate the curie temperature from the low and high temperature side of the ferromagnetic / paramagnetic transition .
estimates of the curie temperature are provided , based on the mean field and the more accurate random phase approximations .
dominant antiferromagnetic exchange interactions for some low values of the lattice parameter for the fm reference states in crs , crse and crte prompted us to look for antiferromagnetic ( afm ) configurations for these systems with energies lower than the corresponding fm and dlm values .
results for a limited number of such afm calculations are discussed , identifying the afm[111 ] state as a likely candidate for the ground state for these cases .
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the physics of elementary particles and forces determined the development of the early universe and thus , of the structure of our world today ( * fig . 1 * ) . according to our present knowledge , three families of quarks and leptons , four fundamental interactions , their respective exchange bosons and a yet - to - discover mechanism to generate particle masses are the ingredients ( * fig . 2 * ) which are necessary to describe our universe , both at cosmic as well as at microscopic scales . three of the four forces are relevant for particle physics at small distances : the strong , the electromagnetic and the weak force . they are described by quantum field theories , quantum chromodynamics ( qcd ) for the strong , quantum - electrodynamics ( qed ) for the electromagnetic and the so - called standard model of the unified electro - weak interactions @xcite . the weakest force of the four , gravitation , is the major player only at large distances where the other three are , in general , not relevant any more : the strong and the weak force are short - ranged and thus limited to sub - nuclear distances , the electromagnetic force only acts between objects whose net electric charge is different from zero . of the objects listed in * fig . 2 * , only the @xmath2-neutrino ( @xmath3 ) , the graviton and the higgs - boson are not explicitly detected to - date . besides these particular points of ignorance , the overall picture of elementary particles and forces was completed and tested with remarkable precision and success during the past few years , and the data from the lep electron - positron collider belong to the major important ingredients in this field . this lecture reviews selected aspects of standard model physics at lep . the frame of this write - up is not a standard and text - book - like presentation , but rather a collection and reproduction of slides , pictures and tables , similar as presented in the lecture itself . since most of the slides are self - explanatory , the collection is only accompanied by a short , connecting text , plus a selection of references where the reader can find more detailed information . a decade of successful operation of the large electron positron collider , lep @xcite ( * fig . 3 * ) , provided a whealth of precision data ( * fig . 4 * ) on the electroweak and on the strong interactions , through a multitude of @xmath0 annihilation final states ( depicted in * fig . 5 * ) which are recorded by four multi - purpose detectors , aleph @xcite , delphi @xcite , l3 @xcite and opal @xcite . in the phase which is called lep - i `` , from 1989 to 1995 , the four lep experiments have collected a total of about 17 million events in which an electron and a positron annihilate into a @xmath4 which subsequently decays into a fermion - antifermion - pair ( see figs . 4 and 5 ) . since 1995 , the lep collider operates at energies above the @xmath4 resonance , @xmath5 ( lep - ii '' ) , up to currently more than 200 gev in the centre of mass system . the different final states of @xmath0 annihilations can be measured and identified with large efficiency and confidence , due to the hermetic and redundant detector technologies realised by all four experiments . an example of a hadronic 3-jet event , originating from the process @xmath6 with subsequent fragmentation of quarks and gluon(s ) into hadrons , as recorded by the opal detector ( * fig . 6 * ) @xcite , is reproduced in * fig . the basic predictions of the standard model of electroweak interactions , for fermion - antifermion production of @xmath0 annihilations around the @xmath4 resonance , are summarised in * fig . 8 * to * fig . 11 * , see @xcite and recent experimental reviews @xcite for more details . cross sections of these processes are energy ( s``- ) dependent and contain a term from @xmath4 exchange , another from photon exchange as well as a @xmath7 '' interference term ( * fig . 8 * ) . measurements of s - dependent cross sections around the @xmath4 resonance provide model independent results for the mass of the @xmath4 , @xmath8 , of the @xmath4 total and partial decay widths , @xmath9 and @xmath10 , and of the fermion pole cross sections , @xmath11 . beyond the lowest order born approximation `` , photonic and non - photonic radiative corrections must be considered ( * fig . 9 * ) ; the latter can be absorbed into running coupling constants '' ( * fig . 10 * ) which , if inserted into the born approximation , make the experimental observables depend on the masses of the top quark and of the higgs boson , @xmath12 and @xmath13 . measurements of the fermion final state cross sections as well as of other observables like differential cross sections , forward - backward asymmetries and final state polarisations of leptons ( * fig . 11 * ) allow to extract the basic electroweak parameters . combined analyses of the data of all 4 lep experiments by the lep electroweak working group `` @xcite provide very precise results ( * fig . 12 * ) : for instance , due to the precise energy calibration of lep @xcite , @xmath8 is determined to an accuracy of 23 parts - per - million , and the number of light neutrino generations ( and thus , of quark- and lepton - generations in general ) is determined to be compatible with 3 within about 1% accuracy . from radiative corrections and a combination of data from lep - i and lep - ii , @xmath12 , @xmath13 , the coupling strength of the strong interactions , @xmath1 , the effective weak mixing angle @xmath14 and the mass of the w - boson , @xmath15 , can be determined with remarkable accuracy ( except for @xmath13 which only enters logarithmically ) . a list of the most recent results @xcite is given in * fig . 13 * , where also the deviations of the experimental fits from the theoretical expectations are given by the number of standard deviations ( pull '' ) . graphical representations of some of these results are given in * fig . 14 * to * fig . 18*. the significance of counting the number of light neutrino families , @xmath16 , from the measurement of the @xmath4 line shape , based on aleph data from the 1990 and 1991 scan period , is displayed in * fig . 14*. the gain in precision of electroweak parameters between 1987 , before the era of lep , and the lep results of 1999 is demonstrated in * fig . 15 * , for the values of the leptonic axial and vector couplings , @xmath17 and @xmath18 . the fit result of the higgs mass , @xmath13 , ist given in * fig . 16 * , calculated using two different input values for the uncertainty of the hadronic part of the running qed coupling constant , @xmath19 @xcite , together with the exclusion limit from direct higgs production searches , @xmath20 gev ( 95% confidence level ) @xcite . the measured cross section for @xmath21 pair production , @xmath22 , is presented in * fig . 17 * , together with the standard model prediction and two toy models " which demonstrate the importance of the @xmath23 triple gauge boson vertex and the @xmath24 exchange diagram , see * fig . 5*. a summary of the available measurements ( top ) and indirect determinations , i.e. through radiative corrections ( bottom ) , of the @xmath21 mass is given in * fig . 18*. more results and graphs are available from @xcite and from the home page of the lep electroweak working group @xcite . a short introduction to the development of hadron physics , from the discovery of the neutron to the development of qcd and the experimental manifestation of gluons , is given in * fig . 19*. the basic properties of qcd - in comparison with qed - are summarised in * fig . 20*. the energy dependence of the strong coupling strength @xmath1 , given by the so - called @xmath25-function in terms of the renormalisation scale @xmath26 and the qcd group structure parameters @xmath27 , @xmath28 and @xmath29 , is described in * fig . 21*. in * fig . 22 * , the anatomy of the process @xmath30 hadrons is illustrated . factorisation is assumed to hold when splitting this process into an electroweak part ( annihilation of @xmath0 into a virtual photon or @xmath4 and subsequent decay into a quark - antiquark pair ) , the development of a parton ( i.e. quark and gluon ) shower described by perturbative qcd , a hadronisation phase which can be modelled using various different fragmentation or hadronisation models , and finally a parametrisation of the decays of unstable hadrons ( according to measured decay modes and branching fractions ) @xcite . a list of the most prominent qcd topics covered by the lep experiments is given in * fig . 23*. for a more detailed introduction to qcd and hadronic physics at high energy particle colliders see e.g. @xcite ; earlier reviews of qcd tests at lep can be found in @xcite . one of the most prominent qcd - related measurements at lep is the determination of @xmath1 from the radiative corrections to the hadronic partial decay width of the @xmath4 , which is summarised in * fig . 24*. the ratio @xmath31 is a totally inclusive quantity which is independent of hadronisation effects , and qcd corrections are available in complete @xmath32 , i.e. in next - to - next - to - leading order qcd perturbation theory @xcite . the determination of @xmath1 from @xmath33 , however , crucially depends on the validity of the predictions of the electroweak standard model . the basic principles of the physics of hadrons jets , which are interpreted as the footprints of energetic quarks and gluons , and the definition of hadron jets are described in * fig . 25*. the most commonly used jet algorithms in @xmath0 annihilations are clustering procedures as first introduced by the jade collaboration @xcite , and variants of this algorithm @xcite as listed in * tab . 1*. for these algorithms , relative production rates of @xmath34-jet events ( @xmath34 = 2 , 3 , 4 , ... ) are predicted by qcd perturbation theory , and are therefore well suited to determine @xmath1 and to prove the energy dependence of @xmath1 , see * fig . 26*. in particular , the relative rate of 3-jet events , @xmath35 , is predicted to be proportional to @xmath1 , in leading order perturbation theory . corrections in complete next - to - leading order , i.e. in @xmath36 , are available for these algorithms @xcite . hadronisation effects , however , may significantly influence the reconstruction of jets . this can be seen in * fig . 27 * , where jet production rates are analysed using qcd model ( jetset ) events of @xmath0 annihilation at @xmath37 gev before and after hadronisation , i.e. at parton- and at hadron - level . the @xmath38 of 3-jet reconstruction , i.e. the number of events which are classified as 3-jet both on parton- and at hadron - level , normalised by the number of events classified as 3-jet on hadron level , is displayed in * fig . 28*. the energy dependence of hadronisation corrections to measurements of 3-jet event production rates at fixed jet resolution @xmath39 is analysed in * fig . 29*. from these studies , the original jade and the durham schemes emerge as the most reliable " algorithms to test qcd in jet production from @xmath0 annihilations ( for a comparative study of the newer cambridge algorithm , see e.g. @xcite ) . especially the jade algorithm exhibits small and almost energy independent hadronisation corrections . this allows to test the energy dependence of @xmath1 and thus , of asymptotic freedom , without actually having to determine numerical values of @xmath1 , see * fig . 30 * @xcite . hadronic event shapes ( * fig . 31 * ) are a common tool to study aspects of qcd , and in particular , to determine @xmath1 . for many of these observables , qcd predictions in next - to - leading order ( @xmath36 ) are available @xcite , and for some of them , the leading and next - to - leading logarithms were resummed to all orders @xcite . the results of one such study , performed by l3 @xcite using event shapes of lep - i and lep - ii data plus radiative events at reduced centre of mass energies , is shown in * fig . 32 * , demonstrating the running of @xmath1 . for more details on the determination of @xmath1 from hadronic event shape and jet related observables , see eg . @xcite . a list of high energy particle processes and observables from which significant determinations of @xmath1 are obtained is given in * fig . 33*. the most recent measurements , as an update to the world summary of @xmath1 from 1998 @xcite , are listed in * fig . 34*. * table 2 * summarises the current status of @xmath1 results . the corresponding values of @xmath40 , where @xmath41 is the typical hard scattering energy scale of the process which was analysed , are displayed in * fig . 35*. the data , spanning energy scales from below 1 gev up to several hundreds of gev , significantly demonstrate the energy dependence of @xmath1 , which is in good agreement with the qcd prediction . evolving these values of @xmath42 to a common energy scale , @xmath43 , using the qcd @xmath25-function in @xmath44 with 3-loop matching at the heavy quark pole masses @xmath45 gev and @xmath46 gev @xcite , results in * fig . 36 * , demonstrating the good agreement between all measurements . from the results based on qcd calculations which are complete to next - to - next - to - leading order ( filled symbols in fig . 36 ; see also table 2 ) , a new world average of @xmath47}\ ] ] is determined . the overall error is calculated using a method @xcite which introduces an common correlation factor between the errors of the individual results such that the overall @xmath48 amounts to 1 per degree of freedom . the size of the resulting overall uncertainty depends on the method and philosophy used to determine the world average of @xmath49 , see @xcite for further discussion . 99 see textbooks on gauge theories and particle physics , as for instance : + e. leader , e. predazzi , _ an introduction to gauge theories and modern particle physics _ , vols . 1 and 2 , cambridge university press , 1996 ; + c. quigg , _ gauge theories of the strong , weak and electromagnetic interactions _ , benjamin / cummings ( 1983 ) . s. myers , e. picasso , contemp . 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( 1991 ) 560 . t. hebbeker , m. martinez , g. passarino and g. quast , phys . b331 ( 1994 ) 165 . jade collab . , w. bartel et al . , z. phys . c33 ( 1986 ) , 23 ; + jade collab . , s. bethke et al . b213 ( 1988 ) , 235 . dokshitzer , contribution to the workshop on jets at lep and hera , durham ( 1990 ) , j.phys.g17 ( 1991 ) . z. kunszt and p. nason [ conv . ] in _ z physics at lep 1 _ ( eds . g. altarelli , r. kleiss and c. verzegnassi ) , cern 89 - 08 ( 1989 ) . s. bethke , z. kunszt , d.e . soper and w.j . stirling , nucl . b370 ( 1992 ) 310 . dokshitzer , g.d . leder , s. moretti and b.r . webber , jhep 9708:001 , 1997 ; hep - ph/9707323 . s. bentvelsen and i. meyer , eur . j c4 ( 1998 ) 623 . s. bethke , _ proc . qcd euroconference 96 _ , montpellier , france , july ( 1996 ) , nucl . ( proc.suppl . ) 54a ( 1997 ) 314 ; hep - ex/9609014 . s. catani , l. trentadue , g. turnock and b.r . webber , nucl . b407 ( 1993 ) 3 . l3 collaboration , m. acciarri et al . b411 ( 1997 ) 339 . movilla - fernandez , o. biebel and s. bethke , hep - ex/9906033 . s. bethke , @xmath51 int . symp . on radiative corrections , barcelona , sept . 8 - 12 , 1998 ; hep - ex/9812026 . chetyrkin et al . , hep - ph/9706430 . m. schmelling , phys . scripta 51 ( 1995 ) 676 .
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selected topics on precision tests of the standard model of the electroweak and the strong interaction at the lep @xmath0 collider are presented , including an update of the world summary of measurements of @xmath1 , representing the state of knowledge of summer 1999 .
this write - up of lecture notes consists of a reproduction of slides , pictures and tables , supplemented by a short descriptive text and a list of relevant references . 0z^0
2d_2 2 ^ 2 l 2q^2 mpi - phe/2000 - 02 + january 2000
| 5,678 | 141 |
the electric transport in semiconductor superlattices is usually dominated by resonances between the localized energy levels inside the wells resulting in peaks in the current - field relation . this may yield complicated current - voltage characteristics exhibiting may branches due to the formation of domains of different electric field inside the sample ( see , e.g. , @xcite and references therein ) . these experimental features could be qualitatively reproduced by theoretical models combining rate equations for the transport between the wells and poisson s equation @xcite . while these approaches assume interwell transition rates which are either fitted or obtained from phenomenological models , we have recently proposed a way to calculate the transport microscopically @xcite . we obtained good quantitative agreement with the experimental data of ref . @xcite for highly doped samples , where the scattering from ionized impurities causes a strong broadening of the levels . here we consider the lower doped sample used in @xcite consisting of a 40 period gaas / alas superlattice ( barrier width @xmath0 nm , well width @xmath1 nm , period @xmath2 , doping @xmath3@xmath4 per well , cross section @xmath5 @xmath4 ) and investigate the impact of interface roughness which both contributes to the broadening and causes nonresonant transitions between the wells . in the case of weakly coupled quantum wells the appropriate basis set is a product of wannier functions @xmath6 of subband @xmath7 localized in well @xmath8 , and plane waves @xmath9 . here the @xmath10 direction is the growth direction and @xmath11 are vectors within the @xmath12 plane . restricting ourselves to the lowest two minibands ( denoted by @xmath13 and @xmath14 ) and coupling between neighbouring wells the hamiltonian in the presence of an electric field @xmath15 is given by @xmath16 , where @xmath17\label{eqham1}\\ \hat{h}_1&=&\sum_{n,{\underline{k } } } \left [ t_1^a a_{n+1}^{\dag}({\underline{k}})a_n({\underline{k } } ) + t_1^b b_{n+1}^+({\underline{k}})b_n({\underline{k } } ) -efr^{ab}_1 a_{n+1}^{\dag}({\underline{k}})b_n({\underline{k } } ) -efr^{ba}_1 b_{n+1}^{\dag}({\underline{k}})a_n({\underline{k } } ) \right]\end{aligned}\ ] ] with @xmath18 ( @xmath19 is the effective mass in the well ) , the couplings @xmath20 , and the miniband width @xmath21 of subband @xmath7 . diagonalizing the hamiltonian @xmath22 leads to renormalized coefficients in @xmath22 and @xmath23@xcite which we use in the following . we calculate the wannier functions in a kronig - penney - type model . following ref . @xcite we model the nonparabolicity by an energy dependent effective mass @xmath24 , where @xmath25 is the effective mass at the conduction band minimum of energy @xmath26 , and @xmath27 is the energy gap . then the usual connection rules hold for the envelope function provided that the momentum matrix element @xmath28 between the conduction and valence band states is identical in both materials . we use the values@xcite @xmath29 , @xmath30 , @xmath31 ev , @xmath32 ev , and the conduction band discontinuity @xmath33 ev . these parameters yield a relation @xmath34 which is in excellent agreement with the band structure of alas@xcite for the energies of interest . is slightly different for the two materials in contrast to the assumption . furthermore , the envelope functions for different energies are not orthogonal as the effective hamiltonian is energy dependent . however , the overlap is small and we neglect these complications . ] we obtain the coefficients @xmath35 mev , @xmath36 176.6 mev , @xmath37 mev , @xmath38 mev , @xmath39 , and @xmath40 . for small couplings between the wells and fast intersubband relaxation the current from subband @xmath7 in well @xmath8 to subband @xmath41 in well @xmath42 is given by the following expression@xcite : @xmath43 \label{eqj}\ , .\end{aligned}\ ] ] here @xmath44 is the spectral function of subband @xmath7 in well number @xmath8 and @xmath45 is the fermi function . the energy @xmath46 is measured with respect to the electrochemical potential @xmath47 in well @xmath8 yielding an implicit dependence of @xmath44 on @xmath47 . we determine @xmath47 from the local electron density @xmath48 . then the difference @xmath49 is equal to @xmath50 for @xmath51 . we obtain @xmath44 in equilibrium from the retarded self - energy @xmath52 neglecting the coupling to the other wells . in ref . @xcite we have calculated the self - energy for scattering from the screened potential of ionized impurities within the self - consistent single - site - approximation . as an additional contribution to the self - energy we study here the impact of interface roughness we consider an interface located at @xmath53 exhibiting thickness fluctuations @xmath54 of the order of @xmath55 ( we use @xmath56 which is one monolayer of gaas ) . we assume @xmath57 and the correlation @xmath58 this can be motivated in the following way : at a given point @xmath59 there is an island of thickness @xmath60 with a probability @xmath61 ( we use an average coverage @xmath62 ) . therefore @xmath63 . provided the island extends from @xmath59 to @xmath64 we assume a constant probability to find a further neighbouring atom beyond @xmath64 yielding the exponential in eq . ( [ eqexpkorr ] ) . following ref . @xcite we model the additional potential by a @xmath65-function at the perfect interface @xmath66 and obtain @xmath67\label{eqhamrough}\end{aligned}\ ] ] with the matrix elements @xmath68 $ ] . the elements @xmath69 contribute to the current from one well to the next via eq . ( [ eqj ] ) . for weakly coupled wells @xmath70 are small and are neglected in the following . the elements @xmath71 result in scattering within the wells . we calculate their contribution to the self - energy within the self - consistent born - approximation @xmath72 where the factor 2 takes into account the two interfaces per well . the calculation for the subband @xmath14 is performed in the same way . with ( [ eqexpkorr ] ) we obtain the matrix element @xmath73 in contrast to this the usual choice ( see , e.g. , @xcite ) @xmath74 yields @xmath75 . in order to compare both expressions we consider the matrix element for scattering at a single circular island of thickness @xmath76 and radius @xmath77 which is given by @xmath78 , where @xmath79 is the bessel function of the first kind . now there are @xmath80 islands and we assume that negative and positive @xmath76 cancel each other . then @xmath81 for large @xmath82 as in eq . ( [ equqexp ] ) with a prefactor differing by @xmath83 . this indicates that the choice ( [ eqexpkorr ] ) captures an essential part of physics not contained in the usual choice . furthermore , the single island result reflects the quality of the born - approximation for the self - energy . the analysis of a @xmath65-potential reveals that diagrams containing multiple scattering at a single island become important if the product @xmath84 is larger than 1 . using the value @xmath85 this yields @xmath86 . ( for our parameters @xmath87 nm . ) this expression can be interpreted easily : the right hand side is the energy the electron gains if it is located at an island exhibiting a larger well width . the left hand side is the quantization energy associated with a length @xmath88 . the localization of the wavefunction takes place if the energy gain due to the larger well width dominates the cost due to the localization . in this case higher order diagrams become important . summing the contributions between different subbands given by eq . ( [ eqj ] ) we obtain the current @xmath89 depending on the field and the electron densities . 1a shows @xmath90 exhibiting a first maximum @xmath91 ma at @xmath92 mev where the current is dominated by @xmath93 transitions and a second peak @xmath94 ma at @xmath95 mev due to the @xmath96 resonance . the interface roughness has two implications : firstly the maxima are broadened and slightly shifted due to the contribution to the self - energies via eq . ( [ eqsigmarough ] ) . secondly there is a nonresonant current due to the @xmath97 transitions dominating the behaviour between the maxima . in order to study the domain formation we consider effective fields @xmath98 between wells @xmath99 and @xmath100 fulfilling the discretized poisson equation @xmath101 . then the local currents @xmath102 have to be equal for all @xmath99 in the steady state . the total voltage across the superlattice is @xmath103 . using the boundary conditions @xmath104 , @xmath105 we obtain the current - voltage characteristic shown in fig . 1b which exhibits the usual sequence of branches @xcite . in contrast to previous theoretical results the maximal current of the branches ( 90 @xmath41a ) is significantly lower than the height of the first resonance . it is almost independent of the boundary conditions , which mainly affect the voltage where the leftmost homogeneous branch breaks up . the maximal current of 90 @xmath41a is between the two experimental results@xcite 60 @xmath41a and 130 @xmath41a for samples # 1 and # 2 , respectively , which both have the nominal sample parameters used here . we can reproduce the difference in current between both samples by assuming barrier widths which differ by just one monolayer , thus suggesting a possible explanation for the experimental discrepancy . the minimal currents of the branches ( 26 @xmath41a ) seem to be lower than the experimental values . they depend strongly on the nonresonant currents , e.g. , we obtain the value 10 @xmath41a ignoring transitions via @xmath106 . this indicates that we have probably underestimated the nonresonant current . the slope of the branches is steeper in our calculation than found experimentally ; this could be due to some additional contact resistance of the order of 1 k@xmath107 which is not included in our calculation . in conclusion , we have presented a microscopic model for sequential tunnelling in weakly coupled quantum wells . the model describes the domain formation in superlattices quantitatively with no fitting parameters . nonresonant transitions due to interface roughness have been shown to strongly affect the current between the resonances and the extent of the current branches . s. h. kwok , h. t. grahn , m. ramsteiner , k. ploog , f. prengel , a. wacker , e. schll , s. murugkar , and r. merlin , phys . b * 51 * , 9943 ( 1995 ) . f. prengel , a. wacker , and e. schll , phys . b * 50 * , 1705 ( 1994 ) , erratum in * 52 * , 11518 ( 1995 ) . g. brozak , e. a. de andrada e silva , l. j. sham , f. derosa , p. miceli , s. a. schwarz , j. p. harbison , l. t. florez , and j. s. allen , phys . lett . * 64 * , 471 ( 1990 ) . j. n. schulman and y .- c . chang , phys . b * 31 * , 2056 ( 1985 ) .
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a microscopic calculation of the perpendicular current in doped multiple quantum wells is presented .
interface roughness is shown to affect the resonant transitions as well as to cause a nonresonant background current .
the theoretical characteristics exhibit several branches due to the formation of electric field domains in quantitative agreement with experimental data .
| 3,339 | 77 |
for well over a century , radial velocities for objects outside the solar system have been determined through spectroscopy , using the ( doppler ) shifts of stellar spectral lines . the advent of high - accuracy ( sub - milliarcsec ) astrometric measurements , both on ground and in space , now permits radial velocities to be obtained by alternative methods , based on geometric principles and therefore independent of spectroscopy . the importance of such _ astrometric radial velocities _ stems from the fact that they are independent of phenomena which affect the spectroscopic method , such as line asymmetries and shifts caused by atmospheric pulsation , surface convection , stellar rotation , stellar winds , isotopic composition , pressure , and gravitational potential . conversely , the differences between spectroscopic and astrometric radial velocities may provide information on these phenomena that can not be obtained by other methods . although the theoretical possibility of deducing astrometric radial velocities from geometric projection effects was noted already at the beginning of the 20th century ( if not earlier ) , it is only recently that such methods have reached an accuracy level permitting non - trivial comparison with spectroscopic measurements . we have analysed three methods by which astrometric radial velocities can be determined ( fig . [ fig : methods ] ) . two of them are applicable to individual , nearby stars and are based on the well understood secular changes in the stellar trigonometric parallax and proper motion . the third method uses the apparent changes in the geometry of a star cluster or association to derive its kinematic parameters , assuming that the member stars share , in the mean , a common space velocity . in sects . [ sec : pidot ] to [ sec : mcm ] we describe the principle and underlying assumptions of each of the three methods and derive approximate formulae for the expected accuracy of resulting astrometric radial velocities . for the first and second methods , an inventory of nearby potential target stars is made , and the second method is applied to several of these . however , given currently available astrometric data , only the third ( moving - cluster ) method is capable of yielding astrophysically interesting , sub - km s@xmath1 accuracy . in subsequent papers we develop in detail the theory of this method , based on the maximum - likelihood principle , as well as its practical implementation , and apply it to a number of nearby open clusters and associations , using data from the hipparcos astrometry satellite . in the following sections , @xmath2 , @xmath3 and @xmath4 denote the trigonometric parallax of a star , its ( total ) proper motion , and its radial velocity . the components of @xmath3 in right ascension and declination are denoted @xmath5 and @xmath6 , with @xmath7 . the dot signifies a time derivative , as in @xmath8 . the statistical uncertainty ( standard error ) of a quantity @xmath9 is denoted @xmath10 . ( we prefer this non - standard notation to @xmath11 , since @xmath9 is itself often a subscripted variable . ) @xmath12 is used for the physical velocity dispersion in a cluster . @xmath13 km is the astronomical unit ; the equivalent values @xmath14 km yr s@xmath1 and @xmath15 mas km yr s@xmath1 are conveniently used in equations below ( cf . table 1.2.2 in vol . 1 of esa @xcite ) . other notations are explained as they are introduced . in estimating the potential accuracy of the different methods , we consider three hypothetical situations : * case a : a quasi - continuous series of observations over a few years , resulting in an accuracy of @xmath16 mas ( milliarcsec ) for the trigonometric parallaxes and @xmath17 mas yr@xmath1 for the proper motions . * case b : similar to case a , only a thousand times better , i.e. @xmath16 @xmath3as ( microarcsec ) and @xmath17 @xmath3as yr@xmath1 . * case c : _ two _ sets of measurements , separated by an interval of 50 yr , where each set has the same accuracy as in case b. the much longer - time baseline obviously allows a much improved determination of the accumulated changes in parallax and proper motion . the accuracies assumed in case a are close to what the hipparcos space astrometry mission ( esa @xcite ) achieved for its main observation programme of more than 100@xmath18000 stars . current ground - based proper motions may be slightly better than this , but not by a large factor . this case therefore represents , more or less , the state - of - the - art accuracy in optical astrometry . accuracies in the 1 to 10 @xmath3as range are envisaged for some planned or projected space astrometry missions , such as gaia ( lindegren & perryman @xcite ) and sim ( unwin et al . @xcite ) . the duration of such a mission is here assumed to be about 5 years . using the longer - time baselines available with ground - based techniques , similar performance may in the future be reached with the most accurate ground - based techniques ( pravdo & shaklan @xcite ; shao @xcite ) . case b therefore corresponds to what we could realistically hope for within one or two decades . case c , finally , probably represents an upper limit to what is practically feasible in terms of long - term proper - motion accuracy , not to mention the patience of astronomers . the most direct and model - independent way to determine radial velocity by astrometry is to measure the secular change in the trigonometric parallax ( fig . [ fig : methods]a ) . the distance @xmath19 ( from the solar system barycentre ) is related to parallax @xmath2 through @xmath20 . since @xmath21 , the radial velocity is @xmath22 where @xmath23 is the astronomical unit ( sect . [ sec : not ] ) . the equivalent of eq . ( [ eq : pidot ] ) was derived by schlesinger ( @xcite ) , who concluded that the parallax change is very small for every known star . however , although extremely accurate parallax measurements are obviously required , the method is not as unrealistic as it may seem at first . to take a specific , if extreme , example : for barnard s star ( gl 699 = hip 87937 ) , with @xmath24 mas and @xmath25 km s@xmath1 , the expected parallax rate is @xmath26as yr@xmath1 . according to our discussion in sect . [ sec : acc ] this will almost certainly be measurable in the near future . it can be noted that the changing - parallax method , in contrast to the methods described in sect . [ sec : mudot ] and [ sec : mcm ] , does not depend on the object having a large and uniform space motion , and would therefore be applicable to all stars within a few parsecs of the sun . lrcc @xmath2 & @xmath27 & + [ mas ] & & case b & case c + 740 & 3 & 1.2 & 0.05 + 300 & 14 & 7.5 & 0.3 + 200 & 60 & 17 & 0.7 + 100 & 326 & 68 & 2.8 + [ tab : epspidot ] the accuracy in @xmath4 is readily estimated from eq . ( [ eq : pidot ] ) for a given accuracy in @xmath28 , since the contribution of the parallax uncertainty to the factor @xmath29 is negligible . the achievable accuracy in @xmath28 depends both on the individual astrometric measurements and on their number and distribution in time . concerning the temporal distribution of the measurements we consider two limiting situations : _ quasi - continuous observation . _ the measurements are more or less uniformly spread out over a time period of length @xmath30 centred on the epoch @xmath31 . this is a good approximation to the way a single space mission would typically be operated ; for example , hipparcos had @xmath32 yr and @xmath33 . in such a case there exist simple ( mean ) relations between how accurately the different astrometric parameters of the same star can be derived , depending on @xmath30 . for instance , @xmath34 , if @xmath35 is the accuracy of the proper motion in declination and @xmath36 that of the declination at @xmath31 . this approximation is applicable to case a and b as defined in sect . [ sec : acc ] . _ two - epoch observation . _ two isolated parallax or proper - motion measurements are taken , separated by a long time interval ( say , @xmath37 years ) during which no observation takes place . each measurement must actually be the result of a series covering at least a year or so , but the duration of each such series is assumed to be negligible compared with @xmath37 . this could be two similar space missions separated by several decades and is applicable to case c in sect . [ sec : acc ] . for quasi - continuous observation we may assume that the parallax variation is linear over the observation period @xmath30 . thus , @xmath38 , where @xmath39 and @xmath28 are two parameters to be determined from the observations . there exists an approximate relation between the accuracies of these two parameters that is similar to that between the proper motion and the position at the mean epoch , viz . moreover , the estimates of the two parameters are uncorrelated , so @xmath41 equals the accuracy @xmath42 of a parallax determination in the absence of the parallax - change term ; thus @xmath43 in the case of a two - epoch observation , let us assume that independent parallax measurements @xmath44 and @xmath45 are made at epochs @xmath46 and @xmath47 . the estimated rate of change is @xmath48 . with @xmath49 and @xmath50 denoting the accuracies of the two measurements , we have @xmath51^{1/2}/t$ ] and consequently @xmath52^{1/2 } \ , . \label{eq : epspidotb}\ ] ] for given observational errors we find , from both eq . ( [ eq : epspidota ] ) and ( [ eq : epspidotb ] ) , that the radial - velocity error is simply a function of distance . the number of potential target stars for a certain maximum radial - velocity uncertainty is therefore given by the total number of stars within the corresponding maximum distance . table [ tab : epspidot ] gives the actual numbers of such stars , and the observational accuracies that may be reached . to a good approximation , single stars move with uniform linear velocity through space . for a given linear tangential velocity , the angular velocity ( or proper motion @xmath3 ) , as seen from the sun , varies inversely with the distance to the object . however , the tangential velocity changes due to the varying angle between the line of sight and the space - velocity vector ( fig . [ fig : methods]b ) . as is well known ( e.g. van de kamp @xcite , murray @xcite ) the two effects combine to produce an apparent ( perspective ) acceleration of the motion on the sky , or a rate of change in proper motion amounting to @xmath53 . with @xmath54 we find @xmath55 schlesinger ( @xcite ) derived the equivalent of this equation , calculated the perspective acceleration for kapteyn s and barnard s stars ( cf . table [ tab : epsmudot ] ) and noted that , if accurate positions are acquired over long periods of time , `` we shall be in position to determine the radial velocities of these stars independently of the spectroscope and with an excellent degree of precision '' . the equation for the perspective acceleration was earlier derived by seeliger ( @xcite ) and used by ristenpart ( @xcite ) in an ( unsuccessful ) attempt to determine @xmath56 observationally for groombridge 1830 . a major consideration for ristenpart seems to have been the possibility to derive the parallax from the apparent acceleration in combination with a spectroscopic radial velocity . such a determination of ` acceleration parallaxes ' was also considered by eichhorn ( @xcite ) . lrrlrccccl cns3 & hd & hip & sp & @xmath57 & @xmath19 & @xmath58 & & remark + & & & & & [ pc ] & [ arcsec@xmath59 yr@xmath1 ] & case b & case c + gl 699 & & 87937 & sdm4 & 9.5 & 1.8 & 5.69 & 0.13&0.01 & barnard s star + gl 551 & & 70890 & m5ve & 11.0 & 1.3 & 2.98 & 0.25&0.01 & @xmath61 cen c ( proxima ) + gl 559b & 128621 & 71681 & k0v & 1.4 & 1.3 & 2.76 & 0.27&0.01 & @xmath61 cen b + gl 559a & 128620 & 71683 & g2v & 0.0 & 1.3 & 2.75 & 0.28&0.01 & @xmath61 cen a ( ab : @xmath62 yr ) + gl 191 & 33793 & 24186 & m0v & 8.9 & 3.9 & 2.21 & 0.34&0.01 & kapteyn s star + gl 887 & 217987 & 114046 & m2ve & 7.4 & 3.3 & 2.10 & 0.36&0.01 & + gl 406 & & & m6 & 13.5 & 2.4 & 1.96 & 0.38&0.01 & wolf 359 + gl 411 & 95735 & 54035 & m2ve & 7.5 & 2.5 & 1.88 & 0.40&0.01 & + gl 820a & 201091 & 104214 & k5ve & 5.2 & 3.5 & 1.52 & 0.50&0.02 & 61 cyg a + gl 820b & 201092 & 104217 & k7ve & 6.1 & 3.5 & 1.48 & 0.51&0.02 & 61 cyg b ( ab : @xmath63 yr ) + gl 1 & 225213 & 439 & m4v & 8.6 & 4.4 & 1.40 & 0.54&0.02 & + gl 845 & 209100 & 108870 & k5ve & 4.7 & 3.6 & 1.30 & 0.58&0.02 & @xmath64 ind + gl 65a & & & dm5.5e & 12.6 & 2.6 & 1.28 & 0.59&0.02 & + gl 65b & & & dm5.5e & 12.7 & 2.6 & 1.28 & 0.59&0.02 & uv cet ( ab : @xmath65 yr ) + gl 273 & & 36208 & m3.5 & 9.8 & 3.8 & 0.98 & 0.77&0.03 & luyten s star + gl 866ab & & & m5e & 12.3 & 3.4 & 0.96 & 0.79&0.03 & @xmath66 yr + gl 412a & & 54211 & m2ve & 8.8 & 4.8 & 0.93 & 0.81&0.03 & + gl 412b & & & m6e & 14.4 & 5.3 & 0.86 & 0.88&0.03 & wx uma + gl 825 & 202560 & 105090 & m0ve & 6.7 & 3.9 & 0.88 & 0.86&0.03 & + gl 15a & 1326 & 1475 & m2v & 8.1 & 3.6 & 0.82 & 0.92&0.03 & gx and + gl 15b & & & m6ve & 11.1 & 3.6 & 0.84 & 0.90&0.03 & gq and ( ab : @xmath67 yr ) + gl 166a & 26965 & 19849 & k1ve & 4.4 & 5.0 & 0.81 & 0.94&0.03 & 40 eri a ( @xmath68 eri ) + gl 166b & 26976 & & da4 & 9.5 & 4.8 & 0.84 & 0.90&0.03 & 40 eri b ( bc : @xmath69 yr ) + gl 166c & & & dm4.5e & 9.5 & 4.8 & 0.84 & 0.90&0.03 & dy eri + gl 299 & & & dm5 & 12.8 & 6.8 & 0.77 & 0.98&0.04 & ross 619 + gl 451a & 103095 & 57939 & g8vi & 6.4 & 9.2 & 0.77 & 0.98&0.04 & groombridge 1830 + gl 451b & & & & 12 & 9.2 & 0.82 & 0.92&0.03 & cf uma ( non - existent star ? ) + gl 35 & & 3829 & dz7 & 12.4 & 4.4 & 0.68 & 1.1&0.04 & van maanen 2 + gl 725b & 173740 & 91772 & dm5 & 9.7 & 3.6 & 0.66 & 1.2&0.04 & ab : @xmath70 yr + gl 725a & 173739 & 91768 & dm4 & 8.9 & 3.6 & 0.63 & 1.2&0.04 & + gl 440 & & 57367 & dq6 & 11.5 & 4.6 & 0.58 & 1.3&0.05 & + gl 71 & 10700 & 8102 & g8vp & 3.5 & 3.6 & 0.53 & 1.4&0.05 & @xmath71 cet + gl 754 & & & m4.5 & 12.2 & 5.7 & 0.52 & 1.5&0.05 & + gl 139 & 20794 & 15510 & g5v & 4.3 & 6.1 & 0.52 & 1.5&0.05 & 82 eri + gl 905 & & & dm6 & 12.3 & 3.2 & 0.51 & 1.5&0.05 & + gl 244a & 48915 & 32349 & a1v & @xmath72 & 2.6 & 0.51 & 1.5&0.05 & @xmath61 cma ( sirius ) + gl 244b & & & da2 & & 2.6 & 0.51 & 1.5&0.05 & ab : @xmath73 yr + gl 53a & 6582 & 5336 & g5vi & 5.2 & 7.6 & 0.50 & 1.5&0.06 & @xmath3 cas + gl 53b & & & & 11 & 7.6 & 0.51 & 1.5&0.06 & ab : @xmath74 yr + [ tab : epsmudot ] subsequent attempts to determine the perspective acceleration of barnard s star by lundmark & luyten ( @xcite ) , alden ( @xcite ) and van de kamp ( @xcite ) yielded results that were only barely significant or ( in retrospect ) spurious . meanwhile , russell & atkinson ( @xcite ) suggested that the white dwarf van maanen 2 might exhibit a gravitational redshift of several hundred km s@xmath1 and that this could be distinguished from a real radial velocity through measurement of the perspective acceleration . the astrophysical relevance of astrometric radial - velocity determinations was thus already established ( oort @xcite ) . in relatively recent times , the perspective acceleration was successfully determined for barnard s star by van de kamp ( @xcite , @xcite , @xcite , @xcite , @xcite ) ; for van maanen 2 by van de kamp ( @xcite ) , gatewood & russell ( @xcite ) and hershey ( @xcite ) ; and for groombridge 1830 by beardsley et al . ( @xcite ) . among these determinations the highest precisions , in terms of the astrometric radial velocity , were obtained for barnard s star ( corresponding to @xmath75 km s@xmath1 ; van de kamp @xcite ) and van maanen 2 ( @xmath76 km s@xmath1 ; gatewood & russell @xcite ) . our application of the method , combining hipparcos measurements with data in the astrographic catalogue , yielded radial velocities for 16 objects , as listed in table [ tab : ac ] . the accuracy of the radial velocity calculated from eq . ( [ eq : mudot ] ) can be estimated as in sect . [ sec : epspidot ] . it depends on the parallax - proper - motion product @xmath58 . the most promising targets for this method are listed in table [ tab : epsmudot ] , which contains the known nearby stars ranked after decreasing @xmath58 . for quasi - continuous observation during a period of length @xmath30 we may use a quadratic model for the angular position @xmath77 of the star along the great - circle arc : here @xmath79 is the proper motion at the central epoch @xmath31 . the estimates of @xmath79 and @xmath56 are found to be uncorrelated and their errors related by @xmath80 . consequently , @xmath81 where @xmath82 is the accuracy of proper - motion measurements in the absence of temporal changes . we neglect the ( small ) contribution to @xmath83 from the uncertainty in the denominator @xmath58 . for a two - epoch observation , consider proper - motion measurements @xmath84 and @xmath85 made around @xmath46 and @xmath47 . the estimated acceleration is @xmath86 . provided the two observation intervals centred on @xmath46 and @xmath87 do not overlap , the measurements are independent , yielding the standard error @xmath88^{1/2}/t$ ] . for the radial velocity this gives @xmath89^{1/2 } \ , . \label{eq : epsmudotb}\ ] ] based on these formulae , table [ tab : epsmudot ] gives the potential radial - velocity accuracy for the two cases b and c defined in sect . [ sec : acc ] . in a two - epoch observation we normally have , in addition , a very good estimate of the _ mean _ proper motion between @xmath46 and @xmath87 , provided the positions @xmath90 and @xmath91 at these epochs are accurately known . in the previous quadratic model we may take the reference epoch to be @xmath92 and find @xmath93 with standard error @xmath94^{1/2}/t$ ] . the three proper - motion estimates @xmath79 , @xmath84 and @xmath85 ( referred to @xmath31 , @xmath46 and @xmath87 ) are mutually independent and may be combined in a least - squares estimate of @xmath56 . if @xmath95 ( equal weight at @xmath46 and @xmath87 ) , then it is found that @xmath79 does not contribute at all to the determination of @xmath56 , and the standard error is still given by eq . ( [ eq : epsmudotb ] ) . if , on the other hand , the two observation epochs are not equivalent , then some improvement can be expected by introducing the position measurements . an important special case is when there is just a position ( no proper motion ) determined at one of the epochs , say @xmath46 . this is however equivalent to the two independent proper - motion determinations @xmath79 at @xmath92 , and @xmath85 at @xmath87 , separated by @xmath96 . applying eq . ( [ eq : epsmudotb ] ) on this case yields @xmath97^{1/2 } \ , . \label{eq : epsmudotc}\ ] ] this formula is applicable on the combination of a recent position and proper - motion measurement ( e.g. by hipparcos ) with a position derived from old photographic plates ( e.g. the astrographic catalogue ) . taking @xmath98 , @xmath99 mas as representative for the astrographic catalogue , and @xmath100 , @xmath101 mas , @xmath102 mas yr@xmath1 for hipparcos , we find @xmath103 . with such data , moderate accuracies of a few tens of km s@xmath1 can be reached for several stars ( sect . [ sec : pm - results ] ) . the perspective - acceleration method depends critically on the assumption that the star moves with uniform space motion relative the observer . the presence of a real acceleration of their relative motions , caused by gravitational action of other bodies , would bias the calculated astrometric radial velocity by @xmath104 , where @xmath105 is the tangential component of the relative acceleration . the acceleration towards the galactic centre caused by the smoothed galactic potential in the vicinity of the sun is @xmath106 km s@xmath107 . for a hypothetical observer near the sun but unaffected by this acceleration , the maximum bias would be 0.06 km s@xmath1 for barnard s star , and 0.17 km s@xmath107 for proxima . however , since real observations are made relative the solar - system barycentre , which itself is accelerated in the galactic gravitational field , the observed ( differential ) effect will be very much smaller . in the case of proxima the acceleration towards @xmath61 cen ab is of a similar magnitude as the galactic acceleration . for the several orbital binaries in table [ tab : epsmudot ] the curvature of the orbit is much greater than the perspective acceleration . application of this method will therefore require careful correction for all known perturbations : the possible presence of long - period companions may introduce a considerable uncertainty . among other effects which may have to be considered are light - time effects which , to first order in @xmath108 , may require a correction of @xmath109 on the right - hand side of eq . ( [ eq : mudot ] ) , where @xmath110 is the tangential velocity . for typical high - velocity ( population ii ) stars the correction is 0.10.2 km s@xmath1 . at this accuracy level , the precise definition of the radial - velocity concept itself requires careful consideration ( lindegren et al . @xcite ) . llcccl hip & & & remark + & no . & epoch & astrom . & spectr . & + 439 & 3152964 & 1912.956 & @xmath111 & @xmath112 & + 1475 & 1406215 & 1898.435 & @xmath113 & @xmath114 & gx and + 5336 & 1721511 & 1913.868 & @xmath115 & @xmath116 & @xmath3 cas + 15510 & 3488626 & 1901.018 & @xmath117 & @xmath118 & 82 eri + 19849 & 2125614 & 1892.970 & @xmath119 & @xmath120 & 40 eri + 24186 & 3505363 & 1899.058 & @xmath121 & @xmath122 & kapteyn s star + 36208 & 282902 & 1908.859 & @xmath123 & @xmath124 & luyten s star + 54035 & 1340883 & 1930.895 & @xmath125 & @xmath126 & + 54211 & 1463341 & 1895.620 & @xmath127 & @xmath128 & + 57367 & 4195112 & 1924.492 & @xmath129 & & + 57939 & 1342199 & 1930.260 & @xmath130 & @xmath116 & groombridge 1830 + 87937 & 146626 & 1905.979 & @xmath131 & @xmath132 & barnard s star + 104214/217 & 1382645/649 & 1921.699 & @xmath133 & @xmath134 & 61 cyg + 104214/217 & 1382645/649 & 1921.699 & @xmath135 & & + 105090 & 3462277 & 1905.316 & @xmath136 & @xmath137 + 108870 & 4384302 & 1901.189 & @xmath138 & @xmath139 & @xmath64 ind + 114046 & 3355101 & 1913.368 & @xmath140 & @xmath141 & + past determinations of the perspective acceleration , e.g. by van de kamp ( @xcite ) and gatewood & russell ( @xcite ) , were based on photographic observations collected over several decades , in which the motion of the target star was measured relative to several background ( reference ) stars . one difficulty with the method has been that the positions and motions of the reference stars are themselves not accurately known , and that small errors in the reference data could cause a spurious acceleration of the target star ( van de kamp @xcite ) . the hipparcos catalogue ( esa @xcite ) established a very accurate and homogeneous positional reference frame over the whole sky . using the proper motions , this reference frame can be extrapolated backwards in time . it is then possible to re - reduce measurements of old photographic plates , and express even century - old stellar positions in the same reference frame as modern observations . this should greatly facilitate the determination of effects such as the perspective acceleration , which are sensitive to systematic errors in the reference frame . as part of the _ carte du ciel _ project begun more than a century ago , an astrographic programme to measure the positions of all stars down to the 11th magnitude was carried out and published as the astrographic catalogue , ac ( see eichhorn @xcite for a description ) . after transfer to electronic media , the position measurements have been reduced to the hipparcos reference frame ( nesterov et al . @xcite , urban et al . the result is a positional catalogue of more than 4 million stars with a mean epoch around 1907 and a typical accuracy of about 200 mas . we have used the version known as ac 2000 ( urban et al . @xcite ) , available on cd - rom from the us naval observatory , to examine the old positions of all the stars with hip identifiers in table [ tab : epsmudot ] . for the stars in table [ tab : ac ] we successfully matched the ac positions with the positions extrapolated backwards from the hipparcos catalogue and hence could calculate the astrometric radial velocities . other potential targets in table [ tab : epsmudot ] were either outside the magnitude range of ac 2000 ( e.g. @xmath61 cen and proxima ) or lacked an accurate proper motion from hipparcos ( e.g. van maanen 2 and hip 91768 + 91772 ) . the basic procedure was as follows . the rigorous epoch transformation algorithm described in sect . 1.5.5 of the hipparcos catalogue , vol . 1 , was used to propagate the hipparcos position and its covariance matrix to the ac 2000 epoch relevant for each star . this extrapolated position was compared with the actual measured position in ac 2000 , assuming a standard error of 200 mas in each coordinate for the latter . a @xmath142 goodness - of - fit was then calculated from the position difference and the combined covariance of the extrapolated and measured positions . the epoch transformation algorithm requires that the radial velocity is known . the radial velocity was therefore varied until the @xmath142 attained its minimum value . the @xmath1431@xmath144 confidence interval given in the table was obtained by modifying the radial velocity until the @xmath142 had increased by one unit above the minimum . for some of the stars , data had to be corrected for duplicity or known orbital motion . the solutions for the resolved binary 61 cyg ( hip 104214 + 104217 ) refer to the mass centre , assuming a mass ratio of @xmath145 , as estimated by means of standard isochrones from the absolute magnitudes and colour indices of the components ( sderhjelm , private communication ) . for the astrometric binary @xmath3 cas ( hip 5336 ) the hipparcos data explicitly refer to the mass centre using the orbit by heintz & cantor ( @xcite ) ; the same orbit was used to correct the ac position of the primary to the mass centre . no correction for orbital motion was used for gx and and 40 eri . table [ tab : ac ] gives two solutions for 61 cyg . the first solution was obtained as described above , using only the hipparcos data plus the ac positions for the two components . the second solution , marked with an asterisk in the table , was derived by including also the observations by bessel ( @xcite ) from his pioneering determination of the star s parallax . bessel measured the angular distances from the geometrical centre ( half - way between the components ) of 61 cyg to two reference stars , called @xmath146 and @xmath19 in his paper . after elimination of aberration , proper motion and parallax , he found the distances @xmath147 arcsec and @xmath148 arcsec for the beginning of year 1838 ( b1838.0 = j1838.0022 ) . the uncertainties are our estimates ( standard errors ) based on the scatter of the residuals in bessel s solution ` ii ' . we identified the reference stars in ac 2000 and in the tycho catalogue ( esa @xcite ) as @xmath146 = ac 1382543 = tyc 3168 708 1 and @xmath19 = ac 1382712 = tyc 3168 1106 1 . extrapolating the positions from these catalogues back to b1838 allowed us to compute the position of the geometrical centre of 61 cyg in the hipparcos / tycho reference frame . this could then be transformed to the position of the mass centre , using bessel s own measurement of the separation and position angle in 61 cyg and the previously assumed mass ratio . actually , all the available data were combined into a @xmath142 goodness - of - fit measure and the radial velocity was varied in order to find the minimum and the @xmath1431@xmath144 confidence interval . this gave @xmath149 km s@xmath1 . table [ tab : ac ] also gives the spectroscopic radial velocities when available in the literature . a comparison between the astrometric and spectroscopic radial velocities is made in fig . [ fig : res ] . given the stated confidence intervals , the agreement is in all cases rather satisfactory . the exercise demonstrates the basic feasibility of this method , but also hints at some of the difficulties in applying it to non - single stars . the moving - cluster method is based on the assumption that the stars in a cluster move through space essentially with a common velocity vector . the radial - velocity component makes the cluster appear to contract or expand due to its changing distance ( fig . [ fig : methods]c ) . the relative rate of apparent contraction equals the relative rate of change in distance to the cluster . this can be converted to a linear velocity ( in km s@xmath1 ) if the distance to the cluster is known , e.g. from trigonometric parallaxes . in practice , the method amounts to determining the space velocity of the cluster , i.e. the convergent point and the speed of motion , through a combination of proper motion and parallax data . once the space velocity is known , the radial velocity for any member star may be calculated by projecting the velocity vector onto the line of sight . the method can be regarded as an inversion of the classical procedure ( e.g. binney & merrifield @xcite ) by which the distances to the stars in a moving cluster are derived from the proper motions and ( spectroscopic ) radial velocities : if instead the distances are known , the radial velocities follow . the first application of the classical moving - cluster method for distance determination was by klinkerfues ( @xcite ) , in a study of the ursa major system . the possibility to check spectroscopic radial velocities against astrometric data was recognised by klinkerfues , but could not then be applied to the ursa major cluster due to the lack of reliable trigonometric parallaxes . this changed with hertzsprung s ( @xcite ) discovery that sirius probably belongs to the ursa major moving group . the relatively large and well - determined parallax of sirius , combined with its considerable angular distance from the cluster apex , could lead to a meaningful estimate for the cluster velocity and hence for the radial velocities . rasmuson ( @xcite ) and smart ( @xcite ) appear to have been among the first who actually made this computation , although mainly as a means of verifying the cluster method for distance determination . later studies by petrie ( @xcite ) and petrie & moyls ( @xcite ) reached formal errors in the astrometric radial velocities below 1 km s@xmath1 . the last paper concluded `` there does not appear to be much likelihood of improving the present results until a substantial improvement in the accuracy of the trigonometric parallaxes becomes possible . '' one of the purposes of the petrie & moyls study was to derive the astrometric radial velocities of spectral type a in order to check the victoria system of spectroscopic velocities . the method was also applied to the hyades ( petrie @xcite ) but only with an uncertainty of a few km s@xmath1 . given the expected future availability of more accurate proper motions and trigonometric parallaxes , petrie ( @xcite ) envisaged that one or two moving clusters could eventually be used as primary radial - velocity standards for early - type spectra . such astrometric data are now in fact available . in sect . [ sec:3prec ] we derive a rough estimate of the accuracy of the method and survey nearby clusters and associations in order to find promising targets for its application . an important consideration is to what extent systematic velocity patterns in the cluster , in particular cluster expansion , will limit the achievable accuracy . this is discussed in sect . [ sec:3syst ] and appendix a. in sect . [ sec : indpar ] we briefly consider the improvement in the distance estimates for individual stars resulting from the moving - cluster method . the present discussion of the moving - cluster method is only intended to highlight its theoretical potential and limitations . its actual application requires a more rigorous formulation , which is developed in a second paper . lcccccccccccc name & iau & @xmath27 & age & @xmath150 & @xmath151 & @xmath4 & @xmath110 & & & & @xmath152 + & designation & & [ myr ] & [ arcmin ] & [ pc ] & & & ( a ) & ( b ) & & [ km s@xmath1 ] + cassiopeia taurus & & 83 & 25 & 1800 & 190 & @xmath1536&21 & & 0.24 & 0.06 & & @xmath1547.3 + upper centaurus lupus & & 221 & 13 & 670 & 140 & @xmath1535&21 & & 0.25 & 0.09 & & @xmath15410 + ursa major & & 40 & 300 & 4300 & 25 & @xmath15411&5 & & 0.11 & 0.10 & & @xmath1540.08 + lower centaurus crux & & 180 & 10 & 560 & 118 & @xmath15312&19 & & 0.30 & 0.12 & & @xmath15411 + hyades & c 0424@xmath153157 & 380 & 625 & 560 & 46 & @xmath15343&25 & & 0.19 & 0.14 & & @xmath1540.07 + perseus ob3 ( @xmath61 per ) & c 0318@xmath153484 & 186 & 50 & 350 & 180 & @xmath1541&29 & & 0.64 & 0.18 & & @xmath1543.4 + ` hip 98321 ' & & 59 & 60 & 740 & 300 & @xmath15417&4 & & 1.1 & 0.19 & & @xmath1544.8 + upper scorpius & & 120 & 5 & 325 & 145 & @xmath1545&18 & & 0.71 & 0.24 & & @xmath15428 + lacerta ob1 & & 96 & 16 & 350 & 370 & @xmath15413&8 & & 1.8 & 0.26 & & @xmath15422 + collinder 121 & & 103 & 5 & 290 & 540 & @xmath15326&15 & & 3.3 & 0.32 & & @xmath154103 + collinder 70 & c 0533@xmath154011 & 345 & & 140 & 430 & & & & 2.7 & 0.33 & & + cepheus ob2 & & 71 & 5 & 320 & 615 & @xmath15421&12 & & 4.1 & 0.34 & & @xmath154120 + vela ob2 & & 93 & 20 & 260 & 415 & @xmath15318&20 & & 2.8 & 0.36 & & @xmath15420 + perseus ob2 & & 41 & 7 & 340 & 300 & @xmath15320&14 & & 2.5 & 0.43 & & @xmath15446 + pleiades & c 0344@xmath153239 & 277 & 130 & 120 & 125 & @xmath1537&29 & & 1.1 & 0.43 & & @xmath1540.92 + coma berenices & c 1222@xmath153263 & 273 & 460 & 120 & 87 & 0&9 & & 0.84 & 0.43 & & @xmath1540.18 + ngc 3532 & c 1104@xmath154584 & 677 & 290 & 50 & 480 & @xmath1537&27 & & 6.0 & 0.66 & & @xmath1541.6 + praesepe & c 0837@xmath153201 & 161 & 830 & 70 & 160 & @xmath15333&26 & & 3.1 & 0.98 & & @xmath1540.18 + ngc 2477 & c 0750@xmath154384 & 1911 & 1260 & 20 & 1150 & @xmath1537&17 & & 21 & 0.98 & & @xmath1540.87 + ic 4756 & c 1836@xmath153054 & 466 & 830 & 39 & 390 & @xmath15418&5 & & 7.6 & 1.0 & & @xmath1540.45 + ic 4725 & c 1828@xmath154192 & 601 & 41 & 29 & 710 & @xmath1533&20 & & 16 & 1.2 & & @xmath15417 + trumpler 10 & & 23 & 15 & 150 & 370 & @xmath15321&26 & & 8.6 & 1.2 & & @xmath15424 + cepheus ob6 & & 20 & 50 & 150 & 270 & @xmath15420&22 & & 6.8 & 1.3 & & @xmath1545.2 + ngc 752 & c 0154@xmath153374 & 77 & 3300 & 75 & 360 & @xmath1543&20 & & 9.0 & 1.3 & & @xmath1540.10 + ngc 6618 & c 1817@xmath154162 & 660 & & 25 & 1500 & &26 & & 38 & 1.3 & & + ngc 2451 & c 0743@xmath154378 & 153 & 41 & 50 & 315 & @xmath15327&27 & & 8.4 & 1.4 & & @xmath1547.3 + ngc 7789 & c 2354@xmath153564 & 583 & 1780 & 25 & 1800 & @xmath15454&27 & & 49 & 1.4 & & @xmath1541.0 + ngc 2099 & c 0549@xmath153325 & 1842 & 200 & 14 & 1300 & @xmath1538&45 & & 35 & 1.4 & & @xmath1546.4 + ngc 6475 & c 1750@xmath154348 & 54 & 130 & 80 & 240 & @xmath15412&7 & & 6.8 & 1.5 & & @xmath1541.8 + ngc 2264 & c 0638@xmath153099 & 222 & 10 & 39 & 800 & @xmath15322&17 & & 23 & 1.5 & & @xmath15476 + stock 2 & c 0211@xmath153590 & 166 & 170 & 45 & 300 & @xmath1532&30 & & 8.6 & 1.5 & & @xmath1541.7 + ic 2602 & c 1041@xmath154641 & 33 & 29 & 100 & 150 & @xmath15322&14 & & 4.6 & 1.5 & & @xmath1545.0 + [ tab : clus ] the accuracy of the astrometric radial velocity potentially achievable by the moving - cluster method can be estimated as follows . let @xmath19 be the ( mean ) distance to the cluster and consider a star at angular distance @xmath155 from the centre of the cluster , as seen from the sun . the projected linear distance of the star from the centre of the cluster is @xmath156 , provided the angular extent of the cluster is not very large . as the cluster moves through space , its linear dimensions remain constant , so that @xmath157 . putting @xmath158 ( the proper motion relative to the cluster centre ) , @xmath159 , and @xmath54 , gives @xmath160 . now suppose that the parallaxes and proper motions of @xmath27 cluster stars are measured , each with uncertainties of @xmath42 and @xmath82 . standard error propagation formulae give the expected accuracy in @xmath4 as @xmath161^{1/2 } \ , \label{eq : epsmc0}\ ] ] where @xmath150 is in radians ; @xmath23 is the astronomical unit ( sect . [ sec : not ] ) . the expression within the square brackets derives from the uncertainty in the mean cluster distance , by which the derived radial velocity scales . for the type of ( space ) astrometry data considered here ( case a and b ) , @xmath162 is on the order of a few years ( for hipparcos the mean ratio is @xmath163 yr ) . the factor in brackets can then be neglected except for the most extended ( and nearby ) clusters . under certain circumstances it is not the accuracy of proper - motion measurements that defines the ultimate limit on @xmath83 , but rather internal velocity dispersion among the cluster stars . assuming isotropic dispersion with standard deviation @xmath12 in each coordinate , one must add @xmath164 quadratically to the measurement error @xmath82 in eq . ( [ eq : epsmc0 ] ) . thus @xmath165^{1/2 } \nonumber \\ & & \times\left [ 1 + \left(\frac{v_r\rho_{\rm rms}\epsilon(\pi ) } { a\epsilon(\mu)}\right)^2 \right]^{1/2 } \ , , \label{eq : epsmc1}\end{aligned}\ ] ] is the accuracy achievable for the radial velocity of the cluster centroid . for the radial velocity of an individual star this uncertainty must be increased by the internal dispersion . the internal velocity dispersion will dominate the error budget for nearby clusters , viz . if @xmath166 . assuming a velocity dispersion of 0.25 km s@xmath1 and a proper - motion accuracy of 1 mas yr@xmath1 ( as for hipparcos ) , this will be the case for clusters within 50 pc of the sun . for an observational accuracy in the 110 @xmath3as yr@xmath1 range the internal dispersion will dominate in practically all galactic clusters and eq . ( [ eq : epsmc1 ] ) can be simplified to @xmath167 . in this case the achievable accuracy becomes independent of the astrometric one . table [ tab : clus ] lists some nearby clusters and associations , with estimates of the achievable accuracy in the radial velocity of the cluster centroid , assuming current ( hipparcos - type ) astrometric performance ( case a in sect . [ sec : acc ] ) as well as future ( microarcsec ) expectations ( case b ) . as explained above , increasing the astrometric accuracy still further gives practically no improvement ; this is why case c is not considered in the table . the entry ` hip 98321 ' refers to the possible association identified by platais et al . ( @xcite ) and named after one of its members . of dubious status , it was included as an example of the extended , low - density groups that may exist in the general stellar field , but are difficult to identify with existing data . blaauw ( @xcite ) showed that the proper motion pattern for a linearly expanding cluster is identical to the apparent convergence produced by parallel space motions . astrometric data alone therefore can not distinguish such expansion from a radial motion . if such an expansion exists , and is not taken into account in estimating the astrometric radial velocity , a bias will result , as examined in appendix a ( eq . [ eq : bias ] ) . the gravitationally unbound associations are known to expand on timescales comparable with their nuclear ages ( de zeeuw & brand @xcite ) . but also for a gravitationally bound open cluster some expansion can be expected as a result of the dynamical evolution of the cluster ( see mathieu @xcite and wielen @xcite for an introduction to this complex issue ) . in either case the inverse age of the cluster or association may be taken as a rough upper limit on the cluster s relative expansion rate @xmath168 [ yr@xmath1 ] . ( [ eq : bias ] ) then gives @xmath169}{\rm age~[myr]}~{\rm km}~{\rm s}^{-1}\ , \label{eq : deltaexp}\ ] ] for the bias of a star near the centre of the cluster . ( for an expanding cluster , @xmath170 is always negative . ) resulting values , in the last column of table [ tab : clus ] , are adequately small for a few nearby , relatively old clusters . in other cases the potential bias is very large and will certainly limit the applicability of the method . the ob associations are particularly troublesome , not only because they are young objects ( implying large values of @xmath168 ) , but also because they sometimes appear to expand significantly faster than their photometric ages would suggest ( de zeeuw & brand @xcite ) . however , it should be remembered that the ultimate limitation set by cluster expansion depends on how accurately the expansion rate @xmath168 can be estimated by some independent means . for instance , if @xmath168 can somehow be estimated to within 10 per cent of its value , then the residual biases would still be on the sub - km s@xmath1 level for most of the objects in table [ tab : clus ] . numerical simulation of the dynamical evolution of clusters might in principle provide such estimates of @xmath168 , as could the spectroscopic radial velocities as function of distance . the use of spectroscopic data would not necessarily defeat the purpose of the method , i.e. to determine _ absolute _ radial velocities , since the expansion is revealed already by _ relative _ measurements . in a rigorous estimation of the space motion of a moving cluster , such as will be presented in a second paper , the distances to the individual member stars of the cluster appear as parameters to be estimated . a by - product of the method is therefore that the individual distances are improved , sometimes considerably , compared with the original trigonometric distances ( dravins et al . @xcite ; madsen @xcite ) . the improvement results from a combination of the trigonometric parallax @xmath171 with the kinematic ( secular ) parallax @xmath172 derived from the star s proper motion @xmath3 and tangential velocity @xmath110 , the latter obtained from the estimated space velocity vector of the cluster . the accuracy of the combined parallax estimate @xmath173 can be estimated from @xmath174 . in calculating @xmath175 we need to take into account the observational uncertainty in @xmath3 and the uncertainty in @xmath110 from the internal velocity dispersion . the result is @xmath176^{-1/2 } \ , . \label{eq : epspihat}\ ] ] from the data in table [ tab : clus ] we find that , in case a , the moving - cluster method will be useful to resolve the depth structures of the hyades cluster and of the associations cassiopeia taurus , upper centaurus lupus , lower centaurus crux , perseus ob3 and upper scorpius . in case b , all the clusters and associations are resolved by the trigonometric parallaxes , so the kinematic parallaxes will bring virtually no improvement . calculation of kinematic distances to stars in moving clusters is of course a classical procedure ( e.g. , klinkerfues @xcite and van bueren @xcite ) ; what is new in our treatment is that such distances are derived without recourse to spectroscopic data . with improved astrometric data , further methods for radial - velocity determinations may become feasible . the moving - cluster method could in principle be applied to any geometrical configuration of a fixed linear size . to reach an accuracy of 1 km s@xmath1 in the astrometric radial velocity of an object at 10 pc distance requires a dimensional ` stability ' of the order of @xmath177 yr@xmath1 ; at a distance of 1 kpc the requirement is @xmath178 yr@xmath1 . since these numbers are greater than or comparable with the inverse dynamical timescales of many types of galactic objects , there is at least a theoretical chance that the method could work , given a sufficient measurement accuracy . we consider briefly two such possibilities . according to the previous argument it would be possible to ignore the relative motion of the components in a binary if the period is longer than some @xmath179 yr . this implies an linear separation of at least some 50000 astronomical units , or a few degrees on the sky at 10 pc distance . in principle , then , this case is equivalent to a moving cluster with @xmath180 stars . in the opposite case of a ( relatively ) short - period binary , the radial velocity might be obtained from apparent changes of the orbit . the projected orbit will not be closed , but form a spiral on the sky : slightly diverging if the stars are approaching , slightly converging if they recede . for a system at a distance of 10 pc , say , with a component separation of 10 astronomical units , a radial velocity of 100 km s@xmath1 will change the apparent orbital radius of 1 arcsec by 10 @xmath3as per year . the relative positions would need to be measured during at least a significant fraction of an orbital period , or some 20 years in our example , resulting in an accumulated change by about 0.2 mas . since only relative position measurements between the same stars are required , the observational challenges are not as severe as in some other cases . for a binary with a few arcsec separation , the isoplanatic properties of the atmosphere imply that the cross - correlation distance between the speckle images of both stars should be stable to better than one mas . averaging very many exposures should reduce the errors into the @xmath3as range , with practical limits possibly set by differential refraction ( mcalister @xcite ) . the moving - cluster method ( sect . [ sec : mcm ] ) could in principle be applied also to globular clusters . globular clusters differ markedly from open clusters in that ( potentially ) many more stars could be measured , and through a much larger velocity dispersion ( @xmath181 km s@xmath1 ; peterson & king @xcite ) . the higher number of stars partly compensates the larger dispersion . however , all globular clusters are rather distant , making their angular radii small . as discussed in sect . [ sec:3prec ] the approximate formula @xmath182 applies in the case when the internal motions are well resolved . taking @xmath18320 arcmin as representative for the angular radii of comparatively nearby globular clusters , we find that averaging over some @xmath184 to @xmath185 member stars is needed to reach a radial - velocity accuracy comparable with @xmath12 , i.e. several km s@xmath1 . furthermore , in view of the discussion in sect . [ sec:3syst ] , it is not unlikely that the complex kinematic structures of these objects ( e.g. norris et al . @xcite ) would bias the results . thus , globular clusters remain difficult targets for astrometric radial - velocity determination . the theoretical possibility to use astrometric data ( parallax and proper motion ) to deduce the radial motions of stars has long been recognised . with the highly accurate ( sub - milliarcsec ) astrometry already available or foreseen in planned space missions , such radial - velocity determinations are now also a practical possibility . this will have implications for the future definition of radial - velocity standards , as the range of geometrically determined accurate radial velocities , hitherto limited to the solar system and to solar - type spectra , is extended to many other stellar types represented in the solar neighbourhood . we have identified and analysed three main methods to determine astrometric radial velocities . the first method , using the changing annual parallax , is the intuitively most obvious one , but requires data of an accuracy beyond current techniques . it is nevertheless potentially interesting in view of future space missions or long - term observations from the ground . the second method , using the changing proper motion or perspective acceleration of stars , has a long history , and was previously applied to a few objects , albeit with modest precision in the resulting radial velocity . new results for a greater number of stars , obtained by combining old and modern data , were given in table [ tab : ac ] and fig . [ fig : res ] , thus proving the concept . however , to realise the full potential of the method again requires the accuracies of future astrometry projects . in both these methods the uncertainty in the astrometric radial velocity increases , statistically , with distance squared . they are therefore in practice limited to relatively few stars very close to the sun and , in the second method , to stars with a large tangential velocity . in the general case , the two methods could actually be combined to yield a somewhat higher accuracy , but at least for the stars considered in tables [ tab : epspidot ] and [ tab : epsmudot ] this would only bring a marginal improvement . the third method , using the changing angular extent of a moving cluster or association , is an inversion of the classical moving - cluster method , by which the distance to the cluster was derived from its radial velocity and convergence point . if the distance is known from trigonometric parallaxes , one can instead calculate the radial velocities . it appears to be the only method by which astrophysically interesting accuracies can be obtained with existing astrometric data . in future papers we will develop and exploit this possibility in full , using data from the hipparcos mission . a by - product of the method is that the distance estimates to individual cluster stars may be significantly improved compared with the parallax measurements . one would perhaps expect the moving - cluster method to become extremely powerful with the much more accurate data expected from future astrometry projects . unfortunately , this is not really the case , as internal velocities ( both random and systematic ) become a limiting factor as soon as they are resolved by the proper - motion measurements . nevertheless , even the limited number of clusters within reach of such determinations contain a great many stars spanning a wide range in spectral type and luminosity . this project is supported by the swedish national space board and the swedish natural science research council . we thank dr . s. sderhjelm ( lund observatory ) for providing information on double and multiple stars , prof . de zeeuw and co - workers ( leiden observatory ) for advance data on nearby ob associations , and the referee , prof . a. blaauw , for stimulating criticism of the manuscript . in this appendix we examine how sensitive the moving - cluster method is to _ systematic _ velocity patterns in the cluster , and to what extent such patterns can be determined independently of the astrometric radial velocity . for this purpose we may ignore the random motions as well as the observational errors and we consider only a linear ( first - order ) velocity field . let @xmath186 be the position of the cluster centroid relative to the sun and @xmath187 the position of a member star relative to the centroid . the space velocity of the star is @xmath188 , where @xmath189 is the peculiar velocity . the velocity field is described by the tensor @xmath190 such that @xmath191 . in cartesian coordinates the components of this tensor are simply the partial derivatives @xmath192 for @xmath193 . these nine numbers together describe the three components of a rigid - body rotation , three components of an anisotropic expansion or contraction , and three components of linear shear . it is intuitively clear that certain components of the linear velocity field , such as rotation about the line of sight , can be determined purely from the astrometric data . if the corresponding components of @xmath190 are included as parameters in the cluster model , they can be estimated and will not produce a systematic error ( bias ) in the astrometric radial velocity derived from the model fitting . such components of @xmath190 are ` observable ' and in principle not harmful to the method . let us now examine more generally the extent to which @xmath190 is observable by astrometry . suppose there exists a non - zero tensor @xmath190 such that the velocity fields @xmath194 and @xmath195 produce identical observations for some vector @xmath196 . since the cluster velocity @xmath197 is already a parameter of the model , the observational effects of the velocity field @xmath190 could then entirely be absorbed by adjusting @xmath197 . in this case @xmath190 would be a non - observable component of the general velocity field . moreover , if there exists such a component in the real velocities , then the estimated cluster velocity will have a bias equal to @xmath196 . we now need to calculate the effect of the arbitrary field @xmath190 on the observables . since the parallaxes are not affected , only the proper motion vector @xmath198 needs to be considered . in this equation @xmath199 is the unit vector from the sun towards the star , @xmath200 is the rate of change of that direction , and @xmath201 is the tensor representing projection perpendicular to @xmath199 [ @xmath202 is the unit tensor ; thus @xmath203 is the tangential component of the general vector @xmath204 . with @xmath205 we can write @xmath206 . @xmath190 is non - observable if the space velocities @xmath194 and @xmath207 produce identical tangential velocities for every star , i.e. if @xmath208 = ( \vec{i } - \vec{r}\vec{r}^\prime)(\vec{v}_0+\delta\vec{v})\ ] ] for all directions @xmath199 and distances @xmath19 . this is equivalently written @xmath209 in order that this should be satisfied for all @xmath19 , it is necessary that @xmath210 and @xmath211 are separately satisfied for all unit vectors @xmath199 . the latter equation implies that @xmath212 the former equation can be written @xmath213 , which shows that @xmath199 is an eigenvector of @xmath190 ( with eigenvalue @xmath214 ) . but the only tensor for which every unit vector is an eigenvector is the isotropic tensor , @xmath215 for the arbitrary scalar @xmath168 . it follows that the only non - observable component of @xmath190 is of the form @xmath216 , parametrised by the single scalar @xmath168 , and that consequently eight linearly independent components of @xmath190 can , in principle , be determined from the astrometric observations . the non - observable field @xmath215 describes a uniform isotropic expansion ( @xmath217 ) or contraction ( @xmath218 ) of the cluster with respect to its centroid . these effects are observationally equivalent to an approach or recession of the cluster , i.e. to a different value of its radial velocity . @xmath196 is the bias for the centroid velocity . for any given star , the bias vector @xmath219 is the difference between the derived ( apparent ) space velocity vector @xmath220 and the true vector @xmath221 . using @xmath187 and eq . ( [ eq : deltav ] ) we find @xmath222 the resulting bias in the astrometric radial velocity is @xmath223 isotropic expansion ( @xmath224 ) , in particular , gives the bias @xmath225 for a uniformly expanding cluster @xmath226 equals the expansion age , i.e. the time elapsed since all the stars were confined to a very small region of space . gliese w. , jahrei h. , 1991 , preliminary version of the third catalogue of nearby stars , astron . rechen - institut , heidelberg ( available from centre de donnes astronomiques de strasbourg , http://vizier.u-strasbg.fr/ ) nesterov v. , gulyaev a. , kuimov k. , et al . , 1998 , in : mclean b.j . , golombek d.a . , hayes j.j.e . , payne h.e . iau symp . 179 , new horizons from multi - wavelength sky surveys . kluwer , dordrecht , p. 409
|
high - accuracy astrometry permits the determination of not only stellar tangential motion , but also the component along the line - of - sight . such non - spectroscopic
( i.e. astrometric ) radial velocities are independent of stellar atmospheric dynamics , spectral complexity and variability , as well as of gravitational redshift .
three methods are analysed : ( 1 ) changing annual parallax , ( 2 ) changing proper motion and ( 3 ) changing angular extent of a moving group of stars .
all three have significant potential in planned astrometric projects .
current accuracies are still inadequate for the first method , while the second is marginally feasible and is here applied to 16 stars .
the third method reaches high accuracy ( @xmath0 km s@xmath1 ) already with present data , although for some clusters an accuracy limit is set by uncertainties in the cluster expansion rate .
| 18,574 | 229 |
acoustic wave modes in dusty plasma have received a great deal of attention since the last decade @xcite . depending on different time scales , there can exists two or more acoustic waves in a typical dusty plasma . dust acoustic ( da ) and dust ion - acoustic ( dia ) waves are two such acoustic waves in a plasma containing electrons , ions , and charged dust grains . shukla and silin @xcite were the first to show that due to the quasi neutrality condition @xmath15 and the strong inequality @xmath16 ( @xmath17 , @xmath18 , and @xmath19 are , respectively , the number density of electrons , ions , and dust particles , where @xmath20 is the number of electrons residing on the dust grain surface ) , a dusty plasma ( with negatively charged static dust grains ) supports low - frequency dia waves with phase velocity much smaller ( larger ) than electron ( ion ) thermal velocity . in case of long wavelength limit the dispersion relation of dia wave is similar to that of ion - acoustic ( ia ) wave for a plasma with @xmath21 and @xmath22 , where @xmath23 is the average ion ( electron ) temperature . due to the usual dusty plasma approximations ( @xmath16 and @xmath24 ) , a dusty plasma can not support the usual ia waves , but the dia waves of shukla and silin @xcite can . thus dia waves are basically ia waves , modified by the presence of heavy dust particulates . the theoretical prediction of shukla and silin @xcite was supported by a number of laboratory experiments @xcite . the linear properties of dia waves in dusty plasma are now well understood @xcite . dust ion - acoustic solitary waves ( diasws ) have been investigated by several authors . bharuthram and shukla @xcite studied the diasws in an unmagnetized dusty plasma consisting of isothermal electrons , cold ions , in both static and mobile dust particles . employing reductive perturbation method , mamun and shukla @xcite investigated the cylindrical and spherical diasws in an unmagnetized dusty plasma consisting of inertial ions , isothermal electrons , and stationary dust particles . they @xcite have also investigated the condition for existence of positive and negative potential diasws . _ @xcite have shown that in the dust - modified ion acoustic regime , negative structures can also be generated , beside positive potential soliton if the polytropic index @xmath25 for electrons . the effect of ion - fluid temperature on diasws structures have been investigated by sayed and mamun @xcite in a dusty plasma containing adiabatic ion - fluid , boltzmann electrons , and static dust particles . in most of the earlier works , maxwellian velocity distribution function for lighter species of particles has been used to study diasws and dia double layers ( diadls ) . however the dusty plasma with nonthermally / suprathermally distributed electrons observed in a number of heliospheric environments @xcite . therefore , it is of considerable importance to study nonlinear wave structures in a dusty plasma in which lighter species ( electrons ) is nonthermally / suprathermally distributed . berbri and tribeche @xcite have investigated weakly nonlinear dia shock waves in a dusty plasma with nonthermal electrons . recently baluku _ et al . _ @xcite have investigated diasws in an unmagnetized dusty plasma consisting of cold dust particles and kappa distributed electrons using both small and arbitrary amplitude techniques . in the present investigation we have considered the problem of existence of diasws and diadls in a plasma consisting of negatively charged dust grains , adiabatic positive ions and nonthermal electrons . three basic parameters of the present dusty plasma system are @xmath10 , @xmath26 and @xmath27 , which are respectively the ratio of unperturbed number density of nonthermal electrons to that of ions , the ratio of average temperature of ions to that of nonthermal electrons , a parameter associated with the nonthermal distribution of electrons . nonthermal distribution of electrons becomes isothermal one if @xmath28 . the main aim of this paper is to investigated diasws and diadls thoroughly , giving special emphasis on the followings : + ( * a * ) to study the nonlinear properties of dia waves in a dusty plasma with nonthermal electrons . ( * b * ) to find the exact bounds ( lower and upper ) of the mach number @xmath3 for the existence of solitary wave solutions . ( * c * ) as double layer solution plays an important role to restrict the occurrence of at least one sequence of solitary waves of same polarity , we set up an analytical theory to find the double layer solution of the energy integral , which help us to find the mach number at which double layer occurs and also , to find the amplitude of that double layer solution . ( * d * ) on the basis of the analytical theory for the existence of solitary waves and double layers , the present plasma system has been analyzed numerically . actually , analyzing the sagdeev potential , we have found qualitatively different solution spaces or the compositional parameter spaces showing the nature of existing solitary structures of the energy integral . from these solution spaces , the main observations are the followings . ( * d1 * ) for isothermal electrons , the present plasma system does not support any double layer solution in both cold and adiabatic cases . for nonthermal electrons , the present plasma system does not support any positive potential double layer ( ppdl ) solution , whereas negative potential double layers ( npdls ) start to occur whenever the nonthermal parameter exceeds a critical value . however this npdl solution is unable to restrict the occurrence of all negative potential solitary waves ( npsws ) of the present system , i.e. , npdl solution is not the ultimate solution of the energy integral in the negative potential side . actually , we have observed two different types of npsws , in which amplitude of first type of npsw is restricted by the amplitude of npdl whereas the amplitude of npdl is unable to restrict the amplitude of the second type npsws . as a result , we have observed a finite jump in amplitudes between two different types of npsws separated by a npdl . this fact has also been observed recently by verheest@xcite and baluku _ _ et al.__@xcite for ion - acoustic solitary wave with different plasma constituents . ( * d2 * ) for any physically admissible values of the parameters of the system , specifically , for any value of @xmath10 and any value of @xmath29 , npsw exists for all @xmath4 except @xmath30 , where @xmath2 is the lower bound of mach number @xmath3 , i.e. , solitary structures start to exist for @xmath4 and @xmath5 is the mach number corresponding to a npdl solution . however , if the parameter @xmath10 exceeds a critical value @xmath31 , ppsws exist for all @xmath32 whenever the mach number lies within the interval @xmath33 , where @xmath34 is a physically admissible upper bound of @xmath10 and @xmath13 is the upper bound of @xmath3 for the existence of ppsws only , i.e. , there does not exist any ppsw if @xmath12 . therefore , the coexistence of both ppsws and npsws is possible for all @xmath35 whenever @xmath33 , but npsws still exist for @xmath12 . ( * d3 * ) for nonthermal electron species , we have investigated the entire solution space of the energy integral with respect to the nonthermal parameter @xmath29 and we have found four qualitatively different solution spaces depending on the cut off values of @xmath10 . actually , here we are able to define three cut off values @xmath31 , @xmath36 and @xmath37 of @xmath10 such that @xmath38 for any given value of @xmath26 , and consequently , we can partition the entire range of @xmath10 in the following four disjoint subintervals : @xmath39 , @xmath40 , @xmath41 and @xmath42 . for these four disjoint subintervals of @xmath10 , we have four different solution spaces of the energy integral with respect to nonthermal parameter @xmath29 . these solution spaces can define all types of solitary structures of the present system . ( * d4 * ) finally , considering any solution space , one can get new results and physical ideas for the formation of solitary structures if he moves the solution space through the family of curves parallel to the curve @xmath14 . if we move the solution space through the family of curves parallel to the curve @xmath14 , it is simple to understand the mathematics as well as physics for the formation of double layer solution and it is also simple to understand the relation between solitons and double layer solution . the present paper is organized as follows : basic equations are given in . derivation of energy integral along with sagdeev potential is given in . physical interpretation for the existence of solitary structures of the energy integral is given in . the lower and upper bounds of the mach number for the existence of solitary structures are given in . in , the analytical method to find the upper bound of the mach number for the existence of ppsws is given . in , we find that the existence of npdl solution may restrict occurrence of npsws having amplitude less than the amplitude of npdl . in , an analytical theory to find the double layer solution of the energy integral has been provided . in , an algorithm has been provided to find the value of mach number at which double layer solution exists and also the amplitude of that double layer . following a logical sequence of numerical scheme based on the theoretical discussions as given in and , the solution spaces have been constructed in . finally , we have concluded our findings in . the governing equations describing the nonlinear behavior of dia waves , propagating along @xmath43-axis , in collisionless , unmagnetized dusty plasma consisting of negatively charged immobile dust grains are the following : @xmath44 @xmath45 @xmath46 @xmath47 where the parameter @xmath48 . here we have used the notation @xmath49 or @xmath50 for @xmath51 and @xmath52 , @xmath53 , @xmath54 , @xmath55 , @xmath56 , @xmath43 and @xmath57 are , respectively , the ion number density , electron number density , ion velocity , ion pressure , electrostatic potential , spatial variable and time , and they have been normalized by @xmath18 ( unperturbed ion number density ) , @xmath17 ( unperturbed electron number density ) , @xmath58(@xmath59 ) ( ion - acoustic speed ) , @xmath60 , @xmath61 , @xmath62(debye length ) , and @xmath63 ( ion plasma period ) . here @xmath64 is the adiabatic index , @xmath65 is the boltzmann constant , @xmath66 and @xmath67 are , respectively , the average temperatures of ions and electrons , @xmath68 is the mass of an ion , @xmath19 is the dust number density , @xmath20 is the number of negative unit charges residing on dust grain surface , and @xmath69 is the charge of an electron . the above equations are supplemented by nonthermally distributed electrons as prescribed by cairns _ @xcite for the electron species . nonthermal distribution of any lighter species of particles ( as prescribed by cairns _ @xcite for the electron species ) can be regarded as a modified boltzmannian distribution , which has the property that the number of particles in phase space in the neighborhood of the point @xmath70 is much smaller than the number of particles in phase space in the neighborhood of the point @xmath70 for the case of boltzmann distribution , where @xmath71 is the velocity of the particle in phase space . under the above mentioned normalization of the dependent and independent variables , the normalized number density of nonthermal electrons can be written as @xmath72 where @xmath73 with @xmath74 . here @xmath75 is the parameter associated with nonthermal distribution of electrons and this parameter determines the proportion of fast energetic electrons . from ( [ beta1 ] ) and the inequality @xmath74 , it can be easily checked that the nonthermal parameter @xmath29 is restricted by the following inequality : @xmath76 . however we can not take the whole region of @xmath29 ( @xmath77 ) . plotting the nonthermal velocity distribution of electrons against its velocity ( @xmath71 ) in phase space , it can be easily shown that the number of electrons in phase space in the neighborhood of the point @xmath70 decreases with increasing @xmath29 and the number of electrons in phase space in the neighborhood of the point @xmath70 is almost zero when @xmath78 . therefore , for increasing @xmath29 distribution function develops wings , which become stronger as @xmath29 increases , and at the same time the center density in phase space drops , the latter as a result of the normalization of the area under the integral . consequently , we should not take values of @xmath79 since that stage might stretch the credibility of the cairns model too far @xcite . so , here we consider the effective range of @xmath29 as follows : @xmath80 , where @xmath81 . the charge neutrality condition , @xmath82 can be written as @xmath83 where @xmath84 and consequently , the poisson equation ( [ poisson eq without mu ] ) assumes the following form : @xmath85 we note from ( [ charge neutrality with mu ] ) that @xmath86 must be greater than zero , i.e. , @xmath87 . when @xmath88 , the effect of negatively charged dust grains on dia wave is negligible and so , we restrict @xmath10 by the inequality @xmath89 , where @xmath90 is strictly less than 1 . to study the arbitrary amplitude time independent diasws and diadls we make all the dependent variables depend only on a single variable @xmath91 , where the mach number @xmath3 is normalized by @xmath58 . thus in the steady state , ( [ continuity of ions ] ) - ( [ pressure eq ] ) and ( [ poisson eq ] ) can be written as @xmath92 @xmath93 @xmath94 @xmath95 using the boundary conditions , @xmath96 and solving ( [ continuity of ions steady state ] ) - ( [ pressure eq steady state ] ) , we get a quadratic equation for @xmath97 and following the same argument as given in das _ @xcite to find the expression of dust density with exact bounds , we get the following expression of @xmath52 . @xmath98 where @xmath99 from ( [ n_i with phi_m psi_m 2 ] ) , we see that this equation gives both theoretically and numerically correct expression of @xmath52 even when @xmath100 if @xmath101 . now integrating ( [ poisson eq steady state ] ) with respect to @xmath56 and using the boundary conditions ( [ boundary cond ] ) , we get the following energy integral with @xmath102 as sagdeev potential or pseudo - potential . @xmath103 where @xmath104 @xmath105 @xmath106 here , @xmath107 is same as @xmath102 , i.e. , the mach number @xmath3 is omitted from the notation @xmath107 when no particular emphasis is put upon it . the energy integral ( [ energy int ] ) can be regarded as the one - dimensional motion of a particle of unit mass whose position is @xmath56 at time @xmath108 with velocity @xmath109 in a potential well @xmath102 . the first term in the energy integral ( [ energy int ] ) can be regarded as the kinetic energy of a particle of unit mass at position @xmath56 and time @xmath108 whereas @xmath102 is the potential energy at that instant . since kinetic energy is always a non - negative quantity , @xmath110 for the entire motion , i.e. , zero is the maximum value for @xmath102 . again from ( [ energy int ] ) , we find @xmath111 , i.e. the force acting on the particle of unit mass at the position @xmath56 is @xmath112 , where `` @xmath113 '' indicates a derivative with respect to @xmath56 . now , it can be easily checked that @xmath114 , and consequently , the particle is in equilibrium at @xmath115 because the velocity as well as the force acting on the particle at @xmath115 are simultaneously zero . now if @xmath116 can be made an unstable position of equilibrium , the energy integral can be interpreted as the motion of an oscillatory particle if @xmath117 for some @xmath118 , i.e. , if the particle is slightly displaced from its unstable position of equilibrium then it moves away from its unstable position of equilibrium and it continues its motion until its velocity is equal to zero , i.e. , until @xmath56 takes the value @xmath119 . now the force acting on the particle of unit mass at position @xmath120 is @xmath121 . for @xmath122 , the force acting on the particle at the point @xmath123 is directed towards the point @xmath115 if @xmath124 , i.e. , if @xmath125 . on the other hand , for @xmath126 , the force acting on the particle at the point @xmath123 is directed towards the point @xmath115 if @xmath127 , i.e. , if @xmath128 . therefore , if @xmath128 ( for the positive potential side ) or if @xmath125 ( for the negative potential side ) then the particle reflects back again to @xmath116 . again , if @xmath129 then the velocity @xmath130 as well as the force @xmath131 both are equal to zero at @xmath120 . consequently , if the particle is slightly displaced from its unstable position of equilibrium ( @xmath116 ) it moves away from @xmath116 and it continues its motion until the velocity is equal to zero , i.e. , until @xmath56 takes the value @xmath120 . however it can not be reflected back again at @xmath116 as the velocity and the force acting on the particle at @xmath120 vanish simultaneously . actually , if @xmath128 ( for @xmath126 ) or if @xmath125 ( for @xmath122 ) the particle takes an infinite long time to move away from the unstable position of equilibrium . after that it continues its motion until @xmath56 takes the value @xmath119 and again it takes an infinite long time to come back its unstable position of equilibrium . therefore , for the existence of a positive ( negative ) potential solitary wave solution of the energy integral ( [ energy int ] ) , we must have the following : ( a ) @xmath115 is the position of unstable equilibrium of the particle , ( b ) @xmath117 , @xmath132 @xmath133 for some @xmath134 @xmath135 , which is nothing but the condition for oscillation of the particle within the interval @xmath136 and ( c ) @xmath137 for all @xmath138 @xmath139 ) , which is the condition to define the energy integral ( [ energy int ] ) within the interval @xmath136 . for the existence of a positive ( negative ) potential dl solution of the energy integral ( [ energy int ] ) , the conditions ( a ) and ( c ) remain unchanged but here ( b ) has been modified in such a way that the particle can not be reflected again at @xmath116 , i.e. , the condition ( b ) assumes the following form : @xmath140 , @xmath141 for some @xmath134 @xmath142 ) . for the existence of solitary structures , we must have @xmath143 and @xmath144 . now it can be easily verified that the first two conditions , i.e. , @xmath145 and @xmath146 are trivially satisfied whereas , the condition @xmath144 gives @xmath4 , where @xmath2 is given by the following equation . @xmath147 now , for @xmath2 to be real and positive , we must have @xmath148 and @xmath149 . as the effective range of @xmath29 is @xmath80 , where @xmath81 , @xmath2 is well - defined as a real positive quantity for all @xmath150 and @xmath80 . consider the existence of a ppsw for some value of @xmath151 therefore , there exists a @xmath126 such that @xmath152 now as @xmath102 is real for @xmath153 we must have @xmath154 , otherwise @xmath155 is not a real quantity . therefore , @xmath156 defines a large amplitude ppsw , which is in conformity with ( [ ppsw ] ) , if @xmath157 and @xmath158 . again let @xmath13 be the maximum value of @xmath3 up to which solitary wave solution can exist . as @xmath159 increases with @xmath3 then @xmath160 . therefore , @xmath161 defines the largest amplitude ppsw if @xmath162 and @xmath163 . therefore , for the existence of the ppsws , the mach number @xmath3 is restricted by the following inequality : @xmath164 , where @xmath13 is the largest positive root of the equation @xmath165 subject to the condition @xmath166 @xmath167 @xmath168 . now if at @xmath169 ( @xmath170 ) , one can get a ppdl solution of the energy integral ( [ energy int ] ) then for the existence of the ppsws , the mach number @xmath3 is restricted by the inequality : @xmath171 and also @xmath172 . on the other hand if @xmath173 , then for the existence of ppsws , the mach number @xmath3 is restricted by the inequality : @xmath171 . we have seen earlier that @xmath102 is real if @xmath174 , where @xmath159 is strictly positive . for npsws or npdls , we have @xmath175 along with the other conditions stated in section [ sec : physics ] for the existence of npsws or npdls . as @xmath159 is strictly positive and for npsws or npdls , @xmath176 , the condition @xmath177 is automatically satisfied and consequently , for these two cases @xmath102 is well defined for all @xmath176 without imposing extra condition . since there is no such restriction on @xmath56 , we can not use the same definition as in the case of ppsws to find the upper bound of mach numbers for the existence of npsws . for the case of npsws , to find an upper limit or upper bound of @xmath3 , up to which npsw can exist , we shall first of all find a value @xmath178 of @xmath3 for which energy integral ( [ energy int ] ) gives a npdl solution at @xmath30 with amplitude @xmath179 . now if at @xmath30 , @xmath180 is the only root ( double root ) of the equation @xmath181 , i.e. , @xmath182 and @xmath183 , then the npdl solution is the ultimate solution of the energy integral ( [ energy int ] ) and in this case , no npsw solution can be obtained if @xmath1 , i.e. , for the occurrence of npsws , the mach number @xmath3 is restricted by the inequality @xmath0 . on the other hand if there exists an inaccessible simple root @xmath184 of @xmath56 such that @xmath185 , i.e. , @xmath186 along with @xmath182 , @xmath183 and @xmath185 , then there exists a npsw solution of the energy integral ( [ energy int ] ) for at least one value of @xmath1 . hence in the later case double layer solution is unable to restrict the occurrence of npsws for @xmath1 . from this consideration , it is also clear that the occurrence of npsws is restricted by @xmath0 , provided that there exists one and only one double root of the equation @xmath187 for the unknown @xmath188 , otherwise npsws can exist for all @xmath189 . therefore , the double layer solution of the energy integral ( [ energy int ] ) plays an important role to determine the upper bound of the mach number for existence of either ppsws or npsws . for the present problem , ppsw exists if @xmath164 but from the discussion of it is not clear whether the energy integral ( [ energy int ] ) provides a ppdl solution for some @xmath169 such that @xmath190 . similarly , from the discussion of [ subsec : negative ] it is not clear whether the energy integral ( [ energy int ] ) provides a npdl solution for some @xmath30 . so , in the next subsection , we shall analytically investigate the existence of double layer solution of the energy integral ( [ energy int ] ) . from the condition ( b ) as given in for the existence of double layer solution of the energy integral ( [ energy int ] ) , we must have a non - zero @xmath56 ( @xmath191 ) such that following conditions are simultaneously satisfied : @xmath192 using ( [ v(phi ) ] ) - ( [ v_i ] ) , the first equation , the second equation and the third inequality of ( [ dlb1 ] ) can be written , respectively , as @xmath193 @xmath194 @xmath195 where @xmath196 eliminating @xmath52 from ( [ redlb1 ] ) and ( [ redlb2 ] ) , we get @xmath197\nonumber \\- \alpha.\end{aligned}\ ] ] it can be easily checked that @xmath115 if and only if @xmath198 , and consequently , for non - zero @xmath56 , ( [ mdl ] ) can be written as @xmath199 where @xmath200- \alpha}{\mu(1-n_{e})}.\end{aligned}\ ] ] using ( [ remdl ] ) , from ( [ redlb2 ] ) and ( [ redlb3 ] ) , we , respectively , get @xmath201 @xmath202 where @xmath203 now suppose @xmath204 solves ( [ dlamp ] ) , then the double layer solution of the energy integral ( [ energy int ] ) exists at @xmath205 having amplitude @xmath206 if the following conditions are satisfied : @xmath207 @xmath208 @xmath209 to derive the condition ( [ con2 ] ) , we have used the following restriction on @xmath56 : @xmath210 . again the condition ( [ con3 ] ) states that the function @xmath211 decreases in a neighbourhood of @xmath212 and @xmath213 . therefore if @xmath214 , @xmath211 changes sign from positive to negative when it crosses the point @xmath212 from left to right . however , for @xmath215 , @xmath211 changes sign from positive to negative when it crosses the point @xmath212 from right to left . in any case , @xmath211 vanishes at @xmath216 . from this theoretical discussion , it is simple to make a numerical scheme to test whether the energy integral provides a double layer solution . the simple algorithm for the existence of double layer solution can be written as follows . + * step - 1 : * find @xmath217 defined by ( [ hphi ] ) for @xmath218 ( @xmath219 ) . if @xmath217 is negative for all @xmath218 ( @xmath219 ) then the system does not support negative ( positive ) potential double layer . else follow the next step . + * step - 2 : * if @xmath220 be real for @xmath218 ( @xmath219 ) then go to next step . otherwise , the system does not support negative ( positive ) potential double layer . + * step - 3 : * set up a numerical scheme to find all possible real negative ( positive ) roots of @xmath56 of ( [ dlamp ] ) for all admissible values of the parameters . to find these roots of @xmath56 at some fixed values of the parameters , let @xmath56 free and take all those @xmath56 s where the function @xmath211 changes its sign from positive to negative when it crosses the point @xmath56 from right to left ( left to right ) and denote the set of all these @xmath56 s as @xmath221 ( @xmath222 ) . obviously , there may exist other roots of @xmath56 of ( [ dlamp ] ) for unknown @xmath56 such that @xmath211 changes its sign from negative to positive , however , the theoretical discussions suggest that these roots are beyond the scope of the present numerical scheme . + * step - 4 : * using ( [ remdl ] ) find the mach numbers @xmath3 at each @xmath56 of @xmath221 ( @xmath222 ) . + * step - 5 : * find the largest ( smallest ) value @xmath223 ( @xmath224 ) of @xmath56 from @xmath221 ( @xmath222 ) such that conditions ( [ con1 ] ) - ( [ con2 ] ) hold good . then , obviously @xmath223 ( @xmath224 ) is the amplitude of npdl ( ppdl ) solution . next use ( [ remdl ] ) to obtain the mach number @xmath178 ( @xmath225 ) corresponding to npdl ( ppdl ) solution having amplitude @xmath226 ( @xmath224 ) . with the help of this algorithm , we want to investigate numerically the following facts for the present system : + ( i ) whether the present system supports ppdl and/or npdl solutions . ( ii ) whether the double layer solution , if exists , can restrict the occurrence of all solitons of same polarity , i.e. , whether the double layer solution is the ultimate solution of all solitons of same polarity of the present system . in this connection , it is important to remember that for any double layer solution , there must exists at least one sequence of solitons of same polarity converging to the double layer solution , i.e. , the amplitude of any double layer solution acts as an exact upper bound of the amplitudes of at least one sequence of solitary waves of same polarity . to investigate the existence of ppdl , @xmath211 is plotted against @xmath56 for @xmath227 in for different values of @xmath10 and @xmath29 . ( a ) and ( b ) show that @xmath211 is either remain positive throughout @xmath219 or changes sign from negative to positive when it crosses positive @xmath56 axis from left to right . in ( a ) , @xmath211 corresponding to @xmath228 changes sign from negative to positive when it crosses positive @xmath56 axis from left to right . similar facts have been observed in ( a ) for @xmath211 corresponding to @xmath229 and in ( b ) for @xmath211 corresponding to @xmath229 . consequently , there does not exist any ppdl solution . for any admissible values of the parameters involved in the system , it can be easily verified that the system does not support any ppdl solution with the help of plotting @xmath211 against @xmath56 for @xmath227 . since the system does not support any ppdl solution , we can conclude that @xmath13 is the only upper bound of @xmath3 for the occurrence of ppsws . thus ppsws exist whenever @xmath11 . to investigate the existence of npdl , @xmath211 is plotted against @xmath56 for @xmath230 in . in ( a ) , @xmath211 corresponding to @xmath231 and @xmath228 remain positive throughout @xmath218 and thus there does not exist npdl solution for @xmath231 and also for @xmath228 with @xmath232 , @xmath233 . it can be easily checked that there does not exist any npdl solution for @xmath234 with @xmath232 , @xmath233 by simply drawing @xmath211 against @xmath56 . however @xmath211 corresponding to @xmath229 changes sign from positive to negative when it crosses negative @xmath56 axis from right to left and fulfill all the conditions of our algorithm for the existence of npdl solution . a similar interpretation of ( b ) shows that npdls exist for @xmath235 with @xmath232 , @xmath236 . thus for larger values of @xmath10 , npdl solution is expected at lower values of @xmath29 . therefore , for denser electrons in the background plasma , we require less amount of fast energetic electrons to get npdl . in other words , electron density depletion restrict the occurrence of npdl . still it is not clear whether the existence of npdl can restrict all npsws of the present system , or in other words , whether @xmath178 is the upper limit of @xmath3 for the existence of all npsws of the present system . from , one can find three set of values of the parameters @xmath26 , @xmath10 and @xmath29 such that npdl exist . it is easy to find @xmath223 , at which @xmath211 changes sign from positive to negative when it crosses negative @xmath56 axis from right to left and using ( [ remdl ] ) we can find three values of @xmath178 corresponding to three different values of @xmath223 for three set of values of the parameters @xmath26 , @xmath10 and @xmath29 as shown in . for clarity we have tabulate these values in . our algorithm will be verified if we can confirm the occurrence of npdl solutions at those values of parameters . @cccccc & @xmath26&@xmath10&@xmath29&@xmath178&@xmath223 + @xmath237:&0.9&0.2&0.6&5.10476181&-1.9397924 + @xmath238:&0.9&0.5&0.4&2.49995351&-1.5418283 + @xmath239:&0.9&0.5&0.6&3.25924103&-1.5492367 + in ( a ) , @xmath102 is plotted against @xmath56 for three set of values ( denoted as @xmath237 , @xmath238 and @xmath239 in ) of @xmath26 , @xmath10 , @xmath29 and @xmath178 . our aim is to show the amplitudes obtained from ( a ) are exactly the same as obtained from . each of the curves in ( a ) shows the existence of a npdl solution . moreover , the amplitudes of these double layers are exactly the same as obtained in . hence our algorithm regarding the double layer solution is correct . from , we have some typical observations regarding the amplitude of npdls . for fixed @xmath26 and @xmath10 the amplitude of double layer increases with @xmath29 . again for fixed values of @xmath26 and @xmath29 , the amplitude of double layer decreases with increasing @xmath10 . in other words , npdl gets stronger with fast energetic electrons . however , for any fixed non - zero value of @xmath29 and for any value of @xmath26 , one can get stronger npdl by adding more electrons on the dust grain surface . now we are in a position to investigate whether the npdl solution can restrict the occurrence of all npsws of the present system . for this purpose , we explore ( a ) beyond @xmath223 and obtain ( b ) . in this figure , we have drawn the same set of curves as in ( a ) with an exception that we have extended the range of @xmath56 axis far away from @xmath116 . we see that after making a double root at @xmath179 , there exists an @xmath185 such that @xmath240 . thus according to our theoretical discussions in , there exists a @xmath1 such that npsw exists . therefore , the npdl solution can not restrict the occurrence all npsws of the present system and consequently , @xmath178 can not act as an upper bound of @xmath3 for the existence of all npsws of the present system . actually , the present system supports very large amplitude npsw for all @xmath1 . to justify this fact , in ( a ) and ( b ) , @xmath102 is plotted against @xmath56 for three different values of @xmath3 , viz . , @xmath178 , @xmath241 and @xmath242 . ( b ) shows that at @xmath30 , there exists a npdl of amplitude @xmath226 whereas , for @xmath243 , there exists a npsw of amplitude less than @xmath226 . however , ( b ) shows that the equation @xmath244 has no real root of @xmath56 in the neighborhood of @xmath180 . from ( a ) , we see that @xmath107 again vanishes at @xmath245 , @xmath246 and @xmath247 , respectively , for @xmath243 , @xmath30 , and @xmath248 . however , the roots @xmath245 and @xmath246 of @xmath249 corresponding to @xmath243 and @xmath30 are unable to give any solitary wave solution , whereas the root @xmath247 of @xmath250 gives a npsw of amplitude much greater than that of npdl at @xmath30 as well as npsw at @xmath243 . thus there is a finite jump in amplitudes between two npsws at @xmath243 and at @xmath248 separated by the npdl at @xmath30 . this is not a new result , the same result has also been observed in some recent works @xcite with different plasma environments . mathematically , it is simple to prove the following property : property : : : if there exists two types of npsws ( ppsws ) separated by a npdl ( ppdl ) then there is a finite jump between the amplitudes of two types of npsws only when @xmath251 for all @xmath252 and for all @xmath230 ( @xmath227 ) . for the present problem , it is easy to check that @xmath253 for all @xmath252 and for all @xmath188 . thus all the conditions of the property are satisfied but in the positive potential side , there does not exist any jump in amplitudes between two solitary waves . more specifically , we have not found any ppdl solution which separates two types of ppsws . in ( c ) , profiles of npdl has been shown at @xmath30 . in ( d ) , profiles of npsws have been shown at @xmath243 and at @xmath248 , respectively . the profile in ( c ) corresponding to @xmath30 is an usual double layer profile . however the solitary wave profile in ( d ) corresponding to @xmath248 is an unusual one ; its like a dais - type solitary wave profile . the jump between the amplitudes of two npsws separated by the npdl is much prominent here . in the above discussions , we have demonstrated the possible existence of solitary structures for some particular values of the parameters of the problem without making any delimitation of the compositional parameter space for the existence of such nonlinear structures and consequently , we are unable to produce complete scenario of the present problem . so , it is desirable to construct the entire solution space or compositional parameter space showing the nature of different solitary structures present in the system . in the next section , we have considered different solution spaces of the energy integral ( [ energy int ] ) with respect to @xmath29 . - are the different compositional parameter spaces with respect to @xmath29 showing nature of solitary structures and all these figures are aimed to show the solution spaces of the energy integral ( [ energy int ] ) with respect to @xmath29 . to interpret - , we have made a general description as follows : solitary structures start to exist just above the lower curve @xmath254 . for any admissible range of the parameters there always exists at least one @xmath4 such that npsw exists thereat . @xmath13 is the upper bound of @xmath3 for the existence of ppsws , i.e. , there does not exist any ppsw if @xmath12 . more explicitly , if we pick a @xmath29 and goes vertically upwards , then all intermediate @xmath3 bounded by @xmath254 and @xmath255 would give ppsws . the curve @xmath255 also restrict the coexistence of both npsws and ppsws , however the curve @xmath255 is unable to restrict the occurrence of all npsws of the present system , i.e. , there exists npsw for all @xmath12 . at any point on the curve @xmath30 there exists a npdl solution . but this npdl solution is unable to restrict the occurrence of all npsws of the present system . as a result , we get two different types of npsws separated by the npdl solution , in which occurrence of first type of npsw is restricted by @xmath0 whereas the second type npsw exists for all @xmath1 . we have also observed a finite jump between the amplitudes of npsws at @xmath6 and at @xmath7 , where @xmath8 and @xmath9 , i.e. , there is a finite jump in amplitudes of the npsws above and below the curve @xmath30 . now we want to define the cut off values of @xmath10 and @xmath29 , which are responsible to delimit the solution space . @xmath256 : : : @xmath257 is a cut - off value of @xmath29 such that npdl starts to exist whenever @xmath258 for any value of @xmath10 lies within the interval @xmath150 , i.e. , @xmath259 is the lower bound of @xmath29 for the existence of npdl solution . thus , @xmath257 is the minimum proportion of fast energetic electrons such that maximum potential difference occurs in the system and the value of @xmath257 depends on the number of electrons residing on dust grain surface . @xmath31 : : : @xmath31 is a cut of value of @xmath10 such that @xmath13 does not exist for any admissible value of @xmath29 if @xmath10 lies within the interval @xmath39 , i.e. , if @xmath260 , there exists a value @xmath261 of @xmath29 such that @xmath13 exists at @xmath262 , moreover , if @xmath263 , then @xmath13 exists for all @xmath29 lies within the interval @xmath264 . @xmath265 : : : @xmath265 is a cut - off value of @xmath29 such that @xmath13 exists for all @xmath266 whenever @xmath260 . consequently , @xmath267 is the upper bound of @xmath29 for the existence of ppsw . now , if @xmath268 , then there exists an interval @xmath269 in which neither @xmath178 nor @xmath13 exist and consequently , we can define cut - off values @xmath36 and @xmath37 of @xmath10 as follows : @xmath36 : : : @xmath36 is another cut - off value of @xmath10 such that for all @xmath270 , neither @xmath13 nor @xmath178 exist whenever @xmath271 , i.e. , for all @xmath270 and for all @xmath271 only npsws exist for all @xmath4 . @xmath37 : : : @xmath37 is another cut - off value of @xmath10 such that for all @xmath272 , the curve @xmath30 tends to intersect the curve @xmath14 at the point @xmath259 . from the definition of @xmath31 , @xmath36 and @xmath37 , we can numerically find the values of @xmath31 , @xmath36 and @xmath37 for any value of @xmath26 . the numerical solution is shown graphically in . from this figure we see that for any value of @xmath26 , we can partition the entire interval of @xmath10 in the following four subintervals : ( i ) @xmath273 , ( ii ) @xmath270 , ( iii ) @xmath274 and ( iv ) @xmath275 . in these subintervals of @xmath10 , we have qualitatively different solution space of the energy integral ( [ energy int ] ) with respect to @xmath29 . the solution spaces have been shown through - for four different subintervals of @xmath10 . before going to discuss the solution spaces in details , the variations of the curves @xmath267 ( ) and @xmath276 have been plotted against @xmath10 in to demonstrate the solution spaces in true physical sense . in this figure , actually we have shown those two curves , viz . , @xmath267 and @xmath259 which are responsible to divide @xmath10 into several subintervals . a closer look of the - suggests that effectively defined all the solution spaces as shown through - in more compact form provided that we have sound knowledge regarding the appropriate bounds of the mach number for the occurrence of different types ( nature ) of solitary structures of the present system . using the theory as presented in , one can easily set a numerical scheme to find the appropriate bounds for the occurrence of different types ( nature ) of solitary structures of the present system . in , we have used the following terminology . c - n : region of coexistence of both ppsws and npsws for @xmath164 and only npsws whenever @xmath12 ; c - n - d : region of coexistence of both ppsws and npsws for @xmath164 and only npsws whenever @xmath12 with a npdl at some @xmath277 ; n - d : region of existence of only npsws whenever @xmath4 with a npdl at some @xmath277 . we have found @xmath31 , @xmath36 and @xmath37 lies in the neighborhood of @xmath278 , @xmath279 and @xmath280 , respectively for @xmath281 . from , we have the following observations . for @xmath28 , i.e. , for isothermal electrons , for @xmath39 , only npsws exist for all @xmath4 and coexistence of both npsws and ppsws are possible for @xmath282 whenever @xmath164 , whereas npsws exist for all @xmath12 . in presence of isothermal electrons the system does not support any double layer solution . similar facts can also be observed by considering - at the point @xmath28 . so , if @xmath10 exceeds the critical value @xmath31 , ppsw starts to exist for @xmath164 and attains its maximum amplitude at @xmath283 but even for increasing @xmath10 for @xmath282 , the ppsw can not acquire enough strength to make a ppdl even at @xmath283 . actually , from the charge neutrality condition ( [ charge neutrality without mu ] ) , we see that @xmath18 is a constant . consequently , we can not inject positive charge from outside or we can not increase the equilibrium ion number density @xmath18 and this is the reason that ppsw can not acquire enough strength to make a ppdl even at @xmath283 . on the other hand , we can increase or decrease the quantities @xmath20 , @xmath19 , @xmath17 in such way that the charge neutrality condition ( [ charge neutrality without mu ] ) holds good for constant equilibrium ion number density @xmath18 . but in any case , the amplitude of the npsw steadily increasing for increasing mach number @xmath4 . actually , we are unable to restrict the occurrence of npsw for the present system , i.e. , we have not found any upper bound of the mach number which can restrict the occurrence of npsw and this is the reason that npsw can not make a npdl at any point of the compositional parameter space . however , to discuss the formation of double layer from physical point of view , we consider the following simple mathematics . suppose @xmath284 is the amplitude of ppsw at any point of the compositional parameter space , where we have used the following terminology : if there does not exist any ppsw at some point of the compositional parameter space , then @xmath285 . therefore , @xmath284 is well defined as the amplitude of ppsw at any point of the compositional parameter space . similarly , one can define @xmath286 , as the amplitude of npsw at any point of the compositional parameter space , i.e. , if there does not exist any npsw at some point of the compositional parameter space , then @xmath287 . from simple mathematics , we get @xmath288 from inequality ( [ phi_pn ] ) , it is clear that one can get a ppdl solution at a point of the compositional parameter space if @xmath289 is maximum with @xmath290 whereas one can get a npdl solution at a point of the compositional parameter space if @xmath291 is maximum with @xmath292 . for isothermal electrons , it can be easily checked that the potential difference ( @xmath293 with @xmath290 ) for the formation of ppdl can not attain any maximum value at any point of the compositional parameter space . similarly , the potential difference ( @xmath294 with @xmath292 ) for the formation of npdl can not attain any maximum value at any point of the compositional parameter space . so , for isothermal electrons , the present system does not support any double layer solutions . for non - zero @xmath29 , the solution space as obtained in can be partitioned as follows : ( i ) @xmath39 : for @xmath295 , only npsws are possible for all @xmath4 , whereas for @xmath296 , npsws are possible for all @xmath4 except @xmath297 . ( ii ) @xmath40 : for @xmath298 , coexistence of both npsws and ppsws are possible whenever @xmath164 and only npsws exist for all @xmath12 . for @xmath271 , only npsws are possible for all @xmath4 . for @xmath299 , npsws are possible for all @xmath4 except @xmath297 . ( iii ) @xmath300 : for @xmath295 , coexistence of both npsws and ppsws are possible whenever @xmath164 and only npsws exist for all @xmath12 . for @xmath301 , coexistence of both npsws and ppsws are possible whenever @xmath164 and only npsws exist for all @xmath12 except the point @xmath302 . for @xmath303 , npsws are possible for all @xmath4 except @xmath297 . ( iv ) @xmath275 : for @xmath295 , coexistence of both npsws and ppsws are possible whenever @xmath164 and only npsws exist for all @xmath12 . for @xmath304 , coexistence of both npsws and ppsws are possible whenever @xmath164 and only npsws exist for all @xmath12 except @xmath305 . for @xmath306 , coexistence of both npsws and ppsws are possible whenever @xmath164 and only npsws exist for all @xmath12 except the point @xmath302 . for @xmath303 , npsws are possible for all @xmath4 except @xmath297 . in all these solution spaces , whenever @xmath307 or @xmath308 , at the point @xmath30 , one can always find a npdl , whereas one can find the coexistence of a npdl and a ppsw at the point @xmath30 whenever @xmath309 . therefore , from the above discussions , it is clear that for any physically admissible value of @xmath10 , i.e. , @xmath310 , there exists a non - zero value of @xmath29 such that the present system supports a npdl solution for some @xmath277 . so , again from charge neutrality condition ( [ charge neutrality without mu ] ) , we see that if the density of electrons increases up to a certain value @xmath37 ( see ) , minimum energetic electrons ( small value of @xmath29 ) can produce npdl whereas if the density of electrons tends to zero ( almost depletion of electrons ) , more energetic electrons ( higher value of @xmath29 ) are required to form a npdl solution . so , we see that @xmath257 exists for any physically admissible value of @xmath10 , i.e. , @xmath310 and consequently , the present system supports npdl solution for some @xmath277 if @xmath311 . again , there does not exist any @xmath265 for @xmath39 and consequently , coexistence of both npsws and ppsws are not possible even when nonthermal distribution of electrons becomes isothermal one , i.e. , when @xmath28 . it is also important to remember that if the value of @xmath29 increases , negative potential is stronger than positive potential and consequently , instead of getting ppsw , from inequality ( [ phi_pn ] ) , one can get a npdl solution . for @xmath312 , @xmath265 always exists and increases with increasing @xmath10 . consequently , for this interval of @xmath10 , solitary structures of both polarities exist provided that @xmath313 and the region of coexistence of both npsws and ppsws with respect to the nonthermal parameter increases with increasing @xmath10 lying within @xmath312 . moreover , for the values of @xmath10 lying within @xmath314 , in fact , @xmath275 , if @xmath315 , the npsw is stronger than ppsw for @xmath316 and at @xmath317 , inequality ( [ phi_pn ] ) holds good for @xmath318 and consequently , we have not only get a npdl solution a @xmath317 but also get a weaker ppsw at the same point of the compositional parameter space when @xmath30 . for @xmath319 , @xmath257 is always greater than @xmath265 . consequently , for @xmath320 , only npsws exist for all @xmath4 whereas for @xmath321 , @xmath257 is always less than @xmath265 and all types of solitary structures are possible for the present system . all the facts can also be verified by considering - . the physical interpretation for the formation of solitary structures in this case can be demonstrated through charge neutrality condition ( [ charge neutrality without mu ] ) , inequality ( [ phi_pn ] ) and either considering the or more explicitly the figures - . therefore , this is actually the graphical presentation of different solitary structures with respect to different subintervals of @xmath10 within the admissible interval of the nonthermal parameter @xmath29 . finally , considering any solution space , we can get new results and physical ideas for the formation of solitary structures if we move in the solution space along the family of curves parallel to the curve @xmath14 . for example , we shall consider the solution space with respect to the nonthermal parameter @xmath29 for @xmath42 and if we move in the solution space along the family of curves parallel to the curve @xmath14 , it is simple to understand the mathematics as well as physics for the formation of double layer solution and it is also simple to understand the relation between solitons and double layers . to be more specific , solution space for the present system with respect to @xmath29 for @xmath42 has been presented in fig . [ fig : sol space amp](a ) , in which the curve @xmath14 is omitted from the solution space as presented in . now consider the family of curves parallel to @xmath14 . for instance , consider one such parallel curve for @xmath322 as shown in fig . [ fig : sol space amp](a ) . in this figure , @xmath323 is the value of @xmath29 where the curve @xmath30 intersects the curve @xmath322 , whereas @xmath324 is the value of @xmath29 where the curve @xmath255 intersects the curve @xmath322 . [ fig : sol space amp](a ) can be interpreted in the same way of with @xmath257 replaced by @xmath323 and @xmath265 replaced by @xmath324 . however , in fig . [ fig : sol space amp](a ) the solitary structures exist along the curve @xmath322 , specifically , ( i ) both npsw and ppsw coexist for @xmath325 , ( ii ) only npsw exists for @xmath326 and ( iii ) at @xmath327 , a ppsw coexists with a npdl . the variation of amplitude of those solitary waves along the curve @xmath322 for @xmath328 have been shown in fig . [ fig : sol space amp](b ) . this figure shows that the amplitude of npsw decrease with increasing @xmath29 for @xmath329 and the amplitudes of npsws are bounded by the amplitude of npdl at @xmath327 . again , the amplitude of ppsw increases with increasing @xmath29 for @xmath330 having minimum amplitude at @xmath327 . moreover , the fig . [ fig : sol space amp](b ) shows that along the curve @xmath322 , the amplitude of npsw increases with decreasing @xmath3 along the curve @xmath322 for @xmath329 and ultimately , these npsws end with a npdl at @xmath331 . therefore , the solitons and double layer are not two distinct nonlinear structures , i.e. , double layer solution , if exists , must be the limiting structure of at least one sequence of solitons of same polarity . more specifically , existence of double layer solution implies that there must exists at least one sequence of solitary waves of same polarity having monotonically increasing amplitude converging to the double layer solution , i.e. , the amplitude of the double layer solution acts as an exact upper bound or least upper bound ( _ lub _ ) of the amplitudes of the sequence of solitary waves of same polarity . however , we have seen in the literature that when all the parameters involved in the system assume fixed values in their respective physically admissible range , the amplitude of solitary wave increases with increasing @xmath3 and these solitary waves end with a double layer of same polarity , if exists . here it is important to note that @xmath3 is not a function of the parameters involved in the system but is restricted by the inequality @xmath0 , where @xmath30 corresponds to a double layer solution . so we can not compare this case with the case of @xmath322 , since @xmath2 is a function of the parameters involved in the system and consequently , monotonicity of @xmath2 entirely depends on a parameter when the other parameters assume fixed values in their respective physically admissible range . but the solitons and double layer are not two distinct nonlinear structures . therefore , double layer solution , if exists , must be the limiting structure of at least one sequence of solitons of same polarity . in fig . [ fig : sol space amp](b ) , by the vertical line with both sided arrow , we mean , the amplitude of npdl at @xmath327 for @xmath322 . for @xmath232 and @xmath332 , the critical values @xmath323 and @xmath324 lies in the neighborhood of @xmath333 and @xmath334 , respectively . now , for the formation of npdl solution on the curve @xmath322 at @xmath327 , the negative potential ( absolute value ) must dominate the positive potential in the neighborhood of the point @xmath327 and the potential difference ( with respect to negative potential ) must be maximum thereat . from fig . [ fig : sol space amp](b ) , it is clear that the negative potential ( absolute value ) dominates the positive potential in a right neighborhood of @xmath327 and the potential difference ( @xmath291 along with @xmath292 ) is maximum thereat . this figure shows the existence of npdl solution at @xmath327 , which is already confirmed in fig . [ fig : sol space amp](a ) . more specifically , from the inequality ( [ phi_pn ] ) , it is clear that one can get a ppdl solution at a point of the compositional parameter space if @xmath289 is maximum with @xmath290 whereas one can get a npdl solution at a point of the compositional parameter space if @xmath291 is maximum with @xmath292 . from ( b ) , we have found that @xmath291 is maximum with @xmath292 at @xmath327 , and consequently , we can have a npdl solution at @xmath331 for @xmath322 . next we consider the curve @xmath335 parallel to the curve @xmath14 as shown in fig . [ fig : sol space amp1](a ) . here also , @xmath323 and @xmath324 are defined in the same way as in fig . [ fig : sol space amp](a ) . however , from fig . [ fig : sol space amp1](a ) , we see that there does not exist any ppsw along the curve @xmath335 for @xmath336 . so according to the terminology @xmath285 along the curve @xmath335 for @xmath336 . consequently , from inequality ( [ phi_pn ] ) , we have npdl solution at @xmath337 for @xmath335 . this fact is clear from ( b ) . in the present paper , we have investigated diasws and diadls in a dusty plasma system consisting of adiabatic ions , nonthermal electrons and negatively charged dust grains . investigations have been made by going through the entire solution space of the energy integral by considering the entire range of the parameters involved in the system . our aim is to delimit the parameter @xmath29 depending on the nature of existence of diasws and diadls . therefore , for any physically admissible values of the parameters of the system , specifically , for any value of @xmath10 and any value of @xmath29 , npsw exists for all @xmath4 except @xmath30 , where @xmath2 is the lower bound of mach number @xmath3 , i.e. , solitary wave and/or double layer solutions of the energy integral start to exist for @xmath4 and @xmath178 is the mach number corresponding to a npdl solution . however , if the parameter @xmath10 exceeds a critical value @xmath31 , ppsws exist for all @xmath338 whenever the mach number lies within the interval @xmath33 , where @xmath13 is the upper bound of @xmath3 which is well - defined only when the system supports ppsws . therefore , the coexistence of both ppsws and npsws is possible for all @xmath339 whenever @xmath33 , but npsws still exist for @xmath12 . for nonthermal electrons , npdl starts to occur whenever the nonthermal parameter exceeds a critical value . however this double layer solution is unable to restrict the occurrence of npsws . as a result , two different types of npsws have been observed , in which occurrence of first type of npsw is restricted by @xmath0 whereas the second type npsw exists for all @xmath1 , where @xmath178 is the mach number corresponding to a npdl . a finite jump between the amplitudes of npsws at @xmath340 and at @xmath341 has been observed , where @xmath342 is a sufficiently small positive quantity . the amplitude of npsw for @xmath1 is much greater than the amplitude of the npdl solution at @xmath30 as well as the amplitude of npsw for @xmath343 , i.e. , there is a jump in amplitudes of the npsws above and below the curve @xmath30 . however , there is no jump in amplitudes of npsws above and below the curve @xmath255 . in most of the earlier works , dust ion acoustic solitary structures have been investigated with the help of maxwellian velocity distribution function for electrons . however , the dusty plasma with nonthermally / suprathermally distributed electrons observed in a number of heliospheric environments @xcite . therefore , the present paper gives the complete scenario of dust ion acoustic solitary structures in a dusty plasma system in which lighter species ( 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and k. p. das , _ phys . plasmas _ * 16 * , 073703 ( 2009 ) , * 17 * , 014503 ( 2010 ) . f. verheest , _ phys . plasmas _ * 16 * , 013704 ( 2009 ) . f. verheest , _ phys . plasmas _ * 17 * , 062302 ( 2010 ) . @xmath211 is plotted against @xmath56 for @xmath281 with @xmath233 in ( a ) and @xmath236 in ( b ) . in each of ( a ) and ( b ) , three curves have been drawn corresponding to three different values of @xmath29 , viz . , @xmath29 are 0.2[- - - ] , 0.4[ ] and 0.6[@xmath345 . ] @xmath211 is plotted against @xmath56 for @xmath281 with @xmath233 in ( a ) and @xmath236 in ( b ) . in each of ( a ) and ( b ) , three curves have been drawn corresponding to three different values of @xmath29 , viz . , @xmath29 are 0.2[- - - ] , 0.4[ ] and 0.6[@xmath345 . ] @xmath102 is plotted against @xmath56 for three set of values @xmath237 , @xmath238 and @xmath239 presented in . in figure ( a ) all the curves ( @xmath237 , @xmath238 and @xmath239 ) show the existence of npsw solutions . in figure ( b ) the same curves have been plotted in an interval of @xmath56 close to @xmath116 . ] @xmath102 is plotted against @xmath56 for three different values of @xmath3 , viz . , @xmath178[ ] , @xmath241[- - - ] and @xmath242[@xmath345 in ( a ) and ( b ) . in ( b ) we have shown the region of @xmath56 from @xmath346 to @xmath116 , whereas in ( a ) a region of @xmath56 from @xmath347 to @xmath348 has been shown . the npdl profile corresponding to @xmath30 has been shown in ( c ) , whereas the profiles of npsws corresponding to @xmath243 and @xmath248 have been drawn in ( d ) . profiles in ( d ) show a finite jump between the amplitudes of solitary wave by going from @xmath243 to @xmath248 . in all these four figures we have used @xmath281 , @xmath236 and @xmath228 . ] a graphical presentation of different solitary structures have been given with respect to two different subintervals of @xmath29 within the admissible interval of the mach number @xmath3 for a particular value of @xmath10 which lies in @xmath350 . the curves @xmath254 ( ) and @xmath351 are responsible for the occurrence of two subintervals of @xmath29 . at any point @xmath30 one can always find a npdl solution . this solution space has been drawn for @xmath281 and @xmath352 . for @xmath281 , we have found @xmath31 lies in the neighborhood of @xmath278 and @xmath353 . ] a graphical presentation of different solitary structures have been given with respect to three different subintervals of @xmath29 within the admissible interval of the mach number @xmath3 for a particular value of @xmath10 which lies in @xmath354 . the curves @xmath254 ( ) , @xmath355 ( - @xmath356 - ) and @xmath351 are responsible for the occurrence of three subintervals of @xmath29 . at any point @xmath30 one can always find a npdl solution . the region in @xmath266 has been shown in the inset . this solution space has been drawn for @xmath281 and @xmath357 . for @xmath281 , we have found @xmath358 and @xmath359 with @xmath360 and @xmath361 . ] a graphical presentation of different solitary structures have been given with respect to four different subintervals of @xmath29 within the admissible interval of the mach number @xmath3 for a particular value of @xmath10 which lies in @xmath362 . the curves @xmath254 ( ) , @xmath355 ( - @xmath356 - ) and @xmath351 are responsible for the occurrence of three subintervals of @xmath29 . at any point @xmath30 one can always find a npdl solution . this solution space has been drawn for @xmath281 and @xmath363 . for @xmath281 , we have found @xmath364 and @xmath365 with @xmath366 and @xmath367 . ] a graphical presentation of different solitary structures have been given with respect to four different subintervals of @xmath29 within the admissible interval of the mach number @xmath3 for a particular value of @xmath10 which lies in @xmath368 . the curves @xmath254 ( ) , @xmath355 ( - @xmath356 - ) and @xmath351 are responsible for the occurrence of four subintervals of @xmath29 . at any point @xmath30 one can always find a npdl solution . this solution space has been drawn for @xmath281 and @xmath236 . for @xmath281 , we have found @xmath37 lies in the neighborhood of @xmath280 , and @xmath369 , @xmath370 , @xmath371 . ] here @xmath267 ( ) and @xmath276 have been plotted against @xmath10 for @xmath281 . in this figure , actually we have shown those two curves , viz . , @xmath267 ( ) and @xmath276 which are responsible to divide @xmath10 into several subintervals . this figure is actually the graphical presentation of different solitary structures with respect to different subintervals of @xmath10 within the admissible interval of the nonthermal parameter @xmath29 . ] has been presented , where the curve @xmath14 is omitted from the solution space as presented in . ( b)variation in amplitude ( absolute value ) of both npsw and ppsw have been shown along the curve @xmath322 for @xmath372 . [ fig : sol space amp ] ] has been presented , where the curve @xmath14 is omitted from the solution space as presented in . ( b)variation in amplitude ( absolute value ) of npsw have been shown along the curve @xmath335 for @xmath373 . [ fig : sol space amp1 ] ]
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dust ion acoustic solitary structures have been investigated in an unmagnetized nonthermal plasma consisting of negatively charged dust grains , adiabatic positive ions and nonthermal electrons . for isothermal electrons ,
the present plasma system does not support any double layer solution , whereas for nonthermal electrons , negative potential double layer starts to occur whenever the nonthermal parameter exceeds a critical value .
however this double layer solution is unable to restrict the occurrence of all negative potential solitary waves of the present system . as a result ,
two different types of negative potential solitary waves have been observed , in which occurrence of first type of solitary wave is restricted by @xmath0 whereas the second type solitary wave exists for all @xmath1 , where @xmath2 is the lower bound of mach number @xmath3 , i.e. , solitary structures start to exist for @xmath4 and @xmath5 is the mach number corresponding to a negative potential double layer . a finite jump between the amplitudes of negative potential of solitary waves at @xmath6 and at @xmath7 has been observed , where @xmath8 and @xmath9 . as double layer solution plays an important role for the present system ,
an analytical theory for the existence of double layer has been presented .
a numerical scheme has also been provided to find the value of mach number at which double layer solution exists and also the amplitude of that double layer .
the solitary structures of both polarities can coexist whenever @xmath10 exceeds a critical value , where @xmath10 is the ratio of the unperturbed number density of electrons to that of ions .
although the occurrence of coexistence of solitary structures of both polarities is restricted by @xmath11 , only negative potential solitary wave still exists for all @xmath12 , where @xmath13 is the upper bound of @xmath3 for the existence of positive potential solitary waves only .
qualitatively different solution spaces , i.e. , the compositional parameter spaces showing the nature of existing solitary structures of the energy integral have been found .
these solution spaces are capable of producing new results and physical ideas for the formation of solitary structures whenever one can move the solution spaces through the family of curves parallel to the curve @xmath14 .
| 21,809 | 575 |
silica aerogel is a unique material with a refractive index ( @xmath6 ) in the range between gases and liquids or solids . its refractive index can be easily controlled from @xmath7 to @xmath8 . as a result , the refractive index of the aerogel can be chosen such that for a given momentum interval in the few gev/@xmath3 region charged pions radiate cherenkov photons , while kaons stay below the cherenkov radiation threshold @xcite . in the belle experiment at kek @xcite , a threshold type cherenkov detector ( belle - acc ) @xcite which uses aerogel as a radiator , is operated , providing at @xmath9 a kaon identification efficiency of @xmath10% with a pion missidentification probability of @xmath11% @xcite . a new production method of hydrophobic aerogel with a high transmission length and @xmath6 in the interval between @xmath12 and @xmath13 was developed during the construction period of belle - acc @xcite . the improvement in quality allows the use of an aerogel radiator in a ring imaging cherenkov counter ( rich ) @xcite . in the hermes experiment at desy , a rich counter is used with a dual - radiator ( aerogel and gas ) , and mirrors to focus the cherenkov photons @xcite . a similar detector is also designed for the lhcb experiment at cern @xcite . we are studying the feasibility of a rich counter with an aerogel radiator for the belle - acc in the forward end - cap region @xcite . since this part is now optimized for the pion / kaon separation needed for tagging of the @xmath14 flavor , and covering the momentum range below @xmath15 , separation at high - momentum region of around @xmath5 is not adequate . this kinematic region is , however , very important for the studies of two - body decays such as @xmath16 , @xmath17 . in order to achieve a @xmath18/@xmath19 separation for a wider momentum range , a ring imaging - type of detector is needed . due to spatial restrictions , such a counter has to be of the proximity focusing type . to cover the identification in the lower momentum region ( around @xmath20 ) as well as in the region up to @xmath5 , the aerogel has to have a refractive index around @xmath21 . the first beam test of such a detector was carried out in 2001 at the kek - ps @xmath2 beam line @xcite . these tests used an array of multi - anode pmts ( hamamatsu r5900-m16 ) for photo - detection . the detected number of photoelectrons was @xmath22 per ring for a @xmath23 thick aerogel tile with @xmath21 , and the cherenkov angle resolution per photon was @xmath24 . these results were consistent with expectations . the number of detected photons was , however , rather low , partly because only @xmath25% of the detector surface was covered by the photo - cathodes , and partly because the transmission length of the aerogel with @xmath21 could not be made large enough . for the second beam test , we improved the aerogel transmission by optimizing the materials used in the production process . the active area fraction of the photon detector was increased by employing recently developed flat - panel pmts , hamamatsu h8500 . although this type of pmt is not immune to magnetic field , and therefore can not be applied in the belle spectrometer , we consider this device as an intermediate step in our development . the paper is organized as follows . we first present the experimental set - up with flat - panel pmts , briefly review the improvement in aerogel production , describe the measurements , and finally discuss the results . the photon detector for the tested prototype rich counter employed @xmath26 channel multi - anode pmts ( hamamatsu h8500 , so called flat - panel pmt ) because of their large effective area . 16 pmts were used in a @xmath27 array and aligned with a @xmath28 pitch , as shown in figure [ fig : flat - panel - pmt ] . the surface of each pmt is divided into @xmath26 ( @xmath29 ) channels with a @xmath30 pixel size . therefore , the effective area of photon detection is increased to @xmath31 . at the back of each pmt , an analog memory board is attached to read out multi - channel pmt signals , as described below . among @xmath32 pmts , @xmath11 pmts were delivered in january , 2002 , and the remaining pmts were delivered in october , 2002 . since the manufacture method of the pmt was still under development , they exhibit a large variation in quantum efficiency and gain . the quantum efficiency at 400 nm varies between @xmath32% and @xmath33% ; the gain varies from @xmath34 to @xmath35 when the maximal allowed high voltage of @xmath36 v is applied to the photo - cathode @xcite . the pmts from the later batch show a slightly better performance . the hydrophobic form of the aerogel radiator from novosibirsk @xcite is known to have a long transmission length . however we prefer hydrophobic aerogel than hydrophilic one for the application to a collider experiment . with a low refractive index ( @xmath37 ) , such an aerogel was developed for the belle - acc , and is characterized by a high transmission length ( @xmath38 mm at a wave length of 400 nm ) which was not achieved before . however , the transmission length of aerogel with a higher refractive index of @xmath21 fell below one half the value compared to the aerogel with @xmath39 . therefore , we reexamined the aerogel production technique in a joint development with matsushita electric works ltd . as a result , we found that the important factors determining the transmission length are the solvent and selection of the precursor to be used for its production . originally , we used methyl - alcohol for the solvent , and methyl - silicate as a precursor @xcite . when we applied di - methyl - formamide ( dmf ) @xcite , and changed the supplier of the precursor , we could improve the transmission length of the aerogel . figure [ fig : aero ] shows the refractive indices of aerogel and the relation to transmission length for samples which were used in this beam test . the refractive index was determined by measuring the deflection angle of laser light ( laser diode : @xmath40 ) at a corner of each aerogel tile ; the transmission length was measured with a photo - spectrometer ( hitachi u-3210 ) . in addition to the samples produced with the new technique at matsushita electric works ltd . and chiba university , samples from binp ( novosibirsk ) were tested @xcite ; for comparison , we also tested the samples used in the previous beam test . the thicknesses of the prepared aerogel samples ranged from @xmath41 to @xmath42 . various thickness of up to about @xmath43 were tested by stacking these samples . note that in the production of the aerogel samples at binp propenol was used as the solvent , and the resulting aerogel was hydrophilic . also note that the matsushita aerogel samples produced with the new technique have a very similar transmission length as the binp samples . the transmission length for @xmath44 samples used in the first beam test was around @xmath45 mm , but was increased to @xmath46 mm for matsushita s sample with the new production method . for the beam test , pions with momenta between @xmath47 and @xmath5 were used . beside the rich detector under study , counters for triggering , tracking and particle identification were employed . the set up of the aerogel rich is shown in figure [ fig : rich ] . two rich counters were placed in a light - shield box and tested simultaneously . each rich was composed of a layer of aerogel radiator and a photo - detection plane , parallel to the radiator face at a distance of @xmath48 . the upstream cherenkov counter was the detector under study ; the downstream counter was the one employed in the previous beam test . since the latter uses a well - known photo - detector , multi - anode pmts hamamatsu r5900 , we regarded it as a reference . particle identification was done to remove particles other than pions . two co@xmath49 gas cherenkov counters in the beam line were used to exclude electrons . also , an aerogel counter was equipped and used to exclude protons for the high - momentum region . this detector was also used to exclude muons for the low - momentum region around @xmath47 . the particle trajectories were measured with multi - wire proportional chambers ( mwpc ) at the upstream and downstream ends of the light - shield box . these @xmath50 mwpcs , with @xmath51 m diameter , gold - plated tungsten anode wires at @xmath52 pitch and with @xmath53 ar + @xmath54 @xmath55 gas flow , were read out by delay lines on the x and y cathode strips . the trigger signal was generated as a coincidence of signals from several @xmath50 plastic scintillation counters and anode signals from the mwpcs to ensure valid tracking information . for the beam test , a new read - out system was designed by using analog memory chips . the analog memory chip is based on a chip developed by h. ikeda @xcite for a cosmic - ray experiment . we borrowed the chips from nasda ( national space development agency of japan ) , and developed the chip control system . in the analog memory chip , the signals of 32 channels are preamplified , sampled in @xmath56s intervals , and stored in an 8 steps deep analog pipeline . figure [ fig : rich - readout ] shows a schematic view of the readout system with these analog memories . two 32 channel analog memories are attached to each 64 channel pmt . the memories corresponding to four pmts are controlled by a 256 channel memory controller . when the gate pulse is formed from the trigger signal , a control signal is sent from the controller to the analog memories . the difference in the value of the analog memory between the latest and the first memory content is fed to the output . the obtained output values of 256 channels are clocked into one signal train with a period of 10 @xmath57s per channel . each analog memory controller outputs the serial signal together with synchronized control signals . these signals are then read by a 12-bit vme adc ( dsp8112 , mtt co. ) with a conversion time of 5 @xmath57s . a reference rich was instrumented with multi - anode pmts , hamamatsu r5900-m16 , the same photon detector as used in the previous test @xcite . the quantum efficiency of the pmts is around 26% ( at 400 nm ) , and the gain was around @xmath35 with @xmath58 v applied to the photo - cathode . the pmts were grouped in a @xmath59 array at a @xmath60 mm pitch . due to a limited number of available pmts and read - out channels , only a part of the cherenkov ring was covered with photon detectors . most of the test measurements were performed with a @xmath61 beam at @xmath62 . to systematically evaluate the detector performance , data were taken with different aerogel samples with various transmission lengths and thicknesses . data were also taken by varying the @xmath61 momentum in the range from @xmath47 to @xmath63 . a few typical events are displayed in figure [ fig : rich - eventdisplay ] . the hits on pmts can be associated with the expected position of the cherenkov ring . the hit near the center of the ring is due to cherenkov radiation generated by the beam particle in the pmt window . the distribution of accumulated hits is shown in figure [ fig : typical1 ] . cherenkov photons from the aerogel radiator are clearly seen with a low background level . the background hit distribution on the photon detector is consistent with the assumption that it originates from cherenkov photons which were rayleigh scattered in the radiator . the pulse - height distribution of the cherenkov photons detected in one of the flat - panel pmt is shown in figure [ fig : adc ] . the raw data were corrected as follows . a common - mode fluctuation of the base line was subtracted and signals due to cross - talk in the read - out electronics were removed . the signal mainly containing one photoelectron is clearly separated from the pedestal peak . note , however , that this distribution differs considerably from tube to tube because of the large variation in performance , as described before . for further analysis we also applied a threshold cut to suppress the pedestal noise contribution . figure [ fig : typical2](a ) shows a typical distribution of the cherenkov - angle for single photons . the angular resolution was obtained from a fit of this distribution with a gaussian signal and a linear function for the background . figure [ fig : s2n ] shows the resolution in the cherenkov angle for the @xmath61 beam at @xmath62 and @xmath64 mm thick aerogel samples . the resolution was around @xmath65 mrad , independent of the refractive index . the main contributions to the resolution of the cherenkov angle come from the uncertainty in the emission point and from the pixel size of the pmt . the first contribution is estimated to be @xmath66 , where @xmath67 is the aerogel thickness , @xmath68 is the cherenkov angle and @xmath69 is the distance from an average emission point in the aerogel to the surface of the pmt . the second contribution is @xmath70 , where @xmath71 is the pixel size . the measured variation of the resolution with the thickness of aerogel is shown in figure [ fig : s2d ] . by comparing the measured resolution and the expected values , we observed a rather good agreement . there was , however , a discrepancy between the two , which can be accounted for by a contribution of about @xmath72 mrad . the discrepancy could arise from the effect of aerogel ( non - flat aerogel surface and possible non - uniformities in the refractive index due to position variation and chromatic dispersion ) , which are subject to further investigation . the uncertainty in the track direction is expected to be negligible at @xmath62 , but increases considerably at low momenta ( @xmath47 ) due to the effect of multiple - scattering , as can be seen in figure [ fig : s2p ] . figure [ fig : typical2](b ) shows a typical distribution of the number of hits within @xmath73 from the average cherenkov angle . the number of hits for the signal region was estimated by subtracting the background from the fits to the cherenkov - angle distribution . the number of detected photons ( @xmath74 ) depends on the aerogel thickness and the effect of scattering . it is expressed as + @xmath75 , where @xmath76 is the transmission length of the aerogel at an average wave length of @xmath77 nm and @xmath78 is quantum efficiency of the pmt . figure [ fig : n2d ] shows the dependence of the number of detected photons on the aerogel thickness . as expected from the above expression , the number of photons does not linearly increase with the aerogel thickness , but saturates due to the scattering effect in aerogel . figure [ fig : n2tr ] shows the dependence of the number of photons with transmission length . from the figure the benefit of the improvement in the transmission length of the @xmath1 aerogel from around @xmath45 mm , as used in the previous beam test , to @xmath46 mm using the new production technique becomes evident . the dependence on the pion momentum , displayed in figure [ fig : n2p ] , is fitted with the form expected from the cherenkov relations , and shows a good agreement . for pions with momenta above @xmath79 , the number of detected cherenkov photons was typically around 6 for aerogel samples with @xmath1 . the performance of the rich counter under study was compared in the same set - up with the performance of the reference counter with a well - known photon detector , hamamatsu r5900-m16 multi - anode pmts . since the two counters have a different active area fraction ( @xmath80% for the flat - panel pmts , and @xmath25% for the r5900-m16 pmts ) and a different acceptance , the comparison of the photon yields was made by normalizing to the full active surface . while the flat - panel yield for a particular case was @xmath81 , which resulted in @xmath82 if extrapolated to the full active area , the corresponding number for the r5900-m16 was @xmath83 . it appears that this difference is mainly due to the rather low quantum efficiency and amplification of some of the flat - panel tubes employed . this , in turn , causes inefficiencies in single photon detection with a given threshold . if the best tube in the set is normalized to the full acceptance , the corresponding number increases to @xmath84 , and we would expect about 8 photons per ring . finally , we estimate the performance of pion / kaon separation in the momentum range of around @xmath5 , which is of considerable importance for the belle experiment . if we take into account a typical measured value for the single - photon angular resolution , @xmath85 mrad , and the number of detected photons @xmath86 , typical for 20 mm thick aerogel samples with @xmath21 , we can estimate the cherenkov angle resolution per track to be @xmath87 mrad . this naive estimate is also confirmed by the direct measurement shown in figure [ fig : sep_pik_4gev ] . here , the track - by - track cherenkov angle is calculated by taking the average of the angles measured for hits around the predicted position of the cherenkov ring . from this we can deduce that at @xmath5 , where the difference of cherenkov angles for pions and kaons is @xmath88 mrad , a @xmath89 separation between the two is possible . as an additional cross check , we have also collected data with the pion beam of @xmath90 , which can be used to represent a kaon beam of @xmath5 ( apart from a slightly larger sigma due to multiple scattering ) . as can be seen from figure [ fig : sep_pik_4gev ] , the two peaks are well separated . thus , the proximity focusing aerogel rich seems to be promising for the upgrade of the belle pid system at the forward region . we report on the test beam results of a proximity focusing rich using aerogel as the radiator . to obtain larger photoelectron yields , we used flat - panel multi - anode pmt with a large effective area , and aerogel samples produced with a recently developed method which have a higher transmission length than before . we also developed a multi - channel pmt read - out system using analog memory chips . a clear cherenkov ring from the aerogel radiator could be observed , and the number of photons was enhanced compared to that in previous tests . we performed a systematic study of the detector using various samples of the aerogel . the typical angular resolution was around @xmath65 mrad and the number of detected photoelectrons was around @xmath72 . the pion / kaon separation at @xmath5 is expected to be around @xmath89 . however , we still have some issues which have to be solved for implementation in the belle spectrometer . the most important item is the development of a pmt which can be operated under a strong magnetic field ( 1.5 t ) . an example of a candidate for such a device is a multi - anode hybrid photodiode ( hpd ) or hybrid avalanche photodiode ( hapd ) . of course , for a good candidate , its ability to efficiently detect single photons on a large active area has to be demonstrated . the other item is mass production of the aerogel tiles . while we have demonstrated that the new production method significantly increases the transmission length of the @xmath21 aerogel , the production method has to be adapted to stable good - quality manufacturing . we will study these items at the next stage towards construction of a real detector . we would like to thank the kek - ps members for operation of accelerator and for providing the beam to the @xmath2 beam line . we also thank h. ikeda ( kek ) and the meisei co. for their help in preparing the read - out electronics , the matsushita electric works ltd . for the good collaboration in developing the new aerogel type , and hamamatsu photonics k.k . for their support in equipping the photon detector . we also thank a.bondar ( binp , novosibirsk ) for providing us excellent aerogel samples , and dr . t.goka of nasda for providing us their read - out chips . one of the authors ( t.m . ) is grateful to fellowships of the japan society for the promotion of science ( jsps ) for young scientists . this work was supported in part by a grand - in - aid for scientific research from jsps and the ministry of education , culture , sports , science and technology under grant no . 13640306 , 14046223 , 14046224 , and in part by the ministry of education , science and sports of the republic of slovenia . m.cantin et al . instr . meth . * 118 * ( 1974 ) 177 - 182 a.abashian et al . instr . and meth . a * 479 * ( 2002 ) 117 t.sumiyoshi et al . instr . and meth . a * 433 * ( 1999 ) 385 - 391 t.iijima et al . instr . and meth . a * 453 * ( 2000 ) 321 i. adachi et al . , instr . and meth . a * 355 * ( 1995 ) 390 ; t. sumiyoshi et al . , j. non - cryst . solids * 225 * ( 1998 ) 369 d.e.fields et al . instr . meth . a * 349*(1994 ) 431 - 437 ; r. de leo et al . , instr . and meth . a * 401 * ( 1997 ) 187 n.akopov et al . instr . meth . a * 479 * ( 2002 ) 511 - 530 t.ypsilantis and j.seguinot , nucl . meth . a * 368 * ( 1995 ) 229 - 233 t.iijima , `` aerogel cherenkov counter in imaging mode '' , jps meeting , tokyo metropolitan university , september 1997 . i. adachi et al . , `` test of a proximity focusing rich with aerogel as radiator '' , proceedings for the ieee nuclear science symposium , norfolk , va , november 10 - 15 , 2002 , trans . 50 ( 2003 ) 1142 , hep - ex/0303038 , ; t.iijima et al . instr . meth . a * 502*(2003 ) 231 - 235 hamamatsu photonics k.k . matsushita electric works ltd . has a japanese patent ( no . 2659155 ) for usage of dmf as solvent to make aerogel . buzykaev et al . instr . meth . a * 433 * ( 1999 ) 396 - 400 h.ikeda et al . a * 372 * ( 1996 ) 125 - 134 of the mean cherenkov angle . data were corrected with the procedure described in the text . in this figure , the pulse - height distribution for the high sensitive pmt is shown . for further analysis , we used the hits above a threshold adc value , 120 . , width=302 ] ) for @xmath64 mm thick aerogel samples as a function of the transmission length . @xmath74 is corrected for the refractive index to @xmath21 and @xmath39 respectively . the symbols correspond to the data and the curves are fits described in the text.,width=302 ] and @xmath90 . pions at @xmath90 are used to represent the kaon beam of @xmath5 . the angular resolutions for @xmath63 and @xmath90 are @xmath92 mrad and @xmath93 mrad and two peaks are separated by @xmath94 . , width=302 ]
|
a proximity focusing ring imaging cherenkov detector using aerogel as the radiator has been studied for an upgrade of the belle detector at the kek - b - factory .
we constructed a prototype cherenkov counter using a @xmath0 array of 64-channel flat - panel multi - anode pmts ( hamamatsu h8500 ) with a large effective area .
the aerogel samples were made with a new technique to obtain a higher transmission length at a high refractive index ( @xmath1 ) .
multi - channel pmts are read - out with analog memory chips .
the detector was tested at the kek - ps @xmath2 beam line in november , 2002 . to evaluate systematically the performance of the detector ,
tests were carried out with various aerogel samples using pion beams with momenta between 0.5 gev/@xmath3 and 4 gev/@xmath3 .
the typical angular resolution was around 14 mrad , and the average number of detected photoelectrons was around 6 .
we expect that pions and kaons can be separated at a 4@xmath4 level at @xmath5 .
aerogel , flat - panel pmt , ring imaging cherenkov counter , proximity focusing , particle identification , belle 29.40.ka
| 6,622 | 332 |
equations that describe ideal fluid and plasma dynamics in terms of eulerian variables are hamiltonian in terms of noncanonical poisson brackets , degenerate brackets in noncanonical coordinates . because of degeneracy , such poisson brackets possess casimir elements , invariants that have been used to construct variational principles for equilibria and stability . however , early on it was recognized that typically there are not enough casimir elements to obtain all equilibria as extremal points of these variational principles . in @xcite it was noted that this casimir deficit is attributable to rank changing of the operator that defines the noncanonical poisson bracket . thus , a mathematical study of the kernel of this operator is indicated , and this is the main purpose of the present article . recognizing a hamiltonian flow as a differential operator , the point where the rank of the poisson bracket changes is a singularity , from which singular ( or intrinsic ) solutions stem . when we consider a hamiltonian flow on a function space , the problem is an infinite - dimension generalization of the theory of singular differential equations ; the derivatives are functional derivatives , and the construction of singular casimir elements amounts to integration in an infinite - dimension space . in order to facilitate this study , it is necessary to place the noncanonical hamiltonian formalism on a more rigorous footing , and this subsidiary purpose is addressed in the context of euler s equation of fluid mechanics , although the ideas presented are of more general applicability than this particular example . we start by reviewing finite - dimensional canonical and noncanonical hamiltonian mechanics , in order to formulate our problem . these dynamical system have the form @xmath0 where @xmath1 denotes a set of phase space coordinates , @xmath2 is the hamiltonian function with @xmath3 its gradient , and the @xmath4 matrix @xmath5 ( variously called e.g. the cosymplectic form , poisson tensor , or symplectic operator ) is the essence of the poisson bracket and determines important lie algebraic properties@xcite ( see also remark[remark : lie - poisson ] ) . for canonical hamiltonian systems of dimension @xmath6 the matrix @xmath5 has the form @xmath7 _ noncanonical _ hamiltonian systems allow a @xmath8-dependent @xmath9 ( assumed here to be a holomorphic function ) to have a kernel , i.e. @xmath10 may be less than @xmath11 and may change as a function of @xmath12 . from ( [ hamilton_eq_1 ] ) it is evident that equilibrium points of the dynamics , i.e. points for which @xmath13 , satisfy @xmath14 however , in the noncanonical case these may not be the only equilibrium points of a given hamiltonian @xmath15 , because degeneracy gives rise to _ casimir elements _ @xmath16 , nontrivial ( nonconstant ) solutions to the differential equation @xmath17 given such a @xmath16 , replacement of the hamiltonian by @xmath18 does not change the dynamics . thus , an extremal point of @xmath19=0 \label{stationary_points}\ ] ] will also give an equilibrium point . note , in light of the homogeneity of ( [ casmir-1 ] ) an arbitrary multiplicative constant can be absorbed into @xmath20 and so ( [ stationary_points ] ) can give rise to families of equilibrium points . if @xmath21 , ( [ casmir-1 ] ) has only the trivial solution @xmath22 constant . if @xmath23 and @xmath24 is constant , then ( [ casmir-1 ] ) has @xmath25 functionally independent solutions ( lie - darboux theorem ) . the problem becomes more interesting if there is a _ singularity _ where @xmath10 changes : in this case we have a singular ( hyperfunction ) casimir element ( see fig.[fig : foliation ] ) . for example , consider the one - dimensional system where @xmath26 ( @xmath27 ) . at @xmath28 @xmath29 drops to 0 , and this point is a singular point of the differential equation @xmath30 . the singular casimir element is @xmath31 , where @xmath32 is the heaviside step function . the codimension of the casimir foliation changes , resulting in a singular casimir leaf ( with the determining casimir element being a _ hyperfunction _ ) . in the figure the codimension of the casimir leaf at the singular point is two , hence the singular point is an equilibrium point . in higher - dimensional ( infinite - dimensional ) phase space , a singular point may have far richer structure . in sec . [ sec : casimirs ] , we delineate how a singular point is created in an infinite - dimensional phase space by examining an infinite - dimensional hamiltonian system of eulerian fluid ( to be formulated in sec . [ sec : euler_equation ] ) . in sec . [ sec : equilibrium ] , we will study the structure of the singular point ( which is still infinite - dimensional ) by examining equilibrium points.,scaledwidth=80.0% ] we generalize ( [ hamilton_eq_1 ] ) further to include infinite - dimension systems . let @xmath33 be the state variable , where for now @xmath34 is some unspecified function space , @xmath35 be a linear antisymmetric operator in @xmath34 that generally depends on @xmath36 ( for a fixed @xmath36 , @xmath35 may be regarded as a linear operator @xmath37 see remark [ remark : j_operator ] below ) , and @xmath38 be a functional @xmath39 . introducing an appropriate functional derivative ( gradient ) @xmath40 , we consider evolution equations of the form @xmath41 where @xmath42 . a casimir element @xmath43 ( a functional @xmath39 ) is a nontrivial solution to @xmath44 we may solve ( [ casmir-2 ] ) by two steps : 1 . find the kernel of @xmath35 , i.e. , solve a `` linear equation '' ( cf . remark [ remark : j_operator ] ) @xmath45 to determine @xmath46 for a given @xmath36 , which we write as @xmath47 . 2 . `` integrate '' @xmath47 with respect to @xmath36 to find a functional @xmath43 such that @xmath48 . as evident in the above finite - dimension example , _ step-1 _ should involve `` singular solutions '' if @xmath35 has _ singularities_. then , _ step-2 _ will be rather nontrivial for _ singular casimir elements _ , we will need to generalize the notion of _ functional derivative_. as mentioned above , the present paper is devoted to such an extension of the notion of casimir elements in infinite - dimensional noncanonical hamiltonian systems . specifically , we invoke the euler equation of ideal fluid mechanics as an example , but much carries over to other systems since many fluid and plasma systems share the same operator @xmath49 . in sec . [ sec : euler_equation ] , we will describe the hamiltonian form of euler s equation , which places on a more rigorous footing the formal calculations of @xcite . in sec . [ subsec : kernel ] , we will analyze the kernel of the corresponding poisson operator @xmath35 and its singularity . a singular casimir element and its appropriate generalized functional derivative ( gradient ) will be given in sec . [ subsec : casimirs ] . the relation between the ( generalized ) casimir elements and equilibrium points ( stationary ideal flows ) will be discussed in sec . [ sec : equilibrium ] . generalizing ( [ stationary_points ] ) to an infinite - dimensional space , we may find an extended set of equilibrium points by solving @xmath50=0 . \label{stationary_points2}\ ] ] we note , however , it is still uncertain whether or not every equilibrium point can be obtained from casimir elements in this way . for example , let us consider a simple hamiltonian @xmath51 ( in appendix a , the hamiltonian of the euler equation is given in terms of the velocity field @xmath52 , which here corresponds to the state variable @xmath36 ) . then , @xmath53 and ( [ hamilton_eq_2 ] ) reads @xmath54 the totality of nontrivial equilibrium points is @xmath55 . for @xmath56 to be characterized by ( [ stationary_points2 ] ) , which now simplifies to @xmath57 , we encounter the `` integration problem , '' i.e. , we have to construct @xmath43 such that @xmath48 for every @xmath58 this may not be always possible . on the other hand , even for a given @xmath43 , the `` nonlinear equation '' @xmath57 does not necessarily have a solution in sec.[subsec : no - solution ] we will show some examples of no - solution equations . while we leave this question ( the `` integrability '' of all equilibrium points ) open , the present effort shows it is sometimes possible and provides a more complete understanding of the stationary states of infinite - dimensional dynamical systems . in sec . [ sec : conclusion ] we give some concluding remarks . [ remark : lie - poisson ] we endow the phase space @xmath34 of state variable @xmath36 ( @xmath12 if @xmath34 is finite - dimensional ) with an inner product @xmath59 . let @xmath60 be an arbitrary smooth functional ( function if @xmath34 is finite - dimensional ) . if @xmath36 obeys ( [ hamilton_eq_2 ] ) in a hilbert space ( or ( [ hamilton_eq_1 ] ) for finite - dimensional systems ) , the evolution of @xmath61 obeys @xmath62 $ ] , where @xmath63 = ( \partial_{{u } } f({u}),\mathcal{j}({{u } } ) \partial_{{u } } h({u}))\ ] ] is an antisymmetric bilinear form . if this bracket @xmath64 $ ] satisfies the jacob identity , it defines a poisson algebra , a lie algebra realization on functionals . a casimir element @xmath20 is a member of the _ center _ of the poisson algebra , i.e. , @xmath65=0 $ ] for all @xmath66 . the bracket defined by the poisson operator of sec.[subsec : poisson ] satisfies the jacobi s identity @xcite . the jacobi identity is satisfied for all lie - poisson brackets , a class of poisson brackets that describe matter that are built from the structure constants of lie algebras ( see , e.g. , @xcite ) . for finite - dimensional systems there is a beautiful geometric interpretation of such brackets where phase space is the dual of the lie algebra and surfaces of constant casimirs , coadjoint orbits , are symplectic manifolds . unfortunately , in infinite - dimensions , i.e. , for nonlinear partial differential equations , there are functional analysis challenges that limit this interpretation ( see , e.g. , @xcite ) . ( for example , for the incompressible euler fluid equations the group is that of volume preserving diffeomorphisms . ) in terms of this interpretation , the analysis of the present paper can be viewed as a local probing of the coadjoint orbit . [ remark : j_operator ] in ( [ hamilton_eq_2 ] ) , the operator @xmath35 must be evaluated at the common @xmath36 of @xmath67 , thus @xmath68 is a nonlinear operator with respect to @xmath36 . however , the application of @xmath35 ( or @xmath69 ) may be regarded as a linear operator in the sense that @xmath70 or @xmath71 = a\mathcal{j}(u)\partial_{u } f(u)+b\mathcal{j}(u)\partial_{u } g(u).\ ] ] note that @xmath72 ( for @xmath73 ) is _ not _ @xmath74 . investigation of the hamiltonian form of ideal fluid mechanics has a long history . its essence is contained in lagrange s original work @xcite that described the fluid in terms of the `` lagrangian '' displacement . important subsequent contributions are due to clebsch@xcite and kirchhoff @xcite . in more recent times , the formalism has been addressed in various ways by many authors ( e.g.@xcite ) . here we follow the noncanonical poisson bracket description as described in @xcite . analysis of the kernel of @xmath75 requires careful definitions . for this reason we review rigorous results about euler s equation in sec . [ subsec : vorticity_equation ] , followed by explication of various aspects of the hamiltonian description in secs . [ subsec : hamiltonian ] [ subsec : hamiltonian - form ] . this places aspects of the noncanonical poisson bracket formalism of@xcite on a more rigorous footing ; of particular interest , of course , is the poisson operator @xmath75 that defines the poisson bracket . euler s equation of motion for an incompressible inviscid fluid is @xmath76 where @xmath77 is a bounded domain in @xmath78 ( @xmath79 or 3 ) with a sufficiently smooth ( @xmath80-class ) boundary @xmath81 , @xmath82 is the unit vector normal to @xmath81 , @xmath52 is an @xmath24-dimensional vector field representing the velocity field , and @xmath83 is a scalar field representing the fluid pressure ( or specific enthalpy ) ; all fields are real - valued functions of time @xmath84 and the spatial coordinate @xmath85 . we may rewrite ( [ euler-1 ] ) as @xmath86 where @xmath87 is the vorticity and @xmath88 is the total specific energy . the curl derivative of ( [ euler-2 ] ) gives the _ vorticity equation _ we prepare basic function spaces pertinent to the mathematical formulation of the euler equation . let @xmath90 be the hilbert space of lebesgue - measurable and square - integrable real vector functions on @xmath77 , which is endowed with the standard inner product @xmath91 and norm @xmath92 . we will also use the standard notation for sobolev spaces ( for example , see@xcite ) . we define @xmath93 where @xmath94 denotes the trace of the normal component of @xmath52 onto the boundary @xmath95 , which is a continuous map from @xmath96 to @xmath97 . we have an orthogonal decomposition @xmath98 every @xmath99 satisfies the conditions ( [ incompressibility ] ) and ( [ bc ] ) , thus we will consider ( [ euler-2 ] ) to be an evolution equation in the function space @xmath100 ( cf . appendix a ) . hereafter , we assume that the spatial domain has dimension @xmath101 , and @xmath102 is a smoothly bounded and simply connected ( genus=0 ) region . from @xmath103 , i.e. express @xmath104 , where the dimension of the subspace @xmath105 is equal to the genus of @xmath77 . then , the projection of @xmath52 , which obeys ( [ euler-1])-([bc ] ) , is shown to be constant throughout the evolution , whence we may regard ( [ euler-2 ] ) as an evolution equation in @xmath106 . ] for convenience in formulating equations , we immerse @xmath102 in @xmath107 by adding a `` perpendicular '' coordinate @xmath8 , and we write @xmath108 . [ lemma : clebsch_representation ] every two - dimensional vector field @xmath52 satisfying the incompressibility condition ( [ incompressibility ] ) and the vanishing normal boundary condition ( [ bc ] ) can be written as @xmath109 with a single - value function @xmath110 such that @xmath111 , i.e. , @xmath112 for the convenience of the reader , we sketch the proof of this frequently - used lemma . is sometimes called a clebsch potential . to represent an incompressible flow of dimension @xmath24 , we need @xmath113 _ clebsch potentials _ @xmath114 , where each @xmath115 does not have a uniquely determined boundary condition@xcite . hence , the vorticity representation is not effective in higher dimensions . in appendix a , we invoke another method to eliminate the pressure term and formulate the problem in an alternative form , which applies in general space dimension . see e.g. @xcite for discussions of various potential representations . ] evidently , @xmath116=0 $ ] , and @xmath117 if @xmath118 . thus , the linear space @xmath119 is contained in @xmath100 . and , the orthogonal complement of @xmath34 in @xmath103 contains only the zero vector : suppose that @xmath120 satisfies @xmath121 by the generalized stokes formula , we find @xmath122 . since @xmath123 has only the @xmath124 component , ( [ clebsch-1 ] ) implies @xmath125 . since @xmath120 , we also have @xmath126 and @xmath127 . in a simply connected @xmath77 , the only such @xmath52 is the zero vector . hence , we have ( [ clebsch_representation ] ) . using the representation ( [ clebsch_2form ] ) , we may formally calculate @xmath128 the vorticity equation ( [ vorticity_eq ] ) simplifies to a single @xmath124-component equation : , the two - dimensional vorticity equation ( [ vorticity_eq-2d ] ) has the form of a nonlinear liouville equation . the corresponding hamiltonian equations ( characteristic odes ) are given in terms of a streamfunction @xmath110 as @xmath129 . by the boundary condition @xmath130 , the characteristic curves are confined in @xmath77 . hence , we do not need ( or , can not impose ) a boundary condition on @xmath131 , and the single equation ( [ vorticity_eq-2d ] ) determines the evolution of @xmath131 and the velocity field is obtained from @xmath132 . ] @xmath133 where @xmath134 and @xmath135 is the inverse map of @xmath136 with the dirichlet boundary condition , i.e. , @xmath137 gives the solution of the laplace equation @xmath138 as is well - known , @xmath139 is a self - adjoint compact operator . for @xmath118 , we define @xmath140 as a member of @xmath141 , the dual space of @xmath142 with respect to the inner - product of @xmath90 . the inverse map ( weak solution ) , then , defines @xmath143 . [ lemma : weak ] we regard the vorticity equation ( [ vorticity_eq-2d ] ) as an evolution equation in @xmath141 , i.e. , we consider the weak form : @xmath144 by the relations @xmath145 , @xmath146 , and @xmath147 , ( [ vorticity_eq-2d - math ] ) is equivalent to the euler equation ( [ euler-2 ] ) as an evolution equation in @xmath103 . in the topology of @xmath103 , the euler equation ( [ euler-2 ] ) reads as @xmath148 by ( [ decomposition_of_l2 ] ) , the left - hand side of ( [ euler-2 ] ) reduces to @xmath149 . by lemma[lemma : clebsch_representation ] , we may put @xmath150 with @xmath151 . finally , plugging this representation into ( [ euler-2 ] ) , we obtain @xmath152 hence , ( [ euler-2 ] ) is equivalent to ( [ vorticity_eq-2d - math ] ) . now we consider the hamiltonian form of the vorticity equation ( [ vorticity_eq-2d ] ) to be precise , its `` weak form '' ( [ vorticity_eq-2d - math ] ) ( cf . appendix a which treats the euler equation of ( [ euler-2 ] ) ) . first we note that the natural choice for the hamiltonian is @xmath153 , the `` energy '' of the flow @xmath52 . using @xmath146 , we may rewrite @xmath154 . selecting the vorticity @xmath131 as the state variable , we define ( by relating @xmath145 ) @xmath155 which is a continuous functional on @xmath141 . this is equivalent to the square of the norm of @xmath141 , i.e. , the _ negative norm _ induced by @xmath142 . next , we consider the gradient of a functional in function space . let @xmath156 be a functional defined on a hilbert space @xmath34 . a small perturbation @xmath157 ( @xmath158 , @xmath159 ) will induce a variation @xmath160 . if there exists a @xmath161 such that @xmath162 for every @xmath163 , then we define @xmath164 , and call it the gradient of @xmath156 . evidently , the variation @xmath165 is maximized , at each @xmath36 , by @xmath166 . the notion of gradient will be extended for a class of `` rugged '' functionals , which will be used to define singular casimir elements in sec.[subsec : casimirs ] . as for the hamiltonian , however , we may assume it to be a smooth functional . the pertinent hilbert space is @xmath90 , on which the hamiltonian @xmath167 is differentiable ; using the self - adjointness of @xmath135 , we obtain @xmath168 note that the gradient @xmath169 may be evaluated for every @xmath170 with the value in @xmath142 . finally , we describe the noncanonical poisson operator @xmath171 of @xcite . formally , we have @xmath172 \psi = \ { \omega , \psi \}\ , , \label{vorticity_eq - j}\ ] ] which indicates @xmath131 must be a `` differentiable '' function . however , we will need to reduce this regularity requirement on @xmath131 . thus , we turn to the _ weak formulation _ that is amenable to the interpretation of the evolution in @xmath141 ( see lemma[lemma : weak ] ) . formally , we may calculate @xmath173 with the right - hand - side finite ( well - defined ) for @xmath174 and @xmath175 . in fact , @xmath176 where @xmath177 . hence , we may consider the right - hand - side of ( [ j - operator-1 ] ) to be a bounded linear functional of @xmath151 , with @xmath131 and @xmath178 acting as two parameters . we denote this by @xmath179 . by this functional , we `` define '' @xmath180 on the left - hand - side of ( [ j - operator-1 ] ) as a member of @xmath181 , i.e. , we put @xmath182 for a given @xmath174 , we may consider that @xmath171 is a bounded linear map operating on @xmath178 , i.e. , @xmath183 . evidently , @xmath184 , i.e. , @xmath171 is antisymmetric . combining the above definitions of the hamiltonian @xmath167 , the gradient @xmath185 , and the noncanonical poisson operator @xmath171 , we can write the vorticity equation ( [ vorticity_eq-2d ] ) in the form @xmath186 as remarked in lemma[lemma : weak ] , ( [ vorticity_eq-2dh ] ) is an evolution equation in @xmath141 ( cf . appendix a for the @xmath187 formulation ) . for every fixed @xmath174 , @xmath171 may be regarded as a bounded linear map of @xmath188 , where the bound changes as a function of @xmath131 . and @xmath169 is a bounded linear map of @xmath189 . hence , each element composing the right - hand side ( generator ) of the evolution equation ( [ vorticity_eq-2dh ] ) is separately regular . however , their nonlinear combination can create a problem : as noted in remark [ remark : j_operator ] , we must evaluate the operator @xmath171 at the common @xmath131 of @xmath169 . while @xmath169 can be evaluated for every @xmath170 with its range @xmath190 = domain of @xmath171 , _ if _ @xmath171 is defined ; however , we can define the operator @xmath171 only for @xmath174 . the difficulty of this nonlinear system is now delineated by the singular behavior of the poisson operator @xmath171 as a function of @xmath131 if the orbit @xmath191 ( in the function space @xmath141 ) runs away so as to increase @xmath192 , the evolution equation ( [ vorticity_eq-2dh ] ) will breakdown . to match the combination of @xmath171 and @xmath169 , the domain of the total generator @xmath193 must be restricted in @xmath194 . fortunately , this domain is not too small ; the regular ( classical ) solutions for an appropriate initial condition lives in this domain , i.e. , if a sufficiently smooth initial condition is given , the orbit stays in the region where @xmath192 is bounded @xcite . we begin with a general representation of the kernel of the noncanonical poisson operator @xmath171 , which will be a subset of its domain @xmath142 ( see sec . [ subsec : poisson ] ) . [ lemma : kernel ] for a given @xmath174 , @xmath195 is an element of @xmath196 , iff there is @xmath197 such that @xmath198 this implies that @xmath199 by the definition ( [ vorticity_eq - j ] ) , @xmath200 implies @xmath201 in the topology of @xmath141 , i.e. , @xmath202 by lemma[lemma : clebsch_representation ] , ( [ kernel ] ) implies that @xmath203 remembering ( [ decomposition_of_l2 ] ) , we obtain ( [ kernel - general ] ) and ( [ kernel - general ] ) . to construct a casimir element from @xmath204 , we will need a more `` explicit '' relation between @xmath131 and @xmath178 . we will show how such a relation is available for a sufficiently regular @xmath131 . let us start by assuming @xmath205 . then , we may evaluate @xmath180 as @xmath206 . therefore , @xmath195 belongs to @xmath207 , iff @xmath208 . \label{kernel - general''}\ ] ] equation ( [ kernel - general ] ) implies that two vectors @xmath209 and @xmath210 must align almost everywhere in @xmath77 , excepting any region " in which one of them is zero . such a relationship between @xmath131 and @xmath178 can be represented , by invoking a certain scalar @xmath211 , as @xmath212 the simplest solution is given by @xmath213 ( i.e. , @xmath214 identity ) . in later discussion , we shall invoke a nontrivial @xmath215 to represent a wider class of solutions . we note that the condition @xmath195 implies the boundary condition @xmath216 . if @xmath217 constant on some @xmath218 , integrating ( [ kernel - general ] ) with this boundary condition yields @xmath219 along every contour of @xmath131 which intersects @xmath220 [ the contours of @xmath131 are the cauchy characteristics of ( [ kernel - general ] ) , and @xmath221 poses a non - characteristic _ initial condition _ ] . we denote by @xmath222 the largest region in @xmath77 ( not necessarily a connected set ) which is bounded by a level set ( contour ) of @xmath131 . see fig . [ fig : schematic ] . if @xmath223 constant , @xmath222 is smaller than @xmath77 , and then , every level set of @xmath131 in @xmath224 intersects @xmath95 . hence , @xmath225 closure of @xmath226 . we shall assume that @xmath227 for the existence of nontrivial @xmath204 . , the largest subset of @xmath77 bounded by an @xmath131 level set . note , the complement of @xmath222 contains level sets that intersect @xmath81 . also depicted is the plateau set @xmath228 , a region where @xmath131=constant . , scaledwidth=50.0% ] now we make a moderate generalization about the regularity : suppose that @xmath131 is lipschitz continuous , i.e. , @xmath229 . then , @xmath131 has a classical gradient @xmath230 almost everywhere in @xmath77 ( see e.g. @xcite ) , and @xmath231 is bounded . note , @xmath131 may fail to have a classical @xmath230 on a measure - zero subset @xmath232 , but for @xmath233 we may define a set - valued _ generalized gradient _ ( see @xcite ) . with a lipschitz continuous function @xmath234 we can solve ( [ kernel - general ] ) by @xmath235 where @xmath236 . to meet the boundary condition @xmath216 , @xmath237 must satisfy @xmath238 however , the solution ( [ kernel-1 ] ) omits a different type of solution that emerges with a _ singularity _ of @xmath171 : if @xmath131 has a `` plateau , '' i.e. , @xmath239 ( constant ) in a finite region @xmath240 ( see fig . [ fig : schematic ] ) , the operator @xmath171 trivializes as @xmath241 in @xmath242 ( i.e. , the `` rank '' drops to zero ; recall the example of sec . [ sec : introduction ] ) , and within @xmath242 we can solve ( [ kernel - general ] ) by an arbitrary @xmath178 with @xmath243 . notice that the solution ( [ kernel-1 ] ) restricts @xmath244 = constant in @xmath242 . to remove this degeneracy , we abandon the continuity of @xmath237 and , for simplicity , assume that @xmath131 has only a single plateau . first we invoke the reversed form [ cf . ( [ kernel - general-solution ] ) ] : @xmath245 where we assume that @xmath215 is a lipschitz continuous monotonic function . denoting @xmath246 , we may write @xmath247 , where the gradients on both sides evaluate classically almost everywhere in @xmath77 if @xmath178 is lipschitz continuous . if the function @xmath248 is flat on some interval , @xmath249 = constant for @xmath250 , a plateau appears in the distribution of @xmath131 . see figs . [ fig : schematic ] and [ fig : plateau](a ) . since the present mission is to find @xmath178 for a given @xmath131 , we transform ( [ kernel-2 ] ) back to ( [ kernel-1 ] ) with the definition @xmath251 . ( in sec . [ sec : equilibrium ] , however , we will seek an equilibrium @xmath131 that is characterized by a casimir element , and then , the form ( [ kernel-2 ] ) will be invoked again ) . a plateau in the graph of @xmath215 will , then , appear as a `` jump '' in the graph of @xmath237 . see fig . [ fig : plateau](b ) . and @xmath131 . ( a ) the relation @xmath252 with the plateau singularity . ( b ) the singular ( discontinuous ) function @xmath213 illustrating the kernel element that stems from a plateau of @xmath131 . , scaledwidth=80.0% ] we now allow the function @xmath253 to have a jump at @xmath254 . formally we write @xmath255 , with @xmath256 a lipschitz continuous function , @xmath257 the step function , and @xmath258 a constant determining the width of the jump . we connect the graph of the step function by filling the gap between @xmath259 and @xmath260 . see fig . [ fig : plateau](b ) . since @xmath253 is multi - valued at @xmath239 , @xmath261 is arbitrary in the range of @xmath262 $ ] , i.e. in @xmath242 . choosing a sufficiently smooth @xmath178 in @xmath242 , we may assume @xmath263 . summarizing the above discussions ( and making an obvious generalization ) , we have [ theorem : kernel2 ] suppose that @xmath229 and @xmath227 . then @xmath207 contains nontrivial elements , and a part of them can be represented as @xmath264 where @xmath265 is an arbitrary function that satisfies the boundary condition ( [ kernel - summary - b ] ) and such that @xmath266 where @xmath267 denotes the height of a plateau of @xmath131 , @xmath268 is the `` filled '' step function , and @xmath269 is a constant . [ remark : singular_casimir ] notice that @xmath207 is enlarged by the singular components @xmath270 stemming from @xmath131 ( the singular point in the phase space @xmath141 ) that has plateaus @xmath271 . away from the singular point , the plateaus shrink to zero - measure sets in @xmath77 and , then , @xmath213 can no longer be a member of the domain of @xmath171 . however , it is still a `` hyperfunction solution '' of @xmath272 , since @xmath230 parallels the delta - function @xmath273 at @xmath271 . the corresponding casimir element , to be constructed in sec . [ subsec : casimirs ] , is what we call a `` singular casimir element '' ( remember the elementary example discussed in sec . [ sec : introduction ] ; see also fig . [ fig : foliation ] ) . [ remark : j_operator-2 ] in ( [ g - form ] ) , the function @xmath265 may be chosen arbitrarily to define an infinite number of kernel elements @xmath213 satisfying ( [ kernel - general ] ) . to find kernel elements ( and in the following step of finding casimir elements ) , we solve a linear equation @xmath274 ( and @xmath275 ) for a given @xmath131 ; see remark[remark : j_operator ] . in the analysis of `` equilibrium points '' ( sec.[sec : equilibrium ] ) , however , we will relate @xmath204 and @xmath131 by another defining relation @xmath276 so as to make @xmath178 the clebsch potential of @xmath131 ( see ( [ operator - k ] ) ) . then , @xmath265 is provided as a data specifying a casimir leaf on which we seek an equilibrium point . [ remark : more_general_solutions ] clearly the form ( [ g - form ] ) of @xmath253 is rather restrictive : \(i ) in the plateau region , ( [ kernel - general ] ) has a wider class of solutions . in fact , @xmath178 may be an arbitrary @xmath277-class function whose range may exceed the interval @xmath262 $ ] . in this case , the graph of @xmath253 has a `` thorn '' at @xmath278 , and we may not integrate such a function to define a casimir element @xmath279 . ( see sec . [ subsec : casimirs ] . ) \(ii ) in ( [ g - form ] ) , we restrict the continuous part @xmath280 to be lipschitz continuous , by which @xmath213 ( @xmath229 ) is assured of lipschitz continuity ( thus , @xmath263 ) . however , this condition may be weakened , depending on the specific @xmath131 , i.e. , it is only required that @xmath281 . our next mission is to `` integrate '' the kernel element @xmath204 as a function of @xmath131 , and define a casimir element @xmath282 , i.e. , to find a functional @xmath282 such that @xmath283 . to this end , the parameterized @xmath178 of ( [ kernel - summary ] ) will be used , where the function @xmath253 may have singularities as described in theorem[theorem : kernel2 ] . the central issue of this section , then , will be to consider an appropriate `` generalized functional derivative '' by which we can define `` singular casimir elements '' pertinent to the singularities of the noncanonical poisson operator @xmath171 . let us start by considering a _ regular _ casimir element generated by @xmath284 : @xmath285 where @xmath286 . the gradient of this functional can be readily calculated with the definition of sec . [ subsec : gradient ] : perturbing @xmath131 by @xmath287 results in @xmath288 hence , we obtain @xmath289 , proving that @xmath290 of ( [ omega - casimir ] ) is the casimir element corresponding to @xmath291 . now we construct a _ singular _ casimir element corresponding to a general @xmath253 that may have `` jumps '' at the singularity of @xmath171 , i.e. , the plateaus of @xmath131 . the formal primitive function @xmath292 of such a @xmath265 has `` kinks '' where the classical differential does not apply this problem leads to the requirement of an appropriately generalized gradient of the functional @xmath293 generated by a kinked @xmath292 . here we invoke the _ _ clarke gradient__@xcite , which is a generalized gradient for lipschitz - continuous functions or functionals . specifically , if @xmath294 , then the clarke gradient of @xmath295 at @xmath296 , denoted by @xmath297 , is defined to be the convex hull of the set of limit points of the form @xmath298 evidently , @xmath297 is equivalent to the classical gradient , @xmath299 , if @xmath300 is continuously differentiable in the neighborhood of @xmath296 . also , it is evident that a `` kink '' in @xmath295 yields @xmath297 with a graph that has a `` jump '' with the gap filled as depicted in fig . [ fig : plateau](b ) . when @xmath61 is a convex functional on a hilbert space @xmath34 , i.e. @xmath301 , then @xmath302 is equal to the _ sub - differential _ : @xmath303 which gives the _ maximally monotone _ ( i.e. , the gap - filled ) function @xcite . for the purpose of sec . [ sec : equilibrium ] , a monotonic @xmath304 will be sufficient . from the above , the following conclusion is readily deducible : [ corollary : casimir ] suppose that @xmath229 and @xmath227 . by @xmath265 satisfying ( [ kernel - summary - b ] ) and ( [ g - form ] ) , we define @xmath292 such that @xmath304 . then , @xmath305 is a generalized casimir element , i.e. , @xmath306 . the casimir element @xmath290 naturally extends the set of extremal equilibrium points , the set of solutions of the dynamical system ( [ vorticity_eq-2dh ] ) that are both equilibrium solutions and energy - casimir extremal points . this is done by including extremal points satisfying [ cf . ( [ stationary_points2 ] ) ] @xmath307 \ni 0 , \label{fixed_ponit}\ ] ] which explicitly gives @xmath308 here we assume that @xmath304 is a monotonic function , i.e. , @xmath292 is convex . then , we may define a single - valued continuous function @xmath309 and rewrite ( [ kernel-1-fided_point ] ) , denoting @xmath145 , as @xmath310 if @xmath311 , ( [ kernel-2-fixed_point ] ) will determine a nontrivial ( @xmath312 ) equilibrium extremal point . notice that @xmath313 ( or @xmath314 or @xmath315 ) is a given function that specifies a casimir leaf on which we seek an equilibrium point ( see remark [ remark : j_operator-2 ] ) . here we prove the following existence theorem : [ theorem : fixed - point ] there exists a finite positive number @xmath316 , determined only by the geometry of @xmath77 , such that if @xmath317 then ( [ kernel-2-fixed_point ] ) has a solution @xmath118 . we show the existence of the solution by schauder s fixed - point theorem ( see for example @xcite and @xcite ) . first we rewrite ( [ kernel-2-fixed_point ] ) as @xmath318 where @xmath135 is a compact operator on @xmath90 ( cf . ( [ operator - k ] ) ) . since @xmath319 , @xmath320 is a continuous compact map on @xmath90 . we consider a closed convex subset @xmath321 where @xmath322 is the norm of @xmath90 and the parameter @xmath323 will be determined later . we show that the compact map @xmath320 has a fixed point in @xmath324 . by poincar s inequality , we have , for @xmath325 , @xmath326 where @xmath327 is a positive number that is determined by the geometry of @xmath77 . for @xmath328 , we have @xmath329 where @xmath330 is also a positive number determined by the geometry of @xmath77 . here @xmath331 and @xmath332 . hence , denoting @xmath333 , we have @xmath334 by sobolev s inequality , we have @xmath335 where @xmath336 is again a positive number determined by the geometry of @xmath77 . summarizing these estimates , we have , for @xmath337 , @xmath338 for an arbitrary positive number @xmath339 , there is a finite number @xmath340 such that @xmath341 . if we choose @xmath342 , then by the monotonicity of @xmath343 , we find @xmath344 and , upon denoting @xmath345 , we obtain @xmath346 where @xmath347 if @xmath348 , we estimate @xmath349 , and thus @xmath320 maps @xmath324 into @xmath324 . therefore , @xmath320 has a fixed - point in @xmath324 . if @xmath324 contains a fixed point , then an arbitrary @xmath350 with @xmath351 contains a fixed - point . since @xmath339 is arbitrary , @xmath352 is the sufficient condition for the existence of a fixed point . [ corollary : fixed - point ] if @xmath353 for some finite @xmath340 , then we may assume @xmath354 in ( [ estimate-1 ] ) , and the solvability condition ( [ fixed - point_condition ] ) can be modified to be @xmath355 . the bound @xmath316 defined by ( [ estimate - m ] ) is related to the eigenvalue of @xmath356 . on the other hand , the constant @xmath357 defined by ( [ fixed - point_condition ] ) is a property of the casimir element . as noted above , because of the homogeneity of ( [ casmir-1 ] ) , there is at least a one - parameter family of casimir elements and the choice of a multiplicative constant reciprocally scales @xmath357 . this provides an arbitrary parameter multiplying the casimir element in the determining equation ( [ fixed_ponit ] ) , and such a parameter may be considered to be an `` eigenvalue '' characterizing the stationary point . the no - solution example of the next section will reveal the `` nonlinear property '' of this eigenvalue problem . here we present a class of examples that violate the solvability condition proposed in theorem[theorem : fixed - point ] . this demonstrates that finding casimir elements that generalize the extremal equation ( [ kernel-2-fixed_point ] ) , does not automatically extend the set of extremal equilibria , since the resulting elliptic equation may not have a solution . this is demonstrated by the following two propositions . [ proposition : nosolution ] let @xmath358 have the form @xmath359 where @xmath360 is the first eigenvalue of @xmath356 and @xmath361 . then , ( [ kernel-2-fixed_point ] ) does not have a solution . we denote by @xmath362 the eigenfunction corresponding to the eigenvalue @xmath360 . upon taking the inner product of ( [ kernel-2-fixed_point ] ) with @xmath362 , we obtain @xmath363 the left - hand - side of ( [ ipdel ] ) satisfies @xmath364 , which cancels the first term on the right - hand - side , leaving @xmath365 . however , this is a contradiction , because @xmath366 ( or @xmath367 if otherwise normalized ) on @xmath77 be a bounded domain and @xmath368 the eigenvalues of @xmath136 with zero dirichlet boundary condition . it is well - known that the eigenvalues have the order @xmath369 , with @xmath370 as @xmath371 , and @xmath360 is the unique eigenvalue with corresponding eigenfunction that does not change sign on @xmath77 , i.e. , it is strictly positive or strictly negative on the whole of @xmath77 . furthermore , the dimension of the eigenspace associated with @xmath360 is one . ] and , by the assumption , @xmath372 . the above result is easily generalized as follows : [ proposition : nosolution2 ] suppose that @xmath373 where @xmath374 is any eigenvalue of @xmath136 and @xmath375 , for some @xmath376 and @xmath377 . let @xmath378 be the eigenfunction corresponding to @xmath374 . since @xmath378 may not have a definite sign in @xmath77 , we divide @xmath77 into @xmath379 and @xmath380 , and define @xmath381 if @xmath382 , ( [ kernel-2-fixed_point ] ) does not have a solution for some ranges of @xmath376 and @xmath377 . as shown in the proof of proposition[proposition : nosolution2 ] , a solution of ( [ kernel-2-fixed_point ] ) must satisfy @xmath383 by assumption , we have @xmath384 hence , the inequalities @xmath385 must hold , i.e. , @xmath386 however , if @xmath387 , and @xmath376 and @xmath377 are such that @xmath388 , then there is a contradiction with ( [ contradiction1 ] ) . similarly , if @xmath389 , and @xmath376 and @xmath377 are such that @xmath390 , then there is a contradiction with ( [ contradiction2 ] ) . after establishing some mathematical facts about the hamiltonian form of the euler equation of two - dimensional incompressible inviscid flow , we studied the center of the poisson algebra , i.e. , the kernel of the noncanonical poisson operator @xmath171 . casimir elements @xmath282 were obtained by `` integrating '' the `` differential equation '' @xmath391 for finite - dimensional systems with phase space coordinate @xmath8 , this amounts to an analysis of @xmath392 , a linear partial differential operator , and nontriviality can arise from a _ singularity _ of @xmath393 , whence an inherent structure emerges . recall the simple example given in sec . [ sec : introduction ] where @xmath394 was seen to generate the _ hyperfunction _ casimir @xmath395 . for finite - dimensional systems the theory naturally finds its way to algebraic analysis : in the language of d - module theory , casimir elements constitute @xmath396 , where @xmath397 is the ring of partial differential operators and @xmath295 is the function space on which @xmath393 operates , and @xmath398 is the d - module corresponding to the equation @xmath399 . however , in the present study @xmath131 is a member of an infinite - dimensional hilbert space , thus @xmath393 may be regarded as an infinite - dimensional generalization of linear partial differential operators . from the singularity of such an infinite - dimensional ( or _ functional _ ) differential operator @xmath400 , we unearthed _ singular _ casimir elements , and to justify the operation of @xmath393 on singular elements , we invoked a generalized functional derivative ( clarke differential or sub - differential ) that we denoted by @xmath401 . for infinite - dimensional systems , we can not `` count '' the dimensions of @xmath402 and @xmath403 . it is , however , evident that @xmath404 , if @xmath75 has singularities , i.e. , singularities create `` nonintegrable '' elements of @xmath403 . as shown in theorem[theorem : kernel2 ] , a _ plateau _ in @xmath131 causes a singularity of @xmath171 and generates new elements of @xmath207 , which can be integrated to produce singular casimir elements ( corollary[corollary : casimir ] ) . however , as noted in remark [ remark : more_general_solutions](i ) , more general elements of @xmath207 that are _ not integrable _ may stem from a plateau singularity . moreover , we had to assume lipschitz continuity for @xmath131 to obtain an explicit relation between @xmath204 and @xmath131 otherwise we could not _ integrate _ @xmath178 with respect to @xmath131 to construct a casimir element . in the general definition of @xmath171 , however , @xmath131 may be nondifferentiable ( we assumed only continuity ) , and then , a general @xmath204 may not have an integrable relation to @xmath131 ( see lemma [ lemma : kernel ] ) . in sec . [ sec : equilibrium ] we solved the equation @xmath405 \ni 0\ ] ] for @xmath131 , where the solution @xmath131 gave an equilibrium point of the dynamics induced by a hamiltonian @xmath167 . a singular ( kinked ) casimir yielded a multivalued ( set - valued ) gradient @xmath406 , encompassing an infinite - dimensional solution stemming from the plateau singularity . this arose because in the plateau of @xmath131 , @xmath204 is freed from @xmath131 and may distribute arbitrarily . the component @xmath256 of the casimir @xmath282 of ( [ g - form ] ) represents explicitly the regular dimensions " of @xmath207 . in contrast , the undetermined dimensions pertinent to the singularity @xmath239 are `` implicitly '' included in the step - function component of ( [ g - form ] ) , or in the kink of @xmath282 . however , for a given hamiltonian @xmath167 , i.e. a given dynamics , a specific relation between @xmath407 and @xmath131 emerges . theorem [ theorem : fixed - point ] of sec . [ subsec : fixed - points ] and the nonexistence examples of sec . [ subsec : no - solution ] may not be new results in the theory of elliptic partial differential equations , but they do help delineate the relationship between hamiltonians and casimir elements , viz . that casimir elements alone do not determine the extent of the set of equilibrium points . in the present paper , we did not discuss the bifurcation of the equilibrium points ; the reader is referred to @xcite for a presentation of the actual state of the studies of semilinear elliptic problems and several techniques in nonlinear analysis ( see also @xcite and @xcite ) . we also note that we have excluded nonmonotonic @xmath253 that will make @xmath358 multivalued or , more generally , equations like @xmath408 ; cf . ( [ kernel - general-solution ] ) . for fully nonlinear elliptic partial differential equations , the reader is referred to @xcite . the authors acknowledge informative discussions with yoshikazu giga and are grateful for his suggestions . zy was supported by grant - in - aid for scientific research ( 23224014 ) from mext , japan . pjm was supported by u.s . dept . of energy contract # de - fg05 - 80et-53088 . here we formulate the euler equation , for both @xmath101 and 3 , as an evolution equation in @xmath100 ( see sec.[subsec : vorticity_equation ] ) , and discuss the poisson operator @xmath409 in this space . this differs from the formulation of sec . [ sec : euler_equation ] in that the state variable here is the velocity field @xmath99 instead of the vorticity @xmath131 . as noted in sec.[subsec : vorticity_equation ] , @xmath100 is a closed subspace of @xmath90 , and we have the orthogonal decomposition ( [ decomposition_of_l2 ] ) . we denote by @xmath410 the orthogonal projection onto @xmath100 . upon applying @xmath410 to the both sides of ( [ euler-2 ] ) , we obtain @xmath411 which is interpreted as the evolution equation in @xmath103 , where the incompressibility condition ( [ incompressibility ] ) and the boundary condition ( [ bc ] ) are implied by @xmath120 . for @xmath120 , the hamiltonian is , of course , the kinetic energy @xmath412 fixing a sufficiently smooth @xmath52 acting as a parameter , viz . @xmath413 , we define for @xmath414 the following the noncanonical poisson operator : @xmath415 as a linear operator ( recall remark [ remark : j_operator ] of sec . [ sec : introduction ] ) , @xmath416 consists of the vector multiplication by @xmath417 followed by projection with @xmath410 . evidently , @xmath409 is antisymmetric : @xmath418 in fact , for every fixed smooth @xmath52 , @xmath419 is a self - adjoint bounded operator in @xmath103 . assuming @xmath101 , which we do henceforth , by lemma [ lemma : clebsch_representation ] we may put @xmath423 and @xmath140 with @xmath424 . fixing @xmath174 as a parameter , and putting @xmath425 with @xmath426 , we may write @xmath427 = -p_\sigma \left[\omega \nabla\psi\right].\ ] ] by ( [ decomposition_of_l2 ] ) , @xmath428 iff @xmath429 which is equivalent to ( [ kernel - general ] ) . arguing just like in sec . [ subsec : kernel ] , we find solutions of ( [ kernel - u ] ) of the form @xmath430 the casimir element constructed from ( [ kernel-1-u ] ) is @xmath431 perturbing @xmath52 by @xmath432 and restricting @xmath433 on @xmath95 yields @xmath434 , and @xmath435 hence , upon denoting @xmath436 we obtain @xmath437 by ( [ kernel-1-u ] ) , it is evident that @xmath438 = 0 $ ] , confirming that @xmath439 . v. i. arnold , on an a priori estimate in the theory of hydrodynamic stability , _ amer . math . soc . * 19 * ( 1969 ) , 267269 . v. i. arnold , sur la gometrie diffrentielle des groupes de lie de dimension infinie et ses applications lhydrodynamique des fluides parfaits , _ ann . ( grenoble ) * 16 * ( 1966 ) , 319361 . p. j. morrison , hamiltonian field description of two - dimensional vortex fluids an guiding center plasmas , princeton university plasma physics laboratory report , pppl-1783 ( 1981 ) ; available as american institute of physics document no . paps - pfbpe-04 - 771 - 24 . p. j. morrison , hamiltonian field description of one - dimensional poisson - vlasov equation , princeton university plasma physics laboratory report , pppl-1788 ( 1981 ) ; available as american institute of physics document no . paps - pfbpe-04 - 771 - 14 . p. j. morrison , in _ mathematical methods in hydrodynamics and integrability in related dynamical systems _ , aip conference proceedings no . * 88 * , edited by m. tabor and y. treve ( aip , new york , 1982 ) , p. 13 .
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the problem of the nonequivalence of the sets of equilibrium points and energy - casimir extremal points , which occurs in the noncanonical hamiltonian formulation of equations describing ideal fluid and plasma dynamics , is addressed in the context of the euler equation for an incompressible inviscid fluid .
the problem is traced to a casimir deficit , where casimir elements constitute the center of the poisson algebra underlying the hamiltonian formulation , and this leads to a study of singularities of the poisson operator defining the poisson bracket .
the kernel of the poisson operator , for this typical example of an infinite - dimensional hamiltonian system for media in terms of eulerian variables , is analyzed . for two - dimensional flows ,
a rigorously solvable system is formulated .
the nonlinearity of the euler equation makes the poisson operator inhomogeneous on phase space ( the function space of the state variable ) , and it is seen that this creates a singularity where the nullity of the poisson operator ( the `` dimension '' of the center ) changes .
the problem is an infinite - dimension generalization of the theory of singular differential equations .
singular casimir elements stemming from this singularity are unearthed using a generalization of the functional derivative that occurs in the poisson bracket .
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a surface facing a plasma collects electrons from the plasma more efficiently than it looses electrons due to neutralization of ions and/or de - excitation of radicals . it acquires thus a negative charge triggering in turn an electron - depletion layer in front of it the plasma sheath shielding the plasma from the surface . although known since the beginning of modern plasma physics @xcite a quantitative understanding of electron accumulation by plasma walls is still lacking @xcite . this is only due partly to unresolved materials science aspects , such as , chemical contamination and/or mechanical destruction of the surface by the plasma . it is also because little is known fundamentally about the interaction of electrons with surfaces at the energies relevant for plasma applications . electrons interacting with solid surfaces in the divertor region of fusion plasmas @xcite , dielectric barrier discharges @xcite , dusty plasmas @xcite , hall thrusters @xcite , or electric probe measurements @xcite have typically energies below 10ev , much less than the electron energy used in surface analysis @xcite or materials processing @xcite . the energies there are a few 100ev , an energy range , where the physical processes involved , backscattering and secondary electron emission , are sufficiently well understood @xcite to make these techniques reliable tools of applied science . much less is however known about these processes below 100ev and hence in the energy range relevant for plasmas . in particular , the backscattering probability of a low - energy electron , and closely related to it , the probability with which it is absorbed is basically unknown . although electron absorption ( sticking ) and backscattering are important processes for bounded plasmas there is no systematic effort to determine their probabilities either experimentally or theoretically . the electron sticking probability , for instance , is usually assumed to be close to unity @xcite , irrespective of the energy and angle of incident or the wall material ( perfect absorber assumption @xcite ) . the need to overcome this assumption has been strongly emphasized by mendis @xcite but the model calculations he refers to are based on classical considerations not applicable to electrons . in a recent work @xcite we proposed therefore a quantum - mechanical approach for calculating the electron sticking probability . the method is based on two important facts noticed by cazaux @xcite : ( i ) low - energy electrons do not see the strongly varying short - range potentials of the surface s ion cores but a slowly varying surface potential and ( ii ) they penetrate deeply into the surface . for al@xmath0o@xmath1 , for instance , the average electron penetration depth at a few ev is around 200 @xcite . the sticking probability for an electron approaching the wall of a plasma can thus be expressed by the transmission probability for the long - ranged surface potential times the probability to remain inside the wall despite of internal backscattering . essential for our approach is the invariant embedding principle @xcite . it allows us to extract from the overwhelming number of electron trajectories the few backwardly directed ones most relevant for sticking . so far we applied the method to mgo @xcite obtaining excellent agreement with electron - beam scattering data @xcite . in this work we consider sio@xmath0 finding again good agreement with beam data @xcite . in both cases the sticking probability is energy- and angle - dependent as well as significantly less than unity . the remaining part of the paper is organized as follows . in section [ formalism ] we describe our microscopic approach for calculating electron absorption and backscattering probabilities in more detail than previously @xcite , focusing in particular on the invariant embedding principle and its linearization making the approach numerically very efficient . section [ results ] presents results for sio@xmath0 , an in - depth discussion of the model we proposed for the description of imperfect plasma - wall interfaces , and a calculation of orbital - motion limited grain charges beyond the perfect absorber model for electrons . concluding remarks are given in section [ conclusions ] . the method we developed for calculating the probability with which a low - energy electron is absorbed by a surface is general @xcite . it can be applied to metallic as well as dielectric surfaces . to be specific we consider in this work a dielectric sio@xmath0 surface as an example . for a dielectric wall with @xmath2 ( positive electron affinity ) such as sio@xmath0the potential energy of an electron across the plasma - wall interface has roughly the form @xcite shown , together with other aspects of our approach , in fig . [ cartooneps]a . an electron approaching the interface from the plasma has to overcome the wall potential @xmath3 . once it is inside the wall it occupies the conduction band and sees thus a potential barrier @xmath4 . since it is the kinetic energy of the electron in the immediate vicinity of the interface which determines sticking and backscattering probabilities , while the variation of the wall potential @xmath3 is on the scale of the debye screening length , much larger than the scale on which the surface potential varies , the relevant part of the electron potential energy is essentially a three - dimensional potential step with height @xmath4 and electron mass mismatch @xmath5 , where @xmath6 is the effective electron mass in the conduction band of the wall and @xmath7 is the bare electron mass , as illustrated by the solid red line in fig . [ cartooneps]a . ( solid red line ) . three scattering trajectories ( i)(iii ) due to emission of optical phonons inside the wall symbolized by bullets are shown each having the same number of total but a different number of backscattering events and bringing an electron entering the wall at energy @xmath8 back to the plasma at an energy @xmath9 . the half circles denote the moduli of the electron momenta inside and outside the wall . also shown is the energy distribution @xmath10 the approaching electron may have . ( b ) illustration of eqs . . the potential step leads to a quantum - mechanical transmission probability @xmath11 whereas the emission of phonons yields a quantity @xmath12 to be obtained from the invariant embedding principle shown in fig . [ embedding ] . the pre- and post - collision angles inside @xmath13 and outside @xmath14 the wall are measured with respect to the surface normal @xmath15 . ] ( solid red line ) . three scattering trajectories ( i)(iii ) due to emission of optical phonons inside the wall symbolized by bullets are shown each having the same number of total but a different number of backscattering events and bringing an electron entering the wall at energy @xmath8 back to the plasma at an energy @xmath9 . the half circles denote the moduli of the electron momenta inside and outside the wall . also shown is the energy distribution @xmath10 the approaching electron may have . ( b ) illustration of eqs . . the potential step leads to a quantum - mechanical transmission probability @xmath11 whereas the emission of phonons yields a quantity @xmath12 to be obtained from the invariant embedding principle shown in fig . [ embedding ] . the pre- and post - collision angles inside @xmath13 and outside @xmath14 the wall are measured with respect to the surface normal @xmath15 . ] the potential step gives rise to quantum - mechanical reflection and transmission . for the situation shown in fig . [ cartooneps]a , that is , a wall ( plasma ) occupying the @xmath16 ( @xmath17 ) half space and an energy scale for which @xmath18 , the transmission probability for an electron coming from the plasma , and having thus a kinetic energy @xmath19 , is given by @xcite @xmath20 with @xmath21 and @xmath22 the @xmath23components of the electron momenta outside and inside the wall . in and the formulae below we measure length in bohr radii , energy in rydbergs , and mass in electron masses implying inside the wall the electron mass is simply the mass mismatch @xmath24 . the signs of @xmath25 and @xmath26 in are always the same . we can thus define the direction cosines @xmath27 and @xmath28 referenced , respectively , to the electron momenta outside and inside the wall , by their absolute values : @xmath29 and @xmath30 ( see fig . [ cartooneps]b for the definition of the angles ) . this choice is also convenient for the theoretical description of internal backscattering which we address later . since the potential varies only perpendicularly to the surface the lateral momentum @xmath31 is conserved . together with the conservation of energy , @xmath32 , this leads to @xmath33 connecting the direction cosines @xmath28 and @xmath27 . from follows that an electron approaching the wall with kinetic energy @xmath34 enters it only when @xmath27 is larger than @xmath35 with @xmath36 . for @xmath27 less than @xmath37 the electron is in an evanescent wave with @xmath38 and thus totally reflected @xcite . in addition , the requirement may instantaneously reduce the electron s perpendicular kinetic energy to less then the electron affinity @xmath4 once it crossed the interface from the plasma side , that is , @xmath39 even without inelastic scattering . for mass mismatch @xmath40 , applicable to sio@xmath0 , mgo , al@xmath0o@xmath1 , this happens when @xmath41 . provided the electron can not gain energy by inelastic scattering , as it is the case for dielectric walls at room temperature , it will have no chance to ever come back to the plasma . the transmission probability @xmath42 is not identical with the sticking probability . it captures only the ballistic aspect of electron absorption by the wall and is at best an upper bound to it . once the electron is inside the wall it suffers elastic as well as inelastic scattering . both may push the electron back to the interface and , after successfully traversing the surface potential in the reverse direction , eventually back to the plasma . hence , we expect the sticking probability @xmath43 . to take scattering inside the wall into account we defined @xmath44 as the probability of an electron hitting the wall from the plasma with energy @xmath8 and direction cosine @xmath27 not to return to it after entering the wall and suffering backscattering @xcite , @xmath45~ , \label{stickcoeff}\end{aligned}\ ] ] where @xmath46 is the conditional probability for the electron to escape from the wall after at least one backscattering event . the lower integration limits , @xmath47 and @xmath48 , ensure that only events are counted for which the perpendicular post - collision energy @xmath49 , @xmath50 is the conduction band s density of states , and @xmath51 is the normalized probability @xmath52 for an electron with energy @xmath8 and direction cosine @xmath28 to backscatter after an arbitrary number of internal scattering events to a state with energy @xmath9 and direction cosine @xmath53 . since the function @xmath52 describes backscattering , @xmath54 and @xmath55 , implying @xmath56 as @xmath30 . the energy integrals in and anticipate that at room temperature the electron can not gain energy from a dielectric wall and the functions @xmath57 and @xmath58 are implicitly defined by . . however , an infinitesimally thin layer of the same material can not affect @xmath52 . hence , @xmath52 has to be invariant against the change the paths ( 1)(4 ) induce leading to eq . . ] to obtain the quantity @xmath52 we employ the invariant embedding principle @xcite , the essence of it is shown in fig . [ embedding ] . for our purpose it is extremely powerful since it focuses from the start on the backscattering trajectories . compared to forward scattering backscattering is usually much less likely . constructing thus @xmath52 , for instance , from the monte carlo trajectories mimicking the solution of the electron s boltzmann equation , obtained under suitable initial and boundary conditions , would be numerically very expensive . the principle can be derived as follows @xcite . imagine to add to the half space filled with wall material an infinitesimally thin layer of the same material . as a result the four scattering trajectories shown in fig . [ embedding ] may now additionally contribute to @xmath52 . however , an infinitesimally thin additional layer of the same material can not change the backscattering properties of the half - space . hence , @xmath52 has to be invariant against this change . defining a convolution for functions depending on the variables @xmath59 , and @xmath53 , @xmath60 summing up the four paths ( 1)(4 ) shown in fig . [ embedding ] ( without the transmission / reflection due to the surface potential ) , and enforcing them not to change @xmath52 yields the nonlinear integral equation @xcite @xmath61q(e\eta|e^\prime\eta^\prime)= g^-(e\eta|e^\prime\eta^\prime ) \nonumber\\ & + ( g^+ \!\!\ast q)(e\eta|e^\prime\eta^\prime ) + ( q\ast g^+)(e\eta|e^\prime\eta^\prime ) \nonumber\\ & + ( q\ast g^- \!\!\ast q)(e\eta|e^\prime\eta^\prime)~ , \label{nlinteq}\end{aligned}\ ] ] where the kernels @xmath62 encode forward ( @xmath63 ) or backward ( @xmath64 ) scattering . physically , the lhs describes the reduction of the probability for the electron to follow any one of the old paths , already included in @xmath65 , while the rhs corresponds to the four trajectories shown in fig . [ embedding ] . the kernels @xmath66 are scattering rates per length . the non - trivial parts @xmath67 depend on the scattering process . they can be obtained from the golden rule scattering rate per time dividing it by the pre - collision velocity . below we consider a sio@xmath0 surface , where emission of optical phonons dominates the scattering , leading to @xcite @xmath68^{-1/2}~. \label{kkernel}\end{aligned}\ ] ] the function @xmath69 also entering describes the rate per length to make any collision . for phonon emission it is given by @xmath70 . and @xmath71 the square of the inverse of the momentum transfer @xmath72 as a function of the cosine of the angle between the pre- and post - collision momenta . it defines at the low electron densities considered in this work the matrix element for phonon emission by the electron entering the wall . the strong forward peaking of the matrix element localizes the forward scattering kernel @xmath73 , plotted on the right on a log - scale over the whole range of direction cosines @xmath28 and @xmath53 , around the diagonal @xmath74 . a further consequence of the angle dependence of @xmath72 is that the backscattering kernel @xmath75 ( not shown ) is rather isotropic in the @xmath76-plane . ] and @xmath71 the square of the inverse of the momentum transfer @xmath72 as a function of the cosine of the angle between the pre- and post - collision momenta . it defines at the low electron densities considered in this work the matrix element for phonon emission by the electron entering the wall . the strong forward peaking of the matrix element localizes the forward scattering kernel @xmath73 , plotted on the right on a log - scale over the whole range of direction cosines @xmath28 and @xmath53 , around the diagonal @xmath74 . a further consequence of the angle dependence of @xmath72 is that the backscattering kernel @xmath75 ( not shown ) is rather isotropic in the @xmath76-plane . ] in the general case it is hard to work with . at low energies however most scattering processes are forwardly peaked . the larger the change of the propagation direction the less likely the process . it is thus possible to use the backscattering kernel @xmath77 as an expansion parameter controlling an iterative solution of . following glazov and pzsit @xcite we expand therefore in a first step the solution of in the number of the backscattering events . in a second step we take advantage of the fact that for dielectric surfaces at room temperature scattering arises mainly form the emission of optical phonons with energy @xmath78 . the electron can not gain energy by scattering . it can loose at most the energy it initially had when entering the wall . expanding thus @xmath52 also in the number of forward scattering events yields a double series which terminates after a finite number of terms . from the differential scattering cross section for the materials we are interested in , sio@xmath0 , mgo , and al@xmath0o@xmath1 follows moreover that backward scattering due to emission of optical phonons is at least two orders of magnitude less likely than forward scattering @xcite . for sio@xmath0 this can be seen in the left panel of fig . [ xsection ] . hence , writing @xcite @xmath79 with @xmath80 we can truncate the summation already after a single backward scattering event , that is , after @xmath81 leading to a linear recursion @xcite @xmath82 for the expansion coefficients with @xmath83 , where @xmath84 is the number of forward scattering events at most possible , @xmath85 and an initialization @xmath86 in deriving the recursion we assumed forward scattering not to change the direction cosine at all . this is justified because , as shown in the right panel of fig . [ xsection ] , the kernel @xmath87 is strongly peaked for @xmath74 . the directional change due to forward scattering is thus negligible and integrals over the direction cosine containing @xmath87 can be handled by a saddle - point approximation . forward scattering is then encoded in @xmath88 where we used the symmetry of @xmath87 with respect to interchanging @xmath28 and @xmath53 and defined the functions @xmath89 ^ 2 - 4\sqrt{ee^\prime}(e+e^\prime)\eta + 4ee^\prime\eta^2}\nonumber\\ & + \sqrt{ee^\prime } - ( e+e^\prime)\eta~,\\ q(e|e^\prime;\eta ) & = \sqrt{[e - e^\prime]^2 + 4ee^\prime\eta^2 } - ( e+e^\prime)\eta~.\end{aligned}\ ] ] we thus end up with a model similar in spirit to the oswald , kasper and gaukler model @xcite for multiple elastic backscattering of electrons from surfaces . inserting finally for the backscattering probability into and performing the energy integrals yields @xmath90 with @xmath91 . substituting this expression for @xmath92 into eq . gives the electron sticking probability @xmath44 for a clean , homogeneous dielectric wall with positive electron affinity . besides @xmath44 the backscattering probability is also of interest for plasma modeling . our approach contains two types of backscattering processes : specular quantum - mechanical reflection given by @xmath93 and diffuse backscattering encoded in @xmath94 with @xmath95 defined in . the latter gives the probability for an electron hitting the wall with energy @xmath8 and direction cosine @xmath27 to end up in a state with energy @xmath96 and direction cosine @xmath97 where @xmath53 and @xmath98 are the post - collision direction cosines inside and outside the wall . for a clean sio@xmath0 surface . the left panel shows @xmath44 for the whole range of direction cosines @xmath27 and energies @xmath99 . total reflection takes place in the white region . below the yellow dotted line , indicating @xmath100 , inelastic backscattering has no effect on the sticking probability . in the right panel @xmath44 ( solid line ) and @xmath42 ( dashed line ) are plotted as a function of @xmath8 for representative @xmath27 . the grey area denotes the energy range of the conduction band . ] for a clean sio@xmath0 surface . the left panel shows @xmath44 for the whole range of direction cosines @xmath27 and energies @xmath99 . total reflection takes place in the white region . below the yellow dotted line , indicating @xmath100 , inelastic backscattering has no effect on the sticking probability . in the right panel @xmath44 ( solid line ) and @xmath42 ( dashed line ) are plotted as a function of @xmath8 for representative @xmath27 . the grey area denotes the energy range of the conduction band . ] we now apply our approach to a sio@xmath0 surface characterized by @xmath101 @xcite , @xmath102 @xcite , @xmath103 @xcite , and @xmath104 @xcite . for an actual sio@xmath0 surface the parameters may deviate from these values depending on material science aspects which we do not address in this work . numerical results for @xmath44 are shown in fig . [ angleresolved ] . first , we focus on the left panel showing data over the whole range of direction cosines @xmath27 and energies @xmath8 up to @xmath105 . for @xmath106 our results are only rough estimates since an electron entering the wall at these energies can already create electron - hole pairs across the band gap . this coulomb - driven process is not included . it can be treated in the same spirit leads however to a recursion containing energy integrals making the numerical treatment more demanding . the white area in the plot for @xmath44 indicates the region in the @xmath107-plane where total reflection occurs . it is smaller than for mgo @xcite because @xmath24 is larger for sio@xmath0 . below the dotted yellow line inelastic backscattering due to emission of phonons is irrelevant for sticking because conservation of lateral momentum and total energy force the perpendicular energy of the electron to drop below the potential step @xmath4 once it crossed the interface from the plasma side . it is hence already confined by quantum - mechanical transmission alone . only above the dotted yellow line inelastic backscattering may bring the electron back to the interface and , after traversing the surface potential in the reversed direction , back to the plasma . hence , for sio@xmath0 , as well as any other dielectric with mass mismatch @xmath108 , @xmath109 for @xmath110 and @xmath111 for @xmath112 . the sticking coefficient is thus only for some @xmath8 and @xmath27 equal to the transmission probability . this can be more clearly seen in the right panel of fig . [ angleresolved ] , where @xmath44 ( solid lines ) and @xmath42 ( dashed lines ) are plotted as a function of @xmath8 for some representative @xmath27 . to indicate the efficiency of our approach we mention that we obtained the about @xmath113 data points for @xmath44 in fig . [ angleresolved ] , corresponding each to a sum of trajectories with one backward and ( depending on energy ) up to @xmath114 forward scattering events , with the former interlaced between the latter in all possible ways , in only one hour computing time on a notebook . the angle- and energy - resolved probability for diffuse backscattering @xmath115 introduced in is depicted in fig . [ backscatt ] for @xmath116 and @xmath117 . it is largest for @xmath118 and @xmath119 . post - collision direction cosines @xmath120 are excluded because mass mismatch and conservation of lateral momentum and total energy make them to correspond to internal states with perpendicular energy less than @xmath4 . scattering channels in these directions are thus closed . the maximum of the diffuse backscattering probability is always at the initial energy @xmath8 and direction cosine @xmath27 . had we chosen other values for @xmath8 and @xmath27 the plot would look similar only with a shifted maximum . for an electron hitting a clean sio@xmath0 surface with @xmath116 and @xmath117 to backscatter diffusely into a state with direction cosine @xmath97 and energy @xmath9 . below the dotted yellow line @xmath121 since diffuse backscattering can not lead to post - collision direction cosines @xmath122 . on the right are shown a horizontal and a vertical cut through the data depicted on the left . diffuse backscattering peaks around the entrance energy @xmath116 and the entrance direction cosine @xmath117 . ] for an electron hitting a clean sio@xmath0 surface with @xmath116 and @xmath117 to backscatter diffusely into a state with direction cosine @xmath97 and energy @xmath9 . below the dotted yellow line @xmath121 since diffuse backscattering can not lead to post - collision direction cosines @xmath122 . on the right are shown a horizontal and a vertical cut through the data depicted on the left . diffuse backscattering peaks around the entrance energy @xmath116 and the entrance direction cosine @xmath117 . ] total reflection forces the sticking probability for an electron to vanish if it hits the surface with energy @xmath8 and direction cosine @xmath123 . it is caused by the mass mismatch and the conservation of lateral momentum and total energy . the former holds only for a homogeneous interface . in reality imperfections destroy the homogeneity . lateral momentum will thus not be conserved and total reflection suppressed . to account for this possibility we now include elastic interface scattering along the lines smith and coworkers used in their theoretical treatment of ballistic electron - emission spectroscopy @xcite . central to the approach is the probability for an electron hitting the wall from the plasma with a kinetic energy @xmath19 to make a transition from @xmath107 to @xmath124 due to elastic scattering by any of the interfacial scattering centers , @xmath125 where @xmath126 is a parameter proportional to the density of the scatterers and the square of the modulus of the scattering potential which we assume to be independent of the initial and final scattering states ( hard core scattering potential ) . lacking a detailed knowledge of the structural properties of the interface we use @xmath126 as a fit parameter . the function @xmath127 can be derived from the probability to make a transition due to scattering by a single center given by the ratio of the golden rule scattering rate per time and the rate with which electrons hit the interface taking interference corrections due to other centers into account @xcite . any physical quantity @xmath128 affected by interfacial disorder turns then into @xmath129 \nonumber\\ & + \int_0 ^ 1 d\xi^\prime \int_\chi^\infty \!\!\!\ ! de^\prime \sqrt{e^\prime-\chi}p(e|e^\prime;\xi ) f(e^\prime,\xi^\prime)~,\end{aligned}\ ] ] where the first and second term on the rhs stand , respectively , for trajectories without and with interfacial scattering . using this rule together with @xmath130 yields for the sticking probability of a disordered dielectric wall @xcite @xmath131 \nonumber\\ & + \frac{c/\xi}{1+c/\xi } \int^1_{\xi_c } d\xi^\prime t(e,\xi^\prime)[1-\bar{\cal e}(e,\xi^\prime)]~ , \label{dirty}\end{aligned}\ ] ] where @xmath132 is given by with @xmath42 replaced by @xmath133 and @xmath37 defined by . notice , in the limit @xmath134 we recover from the sticking probability @xmath44 for a clean wall given by while for @xmath135 we obtain the sticking probability for the totally disordered , dirty wall . for an electron hitting a sio@xmath0 surface perpendicularly as obtained from eq . ( solid lines ) . we show data for @xmath136 ( black ) , @xmath137 ( red ) , @xmath138 ( green ) , and @xmath139 ( blue ) with @xmath140 also included for @xmath136 and @xmath141 ( dashed lines ) . symbols are data from electron - beam scattering experiments @xcite . ] the sticking probability @xmath142 of a disordered sio@xmath0 surface is shown in fig . [ expdata ] for @xmath117 ( normal incident ) and @xmath143 and @xmath139 . to indicate that our approach captures essential aspects of electron absorption by a surface we also plot data for two types of sio@xmath0 surfaces obtained from electron - beam scattering experiments @xcite . although the experimental data are in an energy range where electron - hole pair generation already starts to play a role they are nevertheless sufficiently close to the theoretical results to support our modeling approach . they also show that the perfect absorber value , @xmath144 , is not applicable to sio@xmath0 . for mgo experimental data are available for lower energies showing a much better agreement with the theoretical data @xcite . dashed lines show , for comparison , @xmath145 , which is the sticking probability in the absence of backscattering . for @xmath136 , @xmath146 deviates strongly from @xmath145 ( black lines ) , whereas for @xmath141 the two quantities approach each other ( blue lines ) . the reason is the angle - averaging at the dirty surface ( see eq . ) which lessens , for a fixed @xmath27 , the impact of inelastic backscattering compared to the knock - out of propagation directions by total reflection . the kink in @xmath146 at @xmath147 signals the knock - out . comparing the results for mgo ( @xmath148 ) @xcite with the results for sio@xmath0 ( @xmath101 ) indicates moreover that the closer @xmath24 to unity the more affected is @xmath146 by inelastic backscattering . the mass mismatch @xmath24 turns thus out to control @xmath146 . this is not surprising . because it is the effective electron mass which subsumes at low energy the elastic scattering of the electron by the ion cores of the wall . .material parameters for the dielectrics used in this work and the orbital - motion limited charge @xmath149 acquired in an argon plasma with @xmath150 and @xmath151 by dust particles made out of these dielectrics and having radius @xmath152 . the parameter @xmath153 defined in eq . contains the material dependence of the particle charge . as for sio@xmath0 @xcite the material parameters for mgo @xcite and al@xmath0o@xmath1 @xcite can be different for actual particles / surfaces due to material science aspects not addressed in this work . [ cols="^,^,^,^,^,^,^",options="header " , ] at low energies , the electron - wall interaction in a plasma is usually treated within the perfect absorber model @xcite stating that the probability with which an electron is absorbed ( backscattered ) by the wall is close to unity ( vanishes ) . we have seen however that both probabilities are in fact energy- and angle - dependent and deviate from the perfect absorber values . it is thus of interest to work out the consequences of our results for the modeling of bounded plasmas . as a first plasma application we consider the orbital - motion limited ( oml ) charging @xcite of a dielectric particle in a plasma . similar results would be however obtained for other charging models as well @xcite . the grain surface is characterized by @xmath154 leading to an electron capture cross section @xmath155 with @xmath8 and @xmath27 the energy and direction cosine of the incident electron and @xmath156 the grain s floating potential . the capture cross section yields the flux balance , @xmath157 where @xmath158 and @xmath159 are the oml fluxes @xcite and @xmath160 \frac{e-\chi}{kt_e}\langle \bar{s}(e,\xi)\rangle_\xi \label{soml}\end{aligned}\ ] ] with @xmath161 the angle - averaged sticking probability . the ( hard ) perfect absorber assumption @xmath162 gives @xmath163 . for a sio@xmath0 , an al@xmath0o@xmath1 , and a mgo surface . solid and dashed lines indicate , respectively , @xmath164 for a clean ( @xmath136 ) and a dirty interface ( @xmath141 ) . the kink in the data signals the knock - out of propagation directions by total reflection . ] in the limit @xmath165 the grain charge in units of @xmath166 is given by @xmath167 with @xmath168 the root of . table [ omlcharge ] shows results for sio@xmath0 , al@xmath0o@xmath1 , and mgo . besides the expected deviation of the grain charge from the perfect absorber ( pam ) charge we also find a material dependence which could be tested by high precision charge measurements @xcite . figure [ angleave ] finally shows for sio@xmath0 , al@xmath0o@xmath1 , and mgo the angle - averaged sticking probability @xmath164 entering . since we calculate @xmath164 only up to @xmath169 , for @xmath170 we should have included electron - hole pair generation leading to a more complicated recursion scheme containing energy integrals , we approximate in @xmath164 for @xmath170 by its upper bound @xmath171 . the values for @xmath153 given in table [ omlcharge ] are thus only upper bounds . we described a method to calculate from a microscopic model the sticking and backscattering probabilities for a low - energy electron hitting the wall of a plasma . taking advantage of the large penetration depth at low energies the method factorizes sticking and backscattering probabilities into probabilities for quantum - mechanical transmission and internal backscattering . for the description of the latter we employed an invariant embedding principle . it allows us to extract from the great number of electron trajectories the most important ones which are the trajectories bringing the electron back to the plasma after at least one backscattering but as many forward scattering events as are energetically possible . the approach is applicable to metallic as well as dielectric surfaces . for dielectric surfaces at room temperature , where emission of optical phonons is the dominant scattering process , it is sufficient to include only one backscattering event interlaced in all possible ways between the energetically allowed sequence of forward scattering events . in this work we focused on sio@xmath0 and found good agreement with experimental data from electron - beam scattering . contrary to the perfect absorber assumption , @xmath172 , we find energy- and angle - dependent sticking probabilities which can deviate significantly from unity because of internal backscattering due to emission of optical phonons and total reflection due to electron mass mismatch and conservation of total energy and lateral momentum . angle - averaged sticking probabilities @xmath164 for sio@xmath0 , al@xmath0o@xmath1 , and mgo are also less than unity . incorporating the sticking probabilities into orbital - motion limited charging fluxes reduces the grain charge by about 10 percent compared to the perfect absorber value and makes the charge material - dependent . the method is particularly strong for electron energies below @xmath173 . various scattering mechanism can be included as well as imperfections of the interface . it could thus be used to systematically investigate the interaction of electrons with the walls of low - temperature plasmas . this is indeed required . electron sticking and backscattering are not universal in the energy range of interest for plasma applications . they have to be studied for each wall material separately .
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we describe a method for calculating the probability with which the wall of a plasma absorbs an electron at low energy .
the method , based on an invariant embedding principle , expresses the electron absorption probability as the probability for transmission through the wall s long - range surface potential times the probability to stay inside the wall despite of internal backscattering . to illustrate the approach we apply it to a sio@xmath0 surface .
besides emission of optical phonons inside the wall we take elastic scattering at imperfections of the plasma - wall interface into account and obtain absorption probabilities significantly less than unity in accordance with available electron - beam scattering data but in disagreement with the widely used perfect absorber model .
| 9,625 | 171 |
intermediate - mass stars ( ims ) comprise objects with zams masses between 0.8 and 8 , corresponding to spectral types between g2 and b2 . the lower mass limit is the minimum value required for double shell ( h and he ) fusion to occur , resulting in thermal pulsations during the asymptotic giant branch ( agb ) phase and eventually planetary nebula formation . above the upper mass limit stars are capable of additional core - burning stages , and it is generally assumed that these stars become supernovae . a salpeter ( 1955 ) imf can be used to show that ims represent about 4% of all stars above 0.08 , but this may be a lower limit if the imf is flat at low stellar masses ( scalo 1998 ) . ims evolution is an interesting and complex subject and the literature is extensive . a good , complete , generally accessible review of the subject is given by iben ( 1995 ) . shorter reviews focussing on the agb stage can be found in charbonnel ( 2002 ) and lattanzio ( 2002 ) . i will simply summarize here . intermediate - mass stars spend about 10 - 20% of their nuclear lives in post main sequence stages ( schaller et al . fresh off the main sequence , a star s core is replete with h - burning products such as 4 & 14 . the shrinking core s temperature rises , a h - burning shell forms outward from the core , and shortly afterwards the base of the outer convective envelope moves inward and encounters these h - burning products which are then mixed outward into the envelope during what is called the _ first dredge - up_. as a result , envelope levels of 4 , 14 , and rise . externally , the star is observed to be a red giant . as the shrinking he core ignites , the star enters a relatively stable and quiescent time during which it synthesizes and . once core he is exhausted , the star enters the agb phase , characterized by a co core along with shells of h and he - fusing material above it . early in this phase , for masses in excess of 4 , _ second dredge - up _ occurs , during which the base of the convective envelope again extends inward , this time well into the intershell region , and dredges up h - burning products , increasing the envelope inventory of 4 , 14 , and as before . later in the agb phase , however , the he shell becomes unstable to runaway fusion reactions , due to its thin nature and the extreme temperature sensitivity of he burning . the resulting he - shell flash drives an intershell convective pocket which mixes fresh outward toward the h - shell . but as the intershell expands , h - shell burning is momentarily quenched , and once again the outer convective envelope extends down into the intershell region and dredges up the fresh into the envelope , an event called _ third dredge - up_. subsequently , the intershell region contracts , the h shell reignites , and the cycle repeats during a succession of thermal pulses . observational consequences of thermal pulsing and third dredge - up include the formation of carbon stars , mira variables , and barium stars . now , in ims more massive than about 3 - 4 , the base of the convective envelope may reach temperatures which are high enough ( @xmath360 million k ) to cause further h - burning via the cn cycle during third dredge - up . as a result , substantial amounts of are converted to 14 in a process referred to as hot - bottom burning ( renzini & voli 1981 ; hbb ) . hbb not only produces large amounts of 14 but also results in additional neutron production through the @xmath4c(@xmath0,n)@xmath5o reaction , where extra mixing is required to produce the necessary @xmath4c . these additional neutrons spawn the production of s - process elements which are often observed in the atmospheres of agb stars . note that carbon star formation is precluded by hbb in those stars where it occurs . other nuclei that are synthesized during thermal pulsing and hbb include @xmath6ne , @xmath7 mg , @xmath8al , @xmath9na , and @xmath10li ( karakas & lattanzio 2003 ) . the thermal pulsing phase ends when the star loses most of its outer envelope through winds and planetary nebula ( pn ) formation , and thus the main fuel source for the h shell ( and for the star ) is removed and evolution is all but over . note that the pn contains much of the new material synthesized and dredged up into the atmosphere of the progenitor star during its evolution . as this material becomes heated by photoionization , it produces numerous emission lines whose strengths can be measured and used to infer physical and chemical properties of the nebula . models of intermediate mass star evolution are typically synthetic in nature . a coarse grid of models , in which values for variable quantities are computed directly from fundamental physics , is first produced . then interpolation formulas are inferred from this grid which are subsequently used in a much larger run of models , thus reducing the computation time requirements . the models described below are of this type . the major parameters which serve as input for ims models include : stellar mass and metallicity , the value of the mixing length parameter , the minimum core mass required for hbb , the formulation for mass loss , and third dredge - up efficiency . the first substantial study of ims surface abundances using theoretical models was carried out by iben & truran ( 1978 ) , whose calculations accounted for three dredge - up stages including thermal pulsing . renzini & voli ( 1981 ; rv ) introduced hot bottom burning and the reimers ( 1975 ) mass loss rate to their models and explicitly predicted pn composition and total stellar yields . van den hoek & groenewegen ( 1997 ; hg ) introduced a metallicity dependence , heretofore ignored , into their evolutionary algorithms along with an adjustment upwards in the mass loss rate , the latter being a change driven by constraints imposed by the carbon star luminosity function ( see below ) . finally , boothroyd & sackmann ( 1999 ) demonstrated effects of cool bottom processing on the / ratio ; marigo , bressan , & chiosi ( 1996 ) , buell ( 1997 ) , and marigo ( 2001 ; m01 ) employed the mass loss formalism of vassiliadis & wood ( 1993 ) , which links the mass loss rate to the star s pulsation period , to predict yields of important cno isotopes ; and langer et al . ( 1999 ) and meynet & maeder ( 2002 ) studied the effects of stellar rotation on cno yields . table [ t1 ] provides a representative sample of yield calculations carried out over the past two decades . to the right of the author column are columns which indicate the lower and upper limits of the mass and metallicity ranges considered , an indication of whether hot bottom burning or cold bottom processing was included in the calculations ( yes or no ) , the type of mass loss used [ r = reimers ( 1975 ) , vw = vassiliadis & wood ( 1993 ) ] , an indication of whether the calculations included stellar rotation ( yes or no ) , and some important nuclei whose abundances were followed during the calculations . values for @xmath12 in table [ t2 ] are in turn plotted against metallicity in figure [ p ] . the figure legend identifies the correspondence between line type and yield source , where the abbreviations are the same as those defined in the footnote to table [ t2 ] . note that massive star integrated yields are indicated with bold lines , while thin lines signify ims integrated yields . for 4 note that the three ims yield sets predict similar results except at low metallicity , where the m01 yields are higher . according to m01 , this difference is presumably due to the earlier activation and larger efficiency of third dredge - up in her models . it s also clear that ims contribute to the cosmic buildup of 4 at roughtly the 20 - 30% level . the rv yields for tend to be less than those of hg , while those of m01 are greater , due to differences in onset time and average efficiency of third dredge - up . globally , the role of ims in production is therefore ambiguous , because it depends upon which set of massive star yields one uses to compare with the ims yields . for example , ims yields are comparable to the massive star yields of ww , yet significantly less than those of portinari et al . finally , there is a significant difference between the three yields sets where 14 is concerned . the lifetimes of the thermal pulses in the stars at the upper end of the mass range in m01 s calculations are largely responsible for her 14 yields being significantly less than the others. on the other hand , rv s lower mass loss rate lengthens a star s lifetime on the late agb and results in more 14 production . when compared with massive star yields , rv and hg predict that ims will produce several times more 14 , particularly at lower z , when compared to either the ww or p yields . on the other hand , m01 s models predict less 14 . universally speaking , then , ims yield predictions indicate that these stars contribute significantly to 14 production , moderately to productions , and hardly at all to 4 production . remember , though , that these conclusions are heavily based upon model predictions . the strength of these conclusions is only as strong as the models are realistic . the respective roles of ims and massive stars in galactic chemical evolution can be further assessed by confronting observations of abundance gradients and element ratio plots with chemical evolution models which employ the various yields to make their predictions . because there is a time delay of at least 30 myr between birth and release of products by ims , these roles may be especially noticeable in young systems whose ages are roughly comparable to such delay times or in systems which experienced a burst less that 30 myr ago . henry , edmunds , & kppen ( 2000 ; hek ) explored the c / o vs. o / h and n / o vs. o / h domains in great detail , using both analytical and numerical models to test the general trends observed in a large and diverse sample of galactic and extragalactic h ii regions located in numerous spiral and dwarf irregular galaxies . using the ims yields of hg and the massive star yields of maeder ( 1992 ) they were able to explain the broad trends in the data , and in the end they concluded that while massive stars produce nearly all of the in the universe , ims produce nearly all of the 14 . they also illustrated the impact of the star formation rate on the age - metallicity relation and the behavior of the n / o value as metallicity increases in low metallicity systems . in this conference , moll , gaviln , & buell ( 2003 ) report on their chemical evolution models which use the buell ( 1997 ) ims yields along with the ww massive star yields , where the former employ the mass loss rate scheme of vassiliadis & wood ( 1993 ) . their model results confirm those of hek in terms of the star formation rate and the age - metallicity relation . recently pilyugin et al . ( 2003 ) reexamined the issue of the origin of nitrogen and found that presently the stellar mass range responsible for this element can not be clearly identified because of limitations in the available data . chiappini et al . ( 2003 ) have explored the cno question using chemical evolution models to study the distribution of elements in the milky way disk as well as the disk of m101 and dwarf irregular galaxies using the hg and ww yields . like hek , they conclude that 14 is largely produced by ims . however , they find that by assuming that the ims mass loss rate varies directly with metallicity , production in these stars is relatively enhanced at low z. in the end , they conclude that ims , not massive stars , control the universal evolution of , in disagreement with hek . figure [ chiappini ] is similar to fig . 10 in their paper and is shown here to graphically illustrate the effects of ims on the chemical evolution of and 14 , according to their models . each panel shows logarithmic abundance as a function of galactocentric distance in kpc for the milky way disk . besides the data points , two model results are shown in each panel . the solid line in each case corresponds to the best - fit model in their paper , while the dashed line is for the same model but with the ims contribution to nucleosynthesis turned off . as can be seen , ims make roughly a 0.5 dex and 1 dex difference in the case of c and n , respectively , i.e. their effects are sizeable . finally , the question of ims production of nitrogen has become entangled in the debate over the interpretation of the apparent bimodal distribution of damped lyman-@xmath0 systems ( dlas ) in the n/@xmath0@xmath0/h plane ( prochaska et al . 2002 ; centurin et al . 2003 ) . represents elements such as o , mg , si , and s , whose abundances are assumed to scale in lockstep . ] most dlas fall in the region of the `` primary plateau , '' located at a [ n/@xmath0 ] value of @xmath3 - 0.7 and between metallicities of -1.5 and -2.0 on the [ @xmath0/h ] axis . however , a few objects are positioned noticeably below the plateau by roughly 0.8 dex in [ n/@xmath0 ] , although still within the same metallicity range as the plateau objects . the prochaska group proposes that these low n objects ( ln - dlas ) correspond to systems characterized by a top - heavy initial mass function with a paucity of ims , or , in the same spirit , a population of massive stars truncated below some threshold mass . either possibility works through suppressing the ims contribution to nitrogen production by reducing the proportion of these stars in a system s stellar population . the centurin group , on the other hand , suggests that ln - dlas are less evolved than the plateau objects , i.e. star formation occurred within them less than 30 myr ago , so the ln - dlas are momentarily pausing at the low - n region until their slowly evolving ims begin to release their nitrogen . the latter picture , while not needing to invoke a non - standard imf ( an action which causes great discomfort among astronomers ) , does require that the time to evolve from the low - n ledge to the plateau region be very quick , otherwise their idea is inconsistent with the observed absence of a continuous trail of objects connecting these points . this problem is bound to be solved when the number of dlas with measured nitrogen abundances increases , but it nevertheless illustrates an important role that ims play in questions involving early chemical evolution in the universe . intermediate mass stars play an important role in the chemical evolution of , , 14 , and @xmath14li as well as s - process isotopes . stellar models have gained in sophistication over the past two decades , so that currently they include effects of three dredge - up stages , thermal pulsing and hot bottom burning on the agb , metallicity , and mass loss by winds and sudden ejection . generally speaking yield predictions from stellar evolution models indicate that yields increase as metallicity declines , as the mass loss rate is reduced , and when rotation is included . furthermore , observational evidence supports the claim that the lower mass limit for hot bottom burning is between 3 and 4 . integration of yields over a salpeter initial mass function shows clearly that ims have little impact on the evolution of 4 while at the same time playing a dominant role in the cosmic buildup of 14 . the case of is a bit more confused . the issue of 14 production is particularly important in the current discussion of the distribution of damped lyman-@xmath0 systems in the n/@xmath0@xmath0/h plane . finally , what i believe is needed are grids of models which attempt to treat ims and massive stars in a consistent and seamless manner . the role of each stellar mass range would be easier to judge if yield sets of separate origins did not have to be patched together in chemical evolution models . otherwise , it is not clear to what extent the various assumptions which are adapted by stellar evolution theorists impact ( and therefore confuse ! ) the analyses . i d like to thank the organizing committee for inviting me to write this review and to present these ideas at the conference . i also want to thank corinne charbonnel , georges meynet , francesca matteucci , cristina chiappini , jason prochaska , john cowan , and paulo molaro for clarifying my understanding on several topics addressed in this review . finally , i am grateful to the nsf for supporting my work under grant ast 98 - 19123 . charbonnel , c. 2003 , carnegie observatories astrophysics series , vol . 4 : origin and evolution of the elements , ed . a. mcwilliam and m. rauch ( pasadena : carnegie observatories , http://www.ociw.edu/ociw/symposia/series/symposium4/proceedings.html karakas , a.i . , & lattanzio , j.c . 2003 , carnegie observatories astrophysics series , vol . 4 : origin and evolution of the elements , ed . a. mcwilliam and m. rauch ( pasadena : carnegie observatories , http://www.ociw.edu/ociw/symposia/series/symposium4/proceedings.html moll , m. , gaviln , m. , & buell , j.f . 2003 , carnegie observatories astrophysics series , vol . 4 : origin and evolution of the elements , ed . a. mcwilliam and m. rauch ( pasadena : carnegie observatories , http://www.ociw.edu/ociw/symposia/series/symposium4/proceedings.html
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intermediate mass stars occupy the mass range between 0.8 - 8 . in this contribution ,
evolutionary models of these stars from numerous sources are compared in terms of their input physics and predicted yields .
in particular , the results of renzini & voli , van den hoek & groenewegen , and marigo are discussed . generally speaking , it is shown that yields of 4 , , and 14 decrease with increasing metallicity , reduced mass loss rate , and increased rotation rate .
integrated yields and recently published chemical evolution model studies are used to assess the relative importance of intermediate mass and massive stars in terms of their contributions to universal element buildup .
intermediate mass stars appear to play a major role in the chemical evolution of 14 , a modest role in the case of , and a small role for 4 .
furthermore , the time delay in their release of nuclear products appears to play an important part in explaining the apparent bimodality in the distribution of damped lyman-@xmath0 systems in the n/@xmath0@xmath0/h plane .
[ 1996/06/01 ] a&a 4@xmath1he 14@xmath2n
| 4,816 | 281 |
we investigate the vector and scalar self - energies of nucleons in nuclear matter composed by the neutrons and protons , distributed with densities @xmath0 and @xmath1 . we calculate the dependence on the total density @xmath2 and on the asymmetry parameter @xmath3 . the qcd sum rules were invented in paper @xcite to express the hadron parameters through the vacuum expectation values of qcd operators . being initially used for the mesons , the method was expanded in @xcite to the description of the baryons . the approach succeeded in describing the static characteristics as well as some of the dynamical characteristics of the hadrons in vacuum see , the reviews @xcite . later the qcd sum rules were applied for investigation of modified nucleon parameters in the symmetric nuclear matter @xcite . they were based on the borel - transformed dispersion relation for the function @xmath4 describing the propagation of the system with the quantum numbers of the nucleon ( the proton ) in the nuclear matter . considering nuclear matter as a system of @xmath5 nucleons with momenta @xmath6 , one introduces the vector @xmath7 , which is thus @xmath8 in the rest frame of the matter . the function @xmath4 can be presented as @xmath9 with the arbitrary function @xmath10 being kept constant in the dispersion relations in @xmath11 . the general form of the function @xmath12 can thus be presented as @xmath13 the in - medium qcd sum rules are the borel - transformed dispersion relations for the components @xmath14 @xmath15 @xmath16 the spectrum of the function @xmath4 is much more complicated than that of the function @xmath17 describing the propagation of the system with the quantum numbers of the nucleon in the vacuum . the choice of the function @xmath10 is dictated by the attempts to separate the singularities connected with the nucleon in the matter from those connected with the properties of the matter itself . since the latter manifest themselves as singularities in the variable @xmath18 , the separation can be done by putting @xmath19 and by fixing @xcite @xmath20 ( @xmath21 is the nucleon mass ) . by using eq . ( [ 8 ] ) the characteristics of the nucleon in nuclear matter can be expressed through the in - medium values of qcd condensate . the possibility of extension of `` pole + continuum '' model @xcite to the case of finite densities was shown in @xcite@xcite . the lowest order of ope of the lhs of eq . ( [ 8 ] ) can be presented in terms of the vector and scalar condensates @xcite , @xcite . vector condensates @xmath22 of the quarks with the flavor @xmath23 ( @xmath24 denotes the ground state of the matter ) are the linear functions of the nucleon densities @xmath0 and @xmath1 . in the asymmetric matter both su(2 ) symmetric and asymmetric condensates @xmath25 @xmath26 obtain nonzero values . in the rest frame of the matter @xmath27 , @xmath28 , @xmath29 . we can present @xmath30 the values @xmath31 are just the numbers of the valence quarks in the nucleons @xmath32 , @xmath33 , and thus @xmath34 hence , we obtain @xmath35 with @xmath36 , @xmath37 . the lhs of eq . ( [ 8 ] ) contains the su(2 ) symmetric scalar condensate @xmath38 , and su(2 ) asymmetric one @xmath39 . these condensates can be presented as @xmath40 @xmath41 is the vacuum value , and @xmath42 the dots in the rhs of eqs . ( [ 15 ] ) and ( [ 16 ] ) denote the terms , which are nonlinear in @xmath43 . in the gas approximation such terms should be omitted . the su(2 ) invariance of vacuum was assumed in eq . ( [ 16 ] ) . the expectation value @xmath44 is related to the @xmath45 sigma term @xmath46 @xcite . the gluon condensate @xmath47 @xmath48 is the vacuum value and @xmath49 obtained in a model - independent way . we shall analyze the sum rules in the gas approximation . it is a reasonable starting point , since the nonlinear contributions to the most important scalar condensate @xmath50 are relatively small at the densities of the order of the phenomenological saturation density @xmath51 of the symmetric matter @xcite . in the second part of the talk we discuss the role of the radiative corrections . the analysis [ 2 ] included also the most important radiative corrections , in which the coupling constant @xmath52 is enhanced by the large logarithm " @xmath53 . the corrections @xmath54 have been included in to all orders for the leading ope terms . this approach provided us with good results for the nucleon mass and for the other characteristics of nucleons . however , inclusion of the lowest order radiative corrections beyond the logarithmic approximation made the situation somewhat more complicated . a numerically large coefficient of the lowest radiative correction to the leading ope of the polarization operator @xmath17 was obtained in @xcite . a more consistent calculation @xcite provided this coefficient to be about 6 . thus , the radiative correction increases this term by about 50% at @xmath55 , which are actual for the sr analysis . this uncomfortably large correction is often claimed as the most weak point of the sr approach @xcite . the radiative corrections of the order @xmath56 and @xmath52 for the contributions up to @xmath57 have been calculated in @xcite . the further development of the nucleon sr in nuclear matter needs the calculation of the radiative corrections . this work is in progress and now we have present the analysis of the role of the radiative corrections in vacuum @xcite . we present the nucleon propagator in nuclear matter as @xmath58 with the total self - energy @xmath59 . we shall use the qcd sum rules for the calculation of the nucleon characteristics @xmath60 identified with the vector self - energy , dirac effective mass and the effective scalar self - energy see @xcite . @xmath61 with @xmath62 and @xmath63 defined by eq . ( [ 23 ] ) . the new position of the nucleon pole is @xmath64 we present also the result for the single - particle potential energies @xmath65 we trace the dependence of these characteristics on the total density @xmath43 and on the asymmetry parameter . the borel - transformed sum rules take the form @xmath66 with @xmath67 , where @xmath68 is the effective value of the nucleon residue in nuclear matter , @xmath69 is borel mass ( @xmath70gev@xmath71 ) . we shall include subsequently the contributions of three types . the terms @xmath72 @xmath73 stand for the lowest order local condensates . these contributions are similar to simple exchanges by isovector vector and scalar mesons between the nucleon and the nucleons of the matter . the terms @xmath74 are caused by the nonlocalities of the vector condensate . they correspond to the account of the form factors in the vertices between the isovector mesons couple to the nucleons . finally , @xmath75 describes the contributions of the four - quark condensates . they correspond to the two - meson exchanges ( or to exchanges by four - quark mesons , if there are any ) and to somewhat more complicated structure of the meson - nucleon vertices . thus we present the lhs of eqs . ( [ 52])([54 ] ) as @xmath76 actually , we shall solve the sum rules equations , subtracting the vacuum effects @xcite : @xmath77 with @xmath78 , @xmath79 , @xmath80 , while @xmath81 and @xmath82 are the corresponding contributions in the vacuum case . * a. local condensates of the lowest dimensions . * + the terms @xmath83 have the form : @xmath84 with the dependence on @xmath43 and @xmath85 being contained in the factors @xmath86 and the function @xmath87 , given by eqs . ( [ 15 ] ) and ( [ 16 ] ) . the other functions are @xcite @xmath88 the notation @xmath89 means that the functions @xmath90 depend on the ratio @xmath91 . the factor @xmath92 , where @xmath93 gev , @xmath94 gev . * b. inclusion of the nonlocal condensates . * + finally , the higher moments and higher twists of the nucleon structure functions provide the contributions @xmath95 to the @xmath96 of the sum rules eq . ( [ 61 ] ) @xmath97\ , \nn\\ & & \quad u^q_{n,2}(m^2)=\frac{8\pi^2}{3l^{4/9}m}\left [ \frac32\,m^2m^2e_{0m}(\langle\eta^u\alpha\rangle -\langle\eta^d\alpha\rangle ) \right ] ; \label{85 } \\ \nn\\ \nn\\ & & u^p(m^2)\ = \ ( u^p_{n,1}(m^2 ) + \beta u^p_{n,2}(m^2))\rho\ , \nn\\ & & \quad u^p_{n,1}(m^2)=\frac{8\pi^2}{3l^{4/9}}\bigg[-5\left(m^4 e_{1 m } -(s - m^2)m^2 e_{0m}\right)\langle\eta\alpha\rangle \nn\\ & & \quad -\ \frac{12}5 m^2m^2e_{0m}\langle\eta\alpha^2\rangle + \ \frac{18}5m^2m^2e_{0m}\langle\xi\rangle\bigg]\ , \nn \\ & & \quad u^p_{n,2}(m^2)=\frac{8\pi^2}{3l^{4/9}}\bigg [ 3\left(m^4e_{1m}-(s - m^2)m^2e_{0m}\right ) \langle(\eta^u-\eta^d)\alpha\rangle \nn\\ \label{86 m } & & \quad+\ \frac95\,m^2m^2e_{0m}\langle(\eta^u-\eta^d)\alpha^2\rangle -\frac{27}{10}\,m^2m^2e_{0 m } \langle(\xi^u-\xi^d)\rangle \bigg ] ; \\ \nn\\ \nn\\ & & u^i(m^2)\ = \ 0\ . \label{85a}\end{aligned}\ ] ] here we denote @xmath98 , @xmath99 @xcite and @xmath100 . we use the structure functions @xmath101 , obtained in @xcite for the calculation of the terms @xmath102 and @xmath103 . here we denote @xmath104 . * c. inclusion of the four - quark condensates . * + the calculations of the contributions of the four - quark condensates require the model assumption on the structure of the nucleon . the complete set of the four - quark condensate was obtained in @xcite by using the perturbative chiral quark model ( pcqm ) . there are three types of contributions to the four - quark condensate in the framework of this approach . all four operators can act on the valence quarks . also , four operators can act on the pion . there is a possibility that two of the operators act on the valence quarks while the other two act on the pions . using the complete set of the nucleon four - quark expectation values @xcite , we obtain @xmath105 with @xmath106 . the calculations give @xmath107 the contributions of the four - quark condensates to the lhs of the borel transformed sum rules ( [ 61 ] ) can be presented as @xmath108 in fig . 1 we show results for effective nucleon mass @xmath63 ( [ 23 ] ) and for the single - particle potential energies ( [ 24 ] ) in symmetric and asymmetric nuclear matter . during calculation we use the value @xmath109 ( [ 15 ] ) , and @xmath110 ( [ 16 ] ) in eq.(19 ) @xcite . we have made @xcite the analysis of radiative corrections to the nucleon sr in vacuum . the lowest ope terms of the operators @xmath111 and @xmath112 in vacuum can be presented as @xmath113 here the lower indices show the dimensions of the condensates , contained in the corresponding terms , @xmath114 is the contribution of the free quark loop . we consider the radiative corrections to the terms @xmath114 , @xmath115 , and @xmath116 . the radiative corrections to the other terms are not included since the values of the corresponding condensates are known with poor accuracy . we find , following @xcite , for the corresponding contributions from @xmath52-corrections to the borel transformed sr : @xmath117 \nonumber\\ & & -\ \frac{\alpha_s}{\pi}\bigg[m^4w^2\!\left(1+\frac{3w^2}{4m^2 } \right)e^{-w^2/m^2 } \nonumber\\ & & + \ m^6 { \cal e}\!\left(-{\rm w^2/ m^2}\right)\bigg ] , \nonumber\\ & & \hspace*{-0.7 cm } \tilde a_6(m^2,w^2)\ = \ \frac43\,a^2 \\ & & \hspace*{-0.3 cm } \times \left[\!1\!-\frac{\alpha_s}\pi\!\left(\!\frac56+\frac13\ ! \left(\!\ln\frac{w^2}{\nu^2}\!+{\cal e}(-{\rm w^2/m^2})\!\right)\!\right)\!\right]\ ! , \nonumber\\ & & \hspace*{-0.7 cm } \tilde b_3(m^2,w^2)\ = \ 2am^4e_1\left(1+\frac32\,\frac{\alpha_s}\pi\right ) \nonumber\end{aligned}\ ] ] with @xmath118 . some terms in ( 28 ) differ from those in @xcite . the numerical difference is , however , not very important . the parts of equations proportional to @xmath52 are the @xmath52-corrections to main terms . now we compare the nucleon parameters obtained as solutions of nucleon sr without @xmath52-corrections , @xmath119 and with @xmath52-corrections , @xmath120 these results are presented for @xmath52=0.35 . we show that in vacuum the radiative corrections modify mainly the values of the nucleon residue , while that of the nucleon mass suffers minor changes .
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we calculate the nucleon parameters in isospin symmetric and asymmetric nuclear matter using the qcd sum rules . the higher moments of the nucleon structure functions are included .
the complete set of the nucleon expectation values of the four - quark operators is employed .
we analyze the role of the lowest order radiative corrections beyond the logarithmic approximation .
| 4,233 | 91 |
given an input sequence , _ segmentation _ is the problem of identifying and assigning tags to its subsequences . many natural language processing ( nlp ) tasks can be cast into the segmentation problem , like named entity recognition @xcite , opinion extraction @xcite , and chinese word segmentation @xcite . properly representing _ segment _ is critical for good segmentation performance . widely used sequence labeling models like conditional random fields @xcite represent the contextual information of the segment boundary as a proxy to entire segment and achieve segmentation by labeling input units ( e.g. words or characters ) with boundary tags . compared with sequence labeling model , models that directly represent segment are attractive because they are not bounded by local tag dependencies and can effectively adopt segment - level information . semi - markov crf ( or semi - crf ) @xcite is one of the models that directly represent the entire segment . in semi - crf , the conditional probability of a semi - markov chain on the input sequence is explicitly modeled , whose each state corresponds to a subsequence of input units , which makes semi - crf a natural choice for segmentation problem . however , to achieve good segmentation performance , conventional semi - crf models require carefully hand - crafted features to represent the segment . recent years witness a trend of applying neural network models to nlp tasks . the key strengths of neural approaches in nlp are their ability for modeling the compositionality of language and learning distributed representation from large - scale unlabeled data . representing a segment with neural network is appealing in semi - crf because various neural network structures @xcite have been proposed to compose sequential inputs of a segment and the well - studied word embedding methods @xcite make it possible to learn entire segment representation from unlabeled data . in this paper , we combine neural network with semi - crf and make a thorough study on the problem of representing a segment in neural semi - crf . @xcite proposed a segmental recurrent neural network ( srnn ) which represents a segment by composing input units with rnn . we study alternative network structures besides the srnn . we also study segment - level representation using _ segment embedding _ which encodes the entire segment explicitly . we conduct extensive experiments on two typical nlp segmentation tasks : named entity recognition ( ner ) and chinese word segmentation ( cws ) . experimental results show that our concatenation alternative achieves comparable performance with the original srnn but runs 1.7 times faster and our neural semi - crf greatly benefits from the segment embeddings . in the ner experiments , our neural semi - crf model with segment embeddings achieves an improvement of 0.7 f - score over the baseline and the result is competitive with state - of - the - art systems . in the cws experiments , our model achieves more than 2.0 f - score improvements on average . on the pku and msr datasets , state - of - the - art f - scores of 95.67% and 97.58% are achieved respectively . we release our code at https://github.com/expresults/segrep-for-nn-semicrf . figure [ fig : ne - and - cws ] shows examples of named entity recognition and chinese word segmentation . for the input word sequence in the ner example , its segments ( _ `` michael jordan'':per , `` is'':none , `` a'':none , `` professor'':none , `` at'':none , `` berkeley'':org _ ) reveal that `` michaels jordan '' is a person name and `` berkeley '' is an organization . in the cws example , the subsequences ( utf8gkai``/pudong '' , `` /development '' , `` /and '' , `` /construction '' ) of the input character sequence are recognized as words . both ner and cws take an input sequence and partition it into disjoint subsequences . formally , for an input sequence @xmath0 of length @xmath1 , let @xmath2 denote its subsequence @xmath3 . segment _ of @xmath4 is defined as @xmath5 which means the subsequence @xmath6 is associated with label @xmath7 . a _ segmentation _ of @xmath4 is a _ segment _ sequence @xmath8 , where @xmath9 and @xmath10 . given an input sequence @xmath4 , the _ segmentation _ problem can be defined as the problem of finding @xmath4 s most probable _ segment _ sequence @xmath11 . 0.241 0.241 0.241 0.241 semi - markov crf ( or semi - crf , figure [ fig : std ] ) @xcite models the conditional probability of @xmath11 on @xmath4 as @xmath12 where @xmath13 is the feature function , @xmath14 is the weight vector and @xmath15 is the normalize factor of all possible _ segmentations _ @xmath16 over @xmath4 . by restricting the scope of feature function within a segment and ignoring label transition between segments ( 0-order semi - crf ) , @xmath13 can be decomposed as @xmath17 where @xmath18 maps segment @xmath19 into its representation . such decomposition allows using efficient dynamic programming algorithm for inference . to find the best segmentation in semi - crf , let @xmath20 denote the best segmentation ends with @xmath21^th^ input and @xmath20 is recursively calculated as @xmath22 where @xmath23 is the maximum length manually defined and @xmath24 is the transition weight for @xmath25 in which @xmath26 . previous semi - crf works @xcite parameterize @xmath27 as a sparse vector , each dimension of which represents the value of corresponding feature function . generally , these feature functions fall into two types : 1 ) the _ crf style features _ which represent input unit - level information such as `` the specific words at location @xmath28 '' 2 ) the _ semi - crf style features _ which represent segment - level information such as `` the length of the segment '' . @xcite proposed the segmental recurrent neural network model ( srnn , see figure [ fig : rnn ] ) which combines the semi - crf and the neural network model . in srnn , @xmath29 is parameterized as a bidirectional lstm ( bi - lstm ) . for a segment @xmath9 , each input unit @xmath30 in subsequence @xmath31 is encoded as _ embedding _ and fed into the bi - lstm . the rectified linear combination of the final hidden layers from bi - lstm is used as @xmath29 . @xcite pioneers in representing a segment in neural semi - crf . bi - lstm can be regarded as `` neuralized '' _ crf style features _ which model the input unit - level compositionality . however , in the srnn work , only the bi - lstm was employed without considering other input unit - level composition functions . what is more , the _ semi - crf styled _ segment - level information as an important representation was not studied . in the following sections , we first study alternative input unit - level composition functions ( [ sec : alt - inp - rep ] ) . then , we study the problem of representing a segment at segment - level ( [ sec : seg - rep ] ) . besides recurrent neural network ( rnn ) and its variants , another widely used neural network architecture for composing and representing variable - length input is the convolutional neural network ( cnn ) @xcite . in cnn , one or more filter functions are employed to convert a fix - width segment in sequence into one vector . with filter function `` sliding '' over the input sequence , contextual information is encoded . finally , a pooling function is used to merge the vectors into one . in this paper , we use a filter function of width 2 and max - pooling function to compose input units of a segment . following srnn , we name our cnn segment representation as scnn ( see figure [ fig : cnn ] ) . however , one problem of using cnn to compose input units into segment representation lies in the fact that the max - pooling function is insensitive to input position . two different segments sharing the same vocabulary can be treated without difference . in a cws example , utf8gkai`` '' ( racket for sell ) and `` '' ( ball audition ) will be encoded into the same vector in scnn if the vector of utf8gkai`` '' that produced by filter function is always preserved by max - pooling . concatenation is also widely used in neural network models to represent fixed - length input . although not designed to handle variable - length input , we see that in the inference of semi - crf , a maximum length @xmath23 is adopted , which make it possible to use padding technique to transform the variable - length representation problem into fixed - length of @xmath23 . meanwhile , concatenation preserves the positions of inputs because they are directly mapped into the certain positions in the resulting vector . in this paper , we study an alternative concatenation function to compose input units into segment representation , namely the sconcate model ( see figure [ fig : concate ] ) . compared with srnn , sconcate requires less computation when representing one segment , thus can speed up the inference . for segmentation problems , a segment is generally considered more informative and less ambiguous than an individual input . incorporating segment - level features usually lead performance improvement in previous semi - crf work . segment representations in section [ sec : alt - inp - rep ] only model the composition of input units . it can be expected that the segment embedding which encodes an entire subsequence as a vector can be an effective way for representing a segment . in this paper , we treat the segment embedding as a lookup - based representation , which retrieves the embedding table with the surface string of entire segment . with the entire segment properly embed , it is straightforward to combine the segment embedding with the composed vector from the input so that multi - level information of a segment is used in our model ( see figure [ fig : with - seg ] ) . however , how to obtain such embeddings is not a trivial problem . a natural solution for obtaining the segment embeddings can be collecting all the `` correct '' segments from training data into a lexicon and learning their embeddings as model parameters . however , the in - lexicon segment is a strong clue for a subsequence being a correct segment , which makes our model vulnerable to overfitting . unsupervised pre - training has been proved an effective technique for improving the robustness of neural network model @xcite . to mitigate the overfitting problem , we initialize our segment embeddings with the pre - trained one . word embedding gains a lot of research interest in recent years @xcite and is mainly carried on english texts which are naturally segmented . different from the word embedding works , our segment embedding requires large - scale segmented data , which can not be directly obtained . following @xcite which utilize automatically segmented data to enhance their model , we obtain the auto - segmented data with our neural semi - crf baselines ( srnn , scnn , and sconcate ) and use the auto - segmented data to learn our segment embeddings . another line of research shows that machine learning algorithms can be boosted by ensembling _ heterogeneous _ models . our neural semi - crf model can take knowledge from heterogeneous models by using the segment embeddings learned on the data segmented by the heterogeneous models . in this paper , we also obtain the auto - segmented data from a conventional crf model which utilizes hand - crafted sparse features . once obtaining the auto - segmented data , we learn the segment embeddings in the same with word embeddings . a problem that arises is the fine - tuning of segment embeddings . fine - tuning can learn a task - specific segment embeddings for the segments that occur in the training data , but it breaks their relations with the un - tuned out - of - vocabulary segments . figure [ fig : wo - ft ] illustrates this problem . since oov segments can affect the testing performance , we also try learning our model without fine - tuning the segment embeddings . in this section , we describe the detailed architecture for our neural semi - crf model . following @xcite , we use a bi - lstm to represent the input sequence . to obtain the input unit representation , we use the technique in @xcite and separately use two parts of input unit embeddings : the pre - trained embeddings @xmath32 without fine - tuning and fine - tuned embeddings @xmath33 . for the @xmath28th input , @xmath34 and @xmath35 are merged together through linear combination and form the input unit representation @xmath36 + b^\mathcal{i})\ ] ] where the notation of @xmath37 $ ] equals to @xmath38 s linear combination @xmath39 and @xmath40 is the bias . after obtaining the representation for each input unit , a sequence @xmath41 is fed to a bi - lstm . the hidden layer of forward lstm @xmath42 and backward lstm @xmath43 are combined as @xmath44+b^\mathcal{h})\ ] ] and used as the @xmath28^th^ input unit s final representation . given a segment @xmath9 , a generic function scomp@xmath45 stands for the segment representation that composes the input unit representations @xmath46 . in this work , scomp is instantiated with three different functions : srnn , scnn and sconcate . besides composing input units , we also employ the segment embeddings as segment - level representation . embedding of the segment @xmath9 is denoted as a generic function semb@xmath47 which converts the subsequence @xmath48 into its embedding through a lookup table . at last , the representation of segment @xmath19 is calculated as @xmath49+b^\mathcal{s})\ ] ] where @xmath50 is the embedding for the label of a segment . .hyper - parameter settings [ cols= " > , < " , ] table [ tbl : cws - stoa ] shows the comparison with the state - of - the - art cws systems . the first block of table [ tbl : cws - stoa ] shows the neural cws models and second block shows the non - neural models . our neural semi - crf model with multi - level segment representation achieves the state - of - the - art performance on pku and msr data . on ctb6 data , our model s performance is also close to @xcite which uses semi - supervised features extracted auto - segmented unlabeled data . according to @xcite , significant improvements can be achieved by replacing character embeddings with character - bigram embeddings . however we did nt employ this trick considering the unification of our model . semi - crf has been successfully used in many nlp tasks like information extraction @xcite , opinion extraction @xcite and chinese word segmentation @xcite . its combination with neural network is relatively less studied . to the best of our knowledge , our work is the first one that achieves state - of - the - art performance with neural semi - crf model . domain specific knowledge like capitalization has been proved effective in named entity recognition @xcite . segment - level abstraction like whether the segment matches a lexicon entry also leads performance improvement @xcite . to keep the simplicity of our model , we did nt employ such features in our ner experiments . but our model can easily take these features and it is hopeful the ner performance can be further improved . utilizing auto - segmented data to enhance chinese word segmentation has been studied in @xcite . however , only statistics features counted on the auto - segmented data was introduced to help to determine segment boundary and the entire segment was not considered in their work . our model explicitly uses the entire segment . in this paper , we systematically study the problem of representing a segment in neural semi - crf model . we propose a concatenation alternative for representing segment by composing input units which is equally accurate but runs faster than srnn . we also propose an effective way of incorporating segment embeddings as segment - level representation and it significantly improves the performance . experiments on named entity recognition and chinese word segmentation show that the neural semi - crf benefits from rich segment representation and achieves state - of - the - art performance . this work was supported by the national key basic research program of china via grant 2014cb340503 and the national natural science foundation of china ( nsfc ) via grant 61133012 and 61370164 . chris dyer , miguel ballesteros , wang ling , austin matthews , and noah a. smith . transition - based dependency parsing with stack long short - term memory . in _ acl-2015 _ , pages 334343 , beijing , china , july 2015 . acl . wenbin jiang , meng sun , yajuan l , yating yang , and qun liu . discriminative learning with natural annotations : word segmentation as a case study . in _ acl-2013 _ , pages 761769 , sofia , bulgaria , august 2013 . john d. lafferty , andrew mccallum , and fernando c. n. pereira . conditional random fields : probabilistic models for segmenting and labeling sequence data . in _ icml 01 _ , pages 282289 , san francisco , ca , usa , 2001 . daisuke okanohara , yusuke miyao , yoshimasa tsuruoka , and junichi tsujii . improving the scalability of semi - markov conditional random fields for named entity recognition . in _ acl-2006 _ , pages 465472 , sydney , australia , july 2006 . acl . xu sun , yaozhong zhang , takuya matsuzaki , yoshimasa tsuruoka , and junichi tsujii . a discriminative latent variable chinese segmenter with hybrid word / character information . in naacl-2009 _ , pages 5664 , boulder , colorado , june 2009 . yiou wang , junichi kazama , yoshimasa tsuruoka , wenliang chen , yujie zhang , and kentaro torisawa . improving chinese word segmentation and pos tagging with semi - supervised methods using large auto - analyzed data . in _ ijcnlp-2011 _ , pages 309317 , chiang mai , thailand , november 2011 .
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many natural language processing ( nlp ) tasks can be generalized into segmentation problem . in this paper , we combine semi - crf with neural network to solve nlp segmentation tasks .
our model represents a segment both by composing the input units and embedding the entire segment .
we thoroughly study different composition functions and different segment embeddings .
we conduct extensive experiments on two typical segmentation tasks : named entity recognition ( ner ) and chinese word segmentation ( cws ) .
experimental results show that our neural semi - crf model benefits from representing the entire segment and achieves the state - of - the - art performance on cws benchmark dataset and competitive results on the conll03 dataset .
| 5,204 | 174 |
molecules can be assembled from atoms using laser light . this process is termed photoassociation . with the advent of femtosecond lasers and pulse shaping techniques , photoassociation became a natural candidate for coherent control of a binary reaction . coherent control had been conceived as a method to determine the fate of chemical reactions using laser fields.@xcite the basic idea is to employ interference of matter waves to constructively enhance a desired outcome while destructively suppressing all undesired alternatives.@xcite control is exerted by shaping the laser pulses , the simplest control knobs being time delays and phase differences.@xcite over the last two decades , the field of coherent control has developed significantly both theoretically and experimentally.@xcite however , a critical examination of the achievements reveals that successful control has been demonstrated almost exclusively for unimolecular processes such as ionization , dissociation and fragmentation . it is natural to ask why the reverse process of controlling binary reactions@xcite is so much more difficult . the main difference between unimolecular processes and a binary reaction lies in the initial state a single or few well - defined bound quantum states vs an incoherent continuum of scattering states.@xcite for a binary reaction , the nature of the scattering continuum is mainly determined by the temperature of the reactants . as temperature decreases , higher partial waves are frozen out . at the very low temperatures of ultracold gases , the scattering energy of atom pairs is so low that the rotational barrier can not be passed , and the scattering becomes purely @xmath0-wave.@xcite in this regime , the reactants are pre - correlated due to quantum threshold effects@xcite and the effect of scattering resonances is particularly pronounced.@xcite at a temperature of about 100@xmath1k , photoassociation with femtosecond laser pulses has been demonstrated.@xcite coherent transient rabi oscillations were observed as the prominent feature in the pump - probe spectra . the transients are due to long tails of the pulses caused by a sharp spectral cut which is necessary to avoid excitation into unbound states.@xcite this pinpoints to the fact that the large spectral bandwidth of a femtosecond pulse is unsuitable to one - photon photoassociation at ultralow temperatures . in this regime , a narrow - band transition needs to be driven in order to avoid atomic excitation.@xcite the situation changes completely for high temperatures where the scattering states can penetrate rotational barriers due to the large translational kinetic energy . the association process is then likely to happen at short internuclear distance close to the inner turning point and for highly excited rotational states . in this case , the large spectral bandwidth of femtosecond laser pulses is ideally adapted to both the broad thermal width of the ensemble of scattering states and the depth of the electronically excited state potential in which molecules are formed . the disadvantage of this setting is that the initial state is completely incoherent , impeding control of the photoreaction . photoassociation with femtosecond laser pulses was first demonstrated under these conditions , employing a one - photon transition in the uv.@xcite subsequent to the photoassociation , coherent rotational motion of the molecules was observed.@xcite we have recently demonstrated generation of both rotational and vibrational coherences by two - photon femtosecond photoassociation of hot atoms.@xcite this is a crucial step toward the coherent control of photoinduced binary reactions since the fate of bond making and breaking is determined by the vibrational motion . employing multi - photon transitions comes with several advantages : the class of molecules that can be photoassociated by near - ir / visible femtosecond laser pulses is significantly larger for multi - photon than one - photon excitation . femtosecond laser technology is most advanced in the near - ir spectral region . due to the different selection rules , different electronic states become accessible for multi - photon transitions compared to one - photon excitation . control strategies differ for multi - photon and one - photon excitation . in particular , large dynamic stark shifts and an extended manifold of quantum pathways that can be interfered come into play for multi - photon excitation.@xcite the theoretical description needs to account for these strong - field effects . we have constructed a comprehensive theoretical model from first principles to describe the experiment in which magnesium atoms in a heated cell are photoassociated by femtosecond laser pulses.@xcite it is summarized in figure [ fig : scheme ] . magnesium in its electronic ground state is a closed shell atom . its ground electronic potential , x@xmath2 , therefore displays only a weak van der waals attractive well . a femtosecond pulse of 100@xmath3fs transform - limited duration with a central wavelength of @xmath4 nm promotes an electron to the @xmath5 orbital . this two - photon transition is driven since a wavelength of 840@xmath3 nm is far from any one - photon resonance both for magnesium atoms and mg@xmath6 molecules , cf . [ fig : scheme ] . upon excitation , a strong chemical bond is formed in the @xmath7 state with a binding energy of @xmath8ev or , equivalently , @xmath9@xmath10 . a time - delayed femtosecond pulse probes the excited mg@xmath6 molecule by inducing a one - photon transition to a higher excited electronic state ( @xmath11 ) . this state has a strong one - photon transition back to the ground state . the corresponding experimental observable is the intensity of the resulting uv fluorescence ( @xmath12 nm ) , measured as a function of the pump - probe time delay . an oscillating signal is a manifestation of coherent rovibrational dynamics in the @xmath13 state.@xcite the correct description of the thermal initial state is crucial to capture the generation of coherence out of an incoherent ensemble . the density operator , @xmath14 , describing the initial state of hot atom pairs at temperature , @xmath15 , is constructed by a thermal average over suitable basis functions . since no dissipative processes occur on the sub - picosecond timescale of the experiment , the coherent time evolution of the density operator is efficiently carried out by propagating the basis functions . expectation values are obtained by thermally averaging the corresponding operator over the propagated basis functions . a numerically efficient description of the initial thermal ensemble is essential to facilitate the time - dependent simulations . the present work on _ ab initio _ simulation of ultrafast hot photoassociation presents a detailed account of theoretical and numerical components and their integration into a comprehensive framework . the paper is organized as follows . section [ sec : model ] presents the theoretical framework by introducing the hamiltonian describing the coherent interaction of an atom pair with strong femtosecond laser pulses . the relevant electronic states , their potential energy curves , transition matrix elements and non - adiabatic couplings , all obtained employing highly accurate state of the art _ ab initio _ methods , are discussed . section [ sec : thermal ] derives an effective description of the thermal ensemble of translationally and rotationally hot atom pairs in their electronic ground state based on random phase thermal wave functions . we consider three different choices of basis functions , two of them turn out to be practical . convergence of the photoassociation probability is studied in section [ sec : conv ] for the different thermal averaging procedures , and the role of shape resonances is discussed . section [ sec : results ] investigates the generation of coherence in terms of the quantum purity and a dynamical coherence measure . finally , we conclude in section [ sec : concl ] . atomic units are used throughout our paper , unless specified otherwise . the coherent ( 2 + 1 ) three - photon excitation of a pair of magnesium atoms , that collide with rotational quantum number @xmath16 , by a strong femtosecond laser pulse is described by the time - dependent hamiltonian , @xmath17 here @xmath18 is the nuclear hamiltonian of electronic state @xmath19 , @xmath20 with @xmath21 the vibrational kinetic energy , @xmath22 the reduced mass and @xmath23 the potential energy curve of electronic state @xmath19 . @xmath24 and @xmath25 denote the ( one - photon ) transition dipole moments between the @xmath26 state and the first and second @xmath11 states . the hamiltonian neglects ro - vibrational couplings . in a two - photon rotating - wave approximation , the two - photon coupling between the @xmath27 @xmath28 and @xmath29 @xmath30 states is denoted by @xmath31,@xcite @xmath32 with @xmath33 the electric field envelope of the laser pulse , @xmath34 the polarization component ( @xmath35 ) , and @xmath36 the tensor elements of the two - photon electric transition dipole moment between the ground ( @xmath37 ) and excited ( @xmath38 ) states,@xcite @xmath39\,.\ ] ] the summation is carried out over all electronic states @xmath40 , except for the states which are explicitly accounted for in our model , cf . @xmath41 and @xmath42 are the transition frequencies between state @xmath40 and , respectively , state @xmath43 and @xmath44 . note that the two - photon transition moment , @xmath45 , depends on the central laser frequency , @xmath46 . here we keep @xmath47 nm fixed . the strong laser field driving the two - photon transitions may lead to non - negligible dynamic stark shifts @xmath48,@xcite @xmath49 where the tensor elements of the dynamic electric dipole polarizability are given by@xcite @xmath50\,,\ ] ] where the sum runs over all electronic states @xmath40 , except those explicitly accounted for in our model , cf . eq . , and @xmath51 is the transition frequency between states @xmath40 and @xmath19 ( @xmath52 ) . we account only for the isotropic part of the polarizability , neglecting anisotropic terms that occur for open shell states with the projection of the electronic angular momentum not equal to zero.@xcite this corresponds to two - photon transitions with @xmath53 , neglecting transitions with @xmath54 . similarly to the two - photon transition moment , @xmath45 , the dynamic polarizability , @xmath55 , depends on the central laser frequency , @xmath56 . note that resonant transitions , both one - photon and two - photon transitions , are treated in a non - perturbative way while all non - resonant transitions are accounted for within second order perturbation theory . the @xmath26 excited state that is accessed by the two - photon transition is weakly coupled to the @xmath57 , and @xmath58 states due to the spin - orbit interaction . the spin - orbit matrix elements relevant for our work read @xmath59 @xmath60 @xmath61 where @xmath62 is the spin - orbit coupling hamiltonian in the breit - pauli approximation including all one- and two - electron terms . the effect of the spin - orbit coupling was actually observed in the fluorescence signal , but it was so weak that we could neglect the triplet states in the time - dependent calculations . a one - photon transition connects the @xmath26 state to the adiabatic @xmath63 and @xmath64 states that are strongly coupled by the radial nuclear momentum operator . in order to include this non - adiabatic coupling , the diabatic representation is employed , see e.g. ref . . @xmath65 and @xmath66 denote the corresponding diagonal diabatic potentials and @xmath67 the coupling term . analogously , @xmath68 ( @xmath69 ) denote the stark shifts in the diabatic basis . the angle of the rotation matrix transforming adiabatic into diabatic representation is given by@xcite @xmath70 with the nonadiabatic radial coupling @xmath71 consequently , the one - photon transition dipole moments @xmath72 , @xmath73 are calculated from the diabatic molecular wave functions , obtained by rotating the adiabatic @xmath63 and @xmath64 wave functions . .spectroscopic characteristics , i.e. , equilibrium bond lengths , @xmath74 , and well depths , @xmath75 , of our _ ab initio _ potentials . [ tab : spec ] [ cols="<,>,>,>",options="header " , ] state - of - the - art _ ab initio _ techniques have been applied to compute the potential energy curves of the magnesium dimer in the born - oppenheimer approximation . all calculations employed the aug - cc - pvqz basis set of quadruple zeta quality as the atomic basis for mg . this basis set was augmented by the set of bond functions consisting of @xmath76 $ ] functions placed in the middle of the mg dimer bond . all potential energy curves were obtained by a supermolecule method , and the boys and bernardi scheme was used to correct for the basis - set superposition error.@xcite the ground @xmath27 state potential was computed with the coupled cluster method restricted to single , double , and noniterative triple excitations , ccsd(t ) . for the excited @xmath77 and @xmath63 states , linear response theory ( equation of motion approach ) within the coupled - cluster singles and doubles framework , lrccsd , was employed . the potential energy curve of the excited @xmath64 state in the region of the minimum of the potential was also obtained with the lrccsd method . at larger internuclear distances this potential energy curve was represented by the multipole expansion with electrostatic and dispersion terms @xmath78 up to and including @xmath79 . the long - range coefficients @xmath80 were obtained within the multireference configuration interaction method restricted to single and double excitations , mrci , with a large active space . the latter procedure was necessary since the @xmath64 state dissociates into mg@xmath81+mg@xmath81 atoms and can not be asymptotically described by a single slater determinant . the ccsd(t ) and ccsd calculations , including the response functions calculations , were performed with the dalton program,@xcite while the mrci calculations were carried out with the molpro suite of codes.@xcite the energy of the separated atoms was set equal to the experimental value for each electronic state , although the atomic excitation energies obtained from the lrccsd calculations were very accurate and for the lowest @xmath82p state the deviation from the experimental values was approximately 100@xmath3@xmath10 . a high accuracy of the computed potential energy curves is confirmed by an excellent agreement of the theoretical dissociation energy for the ground @xmath27 state ( @xmath83403.1@xmath10 ) with the experimental value ( @xmath83404.1@xmath840.5@xmath10).@xcite moreover , the number of bound vibrational states for @xmath85 supported by the electronic ground state agrees with the experimental number , @xmath86 . spectroscopic parameters of the other experimentally observed state , @xmath87 , also agree with our values , for the well position within 0.07@xmath3bohr , while the binding energy ( @xmath75=9427@xmath10 ) is only 0.4% higher than the experimental value ( @xmath75=9387@xmath10).@xcite the root mean square deviation of the rovibrational levels computed with the potential energy curves from the ccsd(t ) and lrccsd calculations for the ground and @xmath88 states were 1.3@xmath3@xmath10 and 30@xmath3@xmath10 , respectively , i.e. , 0.3% of the potential well depth . such a good agreement of the calculations with the available experimental data strongly suggests that we can expect a comparable level of accuracy for the other computed potential energy curves and molecular properties . the spectroscopic characteristics of the ground and excited electronic states are gathered in table [ tab : spec ] , while the corresponding potential energy curves are reported in fig . [ fig : abinitio ] . inspection of table [ tab : spec ] shows that most of the excited electronic states of mg@xmath6 are strongly bound with the dissociation energies ranging from 5400@xmath3@xmath10 for the @xmath63 state up to 18000@xmath3@xmath10 for the @xmath26 state . only the @xmath89 and @xmath90 states are very weakly bound with binding energies of 49@xmath3@xmath10 and 110@xmath3@xmath10 , respectively . the agreement of our results with data reported by czuchaj and collaborators in 2001@xcite is relatively good , given the fact that their results were obtained with the internally contracted multireference configuration singles and doubles method based on a casscf reference function . indeed , for the @xmath26 , a@xmath91 , @xmath89 , @xmath57 , @xmath90 , and @xmath92 states the computed well depths agree within 600 @xmath10 or better , i.e. , within a few percent , while the equilibrium distances agree within a few tenths of bohr at worst . only for the @xmath63 state we observe a very large difference in the binding energy , 3000 @xmath10 . such a very strong binding in the @xmath63 state is very unlikely , since this state would then show a strong interaction with the spectroscopically observed a@xmath91 state . this interaction would , in turn , show up in the observed @xmath93 spectra as inhomogeneous perturbations of lines . however , such perturbations have not been observed in the recorded spectra,@xcite suggesting our _ ab initio _ potential for the @xmath63 state to be more accurate . and the @xmath57 states , and @xmath94 and @xmath90 states ( left panel ) and radial nonadiabatic coupling between the@xmath63 and @xmath64 states of the mg@xmath6 molecule . ] potential energy curves for all electronic states computed in the present work are reported in fig . [ fig : abinitio ] . they all show a smooth behavior with well defined minima and in some cases maxima due to the first - order resonant interactions . the potential energy curves of the a@xmath91 and @xmath90 states and of the @xmath26 and @xmath89 cross each other . these crossings should experimentally be observed as a perturbation due to the very weak , but non - zero spin - orbit coupling between the singlet and triplet states . indeed , the experimental data on the @xmath93 transitions,@xcite and the uv fluorescence spectra from the @xmath26 state @xcite confirm weak perturbations due to the spin - orbit coupling . the corresponding matrix elements are shown in fig . [ fig : abinitio1 ] . except for small interatomic distances they show a weak @xmath95 dependence and smoothly tend to the atomic value . the @xmath96 and @xmath97 show an avoided crossing , but the gap between the two curves is so large that most probably no homogeneous perturbations will be observed in the spectra . by contrast , the @xmath63 and @xmath64 states show a very pronounced avoided crossing with a gap of a few wavenumbers . this suggests a strong interaction between these states through the radial nonadiabatic coupling matrix element . the shape of this element is shown in the right panel of fig . [ fig : abinitio1 ] . as expected , the nonadiabatic coupling matrix element is a smooth lorenzian - type function , which , in the limit of an inifintely close avoided crossing , becomes a dirac @xmath98-function . the height and width of the curve depends on the strength of the interaction , and the small width of the coupling in fig . [ fig : abinitio1]b suggests a strong nonadiabatic coupling . it is gratifying to observe that the maximum on the nonadiabatic coupling matrix element agrees well with the location of the avoided crossing , despite the fact that two very different methods were employed in the _ ab initio _ calculations . as discussed above , the potential energy curves were shown to be accurate , so we are confident that also the nonadiabatic coupling matrix element are essentially correct . the electric transition dipole moments between states @xmath99 and @xmath100 , @xmath101 , where the electric dipole operator , @xmath102 , is given by the @xmath103th component of the position vector and @xmath104 are the wave functions for the initial and final states , respectively , were computed as the first residue of the lrccsd linear response function for the @xmath27 , @xmath77 , and @xmath105 states . for transitions to the @xmath64 state , the mrci method was employed . the two - photon transition moment , eq . , can in practice be obtained as a residue of the cubic response function.@xcite for transitions between the @xmath27 and @xmath77 states , @xmath106 was computed as a residue of the coupled cluster cubic response function with electric dipole operators and wave functions within the ccsd framework.@xcite the tensor elements of the electric dipole polarizability of the ground @xmath27 state were obtained as the coupled cluster linear response function with electric dipole operators and wave functions within the ccsd framework.@xcite the dynamic polarizabilities of the excited states were computed as double residues of the coupled cluster cubic response function with electric dipole operators and wave functions within the ccsd framework.@xcite the nonadiabatic radial coupling matrix elements as well as the spin - orbit coupling matrix elements have been evaluated with the mrci method . state as obtained from response calculations ( black solid line ) and after eliminating the contribution from @xmath107 state ( red dashed line ) in the region important for the time - dependent calculations ( right panel ) . the inset illustrates the smooth behavior of the dynamic stark shift after elimination of the resonance due to the @xmath107 state . ] the electric transition dipole moments from the ground electronic state to the three lowest singlet states of ungerade symmetry are reported in fig . [ fig : abinitio2 ] . the transition moment to the a@xmath91 state is almost constant over a wide range of interatomic distances @xmath95 , and smoothly approaches the atomic value . the transition moments to the @xmath63 and @xmath64 states show more pronounced variations . in fact , the @xmath95 dependence of these two transition moments reflects the avoided crossing of the corresponding potential energy curves around @xmath108 bohr . also reported in fig . [ fig : abinitio2 ] is the trace of the dynamic stark shift for the @xmath26 state as a function of @xmath95 . as expected from the definition , the dynamic stark shift shows resonances for transition energies close to the laser frequency @xmath56 . since the adiabatic elimination that leads to eq . assumes only non - resonant transitions , the electronic states that cause these resonances need to be included explicitly in the non - perturbative hamiltonian . this eliminates their contribution to the dynamic stark shift . [ fig : abinitio2]b illustrates the procedure , showing the trace of the dynamic polarizability as a function of @xmath95 . the black solid line shows a broad resonance around @xmath109 bohr . this resonance corresponds to transitions from the the @xmath26 to the @xmath107 state . the latter state is explicitly included in our hamiltonian for the time - dependent calculations and eliminated from eq . . once this is done , a smooth and almost constant behavior is obtained , as illustrated in fig . [ fig : abinitio2]b . of course , also the contributions from all other electronic states that are explicitly accounted for in the hamiltonian for the time - dependent calculations are eliminated from eq . . the hamiltonian for the time - dependent calculation , neglecting the weak spin - orbit coupling between the @xmath26 , @xmath89 and @xmath57 states and accounting for states with dipole transitions that are near - resonant to the laser frequency , becomes @xmath110 note that eq . makes use of the two - photon rotating wave approximation , i.e. , the electric field envelope , @xmath111 , in eq . , may be complex , @xmath112 , and a non - zero @xmath113 denotes the relative phase with respect to the central laser frequency s phase . the hamiltonian is represented on an equidistant grid for each partial wave @xmath16 . convergence with respect to the grid size @xmath114 and number of grid points @xmath115 is discussed below in section [ sec : conv ] . the initial state for photoassociation is given by the ensemble of magnesium atoms in the heated cell which interact via the @xmath27 electronic ground state potential . assuming equilibrium , the initial state is represented by the canonical density operator for @xmath116 atoms held in a volume @xmath117 at temperature @xmath15 . due to the moderate density in a heat pipe , the description can be restricted to atom pairs . the density operator for @xmath118 atom pairs is then obtained from that for a single atom pair , @xmath119 , which is expanded into a suitable complete orthonormal basis.@xcite thermally averaged time - dependent expectation values of an observable @xmath120 are calculated according to @xmath121\,.\ ] ] the time evolution of @xmath122 is given by @xmath123 starting from the initial state @xmath124 where @xmath125 is the hamiltonian and @xmath126 $ ] the partition function . for a thermal , i.e. incoherent , initial state , undergoing coherent time evolution , it is not necessary to solve the liouville von - neumann equation for the density operator . instead , the dynamics can be captured by solving the schrdinger equation for each basis function . thermally averaged expectation values are calculated by properly summing over the expectation values obtained from the propagated basis states.@xcite since many scattering states in a broad distribution of rotational quantum numbers are thermally populated , the approach of propagating all thermally populated basis states directly@xcite becomes numerically expensive . alternatively , an effective description of the thermal ensemble of scattering atoms is obtained by averaging over realizations of random phases . it makes use of thermal random wave functions , @xmath127 . here , the index @xmath128 labels a set of random phases and @xmath129 the basis states . choosing an arbitrary complete orthonormal basis , @xmath130 , and given that @xmath131 for random phases @xmath132 , @xmath133 and @xmath116 large , an expansion into random phase wave functions yields a representation of unity,@xcite @xmath134 where @xmath135 and @xmath136 . here , we use a random phase expansion of unity for the vibrational degree of freedom , i.e. , the radial part @xmath95 of the relative motion @xmath137 of the diatom in its electronic ground state . no electronic excitations are excited thermally . a separation of rotational and vibrational dynamics and subsequent expansion into partial waves is natural to make use of spherical symmetry , @xmath138 , i.e. , the vibrational motion is conditioned on @xmath16 . this implies a complete set of vibrational basis functions ( both true vibrational eigenfunctions and scattering states ) , and subsequently , a different set of random phases , for each @xmath16 . in principle , one could apply a random phase expansion of unity also in the rotational degree of freedom this would be useful to study the generation of rotational coherence . it requires a model that accounts for rotational coherence , i.e. , either a full rovibrational hamiltonian or , as a minimal approximation , a generalization of eq . comprising @xmath139 and @xmath140 in addition to @xmath141 . however , in the present work , we focus on the generation of vibrational coherence which is crucial for bond formation . in the following , we discuss three possible bases for the vibrational hamiltonian , from which random phase wave functions are generated . all three possibilities will lead , when averaged , to the initial thermal ensemble corresponding to the experiment . while formally equivalent , convergence of the thermal averages with respect to the number of required basis functions differs significantly for the three representations . the simplest but , as it turns out , most inefficient approach uses , for each partial wave @xmath16 , the coordinate basis of @xmath98-functions localized at each grid point @xmath95 , @xmath142.@xcite a random phase wave function is obtained by multiplying each basis state with a different random phase , @xmath143 , @xmath144 with @xmath128 labeling one realization of @xmath115 random phases , @xmath145 . the resulting wave function , @xmath146 , has constant amplitude and a different random phase at each @xmath95 . the initial density operator is obtained by propagating each basis function @xmath147 under @xmath148 in imaginary time , @xmath149 with @xmath150 , using the chebychev propagator.@xcite this yields the thermal random phase wave functions , @xmath151 and thus the initial density operator , @xmath152 thermal random phase wave functions @xmath153 are propagated in real time , @xmath154 with the hamiltonian as generator . thermally averaged time - dependent expectation values are obtained from eq . , using cyclic permutation under the trace , @xmath155 & = & \frac{1}{n}\sum_{k=1}^n \sum_{r , j}(2j+1 ) \langle \psi^k_{r , j}|{\boldsymbol{\mathsf{\hat{a } } } } { \boldsymbol{\mathsf{\hat{u}}}}(t,0 ) { \boldsymbol{\mathsf{\hat{\rho}}}}_t(t=0){\boldsymbol{\mathsf{\hat{u}}}}^+(t,0)|\psi^k_{r , j}\rangle \nonumber \\ & = & \frac{1}{z}\frac{1}{n}\sum_{k=1}^n \sum_{r , j}(2j+1 ) \langle \psi^k_{r , j}| e^{-\frac{\beta}{2}{\boldsymbol{\mathsf{\hat{h}}}}^j_g } { \boldsymbol{\mathsf{\hat{u}}}}^+(t,0 ) { \boldsymbol{\mathsf{\hat{a } } } } { \boldsymbol{\mathsf{\hat{u}}}}(t,0 ) e^{-\frac{\beta}{2}{\boldsymbol{\mathsf{\hat{h}}}}^j_g } | \psi^k_{r , j}\rangle \nonumber \\ & = & \frac{1}{z}\frac{1}{n}\sum_{k=1}^n \sum_{j=0}^{j_{max}}(2j+1)\ ; _ t\langle \psi^k_{j}(t)|{\boldsymbol{\mathsf{\hat{a}}}}|\psi^k_{j}(t)\rangle_t \,.\end{aligned}\ ] ] obtaining the @xmath156 solutions of the schrdinger equation , @xmath157 , required by eq . requires typically significantly less numerical effort than propagating @xmath158 @xmath159-dimensional density matrices , neglecting the rovibrational coupling , or once the full @xmath160-dimensional density matrix . note that while @xmath153 has zero components on all electronic states except the ground state , @xmath161 will be non - zero for all electronic states due to the interaction with the field . relevant expectation values are the excited state population after the pump pulse , possibly @xmath16-resolved . the corresponding operators are the projectors onto the electronically excited state , @xmath162 , and @xmath163 , i.e. , @xmath164 the convergence of this approach is slow . the number of realizations required to reach convergence was found to be much larger than the number of grid points . the reason for the slow convergence is that there is no preselection of those basis states that are most relevant in the thermal ensemble . we will therefore not use this method and have included it here only for the sake of completeness . a preselection of the relevant states becomes possible by choosing the eigenbasis @xmath165 of @xmath166 and evaluating the trace only for basis states with sufficiently large thermal weights , @xmath167 where @xmath168 is a prespecified error . the eigenfunction - based random phase wave functions are given by @xmath169 where @xmath16 denotes the partial wave and @xmath40 is the vibrational quantum number in the bound part of the spectrum of @xmath166 or , respectively , the label of box - discretized continuum states . it is straightforward to evaluate the representation of the initial density operator in this basis , @xmath170 where @xmath171 denotes an eigenvalue of the partial wave ground state hamiltonian , @xmath166 , and the @xmath156 initial random phase wave functions are given by @xmath172 thermally averaged time - dependent expectation values are calculated analogously to eq . , where the time - dependent wave functions are obtained by propagating the wave functions of eq . instead of the initial states given by eqs . and . for convenience , we use normalized random phase wavefunctions instead of eq . , @xmath173 where @xmath174 and @xmath114 indicates the size of the box . for the thermally averaged time - dependent expectation values , this yields @xmath175 = \frac{1}{n } \sum_{k=1}^n\sum_{j=0}^{j_{max } } p_j \langle\tilde\psi^k_j(t)|{\boldsymbol{\mathsf{\hat{a}}}}|\tilde\psi^k_j(t)\rangle\ ] ] with @xmath176 the weight of the contribution of partial wave @xmath16 . the eigenfunction - based random phase approach requires diagonalization of the @xmath177 partial wave ground state hamiltonians , @xmath166 . depending on the time required for the propagation of each basis state , this effort may very well be paid off by the much smaller number of basis states that need to be propagated . the partition function for the computational box of radius @xmath114 is straightforwardly evaluated in the eigenbasis , @xmath178 where @xmath179 , @xmath180 are chosen such that @xmath181 . since we are interested in high temperatures , it is natural to compare the calculated partition function @xmath182 to its classical approximation , @xmath183 with @xmath184 and @xmath185 . the derivation of @xmath186 is given in appendix [ sec : app ] . for a temperature of 1000@xmath3k , we find @xmath182 and @xmath186 to agree within less than 1% . inserting the classical approximation of @xmath187 and @xmath188 into eq . , we find @xmath189 to roughly correspond to the normalized boltzmann weight at the end of the grid . the third approach avoids diagonalization of the partial wave ground state hamiltonians , @xmath166 , approximating them by the kinetic energy , @xmath190 , only . this approximation is valid at high temperatures where the kinetic energy of the scattering atoms is much larger than their potential energy due to the inter - particle interaction . it starts from a gaussian wave paket positioned sufficiently far from the interaction region . if the width of the wave packet is adjusted thermally , projection onto energy resolved scattering wave functions yields boltzmann weights , @xmath191 the thermal width is given by @xmath192 , and @xmath193 where @xmath194 is the interaction region . the fourier transformed wave packet , @xmath195 corresponds to eigenstates of the kinetic energy with boltzmann weights,@xcite i.e. , we approximate @xmath196 . random phase wave functions can be generated from eq . by real - time propagation under @xmath166 as follows . the time - evolved wave packet at time @xmath197 is written as @xmath198 where expansion into the scattering states of the finite computation box , i.e. , the states @xmath165 with positive energy ( @xmath199 ) , has been used . @xmath200 is an initial phase due to @xmath201 . comparing to eq . , the random phases are given by @xmath202 . for sufficiently large times , @xmath203 and @xmath204 which , with @xmath205 yields @xmath206 , the wave function will spread significantly and fill the interaction region which is a prerequisite to correctly represent the thermal density . a different set of phases is obtained by propagating the gaussian wave packet under @xmath166 for a time @xmath207 . for mg@xmath6 and @xmath208k , the two limits translate into @xmath209fs and @xmath210ps for @xmath211a@xmath212 . for large grids , these numbers grow correspondingly . moreover , to reproduce the boltzmann ensemble not only qualitatively , but also quantitatively , the smallest frequency difference between scatterings states in the computation box needs to be resolved . this translates into even longer propagation times . practically , a coordinate grid based wave packet , eq . , is propagated under @xmath166 by a chebychev propagator where each realization corresponds to a different time @xmath197 . the density operator is constructed by averaging over the times , @xmath197 , chosen randomly . alternatively , if the eigenvalues @xmath171 and eigenstates @xmath165 are known , the thermal random phase wavefunctions can be obtained simply by projection on the @xmath165 , choice of a random phase and reassembly of the gaussian wavepacket from the random - phase projections . this avoids the very long propagation times required to faithfully represent the boltzmann ensemble . the method of choice , diagonalization of the ground state hamiltonians , @xmath166 , and subsequent projection , or propagation under @xmath166 depends on the dimensionality of the problem due to the different scaling of diagonalization and propagation . for a diatomic , the diagonalization approach was found to be more efficient . for larger systems , the propagation approach is expected to take over . the convergence of random phase wave functions built on thermal gaussians with respect to the photoassocation yield is comparatively fast , only a few realizations are sufficient . the drawback of the procedure is that only the free part of the initial wave functions is represented , leaving out the interaction energy of the true scattering states as well as initial population in bound or quasi - bound states . thermal expectation values are obtained according to eq . where @xmath213 is a ( normalized ) gaussian random packet , freely propagated for random times @xmath197 . the probability @xmath189 for partial wave @xmath16 , eq . , needs to account for the fact that the gaussian is initially positioned at @xmath201 , not the edge of the grid . therefore , using the classical approximation , @xmath114 in eq . needs to be replaced by @xmath201 . another variant utilizes @xmath98-functions in momentum space , @xmath214 , and add random phases , @xmath215 , to the momentum components directly , @xmath216 the random phases , @xmath215 , translate into positions of the gaussian , @xmath217 . this procedure reconstructs the correct density in the regions of flat potential but fails in the interaction region and is therefore not employed here . the laser pulse excites a small fraction of the incoherent ensemble of ground state atom pairs to the @xmath26 state and further to the first and second @xmath11 state . this action corresponds to distillation and leads to higher purity and coherence of the photoassociated molecules.@xcite in order to study the purity of the subensemble of diatoms in the excited electronic state , @xmath218\,,\ ] ] the normalized density operator of electronic state @xmath219 , is formally constructed , @xmath220 in the grid representation using @xmath115 grid points , @xmath221 becomes a matrix of size @xmath222 , @xmath223 where @xmath224 is the excited state projection of the ( @xmath225)th propagated thermal random phase wave function , @xmath226 . since we expect to populate only a limited number of @xmath26 state eigenfunctions , say @xmath227 , it is computationally advantageous to transform the excited state component of the propagated thermal wave functions into the rovibrational eigenbasis , @xmath228 , of the @xmath26 state , @xmath229 with @xmath230 the resulting density matrix , @xmath231 is only of size @xmath232 and can more efficiently be squared to obtain the purity . moreover , this representation lends itself naturally to the evaluation of the dynamical coherence measure . in the eigenbasis , we can decompose the density operator into its static ( diagonal ) and dynamic ( off - diagonal ) part , @xmath233 . such a decomposition has been motivated in the study of dissipative processes , in particular by the fact that pure dephasing does not alter the static part.@xcite the dynamical coherence measure , @xmath234\,,\ ] ] captures the part of the purity that arises from the dynamical part of the density operator.@xcite the purity of the excited state subensemble after the pump pulse , @xmath235 , shall be compared to the initial purity of the whole ensemble ( in the electronic ground state ) , @xmath236\,.\ ] ] to this end , but also to determine the photoassociation probability , cf . eq . , the partition function @xmath237 needs to be determined explicitly . we need to take into account that our computation box represents only a small part of the experimental volume . the total partition function is therefore given by @xmath238 , where @xmath239@xmath240 for @xmath241a@xmath212 and @xmath117 the experimental volume . alternatively , the probability of a single atom in our computation box is @xmath242 with @xmath243 the experimental density , @xmath244 atoms/@xmath245 . the probability of finding two atoms in the box is then simply @xmath246 . using eq . with @xmath247 , the purity of the initial state is obtained as @xmath248 taking @xmath249 in eq . when evaluating @xmath189 . the interaction of the atom pair with the laser field is simulated by solving @xmath156 time - dependent schrdinger equations , @xmath250 for @xmath251 and @xmath252 , with a chebychev propagator@xcite and thermally averaging the solutions according to eq . . to reduce the computational effort , we evaluate all sums over @xmath16 in steps of five . of ground state atom pairs ( calculated using eq . with 200 realizations for each @xmath16 and excluding bound states and shape resonances from the sum over @xmath128 ) . ] we first study the initial thermal density of atom pairs , cf . eq . , that is excited by the laser pulse . it sets a limit to the excitation yield since thermalization occurs over timescales larger than that of the experiment . the initial thermal density of atom pairs is shown as a function of interatomic distance in fig . [ fig : rho_ini ] for random phase wave functions built from eigenfunctions , cf . eq . and built from gaussians , cf . eq . . for photoassociation , distances smaller than @xmath253 are relevant . the thermal density is converged in this region by including rotational quantum numbers up to @xmath254 . the contribution of higher partial waves only ensures a constant density at large interatomic distances . the long - distance part naturally converges very slowly but this is irrelevant for the dynamical calculations . the peak at short interatomic separations is due to bound levels , shape resonances and the classical turning point of the scattering states at the repulsive barrier of the potential : the difference between the red and orange curves in fig . [ fig : rho_ini ] indicates the contribution of bound levels , the difference between the orange and the purple curve that of shape resonances . the gaussian method requires much larger grids than the eigenfunction based method to converge the initial thermal density since it is based on the assumption that the effective potential is zero at the position of the gaussians . however , for large values of @xmath16 , the rotational barrier is non - zero even at comparatively large internuclear separations . this leads to a spurious trapping of probability amplitude , cf . the dashed curves in fig . [ fig : rho_ini ] . at short internuclear separations , the pair density calculated from thermal gaussians in the upper panel of fig . [ fig : rho_ini ] shows the same behavior as the purple curve in the lower panel of fig . [ fig : rho_ini ] ( up to scaling which is due to the accumulation of amplitude at large internuclear separations ) . this indicates that random phase wave functions built from thermal gaussians do not capture bound states and shape resonances . state , calculated as @xmath255 , vs initial partial wave @xmath16 , averaged over 200 realizations of random phase wave functions , for three different grid sizes @xmath114 and a transform - limited pulse with @xmath256w/@xmath257 and @xmath258fs ( excluding the bound states and shape resonances from the sum over @xmath128 in eq . ) . the light - blue shaded part indicates the contribution of scattering states , i.e. , photoassociation , for @xmath241a@xmath212 . the contributions of the shape resonances is given by the difference between the blue solid and dashed curves . propagating random phase wave functions calculated from thermal gaussians , cf . eq . , does not capture excitation of bound levels and shape resonances ( dotted curves ) . bottom panel : ratio of the excitation yields calculated from random phase wavefunctions based on eigenfunctions and thermal gaussians . for large @xmath16 and sufficiently large grid size , the two methods coincide as expected . ] these two features of the gaussian random phase wave functions show also up in the population transferred from the initial incoherent ensemble to the @xmath259 state , shown in fig . [ fig : pumpfree ] . the thermal averaging procedure has been repeated for increasing initial rotational quantum number , @xmath16 . that is , for each rotational barrier , eigenfunction - based and gaussian random phase wave functions are propagated in real time with the full , time - dependent hamiltonian , @xmath260 . expectation values , such as the population of the @xmath259 state after the pump pulse is over , are calculated for each random phase realization , @xmath128 , and averaged over , including the rotational degeneracy factor @xmath261 , cf . eq . . for large grids ( @xmath241a@xmath212 , @xmath262a@xmath212 ) and large @xmath16 , random phase wave functions built from eigenfunctions and built from thermal gaussians yield the same results due to the trapping of probability amplitude at large internuclear separations , for small grids ( @xmath263a@xmath212 ) and large @xmath16 , the gaussian method underestimates the excitation yield . since our random phase wave functions are normalized in the computation box , this results in an initial thermal density which is too small at the internuclear separations , @xmath264a@xmath265a@xmath212 , that are relevant for the laser excitation . this is illustrated by the black curve in the lower panel of fig . [ fig : pumpfree ] which deviates from the blue and green curves even for large @xmath16 . for @xmath266 , the potential supports bound levels which are not captured by the gaussian random phase wave functions . once the bound levels and shape resonances are removed from the eigenfunction based approach ( solid blue curve ) , the eigenfunction - based approach roughly agrees with the gaussian approach ( blue dotten curve ) . this comparison allows for estimating the contribution of the bound levels . for @xmath267 the ground state potential does not support any bound levels due to the high centrifugal barrier . the total contribution of the bound part of the spectrum to the excitation of @xmath77 population amounts to about 20% . the differences between @xmath268 to @xmath269 are attributed to insufficient sampling of the free propagation method . qualitatively , however , the two approaches yield the same result with a steep rise at low @xmath16-values , a peak at intermediate @xmath16 and an exponential tail for @xmath270 . the peak is shifted toward larger @xmath16 for the gaussian method since it can not capture the excitation of bound levels and shape resonances . each random phase approach represents a statistical sampling of the photoassociation yield . the deviation of an expectation value from its mean scales as @xmath271 where @xmath116 is the number of realizations . this was checked for @xmath272 and @xmath273 the pre - factor @xmath274 is estimated as @xmath275 for the free propagation method and @xmath276 for the eigenvalue method ( note that @xmath277 for the grid based method ) . this makes the eigenvalue method converge fastest . . bottom panel : lifetimes in nanoseconds of the high - lying shape resonances vs partial wave @xmath16 . ] the shape resonances are analyzed in fig . [ fig : shaperes ] which displays their position in energy ( top panel ) and the lifetimes of those shape resonances that are sufficiently short lived , i.e. , sufficiently broad , to contribute to the photoassociation process ( bottom panel ) . the shape resonances were calculated using a complex absorbing potential.@xcite shape resonances are found for @xmath278 . the positions of the short - lived resonances lie between 15@xmath3k and 300@xmath3k . with a sample temperature of 1000@xmath3k i.e. , at least the higher lying of these resonances are thermally populated . it is therefore not surprising that a contribution of the shape resonances is observed for @xmath279 , cf . the difference between the solid and dashed curves in fig . [ fig : pumpfree ] . the contribution is easily rationalized by the shape resonances representing quasi - bound states that are ideally suited for photoassociation.@xcite they give structure to the continuum of scattering states which otherwise is completely flat at high temperature . this can be utilized for generation of coherence and control . in conclusion , the random phase wave functions built from thermal gaussians can be used if a rough estimate of the photoassociation yield is desired . when further refinement is required the eigenvalue approach converges faster by a factor of 2 . since the gaussian approach excludes the bound part of the spectrum and the resonances , it comes with error bars of about 20% . if more accurate results are desired , the eigenfunction based random phase approach is the method of choice . the eigenfunction based method is also best suited to capture the contribution of bound states and shape resonances to the photoassociation yield . , and coherence measure , @xmath280 , cf . eqs . and , vs pulse intensity for the subensemble of photoassociated molecules . bottom panel : photoassociation yield for @xmath272 as a function of peak pulse intensity . all calculations employ transform - limited pulses of 100@xmath3fs full - width at half maximum . ] the degree of distillation of coherence out of an incoherent initial ensemble can be rationalized and quantified by considering the enhancement of quantum purity @xmath281 $ ] and coherence . the experimental signature is the periodic modulation of the pump probe signal.@xcite the purity for an incoherent ensemble is inversely proportional to the number of occupied quantum states of a pair of atoms . this number is completely determined by the temperature @xmath208k , and density , @xmath282atoms/@xmath245 , in the experiment . the bandwidth of our pulse plus stark shift largely exceeds the thermal width . this implies that all atom pairs within the franck - condon window are excited on equal ground , irrespective of their collision energy . we partition the total volume into identical smaller volumes containing exactly one pair of atoms such that the smaller volume corresponds to our computation box.@xcite the initial purity of the atom pair in our computation box of volume @xmath283 is given by the ground state purity , @xmath284 , multiplied by the probability for two atoms to occupy this box , @xmath285 , cf . section [ subsec : rhosq ] . the initial ground state purity is bounded from below by the purity of a maximally mixed state represented in our box . for @xmath286 random phase realizations and @xmath287 we estimate the lower bound to be @xmath288 . evaluating @xmath284 using the classical approximation for @xmath189 in eq . we obtain @xmath289 for @xmath241a@xmath212 and @xmath208k and thus @xmath290 for the initial purity . for the ensemble of molecules in the electronically excited @xmath77 state , the density operator is given by eq . and the purity within the computational box by eq . . the actual excited state purity is obtained by multiplying eq . by the probability for finding two atoms to be in the computational box , @xmath291 . note that the excited state density operator is normalized with respect to the excitation yield , @xmath292 . we obtain a purity @xmath293 for the molecular sub - ensemble in the @xmath77 excited state for the experimental pulse parameters . we thus observe a significant increase in the quantum purity , @xmath294 $ ] , induced by the femtosecond laser pulse . the underlying physical mechanism can be viewed as `` franck - condon filtering '' : for a given initial @xmath16 value there is only a limited range of collision energies that allow the colliding pair to reach the franck - condon window for pa located at short internuclear distances@xcite . in order to obtain a quantitative estimate of the degree of distillation achieved by the femtosecond photoassociation process , we have calculated the purity of the ensemble of photoassociated molecules in the @xmath295 state for a range of laser intensities , cf . [ fig : intensity_j ] . for weak fields , the purity is roughly constant as a function of intensity and about three times larger than the purity obtained for the intensity of @xmath296w/@xmath257 used in the experiment . as intensity is increased , a drop in the purity is observed which levels off at large intensities . we attribute this drop to power broadening for strong fields , which brings more atom pairs into the franck - condon window for pa . the purity of the photoassociated sub - ensemble is less than the inverse of the number of occupied energy states due to coherence . to quantify this effect we separate static and dynamic contributions , @xmath297 . expressing the density operator @xmath298 in the energy representation the static part corresponds to the diagonal matrix elements and the dynamical coherence to the off - diagonal elements . the dynamical contributions are quantified by the coherence measure @xmath299$].@xcite . figure [ fig : intensity_j ] shows the coherence measure of the excited state , @xmath300 , as a function of laser intensity ( red diamonds ) . it is found to be about one order of magnitude smaller than the purity . this is rationalized by the change in franck - condon points with different @xmath16 which degrade the vibrational coherence . within a sub - ensemble for a given angular momentum @xmath16 the difference between @xmath301 and @xmath302 is less than an order of magnitude . we have described two - photon femtosecond photoassociation of magnesium atoms from first principles using state of the art _ ab initio _ methods and quantum dynamical calculations . highly accurate potential energy curves were obtained using the coupled cluster method and a large basis set . two - photon couplings and dynamic stark shifts are important to correctly model the interaction of the atom pairs with the strong field of a femtosecond laser pulse . they were calculated within the framework of the equation of motion ( response ) coupled cluster method . the photoassociation dynamics were obtained by solving the time - dependent schrdinger equation for all relevant partial waves , accounting for the laser - matter interaction in a non - perturbative way , and performing a thermal average . we have developed an efficient numerical method to describe the incoherent thermal ensemble that is the initial state for photoassociation at high temperatures . it is based on random phase wave packets which can be built from eigenfunctions of the grid , the hamiltonian , or the kinetic energy . the latter can provide a rough estimate which is sufficient to yield qualitatively correct results . it neglects , however , the contribution from bound levels and long - lived shape resonances and therefore comes with error bars of about 20% . the best compromise between high accuracy and convergence is found for the eigenfunction - based method where random phase realizations are built from the eigenfunctions of the electronic ground state hamiltonian . about 200 partial waves and 200 realizations for each partial wave are required for converged photoassociation dynamics . time - dependent thermal averages are obtained by propagating each of the random phase wave functions and incoherently summing up all single expectation values . the random phase approach allows for constructing the thermal atom pair density as a function of interatomic separation for high temperatures . this is important to highlight the difference between hot and cold photoassociation.@xcite in the cold regime , the largest density is defined by the quantum reflection and resides in the long - distance , downhill part of the potential . the opposite is true in the hot regime : here , the largest density is found in the repulsive part of the ground state potential . this is due to the many partial waves that are thermally populated and the colliding atom pairs having sufficiently high kinetic energy to overcome the rotational barriers . for specific partial waves , shape resonances are found to play a role . this is not surprising since they represent quasi - bound states that are ideally suited for photoassociation.@xcite at very low temperatures , most partial waves are frozen out and the scattering is almost exclusively @xmath0-wave . the role of the rotational quantum numbers @xmath16 is less important in the electronically excited state but it is still detectable in form of quantum beats.@xcite both hot and cold photoassociation come with advantages as well as drawbacks . in the hot regime , molecules with much shorter bond length than in the cold regime are formed . however , the quantum purity and coherence of the created molecules is much larger in the cold regime where dynamical correlations exist prior to photoassociation . these correlations indicate pre - entanglement of the atom pair . making a molecule corresponds to entangling two atoms , and photoassociation amounts to filtering out an entangled subensemble both in the hot and cold regime . our work has opened up the possibility to study femtosecond photoassociation and its control at high temperatures and to investigate systematically the generation of coherence out of an incoherent initial state . future efforts will address the efficient theoretical description of the probe step . the theme of coherent control of binary reactions requires a sound theoretical basis to which our current study lays the ground work . this study was supported by the israeli science foundation isf grant no . 1450/10 , by the deutsche forschungsgemeinschaft and in part by the national science foundation under grant no . nsf phy11 - 25915 . cpk , dmr , mt , rk and rm enjoyed hospitality of kitp . rm and mt would like to thank the polish ministry of science and higher education for the financial support through the project n n204 215539 . mt was supported by the project operated within the foundation for polish science mpd programme co - financed by the eu european regional development fund . the classical approximation of the partition function is obtained starting from the standard definition , @xmath303 performing the integral over angles and introducing polar momentum coordinates , we find @xmath304 where we made use of @xmath305 . carrying out the integral over the radial momentum yields @xmath306 approximating the potential @xmath307 , the integral over the computational box of size @xmath114 can be performed , @xmath308 with @xmath309 \\ & & + r_{max } e^{-\beta\frac{j^2}{2mr_{max}^2}}\,.\end{aligned}\ ] ]
|
two - photon photoassociation of hot magnesium atoms by femtosecond laser pulses , creating electronically excited magnesium dimer molecules , is studied from first principles , combining _ ab initio _ quantum chemistry and molecular quantum dynamics .
this theoretical framework allows for rationalizing the generation of molecular rovibrational coherence from thermally hot atoms [ l. rybak _
et al . _ ,
phys .
rev .
lett . * 107 * , 273001 ( 2011 ) ] .
random phase thermal wave functions are employed to model the thermal ensemble of hot colliding atoms . comparing two different choices of basis functions , random phase wavefunctions built from eigenstates
are found to have the fastest convergence for the photoassociation yield .
the interaction of the colliding atoms with a femtosecond laser pulse is modeled non - perturbatively to account for strong - field effects .
| 16,307 | 235 |
schwarz maps for hypergeometric systems in single and several variables are studied by several authors ( cf . @xcite ) for more than hundred years . these systems treated were irreducible , maybe because specialists believed that reducible systems would not give interesting schwarz maps . we study in this paper appell s hypergeometric system @xmath0 of rank four when its parameters satisfy @xmath2 or @xmath3 . in this case , the system @xmath0 is reducible , and has a @xmath4-dimensional subsystem isomorphic to appell s @xmath5 ( proposition [ prop : s2 ] ) . if @xmath6 then @xmath0 has two such subsystems . by proposition [ prop : s2 g ] , the intersection of these subsystems is equal to the gauss hypergeometric equation . as a consequence , we have inclusions on @xmath0 , two @xmath5 s and @xmath7 ( theorem [ matome ] ) . we give the monodromy representation of the system @xmath0 which can be specialized to the case @xmath6 in theorem [ th : monod - rep ] . as for explicit circuit matrices with respect to a basis @xmath8 , see corollary [ cor : monod - matrix ] . we further specialize the parameters of the system @xmath0 as @xmath9 in [ schmap ] . in this case , the restriction of its monodromy group to the invariant subspace is arithmetic and isomorphic to the triangle group of type @xmath10 $ ] . we show that its schwarz map admits geometric interpretations : the map can be considered as the universal abel - jacobi map of a 1-dimensional family of curves of genus 2 in theorem [ th : gen - schwarz ] . the system @xmath0 is equivalent to the restriction of a hypergeometric system @xmath11 to a two dimensional stratum in the configuration space @xmath12 of six lines in the projective plane . in appendix [ 3-dim - s ] , we study a system of hypergeometric differential equations in three variables , which is obtained by restricting @xmath11 to the three dimensional strata corresponding to configurations only with one triple point . the methods to prove proposition [ prop : s2 ] are also applicable to this system under a reducibility condition . in appendix [ genus2 ] , we classify families of genus @xmath13 branched coverings of the projective line , whose period maps yield triangle groups . in a forthcoming paper @xcite , we study this schwarz map using period domains for mixed hodge structures . moreover , we explicitly give its inverse in terms of theta functions . gauss hypergeometric series @xmath14 where @xmath15 , admits an integral representation : @xmath16 the function @xmath17 is a solution of the hypergeometric equation @xmath18 where @xmath19 the collection of solutions is denoted by @xmath20 . appell s hypergeometric series @xmath21 admits an integral representation : @xmath22 the function @xmath23 is a solution of the hypergeometric system @xmath24 ( d'(c-1+d+d')-y(a+d+d')(b'+d'))z=0 , \end{array } \right.\ ] ] where @xmath25 , which can be written as @xmath26 where @xmath27 & q_1(a , b , b',c;x , y)=y(1-y)\partial_{yy}+x(1-y)\partial_{yx } + ( c-(a+b'+1)y)\partial_y - b'x\partial_x - ab ' , & \\[2 mm ] & r_1(a , b , b',c;x , y)=(x - y)\partial_{xy}-b'\partial_x+b\partial_y , \end{aligned}\ ] ] and @xmath28 , etc . the last equation @xmath29 is derived from the integrability condition of the first two equations . the collection of solutions is denoted by @xmath30 . appell s hypergeometric series @xmath31 admits an integral representation : @xmath32@xmath33 the function @xmath34 satisfies the system @xmath35 where @xmath36 & & q_2(a , b , b',c , c';x , y)=d'(c'-1+d')-y(a+d+d')(b'+d ) . \end{aligned}\ ] ] the collection of solutions is denoted by @xmath37 . as for the reducibility of the systems @xmath0 and @xmath5 , the following is known : [ redf2]@xmath38@xcite@xmath39 appell s system @xmath40 is reducible if and only if at least one of @xmath41 is an integer . [ redf1]@xmath38@xcite@xmath39 appell s system @xmath42 is reducible if and only if at least one of @xmath43 is an integer . the system @xmath40 is reducible when @xmath44 , fact [ redf2 ] . in fact , we see that the system @xmath45 is a subsystem of @xmath46 ; precisely , we have [ prop : s2 ] @xmath47 we give three `` proof ' 's : one using power series , subsection [ subsec : power ] , one using integral representations , subsection [ subsec : integ ] , and one manipulating differential equations , subsection [ subsec : equat ] . the former two are valid only under some non - integral conditions on parameters , which we do not give explicitly . though the last one is valid for any parameters , it would be not easy to get a geometric meaning . the following fact explains the inclusion in proposition [ prop : s2 ] . [ bailey1 ] @xmath48 we consider the integral @xmath49 which is a solution of the system @xmath50 . we change the coordinate @xmath51 into @xmath52 as @xmath53 which sends @xmath54 the inverse map is @xmath55 since @xmath56 we have @xmath57 this implies , if @xmath58 , then the double integral above becomes the product of the beta integral @xmath59 and the integral @xmath60 which is an element of the space @xmath61 . this shows @xmath62 which is equivalent to @xmath63 the bi - rational coordinate change @xmath64 is so made that the lines defining the integrand of the integral @xmath65 may become the union of vertical lines and horizontal lines in the @xmath66-space . actual blow - up and down process is as follows ( see figure [ st ] ) . name the six lines in the @xmath67-projective plane as : @xmath68 blow up at the 4 points ( shown by circles ) @xmath69 and blow - down along the proper transforms of the line @xmath70 and two lines : @xmath71 these three lines are dotted . this takes the @xmath67-projective plane to @xmath72 . in the figure , lines labeled @xmath73 stand for @xmath74 , and the lines labeled @xmath75 on the right are the blow - ups of the intersection points @xmath76 , respectively . the line obtained by blowing up the point @xmath77 is the line defined by @xmath78 , which should be labeled by @xmath79 . , width=491 ] a proof of the inclusion in proposition [ prop : s2 ] that is valid for any parameters is done as follows . let @xmath80 be a solution of the system @xmath81 . then , the system @xmath82 yields @xmath83-linear expressions of @xmath84 and @xmath85 in terms of @xmath86 and @xmath80 . substitute these expressions into the system @xmath87 . then , we get two linear forms in @xmath86 and @xmath80 . we now have only to see their coefficients vanish for the given parameters after a change of coordinates and a change of the unknown by multiplying a simple factor . we do not here present the actual computation , because if we put @xmath88 in the proof of proposition [ x3fd ] in subsection [ secondproof ] , manipulating differential equations , it gives essentially a proof of proposition [ prop : s2 ] . when @xmath90 , applying proposition 1.3 also for @xmath91 , we see that the system @xmath92 has two subsystems isomorphic to @xmath5 . the intersection of the two @xmath5 s would be the gauss hypergeometric equation . in fact , we have the following proposition . [ prop : s2 g ] @xmath93 similar to the argument of the previous subsection , we can give three `` proof ' 's : one using power series , one using integral representations , and one manipulating differential equations . we give a sketch of them in the following . the following identity explains the inclusion above . [ bailey2 ] @xmath94 we continue the argument in [ subsec : integ ] . in the integral @xmath95 in [ subsec : integ ] above , change the coordinate from @xmath96 to @xmath97 as @xmath98 sending @xmath99 since @xmath100 we have @xmath101 this implies , if @xmath91 , then @xmath102 this shows @xmath103 which of course implies the inclusion relation in proposition [ prop : s2 g ] by combination with that in proposition [ prop : s2 ] . put @xmath104 we have @xmath105 and so on . assume that @xmath106 . the equation @xmath107 gives a linear expression of @xmath108 in @xmath109 and @xmath110 . substitute these expressions in @xmath111 and we get the product of @xmath112 and a @xmath83-linear combination of @xmath110 and @xmath109 . the coefficients vanish if @xmath91 . if we do the same for @xmath113 , then we find that it vanishes when @xmath58 . we show that some indefinite integrals solve the system @xmath92 . we begin with some well - known facts . [ wellknown1 ] @xmath114 where @xmath115 note that @xmath116 this implies @xmath117 since @xmath118 we have @xmath119 & = \dsp b x(d_s+1)\frac{1-s}{x - s}\phi \\[4 mm ] & = \dsp b x\frac{\partial}{\partial s}\left(\frac{s(1-s)}{x - s}\phi\right ) . \end{array}\ ] ] [ wellknown2 ] the indefinite integral @xmath120 solves @xmath121 . in particular , @xmath122 . since @xmath123 , we have @xmath124 lemma [ wellknown1 ] leads to @xmath125 let @xmath126 and @xmath127 be the operators generating the system @xmath121 : @xmath128 q_1(a,0,b , c;s , t ) & = p(a , b , c;t)/t+s(1-t)\partial_{st}-bs\partial_s,\\[2 mm ] r_1(a,0,b , c;s , t)&=(s - t)\partial_{st}-b\partial_s;\end{array}\ ] ] refer to section 1 . note that @xmath129 . by using the above identities , we have @xmath130 and @xmath131 @xmath132 furthermore , for @xmath133 , the forms of operators above imply that @xmath80 lies in @xmath134 . we now use the following fact : @xcite . [ bailey3 ] @xmath135 from this fact we get , when @xmath136 , @xmath137 if we put @xmath138 then @xmath139 thus we have @xmath140 & = ( 1-y)^{-b'}(1-x)^{-b}s_1\left(b,0,b',a;\dsp\frac{-x}{1-x},\frac{xy}{(1-x)(1-y)}\right)\\[3 mm ] & \supset ( 1-y)^{-b'}(1-x)^{-b } s\left(b , b',a;\dsp\frac{xy}{(1-x)(1-y)}\right ) . \end{array}\ ] ] this agrees with the inclusion in proposition [ prop : s2 g ] . in particular , by the inclusion @xmath141 and lemma [ wellknown2 ] we get solution of @xmath142 represented by the indefinite integral : @xmath143 starting point of the path of integration can be any point @xmath144 , so we choose @xmath145 , just for simplicity . by exchanging the role of @xmath146 and @xmath147 , we get an inclusion @xmath148 and another solution of @xmath142 represented by the indefinite integral : @xmath149 after the change @xmath150 , it can be also expressed as @xmath151 thus , we have : [ matome ] we have the following inclusions of the spaces of solutions : @xmath152 moreover , the collection of solutions @xmath153 is spanned by @xmath154 and @xmath155 this will play a key role to understand the schwarz map of a system @xmath0 with specific parameters , which will be introduced in [ e36 ] . in this section , we study the monodromy representation of @xmath156 . though it is assumed in @xcite that the parameters satisfy the irreducibility condition in fact 1.1 : @xmath157 in this section , we only assume the weaker condition @xmath158 we modify theorem 7.1 in @xcite so that the statements remain valid for these parameters . we will apply the result of this section in [ special ] . let @xmath159 be the complement of the singular locus of @xmath0 . for each @xmath160 , we consider a multi - valued function @xmath161 on @xmath162 as in 1.6 and 1.7 of chapter 2 in @xcite , we define the twisted homology group @xmath163 associated with @xmath164 and locally finite one @xmath165 . under some genericity condition , the integral of @xmath164 over a twisted cycle gives a solution of @xmath0 . if @xmath166 then the natural map @xmath167 is bijective , and the inverse map @xmath168 is called the regularization . in general , the map @xmath169 is neither injective nor surjective , however we still have the isomorphism @xmath170 under condition ( 3.1 ) , thanks to the vanishing theorem of cohomology groups in @xcite , rank of @xmath165 and @xmath163 are equal to the euler number @xmath171 of @xmath172 , and the bilinear form _ the intersection form _ @xmath173 is non - degenerate . let @xmath174 be a small simply connected domain in @xmath175 . we can identify the local solution space to @xmath0 on @xmath174 with the trivial vector bundle @xmath176 via the euler type integral representation of solutions to @xmath0 . the monodromy representation of @xmath0 is equivalent to that of the local system @xmath177 over @xmath175 . we also consider a local system @xmath178 over @xmath175 . we fix a small positive real number @xmath179 , and let @xmath180 a base point in @xmath175 . denote the germs at this point of the local systems @xmath181 and @xmath182 by @xmath183 respectively . let @xmath184 be the monodromy representations of @xmath181 and @xmath182 with respect to @xmath180 . @xmath185 and @xmath186 are called the circuit transformations along @xmath187 . [ prop : inv - subspace ] * the image @xmath188 of the natural map @xmath189 is invariant under the monodromy representation @xmath190 . * the kernel @xmath191 of the natural map @xmath192 is invariant under the monodromy representation @xmath193 . it is clear that @xmath194 and @xmath195 are invariant under @xmath190 and @xmath193 , respectively . we have only to note that the natural maps @xmath196 and @xmath197 commute with @xmath190 and @xmath193 . we will see that if @xmath2 or @xmath3 , then both of @xmath188 and @xmath191 are proper subspaces . thus monodromy representations @xmath190 and @xmath193 are reducible in this case . [ lem : duality ] 1 . let @xmath198 and @xmath199 be elements of @xmath200 and @xmath195 , respectively . then we have @xmath201 2 . suppose that @xmath202 is decomposed into the direct sum of the eigenspaces @xmath203 of @xmath204 of eigenvalues @xmath205 . then @xmath206 is decomposed into the direct sum of the eigenspaces @xmath207 of @xmath208 of eigenvalues @xmath209 . the eigenspace @xmath210 is characterized as @xmath211 \(1 ) the intersection number @xmath212 is stable under small deformations of @xmath198 and @xmath199 . \(2 ) since @xmath204 belongs to the general linear group , we have @xmath213 . let @xmath214 be any element of @xmath215 @xmath216 and let @xmath217 be any element of @xmath218 . since we have @xmath219 for @xmath220 , @xmath221 belongs to the space @xmath222 . thus @xmath208 induces a linear transformation of @xmath222 . let @xmath223 be an eigenvalue of the restriction of @xmath208 to @xmath222 , and let @xmath224 be an eigenvector of @xmath223 . since the intersection from @xmath225 is non degenerate , there exists @xmath226 such that @xmath227 . note that @xmath228 hence we have @xmath229 , i.e. , @xmath230 and @xmath231 . similarly , we have @xmath232 for @xmath233 . to show @xmath234 , we consider the restriction of @xmath208 to @xmath235 , where @xmath236 take its eigenvector @xmath237 and repeat the above argument . in this way , we have @xmath238 since @xmath239 we have @xmath240 for @xmath241 . let @xmath242 @xmath243 be locally finite chains shown in figure [ fig : basis ] . we specify a branch of @xmath164 on each chain by the assignment of @xmath244 on it as in table [ tab : arg ] , where @xmath245 and load @xmath164 to get the locally finite twisted cycles @xmath246 . -cycles , width=377 ] .list of @xmath244 on @xmath247 [ cols="^,^,^,^,^,^",options="header " , ] it will be shown that @xmath248 form a basis of @xmath200 in corollary [ cor : basis ] . we choose elements in @xmath195 as @xmath249 @xmath250 where @xmath251 and @xmath252 note that in terms of @xmath253 s the irreducible condition in fact 1.1 is @xmath254 the condition ( 3.1 ) is @xmath255 and ( see figure [ p2fig ] ) @xmath256 and that @xmath257 satisfies ( 3.1 ) and @xmath258 for any @xmath259 . explicitly , @xmath260 and @xmath261 can be written as : @xmath262- \frac{\circlearrowleft_{0}^-}{\mu_1^{-1}-1}\big)\times \big(\frac{\circlearrowleft_0^+}{\mu_3^{-1}-1 } + [ \delta,1-\delta]- \frac{\circlearrowleft_{1}^-}{\mu_4^{-1}-1}\big)\big]^{\psi^{-1}},\ ] ] @xmath263- \frac{\circlearrowleft_{1}^-}{\mu_2^{-1}-1}\big ) \times \big(\frac{\circlearrowleft_\infty^+}{\mu_{345}-1 } + [ \frac{-1}{\delta},-\delta]- \frac{\circlearrowleft_{0}^-}{\mu_3^{-1}-1}\big)\big]^{\psi^{-1}},\ ] ] where @xmath264 is a small positive real number , @xmath265 $ ] and @xmath266 $ ] are closed intervals , @xmath267 is the negatively oriented circle of which radius , center and terminal are @xmath268 , @xmath269 and @xmath270 , @xmath271 @xmath272 is the positively oriented circle of which radius , center and terminal are @xmath264 , @xmath273 and @xmath274 . notice that the definition of the twisted cycles @xmath275 @xmath276 make sense even in the case @xmath2 or @xmath3 . indeed , this specialization gives no harm to @xmath277 @xmath278 , and thanks to the above expression , when @xmath279 and @xmath280 , we have @xmath281- \frac{\circlearrowleft_{1}^-}{\mu_4^{-1}-1}\big)\big]^{\psi^{-1 } } , \\ { { \widetilde{\delta}}_4}&= & \big [ \big(\frac{\circlearrowleft_0^+}{\mu_1^{-1}-1 } + [ \delta,1-\delta]- \frac{\circlearrowleft_{1}^-}{\mu_2^{-1}-1}\big ) \times \circlearrowleft_\infty^+\big]^{\psi^{-1}},\end{aligned}\ ] ] respectively . [ rem : kernel ] suppose that @xmath282 . then the twisted cycles @xmath283 and @xmath284 are homologous to @xmath269 in @xmath285 , since they are the boundary of locally finite @xmath4-chains given by the replacement @xmath286 in their expressions , where @xmath287 is the annulus @xmath288 . they belong to @xmath191 . by proposition [ prop : int - mat ] , it turns out that @xmath188 is spanned by @xmath289 and @xmath290 . [ prop : int - mat ] the intersection matrix @xmath291 for @xmath292,@xmath293,@xmath294 and @xmath295,@xmath293,@xmath296 is given by @xmath297 its determinant is @xmath298 which does not vanish under the assumption @xmath299 . follow 3 of chapter viii in @xcite for the computation of the intersection numbers . by a straightforward calculation , we have its determinant . proposition [ prop : int - mat ] yields the following corollary . [ cor : basis ] the twisted cycles @xmath292,@xmath293,@xmath300 and @xmath295,@xmath293,@xmath301 form a basis of @xmath200 and that of @xmath195 , respectively . we express the twisted cycles @xmath302 and @xmath303 as linear combinations of @xmath304 . [ lem : squares ] we have @xmath305 set @xmath306 and compute the intersection numbers @xmath307 . then we have @xmath308 which yields the expression of @xmath302 . since @xmath309 we have the expression of @xmath303 . in @xcite , we take a basis @xmath292 , @xmath310 , @xmath302 and @xmath303 of @xmath200 . if @xmath6 then each of @xmath302 and @xmath303 is a scalar multiple of @xmath292 by lemma [ lem : squares ] . similar to lemma [ lem : squares ] , we have the following . [ lem : dual - squares ] we have @xmath311 we give generators of the fundamental group @xmath312 . let @xmath313 ( and @xmath314 , @xmath315 ) be a loop in @xmath316 starting from @xmath317 , approaching to the point @xmath318,(and @xmath319 , @xmath320 ) with @xmath321 , turning once around the point positively , and tracing back to @xmath180 . let @xmath322 ( and @xmath323 ) be a loop in @xmath324 starting from @xmath317 , approaching to the point @xmath325 , ( and @xmath326 ) with @xmath327 , turning once around the point positively , and tracing back to @xmath180 . it is known that the loops @xmath328 generate @xmath312 . the circuit matrices of @xmath190 and @xmath329 along @xmath330 will be denoted as @xmath331 [ th : monod - rep ] under the assumption ( 3.1 ) , the circuit transformations @xmath332 @xmath333 are given as @xmath334 @xmath335 where @xmath336 is any element of @xmath200 , @xmath337 @xmath338 is the submatrix of @xmath339 consisting of the @xmath340 , @xmath341 , @xmath342 and @xmath343 entries of @xmath339 , and @xmath344 @xmath345 under the condition @xmath346 , the linear transformation @xmath347 satisfies the assumption of lemma [ lem : duality ] ( 2 ) . in fact , a fundamental system of solutions can be given by the hypergeometric series @xmath348 multiplied by the power functions @xmath349 thus the eigenvalues of @xmath347 are @xmath1 and @xmath350 , and each of the eigenspaces is two dimensional . it is easy to see that the locally finite chains @xmath351 and @xmath352 are invariant under the deformation along @xmath313 . hence the twisted cycles @xmath289 and @xmath353 span the eigenspace of @xmath347 of eigenvalue @xmath1 . similarly we can show that the twisted cycles @xmath354 and @xmath355 span the eigenspace of @xmath356 of eigenvalue @xmath1 . by lemma [ lem : duality ] ( 2 ) , the eigenspace of @xmath347 of eigenvalue @xmath357 is @xmath358 it is easy to see that the right hand side @xmath359 of ( [ eq : monod1 ] ) satisfies @xmath360 by proposition [ prop : int - mat ] , @xmath8 form a basis even in the case @xmath361 or @xmath362 . thus the representation matrix @xmath363 of @xmath347 with respect to this basis is continuous on the parameters @xmath364 . on the other hand , the expression @xmath365 is also continuous since the factor @xmath366 in the denominator of @xmath367 \dfrac{\mu_4 - 1}{\mu_{34}(\mu_5 - 1 ) } & \dfrac{\mu_{34}-1}{\mu_{34}(\mu_5 - 1 ) } \end{pmatrix}.\ ] ] cancels by @xmath368 . similarly we have the expression of @xmath369 . to study @xmath370 , we work temporarily under the condition @xmath371 . we decompose @xmath315 into @xmath372 , where @xmath373 is the approach to @xmath374 and @xmath375 is the turning path . we trace the deformation of the triangle @xmath376 made by @xmath377 and @xmath378 along @xmath373 . after the deformation , this becomes a small triangle near the point @xmath379 . we see the argument of @xmath380 on this triangle . since @xmath381 and @xmath382 in @xmath373 , @xmath383 in this triangle , and @xmath384 on the line @xmath385 , @xmath380 varies negative to positive via the upper - half space , i.e , @xmath386 decreases by @xmath387 along @xmath373 . note that this change is compatible with our assignment of @xmath386 on @xmath377 and @xmath378 in table [ tab : arg ] . thus the twisted cycle @xmath388 plays the role of a vanishing cycle as the line @xmath389 approaches the point @xmath390 . since @xmath375 corresponds to the move of @xmath391 turning around the point @xmath392 the cycle @xmath393 is an eigenvector of @xmath370 of eigenvalue @xmath394 . we can similarly show that @xmath395 is an eigenvector of @xmath396 of eigenvalue @xmath397 . on the other hand , we can find three chambers not affected by the move of the line @xmath389 along @xmath375 . for example , @xmath398 , @xmath399 and @xmath400 . hence @xmath370 has three dimensional eigenspace of eigenvalue @xmath1 . lemma [ lem : duality ] ( 2 ) yields that this eigenspace is expressed as @xmath401 so @xmath370 has desired eigenvalues and eigenspaces . since the factor @xmath402 is continuous on @xmath394 at @xmath1 , the expression of @xmath370 is valid even in the case @xmath403 . similarly we have the expressions of @xmath404 and @xmath405 . [ cor : monod - rep - dual ] under the assumption ( 3.1 ) , the circuit transformations @xmath406 @xmath333 are given as @xmath407 @xmath408 where @xmath199 is any element of @xmath195 . let @xmath409 and @xmath410 @xmath333 be the circuit matrices along the loop @xmath330 with respect to the basis @xmath411 of @xmath200 , and to @xmath412 of @xmath195 , respectively . that is , we have transformations @xmath413 by the continuation along @xmath330 . [ cor : monod - matrix ] the circuit matrices are expressed as @xmath414 @xmath415 where @xmath416 @xmath417 @xmath418 @xmath419 their explicit forms are @xmath420 @xmath421 @xmath422 @xmath423 they satisfy @xmath424 we identify @xmath425 with the row and column vectors @xmath426 respectively . note that @xmath427 theorem [ th : monod - rep ] yields these expressions . we have @xmath428 by lemma [ lem : duality ] ( 1 ) . as a result , @xmath409 @xmath333 coincides with the circuit matrix with respect to the basis @xmath8 by theorem 7.1 in @xcite . in this section we introduce a system @xmath430 , and describe its schwarz map , which is the main result of this paper . we introduce in this subsection a system @xmath430 , which is a system @xmath0 with specific parameters , and mention a reason why this system is of special interest . let @xmath12 be the configuration space of six lines @xmath432 in general position in the projective plane @xmath433 . we identify the space @xmath12 with @xmath434 where @xmath70 is the line at infinity in the @xmath435-plane given by @xmath436 . the system @xmath437 is generated by the linear differential equations which annihilate functions on @xmath12 defined by the integral @xmath438 the schwarz map of the system @xmath439 is studied ( cf . @xcite , @xcite ) in two cases @xmath440 and @xmath441 . we have been interested in the case @xmath442 . on the other hand , let @xmath443 be the 2-dimensional stratum defined by @xmath444 , which is the space of six lines such that the three lines @xmath445 meet at a point , the three lines @xmath446 meet at another point , and nothing further special occurs . it is known ( @xcite ) that the restriction of @xmath439 onto @xmath443 is the appell s hypergeometric system @xmath447 , which is projectively equivalent ( multiplying a function to the unknown ) to @xmath448 setting @xmath449 , we define @xmath450 we believe that the first step of understanding @xmath451 is the study of the system @xmath429 . the schwarz map of a system is defined by the ratio of linearly independent solutions . the main objective of this paper is the schwarz map of the hypergeometric system @xmath452 the system @xmath429 admits solutions stated in proposition [ matome ] . the next subsection gives a geometric background of understanding these solutions . consider a family of curves of genus 2 given as triple covers of @xmath453 : @xmath454 branching at four points @xmath455 . we choose two linearly independent holomorphic 1-forms : @xmath456 and put @xmath457 for a fixed @xmath51 , the _ abel - jacobi map _ for the curve @xmath458 is a multi - valued map @xmath459 it is a single - valued map to its jacobian @xmath460 , where @xmath461 is a lattice generated by its periods : integrals over possible loops with base @xmath462 : @xmath463 proposition [ matome ] for @xmath464 implies that after the coordinate change @xmath465 and the change of unknown : @xmath466 , two linearly independent solutions of @xmath467 and the two indefinite integrals @xmath468 form a set of fundamental solutions of @xmath429 . on the other hand , the integral representation of the gauss hypergeometric equation given in section 1 asserts that the integral above along any _ closed _ path gives a solution of @xmath469 . thus we find that the schwarz map of @xmath429 is the totality of the abel - jacobi map of the family @xmath470 after a slight modification ( multiplying @xmath471 to the second coordinate ) . thus we get [ th : gen - schwarz ] if we change the coordinates @xmath472 of @xmath473 as @xmath474 the schwarz map of the system @xmath429 is equivalent to the projectivization of the family of the abel - jacobi map of the family @xmath470 of curves of genus 2 , explicitly given as @xmath475 the latter two @xmath476 and @xmath477 are @xmath478 and @xmath479 in section 2 . the map by means of the former two @xmath480 is the schwarz map of the hypergeometric equation @xmath481 . its image is a disc , and the inverse map of @xmath482 is single - valued automorphic function on the disc with respect to the triangle group of type @xmath10 $ ] ; in other words , the disc is tessellated by schwarz triangles of type @xmath10 $ ] . the image surface under @xmath483 can be regarded as lying in a fiber bundle with the @xmath482-image disc as its base and the jacobian variety of @xmath458 as the fiber on the image point @xmath484 . a triangle of type @xmath485 $ ] is a hyperbolic triangle with angles @xmath486 and @xmath487 ; the above triangle has angles @xmath488 and @xmath269 . the triangle group of type @xmath485 $ ] is the group consisting of the even products of the reflections with the sides of the triangle of type @xmath485 $ ] as axes . it is known that the triangle group of type @xmath10 $ ] is conjugate to the congruence subgroup @xmath489 for arithmetic triangle groups , see @xcite . other than this family of curves , there are two families of curves of genus 2 branching at four points in @xmath453 ; see appendix 2 . from monodromy side , theorem [ th : gen - schwarz ] can be understood as follows . define @xmath490 and @xmath491 @xmath333 by substituting @xmath492 into @xmath409 and @xmath410 defined in [ circuit ] , respectively . they are the circuit matrices for @xmath429 with respect to @xmath493 and those for @xmath494 with respect to @xmath495 where @xmath496{t_1(1-t_1)t_2(1-t_2)(1-t_1x - t_2y ) } } .\ ] ] [ cor : monodromy ] we have @xmath497 @xmath498 they satisfy @xmath499 by proposition [ prop : inv - subspace ] and remark [ rem : kernel ] , the subspace spanned by solutions @xmath500 and @xmath501 is invariant under the monodromy representation . in fact , the top - left @xmath502 block matrices @xmath503 of @xmath490 @xmath333 act on this space . note that @xmath504 , @xmath505 . let @xmath506 be the group generated by @xmath507 , @xmath508 and @xmath509 . the group @xmath506 is isomorphic to the triangle group @xmath10 $ ] , and is contained in the unitary group @xmath510)\mid g h ' \;^t \overline { g}=h'= \begin{pmatrix } -1 & -\omega \\ \omega+1 & 0 \end{pmatrix } \big\}.\ ] ] by a matrix @xmath511 the hermite matrix @xmath512 and circuit matrices @xmath513 @xmath514 are transformed as @xmath515 @xmath516 hence the projectivization of @xmath506 is isomorphic to the congruence subgroup @xmath517 of @xmath518 and the ratio @xmath519 can be regarded as the map @xmath482 in theorem [ th : gen - schwarz ] and as an element of the upper - half space . let @xmath520 denote the stratum defined by @xmath521 and let us restrict the system @xmath439 to this stratum , which is the space of six lines such that the three lines @xmath445 meet at a point , and nothing further special occurs . this system is denoted by @xmath522 or @xmath523 . little is known about this system . before stating the proposition in this section , we briefly recall the appell - lauricella s system @xmath524 @xmath525 where @xmath526 are variables , and @xmath527 and @xmath528 . this is a 3-variable version of the appell s @xmath5 . it admits solutions given by a power series @xmath529 where @xmath530 , and by an integral @xmath531the collection of solutions is denoted by @xmath532 in this section , we prove the following proposition . [ x3fd ] if @xmath533 , then the system @xmath522 is reducible and has a subsystem isomorphic to the appell - lauricella s system @xmath534 in 3 variables with 4 free parameters . more precisely , the collection of the solutions of @xmath522 includes @xmath535 note @xmath536 . if we apply the proposition under the further restriction @xmath88 , we find a subsystem isomorphic to @xmath5 in @xmath0 , which is equivalent to proposition 1.1 . we give three `` proof ' 's : one using power series , one using integral representations , and one manipulating differential equations . it is known that the system @xmath537 has a solution given by the series @xmath538 where @xmath539 ; refer to @xcite . a computation shows that the identity @xmath540 holds if and only if @xmath541 we manipulate the integral @xmath542 the three lines @xmath543 meet at @xmath544 . introduce new coordinate @xmath545 @xmath546 which send @xmath547 to @xmath548 . since @xmath549 @xmath550 we have @xmath551 @xmath552 where @xmath553 . if @xmath554 then the double integral above becomes the product of the beta integral @xmath555 and the integral @xmath556 which can be written as @xmath557 on the other hand , the integral @xmath558 solves @xmath559 . by solving the system @xmath560 we complete the proof of proposition [ x3fd ] we manipulate the system @xmath523 ( given in [ msy , p.24 ] ) : @xmath561 where @xmath562 , @xmath563 , @xmath564 , and @xmath565 , and the appell - lauricella system @xmath566 . we show that the system @xmath567 is a subsystem of @xmath523 when @xmath533 . change the unknown @xmath568 of @xmath523 into @xmath110 by @xmath569 and the variables @xmath570 into @xmath571 as @xmath572 then @xmath523 can be written as @xmath573 where @xmath574 where @xmath575 write the system @xmath567 as @xmath576 , where @xmath577 eliminating the second derivatives in @xmath578 by using @xmath579 , we see that @xmath578 are linear combination , over @xmath580 , of the @xmath579 s if and only if @xmath581 and @xmath582 this completes the proof of proposition [ x3fd ] . * remark : * actually we have , under the condition @xmath583 , @xmath584@xmath585 we encountered a family of curves @xmath458 of genus 2 given as triple covers of @xmath453 . this is the case 3 in the following proposition . indeed , since the @xmath587 fold cyclic cover @xmath588 of @xmath453 branching at four points with indices @xmath589 has euler characteristic @xmath590 if we assume the genus of @xmath588 is two ( euler characteristic of @xmath588 is @xmath591 ) , we have @xmath592 it is easy to see that only three cases above are possible . the three cases can be realized by the following families of curves : @xmath593 { \rm case\ 6:}&c_t^{(6 ) } : & s^6=s^2(1-s)^4(t - s)^3,&\quad t:{\rm \ parameter},\\[2 mm ] { \rm case\ 4:}&c_t^{(4 ) } : & s^4=s^2(1-s)^2(s - t),&\quad t:{\rm \ parameter}. \end{array}\ ] ] note that the double cover of the base space of case 6 branching at the two points of index 2 is equivalent to case 3 . aomoto k. and kita m. , translated by k. iohara , _ theory of hypergeometric functions _ , springer verlag , now york , 2011 . w. n. bailey , _ generalized hypergeometric series _ , cambridge , 1935 e. bod , algebraicity of the appell - lauricella and horn hypergeometric functions , j. diff . equations * 252 * ( 2012 ) , 541566 . k. cho , a generalization of kita and noumi s vanishing theorems of cohomology groups of local system , nagoya math . j. , * 147 * ( 1997 ) , 6369 . k. mimachi and t. sasaki , irreducibility and reducibility of lauricella s system of differential equations @xmath594 and the jordan - pochhammer differential equation @xmath595 , kyushu j. math . * 66*(2012 ) , 6187 . k. matsumoto , t. sasaki and m. yoshida , the monodromy of the period map of a @xmath596-parameter family of k3 surfaces and the hypergeometric function of type @xmath597 , intern . j. math . * 3*(1992 ) , 1164 . k. matsumoto , t. sasaki , n. takayama and m. yoshida , monodromy of the hypergeometric differential equation of type @xmath597 , ii the unitary reflection group of order @xmath598 , annali della scuola normale superiore di pisa ( 4 ) * 20 * ( 1993 ) , 617631 . k. matsumoto and t. terasoma , period maps of reducible hypergeometric equations and mixed hodge structures , in preparation . k. matsumoto and m. yoshida , monodromy of lauricella s hypergeometric @xmath599-system , ann . ( 5 ) , * 13 * ( 2014 ) , 551577 . takeuchi , k , commensurability classes of arithmetic triangle groups , j. fac . tokyo sect . ia math . 24(1977 ) , 201212 . m. yoshida , _ hypergeometric functions , my love _ , vieweg , 1997 .
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we study an appell hypergeometric system @xmath0 of rank four which is reducible and show that its schwarz map admits geometric interpretations : the map can be considered as the universal abel - jacobi map of a @xmath1-dimensional family of curves of genus 2 .
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x - ray reflection off the surface of cold disks in active galactic nuclei ( agn ) and galactic black holes ( gbhs ) has been an active field of research since the work of @xcite . in early studies , the illuminated material was assumed to be cold and non - ionized @xcite . it was soon realized , however , that photoionization of the disk can have a great impact on both the reflected continuum and the iron fluorescence lines . detailed calculations were then carried out by @xcite and @xcite . however , in all of these papers , the density of the illuminated material was assumed to be constant along the vertical direction . this assumption applies only to the simplest version of radiation - dominated shakura - sunyaev disks @xcite , and only for the portion where viscous dissipation is the dominating heating process . for the surface layers , however , photoionization and compton scattering are the major heating sources . therefore the approximation of constant density is not appropriate . moreover , thermal instability allows the coexistence of gas at different phases . these different phases have very different temperatures , and hence different densities to keep the gas in pressure balance . recently @xcite relaxed the simplifying assumption of constant gas density . they determined the gas density from hydrostatic balance solved simultaneously with ionization balance and radiative transfer . they made an important observation that the thomson depth of the hot coronal layer can have great influence on the x - ray reprocessing produced by the deeper , and much cooler disk . in order to simplify the calculation of the vertical structure , though , they ignored thermal conduction and the effects of transition layers between the different stable phases . a discontinuous change in temperature was allowed whenever an unstable phase was encountered . they argued that such transition layers are of little importance because their thomson depths are negligibly small . however , without taking into account the role of thermal conduction , their method of connecting two different stable layers is rather _ ad hoc_. moreover , even though the thomson depths of these transition layers are small , it does not guarantee that the x - ray emission and reflection from such layers are negligible . because the temperature regime where the transition layers exist is not encountered in the stable phases , some of the most important lines can have appreciable emissivity only in these layers . also , since resonance line scattering has much larger cross section than thomson scattering , the optical depths in resonance lines can be significant . including thermal conduction in the self - consistent solution of the vertical structure presents a serious numerical challenge . the difficulties are due to the coupling between hydrostatic balance , radiative transfer and heat conduction . @xcite first studied the phase equilibrium of a gas heated by cosmic rays and cooled by radiation . they found that taking into account heat conduction in the boundary layer allows one to obtain a unique solution of the stable equilibrium . @xcite calculated the full temperature profile for a compton - heated corona , and @xcite calculated the static conditions of the plasma for different forms of heating and cooling . but they did not include much discussion of the spectroscopic signatures resulting from the derived vertical structure . in this paper , we first calculate the temperature structure in the layers above the accretion disk , then calculate the emission lines via radiative recombination ( rr ) and reflection due to resonance line scattering from the derived layers . certain illuminating continua spectra allow more than two stable phases to coexist , with two transition layers connected by an intermediate stable layer . for the transition layer , since the thomson depth is small , the ionizing continuum can be treated as constant ; and since its geometric thickness is smaller than the pressure scale height , the pressure can be treated as constant as well . we can thus obtain semi - analytic solution of the temperature profile by taking into account thermal conduction . for the intermediate stable layer , its thickness is determined by the condition of hydrostatic equilibrium . in our model , the normally incident continuum has a power - law spectrum with an energy index of @xmath0 . we also assume a plane - parallel geometry and that the resonance line scattering is isotropic . the structure of this paper is as follows : in [ sec_structure ] we discuss the existence of the thermal instability and compute the thermal ionization structure of the transition layers ; in [ sec_spectrum ] we calculate the recombination emission lines and the reflection due to resonance line scattering ; in [ sec_summary ] we summarize the important points of the calculations , the validity of various approximations made in the calculations , and the detectability of the recombination emission and reflection lines . the vertical structure of an x - ray illuminated disk at rest is governed by the equations of hydrostatic equilibrium and of energy conservation @xmath1 in the first equation , @xmath2 is the force density due to gravity and radiation pressure . the dependence of the force on the plasma density is included explicitly through the hydrogen density @xmath3 . in the second equation , a time independent state is assumed , @xmath4 is the thermal conductivity , and @xmath5 is the net heating rate depending on the gas state and the incident flux @xmath6 ( differential in energy ) . we neglect the effects of magnetic field and adopt the spitzer conductivity appropriate for a fully ionized plasma , @xmath7 erg @xmath8 s@xmath9 k@xmath9 @xcite . we have used the classical heat flux , @xmath10 , in equation ( [ eq_transition ] ) because the electron mean free path is short compared to the temperature height scale . since the continuum flux may change along the vertical height , in principle , the above two equations must be supplemented by an equation for radiative transfer . a self - consistent solution of such equations is difficult to obtain . in the following , we invoke a few physically motivated approximations , which make the problem tractable . first , in thermally stable regions , the gas temperature is slowly varying , the heat conduction term in the energy balance equation can be neglected . therefore , the temperature can be determined locally with the condition @xmath11 . it is well known @xcite that the dependence of @xmath12 on the gas pressure @xmath13 and the illuminating continuum @xmath6 can be expressed in the form of @xmath14 , where @xmath15 is the electron density , @xmath16 is the net cooling rate per unit volume , and @xmath17 is an ionization parameter defined by @xmath18 where @xmath19 is the total flux of the continuum , and @xmath20 is the speed of light . in figure [ fig_scurve ] , we show the local energy equilibrium curve @xmath21 versus @xmath17 , at @xmath22 calculated with the photoionization code xstar @xcite . these curves are commonly referred to as `` s curves '' due to their appearance . the illuminating continuum is assumed to be a power law with energy index @xmath0 . the solid line labeled with `` s - curve 1 '' corresponds to a low energy cutoff at 1 ev and a high energy cutoff at 150 kev , while the dashed line `` s - curve 2 '' corresponds to a high energy cutoff at 200 kev . the choice of such incident spectra is based on their common appearance in many agns and gbhcs . the region is thermally `` unstable '' where the `` s - curve '' has a negative slope , and `` stable '' where the slope is positive , as indicated in figure [ fig_scurve ] . in the thermally unstable regions , we have @xmath23 where the derivative is taken while the energy balance is satisfied , i.e. , @xmath24 . this condition was shown @xcite to be equivalent to the instability condition discovered by @xcite @xmath25 the thermal instability allows the gas to coexist at different phases . the gas temperature may change by orders of magnitude over a geometric thin region whenever an unstable phase separates two stable ones . this results in enormous temperature gradients and heat conduction . therefore the heat conduction in the energy balance equation should be included in such transition layers between stable phases . on the other hand , the thicknesses of these transition layers are usually smaller than the pressure scale height , so one can safely treat the gas pressure as constant in these regions . moreover , the continuum radiative transfer can be neglected because the compton optical depth is found to be small . the vertical structure of the transition regions is then solely determined by the energy balance equation with heat conduction . the thomson optical depth @xmath26 of such regions is readily estimated by @xmath27 where @xmath28 is the thomson scattering cross section . the transition layer solution is not arbitrary under the steady - state conditions , i.e. , where there is no mass exchange between the two stable phases which the transition layer connects . a similar problem in the context of interstellar gas heated by cosmic rays was well studied by @xcite . we follow their procedure here and define @xmath29 , in order to rewrite equation ( [ eq_transition ] ) in the form : @xmath30 a steady - state requires vanishing heat flux at both boundaries of the transition layer , or @xmath31 where @xmath32 and @xmath33 are the temperatures of the two stable phases which are connected by a transition layer . this condition determines a unique ionization parameter @xmath17 for the transition layer in question , and the integration of equation ( [ eq_dy2 ] ) along the vertical height gives the detailed temperature profile as a function of optical depth . if the disk does not realize the steady - state solution , there are additional enthalpy terms in equation ( [ eq_transition ] ) which require that there be mass flow through the transition region i.e. the cool material in the disk evaporates , or the hot material in the corona condenses @xcite . physically , this corresponds to a movement of the transition layer up or down through the vertical disk structure . however , since the density increases monotonically toward the center of the disk , this `` motion '' stops where the transition layer reaches the steady state value of @xmath17 . thus , in the absence of disk winds or continuous condensation from a disk corona , the steady state solution should generally be obtained . there is a complication for the `` s - curves '' shown in figure [ fig_scurve ] , because in each curve there exist two unstable regions and therefore there should be two transition layers . for `` s curve 1 '' , condition [ eq_y2 ] can be met for both transition layers with resulting ionization parameters @xmath34 , where @xmath35 and @xmath36 are associated with the transition layer which connects to the lowest temperature phase and highest temperature phase , respectively . for `` s curve 2 '' , the resulting ionization parameters for two transition layers , however , satisfy @xmath37 . such a situation is unphysical , where the ionization parameter of the upper transition layer is smaller than that of the lower one , because in the context of accretion disks , the upper layer receives more ionizing flux and has lower pressure . so in practice , for `` s curve 2 '' the intermediate stable region is skipped and a transition layer connects the lowest temperature phase to the highest temperature phase directly . the ionization parameter of this transition layer is determined the same way by applying equation ( [ eq_y2 ] ) . there is then an intermediate stable layer , of nearly uniform temperature , in between these two transition layers for `` s curve 1 '' , as indicated with bc in figure [ fig_scurve ] . the thickness of the intermediate layer should in principle be obtained by solving the coupled equations of hydrostatic equilibrium , energy balance , and radiative transfer . however , unlike the stable phase at the disk base or that of the corona , which may be compton thick , this intermediate stable layer is generally optically thin , because its optical depth is restricted by the difference between @xmath35 and @xmath36 . furthermore , the temperature in this layer is slowly varying , and therefore heat conduction can be neglected . we shall make another approximation that the variation of the force density @xmath2 in the hydrostatic equation can also be neglected . this may not be a good approximation . however , since our main purpose is to investigate qualitatively the effects of an intermediate stable layer , such a simplifying procedure does capture the proper scaling relations of the problem , and has the advantage of less specific model dependence . writing equation ( [ eq_hydro ] ) in a dimensionless form and parameterizing the force factor by a dimensionless parameter @xmath38 , we obtain @xmath39 the parameter a defined here is identical to the gravity parameter in @xcite in the absence of radiation pressure . the integration of this equation from @xmath35 to @xmath36 gives the thomson depth of the intermediate stable layer . assuming radiation pressure can be neglected , the force factor at a given radius @xmath40 of the disk can be estimated as @xmath41 where @xmath42 is the luminosity of the continuum source in units of the eddington limit , @xmath43 is the half thickness of the disk at radius @xmath40 , @xmath44 is the mass of the central source , @xmath45 is the proton mass , and @xmath46 is the eddington luminosity . in this estimate , we have assumed that @xmath47 . this gives @xmath48 the value : @xmath49 for a typical thin disk , as those present in agn and black hole binaries , one expects @xmath50 . if the luminosity is sub eddington , @xmath51 as in most agns , @xmath48 is of order unity . however , since the disk surface may not be normal to the continuum radiation , @xmath19 may be only a fraction of the value assumed above , which increases @xmath48 by one or two orders of magnitude . on the other hand , as the source approaches the eddington limit , @xmath48 may become smaller than 1 . therefore , we expect @xmath48 to be in the range of 0.1 10 . the exceptional cases of much smaller and larger @xmath48 are discussed in [ sec_summary ] . the temperature profiles and optical depths of the transition layers and possible intermediate stable layer are shown in figures [ fig_transition ] . three labeled curves in solid lines correspond to `` s - curve 1 '' ( in figure [ fig_scurve ] ) with different values of the force factors @xmath48 = 10 , 1 , and 0.1 , respectively . the dashed curve corresponds to `` s - curve 2 '' , where there is no intermediate stable layer . for each of the solid curves , three layers are clearly seen , with two transition layers being connected by an intermediate stable layer , as illustrated for the case of @xmath52 . the smaller @xmath48 produces a more extended intermediate stable layer as expected . because the stable phase with the lowest temperature is almost neutral , and the stable phase at the highest temperature is almost fully ionized , they are not efficient in generating x - ray line emission , except for iron fluorescence lines from the neutral material . only the transition layers and the intermediate stable layer are expected to emit discrete lines in the soft x - ray band . in a photoionized plasma , the temperature is too low for collisional excitation to be an important line formation process . instead , radiative recombination ( rr ) followed by cascades dominates the line emission . the flux of a particular line can be written as @xmath53 where @xmath54 is the density of the ion before recombination , @xmath55 is the line energy , and @xmath56 is the line emissivity defined as in @xcite . in ionization equilibrium where the ionization rate is equal to the recombination rate , we have @xmath57 , where @xmath58 is the recombination coefficient of ion @xmath59 , @xmath60 is the number density of ion @xmath61 , @xmath6 is again the monochromatic incident flux ( differential in energy ) , and @xmath62 is the photoionization cross section of ion @xmath61 . defining the branching ratio @xmath63 , equation ( [ flux_emission ] ) can be rewritten as : @xmath64 where @xmath65 is the fractional abundance of ion @xmath61 with respect to the electron density . as indicated , @xmath66 and @xmath67 depend only on temperature @xmath21 and ionization parameter @xmath17 , which are both functions of optical depth @xmath68 . for convenience , we further define the `` emission equivalent width '' @xmath69 . then if @xmath70 , which depends only on the shape of the incident continuum , @xmath71 can be written as : @xmath72 therefore the emission equivalent width @xmath71 is independent of the density of the medium and the incident flux . it is a unique function of the structure deduced in section [ sec_structure ] . all other variables depend only on @xmath21 and @xmath17 . the numerical values of @xmath73 were provided by d. liedahl ( private communication ) , and were calculated using the models described in @xcite . the values of @xmath74 were computed using xstar @xcite . in figure [ fig_emission ] , we plot the spectra of the recombination emission within the 0.5 1.5 kev band with a spectral resolution @xmath75 ev , which is close to the spectral resolution of the grating spectrometers on _ chandra _ and _ xmm - newton_. the top panel corresponds to the case without an intermediate stable layer , and the bottom panels corresponds to the case with an intermediate layer for @xmath76 and 0.1 , respectively . for clarity , from top to bottom , the flux in each panel is multiplied by a factor indicated in each panel . it appears that the existence of an intermediate stable layer enhances the emission in this energy band . this is not surprising since the ions that are responsible for these lines have peak abundances at temperatures close to that of the intermediate stable layer . in all cases , the equivalent widths ( ews ) of the emission lines are less than 1.0 ev , with respect to the ionizing continuum . the strongest lines are the hydrogen - like and helium - like lines of oxygen , with ews approaching several tenths of an ev . hydrogen - like and helium - like lines from iron outside the plotted energy band are somewhat stronger , with ews reaching a few ev . we note that our low equivalent width values conflicts with those derived by @xcite , who found some lines with equivalent widths as high as 30 ev . however , since they did not consider the appropriate locations for the transition regions in @xmath17-space , their intermediate stable layer subtended a much larger optical depth than we find here . naturally with a thicker layer , they found larger equivalent widths . emission from rr is not the only line formation process in the transition layers and the intermediate stable layer . due to very large cross sections in resonance line scattering , the reflected flux in these lines may be significant . with the computed thermal and ionization structure , the column density in each ion and the optical depth in all resonance lines can be calculated straightforwardly . the cross sections for resonance line scattering depend on the line broadening . we assume thermal doppler effects as the only mechanism . although the gas temperature is a function of depth , we calculate the line width for a temperature where the abundance of each ion peaks as an average , and assume that the resonance scattering cross sections are uniform along the vertical direction . in terms of absorption oscillator strength @xmath77 , this cross section of the resonance line scattering @xmath78 can be written as @xmath79 where @xmath80 is the electron mass , @xmath81 is the electron charge , @xmath82 is the wavelength of the line and @xmath83 is the average line width in wavelength . under the assumption of thermal doppler broadening , @xmath84 where @xmath85 is the temperature at which the ion abundance peaks , and @xmath86 is the ion mass . the resonance scattering optical depth @xmath87 for a line from ion @xmath59 can be estimated as @xmath88 where @xmath89 is the column density of the ion . the radiative transfer in the line is a complicated issue @xcite . a full treatment is beyond the scope of this work . however , since we are only interested in a reasonable estimate of the reflected line flux , a simple approach may be adopted . we assume the resonance line scattering is isotropic and conservative and neglect the polarization dependence . under such conditions , the reflection and transmission contributions by a plane - parallel slab of finite optical depth @xmath90 have been solved by @xcite . for normal incident flux @xmath19 , the angle dependent reflectivity is @xmath91 where @xmath92 , @xmath93 is the reflectivity at @xmath94 , @xmath95 is the reflected intensity , @xmath19 is the incident flux , and @xmath96 is the scattering function defined as @xmath97 where @xmath98 and @xmath99 are two functions that satisfy the following integral equations : @xmath100d\mu^{\prime } \nonumber \\ y(\mu)&= & e^{-\tau /\mu}+\frac{\mu}{2}\int_{0}^{1}\frac{1}{\mu-\mu^{\prime}}[y(\mu)x(\mu^{\prime})-x(\mu)y(\mu^{\prime})]d\mu^{\prime}.\end{aligned}\ ] ] the solutions of these equations may be obtained by an iterative method with the starting point @xmath101 and @xmath102 . the angle integrated reflectivity @xmath103 can be calculated as @xmath104 and is shown in figure [ fig_totref ] as a function of the resonance scattering optical depth @xmath87 . the reflected flux in a line can be written as @xmath105 where @xmath106 is the line width in energy . similarly to the `` emission equivalent width '' @xmath71 , we define a `` reflection equivalent width '' @xmath107 , which results in : @xmath108 this `` reflection equivalent width '' from our numerical results is a few tenths of an ev for strong resonance lines , similar to that of the recombination emission lines . in figure [ fig_reflection ] , we plot the spectra of the resonantly scattered lines in the energy band 0.5 1.5 kev with a spectral resolution @xmath75 ev . the top panel corresponds to the case without an intermediate stable layer , and the bottom panels corresponds to the case with an intermediate layer for @xmath76 and 0.1 , respectively . for clarity , from top to bottom , the flux in each panel is multiplied by a factor of 100 . we see that the equivalent widths of the reflected lines are notably enhanced when there is an intermediate stable layer , but not as significantly as for the recombination emission lines . this is because the optical depths of many strong lines become much larger than unity , and the reflection is saturated . if there are broadening mechanisms other than thermal doppler effects , such as turbulent velocity , the reflected intensity can be further enhanced . in order to gain a crude idea of the relative importance of recombination emission and reflection from the transition layers and intermediate stable layer , we compare them to the `` hump '' produced by compton scattering off a cold surface @xcite . we use the greens function obtained by @xcite to calculate the compton reflection . this method was verified to be accurate with a monte carlo procedure by @xcite . in figure [ fig_spectrum ] , we show the combined spectra including recombination emission and reflection lines from resonance line scattering , and the compton reflection `` hump '' . we now summarize the most important conclusions that can be drawn from the calculations presented in this paper . we also discuss the detectability of the predicted line features . 1 . the unique ionization parameters that characterize the steady - state solutions of the transition layers depend on the shape of the `` s - curve '' . we have shown that two power - law illuminating spectra with different high energy cutoffs produce very different temperature profiles . the harder spectrum only allows one transition layer even though there are two unstable branches in the `` s - curve '' , while the softer one allows two separate transition layers connected by an intermediate stable layer . this is due to the fact that the ionization parameter of the upper transition layer must be larger than that of the lower one , if they are to exist separately in a disk environment . the harder spectrum produces a turnover point of the upper branch of the `` s - curve '' at smaller @xmath17 . therefore the transition layer due to the upper unstable region joins the lower one smoothly without allowing the intermediate stable region to form . the turnover of the upper `` s - curve '' represents the point where compton heating starts to overwhelm bremsstrahlung . the ionization parameter at which this point occurs is related to the compton temperature of the continuum , @xmath109 @xcite . a harder spectrum has larger @xmath110 , therefore the intermediate stable layer tends to disappear for hard incident spectra . although the thomson depths of the transition layers and possible intermediate stable layer are generally negligible , the x - ray emission lines from them may comprise the main observable line features , because the temperatures of these layers are inaccessible to the stable phases , and thus some of the important lines can have appreciable emissivity only in these layers . due to the much larger cross sections for resonance line scattering , reflection due to resonance lines off such transition layers is also important . the strengths of reflected lines are at least comparable with those of the recombination emission lines when there is no intermediate stable layer . because the appearance of the reflected line spectrum is different from that of the recombination emission spectrum , high resolution spectroscopic observations should be able to distinguish these mechanisms . 3 . the justification of the assumption that the ionizing continuum does not scatter in the intermediate layer depends on the magnitude of the parameter @xmath48 . the thomson depth of this layer @xmath68 is given by : @xmath111 for the power - law continuum with high energy cutoff at 150 kev ( `` s curve 1 '' ) , @xmath112 and @xmath113 from our numerical results . therefore , @xmath114 . @xmath68 is much less than unity as long as @xmath48 is greater than 0.1 . for smaller @xmath48 , however , another effect comes into play . @xcite showed that the thomson depth of the coronal layer ( the stable phase with highest temperature ) exceeds unity when the gravity parameter ( identical to @xmath48 defined here when the radiation pressure is neglected ) is @xmath115 0.01 . therefore not much ionizing flux can penetrate this layer , and the reprocessing in the deeper and cooler layers can be neglected completely . as @xmath48 becomes much larger than 10 , the thickness of the intermediate stable layer is negligible compared to the transition layers . therefore , its presence may be ignored . since the recombination rate must equal the photoionization rate in the irradiated gas , recombination radiation is also a form of reflection i.e. the line equivalent widths are independent of the incident flux . they depend only on the structure ( @xmath116 ) deduced from the hydrostatic and energy balance equations . the detectability of these recombination emission and reflection lines depends on whether the primary continuum is viewed directly . when the ionizing continuum is in direct view , our results show that the ews of the strongest lines in the 0.5 1.5 kev band are at most a few tenths of an ev , slightly larger when there is an intermediate stable layer . + the signal to noise ratio ( snr ) in such a line can be written as @xmath117 where @xmath118 is the integration time of the observation , @xmath55 is the energy of the line , @xmath119 is the photon flux in the continuum , @xmath120 and @xmath121 are the effective area and resolving power of the instrument , respectively . for a line at @xmath115 1 kev , with @xmath122 , and with hetgs on board _ chandra _ , we have @xmath123 . a typical seyfert 1 galaxy has a flux of @xmath124 erg @xmath125 s@xmath9 in the energy band of 210 kev . assuming a power law with energy index of 1 , the photon flux at 1 kev would be @xmath126 @xmath125 s@xmath9 kev@xmath9 . for a reasonable integration time of 10 ks , we have snr @xmath127 . when the primary continuum is obscured as in seyfert 2 galaxies , the ews of the emission and reflection lines can be orders of magnitude larger , because the continuum at this energy region is absorbed severely , and the snr can be greatly enhanced , making these lines observable . acknowledgment : smk acknowledges several grants from nasa which partially supported this work , mfg acknowledges the support of a chandra fellowship at mit . we wish to thank m. sako , d. savin and e. behar for several useful discussions . the `` s - curves '' produced by two different incident ionizing spectra . the vertical line indicates the unique solution of @xmath17 which satisfies condition [ eq_y2 ] : two solutions for `` s - curve 1 '' , @xmath35 = 2.82 and @xmath36 = 3.14 ; and only one for `` s - curve 2 '' , @xmath17 = 2.22 . ] the temperature profiles of the transition layers and the intermediate stable layer versus thomson optical depth @xmath68 . the solid curves correspond to `` s - curve 1 '' with different force factors @xmath48 , the dashed line corresponds to the `` s - curve 2 '' . ] the spectra of the emission lines via rr in the transition layers and the intermediate stable layer . the spectral resolution is @xmath128 ev . the top panel corresponds to `` s - curve 2 '' , while bottom panels correspond to `` s - curve 1 '' with different parameter a. ] the spectra of the reflection lines due to resonance scattering in the transition layers and the intermediate stable layer . the spectral resolution is @xmath75 ev . the top panel corresponds to `` s - curve 2 '' , while bottom panels correspond to `` s - curve 1 '' with different parameter a. ] the combined spectrum . emission lines via recombination , red reflection lines due to resonance line scattering , and black the compton reflection hump . the spectral resolution is @xmath129 ev . the top panel corresponds to `` s - curve 2 '' , while bottom panels correspond to `` s - curve 1 '' with different parameter a. ]
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we derive a semi - analytic solution for the structure of conduction - mediated transition layers above an x - ray illuminated accretion disk , and calculate explicitly the x - ray line radiation resulting from both resonance line scattering and radiative recombination in these layers .
the vertical thermal structure of the illuminated disk is found to depend on the illuminating continuum : for a hard power law continuum , there are two stable phases connected by a single transition layer ; while for a softer continuum , there may exist three stable phases connected by two separate transition layers , with an intermediate stable layer in between .
we show that the structure can be written as a function of the electron scattering optical depth through these layers , which leads to unique predictions of the equivalent width of the resulting line radiation from both recombination cascades and resonance line scattering .
we find that resonance line scattering plays an important role , especially for the case where there is no intermediate stable layer .
| 8,082 | 237 |
systematic spectroscopic observations of far - infrared ( ir ) cooling lines in large samples of local star - forming galaxies and active galactic nuclei ( agn ) were first carried out with _ iso _ ( e.g. , @xcite , 2001 ; @xcite ; @xcite ) . these studies showed that is the most intense far - ir emission line observed in normal , star - forming galaxies ( @xcite ) and starbursts ( e.g. , @xcite ; @xcite ) , dominating the gas cooling of their neutral inter stellar medium ( ism ) . this fine - structure line arises from the @xmath11 transition ( @xmath12k ) of singley ionized carbon atoms ( ionization potential=11.26ev and critical density , @xmath13 ; @xmath14 ) which are predominantly excited by collisions with neutral hydrogen atoms ; or with free electrons and protons in regions where @xmath15 ( @xcite ) . ultraviolet ( uv ) photons with energies @xmath166ev emitted by newly formed stars are able to release the most weakly bound electrons from small dust grains via photo - electric heating ( @xcite ; @xcite ) . in particular , polycyclic aromatic hydrocarbons ( pahs ) are thought to be an important source of photo - electrons ( @xcite ) that contribute , through kinetic energy transfer , to the heating of the neutral gas which subsequently cools down via collision with c@xmath17 atoms and other elements in photo - dissociation regions ( pdrs ) ( @xcite ; @xcite ) . the emission accounts , in the most extreme cases , for as much as @xmath18% of the total ir luminosity of galaxies ( @xcite ; @xcite ) . however , the /fir ratio is observed to decrease by more than an order of magnitude in sources with high and warm dust temperatures ( ) . the underlying causes for these trends are still debated . the physical arguments most often proposed to explain the decrease in /fir are : ( 1 ) self - absorption of the c@xmath17 emission , ( 2 ) saturation of the line flux due to high density of the neutral gas , ( 3 ) progressive ionization of dust grains in high far - uv field to gas density environments , and ( 4 ) high dust - to - gas opacity caused by an increase of the average ionization parameter . although self absorption has been used to explain the faint emission arising from warm , agn - dominated systems such as mrk 231 ( @xcite ) , this interpretation has been questioned in normal star - forming galaxies due to the requirement of extraordinarly large column densities of gas in the pdrs ( @xcite , @xcite ) . furthermore , contrary to the or lines , the emission is observed to arise from the external edges of those molecular clouds exposed to the uv radiation originated from starbursts , as for example in arp 220 ( @xcite ) . therefore , self absorption is not the likely explanation of the low /fir ratios seen in most starburst galaxies , except perhaps in a few extreme cases , like ngc 4418 ( @xcite ) . the emission becomes saturated when the hydrogen density in the neutral medium , , increases to values @xmath19 , provided that the far - uv ( @xmath20ev ) radiation field is not extreme ( @xmath21 ; where is normalized to the average local interstellar radiation field ; @xcite ) . for example , for a constant @xmath22 , an increase of the gas density from @xmath23 to @xmath24 would produce a suppression of the emission of almost 2 orders of magnitude due to the rapid recombination of c@xmath17 into neutral carbon and then into co ( @xcite ) . however , pdr densities as high as @xmath25 are not very common . and _ iso _ observations of normal star - forming galaxies and some ir - bright sources confine the physical parameters of their pdrs to a range of @xmath26 and @xmath27@xmath28 ( @xcite ) . on the other hand , the emission can be also saturated when @xmath29 provided that @xmath30 . in this regime , the line is not sensitive to an increase of @xmath31 because the temperature of the gas is well above the excitation potential of the transition . it has also been suggested that in sources where / is high ( @xmath32 ) the line is a less efficient coolant of the ism because of the following reason . as physical conditions become more extreme ( higher / ) , dust particles progressively increase their positive charge ( @xcite ; @xcite ; @xcite ) . this reduces both the amount of photo - electrons released from dust grains that indirectly collisionally excite the gas , as well as the energy that they carry along after they are freed , since they are more strongly bounded . the net effect is the decreasing of the efficiency in the transformation of incident uv radiation into gas heating without an accompanied reduction of the dust emission ( @xcite ; @xcite ; @xcite ) . in a recent work , @xcite have shown that the deficits observed in several far - ir emission lines ( , , , and ) could be explained by an increase of the average ionization parameter of the ism , @xmath33__u__@xmath34 . in `` dust bounded '' star - forming regions the gas opacity is reduced within the region due to the higher @xmath33__u__@xmath34 . as a consequence , a significant fraction of the uv radiation is eventually absorbed by large dust grains before being able to reach the neutral gas in the pdrs and ionize the pah molecules ( @xcite ; @xcite ; @xcite ) , causing a deficit of photo - electrons and hence the subsequent suppression of the line with respect to the total far - ir dust emission . local luminous infrared galaxies ( lirgs : @xmath35 ) are a mixture of single galaxies , disk galaxy pairs , interacting systems and advanced mergers , exhibiting enhanced star formation rates , and a lower fraction of agn compared to higher luminous galaxies . a detailed study of the physical properties of low - redshift lirgs is critical for our understanding of the cosmic evolution of galaxies and black holes since ( 1 ) ir - luminous galaxies comprise the bulk of the cosmic infrared background and dominate star - formation activity between @xmath36 ( @xcite ; @xcite ; @xcite ; @xcite ) and ( 2 ) agn activity may preferentially occur during episodes of enhanced nuclear star formation . moreover , lirgs are now assumed to be the local analogs of the ir - bright galaxy population at @xmath37 . however , a comprehensive analysis of the most important far - ir cooling lines of the ism in a complete sample of nearby lirgs has not been possible until the advent of the _ herschel space observatory _ ( _ herschel _ hereafter ; @xcite ) and , in particular , its photodetector array camera and spectrometer ( pacs ; @xcite ) . in this work we present the first results obtained from _ herschel_/pacs spectroscopic observations of a complete sample of far - ir selected local lirgs that comprise the great observatories all - sky lirg survey ( goals ; @xcite ) . using this complete , flux - limited sample of local lirgs , we are able for the first time to perform a systematic , statistically significant study of the far - ir cooling lines of star - forming galaxies covering a wide range of physical conditions : from isolated disks where star formation is spread across kpc scales to the most extreme environments present in late stage major mergers where most of the energy output of the system comes from its central kpc region . in particular , in this paper we focus on the line and its relation with the dust emission in lirgs . we make use of a broad set of mid - ir diagnostics based on _ spitzer_/irs spectroscopy , such as high ionization emission lines , silicate dust opacities , pah equivalent widths ( ew ) , dust luminosity concentrations , and mid - ir colors , to provide the context in which the observed emission and /fir ratios are best explained . the paper is organized as follows : in [ s : sample ] we present the lirg sample and the observations . in [ s : datared ] we describe the processing and analysis of the data . the results are presented in [ s : results ] . in [ s : highz ] we put in context our findings with recent results from intermediate and high redshift surveys started to be carried out by alma and in the future by ccat . the summary of the results is given in [ s : summary ] . the great observatories all - sky lirg survey ( goals ; @xcite ) encompasses the complete sample of 202 lirgs and ulirgs contained in the _ iras _ revised bright galaxy sample ( rbgs ; sanders et al . 2003 ) which , in turn , is also a complete sample of 629 galaxies with _ iras _ @xmath38jy and galactic latitudes @xmath39 . there are 180 lirgs and 22 ulirgs in goals and their median redshift is z = 0.0215 ( or @xmath40mpc ) , with the closest galaxy being at z = 0.0030 ( 15.9mpc ; ngc 2146 ) and the farthest at z = 0.0918 ( 400mpc ; iras 07251 - 0248 ) . to date , there are many published and on - going works that have already exploited the potential of all the multi - wavelength data available for this sample including , among others , galex uv @xcite , _ hst _ optical and near - ir ( @xcite ; kim et al . 2013 ) , and _ chandra _ x - ray @xcite imaging , as well as _ spitzer_/irs mid - ir spectroscopy ( @xcite ; @xcite ; stierwalt et al . 2013a , b ; inami et al . 2013 ) , as well as a number of ground - based observatories ( vla , carma , etc . ) and soon alma . the rbgs , and therefore the goals sample , were defined based on _ iras _ observations . however , the higher angular resolution achieved by _ spitzer _ allowed us to spatially disentangle galaxies that belong to the same lirg system into separate components . from the 291 _ individual _ galaxies in goals , not all have _ herschel _ observations . in systems with two or more galactic nuclei , minor companions with mips24@xmath4 m flux density ratios smaller than 1:5 with respect to the brightest galaxy were not requested since their contribution to the total ir luminosity of the system is small . because the angular resolution of _ spitzer _ decreases with wavelength , it was not possible to obtain individual mips 24 , 70 and 160@xmath4 m measurements for all goals galaxies , and therefore to derive uniform ir luminosities for them using _ spitzer _ data only . instead , to calculate the individual , spatially - integrated of lirgs belonging to a system of two or more galaxies , we distributed the @xmath41 of the system as measured by _ iras _ ( using the prescription given in @xcite ) proportionally to the individual mips70@xmath4 m flux density of each component when available , or to their mips24@xmath4 m otherwisem fluxes could be obtained . in these cases their irac8@xmath4 m emission was used for scaling the . these lirgs are : mcg+02 - 20 - 003 and vv250a . ] . we will use this measurements of in [ s : highz ] we have obtained far - ir spectroscopic observations for 153 lirg systems of the goals sample using the integral field spectrometer ( ifs ) of the pacs instrument on board _ the data were collected as part of an ot1 program ( ot1_larmus_1 ; p.i . : l. armus ) awarded with more than 165 hours of observing time . in this work will focus mainly on the analysis and interpretation of the observations of our galaxy sample . pacs range spectroscopy of the fine - structure emission line was obtained for 163 individual sources . our observations were complemented with the inclusion of the remaining lirgs in the goals sample for which observations are publicly available in the archive ( as of october 2012 ) from various _ herschel _ projects . the main programs from which these data were gathered are : kpgt_esturm_1 ( p.i . : e. sturm ) , kpot_pvanderw_1 ( p.i . : p. van der werf ) , and ot1_dweedman_1 ( p.i . : d. weedman ) . the total number of lirg systems for which there are data is 200 ( irasf08339 + 6517 and irasf09111 - 1007 were not observed ) . however , because some lirgs are actually systems of galaxies ( see above ) , the number of observed galaxies was 241 . the ifs on pacs is able to perform simultaneous spectroscopy in the @xmath42 or @xmath43 m ( 3rd and 2nd orders , respectively ; `` blue '' camera ) and the @xmath44 m ( 1st order ; `` red '' camera ) ranges . the ifu is composed by a 5@xmath455 array of individual detectors ( spaxels ) each of one with a field of view ( fov ) of @xmath469.4 , for a total of 47@xmath4547 . the physical size of the pacs fov at the median distance of our lirg sample is @xmath4720kpc on a side . the number of spectral elements in each pixel is 16 , which are rearranged together via an image slicer over two 16@xmath4525 ge : ga detector arrays ( blue and red cameras ) . our astronomical observation requests ( aors ) were consistently constructed using the `` range '' spectroscopy template , which allows the user to define a specific wavelength range for the desired observations . our selected range was slightly larger than that provided by default for the `` line '' mode . this was necessary ( 1 ) to obtain parallel observations of the wide absorption feature using the blue camera when observing the line , and ( 2 ) to ensure that the targeted emission lines have a uniform signal - to - noise ratio across their spectral profiles even if they are to be broader than a few hundred . the high sampling density mode scan , useful to have sub - spectral resolution information of the lines ( see below ) , was employed . while we requested line maps for some lirgs of the sample ( from two to a few raster positions depending on the target ) , pointed ( one single raster ) chop - nod observations were taken for the majority of galaxies . for those galaxies with maps , only one raster position was used to obtain the line fluxes used in this work . the chopper throw varied from small to large depending on the source . spectroscopy of the lirgs included in goals but observed by other programs in was not always obtained using the `` range '' mode but some of them were observed using `` linescan '' spectroscopy . the s / n of the data varies not only from galaxy to galaxy but also depending on the emission line considered . we provide uncertainties for all quantities used across the analysis presented here that are based on the individual spectrum of each line , therefore reflecting the errors associated with @xmath48and measured directly on@xmath48 the data . as part of the _ spitzer _ goals legacy , all galaxies observed with _ herschel_/pacs have available _ spitzer_/irs low resolution ( r@xmath49 ) slit spectroscopy ( sl module : @xmath50 m , and ll module : @xmath51 m ) . the 244 irs spectra were extracted using the standard extraction aperture and point source calibration mode in spice . the projected angular sizes of the apertures on the sky are 3.7@xmath4512 at the average wavelength of 10@xmath4 m in sl and 10.6@xmath4535 at the average wavelength of 26@xmath4 m in ll . thus , the area covered by the sl aperture is approximately equivalent ( within a factor of @xmath52 ) to that of an individual spaxel of the ifs in pacs , and so is that of the ll aperture to a 3@xmath453 spaxel box . the observables derived from the irs data that we use in this work are the strength of the 9.7@xmath4 m silicate feature , , and the ew of the , which were presented in stierwalt et al . ( 2013a ) . we refer the reader to this work for further details about the reduction , extraction , calibration , and analysis of the spectra . the _ herschel _ interactive processing environment ( hipe ; v8.0 ) application was used to retrieve the raw data from the _ herschel _ science archive ( hsa ) as well as to process them . we used the script for `` linescan '' observations ( also valid for `` range '' mode ) included within hipe to reduce our spectra . we processed the data from level 0 up to level 2 using the following steps : flag and reject saturated data , perform initial calibrations , flag and reject `` glitches '' , compute the differential signal of each on - off pair of data - points for each chopper cycle , calculate the relative spectral response function , divide by the response , convert frames to pacs cubes , and correct for flat - fielding ( this extra step is included in v8.0 of hipe and later versions , and helps to improve the accuracy of the continuum level ) . next , for each camera ( red or blue ) , hipe builds the wavelength grid , for which we chose a final rebinning with an _ _ oversample__=2 , and an _ _ upsample__=3 that corresponds to a nyquist sampling . the spectral resolution achieved at the position of the line was derived directly from the data and is @xmath46235 . the final steps are : flag and reject remaining outliers , rebin all selected cubes on consistent wavelength grids and , finally , average the nod - a and nod - b rebinned cubes ( all cubes at the same raster position are averaged ) . this is the final science - grade product currently possible for single raster observations . from this point on , the analysis of the spectra was performed using in - house developed idl routines . to obtain the flux of a particular source we use an iterative procedure to find the line and measure its basic parameters . first , we fit a linear function to the continuum emission , which is evaluated at the edges of the spectrum , masking the central 60% of spectral elements ( where the line is expected to be detected ) and without using the first and final 10% , where the noise is large due to the poor sampling of the scanning . then , we fit a gaussian function to the continuum - subtracted spectrum and calculate its parameters . we define a line as not detected when the peak of the gaussian is below 2.5@xmath53 the standard deviation of the continuum , as measured in the previous step . on the other hand , if the line is found , we return to the original , total spectrum and fit again the continuum using this time a wavelength range determined by the two portions of the spectrum adjacent to the line located beyond @xmath54 from its center ( where @xmath55 is the width of the fitted gaussian ) and the following @xmath5615% of spectral elements . we then subtract this continuum from the total spectrum and fit the line again . the new parameters of the gaussian are compared with the previous ones . this process is repeated until the location , sigma and intensity of the line converge with an accuracy of 1% , or when reaching 10 iterations . due to the merger - driven nature of many lirgs , their gas kinematics are extremely complicated and , as a consequence , the emission lines of several sources present asymmetries and double peaks in their profiles . however , despite the fact that the width determined by the fit is not an accurate representation of the real shape of the line , it can be used as a first order approximation for its broadness . therefore , instead of using the parameters of the gaussian to derive the flux of the line , we decided to integrate directly over the final continuum - subtracted spectrum within the @xmath54 region around the central position of the line . the associated uncertainty is calculated as the standard deviation of the latest fitted continuum , integrated over the same wavelength range as the line . absolute photometric uncertainties due to changes in the pacs calibration products are not taken into account ( the version used in this work was pacs_cal_32_0 ) . we obtained the line fluxes for our lirgs from the spectra extracted from the spaxel at which the line + continuum emission of each galaxy peaks within the pacs fov . the _ spitzer_/irs and _ herschel _ pointings usually coincide within @xmath57 . there are a few targets for which the irs pointing is located more than half a spaxel away from that of pacs . in these cases , we decided to obtain the nuclear line flux of the galaxy by averaging the spaxels closest to the coordinates of the irs pointing . these values are used only when pacs and irs measurements are compared directly in the same plot . there is one additional lirg system , iras03582 + 6012 , for which the pacs pointing exactly felt in the middle of two galaxies separated by only 5 . this lirg is not used in the comparisons of the emission to the irs data since the two individual sources can not be disentangled . as mentioned in [ ss : irsobs ] , the angular size of a pacs spaxel is roughly similar ( within a factor of 2 ) to that of the aperture used to extract the _ spizer_/irs spectra of our galaxies . because the pacs beam is under - sampled at 160@xmath4 m ( fwhm@xmath58 compared with the @xmath59 size of the pacs spaxels ) , and most of the sources in the sample are unresolved at 24@xmath4 m in our mips images ( which have a similar angular resolution as pacs at @xmath60 m ) , an aperture correction has to be performed to the spectra extracted from the emission - peak spaxel of each galaxy to obtain their total nuclear fluxes . this was the same procedure employed to obtain the mid - ir irs spectra of our lirgs . the nominal , wavelength - dependent aperture correction function provided by hipe v8.0 works optimally when the source is exactly positioned at the center of a given spaxel . however , in some occasions the pointing of _ herschel _ is not accurate enough to achieve this and the target can be slightly misplaced @xmath61 ( up to 1/3 of a spaxel ) from the center . in these cases , the flux of the line might be underestimated . we explored whether this effect could be corrected by measuring the position of the source within the spaxel . however , some lirgs in our sample show low surface - brightness extended emission , either because of their proximity and/or merger nature , or simply because the gas and dust emission are spatially decoupled . this , combined with the spatial sub - sampling of the pacs / ifs detector and the poor s / n of some sources prevented us from obtaining an accurate measurement of the spatial position and angular width of the emission and therefore from obtaining a more refined aperture correction . thus , we performed only the nominal aperture correction provided by hipe . the _ iras _ far - ir fluxes used throughout this paper were calculated as @xmath62 [ ] , with @xmath63 in [ jy ] . the far - ir luminosities , , were defined as @xmath64 [ ] . the luminosity distances , @xmath65 , were taken from @xcite . this definition of the fir accounts for the flux emitted within the @xmath66 m wavelength range as originally defined in @xcite . the far - ir fluxes and luminosities of galaxies were then matched to the aperture with which the nuclear flux was extracted ( see above ) by scaling the integrated _ iras _ far - ir flux of the lirg system with the ratio of the continuum flux density of each individual galaxy evaluated at 63@xmath4 m in the pacs spectrum ( extracted at the same position and with the same aperture as the line ) to the total _ iras _ 60@xmath4 m flux density of the system . in table [ t : sample ] we present the flux , the /fir ratio , and the continuum flux densities at 63 and 158@xmath4 m for all the galaxies in our sample . future updates of the data in this table processed with newer versions of hipe and pacs calibration files will be available at the goals webpage : http://goals.ipac.caltech.edu . lccccccc ngc0023 & 00h09m53.4s & + 2555m26s & 65.2 & 1.385 @xmath67 0.022 & 4.13 @xmath67 0.09 & 6.17 @xmath67 0.08 & 6.85 @xmath67 0.08 + ngc0034 & 00h11m06.5s & @xmath481206m26s & 84.1 & 0.624 @xmath67 0.018 & 0.85 @xmath67 0.03 & 16.39 @xmath67 0.13 & 9.02 @xmath67 0.06 + arp256 & 00h18m50.9s & @xmath481022m36s & 117.5 & 0.967 @xmath67 0.020 & 2.96 @xmath67 0.07 & 6.70 @xmath67 0.08 & 4.42 @xmath67 0.09 + & & & & & & & [ t : sample ] the far - ir fine structure line emission in normal star - forming galaxies as well as in the extreme environments hosted by ulirgs has been extensively studied for the past two decades . a number of works based on _ iso _ data already suggested that the relative contribution of the line to the cooling of the ism in pdrs compared to that of large dust grains , as gauged by the far - ir emission , diminishes as galaxies are more ir luminous ( @xcite ; @xcite ; @xcite ) . figure [ f : ciifirvsfir ] display the classical plot of the /fir ratio as a function of the far - ir luminosity for our lirg sample . in addition , we also show for reference those galaxies observed with _ iso _ compiled by @xcite that are classified as unresolved and located at redshifts @xmath68 , similar to the distance range covered by goals . as we can see , our _ herschel _ data confirm the trend seen with _ iso _ by which galaxies with @xmath69 show a significant decrease of the /fir ratio . goals densely populates this critical part of phase - space providing a large sample of galaxies with which to explore the physical conditions behind the drop in emission among lirgs . for the 32 galaxies with measurements obtained with both telescopes , the higher angular resolution _ herschel _ observations of the nuclei of lirgs are able to recover an average of @xmath4687% of the total flux measured by _ iso_. figure [ f : ciifirvsf63f158 ] ( upper panel ) shows the /fir ratio for the goals sample as a function of the far - ir pacs @xmath7063@xmath71m/@xmath70158@xmath71 m continuum flux density ratio . we chose to use this pacs - based far - ir color in the x - axis instead of the more common _ iras _ 60/100@xmath71 m color mainly because of two main reasons : ( 1 ) this way we are able plot data from individual galaxies instead of being constrained by the spatial resolution of _ iras _ , which would force us to show only blended sources ; ( 2 ) by using the 63/158@xmath71 m ratio we are probing a larger range of dust temperatures within the starburst ( @xmath72k ) ; with the colder component probably arising from regions located far from the ionized gas - phase , and closer to the pdrs where the emission originates . for reference , we show the relation between the pacs 63/158@xmath71 m and _ iras _ 60/100@xmath71 m colors in the appendix . the ulirgs in the goals sample ( red diamonds ) have a median /fir=@xmath73 , a mean of @xmath74(@xmath75)@xmath76 , and a standard deviation of the distribution of @xmath77 . lirgs span two orders of magnitude in /fir , from @xmath2 to @xmath78 , with a mean of @xmath79 and a median of @xmath80 . the ranges from @xmath0 to 2@xmath1 . our results are consistent with _ iso _ observations of a sample of normal and moderate ir - luminous galaxies presented in @xcite and further analyzed in @xcite . the goals sample , though , populates a warmer far - ir color regime . despite the increase in dispersion at 63/158@xmath71m@xmath81 or /fir@xmath82 ( basically in the ulirg domain ) , the fact that we find the same tight trend independently of the range of ir luminosities covered by the two samples suggests that the main observable linked to the variation of the /fir ratio is the average temperature of the dust ( ) in galaxies . this interpretation agrees with the last physical scenario described in the introduction , in which an increase of the ionization parameter , @xmath33__u__@xmath34 , would cause the far - uv radiation from the youngest stars to be less efficient in heating the gas in those galaxies . at the same time , dust grains would be on average at higher temperatures due to the larger number of ionizing photons per dust particle available in the outer layers of the regions , close to the pdrs . indeed , the presence of dust within regions has been recently observed in several star - forming regions in our galaxy ( @xcite ) . both effects combined can explain the wide range of /fir ratios and far - ir colors we observe in the most warm lirgs . variations in , though , could be responsible for the dispersion in /fir seen at a given 63/158@xmath71 m ratio . to further support these findings , the bottom panel of figure [ f : ciifirvsf63f158 ] shows that the ratio of flux to the monochromatic continuum at @xmath46158@xmath4 m under the line ( the ew ) of the warmest galaxies is only a factor of @xmath83 lower than the average ew displayed by colder sources at 63/158@xmath71m@xmath84 . this implies that the decrease of the /fir ratio seen in our lirgs is primarily caused by a significant increase in warm dust emission ( peaking at @xmath85 m ) , most likely associated with the youngest stars , that is not followed by a proportional enhancement of the emission line . the best fit to the data in figure [ f : ciifirvsf63f158 ] ( upper panel ) yields the following parameters : @xmath86 [ e : defvsfircolorlin ] with a dispersion of 0.28dex . we note that the /fir ratios predicted by the fitted relation for sources with far - ir colors 63/158@xmath71m@xmath87 are probably overestimated , as it is already know that galaxies showing such cool have typical /fir@xmath88 ( e.g. , @xcite ) . the strength of the 9.7@xmath71 m silicate feature is defined as @xmath89 ; with @xmath90 and @xmath91 being the un - obscured and observed continuum flux density measured in the mid - ir irs spectra of our lirgs and evaluated at the peak of the feature , @xmath92 , normally at 9.7@xmath4 m ( see stierwalt et al . 2013a for details on how it was calculated in our sample ) . negative values indicate absorption , while positive ones indicate emission . by definition , measures the apparent optical depth towards the warm , mid - ir emitting dust . figure [ f : ciifirvssiabs ] shows that there is a clear trend ( @xmath93 ) for lirgs with stronger ( more negative ) to display smaller /fir ratios , implying that the dust responsible for the mid - ir absorption is also accountable for the far - ir emission . the formal fit ( solid line ) can be expressed as : @xmath94 [ e : defvslsd ] with a dispersion in the y - axis of 0.30dex . within the context described in the previous section , the contrast between the inner layer of dust that is being heated by the ionizing radiation to @xmath95k and that of the cold dust at @xmath9620k emitting at @xmath97 m would create both : ( 1 ) the silicate absorption seen at 9.7@xmath71 m due to the larger temperature gradient between the two dust components and ( 2 ) the increasingly higher 63/158@xmath71 m ratios seen in figure [ f : ciifirvsf63f158 ] due to the progressively larger amount of dust mass that is being heated to higher temperatures . this scenario is consistent with the physical properties of the ism found in the extreme environments of ulirgs , in which the fraction of total dust luminosity contributed by the diffuse ism decreases significantly , and the emission from dust at @xmath98k arising from optically - thick `` birth clouds '' ( with ages @xmath99myr ) accounts for @xmath10080% of their ir energy output ( @xcite ) . furthermore , our findings are also in agreement with recent results showing that the increase of the silicate optical depth in lirgs is related with the flattening of their radio spectral index ( 1.4 to 8.44ghz ) due to an increase of free - free absorption , suggesting that the dust obscuration must largely be originated in the vicinity and/or within the starburst region ( murphy et al . 2013 ) . there are a few galaxies that do not follow the correlation fitted in figure [ f : ciifirvssiabs ] , showing very large silicate strengths ( @xmath101 ) and small /fir ratios ( @xmath102 ) typical of ulirgs or , in general , warm galaxies ( see color coding or figure [ f : ciifirvsf63f158 ] ) . we would like to note that the trend found for the majority of our sample only reaches values up to around @xmath103 which , interestingly , is only slightly larger than the apparent optical depth limit that a obscuring clumpy medium can explain ( @xcite ) . larger ( more negative ) values of the silicate strength can only be achieved by a geometrically thick ( smooth ) distribution of cold dust , suggesting that the extra dust absorption seen in these few galaxies may not be related with the star - forming region from where the and far - ir emissions arise . but then , what is the origin of this excess of obscuration ? one possibility is that it is caused by foreground cold dust not associated with the starburst . this has been seen in some heavily obscured compton - thick agns , where most of the deep silicate absorption measured in these objects seems to originate from dust located in the host galaxy ( @xcite ; gonzalez - martin et al . alternatively , the presence of an extremely warm source ( different from the star - forming region(s ) that are producing the far - ir and emission ) could contribute with additional emission of hot dust ( t@xmath104k ) to the mid - ir . if at the same time this source is deeply buried ( optically thick ) and embedded in layers of progressively colder dust ( geometrically thick ) , it could produce a cumulative absorption that we would measure via the strength of the silicate feature while still contributing to the emission outside of it ( see @xcite ; @xcite ) . while both explanations are plausible , the second is favored by the fact that these extremely obscured galaxies show mips 24/70@xmath4 m ratios very similar , or even slightly higher than those found for the rest of the lirgs in the sample . if foreground cold dust was the responsible for the excess of obscuration , we would expect these galaxies to show abnormally low 24@xmath4 m luminosities with respect to the far - ir . we find that this is not the case , in agreement with recent results based on radio observations of a sub - sample of lirgs in goals ( murphy et al . furthermore , the existence of an additional hot and obscured dust component in these lirgs is also consistent with the results presented in stierwalt et al . ( 2013a ) , where it is shown that there is a trend for lirgs with moderate silicate strengths ( @xmath105 ) to show higher @xmath7030@xmath71m/@xmath7015@xmath71 m ratios as the becomes stronger ( more negative ) . that is , more obscured lirgs have increasingly larger fluxes at 30@xmath71 m , in agreement with our findings in the previous section . however , galaxies showing the most extreme silicate strengths ( @xmath106 ) do not have proportionally higher 30/15@xmath71 m ratios . on the contrary , they show ratios similar to those of warm lirgs with mild silicate strengths ( or even lower than expected given their extreme ) , supporting the idea that in these particular galaxies the dust producing this additional absorption and excess of mid - ir emission ( @xmath107 m ) represents a component of the overall nuclear starburst activity different than the star - forming regions that drive the far - ir cooling . the compactness of the starburst region of a galaxy has been proven to be related to many of its other physical properties ( @xcite ) . for example , all ulirgs in the goals sample have very small mid - ir emitting regions , with sizes ( measured fwhms ) @xmath108kpc ( @xcite ) . lirgs , on the other hand , span a large range in sizes as well as in how much of their mid - ir emission is extended . the later property is parametrized in @xcite by the fraction of extended emission , fee@xmath109 , which measures the fraction of light emitted by a galaxy that is contained outside of its unresolved component at a given wavelength @xmath110 . the complementary quantity @xmath111fee@xmath109 measures how compact the source is , which in turn is proportional to its luminosity surface density , @xmath112 . we note that in this paper we use the word compactness as an equivalent to light concentration , i.e. , as a measurement of the amount of energy per unit area produced by a source , and not as an absolute measurement of its size . it has been shown that the compactness of the mid - ir continuum emission of lirgs ( evaluated at @xmath113 m ) is related to their merger stage , mid - ir agn - fraction and most importantly , to their far - ir color ( @xcite ) . lirgs with higher _ iras _ @xmath7060@xmath71m/@xmath70100@xmath71 m ratios are increasingly compact . in other words , for a given , the dust in sources with far - ir colors peaking at shorter wavelengths is not only hotter but also confined towards a smaller volume in the center of galaxies . in [ ss : warm ] we found that the /fir ratio is related to the average @xmath114 of our galaxies . thus , we should expect to see a correlation between the deficit and the luminosity surface density and compactness of lirgs in the mid - ir . this is shown in figure [ f : ciifirvscompact ] , where a clear trend is found for galaxies with higher luminosity surface densities at 15@xmath4 m , @xmath115 ( top panel ) , or small fee@xmath116 ( bottom panel ) , i.e , more compact , to show lower /fir ratios , irrespective of the origin of the nuclear power source . excluding those lirgs for which only upper limits on their mid - ir size or are available , we perform a linear fit to the data and obtain the following parameters for the correlation between /fir and @xmath115 : @xmath117 [ e : defvslsd ] with a dispersion in the y - axis of 0.23dex . it is known that the contribution of an agn to the ir emission in lirgs increases with ( @xcite ; @xcite ; @xcite ; @xcite ) . this is most noticeable at mid - ir wavelengths ( @xcite ; @xcite ; @xcite ) but a non - negligible fraction of the far - ir emission of ulirgs can also be powered by an agn . the ew of mid - ir pah features is a simple diagnostic that has been widely used for the detection of agn activity in galaxies at low and high redshifts ( @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; stierwalt et al . broadly speaking , the pah ew decreases as a component of hot dust at t@xmath118k , normally ascribed to an agn , starts to increasingly dominate the mid - ir continuum emission of the galaxy . in addition , the hard radiation field of an agn could be able to destroy a significant fraction of the smallest pah molecules ( @xcite ; @xcite ) . in particular , a galaxy is regarded as mid - ir agn - dominated when its ew @xmath119 , and it is classified as a pure starburst when ew @xmath120 ( although these limits are not strict ) . sources with intermediate values are considered composite galaxies , in which both starburst and agn may contribute significantly to the mid - ir emission . figure [ f : ciifirvslsdpah ] shows the /fir ratio as a function of the ir luminosity surface density , @xmath9 , of the lirgs in goals , color - coded as a function of their ew . as we can see , when only pure star - forming galaxies are considered ( ews @xmath121 m ) , the /fir ratio drops by an order of magnitude , from @xmath7 to @xmath122 . this indicates that the decrease in /fir among the majority of lirgs is not caused by a rise of agn activity but instead is a fundamental property of the starburst itself . it is only in the most extreme cases , when /fir@xmath102 , that the agn could play a significant role . in fact , powerful agn do not always reduce the /fir ratio , as shown also in @xcite . @xcite also find that the agn - powered sources in their high - redshift galaxy sample display small /fir ratios . however , they speculate that except for two blazars , the deficit seen in these sources could be caused compact , nuclear starbursts ( with sizes less than @xmath123kpc ) perhaps triggered by the agn . we will return to this discussion in [ ss : definagn ] . the result obtained above also implies that the line alone is not a good tracer of the sfr in most local lirgs since it does not account for the increase of warm dust emission ( figure [ f : ciifirvsf63f158 ] ) seen in the most compact galaxies that is usually associated with the most recent starburst . in figure [ f : ciifirvslsdpah ] we fit the data to provide a relation between the /fir ratio and the @xmath9 for pure star - forming lirgs . the analytic expression of the fit is : @xmath124 with a dispersion in the y - axis of 0.16dex . the slope and intercept of this trend are indistinguishable ( within the uncertainties ) from those obtained in eq . ( 3 ) , which was derived by fitting all data - points including low ew sources with measured mid - ir sizes . this further supports the idea that the influence of agn activity is negligible among ir - selected galaxies with @xmath125/fir@xmath126 and that the increase in ir luminosity of these sources is due to a boost of their warm dust emission . figure [ f : ciifirvspahew ] shows the /fir ratio as a function of the ew for the lirgs in goals . starburst sources with large pah ews have a mean /fir ratio of @xmath127 with a standard deviation of @xmath80 . as the ew becomes smaller the dispersion increases and we find galaxies with both very small ratios as well as sources with normal values ( or slightly lower than those ) typical of purely star - forming sources ( see also @xcite ) . if agns contribute significantly to the far - ir emission of lirgs and/or suppress pah emission via photo - evaporation of its carriers , we would expect agn - dominated sources to show significantly low /fir ratios and small pah ews . we have used the _ spitzer_/irs spectra of our galaxies to identify which of them are hosting an agn based on several mid - ir diagnostics : ( 1 ) /@xmath128 ; ( 2 ) /@xmath129 ; ( 3 ) @xmath7030@xmath71m/@xmath7015@xmath71 m @xmath130 ( i.e. , @xmath131 for @xmath132 ) ; as well as ( 4 ) the ew itself ( see constraints above ) . all these thresholds are rather restrictive and ensure that the contribution of an agn to the mid - ir luminosity of a galaxy is _ at least _ @xmath133% . in figure [ f : ciifirvspahew ] we mark those sources that have at least two positive indicators of agn activity as red stars . we note that this excludes sources with low ews and no other agn signatures , and is a more conservative cut than applied in @xcite to identify potential agns . strikingly , these sources are not preferentially found at the bottom - left of the parameter space but instead as many as 2/3 show /fir@xmath134 , typical of star - forming sources with large ews . this suggests that the impact of the agn on the far - ir luminosity of these mid - ir dominated agn lirgs is very limited , unless the it contributes to both the and far - ir in the same relative amount as the starburst does . while @xmath135% of our sample appears to have significant agn contribution to the mid - ir emission ( @xcite ) , the fraction in which the agn dominates the bolometric luminosity of the galaxy is much smaller . to investigate this , we use two of the indicators described above , the line and the ew , and the formulation given in @xcite to calculate the bolometric agn fraction of those galaxies with at least two mid - ir agn detections . we find that only four ( 20% ) of these galaxies have contributions @xmath136% in both indicators ( black crosses in figure [ f : ciifirvspahew ] ) . two galaxies have /fir@xmath102 ( 33% ) and two ( 14% ) a larger ratio . to quantitatively asses the relationship between agn activity and the /fir ratio among galaxies hosting an agn , it is important to estimate first the agn contribution to the far - ir flux . if we assume that the ratio of to emission of the star - forming lirgs in goals is constant , as is the case for most normal , lower luminosity galaxies ( @xcite ; @xcite ; @xcite ) , we can calculate the expected evolution of the /fir ratio as a function of the ew if we also assume that pure starbursts have a typical ew@xmath137=0.65@xmath4 m and /fir=@xmath127 ( as shown above ) , and that the average /@xmath138 ratios for pure star - forming galaxies and agns are @xmath4615 and @xmath461 , respectively . the value for star - forming galaxies varies from @xmath139 in our sample while the value for agns has been estimated from the intrinsic agn sed of @xcite . the predicted trend is shown in figure [ f : ciifirvspahew ] as a solid black line , which agrees very well with the location of the agns identified with at least two mid - ir indicators ( red stars ) . under these assumptions , the ew has to be reduced by a factor of @xmath4615 with respect to the ew@xmath137 , i.e. , down to @xmath140 m , before the agn can contribute 50% to the . in fact , 2/3 of galaxies with ews lower than this threshold have been identified as harboring an agn by two or more mid - ir diagnostics . therefore , only when the agn contribution to the far - ir flux is significant do we see a noticeable decrease of the /fir ratio ( always @xmath102 ) . we note however that the contrary might not be necessarily true since there are galaxies with low /fir ratios but with ews @xmath141 m . we emphasize that this prediction does not account for possible destruction of pah molecules due to the agn . however , if the reduction of the pah ew was entirely due to this effect , we would expect a linear correlation between the /fir ratio and the ew , which is described by the dashed line in figure [ f : ciifirvspahew ] . as we can see , the mid - ir agn dominated galaxies do not follow the predicted trend , suggesting that pah destruction is not important in lirgs with /fir@xmath142 at least at the scales probed by _ herschel _ and _ spitzer _ , in agreement with the results obtained in @xcite . nearly half of galaxies with /fir@xmath102 and ew @xmath143 m have no other direct mid - ir diagnostic that reveals the presence of an agn . interestingly , all of them are among the outliers found in figure [ f : ciifirvssiabs ] , showing an excess in the with respect to their observed /fir . we argued in [ ss : siabs ] that these galaxies are probably hosting an extremely warm and compact source , optically and geometrically thick , not associated with the star - forming regions producing the bulk of the and far - ir . the energy source of this component is unknown , though , since both an agn or an ultra compact region could generate such mid - ir signatures . however , the monochromatic @xmath14463/15@xmath71 m ratios displayed by these objects are @xmath145 ( see color - coding in figure [ f : ciifirvspahew ] ) , significantly higher than the typically flat spectrum seen in qsos and pure agn sources ( @xcite ; @xcite ) in which the hot dust emission dominates the mid - ir wavelengths up to @xmath146 m , with @xmath144=constant , and fading beyond . this adds evidence to the result obtained above that this type of deeply embedded objects only dominate the luminosity of the galaxy in the mid - ir . furthermore , we would like to emphasize that the fact that the source of this warm , compact emission does not produce the detected pah or emissions rules out models where pah obscuration is invoked to explain the low pah ews found in these sources , since their observed flux compared to that of the far - ir is also very low , implying that it is not extinction but rather the fact that the pdr emission of the warmest dust component in these lirgs is actually extremely limited . at intermediate redshifts , @xmath147 , it has been found that ir - luminous galaxies span a wide range in /fir ratios : @xmath148 ( @xcite ) . a surprising discovery came from the most luminous systems , and the fact that many of them show values of this ratio similar to those found in local , lower luminosity galaxies ( e.g. , @xcite ; @xcite ; @xcite ; @xcite ) . these results , added to a number of recent findings obtained from the analysis of mid - ir dust features of star - forming galaxies using _ spitzer_/irs spectroscopy ( e.g. , @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; stierwalt et al . 2013a ) are pointing towards an emerging picture in which the local counterparts of the dominant population of ir - bright galaxies at intermediate and high redshifts ( @xmath37 ) are not extremely dusty systems with similar ir luminosities ( i.e. , local ulirgs ) but rather galaxies with more modest sfrs , or @xmath149 ( starbursts and lirgs ) . therefore , since goals is a complete , flux - limited sample of 60@xmath4 m rest - frame selected lirgs systems in the local universe covering an ir luminosity range from @xmath150 to @xmath151 , the empirical relations we find can be used to estimate what might be seen in similar surveys of intermediate and high redshift ir - luminous galaxies . recently , @xcite has found that the majority of star - forming galaxies , from the nearby universe and up to @xmath152 , follow a `` main sequence '' ( ms ) that is depicted by a specific sfr ( ssfr@xmath153 ) that increases with redshift . galaxies in the ms are characterized by producing stars in a quiescent mode , with very low efficiencies ( /@xmath15410/ ) and extended over spatial scales of several kpc ( @xcite ; @xcite ) . on the other hand , galaxies with high ssfrs are currently experiencing a strong and very efficient , but short - lived ( less than few hundred myr ) starburst event , some of them probably consequence of a major merge interaction . the ssfr of local galaxies is anti - correlated with their compactness as measured in the mid - ir , as well as from radio wavelengths ( see also murphy et al . therefore , the trends of /fir with fee@xmath116 , @xmath115 , and @xmath9 found in [ ss : compact ] and [ ss : definsb ] tell us not only about the compactness of the starburst region but also about the main mode of star formation itself . sources with high /fir ratios should belong to the ms while galaxies with low ratios will likely be compact , starbursting sources . because @xmath155% of the uv and optical light of star - forming lirgs and ulirgs is reprocessed by dust into the ir wavelengths ( @xcite ) , the / ratio is equivalent to their ssfr since the sfr is directly proportional to the ir luminosity , with sfr@xmath156/=1.72@xmath157yr@xmath158@xmath158 ( as derived in @xcite ) . using eq . ( 13 ) from @xcite we calculate that the ssfr of ms galaxies in the local universe is ssfr@xmath159gyr@xmath158 . however , this equation does not take into account the dependence of the ssfr@xmath153 on the stellar mass of galaxies . thus , we combine it with the sfr vs. correlation obtained from the sdss sample by @xcite , using a power - law index of 0.8 for the dependence of the sfr on ( their eq . ( 5 ) ) and after normalizing it by the ssfr@xmath153 at @xmath160 . this joint equation assumes that the exponential dependence of the sfr@xmath153 as a function of does not vary with @xmath161 , which is roughly the case at least up to @xmath162 ( see , e.g. , @xcite ) . we have used this normalization factor to derive the excess of ssfr in our galaxies ( also called `` starburstiness '' : ssfr / ssfr@xmath153 ) . if we define starbursting galaxies as those having a ssfr@xmath163ssfr@xmath153 , then @xmath4668% of the galaxies in goals would be classified as such . figure [ f : ciifirvsssfr ] shows the /fir ratio as a function of the integrated ssfr normalized to the representative ssfr@xmath153 of galaxies at @xmath164 for our lirg sample . the stellar mass values are taken from @xcite and were derived directly either from the 2mass k - band or the 3.6@xmath4 m irac luminosities using the @xmath165 conversions from @xcite . the solid line represents a fit to pure star - forming galaxies with ews @xmath121 m . the pearson s test yields @xmath166 ( @xmath167 ) , while the kendall s test provides @xmath168 . the correlation coefficient derived from the robust fit is @xmath169 . we note that the sfr and plotted here represent integrated measurements of our galaxies while the and fir values were obtained from a single pacs spaxel , probing an area of @xmath469.4@xmath459.4 , which at the median distance of our lirgs is equal to a projected physical size of @xmath170kpc on a side ( similar to a 0.5 beam at @xmath152 ) ; an aperture big enough to contain most of the galaxies emission . in fact , if a @xmath171 spaxel aperture is used instead , the fit yield virtually the same parameters as those reported below ( within the uncertainties ) . the trend between /fir and normalized ssfr is not surprising since the ir luminosity appears in both quantities . nevertheless , the correlation is indeed practical in terms of its predictive power . for example , we can see that there are no star - forming ms galaxies with /fir@xmath102 . their median ratio is @xmath172 . on the other hand , starbursting sources show a larger range of ratios , from @xmath7 to @xmath3 , with a median of @xmath173 . the correlation between the /fir and the ssfr / ssfr@xmath153 is given by the following equation : @xmath174 with a dispersion in the y - axis of 0.26dex . if the separation between ms and starbursting galaxies ( high / low /fir ) at any given redshift is related to an increase of the star formation efficiency ( sfe ; higher / ; see , e.g. , @xcite ; @xcite ) then this equation can be applied to predict the luminosity of star - forming galaxies at any @xmath161 for which a measurement of the ( far-)ir luminosity and stellar mass are available as long as their ssfr is normalized to the ssfr@xmath153 at that @xmath161 to account for the increase in gas mass fraction ( / ) and therefore ssfr of ms galaxies at higher redshifts ( @xcite ; @xcite ) . conversely , in future large ir surveys like those projected with the cornell - caltech atacama telescope ( ccat ) , this relation could be used for estimating the ssfr or stellar mass of detected galaxies when their and far - ir luminosities are known . as an example , in figure [ f : ciifirvsssfr ] we show two intermediate redshift @xmath175 galaxies from @xcite ( smmj123633 and 3c368 ) and three high redshift @xmath176 sub - millimeter galaxies ( smgs ) ( less 61.1 and less 65.1 : @xcite ; less 033229.3 : @xcite ; @xcite ) marked as open black squares and diamonds , respectively . to calculate the sfr of the smgs we assumed that their @xmath177 . since eq . ( 13 ) from @xcite is calibrated only up to @xmath178 and the dependence of the ssfr@xmath153 on is also uncertain beyond this redshift , we normalized the ssfr of the smgs by the ssfr@xmath153 at @xmath179 . as we can see , three galaxies follow the correlation suggesting that they are starbursting sources with /fir ratios consistent with their normalized ssfrs . on the other hand , the two remaining high-@xmath161 smgs show significantly lower /fir ratios than the average of lirgs for the same normalized ssfr . in particular , one of them display a /fir more than an order of magnitude lower than the value predicted by the fit to our local galaxy sample . interestingly , both smgs lie in the parameter space where most of the mid - ir identified agn are located ( red stars ) . this may suggest that these two galaxies could harbor agn or unusually week emission for their normalized ssfr . ( 3 ) and ( 4 ) can also be used to predict the mid- and total ir luminosity surface density of star - forming galaxies at high redshifts for which the and far - ir fluxes are known , such as those that may be found in future spectroscopic surveys with the x - spec instrument on ccat . with instantaneous coverage over all the atmospheric windows between 190 and 440 ghz , x - spec will access the line at redshifts from @xmath180 to @xmath181 . moreover , if a measurement of the rest - frame mid - ir luminosity of galaxies is also available , this correlation can be further used to estimate the physical size of the star - forming region in the mid - ir . this is particularly useful for z@xmath182 sources detected with _ herschel _ in deep fields . in this cases , the predicted mid - ir size of galaxies could be compared with direct measurements of the size of their far - ir emitting region as observed with alma on physical scales similar to those we are probing in our goals lirgs with pacs . finally , because goals is a complete flux - limited sample of local lirgs , we are able to predict the contamination of sources hosting agns in future large - scale surveys with both and far - ir measurements . in table [ t : ciilfir ] we provide the percentages of galaxies with mid - ir detected agns classified in different /fir and @xmath7063@xmath71m/@xmath70158@xmath71 m bins . the values provided in the table were computed using two conditions for the detection of the agn that serve as upper and lower limits for the estimated fractions ( columns ( 2 ) and ( 3 ) ) . the first was based on the ew only and the second required of an additional mid - ir diagnostic to classify the galaxy as harboring an agn ( see above ) . for example , we predict an agn contamination ( based on the ew only ) of up to @xmath4670% for /fir@xmath183 , which implies that at least @xmath461/3 of ir - selected sources with extremely low /fir ratios will be powered by starbursts . moreover , at the levels of /fir@xmath184 or 63/158@xmath4m@xmath185 , the agn detection fraction is expected to be @xmath186% . cccc @xmath187 & 4% & 2% & 0.52 + @xmath188 & 15% & 10% & 0.93 + @xmath189 & 18% & 9% & 1.22 + @xmath190 & 72% & 22% & 1.92 + 63/158@xmath71 m & agn - frac & agn - frac & /fir + range & & multi & median + ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) + @xmath191 & 8% & 4% & 7.0@xmath192 + @xmath193 & 9% & 8% & 3.2@xmath192 + @xmath194 & 28% & 14% & 1.5@xmath192 + @xmath195 & 56% & 13% & 5.8@xmath196 [ t : ciilfir ] we obtained new _ herschel_/pacs spectroscopy for 200 lirg systems in goals , a 60@xmath4 m flux - limited sample of all lirgs detected in the nearby universe . a total of 241 individual galaxies where observed in the line . we combined this information together with _ spitzer_/irs spectroscopic data to provide the context in which the observed luminosities and /fir ratios are best explained . we have found the following results : * the lirgs in goals span two orders of magnitude in /fir , from @xmath2 to @xmath3 , with a median of @xmath197 . ulirgs have a median of @xmath73 . the range from @xmath0 to 2@xmath198 for the whole sample . the /fir ratio is correlated with the far - ir @xmath7063@xmath71m/@xmath70158@xmath71 m continuum color . we find that all galaxies follow the same trend independently of their , suggesting that the main observable linked to the variation of the /fir ratio is the average dust temperature of galaxies , which is driven by an increase of the ionization parameter , @xmath33__u__@xmath34 . * there is a clear trend for lirgs with deeper 9.7@xmath4 m silicate strengths ( ) , higher mid - ir luminosity surface densities ( @xmath5 ) , smaller fractions of extended emission ( fee@xmath116 ) and higher ssfrs to display lower /fir ratios . these correlations imply the the dust responsible for the mid - ir absorption must be directly linked to the process driving the observed deficit . lirgs with lower /fir ratios are more warm and compact ( higher mid- and ir luminosity surface densities , @xmath199 ) , regardless of what is the origin of the nuclear power source . however , this trend is clearly seen also among pure star - forming lirgs only , implying that it is the compactness of the starburst , and not agn activity as identified in the mid - ir , that is the main driver for the declining of the to far - ir dust emission . this implies that the luminosity is _ not _ a good indicator of the sfr in lirgs with high or large @xmath9 since it does not scale linearly with the warm dust emission most likely associated to the youngest stars . there are a small number of lirgs that have a larger /fir ratio than suggested by their deep and warm dust emission . the origin of the energy source of these lirgs is unknown , although they likely contain a deeply buried , compact source with little or no pdr emission . * pure star - forming lirgs ( ew @xmath200 ) have a mean /fir=@xmath127 with a standard deviation of @xmath80 , while galaxies with low ews span the entire range in /fir . a significant fraction ( 70% ) of the lirgs in which an agn is detected in the mid - ir have /fir ratios @xmath10 , similar to those of starburst galaxies suggesting that most agns do not contribute substantially to the far - ir emission . thus , only in the most extreme cases when /fir@xmath102 might the agn contribution be significant . * the completeness of the goals lirg sample has allowed us to provide meaningful predictions about the , @xmath5 , and agn contamination of large samples of ir - luminous high - redshift galaxies soon to be observed by alma or ccat . in a far - ir selected survey of high-@xmath161 lirgs we expect to find up to @xmath4670% of agn contamination for /fir@xmath183 , which implies that at least 1/3 of ir - selected sources with extremely low /fir will be powered by starbursts . moreover , above this ratio the agn fraction is expected to be @xmath186% . for deep fields with and far - ir emission measurements we can predict the ir luminosity surface density of galaxies , which could be compared with direct measurements of the size of their far - ir emitting region as observed with alma on physical scales similar to those we are probing in our goals lirgs with pacs . we thank the referee for his / her useful comments and suggestions which significantly improved the quality of this paper . we also thank david elbaz , alexander karim , j. d. smith , moshe elitzur , and j. gracia - carpio for very fruitful discussions . l. a. acknowledges the hospitality of the aspen center for physics , which is supported by the national science foundation grant no . v. c. would like to acknowledge partial support from the eu fp7 grant pirses - ga-2012 - 31578 . this work is based on observations made with the _ herschel space observatory _ , an european space agency cornerstone mission with science instruments provided by european - led principal investigator consortia and significant participation from nasa . the _ spitzer space telescope _ is operated by the jet propulsion laboratory , california institute of technology , under nasa contract 1407 . this research has made use of the nasa / ipac extragalactic database ( ned ) , which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration , and of nasa s astrophysics data system ( ads ) abstract service . in [ ss : warm ] , figure [ f : ciifirvsf63f158 ] , we show the /fir ratio as a function of the pacs - based @xmath7063@xmath71m/@xmath70158@xmath71 m ratio for the galaxies in the goals sample observed with _ herschel _ , and explain the reasons for adopting and plotting this far - ir color instead the more commonly used _ iras_-based @xmath7060@xmath71m/@xmath70100@xmath71 m color . here we show a comparison between both , to provide the reader with a tool for interpreting our results in terms of _ iras _ colors , if necessary . figure [ f : pacsvsiras ] shows that the far - ir ratios correlate well ( @xmath201 , @xmath202 ) , as expected , with a slope of 1.80@xmath2030.08 , an intercept of 0.30@xmath2030.02 , and a dispersion in the y - axis of 0.10dex . we note that part of the scatter is caused by lirg systems comprised by more than one galaxy , for which we have used their individual 63/158@xmath71 m ratios but the same 60/100@xmath71 m ratio , as they correspond to a single , unresolved _ iras _ source . nevertheless , independently of which of these far - ir colors we utilize , the same overall trend seen in figure [ f : ciifirvsf63f158 ] emerges , namely warmer galaxies display smaller /fir ratios . as mentioned in [ ss : warm ] , when we use the _ iras _ far - ir colors of integrated systems , our lirg sample follows the same trend found for normal and moderate ir - luminous galaxies observed by _ iso_. however , when using the 63/158@xmath71 m ratio , the anti - correlation is tighter than that obtained when the emission from entire systems is employed , likely due to the fact that we are able to disentangle the true far - ir colors of individual galaxies . moreover , the observed 63/158@xmath71 m continuum ratios span a much larger dynamical range ( a factor of @xmath4610 , in contrast with the factor of @xmath204 covered by the integrated 60/100@xmath71 m ratios ) , which translates in a more accurate sampling of the average dust temperature of lirgs .
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we present the first results of a survey of the emission line in 241 luminous infrared galaxies ( lirgs ) comprising the great observatories all - sky survey ( goals ) sample , obtained with the pacs instrument on board the _ herschel space observatory_. the luminosities , , of the lirgs in goals range from @xmath0 to 2@xmath1 .
we find that lirgs show a tight correlation of /fir with far - ir flux density ratios , with a strong negative trend spanning from @xmath2 to @xmath3 , as the average temperature of dust increases .
we find correlations between the /fir ratio and the strength of the 9.7@xmath4 m silicate absorption feature as well as with the luminosity surface density of the mid - ir emitting region ( @xmath5 ) , suggesting that warmer , more compact starbursts have substantially smaller /fir ratios .
pure star - forming lirgs have a mean /fir@xmath6 , while galaxies with low equivalent widths ( ews ) , indicative of the presence of active galactic nuclei ( agn ) , span the full range in /fir .
however , we show that even when only pure star - forming galaxies are considered , the /fir ratio still drops by an order of magnitude , from @xmath7 to @xmath8 , with @xmath5 and @xmath9 , implying that the luminosity is not a good indicator of the star formation rate ( sfr ) for most lirgs , for it does not scale linearly with the warm dust emission most likely associated to the youngest stars . moreover , even in lirgs
in which we detect an agn in the mid - ir , the majority ( 2/3 ) of galaxies show /fir@xmath10 typical of high ew sources , suggesting that most agns do not contribute significantly to the far - ir emission .
we provide an empirical relation between the /fir and the specific sfr ( ssfr ) for star - forming lirgs .
finally , we present predictions for the starburst size based on the observed and far - ir luminosities which should be useful for comparing with results from future surveys of high - redshift galaxies with alma and ccat .
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polymers with contour length @xmath1 much larger than the persistence length @xmath2 , which is the correlation length for the tangent - tangent correlation function along the polymer and is a quantitative measure of the polymer stiffness , are flexible and are described by using the tools of quantum mechanics and quantum field theory @xcite-@xcite . if the chain length decreases , the chain stiffness becomes an important factor . many polymer molecules have internal stiffness and can not be modeled by the model of flexible polymers developed by edwards @xcite . the standard coarse - graining model of a wormlike polymer was proposed by kratky and porod @xcite . the essential ingredients of this model are the penalty for the bending energy and the local inextensibility . the latter makes the treatment of the model much more difficult . there have been a substantial number of studies of the kratky - porod model in the last half century @xcite-@xcite ( and citations therein ) . in recent years there has been increasing interest in the theoretical description of semiflexible polymers @xcite-@xcite . the reason for this interest is due to potential applications in biology allemand05 ( and citations therein ) and in research on semicrystalline polymers @xcite . it was found in the recent numerical work by lattanzi et al . lattanzi04 , and studied analytically in @xcite within the effective medium approach , that the transverse distribution function of a polymer embedded in two - dimensional space possesses a bimodal shape for short polymers , which is considered to be a manifestation of the semiflexibility . the bimodal shape for the related distribution function of the 2d polymer was also found in recent exact calculations by spakowitz and wang @xcite . in this paper we study the transverse distribution function @xmath3 of the three dimensional wormlike chain with a fixed orientation @xmath4 of one polymer end using the exact representation of the distribution function in terms of the matrix element of the green s function of the quantum rigid rotator in a homogeneous external field @xcite . the exact solution of the green s function made it possible to compute the quantities such as the structure factor , the end - to - end distribution function , etc . practically exact in the definite range of parameters @xcite , @xcite . our practically exact calculations of the transverse distribution function of the 3d wormlike chain demonstrate that it possesses the bimodal shape in the intermediate range of the chain lengths ( @xmath0 ) . in addition , we present analytical results for short and long wormlike chain based on the exact formula ( [ gtkp ] ) , which are in complete agreement with the previous results obtained in different ways @xcite ( wkb method for short polymer ) , @xcite ( perturbation theory for large chain ) . the paper is organized as follows . section [ sect1 ] introduces to the formalism and to analytical considerations for short and large polymers . section [ numer ] contains results of the numerical computation of the distribution function for polymers with different number of monomers . the fourier - laplace transform of the distribution function of the free end of the wormlike chain with a fixed orientation @xmath5 @xmath6 of the second end is expressed , according to @xcite , in a compact form through the matrix elements of the green s function of the quantum rigid rotator in a homogeneous external field @xmath7 as @xmath8where @xmath9 , and @xmath7 is defined by @xmath10with @xmath11 and @xmath12 being the infinite order square matrices given by @xmath13and @xmath14 . the matrix @xmath11 is related to the energy eigenvalues of the free rigid rotator , while @xmath12 gives the matrix elements of the homogeneous external field . since @xmath7 is the infinite order matrix , a truncation is necessary in the performing calculations . the truncation of the infinite order matrix of the green s function by the @xmath15-order matrix contains all moments of the end - to - end chain distance , and describes the first @xmath16 moments exactly . the transverse distribution function we consider , @xmath3 , is obtained from @xmath17 , which is determined by eqs . ( [ gtkp])-([d ] ) , integrating it over the @xmath18-coordinate , and imposing the condition that the free end of the chain stays in the @xmath19 plane . as a result we obtain @xmath20 is the bessel function of the first kind abramowitzstegun . taking the @xmath18-axis to be in the direction of @xmath21 yields @xmath22 , so that the arguments of the legendre polynomials in eq . ( [ gtkp ] ) become zero , and consequently only even @xmath23 will contribute to the distribution function ( [ gyn ] ) . we now will consider the expansion of ( [ gtkp ] ) around the rod limit @xmath24 , which corresponds to the expansion of @xmath25 in inverse powers of @xmath26 . to derive such an expansion , we write @xmath11 in the equivalent form as@xmath27with @xmath28 and @xmath29 . further we introduce the notation @xmath30 with @xmath31 and @xmath32 defined by@xmath33the iteration of @xmath11 and @xmath34 results in the desired expansion of @xmath32 and consequently of @xmath35 in inverse powers of @xmath26 , which corresponds to an expansion of @xmath36 in powers of @xmath37 . the leading order term in the short chain expansion is obtained by replacing @xmath11 by @xmath38 in eq . ( [ gtkp ] ) as @xmath39 _ { 0l}\sqrt{2l+1}p_{l}(\mathbf{t}_{0}\mathbf{n } ) . \label{gtkp0}\]]the latter coincides with the expansion of the plane wave landau - lifshitz3@xmath40where @xmath41 is the angle between the tangent @xmath4 and the wave vector @xmath42 . the connection of @xmath43 with the plane wave expansion is due to the fact that the kratky - porod chain becomes a stiff rod in the limit of small @xmath37 . we have checked the equivalency between the plane wave expansion ( [ plw ] ) and the distribution function ( [ gtkp0 ] ) term by term expanding ( [ gtkp0 ] ) in series in powers of the wave vector @xmath44 . the arc length @xmath37 is equivalent for stiff rod in units under consideration to the chain end - to - end distance @xmath45 . in the @xmath45 space the plane wave ( [ plw ] ) corresponds to the stiff rod distribution function @xmath46 the iteration of @xmath11 in ( [ dit ] ) and @xmath34 in ( [ y ] ) generates an expansion of @xmath35 in inverse powers of @xmath26 . the corrections to the plane wave to order @xmath47 are obtained as@xmath48 _ { 0l}\sqrt{2l+1}% p_{l}(\mathbf{t}_{0}\mathbf{n})+ .... \end{aligned}\ ] ] the above procedure yields for @xmath49 the short chain expansion of the distribution function of the free kratky - porod chain , which was studied recently in @xcite . unfortunately , we did not succeed yet in analytical evaluation of @xmath50 . such computation would be an interesting alternative to the treatment of the short limit of the wormlike chain by yamakawa and fujii @xcite within the wkb method . nevertheless , following the consideration in stepanow05 we succeeded in computing the anisotropic moments @xmath51 for small @xmath37@xmath52the first - order correction coincide with that obtained in @xcite using the wkb method , while the second - order correction is to our knowledge new . the higher - order terms in ( [ rt_0 ] ) can be established in a straightforward way using the present method . note that the computation of @xmath53 does not require the knowledge of the full distribution function @xmath54 . in studying the end - to - end distribution function for large @xmath37 we utilize the following procedure . we expand first the expression @xmath55 in powers of @xmath56 . the structure of this expansion for @xmath49 is presented in table [ table1 ] . the subseries in powers of @xmath56 in the @xmath57th column are denoted by @xmath58 . thus we have @xmath59the series @xmath58 with small values @xmath23 and @xmath57 possess a simple structure and can be summed up . for example @xmath60 and @xmath61 are given by @xmath62@xmath63while @xmath61 corresponds to the distribution function of the gaussian chain , @xmath64 give the @xmath57th correction to the gaussian distribution . the inspection of the series for @xmath65 and @xmath66 shows that they are expressed by @xmath61 and @xmath67 as@xmath68however , it seems that there is no general recursion relation for @xmath69 . the results of computations of @xmath70 for @xmath71 by taking into account @xmath72 ( @xmath49 ) and @xmath73 ( @xmath74 ) are summarized in table [ table2 ] . . [ cols="^,<",options="header " , ] the inverse laplace - fourier transform of @xmath70 given in the table yields the expansion of the end - to - end distribution function @xmath75 to order @xmath76 as @xmath77 , \label{grt}\end{aligned}\]]where @xmath78 , @xmath79 , and @xmath80 is the angle between @xmath81 and @xmath82 . the latter is in accordance with the result by gobush et . @xcite derived in a different way . the expansion of @xmath83 for large @xmath37 can be extended in a straightforward way to include higher - order corrections . the computation of the distribution function of the polymer with the fixed orientation of one end is performed by truncating the infinite order matrices in ( [ gtkp ] ) with the finite ones , and by taking into account the finite number of terms in the summation over @xmath23 . the inverse laplace transform of ( [ gtkp ] ) is carried out with maple . the results of the calculation of the distribution function @xmath3 , for various chain lengths , computed with @xmath84 matrices . the insets show the distribution function at the onset of bimodality , and in the region of its disappearance . ] using the truncations with @xmath84 order matrices and restricting the summation over the quantum number @xmath23 at @xmath85 , are given in fig . fig - doublepeak . the results show that the distribution function possesses the bimodal shape at intermediate chain lengths within the interval @xmath86 . we also find that the distribution function becomes gaussian for very short and very long chains . at the onset there is a tiny maximum at @xmath87 , which we interpret as remnant of the gaussian behavior of short chains . the maximum at @xmath87 for 3d chain is a rather small effect , which is difficult to be explained in a qualitative way . we now will discuss qualitatively the origin of the bimodal behavior of the projected distribution function of the free end of the wormlike chain . the very short wormlike chain behaves similar to a weakly bending stiff rod , so that the distribution function of the free end is gaussian with the maximum at @xmath87 . the typical conformation of the chain in this regime looks like a bending rod with constant sign of curvature along the chain . for larger contour lengths the curvature fluctuations are small and are still controlled by the bending energy , however with varying sign of curvatures along the chain . the typical conformation of the chain can be imagined as undulations along the average conformation of the polymer . the projected distribution function of the free end in this regime is expected to be roughly uniform within some range of @xmath88 . we expect that the inhomogeneities of curvature fluctuations in the vicinity of the clamped end are the reason for the maximum at @xmath89 . the larger curvatures in the vicinity of the fixed end result in larger displacements @xmath88 of the free end , and therefore contribute preferentially to the maximum at @xmath89 . with further increase of @xmath37 the conformations correspond to undulations around the average conformation of the chain , which is now a meandering line . fluctuations become now less controlled by bending energy , which results in weakening of the bimodality . since the difference between 2d and 3d chain is assumed to be marginal for short chains on the projected distribution function , we will compare the onset of the bimodality in both cases . because the transversal displacement is measured in both cases in units of the contour length , we have to recompute the number of segments for 3d and 2d chain at the onset according to @xmath90using the dependence of the persistence length on dimensionality benetatos05 , @xmath91 , we obtain @xmath92 . according to @xcite @xmath93 at the onset , hence we obtain @xmath94 , which is not far from our numerical result , @xmath95 ( see fig . [ fig - doublepeak ] ) . for chain length @xmath96 . squares : truncation with @xmath97 matrices ; continuous line : truncation with @xmath84 matrices . ] we now will address the issues of accuracy of the calculations , which depends on the size of the matrices @xmath11 and @xmath12 , and the maximal @xmath23 at which the summation over @xmath23 is stopped . in order to check our computation we verified that at large @xmath37 ( @xmath98 ) the numerical evaluation of ( gyn ) gives with very high accuracy the gaussian distribution @xmath99 . the general tendency is such that the sufficient level of matrix truncations and the number of terms in the expansion over the legendre polynomials increase with decreasing @xmath37 . in the limit @xmath24 the whole series over @xmath23 should be taken into account . the accuracy of the computations is demonstrated in fig . [ fig - n0 - 5 ] showing the computation of the distribution function for @xmath96 . the squares and the continuous curve correspond to the truncations by @xmath97 matrices and @xmath84 matrices , respectively . in both cases the summation was stopped at @xmath85 . the corrections due to higher @xmath23-s are negligibly small . for example , the corrections associated with @xmath100 and @xmath101 contribute only in 3rd and 5th decimal digits , respectively . thus , the computations depicted in figs . [ fig - doublepeak ] , [ fig - n0 - 5 ] can be considered as exact . figure [ figure3 ] shows the results of the computation of the 3d distribution functions @xmath102 of the free polymer obtained by performing the inverse laplace - fourier transform of the term @xmath49 in eq . ( [ gtkp ] ) for different chain lengths , and its comparison with the monte carlo simulations @xcite . , @xmath103 , @xmath104 , @xmath105 , @xmath106 ( from left to right ) computed with @xmath107 matrices . the symbols are the monte carlo simulation data extracted from fig . 1 in @xcite . ] our results are in excellent agreement with the numerical data . to conclude , we have studied the transverse distribution function of the free end of the three dimensional wormlike chain with fixed orientation and position of the second end using the exact solution for the green s function of the wormlike chain . within the procedure of truncations of the exact formula with finite order matrices we find that the distribution function @xmath108 for intermediate chain lengths , belonging to the interval @xmath0 , possesses the bimodal shape with the maxima at a finite value of the transverse displacement , which is consistent with the recent studies @xcite and @xcite for the two dimensional chain . in contrast to the 2d wormlike chain , the transverse 1d distribution function of the 3d chain shows only a tiny peak at @xmath87 in the vicinity of the onset of bimodality , which however disappears for larger @xmath37 . we present also results of analytical considerations for short and large polymers which are in complete agreement with the classical works @xcite where these limits were investigated using different methods . the computation of the three dimensional distribution function of a free polymer is in excellent agreement with the monte carlo simulations @xcite .
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we study the distribution function of the three dimensional wormlike chain with a fixed orientation of one chain end using the exact representation of the distribution function in terms of the green s function of the quantum rigid rotator in a homogeneous external field .
the transverse 1d distribution function of the free chain end displays a bimodal shape in the intermediate range of the chain lengths ( @xmath0 ) .
we present also analytical results for short and long chains , which are in complete agreement with the results of previous studies obtained using different methods .
| 4,454 | 124 |
in this paper , @xmath1 will be a simple graph with vertex set @xmath2 and edge set @xmath3 . [ closeddef ] a _ labeling _ of @xmath1 is a bijection @xmath4 = \{1,\dots , n\}$ ] , and given a labeling , we typically assume @xmath5 $ ] . a labeling is _ closed _ if whenever we have distinct edges @xmath6 with either @xmath7 or @xmath8 , then @xmath9 . finally , a graph is _ closed _ if it has a closed labeling . a labeling of @xmath1 gives a direction to each edge @xmath10 where the arrow points from @xmath11 to @xmath12 when @xmath13 , i.e. , the arrow points to the bigger label . the following picture illustrates what it means for a labeling to be closed : @xmath14 ( n1 ) at ( 2,1 ) { $ i\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n2 ) at ( 1,3 ) { $ \rule[-2.5pt]{0pt}{10pt}j$ } ; \node[vertex ] ( n3 ) at ( 3,3 ) { $ k\rule[-2.5pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n1/n2,n1/n3 } \draw[- > ] ( \from)--(\to ) ; ; \foreach \from/\to in { n2/n3 } \draw[dotted ] ( \from)--(\to ) ; ; \end{tikzpicture}&\hspace{30pt } & \begin{tikzpicture } \node[vertex ] ( n1 ) at ( 2,1 ) { $ i\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n2 ) at ( 1,3 ) { $ j\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n3 ) at ( 3,3 ) { $ k\rule[-2.5pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n2/n1,n3/n1 } \draw[- > ] ( \from)--(\to ) ; ; \foreach \from/\to in { n2/n3 } \draw[dotted ] ( \from)--(\to ) ; ; \end{tikzpicture } \end{array}\ ] ] whenever the arrows point away from @xmath11 ( as on the left ) or towards @xmath11 ( as on the right ) , closed means that @xmath12 and @xmath15 are connected by an edge . closed graphs were first encountered in the study of binomial edge ideals . the _ binomial edge ideal _ of a labeled graph @xmath1 is the ideal @xmath16 in the polynomial ring @xmath17 $ ] ( @xmath18 a field ) generated by the binomials @xmath19 for all @xmath20 such that @xmath10 and @xmath13 . a key result , discovered independently in @xcite and @xcite , is that the above binomials form a grbner basis of @xmath16 for lex order with @xmath21 if and only if the labeling is closed . the name `` closed '' was introduced in @xcite . binomial edge ideals are explored in @xcite and @xcite , and a generalization is studied in @xcite . the paper @xcite characterizes closed graphs using the clique complex of @xmath1 , and closed graphs also appear in @xcite . the goal of this paper is to characterize when a graph @xmath1 has a closed labeling in terms of properties that can be seen directly from the graph . our starting point is the following result proved in @xcite . [ hprop ] every closed graph is chordal and claw - free . `` claw - free '' means that @xmath1 has no induced subgraph of the form @xmath22 ( k ) at ( 3,6 ) { $ \bullet$ } ; \node[vertex ] ( j ) at ( 2.1,3.9){$\bullet$ } ; \node[vertex ] ( l ) at ( 3.9,3.9 ) { $ \bullet$ } ; \node[vertex ] ( i ) at ( 3,5){$\bullet$ } ; \foreach \from/\to in { i / l , i / k , i / j } \draw ( \from ) -- ( \to ) ; \end{tikzpicture } \end{array}\ ] ] besides being chordal and claw - free , closed graphs also have a property called _ narrow_. the _ distance _ @xmath23 between vertices @xmath24 of a connected graph @xmath1 is the length of the shortest path connecting them , and the _ diameter _ of @xmath1 is @xmath25 . given vertices @xmath24 of @xmath1 satisfying @xmath26 , a shortest path connecting @xmath27 and @xmath28 is called a _ longest shortest path _ of @xmath1 . [ narrowdef ] a connected graph @xmath1 is _ narrow _ if for every @xmath29 and every longest shortest path @xmath30 of @xmath1 , either @xmath31 or @xmath32 for some @xmath33 . thus a connected graph is narrow if every vertex is distance at most one from every longest shortest path . here is a graph that is chordal and claw - free but not narrow : @xmath34 ( n1 ) at ( 3,1 ) { $ a\rule[-2pt]{0pt}{10pt}$ } ; \node[vertex ] ( n2 ) at ( 2,3 ) { $ b\rule[-2pt]{0pt}{10pt}$ } ; \node[vertex ] ( n3 ) at ( 4,3 ) { $ c\rule[-2pt]{0pt}{10pt}$ } ; \node[vertex ] ( n4 ) at ( 3,5){$e\rule[-2pt]{0pt}{10pt}$ } ; \node[vertex ] ( n5 ) at ( 5,5){$f\rule[-2pt]{0pt}{10pt}$ } ; \node[vertex ] ( n6 ) at ( 1,5){$d\rule[-2pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n1/n2,n1/n3,n2/n3,n2/n4 , n3/n4 , n3/n5 , n4/n5,n2/n6,n4/n6 } \draw ( \from)--(\to ) ; ; \end{tikzpicture } \end{array}\ ] ] narrowness fails because @xmath35 is distance two from the longest shortest path @xmath36 . we can now state the main result of this paper . [ mainthm ] a connected graph is closed if and only if it is chordal , claw - free , and narrow . this theorem is cited in @xcite . since a graph is closed if and only if its connected components are closed @xcite , we get the following corollary of theorem [ mainthm ] . [ cormainthm ] a graph is closed if and only if it is chordal , claw - free , and its connected components are narrow . the independence of the three conditions ( chordal , claw - free , narrow ) is easy to see . the graph is chordal and narrow but not claw - free , and the graph is chordal and claw - free but not narrow . finally , the @xmath37-cycle @xmath38 ( a ) at ( 2,1 ) { $ \bullet$ } ; \node[vertex ] ( b ) at ( 4,1 ) { $ \bullet$ } ; \node[vertex ] ( c ) at ( 4,3 ) { $ \bullet$ } ; \node[vertex ] ( d ) at ( 2,3 ) { $ \bullet$ } ; \foreach \from/\to in { a / b , b / c , c / d , d / a } \draw ( \from)--(\to ) ; ; \end{tikzpicture}\ ] ] is claw - free and narrow but not chordal . the paper is organized as follows . in section [ properties ] we recall some known properties of closed graphs and prove some new ones , and in section [ algorithm ] we introduce an algorithm for labeling connected graphs . section [ characterize ] uses the algorithm to prove theorem [ mainthm ] . in a subsequent paper @xcite we will explore further properties of closed graphs . a path in a graph @xmath1 is @xmath39 where @xmath40 for @xmath41 . a single vertex is regarded as a path of length zero . when @xmath1 is labeled , we assume as usual that @xmath42 $ ] . then a path @xmath43 is _ directed _ if either @xmath44 for all @xmath12 or @xmath45 for all @xmath12 . here is a result from @xcite . [ directed ] a labeling on a graph @xmath1 is closed if and only if for all vertices @xmath46 $ ] , all shortest paths from @xmath11 to @xmath12 are directed . given a vertex @xmath29 , the _ neighborhood _ of @xmath27 in @xmath1 is @xmath47 when @xmath1 is labeled and @xmath48 $ ] , we have a disjoint union @xmath49 where @xmath50 this is the notation used in @xcite , where it is shown that a labeling is closed if and only if @xmath51 and @xmath52 are complete for all @xmath48 $ ] . vertices @xmath53 $ ] with @xmath54 give the _ interval _ @xmath55 = \{k \in [ n ] \mid i \le k \le j\}$ ] . here is a characterization of when a labeling of a connected graph is closed . [ nbdinterval ] a labeling on a connected graph @xmath1 is closed if and only if for all @xmath48 $ ] , @xmath51 is complete and equal to @xmath56 $ ] , @xmath57 . assume that the labeling is closed . then definition [ closeddef ] easily implies that @xmath51 is complete . it remains to show that @xmath51 is an interval of the desired form . pick @xmath58 and @xmath59 $ ] with @xmath60 . a shortest path @xmath61 from @xmath62 to @xmath63 is directed by proposition [ directed ] . since @xmath64 , we have @xmath65 . thus @xmath66 and hence @xmath67 since @xmath68 is complete . since @xmath69 , we have @xmath70 . we now prove by induction that @xmath71 for all @xmath72 . the base case is proved in the previous paragraph . now assume @xmath73 . then @xmath74 since @xmath75 and the labeling is closed . this completes the induction . since @xmath63 , it follows that @xmath76 . then we have @xmath77 with @xmath78 . thus @xmath79 since the labeling is closed , so @xmath80 since @xmath64 . hence @xmath51 is an interval of the desired form . conversely , suppose that @xmath51 is complete and @xmath81 $ ] , @xmath57 , for all @xmath82 . take @xmath6 with @xmath7 or @xmath8 . the former implies @xmath9 since @xmath68 is complete . for the latter , assume @xmath83 . then @xmath84 with @xmath85 . since @xmath86 is an interval containing @xmath87 and @xmath11 , @xmath86 also contains @xmath15 . hence @xmath9 . the following subsets of @xmath2 will play a key role in what follows . [ layerdef ] let @xmath1 be a connected graph labeled so that @xmath88 $ ] . then the _ @xmath89 layer of @xmath1 _ is the set @xmath90 \mid d(i,1 ) = n\}.\ ] ] thus @xmath91 consists of all vertices that are distance @xmath92 from the vertex @xmath93 . note that @xmath94 and @xmath95 . furthermore , since @xmath1 is connected , we have a disjoint union @xmath96 where @xmath97\}$ ] . we omit the easy proof of the following lemma . [ layerlem ] let @xmath1 be a connected graph labeled so that @xmath98 $ ] . then : 1 . if @xmath99 and @xmath10 , then @xmath100 , @xmath91 , or @xmath101 . if @xmath30 is a path in @xmath1 connecting @xmath99 to @xmath102 with @xmath103 , then for every integer @xmath104 , there exists @xmath105 with @xmath106 . [ layerprop ] let @xmath1 be a connected graph with a closed labeling satisfying @xmath42 $ ] . then : 1 . each layer @xmath91 is complete . 2 . if @xmath107 , then @xmath108 . we first show that @xmath109 to see why , take a shortest path from 1 to @xmath110 . this path has length @xmath111 , so appending the edge @xmath112 gives a path of length @xmath113 to @xmath114 . since @xmath115 , this is a shortest path and hence is directed by proposition [ directed ] . thus @xmath116 . for ( 1 ) , we use induction on @xmath117 . the base case is trivial since @xmath118 . now assume @xmath91 is complete and take @xmath119 with @xmath120 . a shortest path @xmath121 from @xmath93 to @xmath122 has a vertex @xmath123 adjacent to @xmath11 , and a shortest path @xmath124 from @xmath93 to @xmath125 has a vertex @xmath126 adjacent to @xmath12 . then @xmath127 and @xmath128 by . if @xmath129 , then @xmath130 , which implies @xmath131 since the labeling is closed . if @xmath132 , then @xmath133 since @xmath91 is complete . assume @xmath134 . then @xmath135 and closed imply @xmath136 . since @xmath137 and @xmath138 , we have @xmath139 by . then @xmath140 and closed imply @xmath131 . hence @xmath101 is complete . we now turn to ( 2 ) . to prove @xmath141 , @xmath142 , take @xmath138 . a shortest path from @xmath93 to @xmath11 will have a vertex @xmath143 such that @xmath144 . then @xmath127 by , hence @xmath145 . also , @xmath146 since @xmath147 . if @xmath148 , then @xmath149 . if @xmath150 , then @xmath151 since @xmath91 is complete . then @xmath152 and closed imply @xmath153 , and then @xmath154 by . thus @xmath149 . to prove the opposite inclusion , take @xmath149 . since @xmath153 and @xmath155 , we have @xmath156 for @xmath157 by lemma [ layerlem ] . if @xmath158 , then would imply @xmath159 , contradicting @xmath160 . if @xmath161 , then @xmath162 , again contradicting @xmath163 . hence @xmath122 . when the labeling of a connected graph is closed , the diameter of the graph determines the number of layers as follows . [ diameters ] let @xmath1 be a connected graph with a closed labeling . then : 1 . @xmath164 is the largest integer @xmath165 such that @xmath166 . if @xmath30 is a longest shortest path of @xmath1 , then one endpoint of @xmath30 is in @xmath167 or @xmath168 and the other is in @xmath169 , where @xmath170 . for ( 1 ) , let @xmath165 be the largest integer with @xmath171 . since points in @xmath169 have distance @xmath165 from @xmath93 , we have @xmath172 . for the opposite inequality , it suffices to show that @xmath173 for all @xmath174 with @xmath175 . we can assume @xmath1 has more than one vertex , so that @xmath176 . suppose @xmath161 and @xmath177 with @xmath178 . if @xmath179 , then @xmath180 and @xmath181 since @xmath102 . also , if @xmath182 , then @xmath183 , so that @xmath184 since @xmath91 is complete by proposition [ layerprop ] . finally , if @xmath185 , let @xmath186 for each integer @xmath187 . by proposition [ layerprop ] , we know that @xmath188 . hence , if @xmath189 , then @xmath190 is a path of length @xmath191 . if @xmath192 , then @xmath193 is a path of length @xmath194 . thus we have a path from @xmath11 to @xmath12 of length at most @xmath191 , so that @xmath195 . for ( 2 ) , let @xmath11 and @xmath12 be the endpoints of the longest shortest path @xmath30 with @xmath161 , @xmath177 and @xmath178 . if @xmath196 , then the previous paragraph implies @xmath197 which forces @xmath198 ( so @xmath199 ) and @xmath200 ( so @xmath201 ) . the remaining cases @xmath202 and @xmath203 are straightforward and are left to the reader . recall from definition [ narrowdef ] that a connected graph @xmath1 is narrow when every vertex is distance at most one from every longest shortest path . narrowness is a key property of connected closed graphs . [ narrowthm ] every connected closed graph is narrow . let @xmath1 be a connected graph with a closed labeling . pick a vertex @xmath204 and a longest shortest path @xmath30 . since @xmath1 is connected , @xmath161 for some integer @xmath92 . by proposition [ diameters ] , the endpoints of @xmath30 lie in @xmath167 or @xmath168 and @xmath169 , @xmath205 . then lemma [ layerlem ] implies that @xmath30 has a vertex @xmath206 in @xmath207 for every @xmath208 . if @xmath209 , then either @xmath210 or @xmath211 , in which case @xmath212 since @xmath91 is complete by proposition [ layerprop ] . on the other hand , if @xmath202 , then @xmath213 , hence @xmath180 . then @xmath214 since @xmath215 . in either case , @xmath11 is distance at most one from @xmath30 . we introduce algorithm [ alg : labeling ] , which labels the vertices of a connected graph . this algorithm will play a key role in the proof of theorem [ mainthm ] . [ alg : input ] [ alg : output ] @xmath216 @xmath217 @xmath218 endpoint of a longest shortest path with minimal degree[alg : possible1 ] label @xmath219 as @xmath11 [ alg : label1 ] @xmath220[alg : function1 ] @xmath221 @xmath222 @xmath223[alg : labelj1 ] the algorithm works as follows . among the endpoints of all longest shortest paths , we select one of minimal degree and label it as @xmath93 . we then go through the vertices in @xmath224 and label them @xmath225 , first labeling vertices with the fewest number of edges connected to unlabeled vertices . this process is repeated for the unlabeled vertices connected to vertex @xmath226 , and vertex @xmath227 , and so on until every vertex is labeled . furthermore , every vertex will be labeled because we first label everything in @xmath228 , then label everything in @xmath229 not already labeled , and so on . since the input graph is connected , this process must eventually reach all of the vertices . hence we get a labeling of @xmath1 . the following lemma explains the function @xmath230 that appears in algorithm [ alg : labeling ] . [ claim : meaningoffunction ] let @xmath1 be a connected graph with the labeling from algorithm [ alg : labeling ] . then : 1 . @xmath231 , and for every @xmath232 $ ] with @xmath233 , @xmath234 . 2 . if @xmath235 , then @xmath236 . algorithm [ alg : labeling ] defines @xmath231 . now assume @xmath233 and let @xmath27 be the vertex assigned the label @xmath11 . by lines [ alg : label2 ] and [ alg : function2 ] of the algorithm , we need to show that when the label @xmath11 is assigned to @xmath27 , the variable @xmath12 equals @xmath237 . this follows because for any smaller value @xmath238 , line [ alg : secondloop ] implies that everything in the neighborhood of @xmath239 is labeled before @xmath239 is incremented . however , lines [ alg : sset][alg : beginsecondloop ] show that @xmath27 is adjacent to @xmath12 and unlabeled at the start of the loop on line [ alg : secondloop ] . hence @xmath27 can not link to any smaller value of @xmath12 , and since @xmath27 has label @xmath11 , @xmath240 follows . \(2 ) suppose that @xmath241 $ ] satisfy @xmath235 . since @xmath242 ( resp . @xmath243 ) is the value of @xmath12 when the label @xmath244 ( resp . @xmath114 ) was assigned in algorithm [ alg : labeling ] , @xmath235 implies that the label @xmath114 was assigned later than @xmath244 in the algorithm . since the labels are assigned in numerical order , we must have @xmath245 . the labeling produced by algorithm [ alg : labeling ] allows us to define the layers @xmath91 . these interact with the function @xmath230 as follows : [ claim : orderingclaim ] let @xmath1 be a connected graph with the labeling from algorithm [ alg : labeling ] . then : 1 . if @xmath246 , then @xmath247 if @xmath248 . 2 . if @xmath246 and @xmath249 with @xmath250 , then @xmath236 . we prove ( 1 ) and ( 2 ) simultaneously by induction on @xmath209 ( the case @xmath202 of ( 2 ) is trivially true ) . the first time algorithm [ alg : labeling ] gets to line [ alg : sset ] , we have @xmath251 . every vertex in @xmath252 , is labeled during the loop starting on line [ alg : firstloop ] , so @xmath253 for all @xmath254 . hence ( 1 ) holds when @xmath255 . also , if @xmath249 with @xmath256 , then the vertex @xmath114 is not labeled at this stage . since labels are assigned in numerical order , we must have @xmath236 for all @xmath254 . hence ( 2 ) holds when @xmath198 . now assume that ( 1 ) and ( 2 ) hold for @xmath111 and every @xmath257 . given @xmath258 , a shortest path from 1 to @xmath244 gives @xmath259 with @xmath260 . since @xmath261 by lemma [ claim : meaningoffunction](1 ) , we have @xmath262 . we have @xmath263 for some @xmath187 . if @xmath264 , then the inductive hypothesis for ( 2 ) would imply @xmath265 , which contradicts @xmath262 . hence @xmath263 for some @xmath266 . but @xmath267 and @xmath268 imply @xmath269 for @xmath270 by lemma [ layerlem](1 ) . hence @xmath271 , proving ( 1 ) for @xmath272 . turning to ( 2 ) , pick @xmath258 and @xmath249 with @xmath273 . we just showed that @xmath274 , and lemma [ layerlem](1 ) implies that @xmath275 , @xmath276 , since @xmath277 . then @xmath278 , so our inductive hypothesis , applied to @xmath279 and @xmath280 , implies @xmath235 . then @xmath236 by lemma [ claim : meaningoffunction](2 ) , proving ( 2 ) for @xmath272 . we now turn to the main result of the paper . theorem [ mainthm ] from the introduction states that a connected graph is closed if and only if it is chordal , claw - free and narrow . one direction is now proved , since closed graphs are chordal and claw - free by proposition [ hprop ] , and connected closed graphs are narrow by theorem [ narrowthm ] . the proof of converse is harder . the key idea that the labeling constructed by algorithm [ alg : labeling ] is closed when the input graph is chordal , claw - free and narrow . thus the proof of theorem [ mainthm ] will be complete once we prove the following result . [ converse ] let @xmath1 be a connected , chordal , claw - free , narrow graph . then the labeling produced by algorithm [ alg : labeling ] is closed . by proposition [ nbdinterval ] , it suffices to show that the labeling produced by algorithm [ alg : labeling ] has the property that for all @xmath281 $ ] , @xmath282 \text { for } r_m = |{n_g^>}(m)|.\ ] ] we will prove this by induction on @xmath283 . in below , we show that holds for @xmath284 , and in below , we show that if holds for all @xmath285 , then it also holds for @xmath283 . thus , we will be done after proving and . after algorithm [ alg : labeling ] runs on a chordal , claw - free and narrow graph @xmath1 , the base case of the induction in the proof of theorem [ converse ] is the following assertion : @xmath286,\ r = |{n_g^>}(1)|,\ \text{and } { n_g^>}(1 ) \text { is complete } .\ ] ] we will first show that @xmath287 $ ] , @xmath288 . the first time through the the loop beginning on line [ alg : firstloop ] in algorithm [ alg : labeling ] , @xmath289 and @xmath290 and @xmath291 . for each vertex in @xmath292 , the loop beginning on line [ alg : secondloop ] labels that vertex @xmath11 , removes it from @xmath292 , and increments @xmath11 . this continues until @xmath293 , at which point every vertex in @xmath292 has been labeled @xmath294 , where @xmath295 is the initial size of @xmath292 . hence @xmath296 $ ] . to prove that @xmath228 is complete , there are several cases to consider . pick distinct vertices @xmath297 and assume that @xmath298 . note that @xmath299 are distance @xmath226 apart and therefore @xmath300 . our choice of vertex @xmath93 guarantees that there is a longest shortest path @xmath30 with 1 as an endpoint . let @xmath301 be the other , so that @xmath302 , @xmath303 and @xmath304 . since @xmath305 is the only vertex of @xmath30 in @xmath168 , @xmath114 and @xmath244 can not both lie on @xmath30 . therefore , either @xmath306 , @xmath307 , or @xmath308 . we will show that each possibility leads to a contradiction , proving that @xmath309 . plus 1 pt minus 1 pt * case 1 . * both @xmath308 . if @xmath114 has distance @xmath310 from @xmath311 , then appending the edge @xmath312 to a shortest path from @xmath114 to @xmath311 gives a longest shortest path @xmath313 from @xmath93 to @xmath311 that contains @xmath114 . replacing @xmath30 with @xmath313 , we get @xmath314 , which is case 2 to be considered below . similarly , if @xmath244 has distance @xmath310 from @xmath311 , then replacing @xmath30 allows us to assume @xmath315 , which is also covered by case 2 below . thus we may assume that neither @xmath114 nor @xmath244 has distance @xmath310 from @xmath311 . since @xmath316 would imply @xmath317 , we conclude that @xmath114 has distance @xmath165 from @xmath311 , and the same holds for @xmath244 . it follows that @xmath318 , since otherwise there is a path shorter than length @xmath165 from @xmath114 or @xmath244 to @xmath311 . since the subgraph induced on vertices @xmath319 can not be a claw , either @xmath320 or @xmath321 or both . we consider each possibility separately . plus 1 pt minus 1 pt * case 1a . * both @xmath322 , @xmath321 , as shown in figure 1(a ) on the next page . then the subgraph induced on @xmath323 is a claw , contradicting our assumption of claw - free . -10pt @xmath324 ( n1 ) at ( 2,1 ) { $ 1\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( s ) at ( 3.5,3 ) { $ s\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( t ) at ( .5,3 ) { $ t\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( v1 ) at ( 2,3 ) { $ v_1\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( v2 ) at ( 2,5 ) { $ v_2\rule[-2.5pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n1/v1,n1/t , n1/s , v1/v2,s / v1,t / v1 } \draw ( \from)--(\to ) ; \node [ right , font = \large ] at ( 4.2,3 ) { $ l_1 $ } ; \node [ right , font=\large ] at ( 4.2,5 ) { $ l_2 $ } ; \node [ right , font = \large ] at ( 4.2,1 ) { $ l_0 $ } ; \node [ right ] at ( 2.5,.3 ) { ( a ) } ; \end{tikzpicture } & \hspace{25pt}\begin{tikzpicture } \node[vertex ] ( n1 ) at ( 2,1 ) { $ 1\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( s ) at ( 3.5,3.2 ) { $ s\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( t ) at ( .5,3.2 ) { $ t\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( v1 ) at ( 2,3.2 ) { $ v_1\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( v2 ) at ( 2,5.4 ) { $ v_2\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( t2 ) at ( .5,5.4 ) { $ t_2\rule[-2.5pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n1/s , n1/t , n1/v1 , v1/v2 , s / v1,t / t2 } \draw ( \from)--(\to ) ; \node [ right , font = \large ] at ( 4.3,3.2 ) { $ l_1 $ } ; \node [ right , font=\large ] at ( 4.3,5.4 ) { $ l_2 $ } ; \node [ right , font = \large ] at ( 4.3,1 ) { $ l_0 $ } ; \node [ right ] at ( 2.5,.3 ) { ( b ) } ; \end{tikzpicture } \end{array}\ ] ] -8pt plus 1 pt minus 1 pt * case 1b . * exactly one of @xmath325 is in @xmath3 . without loss of generality , we may assume @xmath326 and @xmath327 , as shown in figure 1(b ) . recall that @xmath328 and @xmath244 is distance @xmath165 to @xmath311 . since @xmath244 and @xmath93 are both endpoints of longest shortest paths , line [ alg : possible1 ] of algorithm [ alg : labeling ] implies that @xmath329 . since @xmath330 is adjacent to @xmath93 but not @xmath244 , there must be at least one @xmath331 adjacent to @xmath244 but not @xmath93 , i.e. , @xmath332 with @xmath333 . for this @xmath331 , it follows that @xmath334 . we also have @xmath335 since @xmath328 . furthermore , @xmath336 , since otherwise we would have the @xmath37-cycle @xmath337 with no chords as @xmath338 , @xmath339 . similarly , @xmath340 or else we would have the @xmath37-cycle @xmath341 with no chords since @xmath338 , @xmath327 . note also that @xmath342 , since otherwise we would have the @xmath343-cycle @xmath344 with no chords as @xmath345 , @xmath346 , @xmath347 , @xmath348 , @xmath340 , contradicting chordal . hence @xmath331 gives figure 1(b ) as an induced subgraph . since @xmath1 is narrow , either @xmath349 or @xmath331 is adjacent to a vertex of @xmath30 . however , @xmath349 would imply @xmath350 since both lie in @xmath351 , contradicting @xmath352 . thus @xmath353 for some @xmath354 . since @xmath355 and @xmath356 , we have @xmath357 by lemma [ layerlem](1 ) . we just proved @xmath342 , so we must have @xmath358 . this gives the @xmath359-cycle @xmath360 . since figure 1(b ) is an induced subgraph , the only possible chords are @xmath361 , @xmath362 , @xmath363 , but by lemma [ layerlem](1 ) none of these are in @xmath3 since @xmath364 and @xmath365 . hence the @xmath359-cycle has no chords , contradicting chordal . * case 2 . * @xmath314 or @xmath366 . we may assume @xmath367 . arguing as in case 1b , there is @xmath332 with @xmath333 and @xmath334 . we also have @xmath368 , since otherwise the @xmath37-cycle @xmath369 has no chords as @xmath370 , @xmath371 . since @xmath1 is narrow , @xmath331 must either be in @xmath30 or be adjacent to a vertex in @xmath30 . however , @xmath372 would imply @xmath373 since @xmath374 , and the latter would give @xmath375 , which we just showed to be impossible . hence @xmath376 , so that @xmath377 for some @xmath187 . note that @xmath378 by lemma [ layerlem](1 ) . we claim that @xmath379 . to see why , first note that @xmath380 , since otherwise we would have the @xmath37-cycle @xmath381 with no chords as @xmath370 , @xmath382 . we also know that @xmath342 , as otherwise we would have the @xmath343-cycle @xmath383 with no chords since @xmath384 , @xmath338 , @xmath385 , @xmath347 , @xmath386 . see figure 2(a ) . thus we must have @xmath387 . however , this gives a @xmath359-cycle @xmath388 with the same impossible chords as before along with @xmath362 , @xmath361 , @xmath389 , @xmath390 , as in figure 2(b ) . this contradicts chordal , and follows . @xmath391 ( n1 ) at ( 2,1 ) { $ 1\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n2 ) at ( 1,3 ) { $ t\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n3 ) at ( 3,3 ) { $ s\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n4 ) at ( 1,4.5){$t_2\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n5 ) at ( 3,4.5){$v_2\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n6 ) at ( 3,6){$v_3\rule[-2.5pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n1/n2,n1/n3,n2/n4 , n3/n5 , n6/n5 } \draw ( \from)--(\to ) ; ; \foreach \from/\to in { n4/n5 } \draw[dotted ] ( \from)--(\to ) ; \node [ right , font = \large ] at ( 3.7,3 ) { $ l_1 $ } ; \node [ right , font=\large ] at ( 3.7,4.5 ) { $ l_2 $ } ; \node [ right , font = \large ] at ( 3.7,1 ) { $ l_0 $ } ; \node [ right , font = \large ] at ( 3.7,6 ) { $ l_3 $ } ; \node [ right ] at ( 2.5,.3 ) { ( a ) } ; \end{tikzpicture}&\hspace{25pt } & \begin{tikzpicture } \node[vertex ] ( n1 ) at ( 2,1 ) { $ 1\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n2 ) at ( 1,3 ) { $ t\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n3 ) at ( 3,3 ) { $ s\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n4 ) at ( 1,4.5){$t_2\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n5 ) at ( 3,4.5){$v_2\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n6 ) at ( 3,6){$v_3\rule[-2.5pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n1/n2,n1/n3,n2/n4 , n3/n5 , n6/n5 } \draw ( \from)--(\to ) ; ; \foreach \from/\to in { n4/n6 } \draw[dotted ] ( \from)--(\to ) ; \node [ right , font = \large ] at ( 3.7,3 ) { $ l_1 $ } ; \node [ right , font=\large ] at ( 3.7,4.5 ) { $ l_2 $ } ; \node [ right , font = \large ] at ( 3.7,1 ) { $ l_0 $ } ; \node [ right , font = \large ] at ( 3.7,6 ) { $ l_3 $ } ; \node [ right ] at ( 2.5,.3 ) { ( b ) } ; \end{tikzpicture } \end{array}\ ] ] -8pt after algorithm [ alg : labeling ] runs on a chordal , claw - free and narrow graph @xmath1 , we now prove that the resulting labeling satsifies the inductive step in the proof of theorem [ converse ] : @xmath392 $ , $ r_u = |{n_g^>}(u)|$ , $ { n_g^>}(u)$ is complete , $ 1\leq u < m$ , } \\ & \text{then $ { n_g^>}(m ) = [ m+1,m+r_m]$ , $ r_m = |{n_g^>}(m)|$ , and $ { n_g^>}(m)$ is complete . } \end{aligned}\ ] ] for the first assertion of , we know that @xmath393 $ ] is complete , which implies that @xmath394 . by analyzing the loop beginning on line [ alg : secondloop ] at this stage of algorithm [ alg : labeling ] , one finds that every vertex in @xmath292 will be labeled with consecutive integers , starting at @xmath395 and continuing until the final vertex in @xmath396 is labeled @xmath397 , where @xmath295 is the original size of @xmath292 . it follows that @xmath396 is an interval of the desired form . to show that @xmath398 is complete , pick @xmath399 in @xmath398 . let @xmath400 be a shortest path from @xmath303 to @xmath401 , with @xmath356 for all @xmath187 . lemmas [ layerlem](1 ) and [ claim : orderingclaim](2 ) imply that @xmath402 . hence , @xmath114 and @xmath244 are either both distance @xmath403 from 1 , both distance @xmath404 from 1 , or one of @xmath114 and @xmath244 is distance @xmath404 from 1 and the other is distance @xmath403 from 1 . we consider each case separately . plus3pt minus2pt * case 1 . * @xmath405 . then @xmath406 , @xmath407 by lemma [ layerlem](1 ) . since the subgraph induced on @xmath408 can not be a claw , we must have @xmath409 . plus3pt minus2pt * case 2 . * @xmath410 . we can assume @xmath411 and choose a shortest path @xmath412 from @xmath413 to @xmath414 with @xmath415 . then @xmath416 by lemma [ claim : orderingclaim](2 ) , giving @xmath417 . since @xmath418 and @xmath419 is an interval by hypothesis , we have @xmath420 . but then @xmath309 since we are also assuming that @xmath419 is complete . plus3pt minus2pt * case 3 . * we can assume @xmath421 and @xmath422 , so @xmath411 by lemma [ claim : orderingclaim](2 ) . we also have @xmath423 by lemma [ claim : meaningoffunction](2 ) since @xmath424 . we will consider separately the two possibilities that @xmath425 and @xmath426 . plus3pt minus2pt * case 3a . * suppose that @xmath427 . then @xmath428 since @xmath429 by lemma [ claim : meaningoffunction](1 ) . we also have @xmath430 , for otherwise we would have @xmath431 since @xmath261 . then @xmath432 , which implies @xmath236 by lemma [ claim : meaningoffunction](2 ) , contradicting @xmath411 . since the subgraph induced on @xmath433 can not be a claw , we must have @xmath409 . plus3pt minus2pt * case 3b . * suppose that @xmath426 . we will assume @xmath434 and derive a contradiction . the equality @xmath426 means that @xmath283 and @xmath114 were both labeled when @xmath435 in the loop starting on line [ alg : firstloop ] of algorithm [ alg : labeling ] . consider the moment in the algorithm when the label @xmath283 is assigned . since @xmath436 and @xmath435 , this happens during an iteration of the loop on line [ alg : secondloop ] for which @xmath437 . line [ alg : beginsecondloop ] guarantees that the vertices assigned the labels @xmath283 and @xmath114 satisfy @xmath438 . since @xmath114 is not yet labeled at this point and @xmath411 , @xmath244 is also not yet labeled and therefore @xmath439 . it follows that @xmath440 and @xmath441 . but , in order for @xmath442 to hold , there must be @xmath443 with @xmath444 and @xmath445 and @xmath446 . let us study @xmath447 . if @xmath448 , then @xmath449 . but we also have @xmath450 . since @xmath451 , @xmath452 is complete by the hypothesis of , so we would have @xmath453 . this contradicts our choice of @xmath447 . hence @xmath454 . we also have @xmath455 , since otherwise the @xmath37-cycle @xmath456 would have no chords as @xmath457 . also , since @xmath458 , lemma [ claim : orderingclaim](1 ) implies that @xmath459 . we claim that @xmath460 . lemma [ layerlem](1 ) , @xmath461 , and @xmath462 imply that @xmath463 or @xmath464 . if @xmath463 , then @xmath465 by lemma [ claim : orderingclaim](2 ) . from here , @xmath466 implies @xmath467 by lemma [ claim : orderingclaim](2 ) . hence we have @xmath468 . the hypothesis of implies that @xmath469 is complete and is an interval . since @xmath470 , it follows that @xmath471 , which contradicts our choice of @xmath447 . hence @xmath460 and we have figure 3(a ) . @xmath391 ( n1 ) at ( 2,1 ) { $ j\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n2 ) at ( 1,3 ) { $ m\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n3 ) at ( 3,3 ) { $ s\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n4 ) at ( 1,4.5){$t\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n5 ) at ( 3,4.5){$s_2\rule[-2.5pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n1/n2,n1/n3,n2/n3,n2/n4 , n3/n5 } \draw ( \from)--(\to ) ; ; \node [ vertex , font = \large ] at ( 4,3 ) { $ l_q$ } ; \node [ vertex , font=\large ] at ( 4,4.5 ) { $ l_{q+1}$ } ; \node [ vertex , font = \large ] at ( 4,1 ) { $ l_{q-1}$ } ; \node [ right ] at ( 2.5,.3 ) { ( a ) } ; \end{tikzpicture}&\hspace{25pt } & \begin{tikzpicture } \node[vertex ] ( n1 ) at ( 2,1 ) { $ j\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n2 ) at ( 1,3 ) { $ m\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n3 ) at ( 3,3 ) { $ s\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n4 ) at ( 1,4.5){$t\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n5 ) at ( 3,4.5){$s_2\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n6 ) at ( 1,6){$w_{q+2}$ } ; \foreach \from/\to in { n1/n2,n1/n3,n2/n3,n2/n4 , n3/n5 , n4/n6 , n6/n5 } \draw ( \from)--(\to ) ; ; \node [ right , font = \large ] at ( 4,3 ) { $ l_q$ } ; \node [ right , font=\large ] at ( 4,4.5 ) { $ l_{q+1}$ } ; \node [ right , font = \large ] at ( 4,1 ) { $ l_{q-1}$ } ; \node [ right , font = \large ] at ( 4,6 ) { $ l_{q+2}$ } ; \node [ right ] at ( 2.5,.3 ) { ( b ) } ; \end{tikzpicture } \end{array}\ ] ] -10pt let @xmath311 be a vertex of distance @xmath170 from @xmath93 and pick a longest shortest path @xmath472 from @xmath413 to @xmath473 , so @xmath415 . since @xmath1 is narrow , @xmath244 and @xmath447 must each either be in @xmath124 or be adjacent to a vertex in @xmath124 . we will consider each of these cases . _ first _ , suppose that @xmath474 . then @xmath422 implies that @xmath475 . since @xmath476 , there is a path of length @xmath477 connecting @xmath93 to @xmath478 . using @xmath479 , it follows that @xmath480 is a path of length @xmath170 . since @xmath1 is narrow , @xmath447 must be adjacent to some vertex @xmath481 . then @xmath482 , @xmath483 and lemma [ layerlem](1 ) imply that @xmath484 . this gives the @xmath343-cycle @xmath485 with no chords since @xmath482 , @xmath486 , @xmath298 and @xmath487 , @xmath488 since @xmath489 but @xmath490 . see figure 3(b ) . hence we have a contradiction since @xmath1 is chordal . _ second _ , suppose that @xmath491 . then @xmath492 . arguing as in the _ first _ , we arrive at figure 3(b ) with the same @xmath343-cycle with no chords , again a contradiction . _ third _ , suppose that @xmath493 . first note that @xmath124 was an arbitrary longest shortest path starting at @xmath93 . thus the above _ first _ and _ second _ give a contradiction whenever @xmath447 or @xmath244 are on _ any _ longest shortest path starting at @xmath93 . hence we may assume that @xmath447 and @xmath244 are not on any shortest path of length @xmath165 starting at @xmath93 . since @xmath1 is narrow , @xmath460 is adjacent to a vertex of @xmath124 , which must be @xmath494 , @xmath495 , or @xmath496 by lemma [ layerlem](1 ) . however , if @xmath484 , then we would get a path of length @xmath165 from @xmath93 to @xmath311 by taking any shortest path from 1 to @xmath447 , followed by @xmath497 , and then continuing along @xmath124 from @xmath496 to @xmath311 . this longest shortest path starts at @xmath93 and contains @xmath447 , contradicting the previous paragraph . hence @xmath498 and @xmath447 must be adjacent to @xmath494 or @xmath495 , and the same is true for @xmath244 by a similar argument . in fact , we must have @xmath499 , since otherwise @xmath500 and the subgraph induced on @xmath501 would be a claw . a similar argument shows that @xmath502 . since @xmath503 and @xmath504 , this implies that the subgraph induced on @xmath505 is a claw , again contradicting claw - free . this final contradiction completes the proof of , and theorem [ converse ] is proved . [ remark : ching ] in and , the chordal hypothesis is applied only to cycles of length @xmath37 , @xmath343 , or @xmath359 . hence , in theorem [ mainthm ] and corollary [ cormainthm ] , we can replace chordal with the weaker hypothesis that all cycles of length @xmath37 , @xmath343 , or @xmath359 have a chord . theorem [ mainthm ] is based on the senior honors thesis of the second author , written under the direction of the first author . we are grateful to amherst college for the post - baccalaureate summer research fellowship that supported the writing of this paper . thanks also to michael ching for remark [ remark : ching ] . m. crupi and g. rinaldo , _ binomial edge ideals with quadratic grbner bases _ , electron . j. combin . * 18 * ( 2011 ) , paper 211 , 13pp . v. ene , j. herzog and t. hibi , _ cohen - macaulay binomial edge ideals _ , nagoya math . j. * 204 * ( 2011 ) , 5768 . v. ene , j , herzog and t. hibi , _ koszul binomial edge ideals _ , arxiv:1310.6426 [ math.ac ] . v. ene , j. herzog and t. hibi , _ linear flags and koszul filtrations _ , arxiv:1312.2190 [ math.ac ] . v. ene and a. zarojanu , _ on the regularity of binomial edge ideals _ , arxiv:1307.2141 [ math.ac ] . j. herzog , t. hibi , f. hreinsdttir , t. kahle and j. rauh , _ binomial edge ideals and conditional independence statements _ , adv . in appl . * 45 * ( 2010 ) , 317333 . m. ohtani , _ graphs and ideals generated by some 2-minors _ , commun . algebra * 39 * ( 2011 ) , 905917 . j. rauh , _ generalized binomial edge ideals _ , adv . in appl . * 50 * ( 2013 ) , 409414 . s. saeedi madani and d. kiani , _ binomial edge ideals of graphs _ , electron . j. combin . * 19 * ( 2012 ) , paper 44 , 6pp .
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a graph is closed when its vertices have a labeling by @xmath0 $ ] with a certain property first discovered in the study of binomial edge ideals . in this article
, we prove that a connected graph has a closed labeling if and only if it is chordal , claw - free , and has a property we call _ narrow _ , which holds when every vertex is distance at most one from all longest shortest paths of the graph .
| 16,781 | 117 |
properly describing the pm surface motion requires taking into account both the plasma electron and ion dynamics . the response time of electrons to the laser field is much smaller than the optical period , while ions react on a longer time scale due to their larger mass . akin to the born - oppenheimer approximation in molecular physics , this makes it possible to model the system in three steps : ( i ) we first describe the quasi - instantaneous response of electrons to the laser field , considering a given ion background ( fig.[sim1](a ) ) ; ( ii ) we then calculate the slow ion motion , resulting from the combined actions of the laser - field and of the charge separation fields it induces ( fig.[sim1](b ) ) and ( iii ) finally the influence of the slow dynamics on the fast one is included , to determine the surface motion over the entire laser pulse ( fig.[sim1](c ) ) . the derivations of all formulas and their validation by particle - in - cell ( pic ) simulations are provided in the online supplementary information . qualitatively , the plasma electrons respond to the laser field as a spring , being alternatively pushed inside , and pulled outside of the ion background in each optical period @xcite . when pulled outward , they form relativistic electron jets ( red arrow in fig.[sim1](a ) ) , that are responsible for the rom attosecond pulse emission . when pushed inward , a high - density spike is formed at the sharp surface of the electron distribution ( green arrow in fig.[sim1](a ) ) , at a position @xmath11 ( fig.[sim1](e - f ) ) . a detailed analysis of pic simulations ( see supplementary information ) shows that the position of the outgoing electron jet responsible for the emission of an attosecond pulse in each laser cycle is tied to the position of the high - density spike formed in this compression phase , and thus follows the same evolution as the laser intensity changes in time or space . we therefore concentrate on the value of @xmath11 , which can be easily determined by the balance between the pushing force exerted by the laser field , and the restoring force exerted by the ion background . in the relativistic regime , this balance leads to the following expression for the maximum inward excursion @xmath12 of electrons in a given optical period : @xmath13 \label{xe}\ ] ] where @xmath14 is the angle of incidence of the laser on the pm , and @xmath15 is the critical plasma density at the laser frequency . @xmath16 is the ion charge density at the ion - vacuum boundary ( fig.[sim1](e - f ) ) , i.e. the density from which the laser field starts pushing electrons inside the ion background . for this derivation , the ion density gradient at the pm surface has been assumed to be exponential beyond @xmath16 , with a scale length @xmath17 , i.e. @xmath18 for @xmath19 ( fig.[sim1](e - f ) ) . @xmath17 is a crucial parameter of the interaction , which in particular strongly affects the hhg efficiency @xcite . @xmath12 increases for larger values of @xmath17 in eq.([xe ] ) , because the laser field can more easily push electrons inside a smoother ion background . the electron boundary displacement @xmath12 also increases with @xmath20^{1/2}$ ] , the amplitude of the normalized vector potential of the incident laser field : the higher this amplitude , the further electrons get pushed inside the target . for a focused laser pulse , the field envelop is a function of both time and space , @xmath21 . the spatial envelop results in an overall spatial curvature -a denting- of the plasma electron density surface . this laser - cycle - averaged curvature is clearly observed on a spatial map of electron density at @xmath22 corresponding to the laser pulse maximum ( fig.[sim1](a ) ) . it is very well reproduced by the curve @xmath23 $ ] deduced from eq.([xe ] ) and can be attributed to the spatially - inhomogenous ponderomotive force exerted by the laser field . as for the temporal evolution @xmath11 associated to the laser pulse temporal envelop , the prediction of eq.([xe ] ) is shown as a red dashed line in fig.[sim1](d ) , in the case of a fixed ion background : electrons move back to their initial position in the falling edge of the laser pulse , due to their immediate response to the field @xmath24 ( eq.(1 ) ) and the restoring force from the ion background . however , this temporal evolution will be affected when ion motion is taken into account , because @xmath16 then becomes a slow function of time in eq.([xe ] ) . the second step of our model aims at determining @xmath25 . the charge separation induced by the laser field between the electron and ion populations leads to a quasi - electrostatic field in the plasma , which peaks around @xmath12 and tends to accelerate the ion population located around this position @xcite . this acceleration expels the ions from this location , which results in an erosion of the ion density gradient in time . the position @xmath26 of the ion - vacuum boundary thus drifts inward during the laser pulse , and the density @xmath27 increases in time ( fig.[sim1](e - f ) ) . the so - called hole boring velocity @xmath28 of the ion surface can be calculated by writing a momentum flux balance @xcite . the reflection of the laser beam corresponds to a change in momentum of the field , which is compensated by an opposite change in momentum of the plasma particles . to determine how the light momentum is shared between electrons and ions , we use the same approach as developed independently in @xcite , i.e. we also write the energy flux balance , assuming that the absorbed laser intensity @xmath29 ( where @xmath30 is the plasma reflection coefficient for the laser ) is entirely carried away by electrons . the combination of these two balances leads to : @xmath31 with @xmath32 , where @xmath33 , @xmath34 are respectively the average charge state and mass number of the ions , @xmath35 is the proton mass and @xmath36 is the electron mass . the prediction of this equation for @xmath37 at the laser pulse maximum @xmath22 is shown as a blue line in fig.[sim1](b ) , and fits well the surface of the superimposed ion density map obtained from a pic simulation with mobile ions . the derivation of eq.([xpl ] ) shows that this curvature of the ion surface is induced by the spatially - inhomogeneous laser radiation pressure on the pm . the temporal evolution @xmath38 is represented in fig.[sim1](d ) by the blue line . as opposed to @xmath12 , the ion boundary displacement @xmath26 does not return to its initial value at the end of the pulse . this is because @xmath26 depends on the time integral of @xmath39 ( see eq.([xpl ] ) , where @xmath39 corresponds to the envelop of the laser field ) , meaning that it is the cumulated action of the laser field over time that is responsible for the ion dynamics . this results in a progressive change in the ion profile , which in turn affects the electrons dynamics . this coupling is included in our model in a very simple way . due to the erosion of the ion density profile , the laser field now starts pushing the electrons inside the ion background directly from @xmath38 , instead of @xmath40 initially . consequently , the position of the electron boundary @xmath41 when ion motion is taken into account is now given by @xmath42 ( see fig.[sim1](f ) ) . in this equation , the value of @xmath11 is also affected by ion motion , because the restoring force induced by the ions initially located between @xmath43 and @xmath38 is suppressed . as explained before , this second effect is accounted for simply by using @xmath44 in eq.([xe ] ) . the temporal evolution of the electron boundary resulting from these coupled dynamics is illustrated in fig.[sim1](c ) . an excellent agreement is obtained between the pic simulation and the prediction of the full model ( black dots ) . an extensive parametric study of the surface dynamics , using hundreds of pic simulations , confirms the excellent accuracy ( @xmath45 ) of this model over a broad range of physical conditions ( see supplementary information ) . figure [ sim1](d ) uses our model to highlight the relative contributions of ion and electron dynamics in the case of the simulation of fig.[sim1](c ) . despite the brevity of the pulse , the influence of ion motion on the position @xmath46 of the electron boundary becomes significant in the second part of the pulse ( beyond @xmath47 ) . its main effect is to prevent the electron boundary from moving back to its initial position in the falling edge of the laser pulse , which has observable consequences in experiments , as we will see later . as expected intuitively , the influence of ion dynamics on the total pm surface motion is predicted to become more and more significant as the laser pulse duration increases ( fig . [ sim2 ] ) . the laser - induced denting of the pm leads to a curvature of the wavefronts of the reflected light beam , which tends to focus this beam -including the harmonics generated upon reflection- in front of the surface @xcite . this is clearly observed in fig.[sim3 ] on the attosecond pulse train generated by the rom mechanism , which is focused at a distance @xmath48 from the surface , with a magnification ratio @xmath49 . this focusing of the beam naturally tends to increase its divergence . assuming a gaussian intensity profile of width @xmath50 for the @xmath51 harmonic in the source plane , this divergence is given by ( see supplementary information ) : @xmath53 is the pm dimensionless focusing parameter for the @xmath51 harmonic , that characterizes the effect of the pm curvature on all spatial properties of the reflected beam : @xmath54 with @xmath55 the harmonic wavelength . here @xmath56 is defined as @xmath57 ( fig.[sim3](left ) ) , i.e. it is the difference between the surface position at the center of the focal spot @xmath58 , and its position at @xmath59 ( with @xmath60 the half spatial width at @xmath61 of the laser field amplitude ) . in eq.[thetan ] , @xmath62 is the divergence that would be obtained in the absence of surface curvature , i.e. imposed by diffraction from the source plane . it can be expressed as a function of laser divergence @xmath63 . in eq.([thetan ] ) , each term of @xmath64 corresponds to a different physical limit . if @xmath65 ( e.g. @xmath66 or @xmath67 ) , surface curvature has a negligible effect on the spatial properties , which are determined only by the beam diffraction from the source plane . on the opposite , if @xmath68 , the focusing induced by the pm imposes the beam divergence , leading to @xmath69 . @xmath53 is in principle a function of time . however , our model shows that after a fast transient of less than five laser periods only , @xmath70 and hence @xmath53 weakly vary in time ( black dots in fig.[sim1](d ) ) . as a first approximation , we therefore neglect its temporal variation in our study of the spatial properties of the reflected beam . this model for the reflected beam properties has been successfully compared with a series of 2d pic simulations ( see supplementary information ) . in the interaction conditions corresponding to the present state of the art of femtosecond lasers ( @xmath71 , @xmath72 ) , it predicts @xmath73 ( 80 nm for @xmath74 nm ) and @xmath75 typically . the effect of surface curvature thus already becomes significant for harmonic orders @xmath76 . we now turn to an experimental investigation of the spatial properties of such harmonics , to validate the model , and show what insight it provides on hhg , and more generally on the physics of plasma mirrors . the experiment was performed on the uhi100 laser of iramis ( cea , france ) , that delivers 25 fs pulses with a peak power of up to 100 tw and an ultrahigh temporal contrast ( see methods section ) . this beam was focused in @xmath77-polarization to a spot size of 4 @xmath78 on a silica target , reaching an estimated peak intensity of @xmath79 @xmath2 ( @xmath80 ) , thus producing a relativistic plasma mirror . the density gradient scale length @xmath17 at the pm surface was varied by using a small controlled prepulse , intense enough to create a plasma ( @xmath81 @xmath2 ) at an adjustable delay @xmath82 ( @xmath83 @xmath84 ) before the main pulse . the value of @xmath17 was determined experimentally using time - resolved interferometry @xcite . under these conditions , high - order harmonics are produced in the reflected beam by the rom mechanism @xcite , and two diagnostics were used to characterize the spatial properties of the resulting harmonic beam in the far field ( see fig.[exp1](a - b ) and methods section ) . the spectrally - resolved divergence , extracted from images such as shown in fig.[exp1](a ) , is presented in fig.[exp1](c - d ) as a function of harmonic order and of the density gradient @xmath17 at the pm surface . the full lines show the results of the model . the only two unknowns of the model are the plasma reflectivity @xmath30 ( used only to calculate the ionic contribution to the surface curvature ) , and the ratio of harmonic and laser source size @xmath85 ( used to deduce the harmonic divergence from the surface curvature ) . these are however not used as free parameters to fit the data , but are directly extracted from 2d pic simulations performed in the physical conditions of the experiment ( see methods section ) . this provides @xmath86 and @xmath87 for the @xmath88 harmonic . parametric studies ( see supplementary information ) show that these values hardly change over a broad range of interaction conditions ( @xmath39 and @xmath17 ) . in addition , we note that @xmath30 hardly influences the results , since it only affects ion motion and appears in a square - root in eq.2 . these curves are in remarkable agreement with the measurements , thus validating the model and showing it can be used to gain insight on the physics involved in this experiment . note that this agreement was obtained without introducing any additional ` intrinsic phase ' @xmath89 , such as the one described by an der brgge et al @xcite . our model actually suggests that this phase is simply given by @xmath90 where @xmath91 is the electron denting provided by eq.([xe ] ) , and is thus implicitly included in our analysis . this expression exactly predicts the scaling of @xmath89 obtained in @xcite for normal incidence and a step - like plasma surface in the limit of ultra - relativistic intensities . comparing the measured divergences with those that would be obtained by diffraction from a flat pm for the same source size [ black dashed lines in fig.[exp1](c - d ) ] shows that the harmonic divergence is close to this limit when @xmath17 is small , but is then very significantly increased by the pm curvature , here by a factor of up to 3 , for the typical gradients that optimize the rom conversion efficiency ( @xmath92 to @xmath93 ) @xcite . this analysis provides a clear indication of the focusing of the harmonics in front of the pm , due to its surface curvature . the measurements of fig.[exp1](d ) show that this focusing increases with the gradient scale length @xmath17 , as expected from the model , since a longer gradient leads to a softer restoring force from the ion background , and hence to a larger surface denting @xmath56 . the laser pulse duration used in this experiment is so short that ion motion has little influence on the pm curvature , and hence on the harmonic divergence ( white dot on fig.[sim2 ] ) . according to fig.[sim1](d ) , it however significantly changes the temporal dynamics of the surface in the falling edge of the pulse . we now demonstrate that this can lead to observable effects in the experiment , by considering the spectral properties of the harmonics . after a fast initial transient where the denting @xmath56 strongly varies , the temporal evolution of the field envelop @xmath24 only leads to a weak residual drift of pm surface during the pulse ( black dots in fig.[sim1](d ) ) , with typical velocities of the order of @xmath94 according to our model . this motion appears as a slow drift on the femtosecond time scale , that combines with the fast relativistic oscillation of the plasma surface at the laser frequency responsible for hhg ( see fig.[sim1](c ) ) . this results in a doppler shift on the reflected light , which scales linearly with harmonic order @xmath95 , and thus gets measurable for large enough values of @xmath95 . since the ion dynamics affects the temporal evolution of the pm surface , it can potentially influence this doppler effect . this is confirmed by a comparison of pic simulations performed with fixed ( fig.[sim4](a ) ) and moving ions ( fig.[sim4](b ) ) . in the case of fixed ions , the plasma surface moves inward in the rising part of the pulse , leading to a doppler redshift , and then moves outward in the falling part , leading to a doppler blueshift . if strong enough , this effect leads to harmonics with a double peak structure @xcite , clearly observed in fig.[sim4](a ) . in contrast , when ion motion is allowed in the simulation , the irreversible erosion of the ion density gradient prevents the electron boundary from moving back to its initial position when the laser intensity decreases . this naturally suppresses the doppler blueshift , and only a doppler redshift is observed , in the interaction conditions considered here . turning back to the experiment , fig.[exp2](a ) shows a zoom on the spatio - spectral distribution of the @xmath96 harmonic , measured in a typical shot . it is very similar to the pic results of fig.[sim4](b ) , and only a red shift is observed : according to the previous discussion , this is a signature of ion motion . fig.[exp2](b ) shows that the doppler shift at the center of the beam increases with the density gradient scale length @xmath17 , which is consistent with the stronger curvature of the pm for larger @xmath17 . this dependence is quantitatively reproduced by our model , when both ion and electron dynamics are taken into account using the same parameters as in fig.[exp1 ] . thus , although ion dynamics does not affect the spatial properties of harmonics in our experimental conditions , it has a clear signature in the spectral domain , which validates the ionic part of our model . we have presented a simple analytical model for the spatial properties of light beams reflected by relativistic plasma mirrors , in excellent agreement with both pic simulations and experimental results . it provides insight into the respective roles of ion and electron dynamics , and into the spatial and spectral properties of harmonics generated in the reflected beam . combined with this model , these harmonics now constitute a direct and powerful diagnostic of the femtosecond motion of the pm surface , with spatial resolution within the laser focal spot ( fig.[sim4](c ) ) . measurements schemes such as photonic streaking @xcite will potentially also provide temporal resolution within the laser pulse envelop . this model will be instrumental in designing future applications of plasma mirrors , in particular for attosecond science . it can for instance be used to determine what laser pulse duration is required to generate isolated rom attosecond pulses using the lighthouse effect @xcite . in this perspective , as well as in most applications where the reflected beam is manipulated or used in the far - field , being able to control and minimize the attosecond beam divergence is essential @xcite , which requires mitigating the effect of the laser - induced pm curvature . figure [ exp3 ] provides the first experimental demonstration in the relativistic regime of a very simple scheme for such a control @xcite : by using a driving laser - beam with a slightly diverging wavefront on target , the effect of the pm curvature on the attosecond beam can be compensated , leading to a divergence close to the one that would be obtained for a flat mirror , reduced by a factor of more than 2 compared to the one obtained at best focus . in other applications , pm will prove useful to focus the reflected beam , and boost the peak intensity of the fundamental laser frequency @xcite or its harmonics @xcite . this can be achieved using either curved substrates , or the natural light - induced pm curvature described in this work , which typically leads to magnification factors @xmath97 for @xmath98 in the interaction regime considered here . in either case , the understanding of the laser - induced pm surface dynamics provided by this work will be essential . the research leading to these results has received funding from the european research council ( erc grant agreement no . 240013 ) and laserlab - aladin ( grant no . 228334 ) . this work was performed by using hpc resources from genci - ccrt / cines ( grant no . 2012 - 056057 ) . + * methods * + * simulations * + + * experiment * + + * fit of experimental data with the model * + -0.3 cm of the pulse , @xmath99 , when ion motion is taken into account ( @xmath100 ) and when ions are considered as fixed ( @xmath12 , eq.([xe ] ) with a constant @xmath16 ) , as predicted by our model . this is plotted as a function of @xmath39 and pulse duration , for a typical value of the density gradient ( @xmath101 ) . the white dot corresponds to the interaction conditions of the experiment performed with uhi100 ( see experimental section ) . , title="fig:",width=302 ] -0.5 cm
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the advent of ultrahigh - power femtosecond lasers creates a need for optical components suitable to handle ultrahigh light intensities . due to the unavoidable laser - induced ionization of matter
, these components will have to be based on a plasma medium .
an archetype of such optical elements is a plasma mirror , created when an intense femtosecond laser pulse impinges on a solid target .
it consists of a dense plasma , formed by the laser field itself , which specularly reflects the main part of the pulse .
plasma mirrors have major potential applications as active optical elements to manipulate the temporal and spatial properties of intense laser beams , in particular for the generation of intense attosecond pulses of light .
we investigate the basic physics involved in the deformation of a plasma mirror resulting from the light pressure exerted by the ultraintense laser during reflection , by deriving a simple model of this fundamental process , which we validate both numerically and experimentally .
the understanding of this deformation is essential for all future applications of plasma mirrors , especially for the generation of collimated attosecond beams .
we show how its effect on the attosecond beam divergence can be mitigated by using the laser phase , thus providing crucial control for future applications in attosecond science .
+ ultrafast laser technology now makes it possible to study the interaction of femtosecond ( fs ) laser pulses with plasmas in an extreme regime , where the motion of electrons in the laser field is relativistic @xcite . with several facilities aiming at peak powers beyond a petawatt ,
the study of new regimes of quantum electrodynamics should thus become feasible in the near future @xcite . the rapid growth in the number of high - power ultrashort lasers is also driven by the perspective of societal and scientific applications , such as compact laser - driven particle accelerators @xcite .
these laser developments and their prospects call for new types of optical elements , that can be used to manipulate and tailor ultrahigh - power laser beams at very high intensities @xmath0 , both in the temporal and spatial domains . as soon as @xmath1 @xmath2 , any medium gets strongly ionized by the field , making conventional optics inappropriate :
in this regime , optical components will inevitably consist of a plasma medium .
easy to use and versatile , plasma mirrors ( pm ) have a major role to play as high - intensity optical components @xcite , and constitute simple testbeds for models of relativistic laser - plasma interaction .
pm are already routinely used at moderate light intensities ( @xmath3 @xmath2 ) as ultrafast optical switches , to enhance the temporal contrast of femtosecond lasers @xcite .
as @xmath4 @xmath2 , the non - linear response of pms to the laser field results in sub - cycle temporal modulations of the reflected field , associated to high - order harmonic generation ( hhg ) in its spectrum @xcite .
these harmonics , generated through different mechanisms , are associated in the time domain to attosecond pulses @xcite . beyond @xmath5 @xmath2 ,
a key hhg mechanism is the relativistic oscillating mirror ( rom ) , where the laser - driven oscillation of the plasma surface induces a periodic doppler effect on the reflected field @xcite , which can result in harmonic orders of several thousands @xcite .
plasma mirrors thus hold great promise for the generation of intense attosecond pulses of light @xcite , which would break down a major barrier in attosecond science , opening the way to potential ground - breaking applications such as pump - probe experiments on electron dynamics in matter @xcite .
in addition to these temporal effects , the initial solid target on which the pm is created can be geometrically shaped , to also spatially manipulate the reflected beam . at moderate intensities , elliptical pms
have thus recently allowed extremely tight focusing of a high - power laser beam @xcite . in the relativistic regime , curved pms
have been proposed as a way to focus the very high generated harmonic orders to a spot size @xmath6 ( where @xmath7 is the laser wavelength ) @xcite .
combined with their attosecond temporal bunching , this is a promising path to boost the peak intensity of ultrashort lasers , which might help approaching the schwinger limit @xcite @xmath8 @xmath2 , where the light field starts inducing electron - positron pair creation from vacuum @xcite . in these high intensity applications ,
the laser field exerts such a high pressure on the plasma ( typically @xmath9 gbar for @xmath10 @xmath2 ) that it induces a significant motion of the pm surface , even during a femtosecond laser pulse .
any spatial variation of the intensity on target , as generally occurs at or around focus , then leads to a deformation of the pm surface -typically a curvature- which can affect the spatial @xcite and spectral @xcite properties of the reflected beam . beyond its fundamental interest ,
understanding and controlling this intrinsic dynamics of pm is crucial for any of the previous applications .
it in particular determines the divergence of attosecond beams produced from plasma mirrors , which is a key parameter for future experiments . in this article , we elucidate the physics of the light - induced curvature of the pm , with an analytical model of the surface dynamics and its consequences on the reflected light . despite its simplicity , it captures the essential aspects of this process , and disentangles the influences of electron and ion dynamics in the femtosecond regime .
due to their small wavelengths high - order harmonics generated on pm are strongly affected , and thus constitute sensitive probes of its curvature .
we present some of the most exhaustive measurements of the rom harmonic properties performed to date , which we use to validate this model experimentally .
controlling the spatial properties of these harmonics is crucial for future applications in attosecond science .
we finally demonstrate that such a control can be achieved very simply by using the spatial phase of the driving laser .
| 6,005 | 1,497 |
r coronae borealis ( rcb ) stars are hydrogen - deficient carbon ( hdc ) stars that exhibit spectacular ( @xmath1 up to @xmath28 mag ) , aperiodic declines in brightness ( for a review on rcb stars see @xcite ) . the fading occurs rapidly ( @xmath21 to few weeks ) as new dust is formed in the circumstellar environment , and the recovery is slow , sometimes taking several years , as the new dust is dispersed and removed from the line of sight . at maximum light rcb stars are bright supergiants , which in combination with the large - amplitude photometric variability should make them easy to discover . yet , to date there are only @xmath256 known rcb stars in the galaxy @xcite . the rarity of these stars suggests that they reflect a very brief phase of stellar evolution , or a bias in rcb star search methods , or both . the lack of hydrogen and overabundance of carbon in rcb atmospheres implies that rcb stars are in a late stage of stellar evolution , but no consensus has yet emerged regarding their true physical nature . there are two leading theories for explaining the observed properties of rcb stars : the double degenerate ( dd ) scenario and the final helium shell flash ( ff ) scenario ( see e.g. , @xcite ) . the dd scenario posits that rcb stars are the stellar remnant of a carbon oxygen white dwarf ( wd ) and helium wd merger . in the ff scenario , a thin layer of he in the interior of the star begins runaway burning , which leads to the rapid expansion of the photosphere shortly before the star becomes a planetary nebula . there are observational properties of rcb stars that both theories have difficulty explaining @xcite , and conflicting observational evidence supporting aspects of both ( e.g. , @xcite ) . if , as some of the recent observations suggest , the dd scenario proves correct , then a complete census of galactic rcb stars should be able to calibrate population synthesis models of wd binary systems ( e.g. , @xcite ) , which may improve our understanding of these systems as the progenitors of type ia supernovae . in any event , the enigmatic nature of these rare objects , and the opportunity to elucidate the astrophysics of an important late stage of stellar evolution , motivates us to search for additional benchmark exemplars of the class . based on the detection of rcb stars in the large magellanic cloud ( lmc ) , it is argued in @xcite that there should be @xmath23200 rcb stars in the galaxy . with the actual number of known rcb stars in the milky way roughly two orders of magnitude below this estimate , this suggests that either thousands of rcb stars remain undetected or the differing star formation environments / histories in the lmc and the milky way result in highly different rcb populations . an observational bias that preferentially selects warm rcb stars likely contributes to the discrepancy between the predicted and known number of these stars in the galaxy @xcite . indeed , recent discoveries of rcb stars in the galactic bulge and magellanic clouds ( mcs ) have uncovered more cool , @xmath35000 k , rather than warm , @xmath37000 k , rcb stars @xcite . the observed correlation between color and @xmath4 , with bluer rcb stars in the mcs being more luminous @xcite , clearly shows that any magnitude - limited survey will have an observational bias towards discovering the intrinsically rarer warm rcb stars . there may also be a large population of rcb stars that have colder photospheres than the cool rcb stars : there is one known galactic rcb star , dy persei @xcite , that has @xmath33500 k @xcite . recent observations of the mcs have identified several dy persei - like stars ( dypers ) while searching for rcb stars @xcite , while @xcite discovered the second known dyper in the milky way using observations of the galactic bulge . in addition to cooler photospheres , dypers have other properties that differ from rcb stars , which has led to some degree of ambiguity regarding the connection between these two classes ( see e.g. , @xcite ) . dypers and rcb stars both show an overabundance of carbon in their atmospheres and unpredictable , large - amplitude declines in their light curves . several properties differ between the two , however , for instance , dypers : ( i ) have symmetric declines in their light curves , ( ii ) clearly show @xmath5c in their spectra , ( iii ) are on average @xmath210 times fainter than rcb stars , and ( iv ) may have significant h in their atmospheres . a detailed examination of the differences in the mid - infrared excesses of rcb stars and dypers in the mcs led to the conclusion in @xcite that dypers are most likely normal carbon stars that experience ejection events rather than an extension of the rcb phenomenon to lower temperature stars . furthermore , using ogle - iii observations , it is shown in @xcite that several carbon - rich asymptotic giant branch stars ( agbs ) , which have been classified as mira or semi - regular periodic variables on the basis of their light curves , show evidence for dyper - like declines in their light curves . this leads to the conclusion in @xcite that dypers are heavily enshrouded carbon - rich agb stars that are an extension of typical variables rather than a separate class of variable stars . nevertheless , all studies of dypers to date have cited a need for more observations , in particular high resolution spectra to conduct detailed abundance analyses , to confirm or deny the possibility that dypers are the low temperature analogs to rcb stars . over the past decade the decrease in the cost of large ccds , coupled with a dramatic increase in computer processing power and storage capabilities , has enabled several wide - field , time - domain surveys . these surveys will continue to produce larger data sets before culminating near the end of the decade with the large synoptic survey telescope ( lsst ; @xcite ) . this explosion of observations should enable the discovery of the thousands of `` missing '' galactic rcb stars , should they in fact exist . these new discoveries do not come without a cost , however , as the data rates of astronomical surveys are now becoming enormous . while it was once feasible for humans to visually examine the light curves of all the newly discovered variable stars , as the total number of photometric variables grows to 10@xmath610@xmath7 visual inspection by expert astronomers becomes intractable . advanced software solutions , such as machine - learning ( ml ) algorithms , are required to analyze the vast amounts of data produced by current and upcoming time - domain surveys . in an ml approach to classification , data from sources of known science class are employed to train statistical algorithms to automatically learn the distinguishing characteristics of each class . these algorithms generate an optimal predictive model that can determine the class ( or posterior class probability ) of a new source given its observed data . @xcite presented an end - to - end ml framework for multi - class variable star classification , in which they describe algorithms for feature generation from single - band light curves and outline a methodology for non - parametric , multi - class statistical classification . in this paper we present the results of a search for new rcb stars and dypers in the galaxy using version 2.3 of the ml catalog presented in @xcite . in 2 we describe the candidate selection procedure , while 3 describes the new and archival observations of the candidates . our analysis of the photometric and spectroscopic data is contained in 4 . the individual stars are examined in further detail in 5 , while we discuss the results in 6 . our conclusions are presented in 7 . candidate selection of possible rcb stars was performed using version 2.3 of the machine - learned acvs classification catalog ( http://www.bigmacc.info/[macc ] ; @xcite ) of variable sources cataloged from all - sky automated survey ( asas ; @xcite ) . full details of the classification procedure can be found in @xcite and @xcite . briefly , we employ a random forest ( rf ) classifier , which has been shown to provide the most robust results for variable star classification ( see e.g. , @xcite ) , to provide probabilistic classifications for all of the 50,124 sources in asas catalog of variable stars ( acvs ; @xcite ) . the classification procedure proceeds as follows : for each source in the acvs 71 features are computed , 66 from the asas light curves ( e.g. , period , amplitude , skew , etc . ; for the full list of features we refer the reader to @xcite and references therein ) and 5 color features from optical and near - infrared ( nir ) catalogs . a training set , upon which the rf classifications will be based , is constructed using light curves from 28 separate science classes , most of which are defined using well studied stars with high precision light curves from the _ hipparcos _ and ogle surveys ( @xcite ) , as well as some visually classified sources from acvs for a few of the classes that are not well represented in _ hipparcos _ the same 71 features are calculated for all the sources in the training set , and the rf classifier uses the separation of the 28 science classes in the multi - dimensional feature space to assign probabilistic classifications to each source in the acvs . in the end , the probability of belonging to each individual science class is provided for each acvs source and a post - rf procedure is used to calibrate these probabilities ( meaning that a source with @xmath8 has a @xmath250% chance of actually being a mira ) . when searching for rcb stars in time - domain survey data , rf classification provides a number of advantages relative to the more commonly used method of placing hard cuts on a limited set of a few features . many studies have focused on light curves with large amplitude variations and a lack of periodic signal ( e.g. , @xcite ) . a few recent studies have noted that additional cuts on nir and mid - infrared colors can improve selection efficiency @xcite . while these surveys have all proven successful , the use of hard cuts may eliminate actual rcb stars from their candidate lists . hard cuts are not necessary , however , when using a multi - feature rf classifier , which is capable ( in principle ) of capturing most of the photometric behavior of rcb stars ( including the large - amplitude , aperiodic fades from maximum light as well as the periodic variations that occur near maximum light ) . another general disadvantage in the use of hard cuts for candidate selection of rare sources is that the hard cuts are typically defined by known members of the class of objects for which the search is being conducted . any biases present in the discovery of the known members of a particular class will then be encoded into the absolute ( i.e. , hard cuts ) classification schema . this can exclude subclasses of sources that differ slightly from the defining members of a class . furthermore , new discoveries will be unable to refine the selection criteria since , by construction , they will fall within the same portion of feature space as previously known examples . the rf classifier produces an estimate of the posterior probability that a source is an rcb star given its light curve and colors . this allows us to construct a relative ranking of the rcb likelihood for all the sources in acvs . instead of making cuts in feature space , we can search down the ordered list of candidates . in this sense the rf classifier identifies the sources that are closest to the rcb training set relative to the other classes . the rf classifier finds the class boundaries in a completely data - driven way , allowing for the optimal use of known objects to search for new candidates in multi - dimensional feature spaces . this helps to mitigate against biases present in the training set , as classifications are performed using the location of an individual source in the multidimensional feature phase - space volume relative to defined classes in the training set . the macc rcb training set was constructed using high - confidence positional matches between acvs sources and known rcb stars identified in simbad and the literature . in total there are 18 cataloged rcb stars that are included in the acvs , which we summarize in table [ tab : training_set ] . the light curves of the known rcb stars were visually examined for the defining characteristic of the class : sudden , aperiodic drops in brightness followed by a gradual recovery to pre - decline flux levels . all of the known rcb stars but one , asas 054503@xmath96424.4 , showed evidence for such behavior . asas 054503@xmath96424.4 is one of the brightest rcb stars in the lmc ( @xmath10 mag ) , which during quiescence is barely above the asas detection threshold . the light curve for asas 054503@xmath96424.4 does not show a convincing decline from maximum light , and as such we do not include it in the training set . in addition to the 18 rcb stars in acvs , 7 additional rcb stars are detected in asas with the characteristic variability of the class . these sources all have clearly variable asas light curves ; their exclusion from the acvs means there is some bias in the construction of that catalog . in order to keep this bias self - consistent the training set for the macc only included sources from @xcite and supplements from acvs ( see @xcite ) . we note that a future paper to classify all @xmath212 million sources detected by asas will include all asas rcb stars in its training set ( richards et al . , in prep ) . therefore the training set includes 17 rcb stars , which is limited by the coverage and depth of asas , the selection criteria of the acvs , and the paucity of known rcb stars in the galaxy . there are no known dypers in acvs : only two are known in the galaxy and the dypers in the mcs are fainter than the asas detection limits . nevertheless , the similarity in the photometric behavior of rcb stars and dypers allows us to use the rcb training set to search for both types of star . as more galactic rcb stars and dypers are discovered , we will be able to supplement the training set and improve the ability of future iterations of the rf classifier ( see 6.2 ) . lllccr asas 054503@xmath116424.4 & hv 12842 & 220040 & n & 0.010 & 2806 + asas 143450@xmath113933.5 & v854 cen & 240306 & y & 0.914 & 2 + asas 150924@xmath117203.8 & s aps & 241463 & y & 0.934 & 2 + asas 154834 + 2809.4 & r crb & 242999 & y & 0.430 & 34 + asas 162419@xmath115920.6 & rt nor & 244506 & y & 0.629 & 10 + asas 163242@xmath115315.6 & rz nor & 244888 & y & 0.944 & 2 + asas 171520@xmath112905.6 & v517 oph & 247066 & y & 0.169 & 158 + asas 172315@xmath112252.0 & v2552 oph & 247575 & y & 0.105 & 425 + asas 180450@xmath113243.2 & v1783 sgr & 250762 & y & 0.642 & 9 + asas 180850@xmath113719.7 & wx cra & 251121 & y & 0.834 & 2 + asas 181325@xmath112546.9 & v3795 sgr & 251489 & y & 0.726 & 8 + asas 181509@xmath112942.5 & vz sgr & 251638 & y & 0.845 & 2 + asas 181851@xmath114632.9 & rs tel & 251987 & y & 0.951 & 1 + asas 184732@xmath113809.6 & v cra & 254404 & y & 0.804 & 3 + asas 190812 + 1737.7 & sv sge & 256072 & y & 0.053 & 859 + asas 191012@xmath112029.7 & v1157 sgr & 256221 & y & 0.276 & 74 + asas 193222@xmath110011.5 & es aql & 257713 & y & 0.044 & 901 + asas 220320@xmath111637.6 & u aqr & 263740 & y & 0.839 & 3 + [ tab : training_set ] in order to determine our ability to recover known rcb stars using the rf classifier we perform a leave - one - out cross validation ( cv ) procedure . for the 17 sources in the rcb training set , we remove one source and re - run the rf classifier in an identical fashion to that used in @xcite . we then record the rf - determined probability that the removed source belongs to the rcb class , @xmath12 , and the ranked value of @xmath12 relative to all other stars that are not included in the training set , @xmath13 . we repeat the cv procedure for each star included in the training set , and the results are shown in table [ tab : training_set ] . since the training set is being altered in each run of the cv , @xmath13 provides a better measure of the quality of each candidate ; @xmath13 is a relative quantity , whereas the calibration of @xmath12 will differ slightly from run to run . eight of the 17 sources in the training set have @xmath14 , implying that @xmath250% of the training set would be a top three candidate rcb star had we previously not known about it . fifteen of the 17 rcb stars in the training set would be in the top 0.8% of the 50,124 sources in the acvs , while all the known rcb stars in acvs , including asas 054503@xmath116424.4 ( which is not in the training set ) , are in the top @xmath26% of rcb candidates . two sources in the training set , sv sge and es aql , are not listed near the top of @xmath13 ranking during cv . for es aql this occurs because the star is highly active during the asas observations showing evidence for at least six separate declines during the @xmath210 yr observing period . as a result the light curve folds fairly well on a period of @xmath2397 day , and es aql becomes confused with mira and semi - regular periodic variables ( see figure [ fig : hardcuts ] ) . sv sge , on the other hand , shows significant periodicity at the parasite frequency of 1 day , which precludes it from having a high @xmath13 . the cv procedure allows us to roughly tune the efficiency of our selection criteria ; the purity of the selection criteria can not be evaluated until candidates have been spectroscopically confirmed . due to the relative rarity of rcb stars , we elected to generate a candidate list with high efficiency while sacrificing the possibility of high purity . with only @xmath250 galactic rcb stars known to date , every new discovery has the potential to add to our knowledge of their population and characteristics . to generate our candidate list , we selected all sources from the macc with @xmath15 0.1 , which resulted in a total of 472 candidates . the selection criterion was motivated by the cv experiment , which indicates that our candidate list should have an efficiency @xmath1680% . to obtain an efficiency close to 1 would require visual examination of roughly 3000 sources . since the expected purity of our sample is small by design , we examine the light curves of all sources within our candidate list by eye to remove sources that are clearly not rcb stars . these interlopers are typically semi - regular pulsating variables or mira variables , often with minimum brightness levels below the detection threshold . we use the allstars web interface @xcite to examine candidates , which in addition to light curves provides summary statistics ( period , amplitude , color , etc . ) for each source , as well as links to external resources , such as simbad . we also remove any sources from the candidate list that are spectroscopically confirmed as non - carbon stars . following the removal of these stars the candidate list was culled from 472 to 15 candidates we considered likely rcb stars , for which we obtained spectroscopic follow - up observations . the general properties of the 15 spectroscopically observed candidates , including their names , coordinates , and rf probabilities , are summarized in table [ tab : coords ] . finding charts using images from the digitized sky survey ( dss ) for the spectroscopically confirmed rcb and dyper candidates can be found in figure [ fig : rcbfinders ] . six of the selected candidates for spectroscopic observations are known carbon stars listed in the general catalog of galactic carbon stars ( cgcs ; @xcite ; see table [ tab : coords ] ) . lllrrccrr asas 060105 + 1654.7 & v339 ori & 220556 & 06 01 04.65 & + 16 54 40.8 & ... & 0.466 & 25 & n + asas 065113 + 0222.1 & c * 596 & 223100 & 06 51 13.31 & + 02 22 08.6 & 1429 & 0.302 & 73 & d + asas 073456@xmath112250.1 & v455 pup & 225801 & 07 34 56.24 & @xmath1122 50 04.2 & 1782 & 0.123 & 283 & n + asas 095221@xmath114329.7 & iras 09503@xmath114315 & 232170 & 09 52 21.37 & @xmath1143 29 40.5 & ... & 0.617 & 10 & n + asas 153214@xmath112854.4 & bx lib & 242289 & 15 32 13.48 & @xmath1128 54 21.6 & ... & 0.367 & 43 & n + asas 162229@xmath114835.7 & io nor & 244409 & 16 22 28.84 & @xmath1148 35 55.8 & ... & 0.950 & 1 & r + asas 162232@xmath115349.2 & c * 2322 & 244411 & 16 22 32.08 & @xmath1153 49 15.6 & 3685 & 0.391 & 36 & d + asas 165444@xmath114925.9 & c * 2377 & 245841 & 16 54 43.60 & @xmath1149 25 55.0 & 3744 & 0.490 & 22 & r + asas 170541@xmath112650.1 & gv oph & 246478 & 17 05 41.25 & @xmath1126 50 03.4 & ... & 0.702 & 8 & r + asas 180823@xmath114439.8 & v496 cra & 251092 & 18 08 23.05 & @xmath1144 39 46.7 & ... & 0.110 & 389 & n + asas 182658 + 0109.0 & c * 2586 & 252675 & 18 26 57.64 & + 01 09 03.1 & 4013 & 0.115 & 343 & d + asas 185817@xmath113543.8 & iras 18549@xmath113547 & 255280 & 18 58 17.19 & @xmath1135 43 44.7 & ... & 0.127 & 251 & n + asas 191909@xmath111554.4 & v1942 sgr & 256869 & 19 19 09.60 & @xmath1115 54 30.1 & 4229 & 0.543 & 17 & d + asas 194245@xmath112137.0 & ... & 258411 & 19 42 45.05 & @xmath1121 36 59.8 & ... & 0.112 & 376 & n + asas 203005@xmath116208.0 & nsv 13098 & 261023 & 20 30 04.96 & @xmath1162 07 59.2 & ... & 0.340 & 52 & r + [ tab : coords ] with north up and east to the left . the circles show the location of the targets and have @xmath17 which is the typical fwhm for asas images @xcite . the large pixels on the asas camera result in psfs that include the light from several stars , meaning that some asas light curves underestimate the true variability of the brightest star within the psf.,width=302 ] rf classifiers can provide quantitative feedback about the relative importance of each feature used for classification . the rf feature importance measure describes the decrease in the overall classifier performance following the replacement of a single feature with a random permutation of its values ( see @xcite for further details ) . we measure the importance of each feature using the average importance from a one - versus - one classifier whereby the rcb class is iteratively classified against each of the 27 other science classes on an individual basis . this procedure is run five times and the average of all runs is taken to reduce the variance present in any single run . unsurprisingly , we find that ` mplitude is the most import`nt feature . the importance measure does not properly capture the covariance between features and as a result the majority of the important features have to do with amplitude . the second and third most important features that are not highly covariant with amplitude are ` qso_log_chi2nunull_chi2nu ` , statistic , which is defined in @xcite . ] a measure of the dissimilarity between the photometric variations of the source and a typical quasar , and ` req1_harmonics_`req_0 , the best fit period . interestingly , ` req_signi ` , the significance of the best fit period of the light curve , ranks as only the 31st most important out of the 71 features . we summarize the results of these findings with two - dimensional cuts through the multi - dimensional feature space showing amplitude versus period significance , @xmath18 , and period in figure [ fig : hardcuts ] . we also show amplitude versus @xmath19 . in each panel we show the location of the rcb stars in the training set as well as the newly discovered rcbs and dypers presented in this paper , and we use the @xmath19 values from the cv experiment from 2.2 for the rcb stars in the training set . we also show the location of cuts necessary to achieve @xmath280% efficiency ( blue dashed line ) when selecting candidates using only two features , as well as the cuts necessary to achieve @xmath2100% efficiency ( red dashed line ) . as would be expected based on the results presented above , it is clear that @xmath18 and period are far more discriminating than period significance when selecting rcb candidates . to achieve an efficiency near 100% , @xmath19 is vastly superior to any two dimensional slice through feature space . we note that the discretization seen in the distribution of @xmath19 is the result of using a finite number of trials within the rf classifier . the probability of belonging to a class is defined as the total number of times a source is classified within that class divided by the total number of trials . these discrete values are then smeared following the calibration procedure described in 2.1 . many of the known and new rcb stars have very similar measured best periods clustered near @xmath22400 and 5300 days , which for each corresponds to the largest period searched during the lomb - scargle analysis in @xcite . folding these light curves on the adopted periods clearly shows that they are not periodic on the adopted periods , despite the relatively high period significance scores ( see the upper left panel of figure [ fig : hardcuts ] ) , which suggests some peculiarity in the feature generation process for these sources . we are exploring improved metrics for periodicity to be used in future catalogs . nevertheless , despite these spurious period measurements , the ml classifier has correctly identified that this feature tends to be erroneous for rcb stars , and as such it is a powerful discriminant for finding new examples of the class . all optical photometric observations were obtained during asas-3 , which was an extension of asas , conducted at the las campanas observatory ( for further details on asas and asas-3 see @xcite ) . light curves were downloaded from the acvs , and imported into our dotastro.org ( http://dotastro.org ) astronomical light - curve warehouse for visualization and used with internal frameworks @xcite . the acvs provides @xmath20-band measurements for a set of 50,124 pre - selected asas variables , measured in five different apertures of varying size @xcite . for each star in the catalog an optimal aperture selection procedure is used to determine the final light curve , as described in @xcite . the asas-3 @xmath20-band light curves for the eight new rcb stars and dypers are shown in figure [ fig : rcblc ] . optical spectra of the candidate rcb stars were obtained between 2011 sep . and 2012 may with the kast spectrograph on the lick 3-m shane telescope on mt . hamilton , california @xcite , the low - resolution imaging spectrometer ( lris ) on the 10-m keck i telescope on mauna kea @xcite , and the rc spectrograph on the smarts 1.5-m telescope at the cerro tololo inter - american observatory @xcite . all spectra were obtained via long slit observations , and the data were reduced and calibrated using standard procedures ( e.g. , @xcite ) . on each night of observations , we obtained spectra of spectrophotometric standards to provide relative flux calibration for our targets . for queue - scheduled observations on the rc spectrograph , all observations in a single night are conducted with the slit at the same position angle . thus , the standard stars and targets were not all observed at the parallactic angle , leading to an uncertain flux correction @xcite . we note , however , that the uncertainty in the flux correction does not alter any of the conclusions discussed below . a summary of our observations is given in table [ tab : spec ] , while the blue portion of the optical spectra are shown in figures [ fig : rcbbluespec][fig : dyperbluespec ] . lccrrr asas 060105 + 1654.7 & 2011 - 08 - 28.644 & lris & 33005630 & 4 & 60 + asas 060105 + 1654.7 & 2011 - 08 - 28.644 & lris & 58107420 & 2 & 60 + asas 170541@xmath112650.1 & 2011 - 09 - 26.206 & lris & 33005630 & 4 & 60 + asas 170541@xmath112650.1 & 2011 - 09 - 26.205 & lris & 57207360 & 2 & 30 + asas 191909@xmath111554.4 & 2011 - 09 - 26.212 & lris & 33005630 & 4 & 5 + asas 191909@xmath111554.4 & 2011 - 09 - 26.212 & lris & 57207360 & 2 & 2 + asas 162232@xmath115349.2 & 2012 - 01 - 16.375 & rc & 33009370 & 17 & 720 + asas 095221@xmath114329.7 & 2012 - 01 - 19.151 & rc & 33009370 & 17 & 540 + asas 065113 + 0222.1 & 2012 - 02 - 01.204 & kast & 34509850 & 4 & 300 + asas 162229@xmath114835.7 & 2012 - 02 - 06.373 & rc & 33009370 & 17 & 540 + asas 162232@xmath115349.2 & 2012 - 02 - 13.325 & rc & 36605440 & 4 & 2700 + asas 162229@xmath114835.7 & 2012 - 02 - 15.348 & rc & 36605440 & 4 & 2700 + dy per & 2012 - 02 - 23.107 & kast & 34509850 & 4 & 380 + asas 073456@xmath112250.1 & 2012 - 03 - 15.225 & lris & 335010000 & 4 & 60 + asas 165444@xmath114925.9 & 2012 - 03 - 15.651 & lris & 325010000 & 4 & 30 + asas 182658 + 0109.0 & 2012 - 03 - 15.654 & lris & 325010000 & 4 & 30 + asas 185817@xmath113543.8 & 2012 - 03 - 15.667 & lris & 325010000 & 4 & 30 + asas 153214@xmath112854.4 & 2012 - 03 - 15.670 & lris & 325010000 & 4 & 60 + asas 203005@xmath116208.0 & 2012 - 03 - 23.426 & rc & 36605440 & 4 & 2160 + nsv 11154 & 2012 - 04 - 02.524 & kast & 34509850 & 4 & 210 + asas 194245@xmath112137.0 & 2012 - 04 - 23.513 & kast & 34509850 & 4 & 120 + asas 180823@xmath114439.8 & 2012 - 05 - 17.626 & lris & 542010000 & 4 & 16 + [ tab : spec ] while the unique photometric behavior of rcb stars makes them readily identifiable in well sampled light curves taken over the course of several years , there are several examples of high - amplitude variables being classified as rcb stars which are later refuted by spectroscopic observations . most of the misidentified candidates are either cataclysmic , symbiotic or semi - regular variables ( see e.g. , @xcite ) . rcb stars are a subclass of the hdc stars . for an rcb candidate to be confirmed as a true member of the class , its spectrum must show the two prominent features of hdc stars : anomalously strong carbon absorption and a lack of atomic and molecular h features . to confirm the rcb candidates found in the acvs , we obtained low - resolution spectra of the 15 candidates presented in [ sec : cands ] . candidates observable from the northern hemisphere were observed with kast and lris , while those only accessible from the southern hemisphere were observed with the rc spectrograph . for some of the southern hemisphere targets very low resolution spectra were obtained first to confirm the presence of c@xmath21 before slightly higher resolution observations were obtained ( see table [ tab : spec ] ) . we searched the spectra for the presence of strong carbon features , primarily c@xmath21 and cn , and a lack of balmer absorption to confirm the rcb classification for the acvs candidates . we find these characteristics in eight of the spectroscopically observed stars ( see figures [ fig : rcbbluespec][fig : dyperbluespec ] ) , which we consider good rcb and dyper candidates as summarized in table [ tab : observations ] . the remaining candidates were rejected as possible rcb stars based on their spectra , which typically showed strong tio and vo absorption or clear evidence for h. the properties of the rejected candidates are summarized in table [ tab : rejects ] . in the remainder of this paper we no longer consider these stars candidates and restrict our discussion to the eight good candidates listed in table [ tab : observations ] . in addition to the hallmark traits of overabundant carbon and a lack of balmer absorption , rcb stars show a number of other unique spectroscopic characteristics . in particular , they show a very high ratio of @xmath22c/@xmath5c and no evidence for @xmath23 band absorption . to search for the presence of @xmath5c , we examined the spectra for the @xmath244744 band head of @xmath22c@xmath5c , which is typically very weak or absent in the spectra of rcb stars . we find evidence for @xmath22c@xmath5c in asas 191909@xmath111554.4 , asas 162232@xmath115349.2 , asas 065113 + 0222.1 , and asas 182658 + 0109.0 while asas 162232@xmath115349.2 shows possible evidence for the @xmath5c@xmath5c band at @xmath244752 . the presence of @xmath5c suggests that these four stars are likely dypers . we consider these four stars closer analogs to dy per and the dypers found in the lmc and smc @xcite than they are to classical rcb stars . one of the dypers , asas 065113 + 0222.1 , shows weak evidence for ch @xmath244300 ( @xmath23 band ) absorption and possible evidence for h@xmath25 , which is sometimes seen in the spectra of dypers . we note that the signal - to - noise ratio ( s / n ) of all our spectra in the range between @xmath243004350 is relatively low , making definitive statements about the presence or lack of both ch and h@xmath25 challenging . finally , we note that we see evidence for the merrill - sanford bands of sic@xmath21 in three of our candidates : asas 162232@xmath115349.2 , asas 065113 + 0222.1 , and asas 182658 + 0109.0 . to our knowledge this is the first identification of sic@xmath21 in a dyper spectrum , though the presence of this molecule should not come as a surprise as rcb stars are both c and si rich @xcite . lrrrrcccl asas 170541@xmath112650.1 & 11.9 & 1.2 & 31 & 0.04 & weak 4744 ? & none & n & rcb + asas 162229@xmath114835.7 & 10.8 & 2.8 & 83 & 0.03 & none & weak h@xmath25 ? , weak ch & y & rcb + asas 165444@xmath114925.9 & 11.8 & @xmath261.6 & @xmath2648 & 0.03 & none & weak h@xmath25 ? , h@xmath27 ? & y & rcb + asas 203005@xmath116208.0 & 13.2 & @xmath261.4 & @xmath2630 & 0.05 & none & h@xmath25 ? , h@xmath27 ? blends & y & rcb + asas 191909@xmath111554.4 & 6.9 & 1.0 & 20 & 0.05 & y & none & y & dyper + asas 162232@xmath115349.2 & 11.5 & 1.7 & @xmath28256 & @xmath260.007 & y & none & y & dyper + asas 065113 + 0222.1 & 12.4 & 1.0 & @xmath28140 & @xmath260.007 & y & weak h@xmath25 ? , weak ch & y & dyper + asas 182658 + 0109.0 & 12.1 & 1.6 & 960 & 0.002 & y & none & y & dyper + [ tab : observations ] lrl asas 060105 + 1654.7 & 12.3 & no c@xmath21 ; strong h , @xmath23 band + asas 073456@xmath112250.1 & 12.8 & c@xmath21 ; strong h emission + asas 095221@xmath114329.7 & 10.6 & strong tio , vo ; h ? + asas 153214@xmath112854.4 & 12.3 & no c@xmath21 ; h@xmath29 emission , strong @xmath23 band ; srpv + asas 180823@xmath114439.8 & 12.1 & strong tio , vo + asas 185817@xmath113543.8 & 10.9 & strong tio , vo + asas 194245@xmath112137.0 & 12.5 & strong tio , vo + [ tab : rejects ] in addition to spectral differences , rcb stars and dypers show some dissimilarities in their photometric evolution as well . the first order behavior is the same : both show deep , irregular declines in their light curves which can take anywhere from a few months to a several years to recover to maximum brightness . beyond that generic behavior , however , the shape of the decline tends to differ : rcb stars show fast declines with slow recoveries whereas dypers tend to show a more symmetric decline and recovery . the photometric properties of our candidates , including decline rates for the most prominent and well sampled declines , are summarized in table [ tab : observations ] . as previously noted in the caption of figure [ fig : rcbfinders ] , the full amplitude of the variations of these stars are likely underestimated due to the large psf on asas images . this means that the decline rates should be treated as lower limits , since the true brightness of the star may be below that measured in a large aperture . nevertheless , the decline rates for the four rcb stars are relatively fast and consistent with those given in @xcite for rcb stars in the mcs , @xmath20.04 mag day@xmath30 . the most telling feature of the light curves is the shape of the declines , however . for the four spectroscopic rcb stars , asas 170541@xmath112650.1 , asas 162229@xmath114835.7 , asas 165444@xmath114925.9 , and asas 203005@xmath116208.0 the declines are very rapid . while we do not detect asas 165444@xmath114925.9 after its sharp decline on tjd@xmath315000 , the other three show slow asymmetric recoveries to maximum light . the four spectroscopic dypers generally show a slower decline with a roughly symmetric recovery , though we note that the full recoveries of asas 065113 + 0222.1 and asas 162232@xmath115349.2 are not observed . all rcb stars are variable near maximum light , with most and possibly all of the variations thought to be due to pulsation @xcite . typical periods are @xmath240100 days , and the amplitudes are a few tenths of a magnitude . the pulsational properties of dypers are not as well constrained , because the sample is both small and only recently identified . each of the four dypers identified in @xcite shows evidence for periodic variability near maximum light , with typical periods of @xmath2100200 days . to search for the presence of pulsations in our candidate rcb stars , we use a generalized lomb scargle periodogram ( @xcite ) to analyze each star ( see @xcite for more details on our lomb scargle periodogram implementation ) . our analysis only examines data that are well separated from decline phases , and we focus on the portions of light curves where the secular trend is slowly changing relative to the periodic variability . for each star we simultaneously fit for the harmonic plus linear or quadratic long - term trend in the data . the frequency that produces the largest peak in the periodogram , after masking out the 1 day alias , is adopted as the best - fit period . we find evidence for periodicity in the light curves of each star , except for asas 162229@xmath114835.7 . for most of the observing window asas 162229@xmath114835.7 was in or near a decline , and we predict that additional observations of asas 162229@xmath114835.7 , be they historical or in the future , will show periodic variability near maximum light . the trend - removed , phase - folded light curves of the remaining seven stars are shown in figure [ fig : rcbpulse ] . insets in each panel list the range of dates that were included in the lomb - scargle analysis , as well as the best fit period for the data . some rcb stars are known to have more than a single dominant period ( see @xcite and references therein ) . we find evidence for multiple periods in asas 191909@xmath111554.4 , with periods of 120 , 175 , and 221 days that appear to change every @xmath212 years . evidence for multiple periods also appears to be present in asas 165444@xmath114925.9 . the best fit periods in this case are 27 and 56 days , which differ by roughly a factor of two . the longer period may in this case simply be a harmonic of the shorter period . finally , we note that the best fit period for asas 162232@xmath115349.2 , 359 days , is very close to one year , and it is possible that the data are beating against the yearly observation cycle . the folded light curve appears to traverse a full cycle over @xmath2half the full phase cycle . the slight upturn in the folded data around phase 0.15 suggests that the true period is likely @xmath2180 days , half the best fit period . 4925.9 and asas 191909@xmath111554.4 show evidence for multiple dominant periods , which are shown with blue and red points as indicated in their respective legends.,width=340 ] all rcb stars are known to have an infrared ( ir ) excess due to the presence of circumstellar dust @xcite , and all the known dypers in the mcs also show evidence for excess ir emission @xcite . to check for a similar excess in the new acvs rcb stars and dypers , broadband spectral energy distributions ( seds ) were constructed with catalog data obtained from usno - b1 @xcite , the two micron all sky survey ( 2mass ; @xcite ) , the _ wide - field infrared survey explorer _ ( _ wise _ , @xcite ) , _ akari _ @xcite , the _ mid - course space experiment _ ( _ msx _ ; @xcite ) , and the _ infrared astronomical satellite _ ( _ iras _ ; @xcite ) . the usno - b1 catalog contains measurements made on digital scans of photographic plates corresponding roughly to the @xmath32 , @xmath33 , and @xmath34 bands . repeated @xmath32 and @xmath33 plates were taken typically more than a decade apart . to convert the five separate usno - b1 magnitude measurements to the standard @xmath35 system of the sloan digital sky survey ( sdss ; @xcite ) , we invert the filter transformations from ( @xcite ; see also @xcite ) . the two measurements each for the @xmath36 and the @xmath37 band are then averaged to get the reported sdss @xmath36 and @xmath37 magnitudes , unless the two measurements differ by @xmath26 1 mag , in which case it is assumed that the fainter observation occurred during a fading episode of the star . then the final adopted sdss magnitude is that of the brighter measurement . there is a large scatter in the transformations from usno - b1 to sdss @xcite , which , leads us to adopt a conservative 1-@xmath38 uncertainty of 40% in flux density on each of the transformed sdss flux measurements . the 2mass magnitude measurements are converted to fluxes via the calibration of @xcite and the _ wise _ magnitudes are converted to fluxes via the calibration in @xcite . the remaining catalogs provide flux density measurements in jy rather than using the vega magnitude system . the full seds extending from the optical to the mid - infrared for each of the new rcb stars and dypers are shown in figure [ fig : rcbsed ] . all of the candidates but asas 203005@xmath116208.0 saturate the w1 ( 3.5 @xmath39 m ) and w2 ( 4.6 @xmath39 m ) bands of _ wise _ , while of those all but asas 170541@xmath112650.1 and asas 165444@xmath114925.9 saturate w3 ( 11.6 @xmath39 m ) as well . asas 191909@xmath111554.4 and asas 182658 + 0109.0 saturate all three of the 2mass filters and are the only candidates detected at either 60 and/or 100 @xmath39 m by _ iras_. asas 162229@xmath114835.7 and asas 191909@xmath111554.4 were the only candidates detected at 90 @xmath39 m by _ akari_. using rcb stars and dypers in the mcs , @xcite find that the rcb stars typically have seds with two distinct peaks , whereas dypers typically have a single peak . it is argued in @xcite that the seds of both can be understood as emission from a stellar photosphere and surrounding dust shell ; the cooler photospheric temperatures of dypers are less distinct relative to the dust emission leading to a single broad peak rather than two . we caution that the reddening toward each of the new galactic candidates is unknown , which makes a detailed analysis of their seds challenging . furthermore , the observations were not taken simultaneously in each of the various bandpasses . nevertheless , a few interesting trends can be gleaned from the data . the four stars that spectroscopically resemble rcb stars , asas 170541@xmath112650.1 , asas 162229@xmath114835.7 , asas 165444@xmath114925.9 , and asas 203005@xmath116208.0 all show clear evidence for a mid - infrared excess relative to their optical brightness . the peak of emission from asas 191909@xmath111554.4 and asas 182658 + 0109.0 is not well constrained because they saturate the detectors between 1 and 6 @xmath39 m , yet interestingly both show evidence for an infrared excess redward of 50 @xmath39 m . this suggests that there might be some very cool ( @xmath40 100 k ) dust in the circumstellar environment of these stars , which is observed in some of the bright , nearby rcb stars . for instance , _ spitzer _ and _ herschel _ observations of r crb show evidence for a large , @xmath24 pc , cool , @xmath41 k , and diffuse shell of gas that is detected in the far - ir @xcite . asas 162232@xmath115349.2 and asas 065113 + 0222.1 show evidence for a single broad peak in their sed occurring around @xmath22 @xmath39 m , which is similar to the seds of the dypers observed in the mcs . an overlap in the survey fields between 2mass and the deep near infrared souther sky survey ( denis ; @xcite ) allows measurements of the nir variability of four of the newly discovered rcb stars and dypers . photometric measurements from 2mass and the two epochs of denis observations for these stars are summarized in table [ tab : ir_var ] . unfortunately the 2mass and denis observations proceeded the asas monitoring , and so we can not provide contextual information such as the state of the star ( near maximum , on decline , during deep minimum , etc . ) at the time of the nir observations . to within a few tenths of a magnitude , asas 162229@xmath114835.7 is not variable between the 2mass and denis observations . on timescales of a few weeks to months both asas 162232@xmath115349.2 and asas 170541@xmath112650.1 show evidence for variations @xmath16 1 mag in the nir . similar variations have been observed for several of the rcb stars and dypers in the mcs ( e.g. , @xcite ) . the largest variations were observed in asas 203005@xmath116208.0 , which changed by @xmath24 mag in the @xmath42 band during the @xmath24 yr between the denis and 2mass observations . asas 203005@xmath116208.0 also shows a large variation between the denis @xmath34 band measurement and the @xmath34-band measurement from usno - b1 , with @xmath43 mag . this star is clearly a large - amplitude variable , which likely explains its unusual sed . in the denis observations , which provide simultaneous optical and nir measurements , asas 203005@xmath116208.0 is always fainter in the optical , suggesting that the unusual shape to its sed ( see figure [ fig : rcbsed ] ) is the result of non - coeval observations . lrrrrrrrr asas 170541@xmath112650.1 & 2451004.658 & 10.01 & 9.28 & 8.46 & 2451730.639 & 11.09 & 10.18 & 9.13 + & & & & & 2451749.592 & 9.94 & 8.91 & 7.96 + asas 162229@xmath114835.7 & 2451347.541 & 7.26 & 6.47 & 5.73 & 2451387.540 & 9.10 & 6.90 & 5.40 + & & & & & 2451395.492 & 8.98 & 7.13 & 5.45 + asas 162232@xmath115349.2 & 2451347.538 & 7.24 & 6.01 & 5.36 & 2451387.532 & 9.12 & 6.59 & 4.40 + & & & & & 2451395.485 & 9.20 & 6.30 & 4.28 + asas 203005@xmath116208.0 & 2451701.878 & 11.73 & 11.20 & 10.40 & 2450267.768 & 17.62 & 15.66 & 12.06 + & & & & & 2451003.746 & 15.82 & 14.37 & 11.89 + [ tab : ir_var ] we discuss the individual stars and whether they should be considered rcb stars or dypers below . we also use simbad to identify alternate names for these stars and previous studies in the literature ( see also table [ tab : coords ] ) . this star was first identified as a variable source on harvard photographic plates with the name harvard variable 4368 , and was cataloged as a likely long period variable based on the large amplitude of variations from 13.9 mag to below the photographic limit of @xmath216.5 mag @xcite . it was later named gv oph in the general catalog of variable stars ( gcvs ; @xcite ) as a variable of unknown type with rapid variations . the light curve , spectrum , and sed of this star are consistent with it being an rcb star . this star is listed as a mira variable in the gcvs with the name io nor . in @xcite it is identified as a star with an ir excess based on _ msx _ observations . on the basis of its nir and mid - ir colors , it is identified as a rcb candidate in @xcite , and considered a likely rcb star on the basis of its asas light curve .. it has @xmath44 and @xmath45 . the light curve is somewhat similar to es aql , in that it fades below the asas detection limits and it is highly active during the @xmath318 yr it was observed , meaning it that folds decently well on a period of @xmath2450 days . a spectrum will be needed to disambiguate between an rcb and long period variable classification for v653 sco . ] we independently identified io nor as a likely rcb star on the basis of its light curve ( in the macc it is the most likely rcb in acvs ) , and our spectrum confirms that it is a genuine rcb star . the previous classification as a mira variable is likely based on the late spectral type and large amplitude of variability , but figure [ fig : rcblc ] clearly shows that asas 162229@xmath114835.7 is not a long period variable . the variability of this star has not been cataloged to date , and it is listed in the cgcs as c * 2377 @xcite . the spectrum , sed , and pulsations exhibited by this star are consistent with rcb stars . there may be evidence for weak ch absorption , though we caution that the s / n is low near @xmath24300 . the light curve shows a sharp decline , similar to rcb stars , but the recovery is not observed . nevertheless , the evidence points to asas 165444@xmath114925.9 being a new member of the rcb class . this star was first identified as variable by @xcite with a maximum brightness of 14 mag and a minimum @xmath26 18 mag . @xcite assigned it the name an 141.1932 , and it was later cataloged as a possible variable star in the gcvs as nsv 13098 . the light curve , spectrum , and sed are all consistent with an rcb classification . higher resolution and s / n spectra are needed to confirm if h absorption is present , though we note that some rcb stars do show evidence of h in their spectra ( e.g. , v854 cen ; @xcite ) , leading us to conclude that asas 203005@xmath116208.0 is an rcb star . this star is listed as a slow irregular variable of late spectral type with the name v1942 sgr in the gcvs . it is the brightest star among our candidates , and as such it is one of the best studied carbon stars to date . according to simbad asas 191909@xmath111554.4 is discussed in more than 50 papers in the literature . in the cgcs it is listed as c * 2721 . asas 191909@xmath111554.4 is detected by _ ( hip 94940 ) and has a measured parallax of 2.52@xmath460.82 mas @xcite . this corresponds to a distance @xmath47 pc and a distance modulus @xmath48 8 mag . asas 191909@xmath111554.4 is one of the few galactic carbon stars with a measured parallax , and it is important for constraining the luminosity function of carbon stars @xcite . in their spectral atlas of carbon stars , @xcite identify asas 191909@xmath111554.4 as having a spectral type of n5 + c@xmath215.5 . relative abundance measurements by @xcite show that @xmath22c/@xmath5c@xmath49 , which is low relative to classical rcb stars . the proximity of asas 191909@xmath111554.4 allows its circumstellar dust shell to be resolved in _ iras _ images ( e.g. , @xcite ) , and @xcite use asas 191909@xmath111554.4 and other carbon stars with resolved dust shells and 100 @xmath39 m excess to statistically argue that each of these stars must be surrounded by two dust shells , one that is old , @xmath210@xmath50 yr , and the other that is produced by a current episode of mass loss . recent observations by @xcite have shown evidence for the presence of h in the circumstellar shell of asas 191909@xmath111554.4 . the shallow , symmetric fade of the light curve , along with the n type carbon star spectrum and the presence of @xmath5c in the spectrum , leads us to conclude that asas 191909@xmath111554.4 is a dyper . this is the only candidate within our sample for which we can measure the absolute magnitude , since we have the _ hipparcos _ parallax measurement . adopting a maximum light brightness of @xmath51 6.8 mag , we find that asas 191909@xmath111554.4 has @xmath52 mag . this is roughly 0.4 mag fainter than the faintest dypers in the mcs @xcite , suggesting that either the luminosity function extends fainter than that observed in the mcs or there is unaccounted for dust extinction toward asas 191909@xmath111554.4 . the variability of this star has not been cataloged to date , and it is listed in the cgcs as c * 2322 . the relatively slow , symmetric decline and recovery in the light curve of asas 162232@xmath115349.2 lead us to classify it as a dyper variable . the presence of @xmath5c in the spectrum and the single peak in the sed support this classification . the variability of this star has not been cataloged to date , and it is listed in the cgcs as c * 596 . the presence of @xmath5c in the spectrum and the single peak in the sed lead us to classify asas 065113 + 0222.1 as a dyper variable . the variability of this star has not been cataloged to date , and it is listed in the cgcs as c * 2586 . based on weekly averages of _ dirbe _ nir observations taken over 3.6 yr , @xcite list asas 182658 + 0109.0 as a non - variable source . low resolution spectra taken with _ iras _ show 11 @xmath39 m sic dust emission , which typically indicates significant mass loss from a carbon star @xcite . the light curve shows a @xmath256 yr symmetric decline , and there is clear evidence for @xmath5c in the spectrum . the _ iras _ detection at 60 @xmath39 m shows a clear ir excess relative to a single temperature blackbody . while there is no evidence for h@xmath29 , the s / n in our spectrum is low blueward of @xmath24700 . we consider asas 182658 + 0109.0 a likely dyper , though higher s / n spectra are required for a detailed abundance analysis to confirm this classification . lllrrcccc asas 053302 + 1808.0 & iras 05301@xmath531805 & 219583 & 05 33 01.72 & 18 07 59.0 & 980 & 0.339 & 380 & 1 + asas 081121@xmath113734.9 & c * 1086 & 227950 & 08 11 21.39 & @xmath1137 34 54.2 & 2106 & 0.145 & 492 & 1 + asas 125245@xmath115441.6 & ... & 237449 & 12 52 44.92 & @xmath1154 41 37.5 & ... & 0.309 & 394 & 2 + asas 160033@xmath112726.3 & 1rxs j160033.8@xmath11272614 ? & 243486 & 16 00 33.16 & @xmath1127 26 18.5 & ... & 0.142 & 498 & 2,3 + asas 175226@xmath113411.5 & iras 17491@xmath113410 & 249729 & 17 52 25.50 & @xmath1134 11 28.2 & ... & 0.166 & 460 & + asas 200531 + 0427.2 & v902 aql & 259768 & 20 05 30.83 & 04 27 12.8 & ... & 0.382 & 353 & 2,4 + [ tab : new_cands ] as mentioned in [ sec : ml ] , one of the major strengths of ml classification is that new discoveries may be fed back into the machinery in order to improve future iterations of the classifier . in an attempt to recover more rcb stars and dypers in the acvs that were missed in our initial search of the macc , we created an augmented rcb training set by adding the eight new rcb stars and dypers identified in this paper to the 17 sources already included in the training set . this augmented training set should increase the likelihood of discovering new candidates , particularly dypers , of which there were no examples in the original training set . using the augmented training set we re - ran the rf classifier from @xcite on all the acvs light curves to search for any additional good candidates . we focus our new search on candidates with a significant change in @xmath13 , which were not examined in the initial search of the macc . in particular , we visually examine the light curves of all sources with @xmath54 , @xmath55 , and @xmath56 . there are a total of 96 sources that meet these criteria , which were not included in the 472 visually inspected sources from the original macc . of these 96 , we conservatively select six as candidate rcb stars or dypers . one is a highly likely rcb star with multiple declines and asymmetric recoveries , three show evidence for a single decline which is only partially sampled , and two are known carbon stars that are likely semi - regular periodic variables . we list the candidates in table [ tab : new_cands ] with brief comments on each and show their light curves in figure [ fig : aug_lc ] . restricting our search for bright rcb stars to only those sources in the acvs has biased the results of our search . as was mentioned in [ sec : training_set ] , there are seven known rcb stars that show clear variability in their asas light curves yet were not selected for inclusion in the acvs . this suggests that several large - amplitude asas variables are missing from the acvs , presumably including a few unknown rcb stars . this bias can easily be corrected by searching all of asas for rcb stars , however , such a search would include significant new challenges as there are @xmath212 million sources in asas . in addition , our existing classification framework is not designed to deal with a catalog where the overwhelming majority of sources are not in fact variable . nevertheless , both of these challenges must be addressed prior to the lsst era . we have developed frameworks that can ingest millions of light curves and are currently experimenting with methods to deal with non - variable sources , the results of which will be presented in a catalog with classifications for all @xmath212 million sources in asas ( richards et al . , in prep ) . furthermore , it has been shown that the use of mid - infrared colors is a powerful discriminant when trying to select rcb stars @xcite . while our use of nir colors is important for selecting rcb stars ( see e.g. , @xcite ) , adding the mid - infrared measurements from the all - sky _ wise _ survey will dramatically improve our purity when selecting rcb candidates as several of the mira and semi - regular variables that served as interlopers in the current search ( [ sec : macc_cands ] ) would be eliminated with the use of mid - infrared colors . we have used the 71-feature random forest machine - learning acvs classification catalog from @xcite to identify likely dypers and rcb stars in the acvs catalog . the rf classifier provides several advantages over previous methods to search time - domain survey data for rcb stars and dypers . previously successful searches for rcb stars have developed a methodology focused on large amplitude variables that do not show strong evidence for periodicity ( e.g. , @xcite ) . while the rf classifier is capable of capturing the large variations and irregular declines observed in rcb stars , the use of many features allows complex behavior , such as the shape of the decline and recovery , to be captured as well . another advantage of rf classification is that it does not require hard cuts on any individual light curve feature , which can exclude real rcb stars from the final candidate selection . there are a total of 472 stars with @xmath57 0.1 in version 2.3 of the macc , 15 of which were selected as good rcb or dyper candidates after visual inspection and existing spectroscopic information . following spectroscopic observations eight of the good candidates were identified as bona fide rcb stars or dypers . four of these , asas 170541@xmath112650.1 ( gv oph ) , asas 162229@xmath114835.7 ( io nor ) , asas 165444@xmath114925.9 , and asas 203005@xmath116208.0 were confirmed as new rcb stars on the basis of ( i ) their light curves showing irregular , sharp declines of large amplitude ( @xmath1 @xmath58 1 mag ) , ( ii ) carbon rich spectra showing a lack of evidence for h and @xmath5c , and ( iii ) the mid - infrared excess observed in their seds . four of the candidates , asas 191909@xmath111554.4 ( v1942 sgr ) , asas 065113 + 0222.1 , asas 162232@xmath115349.2 , and asas 182658 + 0109.0 appear to be galactic dypers on the basis of ( i ) shallow , symmetric declines in their light curves occurring at irregular intervals , ( ii ) carbon rich spectra resembling carbon n stars showing @xmath5c and weak or no h , and ( iii ) seds that show a single peak , but which are too broad to be explained via a single temperature blackbody . with the exception of asas 170541@xmath112650.1 , all of the new candidates show evidence for periodic variability near maximum light . we incorporate the newly confirmed rcb stars and dypers into the training set to identify six new candidates as likely rcb stars . our effort has increased the number of known galactic dypers from two to six . while the sample size is small , it appears that dypers have pulsations with period @xmath59 days at maximum light , which is longer than the typical timescale for pulsation in rcb stars ( see also @xcite ) . each of the new rcb stars and dypers is bright , @xmath60 12 mag , which will enable high - resolution spectroscopy for future studies of the detailed abundances of these stars . this is particularly important in the case of the four new dypers , as dy per itself is the only member of the class which has been observed at high resolution to confirm the lack of h absorption in the spectrum @xcite . if these stars are shown to be h deficient , it would be strong evidence that dypers are the cool ( @xmath33500 k ) analogs to rcb stars . we view the results presented herein as one culmination of a broader effort to extract novel science from the time - domain survey data deluge . earlier work focused on determining the most suitable ml frameworks for classification and subsequent classification efficiency ( see @xcite for review ) . while production of ml - based catalogs ( e.g. , acvs ; @xcite ) have been the norm for over a decade , we know of no concerted effort to validate the predictions of those catalogs . now having a probabilistic catalog of variable sources @xcite to work with , we can select our demographic priors on classes of interest and also decide just how many false - positives we are willing to tolerate in the name of improved efficiency . in the case of the construction of a new set of very common stars ( e.g. , rr lyrae catalog ) , we might be willing to tolerate a reduced discovery efficiency to preserve a high level of purity . management of the available resources to follow - up the statements made in a probabilistic catalog becomes the next challenge . we were obviously most interested in finding new exemplars of two rare classes and thus tolerated a high impurity . in the discovery and characterization of several bright rcb stars and dypers , the payoff of the efficient use of follow - up resources enabled by probabilistic classification is evident . the classification taxonomy of variable stars clearly conflates phenomenology ( e.g. , `` periodic '' ) within a physical understanding ( `` pulsating '' ) of the origin of what is observed . and while phenomenologically based mining around an envelope of class prototypes can turn up new class members , we have shown that the diversity of rcb stars and dypers demands an expanded approach to discovery . we speculate that the richness and connections of the feature set in the ml search may be also capturing some of the phenomenological manifestations of the underlying physics , however nuanced , that we can not ( yet ) express . we thank the anonymous referee for comments that have helped to improve this paper . we thank alex filippenko for obtaining the lick telescope time , while he and peter nugent contributed to the keck proposal and assisted in the observations . we thank k. clubb , a. morgan , d. cohen and i. shivvers for assisting in the keck and lick observations . we thank fred walter for help with the scheduling and reduction of rc spectrograph data , and we thank r. hernandez , a. miranda , and m. hernandez for carrying out the rc spectrograph observations . is supported by the national science foundation ( nsf ) graduate research fellowship program . j.s.b . and j.w.r . acknowledge support of an nsf - cdi grant-0941742 . some of the work for this study was performed in the cdi - sponsored center for time domain informatics . was partially supported by nsf - aag grant-1009991 . s.b.c . acknowledges generous financial assistance from gary & cynthia bengier , the richard & rhoda goldman fund , nasa/_swift _ grants nnx10ai21 g and go-7100028 , the tabasgo foundation , and nsf grant ast-0908886 . support for k.g.s . is through the vanderbilt initiative in data - intensive astrophysics ( vida ) . some of the data presented herein were obtained at the w. m. keck observatory , which is operated as a scientific partnership among the california institute of technology , the university of california , and nasa . the observatory was made possible by the generous financial support of the w. m. keck foundation . the authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of mauna kea has always had within the indigenous hawaiian community . we are most fortunate to have the opportunity to conduct observations from this mountain . this research has made use of nasa s astrophysics data system bibliographic services , the simbad database operated at cds , strasbourg , france , the nasa / ipac extragalactic database and nasa/ ipac infrared science archive operated by the jet propulsion laboratory , california institute of technology , under contract with nasa , and the vizier database of astronomical catalogs @xcite . feature computations and classifier research and evaluations were performed using ibm - http://citris - uc.org/[citris ] s 280 core linux cluster located at uc berkeley . the digitized sky surveys were produced at the space telescope science institute under u.s . government grant nag w-2166 . the images of these surveys are based on photographic data obtained using the oschin schmidt telescope on palomar mountain and the uk schmidt telescope . the plates were processed into the present compressed digital form with the permission of these institutions . , j. m. , bloom , j. s. , kennedy , r. , & starr , d. l. 2009 , in astronomical society of the pacific conference series , vol . 411 , astronomical data analysis software and systems xviii , ed . d. a. bohlender , d. durand , & p. dowler , 357 , j. p. , bailyn , c. d. , smith , r. c. , et al . 2010 , http://dx.doi.org/10.1117/12.859145[in society of photo - optical instrumentation engineers ( spie ) conference series , vol . 7737 , society of photo - optical instrumentation engineers ( spie ) conference series ]
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we present the results of a machine - learning ( ml ) based search for new r coronae borealis ( rcb ) stars and dy persei - like stars ( dypers ) in the galaxy using cataloged light curves from the all - sky automated survey ( asas ) catalog of variable stars ( acvs ) .
rcb stars a rare class of hydrogen - deficient carbon - rich supergiants are of great interest owing to the insights they can provide on the late stages of stellar evolution .
dypers are possibly the low - temperature , low - luminosity analogs to the rcb phenomenon , though additional examples are needed to fully establish this connection . while rcb stars and dypers are traditionally identified by epochs of extreme dimming that occur without regularity , the ml search framework more fully captures the richness and diversity of their photometric behavior .
we demonstrate that our ml method can use newly discovered rcb stars to identify additional candidates within the same data set .
our search yields 15 candidates that we consider likely rcb stars / dypers : new spectroscopic observations confirm that four of these candidates are rcb stars and four are dypers .
our discovery of four new dypers increases the number of known galactic dypers from two to six ; noteworthy is that one of the new dypers has a measured parallax and is @xmath0 mag , making it the brightest known dyper to date .
future observations of these new dypers should prove instrumental in establishing the rcb connection .
we consider these results , derived from a machine - learned probabilistic classification catalog , as an important proof - of - concept for the efficient discovery of rare sources with time - domain surveys .
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the discovery of hot jupiters that transit in front of their parent stars has advanced our knowledge of extrasolar planets adding a fundamental datum : the planetary radius . there has been considerable activity revising the measured radii , owing to uncertainties in the differential image analysis ( see pont et al . 2006 ) . it is important to obtain accurate radii from photometry , in order to compare these exoplanets with the giant planets of the solar system , and with the models . in addition , if accurate photometry of transits is available , one can use timing for future studies of multiplicity in these systems ( e.g. sartoretti & schneider 1999 , miralda - escude 2002 , holman & murray 2005 , agol et al . 2005 ) . new samples of transiting hot jupiters should become available soon ( see for example fischer et al . 2005 , sahu et al . 2006 ) , but up to now the ogle search has provided the largest number of transiting candidates . in particular , udalski et al . ( 2002 ) discovered very low amplitude transits in the @xmath5 , @xmath6 magnitude star ogle - tr-111 , located in the carina region of the milky way disk , at @xmath7 , @xmath8 . they monitored 9 individual transits , measuring an amplitude @xmath9 mag , and a period @xmath10 days . the period is a near - multiple of a day , therefore , the window for transit observations is restricted to a couple of months per year . the planet ogle - tr-111-b was discovered by pont et al . ( 2004 ) with precise velocity measurements . they measured @xmath11 , @xmath12 , and @xmath13 . they call this planet the `` missing link '' because of the relatively long period , which overlaps with the planets discovered by radial velocity searches . ogle - tr-111-b is one of the least irradiated known transiting extrasolar planets , ( baraffe et al . 2005 , laughlin et al . 2005 ) , and therefore it is also an interesting case to study because it may probe the transition region between strongly irradiated and isolated planets . we have previously carried out a selection of the most promising ogle planetary candidates using low dispersion spectroscopy in combination with optical and near - infrared photometry ( gallardo et al . 2005 ) . this work identified ogle - tr-111 as one of the most likely candidates to host exoplanets . gallardo et al . ( 2005 ) classify ogle - tr-111 as a k - type main sequence star with @xmath14 k , located at a distance @xmath15 pc , with magnitudes @xmath5 , @xmath6 , and @xmath16 , and reddening @xmath17 . their low dispersion spectrum shows strong mgb band characteristic of a metal - rich dwarf . they find that this star is intrinsically fainter ( @xmath18 ) , and smaller ( @xmath19 ) than the sun . based on the high dispersion spectroscopy , pont et al . ( 2004 ) derive similar stellar parameters for ogle - tr-111 : temperature @xmath20 k , gravity @xmath21 , mass @xmath22 , radius @xmath23 , and metallicity @xmath24=0.12 $ ] dex . the stellar parameters were further improved by santos et al . ( 2006 ) , based on high s / n spectra , deriving @xmath25 , @xmath26 , and @xmath24=+0.19 \pm 0.07 $ ] , and assume @xmath27 . the values from these independent studies agree within the uncertainties . the known planetary parameters are in part based on the ogle photometry . there has been recent revisions of the radii of other confirmed ogle planets using high cadence , high s / n photometry with large telescopes ( see pont et al . recently , winn et al . ( 2006 ) presented accurate photometry of two transits for ogle - tr-111 in the @xmath28-band , revising the ephemeris , obtaining a period @xmath29 , and measuring the system parameters , including an accurate stellar radius @xmath30 , and planet radius @xmath31 . this planet radius is @xmath32 larger than the recent value of santos et al . ( 2006 ) . in this paper we present new high cadence @xmath0-band photometry covering a transit of ogle - tr-111 , giving an independent determination of the planetary radius , and deriving an accurate period for the system . the observations and photometry are described by fernndez et al . ( 2006 ) and daz et al . the photometric observations were taken with vimos at the unit telescope 4 ( ut4 ) of the european southern observatory very large telescope ( eso vlt ) at paranal observatory during the nights of april 9 to 12 , 2005 . the vimos field of view consists of four ccds , each covering 7@xmath338 arcmin , with a separation gap of 2 arcmin , and a pixel scale of 0.205 arcsec / pixel . the large field of view of this instrument allows to monitor simultaneously a number of ogle transit candidates , in comparison with fors at the vlt , which has a smaller field of view ( fernndez et al . however , for high precision photometry of an individual candidate fors should be preferred because its finer pixel scale allows better sampling ( e.g. pont et al . 2006 ) . here we report on the observations of ogle - tr-111 , which was located in one of the four monitored fields , and it happened to have a transit during the first night of our run . we used the bessell @xmath0 filter of vimos , with @xmath34 , @xmath35 . the @xmath0-band was chosen in order to complement the ogle light curves which are made with the @xmath28-band filter . in addition , the @xmath0-band is more sensitive to the effects of limb darkening during the transit , and is adequate for the modeling of the transit parameters . we have monitored two fields on april 9 , 2005 , one of which included the star ogle - tr-111 . the fields were observed alternatively with three exposures of 15s before presetting to the next field . for this program we managed to reduce the observation overheads for telescope presets , instrument setups , and the telescope active optics configuration to an absolute minimum . this ensured adequate sampling of the transit : we obtained 224 points during the first night in the field of ogle - tr-111 . the observations lasted for about 9.5 hours , until the field went below 3 airmasses . in order to reduce the analysis time of the vast dataset acquired with vimos , the images of ogle - tr-111 analyzed here are 400@xmath33400 pix , or 80 arcsec on a side . each of these small images contains about 500 stars with @xmath36 that can be used in the difference images , and light curve analysis . the 7 best seeing images ( @xmath37 ) taken near the zenith were selected , and a master image was made in order to serve as reference for the difference image analysis ( see alard 2000 , alard & lupton 1998 ) . the candidate star is not contaminated by faint neighbours , judging from our deep @xmath0 and @xmath38-band images . there is a @xmath39 mag star 2 arcsec south of our target , which does not affect our photometry . the difference image photometry yields a noisier light curve with a low amplitude transit , and we decided to apply the ogle pipeline dia ( udalski et al . this new reduction showed significantly reduced photometric scatter . for ogle - tr-111 , with a mean visual magnitude @xmath5 , we achieved a photometric accuracy of @xmath40 @xmath41 magnitudes . the errors mainly depend on the image quality , which was worse at the beginning and end of the time series , when the target had large airmass . figure 1 shows the light curve for the first night of observations , when the ogle - tr-111 transit was monitored . every single datapoint is shown , no mesurement is discarded . for comparison , figure 1 also shows the phased light curve of the ogle @xmath28-band photometry ( in a similar scale ) . the transit is well sampled in the @xmath0-band , and the scatter is smaller . there are @xmath42 points in our single transit shown in figure 1 , and the minimum is well sampled , allowing us to measure an accurate amplitude . in the case of ogle , the significance of the transits is in part judged by the number of transits detected . in the case of the present study , we compute the signal - to - noise of the single , well sampled transit . for ogle - tr-111 we find the s / n of this transit to be @xmath43 following gaudi ( 2005 ) . however , this does not include systematic effects ( red noise ) , which we consider below . after the ogle 2002 transit campaign , the field of ogle - tr-111 was observed by the ogle survey regularly but less frequently with the main aim of improving the ephemeris . altogether more than @xmath44 new epochs were collected in the observing seasons 20032005 . unfortunately , there were no eclipses observed between 2002 and april 2005 , when ogle started recovering eclipses . this is because this system has a near multiple of a day period . the full ogle photometric dataset covering almost 350 cycles made it possible to significantly refine the ephemeris of ogle - tr-111 : @xmath45 ( udalski et al . 2005 , private communication ) . the vimos transit reported here also agrees with this ephemeris , as discussed in section 5 . in light of the high dispersion follow up of this target by pont et al . ( 2004 ) that confirmed the low mass of ogle - tr-111-b , we can consider that the transit measured here is representative of all transits of ogle - tr-111-b . unfortunately , just a single transit was measured here , and only in the @xmath0-band filter , but it is of value because we used a different passband that previous works and can therefore independently check the parameters for the system . for example , the stellar limb darkening is different from the @xmath28-band , with transits that are shallower at the edges but about @xmath46 deeper in the central parts ( e.g. claret & hauschildt 2003 ) . there are a few published measurements of the radius of the ogle - tr-111 companion , based on the ogle photometric data ( table 1 ) . udalski et al . ( 2002 ) measured @xmath9 mag , and estimated @xmath47 for ogle - tr-111 , and a lower limit of @xmath48 for its companion , arguing this was one of the most promising extrasolar planetary transit candidates . based on a fit to the same ogle data , silva & cruz ( 2006 ) estimated @xmath49 for the companion . pont et al . ( 2004 ) give @xmath50 for ogle - tr-111 , and @xmath51 for its companion , also using on the ogle data . gallardo et al . ( 2005 ) measure @xmath52 for ogle - tr-111 , and @xmath53 for its companion , also based on the ogle amplitude . figure 2 shows our best fit to the transit curve , following mandel & agol ( 2002 ) , using appropriate limb - darkenning coefficients for the @xmath0-band . this fit yields @xmath54 mag , @xmath55 , and @xmath56 . using @xmath57 gives @xmath58 . the uncertainties of the fit parameters were estimated from the @xmath59 surface . as shown by pont ( 2006 ) , the existence of covariance between the observations produces a low - frequency noise which must be considered to obtain a realistic estimation of the uncertainties . to model the covariance we followed gillon et al . ( 2006 ) and obtained an estimate of the systematic errors in our observations from the residuals of the lightcurve . the amplitude of the white ( @xmath60 ) and red ( @xmath61 ) noise can be obtained by solving the equation system presented in their equations ( 5 ) and ( 6 ) . repeating our procedure for similar vlt data on ogle - tr-113 ( daz et al . 2006 ) , we estimated the white noise amplitude , and the low - frequency red noise amplitude , and then , the surface which determines the uncertainty interval . the projections of the @xmath59 surface to estimate the uncertainties of the fit parameters are shown in figure 3 , as done by daz et al . ( 2006 ) for ogle - tr-113 . the light curve was also fitted to obtain the transit time , by fixing the other parameters ( a , r@xmath62 and i ) . for example , the regions for @xmath63 ( only white noise , without systematics ) , and @xmath64 ( with white + red noise , including systematics ) for the fit parameters as function of the transit time are marked in figure 4 . at this point , the major uncertainty on the ogle - tr-111-b planetary radius arises from the uncertainties in the stellar properties . note that we do not fit the star radius simultaneously using the photometry as done by winn et al . ( 2006 ) . adopting @xmath19 from gallardo et al . ( 2005 ) , we obtain @xmath65 . adopting @xmath57 from santos et al . ( 2006 ) , we obtain @xmath66 . the unweighted mean of these two independent determinations is @xmath67 . however , we will adopt the spectroscopic determination of santos et al . ( 2006 ) of @xmath57 , instead of the one from gallardo et al . ( 2005 ) for two main reasons . first , the santos spectroscopic data are more recent and of higher quality , resulting on a complete analysis of the composition and stellar parameters based on high dispersion spectroscopy , while gallardo give an indirect determination based on the surface brightness . second , in order to allow a direct comparison with the results of winn et al . ( 2006 ) , who also adopt the mass from santos et al . ( 2006 ) . table 1 lists the previous estimates of the size for the ogle - tr-111 transiting planet @xmath68 from the literature , and this work , along with the stellar parameters . the agreement of the most recent values to within about @xmath32 implies that the radius of this planet is known . the unweighted mean of the radii measured by santos et al . ( 2006 ) , winn et al . ( 2006 ) , and this work is : @xmath69 with this radius , ogle - tr-111-b does not seem to be oversized for its mass , its gross properties ( mass , radius , mean density ) being similar to the jovian planets of the solar system , as listed for example by guillot ( 2005 ) . ogle - tr-111-b is the least irradiated of the known transiting extrasolar planets , with equilibrium temperature @xmath70 ( baraffe et al . 2005 , laughlin et al . 2005 , lecavelier des etangs 2006 ) . it lies at the cool end of the distribution of the other transiting hot jupiters ( @xmath71 ) , but it is still warmer than the solar system giants ( @xmath72 ) . thus , ogle - tr-111-b is not only a missing link regarding its orbital properties , as suggested by pont et al . ( 2004 ) , but also may probe the transition between strongly irradiated and more isolated planets . it is interesting to compare ogle - tr-111-b with hd209458b , which has a similar mass ( @xmath73 ) , and orbital semimajor axis ( @xmath74 ) . yet the radius of hd209458b is about 40% larger than the radius of ogle - tr-111-b measured here ( laughlin et al . 2005 , baraffe et al . 2005 ) . two main effects could produce this large difference . first , hd209458b is inflated by stellar irradiation , which is smaller for ogle - tr-111-b . the difference in incident flux is a factor of 4 according to baraffe et al . this irradiation difference is mostly because the primary star hd209458 ( @xmath75 ) , is hotter than ogle - tr-111 ( @xmath76 ) . second , the presence of a massive solid core in ogle - tr-111-b might make its radius smaller in comparison with hd209458b . models predict that the presence of a massive solid core should reduce the radius of a giant planet significantly ( saumon et al . 1996 , burrows et al . 2003 , bodenheimer et al . 2003 , sato et al . 2006 , guillot et al . for example , the reduction in radius is @xmath77% for a @xmath78 planet with a @xmath79 core with respect to a planet with a small core . it has been recently realized that it is very important to measure the transit times accurately , because of the exciting possibility of using these times for future studies of multiplicity in these systems ( holman & murray 2005 , agol et al . 2005 ) . winn et al . ( 2006 ) measure two transits accurately , giving an improved period of : @xmath80 the mean transit times measured by winn et al . ( 2006 ) are : @xmath81= 2453787.70854 \pm 0.00035,$ ] and @xmath81= 2453799.75138 \pm 0.00030.$ ] these transits are separated by 3 cycles , and we use their mean along with our own observations to compute a more accurate period . the vlt transit occurred 80.5 cycles before the mean of the transits observed by winn et al . the mean transit time measured at the vlt is : @xmath81= 2453470.56397 \pm 0.00076,$ ] and the final period measured here is : @xmath82 this period is independent of the ogle photometry , but it is consistent with the ogle data . the errors include the systematic errors ( figure 4 ) . therefore , our improved ephemeris for the mean transit times of ogle - tr-111-b is : @xmath83 where the numbers in parenthesis indicate the errors in the last two digits of their respective quantities . adopting the ephemeris of winn et al . ( 2006 ) , the mean predicted time is off from the center of our transit , which occurs about 5 minutes earlier . in fact , the period determined here is more than @xmath84 away from the period measured by winn et al . ( 2006 ) , or @xmath85 away using our larger errorbars ( that include systematics ) . nonetheless , we believe that it is accurate , because it relies on their two well sampled transits as well as our single transit . interestingly , there is a difference with the previous measured periods.such difference can arise in the presence of another massive planet in the system , which is the main motivation for measuring accurate timing of this planet . with the present data , however , it can not be claimed that this is the case : we can not yet rule out a constant period . more transits should be accurately measured in the following seasons to follow up this interesting system and confirm or rule out variations in the mean transit times . due to the period very close to @xmath86 , about 20 consecutive full transits can be observed during a season for a period of three months , and then they are not observable again for a period of about six months . with the new ephemeris we are able to predict that the next observable series of transits of ogle - tr-111 occur around december 2006 , and then again in mid-2008 , as shown in figure 5 . it is evident that the window of opportunity for accurate photometric measurements of the transits for this target is small . an alternative way to track another massive planet in the system is to perform radial velocity follow - up with no critical time of observational windows , even though this method is time consuming and difficult for such a faint candidate . udalski et al . ( 2002 ) discovered low amplitude transits in the main sequence star ogle - tr-111 , which we observed with vimos at the eso vlt . the planet ogle - tr-111-b is a massive planet with mass @xmath2 ( pont et al . we were able to accurately sample one low amplitude transit on this star with high cadence observations , complementing the recent measurement of two transits by winn et al . we improve upon the parameters of the transit , in particular measuring the transit time , @xmath87 hours , the orbital inclination @xmath88 , and the orbital semimajor axis @xmath89 . these data are in agreement with the orbital parameters measured by radial velocities and with the stellar parameters . \(1 ) we measure an accurate radius based on vlt transit photometry and revised stellar parameters . we obtain @xmath66 , which for a mass of @xmath11 , gives a density of @xmath3 . thus , the planet of ogle - tr-111 has a radius similar to jupiter , and a mean density that resembles that of saturn . the ogle - tr-111-b planet does not appear to be significantly inflated by stellar radiation like hd209458-b . \(2 ) using newly available transits from winn et al . ( 2006 ) , we are able to measure accurately the orbital period of this interesting system , @xmath90 as well as to update the ephemeris . the timing is different from previously published values , but there is not yet sufficient data to claim time variations caused by another massive planet in the system . follow up of the ogle - tr-111-b transits is warranted in the next observable transit windows around december 2006 and mid-2008 . llllll udalski + 2002 & @xmath91 & @xmath92 & @xmath93 & lower limit , 9 transits , @xmath28-band phot.pont + 2004 & @xmath94 & @xmath95 & @xmath96 & using ogle phot . , new stellar par.gallardo + 2005& & @xmath97 & @xmath98 & using ogle amp . , new stellar par.silva + 2006 & @xmath99 & & @xmath100 & re - analysis of ogle phot.santos + 2006 & @xmath101 & @xmath102 & @xmath103 & using ogle phot . , new stellar par.winn + 2006 & @xmath101 & @xmath104 & @xmath105 & two transits , @xmath28-band phot.this work & @xmath101 & @xmath104 & @xmath106 & single transit in @xmath0-band
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we present accurate @xmath0-band photometry for a planetary transit of ogle - tr-111 acquired with vimos at the eso very large telescope . the measurement of this transit allows to refine the planetary radius , obtaining @xmath1 .
given the mass of @xmath2 previously measured from radial velocities , we confirm that the density is @xmath3 .
we also revise the ephemeris for ogle - tr-111-b , obtaining an accurate orbital period @xmath4 days , and predicting that the next observable transits would occur around december 2006 , and after that only in mid-2008 .
even though this period is different from previously published values , we can not yet rule out a constant period .
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cosmic rays are elementary particles arriving at the earth from outside that were discovered in the beginning of the 20th century as one of the main sources of natural radiation . the cosmic ray spectrum has been observed as a continuum at all energies since their discovery . throughout this period cosmic rays have always been the source of the highest energy elementary particles known to mankind , and for this reason they have given birth to particle physics . the high energy tail of the spectrum as it is known today corresponds to energies up to 3 @xmath0ev and rates of a few particles per km@xmath1 per century . it is remarkable that the cosmic rays have a quite featureless power law energy spectrum which decreases as approximately the cube of the primary energy . for energies above the few hundred tev the observed flux necessarily requires techniques that take advantage of the extensive air showers that the arriving particles develop as successive secondary particles cascade down into the atmosphere . shower measurements allow the reconstruction of the arrival directions and the shower energy but the nature of the primary particle is extracted by a number of indirect methods . for energies above few tens of gev the detected particles , mainly protons , have arrival directions with a remarkably isotropic distribution . this is understood in terms of diffusive propagation in the galactic magnetic fields . as the energy rises above a given value that depends on the charge of the particle , propagation in the galaxy should cease to be diffusive . such high energy particles are expected to be extragalactic . the observation of high energy cosmic rays has been recently reviewed by nagano and watson @xcite who have shown that there is very good agreement between different experiments including the low and high energy regions of the spectrum . there is increasing evidence for a different component of the high energy end of the cosmic ray spectrum @xcite . combining data of five different experiments , agasa , akeno , haverah park , stereo fly s eye and yakutsk , nagano and watson conclude that there is a clear signal of a change of the spectral slope in the region just above @xmath2ev @xcite . composition studies have also given indications that there is a change to light element composition for energies above @xmath3ev @xcite although this conclusion is model dependent to some extent @xcite . also the small anisotropy ( @xmath4 ) of @xmath2ev cosmic rays in the direction of the galactic anticenter detected with agasa disappears at higher energies @xcite . the highest energy events detected present a serious challenge to theory and little is known about their origin . if they are protons they should attenuate in the cosmic microwave background ( cmb ) over distances of order 50 mpc . such attenuation was predicted to appear in the cosmic ray spectrum as a cutoff , the greisen - zatsepin - kuzmin ( gzk ) cutoff , just above 4 @xmath5gev @xcite . if they are photons or iron nuclei it turns out that interactions with the radio and the infrared backgrounds are respectively responsible for attenuations over similar or even shorter distances . no such features are seen in the observed cosmic ray spectrum . if they are produced sufficiently close to us to avoid the cutoff then the arriving particles should be pointing to their sources . this seems difficult to accommodate because there are very few known astrophysical sources capable of reaching the observed energies and on the other hand there is little evidence for the anisotropy that would result . this article firstly discuses the problem presented by the high energy end of the cosmic ray spectrum with emphasis in the role of composition . then it outlines new progress made in understanding different features of inclined showers illustrating how these showers can contribute to the composition issue reviewing the results obtained by a recent analysis of the inclined data in haverah park . the discovery of events with energies above @xmath6 ev ( 100 eev ) dates back to the 1960 s , to the early days of air shower detection experiments @xcite . since then they have been slowly but steadily detected by different experiments as illustrated in fig . [ uptonow ] . now there is little doubt about the non observation of a gzk cutoff , with over 17 published events above @xmath6 ev and five preliminary new events from hires @xcite . on the contrary the data suggests that the spectrum continues smoothly within the statistical errors , possibly with a change of slope . on the other hand the data show no firm evidence of anisotropy but the significance of such studies is even more limited by the poor statistics . both the details of the spectrum at the cutoff region and the extent to which the arrival directions of these particles cluster in the direction of their sources are very dependent on a number of unestablished issues . these include the source distribution , the distance of the nearest sources , their emission spectra , the intervening magnetic fields and of course on the nature of the the cosmic rays themselves or composition . if these particles are nuclei or photons the observational evidence is suggesting that these are coming from relatively nearby sources compared to the 50 mpc scale . the conclusive power of observations is however strongly limited by both the poor statistics and a complex interrelation of hypotheses , but the situation is bound to change in the immediate future with a new generation of large aperture experiments , some like hires @xcite already in operation , others in construction @xcite and many others in planning @xcite . the complex puzzle that connects particle physics , magnetic fields , and cosmic rays has attracted the attention of many fields in physics . in a conventional approach these particles would be nuclei as the bulk of the cosmic ray spectrum which are accelerated through stochastic acceleration as suggested by fermi in 1949 . this happens every time charged particles cross interfaces between regions that have astrophysical plasmas with different bulk motions , such as shock fronts . transport is assumed to be diffusive in the plasma s magnetic field and on average in these processes a very small fraction of the bulk plasma kinetic energy is transferred as a boost to the individual particles , that typically end up with a power like spectrum . acceleration of a particle of charge @xmath7 to an energy @xmath8 is strictly limited by dimensional arguments to objects that are sufficiently large or have sufficiently large magnetic fields . basically for a particle with momentum @xmath9 to be able to undergo such a boost , propagation must be diffusive , or equivalently the accelerator region @xmath10 must be larger than the larmor radius of the particle , @xmath11 , in its characteristic magnetic field @xmath12 : @xmath13 the requirement is well known by accelerator designers and is the ultimate reason for their high cost . it turns out that few of the known astrophysical objects satisfy the minimum requirements to accelerate particles to @xmath0ev . this is conveniently illustrated in a plot first conceived by michael hillas @xcite which is reproduced in fig . [ hillasplot ] . a number of possible scenarios are being discussed ; they imply acceleration in some objects including young pulsars , gamma ray bursts ( grb ) , our own galaxy , active galaxies and the local group of galaxies @xcite . the power supply needed to keep the observed cosmic rays at the highest energies is consistent with the known power and distributions of these objects @xcite . it is difficult to explain the observed flux spectrum in this conventional approach . a solution in which particles are accelerated nearby has difficulties because there are very few objects which are capable of accelerating particles to the maximum observed energies . moreover many such objects are either too large or too distant for the cosmic ray spectrum detected at the earth not to show the predicted gzk cutoff . if the sources were to be galactic no absorption cutoff would be expected but some spectral features are predicted for primary protons that are produced at a distance of more than a few mpc . on the other hand the non observation of anisotropy complicates the puzzle , because the location of the possible accelerators in our vicinity is pretty well known . primary protons having energies in the @xmath6 ev range are expected to be little deviated in the galactic magnetic fields . our knowledge of extragalactic magnetic fields is poor but bounds on extragalactic magnetic fields also imply that the deviations of protons produced in the few mpc range are not large . there are however possible configurations of the extragalactic magnetic fields that could explain many of the ultrahigh energy events as coming from a single source @xcite . the issue is far from being resolved and knowledge about composition is bound to play a crucial role for future progress in understanding . motivated by particle physics beyond the standard model , many alternatives have been proposed that avoid acceleration and others that postulate different particles or different interactions . these include annihilation of topological defects created in the early universe , heavy relics that survive from the primeval bath , non thermal particles that couple to gravity , or wimpzillas and annihilation of relic neutrinos with messenger neutrinos coming from remote places @xcite . as regards composition two large categories of possible scenarios can be made namely those in which the observed particles are accelerated and those in which they are decay products of other particles . these two classes differ greatly in composition . a knowledge of composition is doubly important because firstly it may decide between these two classes of solutions and secondly because it would simplify the task of interpreting anisotropy measurements . the models that depend on acceleration can reach higher energies if the accelerated particles have large charge @xmath14 . this shows as a different restriction line in fig [ hillasplot ] . the relative composition of different nuclei resulting from such a scenario will depend on the local abundances of the different nuclei and on the energy . depending on distance to the source and the surrounding environment there may be energy losses , absorption and the production of secondary particle fluxes . for instance in active galactic nuclei ( agn ) models the accelerated protons are expected to interact with ambient light or matter to produce pions that decay into photons and neutrinos . the neutrinos can reach the earth unattenuated and provide a signature of proton acceleration . unless the environment becomes opaque to protons the relative fluxes of neutrinos and protons that reach the earth should be a number of order one or smaller , just because neutrinos are secondaries with repect to protons . the relative fluxes of photons and protons would be similar to neutrinos or smaller depending on the photon absorption both at the source and during transport to earth . ratios of the same order of magnitude would apply to most acceleration models . most of the non accelerating alternatives postulate the cosmic rays are products of the decay of other more massive particles produced by different mechanisms . typically these particles of mass of @xmath15ev ( often an @xmath16 particle ) decay into standard model particles which eventually fragment into hadrons , mostly pions and a small fraction of order @xmath17 of nucleons . while neutral pions decay into photons charged pions decay into neutrinos . fragmentation processes , known form accelerator experiments and extrapolated to the high energies , become the common reference point for these mechanisms . for this reason all these models share a very similar composition dominated by photons and neutrinos which typically are about ten times more numerous than nucleons at the production site . depending on the source distribution the relative fluxes of these particles are modified through their interactions with the background radiation fields . the neutrinos are the particles that preserve their production spectrum without being attenuated . protons get attenuated in few tens of mpc in the cosmic microwave background , ( the gzk cutoff ) , while photons are attenuated already in few mps mainly through pair production in the radio background . as a result the ratio of neutrinos to protons can in principle become higher at the earth than when they are produced if the sources are quite distant or cosmologically distributed . many of the proposed mechanisms are expected to cluster in our galactic halo . this possibility is receiving a lot of attention because it would provide a relatively natural explanation for the absence of the gzk cutoff . in that case however the sources will be quite near and the ratio of photons to nucleons should be expected to be of order 10 , close to its value at production . other sources are not expected to cluster and hence the photon to nucleon ratio is expected to drop to values close to one . the ratio of photons to nucleons depends on the source distribution and is rather sensitive to clustering . most air shower detectors in existence consist on arrays of particle detectors that sample the extensive air shower front as it reaches the ground . multiple particle production takes place in the successive high energy interactions produced as the shower penetrates the medium . as a result the number of particles in the shower front increases exponentially . when the average particle energy in the front becomes too low for multiple particle production the shower reaches it maximum number of particles . the development of these showers is typically governed by the radiation length in the material which is of order 36 g @xmath18 in air and shower maximum , which is only logarithmically dependent on the primary particle energy , occurs at a couple of thousand meters for vertical showers of energies of order @xmath6 ev . vertical showers are close to shower maximum when reaching the earth s surface , have pretty good circular symmetry and are less affected by the earth s magnetic field . it is thus not surprising that air showers have traditionally been studied at close to vertical incidence , typically for zenith angles below @xmath19 , in summary because it is much simpler . moreover in most extensive air shower arrays the particle detectors are oriented to have maximum collection area for vertical incidence . since these detectors are often scintillator sheets , they tend to become very inefficient for very inclined showers . as the zenith angle increases the traversed atmospheric depth rises from 1000 to close to 36000 g @xmath18 . as a result the shower maximum is reached in the upper layers of the atmosphere and most of the shower is absorbed before reaching the ground . it has been known for a long time that weakly interacting particles such as neutrinos can induce close to horizontal air showers deep in the atmosphere with particle distributions that are quite similar to vertical showers @xcite . air shower array detectors looking in the close to horizontal direction can thus be sensitive to high energy neutrino fluxes @xcite . in fact most bounds on neutrino fluxes have already been obtained from air shower experiments @xcite . the original motivation of studying inclined showers was to understand the cosmic ray background to the neutrino induced showers . although the electromagnetic part of the air shower induced by an inclined cosmic ray is indeed absorbed before reaching ground level , the shower front however also contains muons which are mainly produced by charge pion decay when the primary particle is a hadron . these muons do travel practically unattenuated all the slant atmospheric depth and produce density patterns on the ground that are much affected by the earth s magnetic field . it has recently become quite clear that such inclined showers can be analysed . this not only nearly doubles the aperture of any air shower array but , when combined with vertical measurements , it has a remarkable potential for the study of primary composition @xcite . much development in this field has been possible by the modelling of the muon density patterns produced by inclined showers under the influence of the earth s magnetic field @xcite . the lateral distributions of muons in inclined showers can be understood in terms of a simple model @xcite in which the magnetic field is firstly neglected . the model stresses two important facts that have been extensively checked with simulations in the absence of a magnetic field @xcite : most of the muons in an inclined shower are produced in a well defined region of shower development which is quite distant from the ground and the lateral deviation of a muon is inversely correlated with its energy . indeed most of the fundamental properties of these inclined showers are governed by the distance and depth travelled by the muons . it is remarkable that the average slant distance travelled by the muons is of order 4 km for vertical showers , becomes 16 km at 60@xmath20 and continues to rise as the zenith angle rises to reach 300 km for a completely horizontal shower . this distance plays a crucial role as a low energy smooth cutoff for the muon energy distribution . for inclined showers the muons must have much more energy at production to reach ground level without decaying than in the vertical case . both the travel time and the muon energy loss become relevant . the model simply assumes that all muons are produced at a given altitude @xmath21 with a fixed transverse momentum @xmath22 that is uniquely responsible for the muon deviation from shower axis . in the transverse plane to the shower at ground level the muon deviation , @xmath23 , is inversely related to muon momentum @xmath9 . the density pattern has full circular symmetry when there is no magnetic field . when the magnetic field effects are considered the muons deviate a further distance @xmath24 in the perpendicular direction to the magnetic field projected onto the transverse plane @xmath25 , given by : @xmath26 where in the last equation @xmath27 is to be expressed in tesla , @xmath21 in m and @xmath28 in gev . as the muon deviations are small compared to @xmath21 they can be added as vectors in the transverse plane and the muon density pattern is a relatively simple transform of the circularly symmetry pattern . the muon patterns in the transverse plane can be projected onto the ground plane to compare with data as well as standard simulation programs . eq . [ alpha ] is telling us that all positive ( negative ) muons that in the absence of a magnetic field would fall in a circle of radius @xmath23 around shower axis , are translated a distance @xmath24 to the right ( left ) of the @xmath29 direction . the dimensionless parameter @xmath30 measures the relative effect of the translation . for small zenith angles @xmath21 is relatively small and @xmath31 so that the magnetic effects are also small , and results into slight elliptical shape of the isodensity curves . for high zeniths however @xmath32 the magnetic translation exceeds the deviation the muons have due to their @xmath28 . in this case _ shadow _ regions with no muons are expected in the muon density profiles . for an approximate @xmath33 mev and @xmath34 t this happens when @xmath21 exceeds a distance of order 30 km , that is for zeniths above @xmath35 . these shadow regions in the transverse plane are indeed an outstanding feature of the ground density profiles at high zeniths as seen in the simulations . the simple model can be actually generalized to account for muon energy distributions as a function of distance to shower axis , and improved using the correlation between the average muon energy and the distance to shower axis as obtained in dedicated simulations . when all this is done the obtained muon density patterns are shown to be accurately reflect those obtained with simulations and this proves to be a very useful tool for the study of inclined showers . for each zenith angle the primary particle energy sets the normalization of the particle densities . for proton primaries the total number of muons in the shower scales with the proton energy @xmath8 as : @xmath36 where @xmath37 is a constant . it is remarkable that the shape of the lateral distribution of the muons does not significantly change for showers of energy spanning over three orders of magnitude . the same happens for heavier nuclei with slightly different parameters . the results are slightly model dependent . two alternative hadronic interaction models have been compared , the quark gluon string model ( qgsm ) and sibyll to give also the same behaviour with also different parameters . table [ nmu.tab ] illustrates these effects . .relationship between muon number and primary energy for proton and irons in two hadronic models ( see equation [ escaling ] ) . [ cols="<,>,^,^",options="header " , ] as a final result the muon distributions can be represented by continuous functions which are analytically obtained once we know the main features of a shower in the absence of magnetic field . in practice this implies that only different zenith angles have to be simulated . different azimuths are obtained by adequate transformations of the showers without magnetic deflections . the algorithm is fast and allows detector simulation and also event by event reconstruction of data obtained by air shower experiments . this powerful technique has been used to analyse the inclined shower data obtained in the haverah park array . the haverah park detector was a 12 km@xmath1 air shower array using 1.2 m deep water erenkov tanks that was running from 1974 until 1987 in northern england which has been described elsewhere @xcite . it is possibly the most appropriate detector for this study because the water erenkov tanks have a uniquely large cross section to sample shower fronts of horizontal air showers . moreover the erenkov technique gives larger signals for muons than for electrons simply because the muons have typically larger energies and travel through the whole detector . a careful study has been made of the energy deposition of signal in water erenkov tanks by horizontal muons using conventional simulation programs for this purpose @xcite . a number of effects have to be considered to interpret the observed data . inclined particles can produce light that falls directly into the phototubes without being reflected in the tank walls . horizontal muons produce more signal through delta rays because on average they have higher energies than in vertical showers . there is a significant signal deposited by electromagnetic particles that arise mainly through muon decay . finally the higher energy muons are more likely to deposit more energy in the tanks because of catastrophic energy losses . the event rate as a function of zenith angle has been simulated with careful treatment of all these effects using the muon distributions obtained as described in the previous section . the qualitative behaviour of the registered rate is well described in the simulation and the normalization is also shown to agree with data to better than @xmath38 using the measured cosmic ray spectrum for vertical incidence , assuming proton primaries and using the qgsm model @xcite . more impressive are the results of fits of the models for muon densities to the observed particle densities sampled by the different detectors on an event by event basis . the nearly 10,000 events recorded with zenith angles above @xmath39 have been analysed for arrival directions , impact point and primary energy in the assumption the primaries are protons . a complex sequence of arrival direction and density fits is performed to minimize the effect of correlations between energy and arrival directions . the analysed date is subject to a set of quality cuts : the shower is contained in the detector ( distance to core less than 2 km ) , the @xmath40 probability of the event is greater than @xmath41 and the downward error in the reconstructed energy is less than @xmath42 . these cuts ensure that the events are correctly reconstructed and exclude all events detected above @xmath43 . examples of reconstructed events compared to predictions are illustrated in fig . [ events.fig ] . two new events with energy exceeding @xmath6 ev have been revealed . the results have been compared to a simulation that reproduces the same fitting procedure and cuts using the cosmic ray spectrum deduced from vertical air shower measurements in reference @xcite . the agreement between the integral rate above @xmath44 ev measured and that obtained with simulation is striking when the qgsjet model is used . sibyll leads to a slight underestimate @xcite . the universality of the muon lateral distribution function is very powerful and once the equivalent proton energy is determined for all events , the corresponding energies in the assumption that the primaries are iron nuclei ( photons ) can be obtained multiplying the proton energy by a factor which is @xmath45 ( 6 ) for @xmath44 ev . as a result when a photon primary spectrum is assumed the simulated rate seriously underestimates the observed data by a factor between 10 and 20 . a fairly robust bound on the photon composition at ultra high energies can be established assuming a two component proton photon scenario . the photon component of the integral spectrum above @xmath46ev ( 4 @xmath44 ev ) must be less than @xmath47 ( @xmath48 ) at the @xmath49 confidence level . details of the analysis are presented in @xcite . the results of this method when applied to a first analysis of inclined showers produced by cosmic rays above @xmath46ev demonstrates that the study of inclined showers not only can double the acceptance of air shower arrays but it can be a very useful tool for the study of photon composition . 9 m. nagano and a.a . watson , _ rev phys . _ 72 ( 2000 ) 689 . t.k gaisser in _ proc . of the int . workshop on observing ultra high eenergy cosmic rays from space and earth _ , ( 2000 ) metepec , puebla , mexico , to be published by aip . d.j . bird _ et al . _ , _ phys . * 71 * ( 1993 ) 3401 . n. hayashida _ et al . _ , _ astropart . * 10 * ( 1999 ) 303 . k. greisen ; _ phys . _ , * 16 * ( 1966 ) 748 . zatsepin and v.a . kuzmin , _ jetp lett . _ , * 4 * ( 1966 ) 78 . j. linsley , _ phys . _ , * 10 * ( 1963 ) 146 . d. kieda _ et al . _ , _ proc . xxvi icrc _ salt lake city ( 1999 ) . design report , auger collab . fermilab - pub-96 - 024 , jan 1996 . for a recent review see _ proc . of the int . workshop on observing ultra high eenergy cosmic rays from space and earth _ , ( 2000 ) metepec , puebla , mexico , to be published by aip . a.m. hillas _ _ , _ proc . of xi icrc _ , budapest ( 1969 ) , acta physica academiae scietiarum hungaricae 29 , suppl . . for recent reviews see for instance , a.v . olinto , _ phys . * 333 - 334 * ( 2000 ) 329 ; g. sigl , lectures at mexican school of astrophysics ( 1999 ) , guanajuato , e - print archive : * astro - ph/0008364*. e. ahn , g. medina - tanco , p.l . biermann , and t. stanev preprint archive : astro - ph/9911123 . berezinsky and g.t . zatsepin , _ yad . fiz . _ * 10 * ( 1969 ) 1228 . [ _ sov . j. nucl . phys _ * 10 * ( 1969 ) 696 ] . e. zas , f. halzen and r.a . vzquez , _ astropart . phys . _ * 1 * ( 1993 ) 297 . j. capelle , j.w . cronin , g. parente , and e. zas , _ astropart . * 8 * ( 1998 ) 321 blanco - pillado , r.a . vzquez , and e. zas , phys . * 78 * ( 1997 ) 3614 . et al . _ , d 31 * , 2192 ( 1985 ) . m. ave , j.a . hinton , r.a . vazquez , a.a . watson , and e. zas , phys . lett . * 85 * , ( 2000 ) 2244 . m. ave , j.a . hinton , r.a . vazquez , a.a . watson , and e. zas , astropart . ( 2000 ) 109 . m. ave , r.a . vzquez , and e. zas , _ astropart . _ 14 ( 2000 ) 91 . tennent , _ proc phys soc _ * 92 * ( 1967 ) 622 . lawrence , r.j.o . reid , and a.a . watson , _ j phys g _ * 17 * ( 1991 ) 733 . j.r.t . de mello neto , wtank : a geant surface array simulation program gap note 1998 - 020 .
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in this article i review the main theoretical problems that are posed by the highest energy end of the observed cosmic ray spectrum , stressing the importance of establishing their composition in order to decide between proposed scenarios .
i then discuss the possibilities that are opened by the detection of inclined showers with extensive air shower arrays .
recent progress in modelling magnetic deviations for these showers has allowed the analysis of inclined showers that were detected by the haverah park experiment .
this analysis disfavours models that predict a large proportion of photons in the highest energy cosmic rays and open up new possibilities for future shower array detectors particularly those , like the pierre auger observatory , using water erenkov detectors .
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barred galaxies constitute a major fraction of all disc galaxies classified in the optical , more than 50% including strong bars and intermediate morphologies ( sellwood & wilkinson @xcite ) . this fraction increases when also near - infrared images are used for classification , thus underlining the importance for the general understanding of the evolution of galaxies . the non - axisymmetric potential has a strong impact on the gas dynamics and the star formation in barred systems . observations reveal a correlation between the radial abundance gradient and the strength of the bar ( martin & roy @xcite ; friedli et al . @xcite ; martinet & friedli @xcite ) . this is interpreted as the result of two effects caused by the bar : a stronger radial gas flow and hence a stronger radial mixing of metals and the efficiency of star formation . the radial mass transfer concentrates gas near the galactic center and at the ends of the bar at corotation . enhanced star formation is the consequence of gas accumulation . the rotating bar potential also heats up the outer disk parts which leads to larger stellar velocity dispersions and a radial diffusion of stars . ( sellwood & wilkinson @xcite ) . galactic bars have also been considered to support the central infall of gas to feed a central `` monster''(e.g . beck et al . several authors have claimed that active galactic nuclei ( agn ) are more likely in barred galaxies than in non - barred ones ( e.g. simkin et al . @xcite ; arsenault @xcite ) . hummel et al . ( @xcite ) note that the fraction of central radio sources in barred spirals is by a factor of 5 higher than in non - barred spirals . other authors doubt that there is a significantly higher number of bars in galaxies harboring an agn ( e.g. balick & heckman @xcite ; ho et al . it appears that the concentration of gas on a scale of @xmath41 kpc at the galactic center required to enhance the central star formation can easily be achieved by a bar potential . it seems much more difficult , however , to accumulate enough gas on a scale of a few pc to tens of pc in order to produce an agn . other effects depending on the environment of the galaxies ( interaction : elmegreen et al . @xcite ; contents : cayatte et al . @xcite ) play an important role in mass distribution , gas flow , and therefore in the formation and evolution of bars and the star formation history in these systems . one of the most famous , closest and most widely studied barred galaxies is ngc 4303 ( m61 ) , member of the virgo cluster , which is observed at an inclination of 27 ( guhathakurta et al . @xcite ) . optical spectra of this galaxy indicate that it consists of a nuclear starburst and a liner or seyfert 2 nucleus ( filippenko & sargent @xcite ; kennicutt et al . @xcite ; colina et al . @xcite ; colina & arribas 1999 , hereafter @xcite ) . indications for a high star formation rate ( sfr ) in ngc 4303 are given by the numerous regions ( hodge & kennicutt @xcite ; martin & roy 1992 , hereafter @xcite ) and three observed supernovae ( van dyk @xcite ) . it also shows strong radio emission distributed over the entire disk ( condon @xcite ) . colina et al . ( @xcite ) and @xcite discussed the question of a starburst agn connection in this barred galaxy , using optical spectroscopy and hst uv images . the data range from a nuclear spiral structure of massive star - forming regions with an outer radius of 225 pc down to the unresolved core of a size @xmath5 8 pc . from the uv data it is not clear if the core is a massive stellar cluster or a pure agn . vla observations ( cayatte et al . @xcite ) show that ngc 4303 is not highly deficient , which can be explained by only slight environmental influences in the outermost region of the virgo cluster . the projected distance to m87 is 82 ( warmels @xcite ) . no significant difference of the abundance gradient in the disk of ngc 4303 compared to non - barred spiral galaxies has been observed ( @xcite ) . martinet & friedli ( @xcite ) discussed the abundance gradient slope in terms of bar age . according to them , a steep gradient in the bar and a flat one in the outer disk are typical for a young bar while a single flat gradient in bar and disk characterizes an old bar . @xcite did not determine the gradient at large radii because of a small number of regions . martinet & friedli ( @xcite ) also claimed that bars in late - type spirals with enhanced star formation like ngc 4303 are expected to be young . probable interaction companions are the nearby galaxies ngc 4303a ( condon @xcite ) and ngc 4292 ( cayatte et al . @xcite ) , at distances of 75 northwest and 10 northeast , respectively . .some basic parameters of ngc 4303 . [ cols="<,^,^",options="header " , ] + col . ( 1) spectral models : bs = thermal bremsstrahlung , rs = raymond - smith , po = power law ( 2) column density in units of 10@xmath6 @xmath7 . ( 3) plasma temperature in units of kev . ( 4) photon index . ( 5) metallicity in units of z@xmath2 . ( 6) scaling factor : for bs in units of ( 10@xmath8/(4@xmath9))@xmath10d@xmath11 , @xmath12 = electron and ion densities ( @xmath13 ) ; for rs in units of ( 10@xmath14/(4@xmath9))@xmath15d@xmath11 , @xmath16 = electron and h densities ( @xmath13 ) ; for po in units of 10@xmath17 photons kev@xmath3 @xmath7 s@xmath3 at 1 kev ( 7) reduced @xmath18 . + col . ( 8) degrees of freedom . ( 9) unabsorbed x - ray flux in units of 10@xmath19 erg @xmath7 s@xmath3 . values in brackets give the contribution of the thermal component . ( 10) x - ray luminosity in units of 10@xmath20 erg s@xmath3 . values in brackets give the contribution of the thermal component . a single power - law model implies the assumption , that the active nucleus of ngc 4303 dominates the x - ray emission . furthermore , the sources detected by the hri in the galactic disk would also have to be described with the same power law . the photon index in this model is @xmath21=3.2@xmath220.2 . the emission of an agn in the rosat energy band is best described by a power law with a photon index of @xmath232.4 ; nevertheless some cases have been observed with @xmath243 ( mcg 5 - 23 - 16 : mulchaey et al . @xcite ; mkn 335 : turner et al . @xcite ) . high - mass x - ray binaries ( hmxb ) found in young star - forming regions in the spiral arms have a similar spectral shape in the 0.12.4 kev energy range with a photon index of @xmath232.7 ( mavromatakis @xcite ) . the column density of the absorbing component amounts to 5.7 @xmath7 , which is by a factor of 3 higher than the galactic foreground column density ( dickey & lockman 1990 , @xcite ) . nevertheless , self - absorption within ngc 4303 must be expected , and small - scale deviations from the observed galactic value by @xcite can not be ruled out and may result in a higher absorption from the milky way . the resulting 0.12.4 kev x - ray luminosity amounts to 1.3 erg s@xmath3 . the flux portion from the sources outside the nuclear region as observed with the hri amounts to @xmath41.4 erg s@xmath3 in the case of a single power - law emission model with @xmath21=3.2 using the corresponding ecf of 1 cts @xmath25 erg@xmath3 . assuming a mean x - ray luminosity of 10@xmath26 erg s@xmath3 for an hmxb , as observed in the milky way ( fabbiano et al . @xcite ; watson @xcite ) would require an unlikely high number of 1400 of these systems to produce the observed x - ray flux . the ratio of ob stars to hmxbs is assumed to be @xmath4500 ( fabbiano et al . this means that a total number of 7 ob stars would be required to account for the hmxb x - ray flux in ngc 4303 . even if we consider to have 10@xmath27 ob stars in ngc 4303 , as observed e.g. in mkn 297 ( benvenuti et al . @xcite ) , it is still a factor of 7 higher than expected . moreover , this is the required number only for the disk sources and would involve almost 1.5 m@xmath2 in massive stars with a salpeter imf and , by this , would require a moderately high sfr of about 15 m@xmath2 yr@xmath3 in the disk . on the other hand , the corresponding supernova typeii ( ) rate ( 0.1 yr@xmath3 ) should contribute to the x - ray emission via hot gas . the single component models bs and rs show similar results . consequently , we only achieve an adequate fit of rs with very low metallicity , e.g. the portion of emission lines to the spectrum is very small . in contrast , it is expected that emission lines of highly ionized elements , like fe and mg , should play an important role in the x - ray spectrum of in starburst regions because of the nucleosynthesis of massive progenitor stars ( woosley & weaver @xcite ) . in both models the column density is about 3 @xmath7 and the plasma temperature is 0.6 kev . for the bs model we get a total x - ray luminosity of 4 erg s@xmath3 , for the rs model it is 3.5 erg s@xmath3 . rs models with different higher metallicities yield unacceptable fits . the fit of the x - ray spectrum with a two - component model ( rs+po ) is only slightly better than the one - component fits . nevertheless , from the points mentioned above and the physical picture discussed in sect . [ discussion ] this model serves as the best explanation for the observed soft x - ray emission . hydrogen column density ( @xmath28=3.3 @xmath7 ) and power - law spectral index ( @xmath21=2.6 ) lie within the expected range ( as discussed for the single power law above ) . the plasma temperature of 0.3 kev fits with the observed values of other galaxies ( e.g. ngc 253 : forbes et al . @xcite ; ngc 1808 : junkes et al . the total 0.12.4 kev luminosity for this model amounts to 4.7 erg s@xmath3 with 13% contribution from the rs component . the spectral fit together with the residuals is plotted in fig . [ rspomodel ] . from the quality of the spectral fits alone there is no significance for favoring a single - component model or a combination of two components . due to the lack of any spatial information in the pspc image there is no possibility to distinguish between different spatial and spectral components simultaneously . so only the combined information from the pspc and hri data allows a more detailed interpretation of the x - ray results . several points speak against the single po component model , as discussed in sect . [ specfit ] . a more likely scenario is a composition of several different emission sources , like an active nucleus , hmxbs , and supernova remnants ( snrs ) . in the following we will therefore discuss a composite emission model and the comparison with the uv and optical observations of the galactic core . as can be discerned from the hri image ( fig . [ hrisrcareas ] and table [ tabhrisources ] ) , most of the x - ray emission of ngc 4303 ( 83% ) comes from the central region of the galaxy . three different pictures are imaginable for the nucleus : a central active nucleus , a central or circumnuclear region with enhanced star formation , or a combination of both phenomena . any of these cases requires a sufficient gas density at the galactic center . this can be achieved by a barred potential which triggers radial gas flow from the outer regions toward the nucleus . on the other hand , from numerical simulations including gas dynamics bar formation has proved to be only a transient phenomenon ( combes @xcite ) . in this picture , the galactic bar will be destroyed by the gas inflow after only a few cycles . a new bar - phase can follow this gas infall due to a subsequent gravitational instability from the accreted central mass . the problem with this picture is the contradiction of a necessary gas inflow to form and feed any nuclear activity ( starburst and/or agn ) and the fact that this gas inflow destroys the bar . it seems that a sufficiently massive black hole can provide for its fuelling ( fukuda et al . @xcite ) . another efficient way for gas to flow further into the center is a second smaller bar embedded into the first one due to a second inner lindblad resonance ( friedli & martinet @xcite ) . in some cases , a gaseous circumnuclear ring is formed at the end of the second bar . the concentrated x - ray emission from the galactic core in ngc 4303 may originate from an agn and/or a nuclear or circumnuclear starburst . a starburst can contribute in two different ways to the x - ray flux . first , the produced star population contains hmxbs , emitting an x - ray radiation in spectral shape similar to an agn . hmxbs can not be distinguished from agn in the rosat data . additionally , high - mass stars ( above @xmath4810 m@xmath2 ) evolve to at an age of @xmath410@xmath29 yr , depending on their initial mass . the from one star cluster form a cumulative expanding superbubble filled with hot gas which can be described by a thermal bremsstrahlung spectrum and additional emission lines and recombination edges of highly ionized heavy elements produced in high - mass stars and released by their explosions , e.g. o , ne , mg , si , and fe . theoretical models for the spectral emission of such a hot diffuse gas are meka ( mewe et al . @xcite ) and the model by raymond & smith ( @xcite ) , which we used in our spectral fit . if we apply a two - component model to describe the x - ray spectrum of ngc 4303 , the comparison of the flux ratio from the nucleus ( 83% in the hri ) and the disk sources ( 17% in the hri ) suggests that the central source can be described by the power - law component ( 87% in the spectral fit ) . the thermal rs emission exclusively originates from the disk sources , indicating ongoing star formation . to consider the possible extension of @xmath425 of the central source , as represented by the lowest contours , it is imaginable that a small fraction of the x - ray flux is emitted by a circumnuclear starburst at a distance of @xmath41 kpc around the core . this would add a thermal component to the non - thermal x - ray nucleus . on the other hand , a fraction of the x - ray flux from the disk sources may come from hmxbs within these star forming regions . fuelling an agn on scales of a few parsec at the center of the galaxy leads to the problem of reducing the angular momentum of the central gas by several orders of magnitude , as dynamical simulations show ( barnes & hernquist @xcite ) . concentration of gas in a ring - like feature around the nucleus with a radius of @xmath41 kpc is dynamically much easier to achieve . the hri image agrees with the picture of an extended x - ray source with a diameter of the order of @xmath42 kpc at the galactic center of ngc 4303 , explained by a circumnuclear starburst region with an additional possible compact nuclear source.the decovered massive rotating circumnuclear disk in ngc 4303 can provide by its spiral - like structure of massive star forming regions an effective mechanism to channel gas from the circumnuclear regions further down to the nucleus to feed the agn . but one has to consider that the spiral structure in the uv has a diamater of only 225 pc , while the extension of the central x - ray source is about 2 kpc in diameter . the spiral feature detected in the uv can not be resolved with the hri . from the analysis of uv and optical magnitudes and colors of the central 250 pc colina & wada ( @xcite ) estimated ages of 525 myr for the star - forming regions . consequently a contribution to the central x - ray flux from snrs and cumulatively expanding hot gas has to be expected . the question remains whether we observe a pure nucleus of massive star - forming clusters or a composition of these star clusters and a low luminous agn . if ngc 4303 contains a non - thermal active nucleus , the x - ray luminosity of 4 erg s@xmath3 points to only a low luminous agn ( liner ) . koratkar et al . ( @xcite ) found a correlation between @xmath0 and @xmath1 for low luminous agns of @xmath0/@xmath30 14 . prez - olea & colina ( @xcite ) investigated the correlation between optical and x - ray luminosities of several agns with circumnuclear star - forming rings , pure agns , and pure starburst galaxies . the pure starbursts in their galaxy sample show @xmath0/@xmath1 values of 0.030.3 , 100 times smaller than for pure agns . if we take the h@xmath31 luminosity of ngc 4303 derived by keel ( @xcite ) and assume that 10% originate from the nucleus , we get log @xmath1(nucleus ) = 39.2 ( adopted for a distance of 16.1 mpc ) . therefore the x - ray - to - h@xmath31 ratio amounts to log(@xmath0/@xmath1 ) = 1.4 , which agrees with the value found by koratkar et al . ( @xcite ) . even the lower @xmath0 value from a single rs model ( @xmath42.5 erg s@xmath3 for the nucleus ) results in log(@xmath0/@xmath1 ) = 1.2 . typical pure starburst galaxies show h@xmath31 luminosities of the order of their x - ray luminosities or higher . at first glance the optical disk of ngc 4303 seems to have the quite symmetrical morphology of a late - type spiral . a closer look reveals that the eastern spiral arm of the galaxy has a much more prominent form with a boomerang - like shape and a lot more bright emission regions than the western counterpart . the northern disk shows a complex structure with many separate features . this asymmetry is more discernible in the h@xmath31 image . the regions are mainly located in the northern part of the disk at the junction of the bar with the eastern spiral arm and along that arm . a close encounter of one or both of the nearby galaxies ngc 4303a and ngc 4292 may have caused these features . the interaction within the virgo cluster is another possible source . infall into the intracluster medium could cause ram pressure effects . nevertheless , ngc 4303 is located at the outer edge of the cluster , which may produce only a moderate disturbance . this agrees with the distribution over the whole optical disk . galaxies lying nearer to the cluster center show deficiencies and concentration of the neutral hydrogen in the central regions , indicating past interactions with the gas been stripped off from the outer disk regions . as a striking indication for accumulation of gas in these regions , the x - ray sources a c and f within the galactic disk of ngc 4303 are located at the ends of the bar . gas dynamical simulations of barred galaxies have shown this accumulation due to mass flows along the bar to the center and to the ends of the bar , respectively ( noguchi @xcite ; englmaier & gerhard @xcite ) . the increased densities lead to enhanced star formation . from the low inclination of ngc 4303 no direct information can be obtained whether the disk is warped or not . but it is striking that source d lies exactly at the bend of the eastern boomerang - shaped arm . this may indicate that this x - ray source is caused by the tidal force leading to a local gas concentration . another indirect hint for a past interaction comes from the spectra of the qso 1219 + 047 ( source no . 7 in fig . [ hrifov ] ) , a qso whose line - of - sight penetrates the outer disk of ngc 4303 . bowen et al . ( @xcite ) detected complex mgii absorption , spanning a velocity range of @xmath4300 km s@xmath3 , despite the low inclined galactic disk . this high velocity is not fully understood . one possible explanation could be the result of interactions between ngc 4303 and the nearby companions . table [ tabhrisources ] lists the observed count rates and derived 0.12.4 kev luminosities for the single x - ray sources in the disk of ngc 4303 . these include the assumptions of a raymond - smith plasma with solar abundances at a temperature of 0.3 kev . a power - law model would increase the values by a factor of 2.5 . a rough estimation of the sfr in the disk from the x - ray luminosities is done by calculating the rate @xmath32 using a snr model by cioffi ( @xcite ) , and assuming a salpeter imf within a mass interval from 0.1 m@xmath2 to 100 m@xmath2 and with all stars with masses above 8 m@xmath2 evolving to . according to cioffi , a snr expanding into an ism with a density of 1 @xmath13 radiates a total energy of 4.7 erg in the soft x - ray regime above 0.1 kev for a time of @xmath410@xmath33 yr . norman & ikeuchi ( @xcite ) investigated the cumulative effect of a number of snrs . the total sfr in the disk of ngc 4303 from the x - ray luminosity amounts to 0.5 m@xmath2 yr@xmath3 . this however is just a very simple estimation , containing several simplifications , as e.g. the sum of the single sn model from cioffi ( @xcite ) for several cumulatively expanding snrs in an evolving ob assoziation , or the derivation of the total disk sfr from single x - ray sources . a more detailed determiation of the sfr , for example using an analytic suberbubble model by suchkov et a. ( @xcite ) , would need information about the extensions and expansion time of the superbubbles , in order to determine the mechanical energy release by the sne and , by this , the sn rate . this can be compared with the observed x - ray luminosity . besides the mentioned restrictions the difference between the sfr estimated from the x - ray flux and the sfr derived by the h@xmath31 flux by kennicutt ( @xcite ) ( 14 m@xmath2 yr@xmath3 ) may be due to several reasons . kennicutt used a miller - scalo imf which increases the sfr by a factor of about 1.5 . the total h@xmath31 luminosity underlies a distance determination with a hubble constant of 50 km s@xmath3 mpc@xmath3 . taking a radial velocity of 1569 km s@xmath3 ( de vaucouleurs et al . @xcite ) leads to a 3.8 times higher luminosity than taking the distance of 16.1 mpc which we used . additionally , the fact that not all regions may emit an adequate x - ray flux or are not strong enough to be detected lowers the estimated sfr from the x - ray luminosity which implies the existence of high - mass stars having been evolved to . possible x - ray emission from diffuse hot gas within the disk may lie below the detection limit of 1.1 cts s@xmath3 arcsec@xmath34 ( 5@xmath35 above background level ) . a very faint component located at the spiral arms can be seen in outlines in fig . [ hrisrcareas ] , but is not detected at a 3@xmath35 level . this limit corresponds to an x - ray flux of 4.6 erg s@xmath3 @xmath7 arcsec@xmath34 ( ecf for a rs model as in table [ tabhrisources ] ) . kennicutt ( @xcite)also admitted to treat the derived h@xmath31 flux and resulting sfr with extreme caution because of only moderate accuracy due to possibly strong extinction effects . the h@xmath31 flux derived by keel ( @xcite ) is by a factor of 30 lower than the one derived by kennicutt , after adopting the same distance . strikingly , the sources b and f both coincide with some of the most h@xmath31 luminous regions ( sources no.s 27 and 69 with log @xmath36=@xmath3712.12 and log @xmath36=@xmath3711.99 , respectively , in @xcite ) . depending on the fraction of the central x - ray flux steming from snrs and superbubbles or from an agn component , the sfr for the core is of the order of 1 m@xmath2 yr@xmath3 . we have analyzed spatial and spectral data from the barred late - type spiral galaxy ngc 4303 in the soft x - ray regime . several separate x - ray sources can be observed in the core and disk of the galaxy . the locations of the sources correspond to several regions and indicate a concentration of gas at the center and at the ends of the galactic bar , in agreement with numerical simulations of gas dynamics in a barred potential . the low spatial resolution of the pspc observation of ngc 4303 does not allow a distinction of several individual x - ray sources within the object . the best fit of the soft x - ray spectrum taking into account the information from the high resolution hri observation is a combination of a rs component with a temperature of 0.3 kev and a power - law component with a spectral index of 2.6 . the total 0.12.4 kev x - ray luminosity amounts to 4.7 erg s@xmath3 , in agreement with other comparable barred galaxies with a nuclear starburst , like e.g. ngc 4569 ( tschke et al . in preparation ) . a pure starburst model for the nucleus of ngc 4303 would require a special explanation for the unusually high @xmath0/@xmath1 ratio . the combination of the flux fraction of the separate sources , the spectral information , and the comparison with the h@xmath31 luminosity from the core leads to the following picture : the soft x - ray emission originates from a composition of several distinct emission regions . the central source consists of a low luminous agn and a circumnuclear starburst . the disk sources are dominated by snrs and superbubbles in star forming regions preferably at the ends of the bar and along the eastern spiral arm . several hmxbs may contribute to the x - ray flux . the disk x - ray sources are coincident with some of the most luminous regions in the galaxy . the estimated total sfr from the x - ray flux is 12 m@xmath2 yr@xmath3 . most regions are not detectable in the x - ray , like most h@xmath31 sources in the eastern boomerang - shaped arm . the enhanced star formation in ngc 4303 may have been caused by some kind of interaction although the morphology of the galaxy does not support very strong perturbation . if a dwarf galaxy has fallen in and merged with ngc 4303 in the past , the bar may have been produced with the subsequent triggering of the star formation at the center and in the spiral arms . the accreted dwarf galaxy would be resolved and not directly detectable . the authors are grateful to dominik bomans for stimulating discussions , and to dr . olga silchenko for her substantial and constructive report . the rosat project is supported by the german bundesministerium fr bildung , wissenschaft , forschung und technologie ( bmbf ) and the max - planck - society . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , caltech , under contract with the nasa . observations made with the nasa / esa hubble space telescope were used , obtained from data archive at stsci . stsci is operated by the association of universities for research in astronomy , inc . 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the late - type galaxy ngc 4303 ( m61 ) is one of the most intensively studied barred galaxies in the virgo cluster .
its prominent enhanced star formation throughout large areas of the disk can be nicely studied due to its low inclination of about 27 .
we present observations of ngc 4303 with the rosat pspc and hri in the soft x - ray ( 0.12.4 kev ) .
the bulk of the x - ray emission is located at the nuclear region .
it contributes more than 80% to the total observed soft x - ray flux .
the extension of the central x - ray source and the @xmath0/@xmath1 ratio point to a low luminous agn ( liner ) with a circumnuclear star - forming region .
several separate disk sources can be distinguished with the hri , coinciding spatially with some of the most luminous regions outside the nucleus of ngc 4303 .
the total star formation rate amounts to 12 m@xmath2 yr@xmath3 .
the x - ray structure follows the distribution of star formation with enhancement at the bar - typical patterns .
the best spectral fit consists of a power - law component ( agn and hmxbs ) and a thermal plasma component of hot gas from supernova remnants and superbubbles .
the total 0.12.4 kev luminosity of ngc 4303 amounts to 5 erg s@xmath3 , consistent with comparable galaxies , like e.g. ngc 4569 .
| 10,735 | 404 |
the afterglow emission of gamma - ray bursts ( grbs ) is generally well described by the blast wave model @xcite . this model details the temporal and spectral behaviour of the emission that is created by external shocks when a collimated ultra - relativistic jet ploughs into the circumburst medium , driving a blast wave ahead of it . the level of collimation , or jet opening angle , has important implications for the energetics of the underlying physical process , progenitor models , and the possible use of grbs as standard candles . the signature of this collimation is an achromatic temporal steepening or ` jet break ' at approximately one day in an otherwise decaying , power - law light curve . since the launch of the _ swift _ satellite it has become clear that this model for grbs can not , in its current form , explain the full complexity of observed light curve features and the lack of observed achromatic temporal breaks . the unexpected features detected , such as steep decays , plateau phases ( e.g. , @xcite ) and a large number of x - ray flares ( e.g. , @xcite ) have revealed the complexity of these sources up to about one day since the initial event , which is yet to be fully understood . these superimposed features also make it difficult to measure the underlying power - law features on which the blast wave model is based , and may lead to misinterpretations of the afterglows . in these proceedings we summarize our interpretation of a sample of 10 _ swift _ grb afterglows which we detail in our paper @xcite . here , we introduce our method of sample selection and analysis , and summarize our main results regarding the constraints we can place on the blast wave parameters : electron energy distribution , @xmath0 , density profile of the circumburst medium , @xmath1 , and the continued energy injection index , @xmath2 . throughout , we use the convention that a power - law flux is given as @xmath3 where @xmath4 is the temporal decay index and @xmath5 is the spectral index . the bursts in our sample were chosen from an inspection of previous literature and from a comparison of the literature of optical data to the pre - reduced _ swift _ x - ray telescope ( xrt ) light curves in the on - line repository up to the end of february 2008 . our sample consists of 10 bursts with x - ray and optical light curves with good enough time coverage to allow for the underlying single power - law , or broken power - law , to be determined . the bursts are also well sampled enough in the x - ray to constrain the spectral indices , @xmath6 . we did not confine our sample to bursts with clear breaks in either the x - ray or optical bands as we wanted to include the possibility of hidden or not very obvious breaks , particularly in the x - ray band @xcite , or late , undetected breaks . light curve analyses were carried out on the pre - reduced , xrt light curves from the on - line repository . for bursts where there was a possible light curve break , x - ray spectra were extracted pre - break and post - break . optical photometric points in various bands were taken from the literature and combined via a simultaneous temporal fit . this fitting allowed us to find the common temporal slope of the optical data and the colour differences between bands . using these colours , the optical data were then shifted to a common magnitude and converted into an arbitrary , scaled flux to produce joint optical and x - ray light curves ( figure[lc ] ) . these light curves were fit with single or broken power - laws , including optical host galaxy contributions where known . data at early times at which the underlying behaviour was ambiguous , or flaring , were excluded from the fit . [ lc ] we use the blast wave model @xcite to describe the temporal and spectral properties of the grb afterglow emission ; we assume on - axis viewing , a uniform jet structure and no evolution of the microphysical parameters . the relations between the temporal and spectral indices and the blast wave parameters that we use are summarised in , e.g. , @xcite . our general method was to estimate the value of the electron energy distribution index , @xmath0 , from the x - ray spectral index and use this to calculate the predicted values of temporal decay . we derive @xmath0 from the spectral index as opposed to the temporal index since for a given spectral index there are only two possible values of @xmath0 , while for a given temporal index there are multiple possible values . spectral slopes are dependent only on @xmath0 and the position of the cooling break . temporal indices , @xmath4 , are dependent on @xmath0 , the position of the cooling break , the circumburst density profile , @xmath1 , and on possible continued energy injection . temporal indices are also prone to being incorrectly estimated from broken power - law fits which may underestimate the post - break indices @xcite . for a given value of the x - ray spectral index , there are two possible values of @xmath0 depending on whether the cooling break , @xmath7 , is below ( @xmath8 ) or above ( @xmath9 ) the x - ray frequency , @xmath10 . if the optical to x - ray sed does not display a break then the cooling break can either be above the x - ray regime or below the optical regime and the blast wave predictions of each @xmath0 are compared to the observed temporal slopes to discern which is correct . if the sed requires a broken power - law it most likely implies that a cooling break lies between the two regimes and is below the x - ray regime . a cooling break requires , or must be consistent with , a difference between the spectral slopes of @xmath11 . however , a break between the two regimes does not necessarily imply a cooling break ; it may be due to the fact that each regime has a different spectral index since they are originating from different emission regions . in this case the spectral break does not have a predictable difference between slopes . for this interpretation to work , one must be able to explain why the emission from each region is only visible in one spectral regime and it s power - law slope does not extend to the other . a cooling break is a more likely explanation in the majority of cases but a comparison of the blast wave predictions of each @xmath0 with the observed light curves is required . after comparing the derived values of @xmath0 and the predictions of the blast wave model to the observed temporal and spectral properties , we find the most likely values of the parameters @xmath0 , @xmath1 and @xmath2 for each burst in our sample . we find that : * 8/10 are consistent with the blast wave model * 6/10 have an unambiguous value of @xmath0 * 6/10 have a calculable value of @xmath1 * 4/10 require energy injection , @xmath2 * 5/10 exhibit a jet break [ p ] after our interpretation of individual bursts . the line represents the most likely value of @xmath0 over the plotted sample.,title="fig : " ] the universality of the electron energy distribution index , @xmath0 , has been examined by several authors @xcite who applied different methods to samples of _ bepposax _ and _ swift _ bursts , all reaching the conclusion that the observed range of @xmath0 values is not consistent with a single central value of @xmath0 but displays a width of the parent distribution . in the studies so far there have been some limitations : the _ bepposax _ sample is limited , both in the number of grbs and the temporal and spectral sampling ; and the only study of _ swift _ bursts for this purpose so far @xcite , only used the x - ray afterglows , which introduces a large uncertainty because the position of the cooling break is unknown . here we examine the universality of @xmath0 given the observed distribution of @xmath0 ( figure[p ] ) from our sample of _ swift _ bursts , using the same methods as described in @xcite . we first find the most likely value of @xmath12 and test that the observed distribution can be obtained from a parent distribution with a single central value of @xmath0 . we do so by generating different synthetic sets of @xmath0 for the bursts in our sample and calculating the most likely value of @xmath0 for each . we conclude that a single value of @xmath0 is rejected at the @xmath13 level and that the width of the parent distribution is @xmath14 at the @xmath15 level . we also tested the values of @xmath0 , for each of the 10 bursts in our sample , that offered the least deviation from the expected canonical value of @xmath16 . in this case , the most likely value of @xmath0 is @xmath17 and the values are still inconsistent with one central value at the @xmath18 level . this result confirms the results from previous studies and has important implications for theoretical particle acceleration studies . some of these ( semi-)analytical calculations and simulations indicate that there is a nearly universal value of @xmath19 ( e.g. @xcite ) , while other studies suggest that there is a large range of possible values for @xmath0 of @xmath20 @xcite . although we find that there is not a universal value of @xmath0 , our values for the width of the parent distribution indicate that it is not as wide as the latter study suggest . our result is comparable to the numbers found by other authors @xcite but is based on a sample with better temporal and spectral sampling per grb , on average . the density structure , or profile , of the circumburst medium is generally given as @xmath21 , or @xmath22 ( number density ) , @xmath23 where @xmath24 is a constant density , or ism - like , medium and @xmath25 is a wind - like medium . the value of @xmath1 has important implications for the study of progenitor models , as the currently favoured model , involving the collapse of a massive , wolf - rayet star , is expected to have an associated strong stellar wind affecting the circumburst environment . however , detailed broadband modeling studies on a small number of grbs @xcite have found that although such a wind is favoured in many cases , a constant density medium is favoured in many other cases . [ k ] after our interpretation of individual bursts . the lines represent @xmath24 ( constant density ) and @xmath25 ( wind like ) media.,title="fig : " ] in our sample of 10 bursts , only 6 have optical or x - ray light curves below the cooling break , @xmath7 , where they are dependent on the circumburst density profile . of our calculated values of @xmath1 ( figure[k ] ) , 2 are consistent ( @xmath26 ) with both @xmath25 and @xmath24 ; 2 are best described with a wind - like medium and are inconsistent ( @xmath18 ) with @xmath24 ; and 2 are consistent with a constant density medium but inconsistent with @xmath25 . our results are hence in agreement with those of the previous , similar studies insofar as the sample requires both constant density and wind driven media to explain the observed broadband emission . [ q ] , including those with @xmath2 set as zero ( no energy injection ) on line , after our interpretation of individual bursts.,title="fig : " ] continued energy injection was proposed as a process to explain the shallower than expected decay indices observed in many _ swift _ bursts @xcite . we assume that it takes the form of @xmath27 where @xmath2 is dependent on the shallowing effect of the injection , @xmath28 , and on the value of @xmath0 . this continued energy injection is either due to @xmath29 a distribution of the lorentz factors of the shells ejected from the central engine which causes those shells with lower lorentz factor to catch up with the blast wave at a later time , or @xmath30 continued activity of the central engine itself . the former has a limit of @xmath31 , where @xmath1 is the density profile of the circumburst medium , which has been suggested as a diagnostic to differentiate between the two sources of continued energy injection @xcite . for 6 of the 10 light curves in our sample we are able to estimate the level of energy injection , or to say that it is consistent with zero ( figure[q ] ) , though in one burst there are two equally valid interpretations , one requiring energy injection while the other does not . excluding this burst , energy injection is only required in 3 out of 6 bursts and each of these has an injection index , @xmath2 , of between 0.65 and 1.0 , consistent with both scenarios of energy injection . clearly a much larger sample of afterglows is required to be able to constrain the sample properties of @xmath2 . throughout this paper we have applied the blast wave model @xcite , assuming on - axis viewing , a standard jet structure and no evolution of the microphysical parameters , to a selection of 10 _ swift _ grbs with well sampled temporal and spectral data . we attempted to constrain three parameters of interest ( @xmath0 , @xmath1 and @xmath2 ) and to test the validity of the blast wave model for light curves observed by _ swift _ , the complexity of which has made their interpretation and that of the blast wave more difficult than in the pre-_swift _ era . we find that the majority of the afterglows are well described within the frame work of the blast wave model and that the parameters derived are consistent with those values found by previous authors . furthermore , we identify , reasonably unambiguously , jet breaks in 5 afterglows out of our sample of 10 . after interpretation within the blast wave model , we are able to confidently estimate the electron energy distribution index , @xmath0 , for 6 of the bursts in our sample . a statistical analysis of the distribution of @xmath0 reveals that , even in the most conservative case of least scatter , the values are not consistent with a single , universal value suggested by some studies ; this has important implications for theoretical particle acceleration studies . in a number of cases , we are also able to obtain values for the circumburst density profile index , @xmath1 , and the index of continued energy injection , @xmath2 . the calculated values of @xmath2 are consistent with both suggested sources of continued energy injection and a much larger sample of afterglows will be required to constrain the sample properties of @xmath2 . the values of @xmath1 , consistent with previous works on the matter , suggest that the circumburst density profiles are not drawn from only one of the constant density or wind - like media populations . we thank p.a . evans & k.l . page for useful discussions on the xrt . pac & ramjw gratefully acknowledge support of nwo under vici grant 639.043.302 . pac & rlcs acknowledge support from stfc . ajvdh was supported by an appointment to the nasa postdoctoral program at the msfc , administered by oak ridge associated universities through a contract with nasa .
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the complex structure of the light curves of _ swift _ grbs has made their interpretation and that of the blast wave caused by the burst , more difficult than in the pre-_swift _ era .
we aim to constrain the blast wave parameters : electron energy distribution , @xmath0 , density profile of the circumburst medium , @xmath1 , and the continued energy injection index , @xmath2 .
we do so by comparing the observed multi - wavelength light curves and x - ray spectra of a _ swift _ sample to the predictions of the blast wave model
. we can successfully interpret all of the bursts in our sample of 10 , except two , within the framework of the blast wave model , and we can estimate with confidence the electron energy distribution index for 6 of the sample .
furthermore we identify jet breaks in half of the bursts .
a statistical analysis of the distribution of @xmath0 reveals that , even in the most conservative case of least scatter , the values are not consistent with a single , universal value .
the values of @xmath1 suggest that the circumburst density profiles are not drawn from only one of the constant density or wind - like media populations .
address = mullard space science laboratory , university college london , holmbury st .
mary , dorking rh5 6nt , uk , altaddress = astronomical institute , university of amsterdam , kruislaan 403 , 1098sj amsterdam , the netherlands address = nasa postdoctoral program fellow ,
nsstc , 320 sparkman drive , huntsville , al 35805 , usa address = department of physics and astronomy , university of leicester , university road , leicester le1 7rh , uk address = astronomical institute , university of amsterdam , kruislaan 403 , 1098sj amsterdam , the netherlands
| 3,983 | 503 |
a great deal of progress was recently achieved in our understanding of the multifragmentation phenomenon @xcite when an exact analytical solution of a simplified version of the statistical multifragmentation model ( smm ) @xcite was found in refs . an invention of a new powerful mathematical method @xcite , the laplace - fourier transform , allowed us not only to solve this version of smm analytically for finite volumes @xcite , but to find the surface partition and surface entropy of large clusters for a variety of statistical ensembles @xcite . it was shown @xcite that for finite volumes the analysis of the grand canonical partition ( gcp ) of the simplified smm is reduced to the analysis of the simple poles of the corresponding isobaric partition , obtained as a laplace - fourier transform of the gcp . this method opens a principally new possibility to study the nuclear liquid - gas phase transition directly from the partition of finite system and without taking its thermodynamic limit . exactly solvable models with phase transitions play a special role in the statistical physics - they are the benchmarks of our understanding of critical phenomena that occur in more complicated substances . they are our theoretical laboratories , where we can study the most fundamental problems of critical phenomena which can not be studied elsewhere . note that these questions _ in principle _ can not be clarified either within the widely used mean - filed approach or numerically . despite this success , the application of the exact solution @xcite to the description of experimental data is limited because this solution corresponds to an infinite system volume . therefore , from a practical point of view it is necessary to extend the formalism for finite volumes . such an extension is also necessary because , despite a general success in the understanding the nuclear multifragmentation , there is a lack of a systematic and rigorous theoretical approach to study the phase transition phenomena in finite systems . for instance , even the best formulation of the statistical mechanics and thermodynamics of finite systems by hill @xcite is not rigorous while discussing the phase transitions . exactly solvable models of phase transitions applied to finite systems may provide us with the first principle results unspoiled by the additional simplifying assumptions . here we present a finite volume extension of the smm . to have a more realistic model for finite volumes , we would like to account for the finite size and geometrical shape of the largest fragments , when they are comparable with the system volume . for this we will abandon the arbitrary size of largest fragment and consider the constrained smm ( csmm ) in which the largest fragment size is explicitly related to the volume @xmath0 of the system . a similar model , but with the fixed size of the largest fragment , was recently analyzed in ref . @xcite . in this work we will : solve the csmm analytically at finite volumes using a new powerful method ; consider how the first order phase transition develops from the singularities of the smm isobaric partition @xcite in thermodynamic limit ; study the finite volume analogs of phases ; and discuss the finite size effects for large fragments . the system states in the smm are specified by the multiplicity sets @xmath1 ( @xmath2 ) of @xmath3-nucleon fragments . the partition function of a single fragment with @xmath3 nucleons is @xcite : @xmath4 , where @xmath5 ( @xmath6 is the total number of nucleons in the system ) , @xmath0 and @xmath7 are , respectively , the volume and the temperature of the system , @xmath8 is the nucleon mass . the first two factors on the right hand side ( r.h.s . ) of the single fragment partition originate from the non - relativistic thermal motion and the last factor , @xmath9 , represents the intrinsic partition function of the @xmath3-nucleon fragment . therefore , the function @xmath10 is a phase space density of the k - nucleon fragment . for ( nucleon ) we take @xmath11 ( 4 internal spin - isospin states ) and for fragments with @xmath12 we use the expression motivated by the liquid drop model ( see details in @xmath13 with fragment free energy @xmath14k + \sigma ( t)~ k^{2/3}+ ( \tau + 3/2 ) t\ln k~ , % \ ] ] with @xmath15 . here mev is the bulk binding energy per nucleon . @xmath17 is the contribution of the excited states taken in the fermi - gas approximation ( @xmath18 mev ) . @xmath19 is the temperature dependent surface tension parameterized in the following relation : @xmath20^{5/4 } , $ ] with @xmath21 mev and @xmath22 mev ( @xmath23 at @xmath24 ) . the last contribution in eq . ( [ one ] ) involves the famous fisher s term with dimensionless parameter @xmath25 . the canonical partition function ( cpf ) of nuclear fragments in the smm has the following form : @xmath26^{n_k}}{n_k ! } \biggr ] { \textstyle \delta(a-\sum_k kn_k)}\ , . % \ ] ] in eq . ( [ two ] ) the nuclear fragments are treated as point - like objects . however , these fragments have non - zero proper volumes and they should not overlap in the coordinate space . in the excluded volume ( van der waals ) approximation this is achieved by substituting the total volume @xmath0 in eq . ( [ two ] ) by the free ( available ) volume @xmath27 , where @xmath28 ( @xmath29 @xmath30 is the normal nuclear density ) . therefore , the corrected cpf becomes : @xmath31 . the smm defined by eq . ( [ two ] ) was studied numerically in refs . this is a simplified version of the smm , e.g. the symmetry and coulomb contributions are neglected . however , its investigation appears to be of principal importance for studies of the liquid - gas phase transition . the calculation of @xmath32 is difficult due to the constraint @xmath33 . this difficulty can be partly avoided by evaluating the grand canonical partition ( gcp ) @xmath34 where @xmath35 denotes a chemical potential . the calculation of @xmath36 is still rather difficult . the summation over @xmath1 sets in @xmath37 can not be performed analytically because of additional @xmath6-dependence in the free volume @xmath38 and the restriction @xmath39 . this problem was resolved @xcite by the laplace transformation method to the so - called isobaric ensemble @xcite . in this work we would like to consider a more strict constraint @xmath40 , where the size of the largest fragment @xmath41 can not exceed the total volume of the system ( the parameter @xmath42 is introduced for convenience ) . the case @xmath43 is also included in our treatment . a similar restriction should be also applied to the upper limit of the product in all partitions @xmath44 , @xmath32 and @xmath45 introduced above ( how to deal with the real values of @xmath46 , see later ) . then the model with this constraint , the csmm , can not be solved by the laplace transform method , because the volume integrals can not be evaluated due to a complicated functional @xmath0-dependence . however , the csmm can be solved analytically with the help of the following identity @xmath47 which is based on the fourier representation of the dirac @xmath48-function . the representation ( [ four ] ) allows us to decouple the additional volume dependence and reduce it to the exponential one , which can be dealt by the usual laplace transformation in the following sequence of steps @xmath49 \theta(v^\prime ) = \nonumber \\ % % & & \hspace*{-0.1cm}\int_0^{\infty}\hspace*{-0.2cm}dv^{\prime } % \int\limits_{-\infty}^{+\infty } d \xi~ \int\limits_{-\infty}^{+\infty } % \frac{d \eta}{{2 \pi } } ~ { \textstyle e^ { i \eta ( v^\prime - \xi ) - \lambda v^{\prime } % + v^\prime { \cal f}(\xi , \lambda - i \eta ) } } \ , . % % % % % { \textstyle e^ { v^\prime { \cal f}(\xi , \lambda - i \eta ) } } ~. % \end{aligned}\ ] ] after changing the integration variable @xmath50 , the constraint of @xmath51-function has disappeared . then all @xmath52 were summed independently leading to the exponential function . now the integration over @xmath53 in eq . ( [ five ] ) can be straightforwardly done resulting in @xmath54 where the function @xmath55 is defined as follows @xmath56\,.\ , % \end{aligned}\ ] ] as usual , in order to find the gcp by the inverse laplace transformation , it is necessary to study the structure of singularities of the isobaric partition ( [ seven ] ) . the isobaric partition ( [ seven ] ) of the csmm is , of course , more complicated than its smm analog @xcite because for finite volumes the structure of singularities in the csmm is much richer than in the smm , and they match in the limit @xmath57 only . to see this let us first make the inverse laplace transform : @xmath58^{-1 } } \ , , % \end{aligned}\ ] ] where the contour @xmath59-integral is reduced to the sum over the residues of all singular points @xmath60 with @xmath61 , since this contour in the complex @xmath59-plane obeys the inequality @xmath62 . now both remaining integrations in ( [ eight ] ) can be done , and the gcp becomes @xmath63^{-1 } } \ , , % % \ ] ] i.e. the double integral in ( [ eight ] ) simply reduces to the substitution @xmath64 in the sum over singularities . this is a remarkable result which can be formulated as the following the simple poles in ( [ eight ] ) are defined by the equation @xmath65 in contrast to the usual smm @xcite the singularities @xmath66 are ( i ) are volume dependent functions , if @xmath46 is not constant , and ( ii ) they can have a non - zero imaginary part , but in this case there exist pairs of complex conjugate roots of ( [ ten ] ) because the gcp is real . introducing the real @xmath67 and imaginary @xmath68 parts of @xmath69 , we can rewrite eq . ( [ ten ] ) as a system of coupled transcendental equations @xmath70 where we have introduced the set of the effective chemical potentials @xmath71 with @xmath72 , and the reduced distributions @xmath73 and @xmath74 for convenience . consider the real root @xmath75 , first . for @xmath76 the real root @xmath77 exists for any @xmath7 and @xmath35 . comparing @xmath77 with the expression for vapor pressure of the analytical smm solution @xcite shows that @xmath78 is a constrained grand canonical pressure of the gas . as usual , for finite volumes the total mechanical pressure @xcite , as we will see in section v , differs from @xmath78 . equation ( [ twelve ] ) shows that for @xmath79 the inequality @xmath80 never become the equality for all @xmath3-values simultaneously . then from eq . ( [ eleven ] ) one obtains ( @xmath81 ) @xmath82 where the second inequality ( [ thirteen ] ) immediately follows from the first one . in other words , the gas singularity is always the rightmost one . this fact plays a decisive role in the thermodynamic limit @xmath57 . the interpretation of the complex roots @xmath83 is less straightforward . according to eq . ( [ nine ] ) , the gcp is a superposition of the states of different free energies @xmath84 . ( strictly speaking , @xmath84 has a meaning of the change of free energy , but we will use the traditional term for it . ) for @xmath81 the free energies are complex . therefore , @xmath85 is the density of free energy . the real part of the free energy density , @xmath86 , defines the significance of the state s contribution to the partition : due to ( [ thirteen ] ) the largest contribution always comes from the gaseous state and has the smallest real part of free energy density . as usual , the states which do not have the smallest value of the ( real part of ) free energy , i. e. @xmath87 , are thermodynamically metastable . for infinite volume they should not contribute unless they are infinitesimally close to @xmath88 , but for finite volumes their contribution to the gcp may be important . as one sees from ( [ eleven ] ) and ( [ twelve ] ) , the states of different free energies have different values of the effective chemical potential @xmath89 , which is not the case for infinite volume @xcite , where there exists a single value for the effective chemical potential . thus , for finite @xmath0 the states which contribute to the gcp ( [ nine ] ) are not in a true chemical equilibrium . the meaning of the imaginary part of the free energy density becomes clear from ( [ eleven ] ) and ( [ twelve ] ) : as one can see from ( [ eleven ] ) the imaginary part @xmath90 effectively changes the number of degrees of freedom of each @xmath3-nucleon fragment ( @xmath91 ) contribution to the free energy density @xmath87 . it is clear , that the change of the effective number of degrees of freedom can occur virtually only and , if @xmath83 state is accompanied by some kind of equilibration process . both of these statements become clear , if we recall that the statistical operator in statistical mechanics and the quantum mechanical convolution operator are related by the wick rotation @xcite . in other words , the inverse temperature can be considered as an imaginary time . therefore , depending on the sign , the quantity @xmath92 that appears in the trigonometric functions of the equations ( [ eleven ] ) and ( [ twelve ] ) in front of the imaginary time @xmath93 can be regarded as the inverse decay / formation time @xmath94 of the metastable state which corresponds to the pole @xmath83 ( for more details see next sections ) . as will be shown further , for @xmath95 the inverse chemical potential can be considered as a characteristic equilibration time as well . this interpretation of @xmath94 naturally explains the thermodynamic metastability of all states except the gaseous one : the metastable states can exist in the system only virtually because of their finite decay / formation time , whereas the gaseous state is stable because it has an infinite decay / formation time . ) for @xmath96 mev and @xmath97 . the l.h.s . ( straight line ) and r.h.s . of eq . ( [ twelve ] ) ( all dashed curves ) are shown as the function of dimensionless parameter @xmath98 for the three values of the largest fragment size @xmath46 . the intersection point at @xmath99 corresponds to a real root of eq . ( [ ten ] ) . each tangent point with the straight line generates two complex roots of ( [ ten ] ) . , width=325,height=226 ] it is instructive to treat the effective chemical potential @xmath100 as an independent variable instead of @xmath35 . in contrast to the infinite @xmath0 , where the upper limit @xmath101 defines the liquid phase singularity of the isobaric partition and gives the pressure of a liquid phase @xmath102 @xcite , for finite volumes and finite @xmath46 the effective chemical potential can be complex ( with either sign for its real part ) and its value defines the number and position of the imaginary roots @xmath103 in the complex plane . positive and negative values of the effective chemical potential for finite systems were considered @xcite within the fisher droplet model , but , to our knowledge , its complex values have never been discussed . from the definition of the effective chemical potential @xmath104 it is evident that its complex values for finite systems exist only because of the excluded volume interaction , which is not taken into account in the fisher droplet model @xcite . as it is seen from fig . 1 , the r.h.s . of eq . ( [ twelve ] ) is the amplitude and frequency modulated sine - like function of dimensionless parameter @xmath105 . therefore , depending on @xmath7 and @xmath106 values , there may exist no complex roots @xmath107 , a finite number of them , or an infinite number of them . in fig . 1 we showed a special case which corresponds to exactly three roots of eq . ( [ ten ] ) for each value of @xmath46 : the real root ( @xmath108 ) and two complex conjugate roots ( @xmath109 ) . since the r.h.s . of ( [ twelve ] ) is monotonously increasing function of @xmath106 , when the former is positive , it is possible to map the @xmath110 plane into regions of a fixed number of roots of eq . ( [ ten ] ) . each curve in divides the @xmath110 plane into three parts : for @xmath106-values below the curve there is only one real root ( gaseous phase ) , for points on the curve there exist three roots , and above the curve there are five or more roots of eq . ( [ ten ] ) . for constant values of @xmath111 the number of terms in the r.h.s . of ( [ twelve ] ) does not depend on the volume and , consequently , in thermodynamic limit @xmath57 only the farthest right simple pole in the complex @xmath59-plane survives out of a finite number of simple poles . according to the inequality ( [ thirteen ] ) , the real root @xmath112 is the farthest right singularity of isobaric partition ( [ six ] ) . however , there is a possibility that the real parts of other roots @xmath113 become infinitesimally close to @xmath77 , when there is an infinite number of terms which contribute to the gcp ( [ nine ] ) . region of one real root of eq . ( [ ten ] ) ( below the curve ) , three complex roots ( at the curve ) and five and more roots ( above the curve ) for three values of @xmath46 and the same parameters as in fig . 1 . , width=325,height=226 ] let us show now that even for an infinite number of simple poles in ( [ nine ] ) only the real root @xmath112 survives in the limit @xmath57 . for this purpose consider the limit @xmath114 . in this limit the distance between the imaginary parts of the nearest roots remains finite even for infinite volume . indeed , for @xmath115 the leading contribution to the r.h.s . of ( [ twelve ] ) corresponds to the harmonic with @xmath116 , and , consequently , an exponentially large amplitude of this term can be only compensated by a vanishing value of @xmath117 , i.e. @xmath118 with @xmath119 ( hereafter we will analyze only the branch @xmath120 ) , and , therefore , the corresponding decay / formation time @xmath121^{-1}$ ] is volume independent . keeping the leading term on the r.h.s . of ( [ twelve ] ) and solving for @xmath122 , one finds @xmath123 where in the last step we used eq . ( [ eleven ] ) and condition @xmath119 . since for @xmath57 all negative values of @xmath67 can not contribute to the gcp ( [ nine ] ) , it is sufficient to analyze even values of @xmath124 which , according to ( [ msixteen ] ) , generate @xmath125 . since the inequality ( [ thirteen ] ) can not be broken , a single possibility , when @xmath126 pole can contribute to the partition ( [ nine ] ) , corresponds to the case @xmath127 for some finite @xmath124 . assuming this , we find @xmath128 for the same value of @xmath35 . substituting these results into equation ( [ eleven ] ) , one gets @xmath129 \ll r_0\ , . % \ ] ] the inequality ( [ mseventeen ] ) follows from the equation for @xmath77 and the fact that , even for equal leading terms in the sums above ( with @xmath130 and even @xmath124 ) , the difference between @xmath77 and @xmath67 is large due to the next to leading term @xmath131 , which is proportional to @xmath132 . thus , we arrive at a contradiction with our assumption @xmath133 , and , consequently , it can not be true . therefore , for large volumes the real root @xmath112 always gives the main contribution to the gcp ( [ nine ] ) , and this is the only root that survives in the limit @xmath57 . thus , we showed that the model with the fixed size of the largest fragment has no phase transition because there is a single singularity of the isobaric partition ( [ six ] ) , which exists in thermodynamic limit . if @xmath46 monotonically grows with the volume , the situation is different . in this case for positive value of @xmath134 the leading exponent in the r.h.s . of ( [ twelve ] ) also corresponds to a largest fragment , i.e. to @xmath135 . therefore , we can apply the same arguments which were used above for the case @xmath136 and derive similarly equations ( [ mfourteen])([msixteen ] ) for @xmath68 and @xmath67 . from @xmath137 it follows that , when @xmath0 increases , the number of simple poles in ( [ eight ] ) also increases and the imaginary part of the closest to the real @xmath59-axis poles becomes very small , i.e @xmath138 for @xmath139 , and , consequently , the associated decay / formation time @xmath140^{-1}$ ] grows with the volume of the system . due to @xmath141 , the inequality ( [ mseventeen ] ) can not be established for the poles with @xmath139 . therefore , in contrast to the previous case , for large @xmath46 the simple poles with @xmath139 will be infinitesimally close to the real axis of the complex @xmath59-plane . from eq . ( [ msixteen ] ) it follows that @xmath142 for @xmath143 and @xmath144 . thus , we proved that for infinite volume the infinite number of simple poles moves toward the real @xmath59-axis to the vicinity of liquid phase singularity @xmath145 of the isobaric partition @xcite and generates an essential singularity of function @xmath146 in ( [ seven ] ) _ irrespective to the sign of chemical potential @xmath35 . _ as we showed above , the states with @xmath147 become stable because they acquire infinitely large decay / formation time @xmath94 in the limit @xmath57 . therefore , these states should be identified as a liquid phase for finite volumes as well . such a conclusion can be easily understood , if we recall that the partial pressure @xmath148 of ( [ meighteen ] ) corresponds to a single fragment of the largest possible size . now it is clear that each curve in fig . 2 is the finite volume analog of the phase boundary @xmath149 for a given value of @xmath46 : below the phase boundary there exists a gaseous phase , but at and above each curve there are states which can be identified with a finite volume analog of the mixed phase , and , finally , at @xmath150 there exists a liquid phase . when there is no phase transition , i.e. @xmath151 , the structure of simple poles is similar , but , first , the line which separates the gaseous states from the metastable states does not change with the volume , and , second , as shown above , the metastable states will never become stable . therefore , a systematic study of the volume dependence of free energy ( or pressure for very large @xmath0 ) along with the formation and decay times may be of a crucial importance for experimental studies of the nuclear liquid gas phase transition . the above results demonstrate that , in contrast to hill s expectations @xcite , the finite volume analog of the mixed phase does not consist just of two pure phases . the mixed phase for finite volumes consists of a stable gaseous phase and the set of metastable states which differ by the free energy . moreover , the difference between the free energies of these states is not surface - like , as hill assumed in his treatment @xcite , but volume - like . furthermore , according to eqs . ( [ eleven ] ) and ( [ twelve ] ) , each of these states consists of the same fragments , but with different weights . as seen above for the case @xmath150 , some fragments that belong to the states , in which the largest fragment is dominant , may , in principle , have negative weights ( effective number of degrees of freedom ) in the expression for @xmath152 ( [ eleven ] ) . this can be understood easily because higher concentrations of large fragments can be achieved at the expense of the smaller fragments and is reflected in the corresponding change of the real part of the free energy @xmath153 . therefore , the actual structure of the mixed phase at finite volumes is more complicated than was expected in earlier works . a similar situation occurs for the real values of @xmath46 . in this case all sums in eqs . ( [ ten]-[thirteen ] ) should be expressed via the euler - maclaurin formula @xmath154 @xmath155 here @xmath156 are the bernoulli numbers . the representation ( [ fourteen ] ) allows one to study the effect of finite volume ( fv ) on the gcp ( [ nine ] ) . the above results are valid for any @xmath46 dependence . however , the linear one , i.e. @xmath157 with @xmath158 , is the most natural . with the help of the parameter @xmath42 it is possible to describe a difference between the geometrical shape of the volume under consideration and that one of the largest fragment . for instance , by fixing @xmath159 it is possible to account for the fact that the largest spherical fragment can not fill completely the cube with the side equal to its two radii , while there is enough space available for small fragments . due to the @xmath160 dependence in the csmm there are two different ways of how the finite volume affects thermodynamical functions : for finite @xmath0 and @xmath161 there is always a finite number of simple poles in ( [ nine ] ) , but their number and positions in the complex @xmath59-plane depend on @xmath0 . to see this , let us study the mechanical pressure which corresponds to the gcp ( [ nine ] ) @xmath162 ^ 2 } % \biggl\ { b^2\,\frac { \partial \lambda _ n}{\partial v } \sum\limits_{k=1}^{k(v ) } \tilde\phi_k \,k^2 % ~{\textstyle e^{\frac{\nu_n\,k}{t } } } + \tilde\phi_{k(v ) } \times % % \nonumber \\ % & & \hspace*{-0.7 cm } % { \textstyle e^{\frac{\nu_n\,k(v)}{t } } } % { \textstyle k(v ) \left [ 1 - \alpha + \frac{\nu_n}{t } \left ( \frac{1}{2 } - \alpha \right ) \right]}+ % o(k(v))\biggr\ } \biggr ] \hspace*{-0.05 cm } , % % % \right\ } % \end{aligned}\ ] ] where we give the main term for each @xmath163 and leading fv corrections explicitly for @xmath164 , whereas @xmath165 accumulates the higher order corrections due to the euler - maclaurin eq . ( [ fourteen ] ) . in evaluation of ( [ fifteen ] ) we used an explicit representation of the derivative @xmath166 which can be found from eqs . ( [ ten ] ) and ( [ fourteen ] ) . the first term in the r.h.s . of ( [ fifteen ] ) describes the constrained grand canonical ( cgc ) complex pressure generated by the simple pole @xmath167 ( due to its free energy density @xmath168 ) weighted with the `` probability '' @xmath169 , whereas the second and third terms appear due to the volume dependence of @xmath46 . note that , instead of the fv corrections , the usage of natural values for @xmath46 would generate the artificial delta - function terms in ( [ fifteen ] ) for the volume derivatives . now it is clear that in case @xmath151 the corrections to the main term will not appear , and the number of poles and their positions will be defined by values of @xmath7 and @xmath35 only . as one can see from ( [ fifteen ] ) , for finite volumes the corrections can give a non - negligible contribution to the pressure because in this case @xmath170 can be positive . the real parts of the partial cgc pressures @xmath171 may have either sign . therefore , if the fv corrections to the pressure ( [ fifteen ] ) are small , then , according to ( [ thirteen ] ) , the positive cgc pressures @xmath172 are mechanically metastable and the negative ones @xmath173 are mechanically unstable compared to the gas pressure @xmath78 . the fv corrections should be accounted for to find the mechanically meta- and unstable states in the general case . however , it is clear that the contribution of the states with @xmath173 into partition and its derivatives is exponentially small even for finite volumes . as we showed earlier in this section , when @xmath0 increases , the number of simple poles in ( [ eight ] ) also increases and imaginary part of the closest to the real @xmath59-axis poles becomes very small . therefore , for infinite volume the infinite number of simple poles moves toward the real @xmath59-axis to the vicinity of liquid phase singularity @xmath174 and , thus , generates an essential singularity of function @xmath146 in ( [ seven ] ) . in this case the contribution of any of remote poles from the real @xmath59-axis to the gcp vanishes . then it can be shown that the fv corrections in ( [ fifteen ] ) become negligible because of the inequality @xmath175 , and , consequently , the reduced distribution of largest fragment @xmath176 and the derivatives @xmath166 vanish for all @xmath7-values , and we obtain the usual smm solution @xcite . its thermodynamics , as we discussed , is governed by the farthest right singularity in the complex @xmath59-plane . in this work we discussed a powerful mathematical method which allowed us to solve analytically the csmm at finite volumes . it is shown that for finite volumes the gcp function can be identically rewritten in terms of the simple poles of the isobaric partition ( [ six ] ) . the real pole @xmath112 exists always and the quantity @xmath177 is the cgc pressure of the gaseous phase . the complex roots @xmath126 appear as pairs of complex conjugate solutions of equation ( [ ten ] ) . as we discussed , their most straightforward interpretation is as follows : @xmath178 has a meaning of the free energy density , whereas @xmath179 , depending on sign , gives the inverse decay / formation time of such a state . the gaseous state is always stable because its decay / formation time is infinite and because it has the smallest value of free energy . the complex poles describe the metastable states for @xmath180 and mechanically unstable states for @xmath181 . we studied the volume dependence of the simple poles and found a dramatic difference in their behavior in case of phase transition and without it . for the former this representation allows one also to define the finite volume analogs of phases unambiguously and to establish the finite volume analog of the @xmath149 phase diagram ( see fig . 2 ) . at finite volumes the gaseous phase exists , if there is a single simple pole , the mixed phase corresponds to three and more simple poles , whereas the liquid is represented by an infinite amount of simple poles at highest possible particle density ( or @xmath95 ) . as we showed , for given @xmath7 and @xmath35 the states of the mixed phase which have different @xmath182 are not in a true chemical equilibrium for finite volumes . this feature can not be obtained within the fisher droplet model due to lack of the hard core repulsion between fragments . this fact also demonstrates clearly that , in contrast to hill s expectations @xcite , the mixed phase is not just a composition of two states which are the pure phases . as we showed , the mixed phase is a superposition of three and more collective states , and each of them is characterized by its own value of @xmath183 . because of that the difference between the free energies of these states is not a surface - like , as hill argued @xcite , but volume - like . for the case with phase transition , i.e. for @xmath184 , we analyzed what happens in thermodynamic limit . when @xmath0 grows , the number of simple poles in ( [ eight ] ) also increases and imaginary part of the closest to the real @xmath59-axis poles becomes vanishing . for infinite volume the infinite number of simple poles moves toward the real @xmath59-axis and forms an essential singularity of function @xmath146 in ( [ seven ] ) , which defines the liquid phase singularity @xmath174 . thus , we showed how the phase transition develops in thermodynamic limit . also we analyzed the finite volume corrections to the mechanical pressure ( [ fifteen ] ) . the corrections of a similar kind should appear in the entropy , particle number and energy density because of the @xmath7 and @xmath35 dependence of @xmath163 due to ( [ ten ] ) @xcite . therefore , these corrections should be taking into account while analyzing the experimental yields of fragments . then the phase diagram of the nuclear liquid - gas phase transition can be recovered from the experiments on finite systems ( nuclei ) with more confidence . a detailed analysis of the isobaric partition singularities in the @xmath185 plane allowed us to define the finite volume analogs of phases and study the behavior of these singularities in the limit @xmath57 . such an analysis opens a possibility to study rigorously the nuclear liquid - gas phase transition directly from the finite volume partition . this may help to extract the phase diagram of the nuclear liquid - gas phase transition from the experiments on finite systems ( nuclei ) with more confidence . s. das gupta , a. majumder , s. pratt and a. mekjian , arxiv : nucl - th/9903007 ( 1999 ) . k. a. bugaev , m. i. gorenstein , i. n. mishustin and w. greiner , phys . rev . * c62 * ( 2000 ) 044320 ; arxiv : nucl - th/0007062 ( 2000 ) . k. a. bugaev , m. i. gorenstein , i. n. mishustin and w. greiner , phys . lett . * b 498 * ( 2001 ) 144 ; arxiv : nucl - th/0103075 ( 2001 ) . p. t. reuter and k. a. bugaev , phys . b * 517 * ( 2001 ) 233 . k. a. bugaev , arxiv : nucl - th/0406033 ( 2004 ) . k. a. bugaev , l. phair and j. b. elliott , arxiv : nucl - th/0406034 ( 2004 ) ; k. a. bugaev and j. b. elliott , arxiv : nucl - th/0501080 ( 2005 ) .
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we discuss an exact analytical solution of a simplified version of the statistical multifragmentation model with the restriction that the largest fragment size can not exceed the finite volume of the system .
a complete analysis of the isobaric partition singularities of this model is done for finite volumes .
it is shown that the real part of any simple pole of the isobaric partition defines the free energy of the corresponding state , whereas its imaginary part , depending on the sign , defines the inverse decay / formation time of this state .
the developed formalism allows us , for the first time , to exactly define the finite volume analogs of gaseous , liquid and mixed phases of this model from the first principles of statistical mechanics and demonstrate the pitfalls of earlier works .
the finite size effects for large fragments and the role of metastable ( unstable ) states are discussed .
numbers : 25.70 .
pq , 21.65.+f , 24.10 . pa
| 9,879 | 226 |
the post - main - sequence evolution of massive stars depends sensitively on the helium core mass and its ratio to the envelope mass , which in turn depends on still poorly understood phenomena such as mixings in the radiative layers ( core overshooting and rotational mixing ) and wind mass loss . recent evolution models with a solar metallicity of @xmath1 by @xcite indicate that a star with a sufficiently large initial mass undergoes a blue - red - blue ( or blue - loop ) evolution before central helium exhaustion ; i.e. , the star ignites he in the center in the blue supergiant ( bsg ) stage , evolves to the red - supergiant ( rsg ) region , and returns to the blue supergiant ( bsg ) region during core he - burning . the lowest initial - mass for the blue - red - blue evolution depends on the degree of mixing in radiative layers and the strength of wind mass loss . @xcite s results indicate the lower bound to be about 20m@xmath0 . the mass limit is lowered if higher mass - loss rates in the rsg phase is assumed @xcite . thus , luminous bsgs consist of two groups having different evolution histories : one group are evolving red - wards just after the termination of main - sequence , while another group have evolved back from the rsg stage . the bsgs belonging to the latter group have significantly reduced envelope mass and the surface is contaminated by the cno - processed matter due to a dredge - up in the rsg stage and a significant mass loss . the fraction of each group depends on the internal mixing in the radiative layers and the strength of stellar wind and metallicity . in other words , if we can distinguish the two kinds of bsgs , it would be very useful for constraining the mixing in radiative layers and wind parameters . furthermore , the fraction relates to the relative frequencies of different types of core - collapse supernovae such as iip , iil , iib , ib and ic ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and the ratio of blue to red supergiants ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? one way to distinguish the two groups is to obtain their surface abundances of the cno elements . this has been pursued intensively by many authors ; e.g. , the vlt - flame survey @xcite , @xcite and @xcite . although the majority of bsgs show enhanced n / c ratios , theoretical interpretations were somewhat hampered by the variety of rotation velocities which yield various degree of internal mixings in the main - sequence stage , and possible effect of close binaries and magnetic fields . we propose , in this paper , another way to distinguish the two groups of bsgs by using stellar pulsation ; i.e. , we will argue that if they show ( radial ) pulsations , they must have been red supergiants before . it is known that many luminous ( @xmath3 ) ba - supergiants in our galaxy and magellanic clouds show micro variations in luminosity and in radial velocities ; they are called @xmath2-cygni variables ( e.g. , * ? ? ? in addition , @xcite found that a fraction of blue supergiants in the galaxy ngc300 are such variables and at least two of those show clear radial pulsation properties . the ngc300 bsgs would be particularly useful for constraining evolutionary models , because of the homogeneity of the data and less ambiguities in luminosity . the pulsation not only provides us with diagnostic means , it might also have effects on stellar winds from massive stars , as @xcite found a relation between episodic changes in mass loss and the 37day pulsation of the luminous blue supergiant hd 50064 . they suggested that the pulsation is a radial strange - mode pulsation , which we confirm in this paper . the paper is organized as follows : evolution models of massive stars and the excitation of radial pulsations in these models are discussed in 2 . the properties of radial and nonradial pulsations and their excitation mechanisms are discussed in 3 . in 4 we compare observed semi - periods of @xmath2-cygni variables with theoretical ones and discuss surface compositions . our conclusion is given in 5 . evolutionary models have been calculated by the geneva evolution code with the same input physics as those described in @xcite . the initial abundances adopted are @xmath4 with a solar mixture for the heavy elements ( * ? ? ? * ; * ? ? ? * for the ne abundance ) . a core overshooting of 0.1 pressure scale height is included . stellar mass loss rate for a given position on the hr diagram and current mass is obtained from the prescriptions described in @xcite ( except for @xmath5 models , see below ) . @lccccc@ name & @xmath6 & @xmath7 & @xmath8 & @xmath7 & ref 15 cma & 4.408 & 0.021 & 4.50 & 0.16 & a @xmath9 cma & 4.40 & 0.04 & 4.45 & 0.20 & b bw vul & 4.358 & 0.028 & 4.29 & 0.14 & c kz mus & 4.415 & 0.012 & 4.22 & 0.20 & d v433 car & 4.425 & 0.012 & 4.20 & 0.2 & d 12 lac & 4.374 & 0.019 & 4.18 & 0.16 & e @xmath10 cet & 4.339 & 0.008 & 4.02 & 0.05 & f @xmath11 eri & 4.360 & 0.022 & 3.89 & 0.29 & g 16 lac & 4.345 & 0.015 & 4.0 & 0.2 & h hd129929 & 4.350 & 0.015 & 3.86 & 0.15 & i [ tab : betcep ] a=@xcite , b=@xcite , c=@xcite , d=@xcite , e=@xcite , f=@xcite , g=@xcite , h=@xcite , i=@xcite + @xmath12this is a very incomplete sample of galactic @xmath9 cep variables collected only for illustrative purpose in fig.[fig : stb ] . fig.[fig : stb ] shows evolutionary tracks up to the central helium exhaustion calculated without including rotational mixing for initial masses of 8 , 9 , 10 , 12 , 14 , 17 , 20 , 25 , 30 , 40 , and 50m@xmath0 . for @xmath13m@xmath0 , the helium burning starts when stars are evolving in the blue supergiant ( bsg ) region after the termination of main - sequence stage . as he burns in the center , they evolve into the red supergiant ( rsg ) stage . stars with @xmath14m@xmath0 evolve back to the bsg region ( blue - loop ) before the helium is exhausted in the center . a star starts a blue - loop when it loses enough mass in the rsg stage ( fig.[fig : mass ] ) . this has an important consequence for the stability of radial pulsations . we have performed a stability analysis of radial pulsations for selected evolutionary models . the method is described in @xcite , where the perturbation of the divergence of convective flux and the effect of rotation is neglected . the latter is justified because rotation is always slow in the envelope of supergiants ( this is also true in our models with rotation ; see appendix ) , where pulsations have appreciable amplitudes . the effect of convection is neglected because the theory for the convection - pulsation coupling is still infant . since the convective flux is less than 50% of the total flux in the convection zones in the envelope of blue supergiants , we do not think that neglecting the convection - pulsation coupling affects significantly our results . the outer boundary is set at an optical depth of @xmath15 . we have adopted the outer mechanical condition that the lagrangian perturbation of the gas pressure goes to zero . the red dots along evolutionary tracks in fig.[fig : stb ] indicate the positions of the models that are found to have at least one excited radial mode . the dashed line in fig.[fig : stb ] indicates the stability boundary of radial low - order pulsations , which are appropriate for the cepheids and @xmath9 cephei variables . for models with @xmath16m@xmath0 , which make blue - red - blue evolution , the part evolving toward the rsg region ( first crossing ) was used to obtain the stability boundary . for comparison , the stability boundary for models with the abundance @xmath17 @xcite with gn93 mixture @xcite is also shown by a dotted line . the nearly vertical ` finger ' of the instability boundary around @xmath18 corresponds to the @xmath9 cephei instability region ( excited by the @xmath19-mechanism at the fe - opacity bump around @xmath20k ) , while the vertical boundary at @xmath21 is the blue edge of the cepheid instability strip , in which pulsations are excited at the second helium ionization zone . ( no red - edge is obtained because our pulsation analysis does not include the coupling between pulsation and convection . ) the boundary for the @xmath9 cephei instability region depends on the metal abundance . the positions of some galactic @xmath9 cephei stars are shown in fig.[fig : stb ] by filled triangles with error bars . comparing the distribution of the @xmath9 cephei variables with the stability boundaries for @xmath1 ( dashed line ) and @xmath17 ( dotted line ) , we see that the most appropriate heavy - element abundance for the galactic @xmath9 cephei variables seems to be slightly larger than @xmath1 . the other part of the instability boundary hardly depends on the metallicity . the instability boundary for radial pulsations by ( weakly non - adiabatic ) @xmath19-mechanisms have steep gradients in the hr diagram as seen in the less luminous part ( @xmath22 ) of fig.[fig : stb ] , where the instability boundaries for @xmath1 and 0.02 are shown by broken lines . this comes from the requirement that the pulsation period should be comparable to the thermal timescale at the zone where the @xmath19-mechanism works ( e.g. , * ? ? ? * ) . at high luminosity , the instability boundary is nearly horizontal . this is associated with the strange modes that occur if @xmath23 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * and discussion below ) . in the bsg models evolving toward the rsg stage no radial modes are excited between the red - boundary of the @xmath9 cephei instability region and the cepheid blue - edge , because @xmath24 is not sufficiently large for the strange mode mechanism to work . models with @xmath14m@xmath0 return to the bsg region ( blue - loop ) from the rsg region before core - helium exhaustion . radial pulsations are excited in the models on the blue - loop ; this is due to the fact that a significant mass is lost in the rsg stage and hence the @xmath24 ratio has increased considerably ( fig.[fig : mass ] ) . we can identify @xmath2 cygni variables ( especially if radial pulsations are involved ) to be core - helium burning stars on the blue - loop returned from the rsg stage . however , the luminosity of the track for @xmath25 is still too high to be consistent with the distribution of @xmath2 cygni variables on the hr diagram . the discrepancy can be solved by taking into account rotational mixing @xcite , or assuming a strong mass loss caused by roch - lobe overflow in the rsg stage . we consider the effect of rotational mixing in this paper . we have calculated evolution models for @xmath26 , and 14m@xmath0 with rotational mixing , and examined the stability of radial pulsations for those models . the ways to treat rotation and the mixing are the same as @xcite ; the rotation speed was assumed to be 40% of the critical one at the zero - age main - sequence stage . the results are shown in fig.[fig : mass ] . the rotational mixing makes helium core and hence luminosity larger in the post main - sequence evolution for a given initial mass . a smaller ratio of the envelope to core mass makes the red - ward evolution faster ; i.e. , less he is consumed in the first bsg stage . ( note that dotted lines in the bottom panel of fig.[fig : mass ] tend to lose more mass as a function of @xmath27 in the first crossing , indicating the evolution to be slow there without rotational mixing ) . also , the higher luminosity enhances stellar winds so that the star starts blue loop earlier , well before the central he exhaustion . we note that the important effect of rotation comes from the mixing that enlarges he core , but not from the centrifugal force . therefore , a similar evolution is possible even without including rotation if a more extensive core overshooting is assumed . for @xmath28 , for example , an extensive blue - loop occurs before he exhaustion if a core overshooting larger than @xmath29 is included without rotational mixings ; if a mass - loss rate is enhanced by a factor of 5 , for example , it occurs for a overshooting larger than @xmath30 . the non - rotating evolutionary track of @xmath31m@xmath0 passes , in the first crossing , around the lower bound of the distribution of the @xmath2 cygni variables ( @xmath3 ) . however , it does not come back to blue region even if rotation is included with our standard parameters . it does make a blue loop as shown in fig.[fig : mass ] , if the rate of cool winds is increased by a factor of five as in @xcite . such an increase is reasonable since there are many theoretical and observational arguments of sustaining higher mass loss rates during the red supergiant phase . from a theoretical point of view , @xcite have studied the consequence of a pulsation driven mass loss during the red supergiant phase . they showed that using empirical relations between the pulsation period and the mass loss rates , the mass loss rates could be increased by quite large factors largely exceeding a factor 5 at least during short periods . from an observational point of view , the circumstellar environment of red supergiants clearly indicates that some stars undergo strong mass loss outbursts . for instance , vy cma ( m2.5 - 5iae ) which has a current mass loss rate of 2 - 4 10@xmath32m@xmath0 per year @xcite is surrounded by a very inhomogeneous nebula likely resulted from a series of episodic mass ejections over the last 1000 yr @xcite . it is estimated that the mass loss rates between a few hundred and 1000 yr ago was 1 - 2 10@xmath33m@xmath0 per year , thus between 2.5 and 10 times greater than the present rate . we also note that mass loss rates obtained by @xcite for dust enshrouded red supergiants are larger by a factor up to 10 compared with the rates estimated from the empirical relations given in @xcite in the present standard computation we used the prescription by @xcite . the arguments above indicate that using mass loss rates increased by a factor 5 is not beyond the uncertainty in our present state of knowledge on the mass loss rates of red supergiants . it is interesting to note that for the case of 14m@xmath0 not all models on the blue - loop excite pulsations . more precisely , no radial modes are excited in the blueward evolution at @xmath34 ( @xmath35 ) . only in the second redward evolution ( third crossing ) , a radial mode is excited around similar effective temperature ; this time , models have slightly higher @xmath24 ( @xmath36 ) . the fact that no @xmath2 cygni variables are observed below a luminosity limit of about @xmath37 does not necessarily means that stars below that limit have no blue - loop . even if they make a blue - loop evolution , their @xmath24 ratio would be too small for pulsations to be excited . the observed properties of @xmath2 cygni variables can be well explained if these stars are core he - burning stars on the blue - loop . this supports the presence of considerable mixing and possibly cool winds stronger than adopted in @xcite . note that an increased mass loss during the rsg phase seems to be also required in order to reproduce the observed positions of @xmath2 cyg variables at high luminosity . ( indeed , the models in @xcite have a mass - loss rate increased by a factor of 3 for models more massive than 20m@xmath0 during the rsg phase ) . from the ratio of the evolution speeds between the first ( red - ward ) and second ( blue - ward ) crossings we can estimate the probability for a bsg to be on the first crossing . at positions of typical bsgs , rigel and deneb ( see table[tab : param ] below ) , for example , the probabilities are 45% and 98% , respectively , for @xmath38 , while they are 15% and 50% or @xmath39 ; in both cases rotational mixings are included . if we use models without rotation , the probabilities are nearly unity for both stars and for both initial masses , because in this case the second crossings occur very swiftly after the core he exhaustion . in this section we discuss the properties and excitations of radial and nonradial pulsations in bsg models with rotational mixing . although these models start with 40% of the critical rotation at the zero - age main - sequence stage , the rotation speeds in the envelopes in all the supergiant models are very low as discussed in appendix , so that we did not include the rotation effects in our pulsation analyses . in our linear pulsation analysis , the temporal dependence of variables is set to be @xmath40 , where @xmath41 is a complex frequency obtained as the eigenvalue for the set of homogeneous differential equations for linear pulsations . the real part @xmath42 gives the pulsation period ( @xmath43 ) and the imaginary part @xmath44 gives the stability of the pulsation mode ( excited if @xmath45 ) . we use the symbol @xmath46 for normalized ( complex ) frequency ; i.e. , @xmath47 with @xmath48 being the gravitational constant and @xmath49 the stellar radius . fig.[fig : omega ] shows @xmath50 , normalized pulsation frequency , for low - order radial pulsation modes in the bsg models of @xmath51 evolving toward the rsg stage ( top panel ) and evolving from the rsg on a blue - loop ( bottom panel ) . filled circles indicate excited modes , while ` @xmath52 ' and ` + ' are damped modes . in the top panel , the normalized frequency of each mode varies regularly as a function of @xmath6 , keeping the order such that the lowest frequency is the fundamental mode ( f ) with no node in the amplitude distribution , next one is the first overtone ( 1ov ) with one node , and so on . ( the ordering is strictly hold only in adiabatic pulsations ) . the fundamental mode in the range @xmath53 is excited by the fe - opacity bump as the models are in the @xmath9 cephei instability region . the other modes in the top panel are all damped . a very low frequency mode which appears in the range @xmath54 of the top panel is a mode associated with thermal ( damping ) wave , so that it is strongly damped . the symbol + is used in fig.[fig : omega ] for strongly damped modes with @xmath55 ( modes with @xmath56 are not plotted ) . in models with high @xmath24 ratios as shown in the bottom panel of fig.[fig : omega ] , the frequencies of thermal modes enter into the frequency range of dynamical pulsations , and @xmath57 decreases ( the damping time becomes longer ) so that the two types of pulsations become indistinguishable . in the bottom panel of fig.[fig : omega ] for models on the blue - loop , at least one mode is excited throughout the @xmath27 range ( three modes are excited in most part ) . the frequency of each mode varies in a more complex way ; the mode ordering rule is lost , additional mode sequences appear , and frequent mode crossings occur , etc . the appearance of such complex behaviors is related with strange modes . the strange modes may be defined as modes which are not seen in adiabatic analyses . with this definition the thermal damping modes also belong to strange modes , but we are more interested in another type of strange modes which are excited by a special instability even when the thermal time goes to zero which was first recognized by @xcite . ( we will discuss the mechanism briefly below ) . sequence s2 in fig.[fig : omega ] corresponds to such a strange mode . sequence s1 , another strange mode , is somehow related with the lowest frequency mode ( @xmath58 ) seen in the range @xmath54 of the top panel . the two sequences is connected in the rsg stage , which is not shown in fig.[fig : omega ] . the s1 mode , as discussed below , seems to be excited mainly by enhanced @xmath19-mechanism at the fe - opacity bump . the main difference between the models in the top and in the bottom panels of fig.[fig : omega ] is the luminosity to mass ratio ; models in the bottom panel ( on the blue - loop ) have @xmath59 , while in the top panel @xmath60 . a higher @xmath24 ratio makes pulsations more nonadiabatic . this can be understood from a linearized form of energy conservation for stellar envelope ; @xmath61 where @xmath10 means the lagrangian perturbation of the next quantity , @xmath62 is the entropy per unit mass , and @xmath63 . the above equation indicates that generally a high value of @xmath24 generates a large entropy change and hence large nonadiabatic effects . a large @xmath24 ratio has also a significant effect on the envelope structure by enhancing the importance of radiation pressure . combining a hydrostatic equation with a radiative diffusion equation for the envelope of a static model , we obtain a relation @xmath64 where @xmath65 is the gas pressure , @xmath66 is the total pressure , @xmath19 is the opacity in units of @xmath67 , and @xmath68 is the local radiative luminosity . this equation indicates that the inward increase of the gas pressure is hampered or inverted if @xmath69 , and the effect is strong where the opacity is large . a radiation pressure dominated zone is formed around an opacity peak in the stellar envelope with a @xmath24 ratio larger than @xmath70 . we note that if the second term on the right hand side of equation ( [ eq : pgas ] ) is sufficiently large , a density inversion is formed . fig.[fig : work3 ] shows the properties of three pulsation modes f , 1ov , and 2ov in a bsg model ( with 23.5m@xmath0 ) before the rsg stage ( left panels ) and s1 , f , and s2 modes in a bsg model ( with 11.6m@xmath0 ) on the blue - loop after the rsg stage ( right panels ) . both models have a similar effective temperature of @xmath71 . each panel shows the runs of fractional displacement , @xmath72 ( solid line ) , work @xmath73 ( dashed line ) , and differential work @xmath74 ( dotted line ) . the work is defined as @xmath75 where @xmath76 and @xmath77 are the lagrangian perturbations of the pressure and the density , respectively , and the superscript @xmath12 indicates the complex conjugate of the quantity . ( in the nonadiabatic linear pulsation analysis , we employ complex representations ) . the layers with @xmath78 ( dotted lines ) contribute to drive the pulsation . the net effect of driving and damping through the stellar interior appears in the surface value of the work , @xmath79 ; if @xmath80 the pulsation mode is excited . the amplitude growth rate ( @xmath81 ) is related with @xmath79 as @xmath82^{-1},\ ] ] where @xmath83 is pulsational displacement vector ( for radial pulsations @xmath84 with @xmath85 being the unit vector in the radial direction ) . we assume the amplitude of an excited radial - pulsation mode to grow to be visible . all modes in the left panels of fig.[fig : work3 ] are damped , while all modes in the right panels are excited . some structure variables are also shown in fig.[fig : work3 ] . note that in the model on the blue - loop ( right panel ) , @xmath86 is extremely small in the range @xmath87 , indicating @xmath88 there . the thermal time @xmath89 is defined as @xmath90 where @xmath91 is the specific heat per unit mass at constant pressure . in the outermost layers with @xmath92 , @xmath89 is shorter than pulsation periods . the pulsations are locally very non - adiabatic there . for the longest period modes shown in the top panels in fig.[fig : work3 ] , driving occurs around the fe - opacity bump ( @xmath93 ) . since the thermal time there is longer than the pulsation periods , the driving can be considered as the ordinary @xmath19-mechanism . roughly speaking , the @xmath19-mechanism works ( under a weak nonadiabatic environment ) if the opacity derivative with respect to temperature increases outward ; i.e. , @xmath94 @xcite . we see that this rule is hold in the model before the rsg stage ( left panel ) , in which the driving is overcome by radiative damping in the upper layers ( where @xmath95 ) so that the mode is damped . for the longest period mode ( s1 ) in the model on the blue - loop ( right panel ) , the driving zone extends out into zones where @xmath96 decreases outward . because of the extension of the driving zone , which is probably caused by small @xmath9 , the mode is excited ; i.e. , the driving effect exceeds radiative damping in the upper layers . this mode is considered to be a strange mode because there is no adiabatic counterpart . however , the excitation is caused by the enhanced @xmath19-mechanism , in accordance with the finding of @xcite . the modes in the middle panels in fig.[fig : work3 ] are ordinary modes ; the first overtone , 1ov , for the model before the rsg stage ( left panel ) and fundamental mode , f , for the model on the blue - loop ( right panel ) . for these modes the driving around @xmath97 has some contribution in addition to the driving around the fe - opacity bump . the mode in the left panel is damped because radiative damping between the two driving zones exceeds the driving effects , while the f mode in the right panel is excited . the driving in the low temperature zone should not be the pure @xmath19-mechanism because the thermal time there is shorter than the pulsation periods . it should be somewhat affected by the strange mode instability , which is discussed below . the mode in the left - bottom panel in fig.[fig : work3 ] , the second overtone ( 2ov ) of the model before the rsg stage is damped because of the lack of appreciable drivings . on the other hand , the s2 mode in the right - bottom panel is excited strongly in the zone ranging @xmath98 ( heii ionization zone ) , where the thermal time @xmath99d is much shorter than the pulsation period ( 10.2d ) . because no heat blocking ( which is essential for the @xmath19-mechanism ) occurs there , the driving mechanism must be the genuine strange - mode instability , which should work even in the limit of @xmath100 ; nar ( nonadiabatic reversible ) approximation introduced by @xcite . in this limit , @xmath101 ( cf . eq.([eq : encons ] ) ) . from this relation with the plane parallel approximation it is possible to derive an approximate relation of @xmath102 @xcite , where @xmath103 is the radiative flux and @xmath104 . this relation indicates that a large phase difference arises between @xmath76 and @xmath105 , which can lead to strong driving ( and damping ) according to the work integral given in eq.([eq : work ] ) ; in the limit of nar approximation , if @xmath106 is an eigenvalue , the complex - conjugate @xmath107 is also an eigenvalue . this explains the strange mode instability ( see * ? ? ? * for a different approach ) . the three dimensional property of a nonradial pulsation of a spherical star is characterized by the degree @xmath108 and the azimuthal order @xmath109 of a spherical harmonic @xmath110 ( @xmath111 for dipole and @xmath112 for quadrupole modes ) . in a non - rotating and non - magnetic spherical star . ] there are two types of nonradial pulsations ; p - modes ( common to radial modes ) and g - mode pulsations . the g - mode pulsations are possible only in the frequency range below the brunt - visl ( or buoyancy ) frequency ( e.g. , * ? ? ? * ; * ? ? ? . we have performed nonradial pulsation analyses based on the method described in @xcite , disregarding the effect of rotation . this is justified because we discuss the modes trapped in the envelope , where the rotation speed is very low as discussed in appendix . the properties and the stability of nonradial pulsations of supergiants are very complex because they have a dense core with very high brunt - visl frequency . all oscillations in the envelope with frequencies less than the maximum brunt - visl frequency in the core can couple with g - mode oscillations in the core through the evanescent zone(s ) laying between the envelope and the core cavities . the coupling strength varies sensitively with the pulsation frequency and the interior structure of the star . depending on the coupling strength , the relative amplitudes in the core and in the envelope vary significantly , and hence the stability changes . excitation by the @xmath19-mechanism and the strange - mode instability works also for nonradial pulsations @xcite . in addition , oscillatory convection mechanism @xcite works in the convection zone associated with the fe - opacity bump in the bsgs . furthermore , the @xmath113-mechanism of excitation at the h - burning shell can excite nonradial modes which are strongly confined to a narrow zone there as shown by @xcite . among those nonradial modes which are excited , we restrict ourselves , in this paper , to possibly observable modes having the following properties : @xmath114 where @xmath115 is the radial component of lagrangian displacement , and the subscripts @xmath116 and @xmath117 indicate the values at the stellar surface and at the maximum amplitude in the interior , respectively . the first requirement selects modes less affected by cancellation on the stellar surface . the second requirement excludes modes highly trapped in the interior ; i.e. , such oscillations hardly emerge to the surface . the second requirement excludes most of the g - modes excited in the core , because they are strongly trapped in the core having small ratios of @xmath118 . fig.[fig : teperi ] shows periods of excited nonradial dipole and quadrupole modes ( @xmath119 ) ( as well as radial modes ) in models evolving toward rsg region ( left panel ) and in models on the blue - loop after the rsg stage ( right panel ) for @xmath120 and 25m@xmath0 cases with rotation . obviously , much more modes are excited in the bsg models after the rsg stage ( on the blue - loop ) in the period ranges of @xmath2 cygni variables . in the bsg models before the rsg stage , excited observable modes are nonradial g - modes and oscillatory convection modes . swarms of modes labeled as g in the left panel of fig.[fig : teperi ] are g - modes excited by the fe - opacity bump . for those oscillations , the amplitude is confined to the envelope by the presence of a shell convection zone above the h - burning shell @xcite , which prevents the oscillation from penetrating into and being damped in the dense core . the red edge for the group , @xmath121 , is bluer by 0.1 dex than that obtained by @xcite for 25m@xmath0 models with @xmath17 . the difference can be attributed to the metallicity difference . these g - modes are probably responsible for the multi - periodic variations of the early bsg hd 163899 ( b2ib / ii ) @xcite observed by the most satellite and some of the relatively less luminous early bsgs studied by @xcite . the sequences labeled as ` c ' in the left panel of fig.[fig : teperi ] are oscillatory convection ( g@xmath122 ) modes associated with the convection zone caused by the fe - opacity peak around @xmath123 ( the presence of such modes is discussed in * ? ? ? the sequences terminate when the requirement of @xmath124 is not met anymore ; i.e. , beyond the termination the modes are trapped strongly in the convection zone . the right panel for models after the rsg stage show many modes excited in the @xmath27period range appropriate for the @xmath2 cygni variables . they are excited by different ways depending on the periods . fig.[fig : nradwork ] presents examples , in which amplitude and work curves are shown for the three dipole ( @xmath111 ) modes excited ( and @xmath125 ; eq.([eq : ratio ] ) ) in a model of @xmath51 at @xmath126 and @xmath127 . ( note that the mass of the model is reduced to 12.3m@xmath0 and hence the @xmath24 ratio is as high as @xmath128 . ) the top panel of fig.[fig : nradwork ] shows the lowest frequency dipole mode excited . this is one of the very rare cases in which the driving effect in the core is comparable with that in the envelope . in the envelope the amplitude is strongly confined to the fe convection zone , which is characteristic of oscillatory convection modes . the envelope mode weakly couples with a core g - mode . although the amplitude in the core is extremely small , the driving effect is comparable or larger than that in the envelope . in the hotter models contribution from the core is negligibly small , while as @xmath27 decreases the core contribution increases rapidly but the ratio @xmath129 soon becomes much smaller than 0.1 . the modes shown in the middle and the bottom panels of fig.[fig : nradwork ] are confined strongly to the envelope without any contribution from the core . the mode in the middle panel with a period of 26 days is excited around fe - opacity bump at @xmath130k and have large amplitude in the convection zone . the mode in the bottom panel with a period of 16.6days corresponds to the s2 strange mode of radial pulsations shown in fig.[fig : work3 ] ( right - bottom panel ) , excited by the strange - mode instability around the he ii ionization . three quadrupole ( @xmath112 ) modes having periods and properties similar to those of dipole modes shown in fig.[fig : nradwork ] are also excited in the same model . however , the amplitudes of the two longer - period quadrupole modes are more strongly confined in the fe - convection zone with amplitude ratios of @xmath131 so that they do not meet the requirement given in eq.([eq : ratio ] ) and hence are not considered to be observable . only the shortest period ( 16.1 days ) mode with @xmath132 satisfies the requirement ; this mode corresponds to the s2 mode . in this model two radial strange modes belonging to s1 and s2 are also excited ( fig.[fig : omega ] ) with periods of 54.1 and 16.3days , respectively . counting these excited observable pulsation modes , we expect a total of six pulsation periods ; 54.1 and 16.3days for the two radial pulsations , 46.3 , 26.0 , and 16.6days for the three dipole modes , and 16.1days for the quadrupole mode . the three very close periods of @xmath133days would yield very long beat periods up to a few years . if these modes are simultaneously excited , the light curve would be extremely complex , which is just consistent with light curves of many @xmath2 cygni variables ( e.g. , * ? ? ? despite the long history of observations for @xmath2 cygni variables , periods of variations are only poorly determined for most of the cases , hampered by complex and long - timescale light and velocity variations . here we compare observed periods of relatively less luminous @xmath2 cygni variables shown in fig.[fig : teperi ] with theoretical models of @xmath134 with rotational mixings . @lcccccc@ hd & @xmath6 & @xmath135 & ref & peri.(d ) & ref34085 ( rigel ) & 4.083 & 5.34 & a & 470 & b 62150 & 4.176 & 5.07 & c & 3643 & d 91619 & 4.121 & 5.31 & e , f & 710 & g 96919 ( hr4338 ) & 4.097 & 5.02 & f & 1080 & g 100262 & 3.958 & 5.02 & h & 1446 & g , i 168607 & 4.013 & 5.32 & c & 3662 & d , j 197345 ( deneb ) & 3.931 & 5.29 & k & 1060 & l 199478 ( hr8020 ) & 4.114 & 5.08 & m & 2050 & n a10 & 3.966@xmath136 & 5.12 & o & 96.1 & p d12 & 3.954@xmath137 & 5.33 & o & 72.5 & p + a = @xcite ; b=@xcite ; c=@xcite ; d=@xcite ; e=@xcite ; f=@xcite ; g=@xcite ; h=@xcite ; i=@xcite ; j=@xcite ; k=@xcite ; l=@xcite ; m=@xcite ; n=@xcite ; o=@xcite ; p=@xcite deneb is the prototype of the @xmath2 cygni variables and has a long history of studies ( see * ? ? ? * for a review and references given ) . although its light - curves are very complex , @xcite obtained radial pulsation features at two short epochs . strangely , however , the two epochs give two different periods ; 17.8days and 13.4days . from the time - series analyses by @xcite and @xcite , we have adopted a period range of @xmath138days . on the hr diagram deneb is close to the blue - loop of @xmath51 ( fig.[fig : teperi ] ) ; the current mass is about 12.7m@xmath0 . our model at @xmath139 predicts the following periods for the excited modes ; 127 & 49days ( radial ) , 42days ( @xmath112 ) . the longest period is longer than the observed period range . the other two periods are in the longer part of the observed range . the periods of excited radial modes are much longer than the periods of radial pulsations , 17.8 and 13.4days , obtained by @xcite . the reason for the discrepancy is not clear . our model can not explain the shorter part of the observed period range . @xcite tried to explain deneb s pulsations by nonradial strange modes and found that if the mass is less than 6.5m@xmath0 , nonradial modes with various @xmath108 are excited . the required mass seems too small for the position of deneb on the hr diagram . recently , @xcite found that nonradial modes of @xmath140 having periods consistent with the observed period range are excited by the h - ionization zone ( @xmath141k ) in models of 25m@xmath0 ( evolving towards the rsg stage ) with effective temperatures similar to the observed one . in our models , however , the h - ionization zone excites relatively short periods modes in cooler models with @xmath142 ( fig.[fig : teperi ] ) , which is inconsistent with deneb . the reason for the difference is not clear ; further investigations are needed . the period range given in table[tab : param ] , @xmath143days , is based on the recent analysis by @xcite . @xcite s analysis indicates a narrower range of @xmath144days . the position on the hr diagram ( fig.[fig : teperi ] ) is consistent with a @xmath51 model on the blue - loop after the rsg stage . our model at @xmath145 predicts excitation of the following visible modes : 40.1 , 18.4 , and 11.2days ( radial ) ; 31.9 , 16.4 , and 11.4days ( @xmath111 ) ; 11.0days ( @xmath112 ) . our model predicts many pulsation modes in the longer part of the observed period range . although no modes with periods shorter than 10 days are excited , those shorter periodicities might be explained by combination frequencies among excited modes . @xcite proposed g - modes ( with periods of longer than 20days ) excited by the @xmath113-mechanism at the h - burning shell . however , we think that those modes are invisible because the amplitude is very strongly confined to a narrow zone close to the h - burning shell . the period range of hd62150 , @xmath146days , is adopted from the analysis by @xcite of the hipparcos observations . in fact they obtained a 36.4 day period and a very small amplitude period of 43.0 days . these periods are consistent with nonradial ( oscillatory convection mode ) pulsations either in the first crossing toward the rsg stage , or on the blue loop of a model with @xmath147 ( fig.[fig : teperi ] ) . hd 62150 is , among the variables considered here , the only star which can be considered as in the first crossing stage . hd100262 has a similar @xmath27 to deneb and has the same problem ; i.e. , the model can not predict the shorter part of the observed period range . the period ranges of other stars , hd 168607 , hr4338 , hr 8020 , and hd 91619 are roughly consistent with predicted periods of the models evolving on blue - loops from the rsg region . @xcite found many @xmath2-cygni type variables among blue supergiants in the galaxy ngc300 . among them , two stars , a10 and d12 show regular light curves with periods of 96.1d and 72.5d , respectively . the light curves look consistent with radial pulsations . hence , according to our stability results , these stars must be on blue - loops after the rsg stage . @xcite obtained heavy element abundances of @xmath148 for a10 and @xmath149 for d12 , which indicate our models with @xmath1 to be appropriate for these stars . the periods of a10 and d12 roughly agree with theoretical periods of radial pulsations obtained for @xmath150 and 25m@xmath0 models ( fig.[fig : teperi ] , right panel ) . these are strange modes excited at the fe - opacity bump ( fig.[fig : work3 ] ) in bsg models after rsg stages . because of the wind mass loss occurred in the previous stages , these models currently have masses of 8.8 and 12.7m@xmath0 , respectively . our models agree with @xcite who concluded that the pulsations of a10 and d12 should be strange modes in models with masses reduced significantly by mass loss . however , fig.[fig : teperi ] ( right panel ) indicates a discrepancy ; in the hr diagram a10 and d12 are close to the evolutionary tracks of @xmath150 ( @xmath151 ) and @xmath152 ( @xmath153 ) , respectively , while the periods correspond to the other way around ; i.e. , the period of a10(96.1d ) is longer than that of d12(72.5d ) . according to the spectroscopic analysis by @xcite , both stars have similar effective temperatures , but the radius of a10 ( 142r@xmath0 ) is smaller than that of d12 ( 191r@xmath0 ) due to the luminosity difference ; i.e. , the smaller star has the longer period . the discrepancy would be resolved if a10 were cooler and d12 were hotter slightly beyond the error - bars . in addition , the folded light - curve of a10 shown in fig.3 in @xcite has considerable scatters , indicating other periods might be involved . further observations and analyses for the two important stars are desirable . @xcite found a 57 day periodicity in hd 50064 from the corot photometry data and interpreted the period as a radial strange mode pulsation . from their spectroscopic analysis they estimated @xmath154k ( @xmath155 ) , @xmath156 , @xmath157 , and @xmath158 . the star seem more luminous than our models . from these parameters we can estimate the normalized frequency corresponding to the observed period , using the relation @xmath159 with @xmath160 being pulsation period . substituting above parameters into the equation we obtain @xmath161 . this value should be considered consistent with either s1 or s2 strange mode ( fig.[fig : omega ] ) , taking into account the possibility of a considerable uncertainty in the value of @xmath162 . this confirms @xcite s interpretation of the 57 day period as a radial strange mode pulsation . @lccc|ccc@ & & @xmath163 & @xmath164 & n / c & n / o & @xmath164 & n / c & n / o 14 ( rot ) & 0.29 & 2.27 & 0.517 & 0.55 & 38.0 & 2.4120 ( rot ) & 0.31 & 2.46 & 0.609 & 0.57 & 39.7 & 2.94 25 ( rot ) & 0.35 & 3.23 & 0.877 & 0.64 & 60.4 & 4.22 + the initial values are n / c@xmath165 and n / o@xmath166 , where @xmath167 means mass fraction of element i. the initial helium mass fraction is 0.266 . [ tab : cno ] as we discussed above , observed periods of many @xmath2 cygni variables are consistent with radial and nonradial pulsations excited in bsg models evolved from the rsg stage . because of the convective dredge - up in the rsg stage , rotational mixing , and wind mass loss , the surface compositions of the cno elements are significantly modified from the original ones . to understand the evolution of the surface composition , we show on fig . [ fig : cno ] a profile of the cno abundance through the stellar envelope , as well as the profile of n / c and n / o ratios for the rotating @xmath168 model when it reaches for the first time the red supergiant branch ( @xmath169 ) around 3.6 , as a function of the lagrangian mass coordinate . the edge of the convective core is on the left ( lagrangian mass coordinate @xmath170 ) , and the stellar surface on the right ( lagrangian mass coordinate @xmath171 ) . the convective zones are shown in transparent gray . the first one ( between @xmath172 and @xmath173 ) is the convective zone developing on top of the hydrogen shell burning zone . the second one is the convective zone below the surface . we see that the region in the first convective zone and above ( up to @xmath174 ) is strongly affected by cno - cycle burning products , with a large @xmath175n mass fraction , and a depletion of @xmath176c and @xmath177o . in this region , both n / c and n / o ratios are very large . only the region near the surface exhibits c , n , and o abundances close to the initial ones , with n / c and n / o ratios less than 1 . in fig . [ fig : cno ] , the red region shows the typical ratios @xmath178 needed for a star to evolve again towards the blue @xcite . this means that the star will become bluer again only if it loses its mass up to that region . the lagrangian mass coordinate corresponding to @xmath179 is @xmath180 , so it means that for the core to contain 60% of the mass , the star must lose about @xmath181 . we can thus expect that a blue star that has evolved from the rsg stage should exhibit very high n / o and n / c ratios . _ the figure shows that what increases a lot the n / c and n / o ratios is not so much the dredge up process due to convection but the mass loss . _ table[tab : cno ] shows surface helium abundance and ratios of cno elements ( mass fraction ) for bsg models ( with rotational mixing ) at @xmath182 before and after the rsg stage . most of the stars listed in table[tab : param ] do not have measured n / c and n / o ratios and thus can not be used to test the above predictions . there are however two stars rigel and deneb for which such data exist . @xcite lists ( y , n / c , n / o)= ( 0.32 , 3.4 , 0.65 ) for deneb and ( 0.37 , 2.0,0.46 ) for rigel . these numbers are rather consistent with models before the rsg stage , in contradiction with our conclusion that they should rather be stars having evolved through a rsg stage . so we are left here with a puzzle : pulsation properties tell us that these two stars should be bsg evolved from a rsg stage , while surface abundances indicate that they should be bsg having directly evolved from the ms phase . in other words , in order to have the excited modes compatible with the observations of deneb and rigel , a high @xmath24 ratio is needed , which , in turn , implies strongly changed n / c and n / o ratios which are not observed . so we are left with a real puzzle here . at this point we can simply mention three directions which may help in resolving this discrepancy : \1 ) are we sure that all the frequency measured correspond to pulsation process ? for instance , some variations could be due to some other type of instabilities at the surface . in this case , the vertical lines associated to rigel and deneb in fig.[fig : teperi ] representing the observed pulsation period ranges might be reduced and be compatible with stars on their first crossing . \2 ) it would be extremely interesting to obtain accurate measurements of the n / c and n / o ratios at the surface of the stars listed in table[tab : param ] other than rigel and deneb to see whether the case of deneb and rigel are representative of all these stars or not . in particular , measurements of surface cno abundances of the two radial pulsators a10 and d12 in ngc300 are most interesting . \3 ) recently some authors @xcite find that the rsg have significantly higher effective temperatures and are hence more compact for a given luminosity . from another perspective , @xcite find that to reproduce the light curve of type ii - p supernovae , the rsg progenitor should be more compact than predicted by current models . the effective temperature of the rsg stars depends on the physics of the convective envelope . for instance , depending whether turbulent pressure is accounted for or not and how the mixing length is computed , much bluer positions for the rsg stars can be obtained ( see e.g. fig . 9 in * ? ? ? one can wonder whether more compact rsg stars would need to lose as much mass as larger rsg stars to evolve to the blue part of the hr diagram . would the blue supergiants resulting from the evolution of more compact rsg stars present the same chemical enrichments as those presented by the current models ? we shall investigate these questions in a forthcoming work . we have studied the pulsation properties of bsg models . in the effective temperature range between the blue edge of the cepheid instability strip and the red - edge of the @xmath9 cephei instability range , where many @xmath2 cygni variables reside , radial ( and most of nonradial ) pulsations are excited only in the models evolving on a blue - loop after losing significant mass in the rsg stage . the observed quasi periods of @xmath2 cygni variables are found to be roughly consistent with periods predicted from these models in most cases . this indicates that the @xmath2 cygni variables are mainly he - burning stars on the blue - loop . however , it is found that the abundance ratios n / c and n / o on the surface seem too high compared with spectroscopic results . further spectroscopic and photometric investigations for bsgs are needed as well as theoretical searches for missing physics in our models . furthermore , it would be interesting to explore the circumstellar environments of those stars that are believed to have evolved from a rsg stage . some of those stars may still have observable relics of the slow and dusty winds that they emitted when they were red supergiants . gm and hs thank vincent chomienne for having computed the first stellar models in the frame of this project . cg acknowledges support from eu - fp7-erc-2012-st grant 306901 . we thank the anonymous referee for useful comments . 99 aerts c. , marchenko s.v . , matthews j.m . , kuschnig r. , guenther d.b . , moffat a.f.j . , rucinski s.m . , sasselov d. , walker g.a.h . , weiss w.w . , 2006 , apj , 642 , 470 aerts c. , christensen - 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hidai m. , 2000 , pasj , 52 , 113 thoul a. , aerts c. , dupret m. a. , scuflaire r. , korotin s. a. , egorova i. a. , andrievsky s. m. , lehmann h. , briquet m. , de ridder j. , noels a. , 2003 , a&a , 406 , 287 unno w. , osaki y. , ando h. , saio h. , shibahashi h. , 1989 , nonradial oscillations of stars , univ . of tokyo press , tokyo vanbeveren d. , de donder e. , van bever j. , van rensbergen w. , de loore c. , 1998 , newa , 3 , 447 vanbeveren c. , mennekens n. , van rensbergen w. , de loore c. , 2012 , arxiv , 1212.4285 van leeuwen f. , van genderen a.m. , zegelaar i. , 1998 , a&as,128 , 117 van loon j. t. , cioni m .- l. , zijlstra a. a. , loup c. , 2005 , a&a , 438 , 273 yoon s .- c . , cantiello m. , 2010 , apj , 717 , l62 yoon s .- c . , grfener g. , vink j.s . , kozyreva a. , izzard r.g . , 2012 , a&a , 544 , l11 for models including rotation effects , the initial rotation speed is assumed to be 40% of the critical rate at the surface of the zero - age main - sequence models . although the rotation speed is considerable during the main - sequence stage , it decreases significantly in the envelopes of supergiant models because of the expansion and mass loss . fig.[fig : rot ] shows runs of angular frequency of rotation in two bsg models having similar effective temperatures ; one ( blue lines ) is evolving toward the red supergiant stage and one ( red lines ) evolving on the blue loop after the red supergiant stage . in this figure , each rotation profile , @xmath183 , is normalized by two different quantities ; @xmath184(solid line ) and @xmath185(dashed line ) . the solid lines indicate that in both models the mechanical effect of rotation on the stellar structure is small because it is much smaller than the local gravity throughout the interior ; i.e. , @xmath186 . the dashed lines in fig.[fig : rot ] indicate rotation frequency itself normalized by the global parameters in the same way as the normalized pulsation frequency shown , e.g. , in fig.[fig : omega ] . although the rotation frequency is very high in the core of a supergiant , in the envelope it is much smaller than the pulsation frequencies ; @xmath187 for radial pulsations ( fig.[fig : omega ] ) and @xmath188 for nonradial oscillatory convection modes in the same normalization . since these pulsations are well confined to the stellar envelope ( @xmath189 , see figs.[fig : work3 ] and [ fig : nradwork ] ) , rotation hardly influence the pulsation property in these supergiant models . this justifies our pulsation analyses without including the effect of rotation .
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a massive star can enter the blue supergiant region either evolving directly from the main - sequence , or evolving from a previous red supergiant stage .
the fractions of the blue supergiants having different histories depend on the internal mixing and mass - loss during the red supergiant stage .
we study the possibility to use diagnostics based on stellar pulsation to discriminate blue supergiants having different evolution histories . for this purpose
we have studied the pulsation property of massive star models calculated with the geneva stellar evolution code for initial masses ranging from 8 to 50m@xmath0 with a solar metallicity of @xmath1 .
we have found that radial pulsations are excited in the blue - supergiant region only in the models that had been red - supergiants before .
this would provide us with a useful mean to diagnose the history of evolution of each blue - supergiant . at a given effective temperature , much more
nonradial pulsations are excited in the model after the red - supergiant stage than in the model evolving towards the red - supergiant .
the properties of radial and nonradial pulsations in blue supergiants are discussed .
predicted periods are compared with period ranges observed in some @xmath2-cygni variables in the galaxy and ngc300 .
we have found that blue supergiant models after the red - supergiant stage roughly agree with observed period ranges in most cases .
however , we are left with the puzzle that the predicted surface n / c and n / o ratios seem to be too high compared with those of deneb and rigel .
[ firstpage ] stars : evolution stars : early - type stars : mass - loss stars : oscillations stars : rotation stars : abundances
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nowadays , digital images and other multimedia files can become very large in size and , therefore , occupy a lot of storage space . in addition , owing to their size , it takes more time to move them from place to place and a larger bandwidth to download and upload them on the internet . so , digital images may pose problems if we regard the storage space as well as file sharing . to tackle this problem , _ image compression _ which deals with reducing the size of an image ( or any other multimedia ) file can be used . image compression actually refers to the reduction of the amount of image data ( bits ) required for representing a digital image without causing any major degradation of the image quality . by eliminating redundant data and efficiently optimizing the contents of a file image , provided that as much basic meaning as possible is preserved , image compression techniques , make image files smaller and more feasible to share and store . the study of digital image compression has a long history and has received a great deal of attention especially with respect to its many important applications . references for theory and practice of this method are @xcite and @xcite , to name but a few . image compression , as well as other various fields of digital image processing , benefits from the theory of linear algebra as a helpful tool . in particular , singular value decomposition ( svd ) is one of the most useful tools for image compression @xcite . the matrix @xmath0 can be written in the form of @xmath1 , where @xmath2 and @xmath3 are unitary matrices , i.e. , @xmath4 , where @xmath5 denotes complex conjugate transpose and @xmath6 is @xmath7 identity matrix . the matrix @xmath8 is an @xmath9 diagonal matrix in such a way that its nonnegative entries are ordered in a non - increasing order ( see for example , theorem 7.3.5 of @xcite ) . with respect to the influences of singular values of @xmath10 in compressing an image , and considering the important point that the singular values of @xmath10 are the positive square roots of the eigenvalues of matrices @xmath11 and @xmath12 , the present study concerns itself with the eigenvalue of the normal matrices @xmath13 and @xmath14 on the purpose of establishing certain techniques for image compression that are efficient , lead to desirable results and need fewer calculations . in the next section , we briefly present some definitions and concepts about normal matrices . section [ comp ] consists of two subsections in which the proposed image compression methods are explained . in section [ exp ] , the validity rates of the presented image compression schemes are investigated and compare their efficiencies by experimental results . in this section , we review the definition and some properties of normal matrices . see @xcite and the references mentioned there as the suggested sources on a series of conditions on normal matrices . in the next section , we will describe the proposed method on the basis of these presented properties . a matrix @xmath15 is called _ normal _ if @xmath16 . assuming @xmath17 as an @xmath18-square normal matrix , there exists an orthonormal basis of @xmath19 that consists of eigenvectors of @xmath17 , and @xmath17 is unitarily diagonalizable . that is , let the scalars @xmath20 , counted according to multiplicity , be eigenvalues of the normal matrix @xmath17 and let @xmath21 be its corresponding orthonormal eigenvectors . then , the matrix @xmath17 can be factored as the following : @xmath22,\ ] ] where the matrix @xmath23 satisfies @xmath24 . maintaining the generality , assume that eigenvalues are ordered in a non - ascending sequence of magnitude , i.e. , @xmath25 . it is to be noticed that , if all the elements of the matrix @xmath17 are real , then @xmath26 , where @xmath27 refers to the transpose of the matrix @xmath17 . a square matrix @xmath17 is called _ symmetric _ if @xmath28 and called _ skew - symmetric _ if @xmath29 . that symmetric and skew - symmetric matrices are normal is easy to see . also , the whole set of the eigenvalues of a real symmetric matrix are real , but all the eigenvalues of a real skew - symmetric matrix are purely imaginary . a general square matrix @xmath17 satisfies @xmath30 , for which the symmetric matrix @xmath31 is called the _ symmetric part _ of @xmath17 and , similarly , the skew - symmetric matrix @xmath32 is called the _ skew - symmetric _ part of @xmath17 . as a consequence , every square matrix may be written as the sum of two normal matrices : a symmetric matrix and a skew - symmetric one . we specially use this point in the proposed image compression techniques . this section consists of two subsections where methods for image compression are presented using normal matrices . to this purpose , the matrix representing the image is transformed into the space of normal matrices . next , the properties of its eigenvalue decomposition are utilized , and some less significant image data are deleted . finally , by returning to the original space , the compressed image can be constructed . let @xmath33 be an @xmath34 matrix to represent the image . two distinct methods are taken into account . first , the symmetric parts of @xmath33 are dealt with to establish an image compression scheme . this procedure can be performed in the same way for the skew - symmetric parts of the matrix @xmath33 . next , another technique is explained using both symmetric and skew - symmetric parts of the matrix @xmath33 . what is noticeable is that finding the eigenvalues and eigenvectors of a matrix requires fewer calculations than finding its singular values and singular vectors . moreover , it is possible to calculate the eigenvalues and eigenvectors of a normal ( especially symmetric or skew - symmetric ) matrix by explicit formulas and , therefore , may yet again need less computation @xcite . in this subsection , a technique of image compression comes into focus on the basis of the eigenvalue decomposition of the symmetric parts of the matrix @xmath33 . this can be performed in the same way for the skew - symmetric parts of @xmath33 . it is to be noted that , for the same image , the results obtained by these two techniques ( i.e. using symmetric or skew - symmetric part ) may be different . assume @xmath35 as the symmetric part of the matrix @xmath33 . the normal matrix @xmath35 can be factored as in the following : @xmath36 now , bearing in mind that the eigenvalues are sequenced in a non - ascending order of magnitude , compress the symmetric part of the image by wiping off the small enough eigenvalues of @xmath35 . if @xmath37 of the larger eigenvalues remains , then there is @xmath38 where the total storage for @xmath39 is @xmath40 . here comes the stage of reconstructing the compressed image @xmath41 from its symmetric part . to this purpose , all the elements located above the main diagonal of the matrix @xmath33 are needed , and this calls for @xmath42 storage spaces . because @xmath41 can be considered as an acceptable approximation of @xmath33 , let us take all the elements above the main diagonal of @xmath41 as the elements of @xmath33 . obviously , the elements located below the main diagonal of the matrix @xmath43 should be determined too . in addition , it is inferred from the fact @xmath44 that the elements located below the main diagonal of @xmath41 can be obtained by subtracting the elements below the main diagonal of @xmath45 from the elements of @xmath43 . see the following equation where @xmath46 and @xmath47 denote the given and unknown entries , respectively . @xmath48}\limits_{\tilde{x } } + \mathop { \left [ { \begin{array}{*{20}c } \times & { } & \times \\ { } & \ddots & { } \\ \checkmark & { } & \times \\ \end{array } } \right]}\limits_{\tilde{x}^t } .\ ] ] it is clear that the main diagonal elements of @xmath41 are the same as those on the main diagonal of @xmath39 . it is also to be noted that , by this procedure , only the elements located below the main diagonal of @xmath33 are modified , and its other elements are remain untouched . moreover , to reconstruct the compressed image @xmath41 , the elements located below the main diagonal of @xmath33 may be reserved instead of those above the main diagonal , and then a procedure similar to what has been performed newly is to be followed . indeed , @xmath33 can be partitioned into several segments , and those segments may be reserved provided that @xmath41 is reconstructed . this may be useful specially when some segments of the image have more significance or unchanging ( or uncompressing ) some partitions of the image is desirable during the image compression . finally , it should be pointed out that reconstruction of the compressed image @xmath41 requires @xmath49 storage spaces when the symmetric part of @xmath33 is dealt with . however , if the skew - symmetric part of @xmath33 is concerned , the diagonal elements of the @xmath41 can not be obtained from @xmath39 and , therefore , they must be reserved . reconstructing the compressed image @xmath41 , thus , requires @xmath50 storage spaces when the skew - symmetric part of @xmath33 is considered for the image compression method . in the next subsection , another image compression technique is provided , for which reconstructing the compressed image demands fewer storage spaces . the previous subsection introduced an image compression scheme dealing with almost half of an image . by that technique , more than half of the elements of the image remained unchanged . this may be of some usages and advantages . compressed images obtained by the presented method have a high quality . however , compressing of just half of the original image may cause the method to lose its reliability as compared to some other image compression schemes . to tackle this problem , a new method is presented here about both symmetric and skew - symmetric parts of the matrix @xmath33 in order to compress the image all over . this new image compression technique is found to be of a remarkably high reliability . as previously mentioned , every matrix equals the sum of its symmetric and skew - symmetric parts . to establish the new method , with the definition of @xmath35 , borne in mind , @xmath51 is used to represent the skew - symmetric part of @xmath33 . the matrix @xmath51 can be written as follows : @xmath52 with respect to the method described in the previous subsection , for an integer @xmath53 , set @xmath54 through ( [ sym ] ) and ( [ ssym ] ) , the compressed image @xmath55 will be @xmath56 as in the case of reserving the matrix @xmath39 , @xmath40 storage spaces are required for saving the matrix @xmath57 . as a result , the total storage requirement for @xmath55 is @xmath58 . in this section , the validity and the influence of the proposed image compression method are examined . let us note that the ideas presented in this paper , can readily be used to establish a block - based compression scheme . this scheme concerns dividing of an image into non - overlapping blocks and compressing of each block . the peak signal to noise ratio ( psnr ) is calculated to measure the quality of the compressed image . in the case of gray scale images of size @xmath59 , whose pixels are represented with 8 bits , psnr is computed as follows : @xmath60 where @xmath61 and @xmath62 refer to the elements of the original and the compressed images respectively . in the above relationship , mse stands for the mean square error between the original image and the compressed image pixels . in addition , compression ratio ( cr ) may be calculated as an important index to evaluate how much of an image is compressed . cr is the amount of bits in the original image divided by the amount of bits in the compressed image ; that is , @xmath63 for the sake of simplicity , method # 1 and method # 2 are branded as image compression techniques which use the symmetric and the skew - symmetric parts of the matrix representing the image , in the image compression schemes , respectively . in addition , let method # 3 denominate the image compression technique described in subsection [ sb2 ] and let method # 4 stand for the image compression method using svd . consequently , the following relationships emerge for cr of these methods : @xmath64 in the experiments conducted in this study , three @xmath65 gray scale images were considered , including ( a ) lena , ( b ) baboon and ( c ) gold hill presented in fig [ fig : images1 ] . the psnr results are shown in tables [ table : lenap ] , [ table : baboonp ] and [ table : goldhillp ] , for some integer values of @xmath37 , for images lena , baboon and gold hill , respectively . also , the cr results are given in table [ table : goldhillc ] for a @xmath65 image . the results obtained by these techniques are compared to those achieved by method # 4 in our tables . furthermore , figures [ fig : compressed50 ] and [ fig : compressed100 ] show the compressed image lena obtained by the proposed techniques as well as image compression method using svd , method # 4 , for @xmath66 and @xmath67 , respectively . .psnr results for lena [ cols="^,^,^,^,^ " , ] [ table : goldhillc ] . ] . ] in this paper , the eigenvalue decomposition of symmetric and skew - symmetric matrices , as two important kinds of normal matrices , and their properties were applied to obtain image compression schemes . the proposed method is straightforward and uncomplicated ones requiring clear and fewer computations as compared to some exiting methods . this image compression method is the first one ever introduced and concerns the symmetric part of an image , which can also be applicable in the case of the skew - symmetric part of the image . the method is capable of keeping half ( or some selected segments ) of an image unchanged , and this feature may be of some usages . however , as observed in this study , just half of the original image could be changed , which causes the compressed image to lose its reliability . this is what makes the technique different from some other image compression techniques . hence , the second image compression scheme using both symmetric and skew - symmetric parts of the original image was proposed . the experimental results show the high reliability of this method . finally , it is to be pointed out that the proposed method can be used to devise block - based compression techniques . j. cullum , r.a . willoughby , computing eigenvalues of very large symmetric matrices an implementation of a lanczos algorithm with no re - orthogonalization , journal of computational physics , 44 ( 1981 ) 329358 . jia , yin - jie , peng - fei xu , xu - ming pei . , an investigation of image compression using block singular value decomposition . , communications and information processing . springer berlin heidelberg , ( 2012 ) . 723 - 731 . tian , w. luo , l.z . liao , an investigation into using singular value decomposition as a method of image compression . , proceedings of the fourth international conference on machine learning and cybernetics , pp . 5200-5204 ( 2005 ) . jia , yin - jie , peng - fei xu , xu - ming pei . , an investigation of image compression using block singular value decomposition . , communications and information processing . springer berlin heidelberg , ( 2012 ) . 723 - 731 .
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in this paper , we present methods for image compression on the basis of eigenvalue decomposition of normal matrices .
the proposed methods are convenient and self - explanatory , requiring fewer and easier computations as compared to some existing methods . through the proposed techniques ,
the image is transformed to the space of normal matrices .
then , the properties of spectral decomposition are dealt with to obtain compressed images .
experimental results are provided to illustrate the validity of the methods .
_ keywords : _ image compression , transform , normal matrix , eigenvalue .
| 4,277 | 141 |
we can study the effect of electromagnetic fields on fluids only if we know the stress induced due to the fields in the fluids . despite its importance , this topic is glossed over in most works on the otherwise well - established subjects of fluid mechanics and classical electrodynamics . the resultant force and torque acting on the body as a whole are calculated but not the density of body force which affects flow and deformation of materials . helmholtz and korteweg first calculated the body force density in a newtonian dielectric fluid in the presence of an electric field , in the late nineteenth century . however , their analysis was criticized by larmor , livens , einstein and laub , who favoured a different expression proposed by lord kelvin . it was later on shown that the two formulations are not contradictory when used to calculate the force on the body as whole and that they can be viewed as equivalent if we interpret the pressure terms appropriately . we refer to bobbio s treatise @xcite for a detailed account of the controversy , the experimental tests of the formulas and their eventual reconciliation . the few published works on the topic like the text books of landau and lifshitz @xcite , panofsky and phillips @xcite and even bobbio @xcite treat fluids and elastic solids separately . further , they restrict themselves to electrically and magnetically linear materials alone . in this paper , we develop an expression for stress due to external electromagnetic fields for materials with simultaneous fluid and elastic properties and which may have non - linear electric or magnetic properties . our analysis is thus able to cater to dielectric viscoelastic fluids and ferro - fluids as well . we also extend rosensweig s treatment @xcite , by allowing ferro - fluids to have elastic properties . let us first see why the problem of finding stress due to electric or magnetic fields inside materials is a subtle one while that of calculating forces on torques on the body as a whole is so straightforward . the standard approach in generalizing a collection of discrete charges @xmath0 to a continuous charge distribution is to replace the charges themselves with a suitable density function @xmath1 and sums by integrals . thus , the expression for force @xmath2 , ( @xmath3 is the electric field at the location of the charge @xmath0 . ) on a body on @xmath4 discrete charges in an electric field @xmath5 , is replaced with @xmath6 , when the body is treated as a continuum of charge , the integral being over the volume of the body . the integral can be written as @xmath7 where @xmath8 is the force density in the body due to an external electric field . it can be shown that @xcite that the same expression for force density is valid even inside the body . if instead , the body were made up of discrete dipoles instead of free charges , then the force on the body as a whole would be written as @xcite @xmath9 where @xmath10 is the dipole moment of the @xmath11th point dipole and @xmath3 is the electric field at its position . if the body is now approximated as a continuous distribution of dipoles with polarization @xmath12 , then the force on the whole body is written as @xmath13 while this is a correct expression for force on the body as a whole , it is not valid if applied to a volume element inside the material . in other words , @xmath14 is not a correct expression for density of force in a continuous distribution of dipoles although @xmath15 is the density of force in the analogous situation for monopoles . we shall now examine why it is so . consider two bodies @xmath16 and @xmath17 that are composed of charges and dipoles respectively . ( the subscripts of quantities indicate their composition . ) let @xmath18 and @xmath19 be volume elements of @xmath16 and @xmath17 respectively . the volume elements are small compared to dimensions of the body but big enough to have a large number of charges or dipoles in them . the forces @xmath20 and @xmath21 on @xmath18 and @xmath19 respectively due to the surrounding body are @xmath22 where @xmath4 is the number of charges or dipoles inside the volume element under consideration . in both these expressions , @xmath3 is the macroscopic electric field at the position of @xmath11th charge or dipole . it is the average value of the microscopic electric field @xmath23 at that location . that is @xmath24 , where @xmath25 denotes the spatial average of the enclosed quantity . the microscopic field @xmath23 can be written as @xmath26 where @xmath27 is the microscopic field due to the charges or dipole outside the volume element and @xmath28 is the field due to charges or dipoles inside the volume element other than the @xmath11th charge or dipole . for the volume element @xmath18 of point charges , @xmath29 where @xmath30 is the microscopic electric field at the position of @xmath11th charge due to @xmath31th charge inside @xmath18 . therefore , @xmath32 newton s third law makes the second sum on the right hand side of the above equation zero . @xmath20 is thus due to charges outside @xmath18 alone for which the standard approach of replacing sum by integral and discrete charge by charge density is valid . therefore , @xmath15 continues to be the volume force density inside the body . if the same analysis were to be done for the volume element @xmath19 of point dipoles , it can be shown that the contribution of dipoles inside @xmath19 is not zero . in fact , the contribution depends on the shape of @xmath19 @xcite . that is the reason why @xmath14 , also called kelvin s formula , is not a valid form for force density in a dielectric material . we would have got the same results for a continuous distribution of magnetic monopoles , if they had existed , and magnetic dipoles . that is @xmath33 is not the correct form of force density of a volume element in a material with magnetization @xmath34 in a magnetic field @xmath35 . the goal of this paper is to develop an expression for stress inside a material with both viscous and elastic properties in the presence of an external electric or magnetic field , allowing the materials to have non - linear electric and magnetic properties . we demonstrate that by making some fairly general assumptions about thermodynamic potentials , it is possible to develop a theory of stresses for materials with fluid and elastic properties . we check the correctness of our results by showing that they reduce to the expressions developed in earlier works when the material is a classical fluid or solid . to our knowledge , there is no theory of electromagnetic stresses in general continua with simultaneous fluid and elastic properties . since we are using techniques of equilibrium thermodynamics for our analysis , we will not be able to get results related to dissipative phenomena like viscosity . deriving an expression for viscosity for even a simple case of a gas requires full machinery of kinetic theory @xcite . developing a theory of electro and magneto viscous effects is a much harder problem and we shall not attempt to solve it in this paper . we begin our analysis in section ( [ sec : thermod ] ) by reviewing expressions for the thermodynamic free energy of continua in electric and magnetic fields . after pointing out the relation between stress and free energy in section ( [ sec : dielectric ] ) , we obtain a general relation for stress in a dielectric material in presence of an electric field . we check its correctness by showing that it reduces to known expressions for stress in newtonian fluids and elastic solids . the framework for deriving electric stress is useful for deriving magnetic stress in materials that are not permanently magnetized . section ( [ sec : magnetic ] ) mentions the expression for stress in a continuum in presence of a static magnetic field . we then point out the assumptions in derivations of ( [ sec : dielectric ] ) and ( [ sec : magnetic ] ) that render the expressions of stress unsuitable for ferro - fluids and propose the one that takes into account the permanent magnetization of ferro particles . we derive expressions for ponderomotive forces in section ( [ sec : ponder ] ) from the expressions for stress obtained in previous sections . most of our analysis rests on framework scattered in the classic works of landau and lifshitz on electrodynamics @xcite and elasticity @xcite generalizing it for continua of arbitrary nature . electromagnetic fields alter thermodynamics of materials only if they are able to penetrate in their bulk . conducting materials have plenty of free charges to shield their interiors from external static electric fields . therefore , the effect of external static electric fields are restricted to their surface alone , in the form of surface stresses . the situation in dielectrics is different - a paucity of free charges allows an external static field to penetrate throughout its interior polarizing its molecules . the external field has to do work to polarize a dielectric . this is akin to work done by an external agency in deforming a body . the same argument applies to a body exposed to a magnetic field . unlike static electric fields that are shielded in conductors , magnetic fields always penetrate in bodies , magnetizing them . the nature of the response depends on whether a body is diamagnetic , paramagnetic or ferromagnetic . in all the cases , magnetic fields have to do work to magnetize them and therefore the thermodynamics of continua is always affected by a magnetic field . we shall develop thermodynamic relations for materials exposed to static electromagnetic fields in this section . at a molecular level , electric and magnetic fields deform matter for which the fields have to do work . the material and the field together form a thermodynamic system . the work done on it is of the form @xmath36 where @xmath37 is an intensive quantity and @xmath38 a related extensive quantity denotes the possibly inexact differential of a quantity @xmath37 . ] . in the case of a dielectric material in a static electric field , the intensive quantity is the electric field @xmath5 and the extensive quantity is the total dipole moment @xmath39 , @xmath12 being the polarization and @xmath40 being the volume of the material . in the case of a material getting magnetized , the intensive quantity is the magnetizing field @xmath35 and the extensive quantity is the total magnetic moment @xmath41 , @xmath34 being the polarization and @xmath40 being the volume of the material . the corresponding work amounts are @xmath42 and @xmath43 respectively . we added a subscript @xmath44 because this is only one portion of the work . the other portion of the work is required to increment the fields themselves to achieve a change in polarization or magnetization . they are @xmath45 and @xmath46 respectively , where @xmath47 is the permittivity of free space and @xmath48 is the permeability of free space respectively and that of a magnetic field is @xmath49 . ] therefore , the total work needed to polarize and magnetize a material , at constant volume , are @xmath50 we have derived these relations for linear materials . we will now show that they are true for any material . imagine a dielectric immersed in an electric field . let the electric field be because of a charge density @xmath51 . let the electric field be increased slightly by changing the charge density by an amount @xmath52 . work done to accomplish this change is @xmath53 where @xmath54 is the electric potential . since @xmath55 , @xmath56 if the charge density is localized then the volume of integration can be taken as large as we like . we do so and also convert the first integral on the right hand side to a surface integral . the first term then makes a vanishingly small contribution to the total and the work done in polarizing a material can be written as @xmath57 let a material be magnetized by immersing it in a magnetic field . the magnetic field can be assumed to be created because of a current density @xmath58 . let the magnetic field be increased slightly by changing the current density . we further assume that the rate of increase of current is so small that @xmath59 at all stages . the source of current has to do an additional work while increasing the amount of current density in order to overcome the opposition of the induced electromotive force . if @xmath60 is the induced emf , then the sources will have to do an additional work at the rate @xmath61 , where @xmath62 is the magnetic flux and the dot over head denotes total time derivative . the amount of work needed is @xmath63 . if @xmath64 is the cross sectional area of the current , then @xmath65 but @xmath66 , therefore , @xmath67 since we assumed the current to be increased at an infinitesimally slow rate , there are no displacement currents and @xmath68 . @xmath69 where we have used the vector identity @xmath70 . we once again assume that the current density is localized and therefore converting the second integral on the right hand side of equation ( [ thermod : e6 ] ) into a surface integral results in an infinitesimally small quantity . the work done in magnetizing a material is therefore , @xmath71 a change in the helmholtz free energy of a system is equal to the work done by the system in an isothermal process , which in turn is related to stresses in the continuum . we will show how stress is related to the helmholtz free energy . let us consider the example of an ideal gas . the change in its helmholtz free energy , is given by @xmath72 . using the first and the second laws of thermodynamics we have @xmath73 . therefore , @xmath74 , which under isothermal conditions means @xmath75 . in this simple system , @xmath76 is the isotropic portion of the stress and @xmath77 is related to the isotropic strain . thus , we can get @xmath76 is we know change in helmholtz free energy and volume . therefore , a first step toward getting an expression for stress is to find the helmholtz free energy . under isothermal conditions , the first law of thermodynamics is @xmath78 where @xmath79 is the total free energy , @xmath80 is the absolute temperature , @xmath81 is the total entropy and @xmath82 is the work done on the system . first law of thermodynamics for polarizable and magnetizable media is @xmath83 where @xmath84 is the mechanical work done on the system . if @xmath85 , @xmath86 and @xmath87 are internal energy , mechanical work and entropy of the media _ per unit volume _ , first law of thermodynamics for polarizable and magnetizable media is @xmath88 the mechanical work done on a material is @xmath89 where @xmath90 is the stress tensor and @xmath91 is the strain tensor in the medium . further , with this substitution , all quantities in equations ( [ fe:4 ] ) and ( [ fe:5 ] ) become exact differentials allowing us to replace @xmath92 with @xmath93 . if @xmath94 is the helmholtz free energy per unit volume , @xmath95 these relations give change in helmholtz free energy in terms of change in @xmath96 and @xmath97 . the @xmath96 field s source is free charges alone while the @xmath97 field s source is all currents . in an experiment , we can control the total charge and free currents . therefore , it is convenient to express free energy in terms of @xmath5 , whose source is all charges - free and bound , and @xmath35 , whose source is free currents . we therefore introduce associated helmholtz free energy function @xmath98 for polarizable media as @xmath99 and for magnetizable media as @xmath100 . equations ( [ fe:6 ] ) and ( [ fe:7 ] ) therefore become @xmath101 if @xmath102 is the deviatoric stress , @xmath103 , where @xmath76 is the hydrostatic pressure . therefore we have , @xmath104 the quantity @xmath105 is the dilatation of the material during deformation . therefore the thermodynamic potential @xmath98 of a polarizable ( magnetizable ) medium is thus , a function of @xmath80 , @xmath40 , @xmath91 and @xmath5(@xmath35 ) . equivalently , it can be considered a function of @xmath80 , @xmath106 , @xmath91 and @xmath5(@xmath35 ) , where @xmath106 is the mass density of the medium . we will now calculate the stress tensor in a polarizable medium . we consider a small portion of the material and find out the work done by the portion in a deformation in presence of an electric field . the portion is small enough to approximate the field to be uniform throughout its extent . we emphasize that through this assumption we are not ruling out non - uniform fields but only insisting that the portion be small enough to ignore variations in it . since a sufficiently small portion of a material can be considered to be plane , the volume element under consideration can be assumed to be in form of a rectangular slab of height @xmath107 . let it be subjected to a virtual displacement @xmath108 which need not be parallel to the normal @xmath109 to the surface . the virtual work done by the medium per unit area in this deformation is @xmath110 , where @xmath111 is the stress _ on _ the portion . if @xmath90 is the stress _ due to _ the portion _ on _ the medium , then @xmath112 . therefore , the virtual work done by the medium on the portion is @xmath113 . further , since both @xmath111 and @xmath90 are symmetric , the virtual work can also be written as @xmath114 . the change in helmholtz free energy during the deformation is @xmath115 per unit surface area . if we assume the deformation to be isothermal , @xmath116 change in height of the slab is @xmath117 [ dielectric : f1 ] the geometry of the problem is described in figure 2 . for an isothermal variation @xmath118 we depart from the convention in thermodynamics , to indicate variables held constant as subscripts to partial derivatives , to make our equations appear neater . we shall also use the traditional notation for partial derivatives . we will now get expressions for each term on the right hand side of equation ( [ dielectric:3 ] ) . 1 . if @xmath119 is the helmholtz free energy in absence of electric field , @xmath120 , where @xmath121 is the permittivity tensor . permittivity is known to be a function of mass density of a material , the dependence being given by clausius - mossotti relation@xcite . electric field is _ usually _ independent of mass density of the material . however , that is not so if the material has a pronounced density stratification like a fluid heated from above . if @xmath122 and @xmath123 are two elements of such a fluid , at the top and bottom respectively , both having identical volume then @xmath122 will have less number of dipoles than @xmath123 . the electric field inside them , due to matter within their confines too will differ . we point out that although divergence of @xmath5 depends only on the density of free charges , @xmath5 itself is produced by all multipoles . therefore , @xmath124 the last term in equation ( [ dielectric:4 ] ) is absent if the material has a uniform temperature . it is not included in the prior works of bobbio@xcite and landau and lifshitz@xcite . if the @xmath125 ( or @xmath126 ) axis is assumed to be along the normal and the deformation is uniform , the displacement of a layer of the volume element can be described as @xmath127 where @xmath125 is the vertical distance from the lower surface . since @xmath108 is fixed , @xmath128 and @xmath129 since the strain tensor is always symmetric , @xmath130 the electric field does not depend on strain but permittivity does . this is because , deformation may change the anisotropy of the material , which determines its permittivity . likewise , permittivity and strain tensors do not depend on electric field ] . therefore they can be pulled out of the integral and @xmath131 3 . from equation ( [ fe:8 ] ) , @xmath132 therefore , the last term is just @xmath133 . using equations ( [ dielectric:4 ] ) , ( [ dielectric:9 ] ) and ( [ dielectric:10 ] ) in ( [ dielectric:3 ] ) , we get @xmath134 substituting ( [ dielectric:11 ] ) and ( [ dielectric:2 ] ) in ( [ dielectric:1 ] ) we get @xmath135 we have gathered terms independent of electric field in the first curly bracket of equation ( [ dielectric:12 ] ) , keeping the contribution of electric field to stress in the rest . we still have to find out the expressions for @xmath136 and @xmath137 . a change in density of a layer depends on the change it its height ( or thickness ) , therefore , @xmath138 or , @xmath139 we will now estimate change in electric field due to deformation . consider a volume element at a point @xmath140 . let it undergo a deformation by @xmath141 . as a result , matter that used to be at @xmath142 now appears at @xmath140 . in a virtual homogeneous deformation , every volume element carries its potential as the material deforms . therefore , the change in potential at @xmath140 is @xmath143 . since @xmath144 ( see equation ( [ dielectric:5 ] ) ) , @xmath145 since @xmath146 , @xmath147 we have used the assumption that the region is small enough to have almost uniform electric field and therefore it can be pulled out of the gradient operator . equation ( [ dielectric:12 ] ) therefore becomes @xmath148 stress in a polarized viscoelastic material at rest is therefore , @xmath149 we can simplify equation ( [ dielectric:16a ] ) by writing the terms in the first curly bracket as familiar thermodynamic quantities . if @xmath150 is the total helmholtz free energy of the substance in absence of electric field and @xmath119 is the helmholtz free energy per unit volume then @xmath151 , where @xmath152 is the volume of the substance , @xmath153 the mass and @xmath106 the density . maxwell relation for pressure in terms of total helmholtz free energy is @xmath154 similarly , the dependence of @xmath119 on strain tensor @xmath155 can be written as @xmath156@xcite , where we have retained only the deviatoric of the strain tensor because the isotropic part is already accounted in hydrostatic pressure of equation ( [ dielectric:16a ] ) . the constant @xmath157 is the shear modulus of the substance . therefore , @xmath158 equation ( [ dielectric:16a ] ) can therefore be written as @xmath159 we will now look at some special cases of ( [ dielectric:16 ] ) , 1 . if there is no matter , terms with pressure , density and strain tensor will not be present . further @xmath160 and equation ( [ dielectric:16 ] ) becomes the maxwell stress tensor for electric field in vacuum . @xmath161 we emphasize that the general expression for stress in a material exposed to static electric field reduces to maxwell stress tensor only when we ignore all material properties . if there is no electric field , all terms in the second and third curly bracket of ( [ dielectric:16 ] ) vanish . further , if the medium is a fluid without elastic properties , @xmath119 , will not depend on @xmath155 and the stress will be @xmath162 thus the stress in a fluid without elastic properties is purely hydrostatic . we do not see viscous terms in ( [ dielectric:17 ] ) because viscosity is a dissipative effect while @xmath163 is obtained from helmholtz free energy which has information only about energy than can be extracted as work . if the material were a solid and if there are no electric fields as well , the stress is @xmath164 it is customary to write the first term of equation ( [ dielectric:18 ] ) in terms of @xmath165 , the bulk modulus as @xmath166 4 . for a fluid dielectric with isotropic permittivity tensor , @xmath119 is independent of @xmath155 and @xmath167 . if the fluid has a uniform density , equation ( [ dielectric:16 ] ) then becomes @xmath168 this expression matches the one obtained in @xcite , after converting to gaussian units , and after accounting for the difference in the interpretation of stress tensor . landau and lifshitz s stress tensor is @xmath169 . 5 . for a fluid dielectric with isotropic permittivity tensor and in which the electric field depends on density equation ( [ dielectric:16 ] ) then becomes @xmath170 6 . for a solid dielectric we can assume that @xmath119 , @xmath5 and @xmath171 are independent of @xmath106 . equation ( [ dielectric:16 ] ) now becomes @xmath172 if the solid is isotropic and remains to be so after application of electric field , @xmath167 and ( [ dielectric:20 ] ) simplifies to @xmath173 where @xmath174 is the part of stress tensor that exists even in absence of electric field . this expression matches with the one in @xcite if one converts it to gaussian units , assumes the constitutive relation @xmath175 and takes into account that their stress tensor is @xmath169 . 7 . for a viscoelastic liquid that is also a linear dielectric with uniform density , @xmath176 in order to calculate stress in a magnetic fluid , we continue to use the physical set up used in section ( [ sec : dielectric ] ) of a small slab of viscoelastic liquid subjected to magnetic field . if there are no conduction and displacement currents , ampere s law becomes @xmath177 , making the @xmath35 field conservative . it can then be treated like the electrostatic field of section ( [ sec : dielectric ] ) . in order to extend the analysis of section ( [ sec : dielectric ] ) to magnetic fluids , we need an additional assumption of magnetic permeability being independent of @xmath35 . although the first assumption , of no conduction and displacement currents , is valid in the case of ferro - viscoelastic fluids , the second assumption of field - independent permeability is not . therefore , this analysis is valid only for the single - valued , linear section of the @xmath97 versus @xmath35 curve of ferro - viscoelastic liquid , giving @xmath178 where @xmath179 is the magnetic permeability tensor . we have omitted the term accounting for dependence of @xmath35 on mass density @xmath106 because we are not aware of a situation where it may happen . the expressions derived in sections ( [ sec : dielectric ] ) and ( [ sec : magnetic ] ) are valid only if permittivity and permeability are independent of electric and magnetic fields respectively . ferro - fluids are colloids of permanently magnetized particles . as the applied magnetic field increases from zero , an increasing number of sub - domain magnetic particles align themselves parallel to the field opposing the random thermal motion leading to a magnetization that increased in a non - linear manner . the magnetic susceptibility and therefore permeability depend on the field . it can not be pulled out of the integral sign . equation ( [ magnetic:1 ] ) should be written as @xmath180 if the elastic effects are negligible , equation ( [ ferro:1 ] ) reduces to @xmath181 if @xmath182 , as is normally for ferro - fluids @xcite , @xmath183 since the applied magnetic field is independent of density , @xmath184 where to get the last equation we have used the relation @xmath185 and the fact that @xmath35 does not depend on @xmath106 . if @xmath186 is the specific volume , that is @xmath187 , equation ( [ ferro:5 ] ) can be written as @xmath188 further , @xmath189 and @xmath185 imply , @xmath190 using equations ( [ ferro:5 ] ) and ( [ ferro:6 ] ) in equation ( [ ferro:3 ] ) , we get @xmath191dh_r\right\}\delta_{ij } - h_i b_j\ ] ] this is same as rosensweig s @xcite equation ( 4.28 ) except that he calculates @xmath111 , which is related to our stress tensor as @xmath169 . we do not know of electric analogues of ferro fluids ( electro - rheological fluids are analogues of magneto - rheological fluids , not ferro fluids ) . however , there are permanently polarized solids , called ferro - electrics . for such materials , the stress is @xmath192 the old term `` ponderable media '' means media that have weight . ponderomotive force is the one that cause motion or deformation in a ponderable medium . in contemporary terms , it is the density of body force in a material . it is related to the stress tensor @xmath90 as @xmath193 we mention a few familiar special cases of this equation for fluids of various kinds . 1 . for incompressible , newtonian fluids the stress tensor is given by ( [ dielectric:17 ] ) and the force density is @xmath194 note that the force density does not include the dissipative component due to viscosity . 2 . for an incompressible , newtonian , dielectric fluid in presence of static electric field , assuming that the electric field inside the fluid is independent of density , the stress tensor is given by equation ( [ dielectric:19 ] ) and the ponderomotive force is @xmath195 where @xmath196 is the density of free charges in the fluid and @xmath197 is its relative permittivity . in deriving equation ( [ ponder:3 ] ) we used gauss law @xmath198 and the fact that we are dealing with an electrostatic field ( @xmath199 ) , for which @xmath200 . in an ideal , dielectric fluid @xmath201 and @xmath202 the relative permittivity is a function of temperature and the term @xmath203 is significant in a single - phase fluid only if there is a temperature gradient . the third term in equation ( [ ponder:4 ] ) is called the electro - striction term and it is present only when the electric field or @xmath204 or both are non - uniform . the derivative of the relative permittivity with respect to mass density is calculated using the clausius - mossotti relation @xcite , @xcite . 3 . continuing with the same fluid as above but now having a situation in which the electric field is a function of mass density @xmath106 , we have an additional term in equation ( [ ponder:4 ] ) given by @xmath205 we come across such a situation when there is a strong temperature gradient in the fluid resulting in a gradient of dielectric constant @xmath197 . since the electric field depends on @xmath197 and @xmath197 depends on mass density through the clausius - mossotti relation , the electric field is a function of mass density and we have to consider this additional term . we hasten to add that it not necessary for there to be a temperature gradient to have such a situation , a gradient of electric permittivity suffices to give rise to such a situation . the derivation for force density in an incompressible , newtonian , diamagnetic or paramagnetic fluid in presence of a static magnetic field is similar except that we use the maxwell s equations @xmath206 and @xmath207 . we also assume the auxiliary magnetic field , @xmath35 , inside the fluid is independent of density . we get @xmath208 the term @xmath209 is the lorentz force term and it is zero if the fluid is not conducting . @xmath210 is the relative permeability of the fluid . the fourth term in equation ( [ ponder:5 ] ) is called the magneto - striction force . it is present only if the magnetic field or @xmath211 or both are non - uniform . the derivative of relative permeability with respect to mass density is calculated using the magnetic analog of the clausius - mossotti relation @xcite . several forms of body force density , all equivalent to each other , can be derived for ferro - fluids from equations ( [ ferro:7 ] ) and ( [ ponder:1 ] ) . we refer to @xcite for more details . [ ponder : i5 ] if the material is dielectric and viscoelastic , we assume that the permittivity depends on the strain . even though the material was isotropic before applying electric field , it may turn anisotropic as its molecules get polarized and align with the field . the scalar permittivity is then replaced with a second order permittivity tensor @xmath171 . following landau and lifshitz s treatment of solid dielectrics @xcite , we assume that the permittivity tensor is a linear function of the strain tensor and write it as @xmath212 where @xmath213 and @xmath214 are constants indicating rate of change of permittivity with strain . we call them @xmath213 and @xmath214 to differentiate them from @xmath215 and @xmath216 used to describe behavior of solid dielectric @xcite . if we assume the material to be incompressible , @xmath217 and @xmath218 for an incompressible material , equation ( [ dielectric:16c ] ) becomes , @xmath219 since @xmath213 and shear modulus @xmath157 are constants , they do not survive in the expression for @xmath220 . the expression for ponderomotive force in a dielectric , viscoelastic fluid is same as that for a dielectric , newtonian fluid . [ ponder : i6 ] the same conclusion follows for a viscoelastic fluid subjected to a magnetic field if we assume that @xmath221 , @xmath222 and @xmath223 being constants , when a fluid is magnetized . time - varying electric fields can penetrate conductors up to a few skin depths that depends on the frequency of the fields and physical parameters of the material like its ohmic conductivity or magnetic permeability . the general problem of response of materials to time - varying fields is quite complicated . however , the results in this paper can be applied for slowly varying fields , that is , the ones that do not significantly radiate . for such fields , the time varying terms of maxwell equations can be ignored . whether a time varying field can be considered quasi - static or not depends on the linear dimension of the materials involved . if @xmath224 is the angular frequency of the fields , the wavelength of corresponding electromagnetic wave is @xmath225 , @xmath226 being the velocity of light in vacuum . if the linear dimension @xmath227 of the materials is much lesser than @xmath228 , for any element @xmath229 of the path of current , there is another within @xmath227 that carries same current in the opposite direction , effectively canceling the effect of current . for power line frequencies , the value of @xmath227 is a few hundred miles and even for low frequency radio waves , with @xmath230 hz , @xmath227 is of the order of @xmath231 m. thus the _ slowly - varying fields _ or _ quasi - static _ approximation @xcite is valid for frequencies up to that of radio waves and our results can be applied under those conditions . 1 . proof of equation ( [ dielectric:8 ] ) . @xmath232 interchange the indices in the first term , @xmath233 since @xmath91 is a symmetric tensor , @xmath234 99 bobbio , s. , electrodynamics of materials : forces stresses , and energies in solids and fluids , academic , new york , chapter 4 , ( 2002 ) . landau l. d. and lifshitz e. m. , electrodynamics of continuous media , pergamon , oxford , ( 1960 ) . panofsky w. k. h. and phillips m. , classical electricity and magnetism , 2nd edition , dover publications inc . , new york , ( 1962 ) . rosensweig r. e. , ferrohydrodynamics , dover publications , mineola , new york , ( 1997 ) . reitz , j. r. , milford , f. j. and christy , r. w. , foundations of electromagnetic theory , 3rd edition , narosa publishing house , new delhi , ( 1990 ) . jackson , j. d. , classical electrodynamics , 3rd edition , john wiley & sons inc , new york , ( 1999 ) . chapman , s. and cowling , t. g. , mathematical theory of non - uniform gases , 3rd edition , cambridge university press , cambridge , ( 1970 ) . landau l. d. and lifshitz e. m. , theory of elasticity , pergamon , oxford , ( 1960 ) . joshi amey , radhakrishna m.c . and rudraiah n , rayleigh - taylor instability in dielectric fluids , physics of fluids , volume 22 , issue 6 , ( 2010 ) .
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a clear understanding of body force densities due to external electromagnetic fields is necessary to study flow and deformation of materials exposed to the fields . in this paper
, we derive an expression for stress in continua with viscous and elastic properties in presence of external , static electric or magnetic fluids .
our derivation follows from fundamental thermodynamic principles .
we demonstrate the soundness of our results by showing that they reduce to known expressions for newtonian fluids and elastic solids .
we point out the extra care to be taken while applying these techniques to permanently polarized or magnetized materials and derive an expression for stress in a ferro - fluid .
lastly , we derive expressions for ponderomotive forces in several situations of interest to fluid dynamics and rheology .
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cold atoms in optical lattices is the application of two formerly distinct aspects of physics : quantum gases from atomic physics @xcite and laser theory from quantum optics @xcite . the optical lattices are artificial crystals of light , that is , a periodic intensity pattern formed by interference of two or more laser beams . as an insight , a pair of these laser beams in opposite directions ( that is , two orthogonal standing waves with orthogonal polarization ) will give a one - dimensional ( 1d ) lattice , two pairs in two opposite directions can be used to create a 2d lattice and a similar three pairs in opposite directions will give a 3d lattice . atoms can be cooled and trapped in these optical lattices . thus in simple form , an optical lattice looks effectively like an egg carton where the atoms , like eggs , can be be arranged one per well to form a crystal of quantum matter @xcite . though the cold atoms in optical lattices was initially used to investigate quantum behaviour such as bloch oscillations , wannier - stark ladders and tunneling phenomena usually associated with crystals in a crystalline solid @xcite , it is the theoretical proposal @xcite and consequent experimental realization @xcite of the superfluid to mott insulator ( sf - mi ) transition which is an important phenomenon in condensed matter physics that has given rise to the possibility of using it as a test laboratory for phenomena in condensed matter physics . the success of the sf - mi transition in turn emanates from the laboratory observation of bose einstein condensation ( bec ) . the history of bec began in 1924 when satyendra nath bose first gave the rules governing the behaviour of photon which is the commonest boson . excited by this work , einstein in the same year extended the rules to other bosons and thereby gave birth to the bose - einstein distribution ( bed ) @xcite . while doing this , einstein found that not only is it possible for two bosons to share the same quantum state at the same time , but that they actually prefer doing so . he therefore predicted that when the temperature goes down , almost all the particles in a bosonic system would congregate in the ground state even at a finite temperature . it is this physical state that is called bose - einstein condensation . thus it has always been considered a consequence of quantum effects from statistical mechanics in many textbooks as the phase transition is achieved without interactions @xcite . the einstein s prediction , however , was considered a mathematical artifact for sometime until fritz london in 1938 while investigating superfluid liquid helium , realized that the phase transition could be accounted for in terms of bec . this analysis , however , suffered a major set back because the helium atoms in the liquid interacted quite strongly . this was why scientists had to move ahead in search of bec in less complicated systems that would be close to the free boson gas model . fortunately , the breakthrough came in 1995 when the first bec was observed in rubidium atoms and this was followed by similar observations in some other cold alkali atoms such as those of lithium and sodium ( see more details in ref . ( @xcite and a guideline to the literature of bec in dilute gases in ref . as stated above , the observation of bec led to the observation of ( sf - mi ) transition and thereby open the possibility to investigate various phenomena in condensed matter physics by mimicking them with ultracold atoms in optical lattices . this possibility has led to a deluge of studies ( see @xcite for a recent review ) as it brings together atomic physicists , quantum opticians and condensed matter physicists . one draw back is that even when there have been theoretical papers investigating these phenomena with fermionic cold atoms @xcite , cold bosons are used in the actual experiments @xcite for testing spin ordering . this has been overcome by the recent observation of the mi with fermonic atoms @xcite . it follows then that the possibility to use cold atoms in optical lattices as a test laboratory for condensed matter physics is no longer a speculative physics . rather , it has become an aspect of physics with its own methods and approaches . therefore , it has reach a stage when it should start having some introductory impact on our curriculum , possibly as applications of optics , atomic physics and simulation of spin ordering hamiltonians @xcite in condensed matter physics . the purpose of this current study is to present a pedagogical study of investigating spin ordering in an isolated double - well - type potential ( simply double well ( dw ) ) which can be adopted for instructional purposes . for the dw is one of the simplest experimental set ups of optical lattices to study spin hamiltonians @xcite . this is because the system can be completely controlled and measured in an arbitrary two - spin basis by dynamically changing the lattice parameters @xcite . on the theoretical side , the dw can be considered as two localized spatial modes separated by a barrier and consequently be investigated as a two - mode approximation @xcite . the dw is a 1d optical lattice in which the transverse directions are in strong confinement and thus the motions of an atom in these directions are frozen out . to create the dw , we start with a standing wave of period @xmath0 ( long lattice ) so that the potential seen by the atoms trapped in it is @xmath1 where @xmath2 is the lattice depth , which is a key parameter for a special lattice potential . + next we superpose a second standing wave with period @xmath3 and depth @xmath4 ( short lattice ) on the first one as in fig . this will lead to a symmetric double - well superlattice ( fig 1c ) with a total optical lattice @xcite @xmath5.\ ] ] the configuration and varying of the parameter space ( i.e. various parameters ) of a hamiltonian to be tested in this superlattice is achieved by manipulating and controlling the depths of the short and long lattices . for example , by increasing the lattice depth of long - lattice @xmath2 , we could reach from superfluid to mott - insulator regime , which is convenient for studying the few particles phenomena in a local double - well cell . and the barrier height of the double - well is controlled by the lattice depth of short - lattice , @xmath4 . the effective double - well is reached if @xmath6 . otherwise the minimal points of the optical lattice are the bottom of the long - lattice . this could be seen clearly from eq . ( [ topotential ] ) after we expand the @xmath7 term , @xmath8 ^ 2+\frac{v_1}{4}\left(2-\frac{v_1}{4v_2}\right),\ ] ] so that @xmath9 results in @xmath10 , that is , the minimum is one of the long - lattice with a lift of @xmath4 . on the other hand , if @xmath11 , an effective double - well is created . the minimum can be found at @xmath12/2 $ ] . making a power series expansion around the potential minimum , then a single atom of mass @xmath13 trapped initially in any of the well will freely tunneling back and forth with the oscillation frequency of @xmath14 thus the frequency depends on not only the lattice depths @xmath2 and @xmath4 , but also on the lattice spacing @xmath0 . usually , the small lattice spacing @xmath0 is preferred as it leads to a large frequency though this could also be restricted by changing the ratio @xmath15 . this preference also lead to the use of the recoil energy of the short lattice as the unit of the depths of the optical lattice @xmath16 where @xmath17 is the wave length of the short lattice . + for example , in the experiment @xcite , the depth of the long lattice is @xmath18 while the depth of the short lattice is about @xmath19 . this gives the oscillation frequency in a range @xmath20 for @xmath21rb and @xmath22 , we can get @xmath23 , which gives @xmath24 . lets define the harmonic oscillator length @xmath25 , then we can readily get @xmath26 . comparing to the period of short lattice @xmath27 , it implies the ground state wave function is rather localized , which ensures the validity of the two - mode approximation . + finally , it is pertinent to describe how to create an asymmetric dw ( fig 1d ) . the potential bias or the tilt @xmath28 of the double - well is introduced by changing the relative phase of the two potentials ( i.e. short and long lattices ) and this can be realized by applying a magnetic field gradient of @xmath29 @xcite . consequently , tuning @xmath30 gives the potential difference between the two potential minima of the dw . we can realize the adiabatic and diabatic operations on the tilt of the dw by controlling the increasing speed of @xmath30 @xcite . and @xmath3 resulting in ( b ) a chain of double wells from which we can study ( c ) an isolated symmetric double well or ( d ) asymmetric double well ] . [ double well ] within the above consideration and assuming atoms confined in an isolated dw , we reach the two - mode approximation represented as a two - site version of the hubbard model @xcite @xmath31+u(n_{\uparrow l}n_{\downarrow l}+n_{\uparrow r}n_{\downarrow r } ) \label{h1}\ ] ] where @xmath32 is the creation operator ( annihilation operator ) for an atom with spin @xmath33 , @xmath34 is the corresponding number operator , @xmath35 ( both j and t are used in the literature though the cold matter community seems to prefer j ) describes the tunneling rate between the two wells , @xmath28 is the potential bias for the double - well and @xmath36 is the two - body interaction when two atoms occupy the same site . + basically , the hubbard model is a single band lattice model supporting a single atomic state which can hold up to two particles . if we consider that these particles have two internal spins , @xmath37 , then the hamiltonian will consist of the superposition of six fock basis states @xcite denoted by @xmath38 and @xmath39 , with a basis state @xmath40 denoting the @xmath41 and @xmath42 wells . the use of the superposition principle which is a fundamental concept in quantum theory @xcite is consistent with both the model hamiltonian and experiment in which the time evolution of the initial states produces coherent superposition of states . thus the wavefunction of the system will be the superposition of all possible states @xmath43 where for convenience , @xmath44 , @xmath45 and @xmath46 , with i , j denoting the sites while @xmath47 and @xmath48 denote singlet and triplet states respectively . from basic physics , the orientation of the two spins in a state can either be singlet @xmath47 if @xmath49 or triplet @xmath48 if @xmath50 @xcite . by tuning the potential bias @xmath51 , we can obtain all the eigenenergies and corresponding eigenstates analytically . this is achieved by directly diagonalizing the hamiltonian in eq . ( [ h1 ] ) to obtain eigenenergies and eigenstates as shown in table ( [ tabeigen1 ] ) . in the weak interacting case , @xmath52 , the state @xmath53 and state @xmath54 have lower energy so that the doubly occupied singlet state , @xmath55 will be the ground state of the system . this can be considered as the signature for a superfluid state for bosonic atom in a double well . however , the strong interaction regime is more interesting to study for spin ordering . in this regime , @xmath56 , the ground state will be singly occupied as the large atomic repulsion energetically suppress the double occupancy . here it is the @xmath57 that are occupied while the @xmath58 are unpopulated @xcite . the populated @xmath47 and @xmath48 of @xmath57 are nearly degenerate because the energy difference between them is about @xmath59 , which is a small quantity . however , when @xmath60 , the ground state approaches @xmath47 while the first excited state is @xmath48 . if we prepare the initial state as antiferromagnetic , @xmath61 , the dynamical evolution involves two frequencies @xcite @xmath62 from the above frequencies , one could get the tunneling rate @xmath63 and the interaction strength @xmath64 respectively . these two frequencies can be obtained from exerimental data and then used to test the validity of the simple two - mode model . this has been done for bosonic atoms in experiment @xcite . the extension to fermionic atoms may be different but the eigenenergies and corresponding eigenstates are the same , which means we can get similar dynamics as long as the interaction between atoms satisfy @xmath65 @xcite . on the other hand , the interaction of fermion could be also attractive generally . it is interesting to identify the ground state in this situation . table ( [ tabeigen1 ] ) works here too . .eigenstates and eigenenergies of hamiltonian ( [ h1 ] ) [ cols="^,^,^,^,^,^,^",options="header " , ] [ tabeigen1 ] when @xmath66 , the ground state does not change too much at weak tunneling . however , the first excited state is not the triplet state @xmath67 anymore but the state @xmath68 . it is interesting that the energy difference is the same @xmath69 although the interaction is attractive and the first excited state is changed . this analysis shows that @xmath68 could be involved in the dynamics if we start from the antiferromagnetic initial state @xmath70 . the two frequencies that can be observed in experiment are @xmath71 here the two - body interaction strength directly relates to @xmath72 , which can be extracted from the measured experimental data . the results show that ultracold atoms trapped in the superlattice not only could be used to simulate the phenomena in condensed matter physics , but also offer the possibility to compare the results with theoretical calculation of model hamiltonians . . left , @xmath74 in the strong interaction regime . middle @xmath75 in the weak interaction regime . right , attractive interaction @xmath76 . , title="fig : " ] . left , @xmath74 in the strong interaction regime . middle @xmath75 in the weak interaction regime . right , attractive interaction @xmath76 . , title="fig : " ] . left , @xmath74 in the strong interaction regime . middle @xmath75 in the weak interaction regime . right , attractive interaction @xmath76 . , title="fig : " ] one of the advantages of the trapped ultracold atoms is that they can be controlled precisely and easily . by tuning the two optical lattices , we can change the bias of the double - well superlattice . the ground state of the two fermions are trapped in the same site in the large potential bias . by slowly reducing the bias , the ground state is followed adiabatically to the singlet state . in this way , we can prepare the initial states either in the state @xmath53 or @xmath54 . so it is interesting to investigate how the bias influences the states . when the potential bias @xmath28 is included , the eigenstates and the eigenenergies have complicated expressions . three of the eigenenergies are @xmath77 always . the others are the roots of the algebra equation @xmath78 we numerically solve the equation and plot the eigen spectra in fig . ( [ spectra ] ) . when @xmath79 , the ground state energy is negative and modified by the presence of potential bias . when @xmath28 is not too big , the ground state energy is close to the one without potential bias . also , the energy difference between the ground state and first excited state is small . at large potential bias @xmath80 , however , the approximate ground state energy reads @xmath81 the two atoms are in the right well and the ground state reaches @xmath54 . on the contrary , the system is degenerate further in the weak interaction regime @xmath82 . if the interaction is attractive , the energy spectra is reversed . the energy difference between the ground state and the first excited state is bigger except at @xmath51 , i.e. an anti - crossing appears . this is shown obviously in fig . ( [ spectra ] ) . the observation from this analysis is that the potential bias can be used to control the energy difference between the singlet and triplet states @xcite . this is why an attempt was made in @xcite to use it to drive eq . [ h1 ] into superexchange interaction observed experimentally . the outcome , however , is that inter - well interactions have to be included to eq . [ h1 ] to get close to the experimental data . thus in the next section , we will consider such an extension . it is obvious from the preceding section that a hamiltonian to study spin ordering in the cold atoms in optical lattices needs to contain long range interactions . it is important to point out that the overlapping of different electronic orbitals gives rise to the interaction between spins in condensed matter but this overlapping is very small in optical lattices @xcite . however , the possibility of the atoms to tunnel through the barrier in quantum mechanics enables the inter - site interactions @xcite . two natural candidates are the inter - site coulombic interaction @xmath83 and exchange interaction @xmath84 . interestingly , the inclusion of these interactions as means of going beyond the standard hubbard hamiltonian to account for ferromagnetism in metals have been proposed @xcite . furthermore , we do not need a potential bias since the spin ordering is induced by these interactions . within these considerations , the extended form of eq . [ h1 ] with @xmath85 is @xmath86 . \label{h2}\ ] ] it is then easy to obtain the ground state energy and wavefunction using the highly simplified correlated variational approach ( hscva ) in @xcite . the beauty of this pedagogical approach is that the ground state energy clearly depicts the physics of the model as one vary the parameter space as in experiments with optical lattices . interestingly , the method allows the decoupling of the kinetic part from the interaction parts so that we can observe the effects of including each of them to the kinetic part . thus the combination of these two factors makes the hscva very suitable to investigate the spin hamiltonian to be tested using cold atoms in optical lattices . + we start with the variational ground state energy @xmath87 where the h is the model hamiltonian and the ket in the hilbert space is the trial wave function ( cf . ( [ wavefunction1 ] ) ) defined as + @xmath88 the x and y in eq . ( [ wavefunction2 ] ) are the variational parameters . it is straightforward to show @xcite that eq . ( [ variational ] ) leads to a 3 x 3 blocked matrix of 2 x 2 and 1 x 1 resulting in the lowest state energies @xcite , @xmath89 for the singlet states @xmath55 or @xmath90 depending on u , @xmath91\ ] ] and @xmath92 for the triplet state @xmath93 , @xmath94 the smallest of these two energies will be the ground state energy of the system . the corresponding eigenvectors are then substituted as the variational parameters in eq . ( [ wavefunction2 ] ) to give the corresponding ground state wavefunctions . thus when @xmath95 , the system will be antiferromagnetic while it will be ferromagnetic otherwise . taking into account this condition and eqs . ( [ singlete ] ) and ( [ triplete ] ) , the critical value of @xmath84 at which there is transition from one state to another is + @xmath96.\ ] ] now to test this hamiltonian in a double well , we need to know how the atomic positions and spin orientations varies with the parameter space . for example , as demonstrated in subsection ( a ) , the ground state of the system will be a mott insulator when the u is very strong . this generally accepted property of the half - filled standard hubbard hamiltonian ( i.e. @xmath97 ) is already achieved with ultracold fermionic atoms @xcite . one of the signatures of the mi state is the decrease in doubly occupied states in the ground state as @xmath36 is increased . this is demonstrated in fig . 3 showing the level of occupation of the states denoted by the variational parameters of the ground state wavefunction with increase in @xmath36 . the inclusion of @xmath83 , however , enhances the double occupancy and is therefore expected to suppress the observation of the mi especially for low values of u. it follows then that when we switch on the @xmath84 , the @xmath36 is likely to drive the system into more singly occupied states and thereby enhancing the transition to a ferromagnetic state while the @xmath83 will suppress it . this is demonstrated in eqs . ( [ singlete ] ) - ( [ criticalj ] ) and then depicted in fig . 4 showing the variation of the antiferromagnetic - ferromagnetic transition critical point of @xmath84 with @xmath36 at various values of @xmath83 . + the above theoretical @xmath83 and @xmath84 can also be compared with the ones obtained from extracted data from the experiments as was done for j and u in the standard hubbard hamiltonian . this is by expressing the possible dynamic evolution frequencies from for the singlet states and triplet states as @xmath98\ ] ] @xmath99 we see immediately that we can recover eq . ( [ frequency ] ) from eq . ( [ freqsinglete ] ) when @xmath97 . taking into account eqs . ( [ frequency ] ) , ( [ freqsinglete ] and ( [ freqtriplete ] ) , we can then estimate @xmath100 $ ] and @xmath101 $ ] . thus we can also obtain the inter - site interaction parameters from the data extracted from the experiments . the increasing advancement on how to prepare , manipulate and detect phenomena in condensed matter physics using cold atoms in optical lattices has reached a stage when it can be used as instructional means . the fact that laser cooling and trapping are now widely used in atomic physics laboratory @xcite means the realization of the double wells experiment can also be achieved . the first investigation is to mimic the mott insulator state . by extracting @xmath102 and @xmath103 from the experiment , the experimental values of j and u can be compared with the ones from their theoretical values . the experiment can then be advanced to determine @xmath83 and @xmath84 and then compare them with their theoretical values as depicted in fig . interestingly , the model hamiltonian studied here has been proposed to account for spin ordering in transition metals . it is hoped therefore , that the testing of this extended hubbard model can easily be compared to available data for the transition metals @xcite after some refining of the approach here . this will also include extending the study to dynamic properties of the model . while there is increase in inter - site states @xmath104 as the on - site coulomb interactions u increases . ] with the on - site coulomb interactions @xmath36 at various values of the inter - site coulomb interactions @xmath83 . a similar graph was obtained by ref . ( @xcite ) using different analytical method ] we acknowledge useful discussions with masud haque , ian spielman and shan - ho tsai . gea acknowledges partial support from afahositech . 99 c. j. foot , atomic physics ( oxford university press , 2005 ) . a. m. fox , quantum optics : an introduction ( oxford university press , 2006 ) . i. bloch,``quantum gases , '' science 319 , 1202 - 1203 ( 2008 ) m. b. dahan , e. peik , j. reichel , y. castin and c. salomon , `` bloch oscillations of atoms in an optical potential , '' phys . 76 , 4508 - 4511 ( 1996 ) . s. r. wilkinson , c. f. bharucha , k. w. madison , q. niu and m. g. raizen `` observation of atomic wannier - stark ladders in an accelerating optical potential , '' phys . 76 , 4512 - 4515 ( 1996 ) . d. jaksch , c. bruder , j. i. cirac , c. w. gardiner and p. zoller , `` cold bosonic atoms in optical lattices , '' phys . rev . 81 , 3108 - 3111 ( 1998 ) . m. greiner , o. mandel , t. esslinger , t. w. hnsch and i. bloch , `` quantum phase transition from a superfluid to a mott insulator in a gas of ultracold atoms , '' nature 415 , 39 - 44 ( 2002 ) . r. l. libboff , introductory quantum mechanics ( adision - wesley publishing co. inc . , 1992 ) s. stringari , `` bose - einstein condensation in ultracold atomic gases , '' phys . a. 347 , 150 - 156 ( 2005 ) . c. a. weiman , `` the richtmyer memorial lecture : bose - einstein condensation in an ultracold gas , '' am . 64 , 847 - 855 ( 1996 ) . d. s. hall , `` resource letter : bec-1 : bose einstein condensates in trapped dilute gases '' am . 71 , 649 - 660 ( 2003 ) . m. lewenstein , a. sanpera , v. ahufinger and b. damski , `` ultracold atomic gases in optical lattices : mimicking condensed matter physics and beyond , '' adv . 56 , 243 - 379 ( 2007 ) . a. rey , v. gritsev , i. bloch , e. demler and m. d. lukin , `` preparation and detection of magnetic quantum phases in optical superlattices , '' phys . . lett . 99 , 140601(1 ) - 140601(4 ) ( 2007 ) . s. trotzky , p. cheinet , s. flling,1 m. feld , u. schnorrberger , a. m. rey , a. polkovnikov , e. a. demler , m. d. lukin and i. bloch1 , `` time - resolved observation and control of superexchange interactions with ultracold atoms in optical lattices , '' science 319 , 295 - 299 ( 2008 ) . r. jrdan , n. strohmaier , k. gnter , h. moritz , t. esslinger , `` a mott insulator of fermionic atoms in an optical lattice , '' nature 455 , 204 - 207 ( 2008 ) s. -h . tsai and d.p . landau , `` computer simulations : a window on the static and dynamic properties of simple spin models , '' am . 76 , 445 - 452 ( 2008 ) . s. flling , s. trotzky , p. cheinet , m. feld , r. saers , a. widera , t. mller and i. bloch1 , `` direct observation of second - order atom tunnelling , '' nature letters 448 , 1029 - 1032 ( 2007 ) m. anderlini , p. j. lee , b. l. brown , j. sebby - strabley , w. d. phillips and j. v. porto , `` controlled exchange interaction between pairs of neutral atoms in an optical lattice , '' nature 448 , 452 - 456 ( 2007 ) . j. sebby - strabley , m. anderlini , p. s. jessen , and j. v. porto , `` lattice of double wells for manipulating pairs of cold atoms , '' phys . a 73 , 033605(1 ) - 033605(9 ) ( 2006 ) . f. h. l essler , h. frahm , f. ghmann , a. klmper , v. e. korepin , the one - dimensional hubbard model ( university press , cambridge , 2005 ) . m. a. parker , physics of optoelectronics ( taylor & francis , 2005 ) . a. t. avelar , t. m. da rocha filho , l. losano , b. baseia , `` preparing fock states of the electromagnetic field via raman interaction , '' physics letters a 340 , 74 - 77 ( 2005 ) . b. h. bransdon , and c. j. joachain , introduction to quantum mechanics , ( john wiley and sons inc . , 1989 ) . a. g. petukhov , j. galan and j.a . ver@xmath105s , `` bound states of two electron decribed by the t - j model , phys . rev . b 46 , 6212 - 6214 ( 1992 ) s. sachdev , r. n. bhatt , ' ' bond - operator representation of quantum spins : mean - field theory of frustrated quantum heisenberg antiferromagnets , `` phys . rev . b 41 , 9323 - 9329 ( 1990 ) . m. lewenstein and a. sanpera , ' ' probing quantum magnetism with cold atoms , `` science 319 , 292 - 293 ( 2008 ) . j. c. amadon and j. e. hirsch , ' ' metallic ferromagnetism in a single - band model : effect of band filling and coulomb interactions , `` phys . b 54 , 6364 - 6375 ( 1997 ) . j. e. hirsch , ' ' metallic ferromagnetism in a band model : intra - atomic versus interatomic exchange , `` phys . b 56 , 11022 - 11030 ( 1997 ) . g.e . akpojotor , ' ' the statistical equivalents of the t - u and t - t - u models , `` in lectures on the physics of strongly correlated systems xii : twelfth training course ( edited by a. avella and f. mancini ) , aip con . 1014 , 251 - 259 ( 2008 ) . e. j. d. vredenbregt , k. a. h. van leeuwen , ' ' laser coling and trapping visualized , " am . 71 , 760 - 765 ( 2003 ) .
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laser cooling and trapping are now widely used in atomic physics laboratory .
interestingly , cold atoms in optical lattices are now used in advanced research to mimic phenomena in condensed matter physics and also as a test laboratory for the models of these phenomena .
it follows then that it is now possible and necessary to advance the atomic physics laboratory by including the use of ultracold atoms in optical lattices for instructional contents of phenomena in condensed matter physics . in this paper , we have proposed how to introduce into the atomic physics laboratory the study of quantum magnetism with cold atoms in a double well optical lattice . in particular , we demonstrates how to compare the theoretical parameters of a spin hamiltonian model with those extracted from spin ordering experiment .
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there are many problems in condensed matter physics and materials science in which the aim is to describe sharp interfaces separating regions with qualitatively different properties . this occurs for instance in solidification , dendritic growth , solid - solid transformations , grain growth , etc . traditionally , two approaches have been followed to tackle these problems . in one of them , the problem is solved for a fixed position of the interface , and based on this , the expected evolution of the interface in the following time step is calculated , and the process repeated . this method has the practical drawback that the different structure of the interface at each time step makes necessary the full solution of a new problem each time . the second approach is sort of brute force , in which the system is modeled to the atomic scale , and evolved according to their ( newtonian ) equations of motion . the problem with this approach is that it is impossible in practice to span the many orders of magnitude between the atomic scale and the relevant macroscopic scale . the diffuse interface technique ( including the so - called phase field models ) is a novel powerful approach to study this kind of problems@xcite . it typically describes the sharp interface by an additional field @xmath1 ( or more than one ) . for instance in a solidification problem @xmath1 can be taken to be 0 in the liquid and 1 in the solid . if the spatial variation of @xmath1 is known the interface can be located . then the problem of keep tracking of the interface is eliminated against having to include also @xmath1 as a new dynamical variable . @xmath1 is coupled ( usually phenomenologically ) to the original degrees of freedom of the problem , and its dynamic evolution is not defined _ a priori _ , but has to be seeded from outside . a key ingredient in phase field models is _ regularization _ of the field @xmath1 . although the sharp interface is the sensible case , in order to implement the theory a smooth transition of @xmath1 between the values on both sides of the interface is necessary . then the interface acquires a fictitious width , which however does not alter the physical behavior of the system if it is much smaller than any other relevant length scale . an additional , very important effect of regularization is to make the properties of the system independent of the underlying numerical mesh used to implement the problem on the computer . regularization is usually achieved by the inclusion in the theory of terms that depend on gradients of @xmath1 , penalizing rapid spatial variations of this quantity . within the field of fracture , the phase field models that have been proposed include those of aranson , kalatsky and vinokur @xcite , karma , kessler and levine @xcite , and eastgate _ @xcite ( see also @xcite ) . all of them use an additional scalar field @xmath1 as a phase field , that is taken to be ( asymptotically ) zero within fractures and one within the intact material . there is now a general consensus that a complete description of the fracture process can not be given only in terms of macroscopic variables . in fact , the divergence of the stress field near to the crack tip implies that physical conditions change a large amount on distances of the order of interatomic separation . then details of the material at the atomic scale can have an effect on the macroscopic behavior of cracks . on the other hand , the roughly similar phenomenology of crack propagation observed in very different materials raises the expectation that a general description with a minimum amount of parameters dependent on microscopic characteristics is feasible . this is in the spirit of the phase field approach to fracture : one may think that the microscopic variables have an effect that translates in the form given to the energy density of the phase field , in the form of the terms coupling the phase field to the elastic degrees of freedom , and in the dynamics assumed for it . except in this effective way , microscopic parameters do not appear in the phase field formalism . the phase field approach is already giving promising results . for instance , it has been shown that crack instabilities , oscillations and bifurcation can be obtained within this scheme @xcite . the sharp interface limit of some phase field models of stress induced instabilities has been studied in @xcite . its possible relevance to fracture is given in @xcite . we are going to present here a diffuse interface approach that has some qualitative difference with previous ones . most importantly , it does not introduce additional variables into the problem , the full set of variables are the components of the strain tensor @xmath0@xcite . description of fracture is achieved by the nonlinear form of the effective free energy density as a function of @xmath0 . actually , our energy is quadratic for small displacements ( and then correctly describes linear elasticity ) and saturates for very large displacements , then describing fracture ( the saturation energy value being related to the fracture energy ) . regularization is provided by terms in the free energy of the generic form @xmath2 . there are a number of reasons to pay attention to this model , both conceptual and from the point of view of implementation . first of all , the absence of additional degrees of freedom makes this model be probably the simplest continuous ( non - atomistic ) description of the fracture process . it is then interesting to know how , and to what extent , fracture phenomenology is captured by the model . from a practical perspective there are two important things to point out . first , an important characteristic is the tensorial nature of the variable describing the occurrence of fractures . in the approaches in which a scalar field @xmath1 is introduced , knowing that @xmath1 has become zero at some point tells that a fracture is passing through that point , but does not tell anything about what direction the fracture follows . in our case fracture is described by @xmath0 itself , and then if we know that a fracture is passing through some point , we can say immediately which direction it has . we believe this is an important computational advantage : a single cell of a computational mesh is sufficient to encode the information about existence and direction of fracture . in models using a scalar field @xmath1 we need a whole neighborhood of a computational cell to encode the same information . second , previous attempts to model fracture through non - linear elasticity have used typically the displacement field as fundamental variable . for those theories regularization is a problematic issue as it typically leads to higher order differential equations which are difficult to solve numerically . our equations contain only second order derivatives of the strain field , and thus they much more smooth to solve numerically . in this paper we concentrate mainly in the presentation and validation of the model , giving only a short presentation of a non trivial application . we have divided the presentation in the following form . in the next section we define the model and give analytically the structure on an infinite straight crack . section iii shows in detail that the model is able to reproduce the griffith s criterion . in section iv we present an example in which the crack paths are determined in a non trivial geometric configuration . in section v we give a perspective of our planned future work with the model . the fundamental variables of the model are the components of the strain tensor @xmath0 : @xmath3 where @xmath4 are the displacements with respect to the unperturbed positions @xcite . our approach follows closely that in ref . @xcite ( see also @xcite ) where it was used to study textures in ferroelastic materials and martensites . for clarity we present the model for a two dimensional geometry , the generalization to three dimensions being conceptually straightforward , although of course more involved . the symmetric tensor @xmath5 has three independent components . for convenience we will choose them in the form @xmath6 which are named respectively the dilation , deviatoric and shear components . these three variables are however not independent . in fact , since @xmath0 is derived from the two displacements @xmath7 , @xmath8 , there is one constraint that has to be fulfilled , leaving only two independent variables . the constraint is known as the st . venant condition @xcite , and it can be written as @xmath9 it is easy to show that this is an identity if the definitions ( [ uno ] ) and ( [ dos ] ) are used@xcite . to define the model we need to know the form of the local free energy density @xmath10 . to correctly describe elasticity for small displacements the limiting form @xmath11 of @xmath12 for small @xmath0 should be @xmath13 where @xmath14 is the fourth rank tensor of elastic constants of the material . we will specialize the expressions to an isotropic material , as this was our first aim to use this kind of model . in the isotropic case @xmath11 reads @xmath15 where @xmath16 and @xmath17 are the two dimensional bulk and shear modulus of the material ( related to the three dimensional values @xmath18 and @xmath19 by @xmath20 , @xmath21 for the case of plane strain , and @xmath20 , @xmath22 $ ] for the case of plain stress ) . the previous expression of the free energy must be extended to large strains to account for fracture . to do this the energy must saturate for large deformation . this is the main requirement , and different choices can be made @xcite . in the simulations presented below for isotropic materials we have chosen the form @xmath23 the limiting value @xmath24 of @xmath12 for @xmath25 is the energy density necessary to impose locally a very large value for at least one component of @xmath0 , and it is then related to the crack energy of the problem ( see below ) . from the present free energy form , we can say that a crack is nucleating when @xmath26 , i.e. , when typical values of @xmath27 s are @xmath28 ( assuming @xmath29 , as it is usually the case ) . cracks in the system are thus characterized as regions where @xmath30 . the second crucial ingredient of the model is regularization . that is provided by gradient terms in the free energy density . the terms we use are typically of the form @xmath31 with numerical coefficients @xmath32 . to retain rotational invariance we have to choose @xmath33 . in certain cases , and to avoid some unphysical behavior , these terms have to be cut off at large values of @xmath34 ( see the next section for justification and details ) , and thus the gradient part of the free energy density is chosen as @xmath35 where @xmath36 goes to 1 for small @xmath34 , and tends to zero at large @xmath34 . the actual form of @xmath36 we use is @xmath37 where @xmath11 is given in ( [ iso ] ) , @xmath38 is a cut off value , and the exponent @xmath39 controls the sharpness of the cut off ( see next section ) . once the full free energy is defined , the equations of motion are obtained by including a kinetic term @xmath40 , which is quadratic in temporal derivatives of the displacements ( i.e. , @xmath41 , for some generic density @xmath42 ) and then it has to be transformed to a function of @xmath34 ( see @xcite for the details ) . the equations to be solved are more conveniently written in fourier space ( we use @xmath43 for the fourier transforms of @xmath34 , and @xmath44 ) . the result is @xmath45 where @xmath46 , @xmath47 , @xmath48 are phenomenological damping coefficients ( that we will take to be equal @xmath49 ) and @xmath50 is a lagrange multiplier depending implicitly on @xmath51 s , chosen to enforce at every moment the st . venant constraint , that in fourier space reads @xmath52 in the examples below , the dynamic equations ( [ ocho ] ) are solved on a rectangular numerical mesh , using periodic boundary conditions in systems up to sizes of 512 @xmath53 512 . in some cases we work with rectangular samples , with the length perpendicular to the crack being two or three times that along the crack , as we have observed that finite size effects are lower in this configuration than in a square sample with the same area . also , the dynamic equations are solved in the overdamped regime , in which the second order time derivative terms in ( [ ocho ] ) are neglected . as a necessary starting point , and also because it is probably the only case that can be treated analytically , we present here the structure of an infinite , straight crack in our model . in this geometry , all quantities depend only on the coordinate @xmath54 ( the crack is assumed to lie along the @xmath55 direction ) , and the model becomes effectively one - dimensional . in fact , assuming the boundary conditions impose no strain in the @xmath55 direction , we have @xmath56 , @xmath57 . as expected , there is a single order parameter in this configuration , that can be taken to be @xmath58 . the free energy of the model restricted to the present case becomes simply @xmath59 where @xmath60 is the system size along the @xmath55 direction , @xmath61 , @xmath62 , and the cut off function @xmath63 is now a function of @xmath58 alone . for the choice in eq . ( [ cf ] ) we get now @xmath64 it is clear that for @xmath65 the solution to eq . ( [ bici ] ) has @xmath66 except at a single point @xmath67 . the position @xmath67 is undetermined . this solution describes the system broken at @xmath67 . the fracture energy per unit length @xmath68 in this case is non - zero only if a finite discretization @xmath69 is used . in this case @xmath70 . to work out the finite regularization case ( @xmath71 ) , it is more convenient to solve the problem under the assumption of an applied stress @xmath72 along the @xmath54 direction . to do this we have to minimize the stress dependent free energy @xmath73 , given by @xmath74 ( the factor of two is due to the fact that in the present case @xmath75 ) let us consider first the case @xmath76 , which implies ( according to ( [ s1d ] ) ) @xmath77 , i.e. , no cut off of the gradient terms at high values of @xmath58 . as in any one dimensional mechanical problem , the solution of eqs . ( [ bici ] ) , ( [ fsigma ] ) is reduced to the evaluation of an integral . the numerical integration gives the profiles shown in fig . an important observation is that now the whole profile is smooth , and this implies that all macroscopic parameters that are numerically calculated will be independent of the mesh discretization @xmath69 if this is small enough . for low @xmath72 , in the region where @xmath78 the following analytical solution is obtained @xmath79 where @xmath80 is order @xmath81 . the sharp fracture of the case @xmath65 transforms now into a smooth object that occupies a finite width @xmath82 in the system . from ( [ e1 ] ) this width can be estimated to be @xmath83 note that @xmath82 measures the width of the fracture in the _ original _ , unstrained reference system . an important parameter to be calculated is the opening of the fracture @xmath84 , defined as @xmath85 . this has the meaning of the strain that the system is able to accommodate due to the existence of the crack . its main contribution for low @xmath72 comes from the central part of the crack , described by eq . ( [ e1 ] ) . we get @xmath86 and from here and ( [ w ] ) @xmath87 then we see that upon increasing the opening @xmath84 of the fracture , the width @xmath82 increases as @xmath88 , and the stress decays only as @xmath89 . then this model crack relieves the stress in the system only asymptotically . however , the decaying of the stress with @xmath84 is too slow to give a finite crack energy . in fact , the crack energy per unit length @xmath68 ( defined as the energy of the system with the crack , minus the energy of the system at the same stress without the crack ) can be estimated as : @xmath90 this means that the present regularization scheme does not provide a crack with a finite energy in the limit of large stretching , i.e. , @xmath91 . in the next section we will argue that this behavior does not invalidate completely the use of the present algorithm with @xmath92 if we are interested in cracks that do not become infinitely long . notwithstanding , if we want a more accurate description of cracks , an implementation in which the energy density of the crack remains finite as its length diverges is mandatory . the following is a possibility to obtain this . in order to have a model that relieves more efficiently the elastic energy , we first remind that gradient terms are necessary for a fracture to propagate without interference of the numerical mesh , therefore , they are important only when fracture is forming , namely , when typical values of @xmath34 are close or lower than @xmath93 . in the regions in which the fracture has nucleated , typical values of @xmath27 s become much larger , and the gradient terms are not necessary any more . then we can weaken their effect in those regions by choosing a non - trivial cut off function as given by eq . ( [ cf ] ) ( or eq . ( [ s1d ] ) in the present one - dimensional case ) . it should be emphasized that this _ ad hoc _ modification of the free energy does not introduces new parameters in the relevant regions that determine crack growth , namely , close to the crack tips . our aim is to choose a value of @xmath39 that generates cracks with finite width and energy in the limit @xmath91 . in order to do this , we first notice that for @xmath94 ( i.e. , where elastic forces become vanishingly small ) , the euler - lagrange equation corresponding to eqs . ( [ bici ] ) , ( [ s1d ] ) , and ( [ fsigma ] ) leads to @xmath95 in particular , if @xmath76 , we re - obtain from here the behavior given in ( [ e1 ] ) . the constant in eq . ( [ 22 ] ) must be calculated matching the present solution for @xmath94 , with that for @xmath96 . assuming @xmath97 for simplicity , it is obtained that this constant is @xmath98 , where @xmath99 is a ( order 1 ) numerical factor . upon integration , equation ( [ 22 ] ) allows to find the full profile of the crack ( in fig . [ f2 ] we can see the results of numerical integration for @xmath100 ) . we concentrate in the case @xmath101 , in which the crack has relaxed completely the applied stress . the crucial result is that in this case , and for @xmath102 , it exists a well defined profile of the crack , given by @xmath103 where the function @xmath104 is defined as @xmath105 for @xmath106 the limiting form is @xmath107 the opening of the crack @xmath108 gives a finite value if @xmath109 . this means that the system has completely relaxed the applied stress with a finite opening of the crack . any further increase @xmath110 of @xmath84 is accommodated in the system at the center of the crack , through a singular term @xmath111 added to ( [ 23 ] ) . for @xmath112 , the value of @xmath84 is divergent due to the non - integrable divergence of @xmath58 around the origin in ( [ 23 ] ) . in this case the stress is relaxed only asymptotically , but still rapidly enough ( compared to the @xmath76 case ) to guarantee a finite width @xmath82 and energy @xmath68 of the fracture . in fact , from ( [ 23 ] ) the width @xmath82 of that part of the system for which @xmath113 can be estimated to be @xmath114 in the same way , the energy of the crack @xmath68 becomes finite , and its value is proportional to @xmath115 from a numerical point of view , the algorithm with @xmath109 is probably too singular to be implemented successfully . we have implemented the case @xmath116 , which we have found is numerically tractable , and provides and almost perfect verification of the griffith s criterion even in the quite small system sizes that we are using . one of the basic cornerstones of fracture physics , described in the very first pages of any fracture book , is the so called griffith s criterion ( gc ) . it states that under the application of a remote stress @xmath72 on a ( effectively two - dimensional ) system with a preexistent crack of length @xmath117 , the crack will extend and eventually break the sample if @xmath72 is larger than some critical value @xmath118 which scales as @xmath119 . a simple justification of this behavior can be given on energetic grounds . upon the application of the stress @xmath72 , the system with the crack stores an elastic energy that is reduced in a quantity @xmath120 with respect to that of the system without the crack ( @xmath121 is proportional to an elastic constant of the material ) . on the other hand , the creation of a crack of length @xmath117 is assumed to have an energy cost of @xmath122 , where @xmath68 defines the fracture energy per unit length . the total energy of the system as a function of @xmath117 is then given by @xmath123 and it has a maximum as a function of @xmath117 at @xmath124 . if @xmath125 , the system relieves sufficient elastic energy upon crack length increase to pay for the energy cost of crack creation , and the crack typically extends abruptly and breaks the material . if @xmath126 there is no sufficient energy in the system to make the crack enlarge . note that according to ( [ e ] ) , the system would prefer to reduce @xmath117 in order to reduce its energy . experimentally this ` healing ' of the crack is prevented by irreversible processes : the crack energy is not recovered upon crack healing , and then the situation is that any @xmath126 is a stable crack . the previous arguments do not depend on the orientation of the crack in the system , assuming the material and the remote applied stress are isotropic . the gc is at the base of the fracture of brittle materials , and has to be satisfied by any model devised to describe such process . in our simulations we control the mean uniform strain @xmath127 this correspond to an isotropic stress @xmath128 . in order to correctly calculate the critical stress @xmath129 for cracks of a given length , we start from an initial spatial distribution of the variables @xmath58 , @xmath130 , @xmath131 that roughly describes a crack , and apply a stabilization procedure through a negative feedback loop in the program , that monitors the length of the crack ( defined using the contour defined by the value of the elastic energy @xmath11=2b ) , and reduces ( increases ) the applied strain when the length increases ( decreases ) . results in figs . [ f3 ] , [ f4 ] , and [ f7 ] below , where obtained with this procedure . the model with no regularization satisfies the gc if we restrict to cracks running in a single direction with respect to the underlying numerical mesh . this is shown in fig . we also see that no noticeable system size effects are observed for the system sizes used . as we use periodic boundary conditions this means that the crack is not influenced by the elastic field of its neighbor images . however , the value of @xmath132 is strongly dependent on the orientation of the crack as shown in fig [ f4 ] . this is one typical drawback of many discrete models of fracture when trying to simulate isotropic materials . moreover , the influence of the numerical mesh has a clearly visible manifestation : as fig . [ f5 ] shows , if a crack is placed at a finite angle with respect to the mesh , when the critical stress is reached , the crack opens along one of the lattice main directions , instead of extending along its original direction as it should do in an isotropic system . the gradient terms are included in the model to solve this problem , and to make the behavior isotropic . the question of the satisfiability of the gc in the presence of a non - zero @xmath32 is not trivial since as we showed in the previous section , the energy of the crack per unit length does not saturate upon increasing strain unless an adequate cut off of the gradient terms is included . let us first study the effect of a finite regularization without cut off ( @xmath92 in eq . ( [ cf ] ) ) . the fact that in this case the crack energy per unit length of an infinite crack is divergent ( as seen in the previous section ) is a manifestation of the fact that the crack energy of a crack of _ finite _ length @xmath117 grows more rapidly than @xmath117 itself . on the basis of the energetic arguments for the gc , we expect here a dependence of @xmath132 on @xmath117 of the form @xmath133 with @xmath134 . numerical simulations first of all confirm that the behavior of the system can be made isotropic by including regularization . in fact fig . [ f4 ] shows that the critical stress become independent of the angle between the crack and the numerical mesh . more than that , the crack extends along the original direction it had , independently of the numerical mesh ( see fig . however , as anticipated , a power low decaying with an exponent lower than 1/2 is obtained for the critical stress as a function of crack length ( see fig . the fitted power for the parameters used is @xmath135 . then the model with finite @xmath32 but infinite @xmath38 gives a slightly incorrect behavior of critical stress as a function of crack length . if this discrepancy can be considered small , then the model with infinite @xmath38 ( which is easier to implement ) can be used . if the previous discrepancy is considered serious ( whether it is serious or not will depend on the particular problem under study ) then a finite @xmath38 formalism has to be implemented . as already discussed , a power @xmath102 in eq . ( [ cf ] ) has to be used to guarantee that the gc is satisfied . in fig . [ f7 ] we show results indicating that a very good fitting to the gc is obtained for @xmath116 and @xmath136 . this value of @xmath38 was chosen to obtain the best verification of gc in the finite system we are using , but in principle any ( finite ) value of @xmath38 should reproduce the 1/2 power dependence in the case of sufficiently large system sizes . fig . [ f8 ] shows the three dimensional profiles of the crack with @xmath137 , stabilized at the critical stress . the prediction of crack trajectories under general loading conditions for bodies of arbitrary shape and possibly with pre - existent cracks is very important in many engineering applications . the present formalism is well suited to study this kind of problems , in particular in those cases in which slight deviations from straight propagation are expected . these cases are particularly difficult to tackle with a non - regularized model . as a very simple an illustrative example of that , we consider a pair of parallel cracks loaded isotropically . we show here some qualitative results , leaving a more detailed quantitative analysis for a forthcoming work . we first stabilized the two parallel cracks by the feedback mechanism already explained , then stop this stabilization , and increase a small amount the stress , and follow the crack evolutions as they grow . snapshots of the system during crack propagation ( figs . [ f9 ] , [ f10 ] ) show that the cracks propagate diverging from straight propagation . this is a non - trivial effect caused by the elastic interaction between the two cracks . note that in the present case , cracks propagate from _ both _ fractures , because of the perfect mirror symmetry of the problem with respect to the middle plane . in a slightly different configuration , we place the two parallel cracks shifted ( fig . [ f11 ] ) . now the propagation of cracks from the internal tips is strongly influenced by the nearby crack , producing a geometrical pattern of curved cracks that is well known ( see for instance @xcite ) . the propagation from the external tips is now almost not influenced by the second crack and then essentially straight . in summary , we have presented a model for the study of cracks propagating in brittle materials . the model does not use additional variables others than the strain tensor , and then is a minimalist continuum model for description of cracks . as a crucial ingredient it includes regularization terms that make the cracks be smoothed . we have shown that in this form the model can describe accurately an isotropic material in conditions in which the non - regularized model fails neatly . we have validated the model showing how it can fit accurately the griffith s law for the critical stress as a function of crack length . we have seen that in order to obtain this result the regularization has to be softened in the interior of the cracks . ad hoc _ modification however leaves intact the model in the neighborhoods of the crack tips , where the processes responsible for crack advance take place . as a first application we have shown how the model can predict the propagation and eventual bending of cracks induced by elastic interactions between them . we believe that the present technique is straightforward to implement and computationally efficient , and that it addresses in a phenomenological way the very important applied problem of predicting crack evolution , without requiring as explicit input any details about the physical conditions in the process zone . the technique can be implemented also in three dimensions , although we feel that this should wait for some increase in computational power before this can be implemented on a desktop computer . we want to indicate a few important direction along which the model can be applied , and in which we have started some work . first of all , all simulations in the present paper were done in the overdamped regime , where dynamical effects are absent . this may be a reasonable choice for the cases in which the cracks are known to grow quasistatically . this may include crack propagation under slowly varying external conditions , as for instance non - uniform thermal stresses . for cases in which dynamical effect are expected to be important the full dynamical equations have to be implemented . preliminary work shows that indeed the implementation of the inertial dynamics gives rise ( under appropriate conditions ) to well known phenomena such as crack oscillation and bifurcation . we expect to report about this soon . another possible interesting application of the model concerns the determination of minimum energy configuration of cracks . in fact , as a result of regularization , our cracks are in principle able to shift laterally , in addition to extend from its tips . this effect is not observed in the simulations presented here as it occurs typically in much longer time scales than the one we were interested in , but it can be enhanced under particular conditions . this shifting of cracks is driven by the tendency of the system to minimize its energy , and then it provides a tool to study cases in which the minimum energy configuration has a physical meaning . an example of this kind of application has been presented in @xcite as a final consideration , we stress that we are describing fracture in a continuum model of a brittle material as a non - linear elastic process . due to the simplicity of the model , the influence of microscopic details at the process zone have only very few places were to leak in the present formalism . one point where this may happen is in the form of the interpolation function between linear elasticity and broken material regime . based on very recent findings,@xcite we expect that particular changes in the form of this function may give rise to different phenomenological behavior of crack propagation . we thank s. r. shenoy for useful comments and discussions . the theory presented by y. m. jin , y. u. wang , and a. g. khachaturyan [ philos . mag . * 83 * , 1587 ( 2003 ) ] uses a phase field of tensorial nature , and then is close to the one we present here . however , it seems to be applicable only when there is a finite number of cleavage planes in the material . we do not include at present the non - linear part of the strain tensor ( @xmath138 ) , and then in its present form our formalism does not properly describe problems in which finite relative rotations of different parts of the body occur . we prefer to work in terms of the three variables @xmath58 , @xmath130 , and @xmath131 , and enforce the compatibility condition by using a lagrange multiplier . notwithstanding , it should be mentioned that another possibility is to use only two truly independent variables , expressing explicitly the third one in terms of the other two using the compatibility equation ( [ stv ] ) . in that case we would obtain a non - local , long range interaction between the two independent variables , as explained in refs . @xcite and @xcite . we expect that the exact form of the expression we use to interpolate between the quadratic dependence at low @xmath0 and the constant value at large @xmath0 captures in the macroscopic model some relevant details of the material at the microscopic scale ( see ) .
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we present a continuum model for the propagation of cracks and fractures in brittle materials .
the components of the strain tensor @xmath0 are the fundamental variables .
the evolution equations are based on a free energy that reduces to that of linear elasticity for small @xmath0 , and accounts for cracks through energy saturation at large values of @xmath0 .
we regularize the model by including terms dependent on gradients of @xmath0 in the free energy .
no additional fields are introduced , and then the whole dynamics is perfectly defined .
we show that the model is able to reproduce basic facts in fracture physics , like the griffith s dependence of the critical stress as a minus one half power of the crack length .
in addition , regularization makes the results insensitive to the numerical mesh used , something not at all trivial in crack modeling .
we present and example of the application of the model to predict the growth and curving of cracks in a non - trivial geometrical configuration .
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the omputation of gravitational radiation from the inspiral and merger of binary black holes poses a difficult boundary value problem . in the geometrically simplest and physically most natural treatment , the black holes are modeled by the gravitational collapse of a pair of stars ( or other astrophysical bodies ) . however , this is a challenging hydrodynamic problem which requires simulating a pair of orbiting bodies for a sufficient time to verify a negligible amount of incoming radiation in the initial conditions , then following their subsequent collapse to black holes and finally computing the outgoing radiation in the exterior spacetime . alternatively , in the purely vacuum approach , the individual black holes form from imploding gravitational waves . this avoids hydrodynamical difficulties at the expense of a globally complicated initial value problem . the imploding waves may emanate either ( i ) from a past singularity or ( ii ) from past null infinity @xmath0 . in case ( i ) , the appropriate boundary condition at @xmath0 is that there be no ingoing radiation but , assuming the time reversed version of cosmic censorship , the past singularity implies a white hole horizon @xmath1 on which boundary data must be specified in some arbitrary manner in order to determine the exterior spacetime . in case ( ii ) , ingoing radiation from @xmath0 is present at early times when the black holes are formed but ingoing radiation must be absent at late times in order for the outgoing radiation to be unambiguously attributed to the merging black holes . in this work , we present a solution to the first stage of a new two - stage global treatment of the vacuum binary black hole problem @xcite . the approach , based upon characteristic evolution , has been carried out in the regime of schwarzschild perturbations where advanced and retarded solutions of the linearized problem can be rigorously identified @xcite . computational experiments are necessary to study the applicability of the approach to the nonlinear regime . from a time - reversed viewpoint , this first stage is equivalent to the determination of the outgoing radiation emitted from the fission of a white hole in the absence of ingoing radiation . this provides the physically correct `` retarded '' waveform for a white hole fission , were such events to occur in the universe . although there is no standard astrophysical mechanism for producing white holes from a nonsingular matter distribution , white holes of primordial or quantum gravitational origin can not be ruled out . this fission problem has a simpler formulation as a characteristic initial value problem than the black hole merger problem . the boundary of the ( conformally compactified ) exterior spacetime contains two null hypersurfaces where boundary conditions must be satisfied : past null infinity @xmath0 , where the incoming radiation must vanish , and the white hole event horizon @xmath2 , which must describe a white hole , which is initially in equilibrium with no ingoing radiation and then distorts and ultimately fissions into two white holes with the emission of outgoing gravitational waves . if we approximate @xmath0 by an outgoing null hypersurface @xmath3 , which intersects @xmath1 at an early time ( approximating past time infinity @xmath4 ) close to the initial equilibrium of the white hole , then data on these two null hypersurfaces , @xmath1 and @xmath3 , constitute a standard double - null initial value problem , whose evolution determines a portion of the exterior spacetime extending to @xmath5 , where the outgoing radiation is computed . in contrast , the corresponding problem for the `` retarded '' waveform from a black hole merger involves two disjoint null hypersurfaces where boundary conditions must be satisfied : past null infinity @xmath0 , where the incoming radiation must vanish , and the future event horizon @xmath6 , which describes the merger of the two black holes and their subsequent approach to equilibrium . in previous work @xcite , we treated the fission problem in the close approximation @xcite as a perturbation of a schwarzschild background . in this paper we present a fully nonlinear treatment that reveals new and interesting strong field behavior . we carry out the evolution of this vacuum double - null problem by means of a characteristic evolution code @xcite , using a recent version of the code which improves accuracy in the highly nonlinear region @xcite . caustics in the ingoing null hypersurfaces used to foliate the exterior spacetime restrict the evolution to the pre - fission stage . we use a conformal horizon model @xcite to supply the necessary null data for a horizon corresponding to a white hole fission . the conformal horizon model provides a stand - alone description of the intrinsic null geometry of the horizon . the algorithm for generating horizon data is constructed to handle a general event horizon representing the fission of a spinning white hole into two outspiraling white holes of non - equal mass @xcite . the specific application in this paper is to the axisymmetric head - on fission into equal mass white holes . ( the necessary data and evolution codes are , however , _ not _ limited to the axial symmetry of a head - on collision . ) the resulting horizon geometry is an upside - down version of the standard trousers - shaped event horizon for a binary black hole merger in the time - reversed scenario . we study a range of models extending from the perturbative close limit , in which the fission occurs in the infinite future , to the highly nonlinear regime . nontrivial global changes , accompanied by dramatic time dependence of the horizon geometry , arise in passing from the perturbative to the highly nonlinear regime @xcite . the existence of a marginally trapped surface divides the horizon into interior and exterior regions , analogous to the division of the schwarzschild horizon by the @xmath7 bifurcation sphere . in passing from the perturbative to the strongly nonlinear regime there is a transition in which the fission occurs in the portion of the horizon visible from @xmath8 . thus these results reveal two classes of binary white hole spacetimes , depending upon whether the crotch in the trousers ( where the fission occurs ) is bare , in the sense that it is visible from @xmath5 , or hidden inside a marginally trapped surface . in this paper we evolve this data using the characteristic code to study the properties of the gravitational radiation produced by this dramatic behavior of a white hole fission . in secs . [ sec : data ] and [ sec : confmod ] we review the formalism and data necessary for a characteristic evolution of the spacetime exterior to a dynamic white hole by means of the characteristic code . in sec . [ sec : wave ] we present the detailed waveforms for a one - parameter family of spacetimes varying from the close approximation to the highly nonlinear regime in which the fission is visible from @xmath5 . we retain the conventions of our previous papers @xcite , with only minor changes where noted in the text . for brevity , we use the notation @xmath9 to denote partial derivatives and @xmath10 to denote retarded time derivatives . we represent tensor fields on the sphere as spin - weighted variables @xcite in terms of a dyad @xmath11 for the unit sphere metric @xmath12 in some standard choice of spherical coordinates , e.g. @xmath13 . ( the numerical code uses two overlapping stereographic coordinate patches . ) we compute angular derivatives of tensor fields in terms of @xmath14 and @xmath15 operators @xcite , e.g. @xmath16 , to compute the gradient of a spin - weight zero ( scalar ) field @xmath17 in terms of the spin - weight @xmath18 field @xmath19 and the spin - weight @xmath20 field @xmath21 . we treat the fission of a white hole by a double null initial value problem based upon the white hole horizon @xmath1 and on an outgoing null hypersurface @xmath22 , which emanates from an early time slice @xmath23 of @xmath1 approximating the initial equilibrium of the white hole . the horizon pinches off in the future where its generators either caustic or cross , producing the ( upside - down ) trousers picture of a fissioning white hole . the double - null problem , first formulated by sachs @xcite , is most conveniently described in sachs coordinates consisting of ( i ) an affine null parameter @xmath24 along the generators of @xmath2 , which foliates @xmath1 into cross sections @xmath25 and labels the corresponding outgoing null hypersurfaces @xmath26 emanating from the foliation , ( ii ) angular coordinates @xmath27 which are constant both along the generators of @xmath1 and along the outgoing null rays of @xmath26 , and ( iii ) an affine parameter @xmath28 along the outgoing rays normalized by @xmath29 , with @xmath30 on @xmath1 . in these @xmath31 sachs coordinates , the metric takes the form @xmath32 we set @xmath33 , where @xmath34 , with @xmath35 the unit sphere metric . we represent the conformal metric @xmath36 by the complex spin - weight 2 field @xmath37 . the remaining dyad component , given by the real function @xmath38 , is fixed by the determinant condition @xmath39 . the requirement that the horizon be null implies that @xmath40 on @xmath2 . in addition , we fix the gauge freedom corresponding to the shift on @xmath1 so that @xmath41 is tangent to the generators , implying that @xmath42 on @xmath1 . the choice of lapse that @xmath24 is an affine parameter implies @xmath43 on @xmath1 . we further fix the affine freedom by specifying @xmath44 on the early slice @xmath23 approximating the asymptotic equilibrium of the white hole in the past . the outgoing null hypersurface @xmath22 emanating from @xmath23 approximates past null infinity @xmath0 . the affine tangent to the generators of @xmath1 , @xmath45 satisfies the geodesic equation @xmath46 and the hypersurface orthogonality condition @xmath47}=0 $ ] . following the approach of refs . @xcite , we project four - dimensional tensor fields into the tangent space of @xmath1 using the operator @xmath48 where @xmath49 . the evolution proceeds along the outgoing null hypersurfaces @xmath50 emanating from the foliation of @xmath1 and extending to ( compactified ) @xmath5 . in this problem , the complete ( and unconstrained ) characteristic data on @xmath1 are its ( degenerate ) intrinsic conformal metric @xmath51 , or equivalently @xmath52 , expressed in terms of the affine parameter @xmath24 . in addition , the characteristic data on @xmath22 are its intrinsic conformal metric @xmath53 , or @xmath54 expressed in terms of its affine parameter @xmath28 . the remaining data necessary to evolve the exterior spacetime consist of the intrinsic metric and extrinsic curvature of @xmath23 ( subject to consistency with the characteristic data ) @xcite . this additional data consist of the surface area @xmath55 , the inward expansion @xmath56 , the outward expansion @xmath57 and the twist @xcite @xmath58 ( our choice of shift implies that @xmath59 . ) the twist is an invariantly defined extrinsic curvature property of the @xmath60 cross sections of @xmath1 , independent of the boost freedom in the extensions of @xmath61 and @xmath62 subject to the normalization @xmath63 . we use the conformal horizon model to supply data @xmath64 on @xmath1 corresponding to a fissioning white hole for a sequence of models ranging from the close approximation to the highly nonlinear regime ( see sec . [ sec : confmod ] ) . on @xmath22 , we set @xmath65 to model the absence of ingoing radiation . in carrying out the evolution computationally , the first step is to propagate the data on @xmath23 along the generators of @xmath66 @xcite so that it can be supplied as boundary data for the exterior characteristic evolution code . einstein s equations for the double - null problem decompose into ( i ) hypersurface equations intrinsic to the null hypersurfaces @xmath26 , which determine auxiliary metric quantities in terms of the conformal metric @xmath36 ; ( ii ) evolution equations which determine the rate of change @xmath67 of the conformal metric of @xmath26 ; and ( iii ) propagation equations which are constraints that need only be satisfied on @xmath1 @xcite . the bianchi identities ensure that the propagation equations will be satisfied in the exterior spacetime as a result of the hypersurface and evolution equations . integration of the propagation equations determines the additional horizon data necessary for the characteristic evolution . one of the propagation equations is the ingoing raychaudhuri equation @xmath68 , which propagates the surface area variable @xmath55 along the generators of @xmath1 in terms of initial conditions on @xmath23 . the value of @xmath56 on @xmath1 determines the convergence of the ingoing null rays . once the intrinsic geometry @xmath64 and the area coordinate @xmath55 are known , the vacuum equation @xmath69 is used to propagate the twist @xmath70 along the generators of @xmath1 . this determines @xmath71 in terms of its initial value at @xmath72 . the @xmath73 vacuum equations propagate the outward expansion and shear of the foliation @xmath74 along @xmath1 . the trace part propagates the outgoing expansion determined by @xmath57 . the trace - free part is an evolution equation for @xmath75 , which describes the shear of the outgoing rays . in summary , the data for the double null problem include the conformal metric @xmath36 on @xmath1 and @xmath22 and the quantities @xmath55 , @xmath56 , @xmath71 and @xmath57 on @xmath76 . equations ( a1 ) , ( 2.29 ) , ( 2.30 ) , and ( 2.34 ) of ref . @xcite propagate the data on @xmath23 to all of @xmath1 . the data required on @xmath23 can be inferred from the asymptotic properties of the white hole equilibrium at @xmath4 . the propagation equations , given in spin - weighted form in @xcite , are implemented numerically with a second order runge - kutta scheme @xcite . complete details can be found in ref . @xcite . the null code is based upon the bondi - sachs version of the characteristic initial value problem @xcite . it is designed to evolve forward in time along a foliation of spacetime by outgoing null hypersurfaces . the bondi - sachs coordinates differ from the sachs coordinates ( [ eq : amet ] ) by the use of a surface area coordinate @xmath55 along the outgoing cones rather than the affine parameter @xmath28 . because a generic horizon is not a hypersurface of constant @xmath55 , it is advantageous to first discuss the necessary data in terms of sachs coordinates and then transform to the @xmath55 coordinate . in bondi - sachs variables , the metric takes the form @xmath77 the field @xmath78 is represented in spin - weighted form @xmath79 . the evolution of the exterior spacetime requires a transformation of the data on @xmath1 from sachs coordinates @xmath80 to bondi - sachs coordinates @xmath81 , as described in detail in ref . @xcite . since @xmath55 is not in general constant on @xmath1 , the horizon does not lie precisely on radial grid points @xmath82 . consequently , an accurate prescription of boundary values on the @xmath82 grid points nearest the horizon requires a taylor expansion of the horizon data . restricted to @xmath1 , the metric variables @xmath55 and @xmath64 have the same values in both sachs and bondi - sachs coordinates and our choices of lapse and shift imply that the bondi - sachs variables @xmath83 , @xmath84 and @xmath85 are related to sachs variables by eqs . ( 2.15 ) , ( 2.31 ) , and ( 2.32 ) of ref . @xcite . a similar construction @xcite provides their first @xmath55 derivatives in terms of known sachs variables . we obtain @xmath86 from eq . ( 2.33 ) of ref . @xcite , @xmath87 from the @xmath88 raychaudhuri equation for the outgoing null geodesics , eq.(2.37 ) of ref . @xcite , and the value of @xmath89 on the horizon from the auxiliary field @xmath90 , where @xmath91 here @xmath92 is obtained from the twist and other sachs variables as per eq . ( 2.39 ) of @xcite . finally , we compute @xmath93 by obtaining @xmath94 from the @xmath28 derivative of eq . ( 2.32 ) of ref . @xcite . the values of each metric function @xmath95 and its first radial derivative can be used to consistently initialize field values at the @xmath82-grid points near the horizon . boundary values for the code are then provided , to second order accuracy , on grid points bracketing the horizon . for each outgoing null ray the value of the area coordinate @xmath96 on the horizon is known . this value is bracketed by the nearest grid points @xmath97 , where the metric values are computed by the second order accurate taylor expansion @xmath98 after the metric quantities have been computed at the grid points neighboring the horizon , the code can evolve the spacetime exterior to the horizon . the system of equations that determines the exterior spacetime forms a hierarchy @xcite . alternative formulations are available ; see for instance ref . here we use a recent formulation @xcite , which was specifically developed to handle the extremely nonlinear post - merger regime of binary black hole collisions . it reduces all angular derivatives to first order by introducing the auxiliary variables @xmath99 , @xmath100 and @xmath101 . this results in the hierarchy of hypersurface equations and one complex evolution equation for the conformal metric function @xmath64 given by eqs . ( 16)(25 ) of ref . we note that the terms on the right hand side of the evolution equation , eq . ( 16 ) of @xcite , have been grouped into hypersurface terms @xmath102 which vanish for linear perturbations of a spherically symmetric spacetime and a term @xmath103 which isolates the only nonlinear term containing a ( retarded ) time derivative of @xmath64 . the introduction of the auxiliary variables @xmath104 , @xmath105 , and @xmath106 in @xcite eliminates all second angular derivatives from the hypersurface and evolution equations . this leads to substantially improved numerical behavior @xcite over the standard characteristic formulation @xcite in the extremely nonlinear post - merger regime of binary black hole collisions . details are given in ref . @xcite . we retain the radial and time integration schemes of the standard characteristic formulation . the radial integration algorithm is explained in detail in refs . @xcite . in a departure from ref . @xcite , we use a three - step iterative crank - nicholson scheme @xcite to ensure stability of the time evolution @xcite . the evolution code first integrates the hypersurface equations [ eqs . ( 17)(23 ) of @xcite ] on the initial null hypersurface @xmath22 in the region exterior to the horizon . the evolution equation , eq . ( 16 ) of @xcite is then used to advance to the next hypersurface in the region exterior to the horizon , etc . , advancing the computation of the exterior spacetime in the region from the horizon to @xmath5 . the evolution continues as long as the coordinate system remains well behaved . the bondi news function @xmath107 @xcite is an invariantly defined field on @xmath5 which gives the amplitude of the flux of gravitational wave energy . its computation requires evaluation of the conformal factor @xmath108 necessary to compactify the spacetime in a conformal bondi frame @xcite , which is an asymptotic inertial frame in which the slices of @xmath5 have unit sphere geometry . following the approach of ref . @xcite , we set @xmath109 , where @xmath110 is a smooth non - vanishing field at @xmath5 . at the early time @xmath111 at which we initiate the evolution , the conformal metric @xmath36 of the outgoing null hypersurface approaches the unit sphere metric @xmath35 at @xmath5 so that @xmath112 . however , as the evolution proceeds , the @xmath113 coordinates , which are naturally adapted to @xmath1 , become non - inertial at @xmath8 . this results in the time dependence @xmath114 where @xmath115 is the asymptotic value of @xmath78 at @xmath8 @xcite . in addition , a transformation to conformal bondi coordinates @xmath116 on @xmath5 is necessary to determine the bondi news @xmath117 as measured in an inertial frame . the relevant expressions are given in sec . iv and appendix b of ref . inspection of eqs . ( b1)(b6 ) of ref . @xcite reveals that second angular derivatives of the conformal factor @xmath118 enter the calculation . for improved accuracy , we remove these second derivatives by introducing @xcite the auxiliary variable @xmath119 . since @xmath118 is defined solely on @xmath120 , we use the consistency relation @xmath121 to propagate @xmath122 along the generators of @xmath120 , initializing it by @xmath123 . we use a combination of second - order runge - kutta and mid - point rule schemes @xcite for the time integration of eq . ( [ eq : conf ] ) . the conformal horizon model @xcite supplies the conformal metric @xmath36 constituting the null data for a binary black or white hole . for the case of a head - on fission of a white hole , the model is based upon the flat space null hypersurface @xmath66 emanating from a prolate spheroid @xmath124 of eccentricity @xmath125 and semimajor axis @xmath126 , which is embedded at a constant inertial time @xmath127 in minkowski space . traced back into the past , @xmath66 expands to an asymptotically spherical shape . traced into the future , @xmath66 pinches off at points where its null rays cross or at caustic points where neighboring null rays focus . with the dyad choice @xmath128 , the intrinsic conformal metric @xmath36 of @xmath66 as a flat space null hypersurface is determined by the spin - weight-2 field @xmath129 where @xmath130 and @xmath131 are the principle radii of curvature of the spheroid . the metric of the spheroid induced by its embedding in a flat space is @xmath132 , where @xmath133 is the local surface area . the white hole horizon shares the same manifold @xmath66 and the same ( degenerate ) conformal metric as its minkowski space counterpart and eq . ( [ eq : headonj ] ) provides the conformal null data for the white hole horizon in the @xmath134 foliation . but the surface area and affine parametrization of the white hole horizon and its flat space counterpart differ . as a white hole horizon , @xmath66 extends infinitely far to the past of @xmath135 to an asymptotic equilibrium with finite surface area . the intrinsic metric of the white hole horizon is given by @xmath136 , and has local surface area @xmath137 where @xmath138 . the conformal factor @xmath139 is designed to stop the expansion of the white hole in the past so that the surface area radius asymptotically hovers at a fixed value @xmath140 . the conformal factor is given by @xcite @xmath141 where @xmath142 is a model parameter , @xmath143 is the difference between the principal curvature radii , and @xmath144 is a flat space affine parameter along the generators of @xmath66 with the same scale as @xmath134 but with its origin shifted along each ray by @xmath145 . for an initially schwarzschild white hole of mass @xmath146 , @xmath147 . smoothness of the white hole requires that the parameter @xmath148 , where @xmath149 is the maximum value of @xmath150 attained on @xmath135 . ( for the prolate spheroid considered here , @xmath151 . ) both as a flat space null hypersurface and as a white hole horizon , @xmath66 must obey the raychaudhuri equation , which governs the second derivative of the surface area with respect to the affine parameter . because the focusing power determined by their conformal geometries are the same , the raychaudhuri equation implies @xmath152 consequently , the different behavior of the surface area coordinates @xmath55 and @xmath153 due to the conformal factor @xmath139 implies that the affine parametrization @xmath154 on the white hole horizon is related to its flat space counterpart @xmath144 according to @xcite @xmath155 where @xmath156 and where the affine scale is fixed by the condition @xmath157 as @xmath158 . the scale of the particular affine parameter @xmath24 used for the time step in the evolution code is related to @xmath154 by @xmath159 ( chosen so that the null data @xmath64 has early time behavior of the pure spin - weight 2 quadrupole form @xmath160 ) . equation ( [ eq : lampr ] ) determines the rate of deviation of a slicing adapted to an affine parameter @xmath24 of the white hole horizon from the @xmath134 slicing given by the minkowski embedding . the build up of a large relative angular dependence between the @xmath24 and @xmath134 foliations leads to the change in topology of the fissioning white hole and the associated pair - of - pants shaped horizon . we begin our evolution at an early slice @xmath23 , at @xmath44 , where the horizon is close to equilibrium as a schwarzschild white hole . initially , the @xmath24 and @xmath134 foliations behave quite similarly . the time at which nonlinear behavior becomes significant is controlled by the location of the minkowski spheroid @xmath135 , at @xmath127 , with respect to the startup slice @xmath23 . we use the characteristic code @xcite to evolve the exterior spacetime of an axisymmetric , head - on white hole fission and compute the detailed waveforms for a sequence of spacetimes in the range @xmath161 , varying from the perturbative to the highly nonlinear regime . the object is to search for new gravitational wave physics resulting from the highly nonlinear behavior of a binary horizon . for that purpose , the simulations reported here were carried out on a grid of @xmath162 ( stereographic patch @xmath163 radial ) points . at this resolution , a higher eccentricity evolution ( @xmath164 ) requires approximately one hour on a single pentium ii 750 mhz processor . since the numerical code has been demonstrated to be second order convergent , more accurate results , especially concerning late time behavior near the point of fission can be attained by increasing the numerical grid resolution . even though the characteristic code @xcite is not parallel , parameter search runs of the type described here are entirely feasible on computing clusters with the present code . we initialize the simulations according to the specifications in ref . @xcite , where the horizon geometry for this data was computed and studied . we begin the evolution at @xmath165 with the initial mass @xmath166 and conformal model parameters @xmath167 ( which fixes the location of the spheroid @xmath124 ) , @xmath168 ( the semi - major axis of the spheroid ) and @xmath169 ( just above the minimum value of @xmath142 allowed by regularity of the conformal model for this range of eccentricities @xcite . on @xmath22 ( @xmath44 ) , we set @xmath65 to model the absence of ingoing radiation in an initially schwarzschild white hole . the limit @xmath170 yields the schwarzschild solution and small eccentricity corresponds to the close approximation where the fission takes place far in the future behind a marginally trapped surface so that it is hidden from external observers . for sufficiently small @xmath125 , the close approximation is valid in the entire exterior kruskal quadrant but for large @xmath125 it is valid only at very early times . for large @xmath125 , the fission occurs in the region of spacetime visible from @xmath5 . for a critical value @xmath171 there is a transition between these two regimes . for @xmath172 , the evolution terminates when the expansion of the outgoing null hypersurfaces vanishes along some ray . ( in the schwarzschild limit this occurs simultaneously on all rays on the black hole horizon . ) for @xmath173 , the evolution extends to the point of fission when @xmath174 on the equator of the white hole . figure [ fig : mloss ] shows the time dependence of the fractional mass loss @xmath175/m_-$ ] for a sequence of runs . a negligible amount of mass is radiated at startup , confirming that in all cases the initial geometry is very close to scharzschild . the mass loss remains small until @xmath176 , near the location of @xmath124 on @xmath1 , at which time a rapid change in @xmath177 occurs , and the mass loss rises sharply . following the sharp rise in @xmath178 , a transient period is observed in all the curves . the largest contribution to the radiated energy comes from the last stage of the evolution . the total mass loss remains well below the 1% level until eccentricities of about @xmath179 are reached , when the total mass loss at the end of the simulation approaches 2% of the initial white hole mass . at higher eccentricities , i.e. at @xmath180 , most of the initial mass is radiated away during the later stages of the simulation . the rate of mass loss in the final stage is particularly dramatic in the cases @xmath181 where the fission is visible from @xmath5 . figure [ fig : mloss_scaled ] shows again the mass loss profiles of fig . [ fig : mloss ] , but now scaled by the relative amplitude @xmath182 , with @xmath183 . the agreement seen clearly indicates that the behavior is @xmath184 in the range where they overlap . figure [ fig : newseq2 - 6 ] shows the time dependence of the news measured at the equator for @xmath185 to @xmath186 . here we have also overlayed the plots , rescaling them by @xmath187 , with @xmath183 . we plot the real part of the bondi news at a point on the equator , as a function of bondi time @xmath188 . due to axisymmetry , the imaginary part of the bondi news is zero to within discretization error . the sharp pulse clearly visible in the news at @xmath189 results from the choice of parameters in the white hole horizon data which control the rapidity of the fission . here , in order that the model yield a bare fission , the parameters have been chosen so that the fission occurs on an extremely rapid time scale . subsequent to this sharp pulse , the news undergoes a damped oscillation ( which is too early in the final ringdown to be associated with a quadrupole quasinormal mode ) . the bondi news as computed by the full code osculates the results obtained in the perturbative limit @xcite , in the regime in which the perturbative and nonlinear codes are clearly solving the same problem ( up to @xmath190 ) . at later times ( @xmath191 ) , the fully nonlinear news calculation deviates from the perturbative results as a consequence of the appearance of higher harmonics . these higher harmonics , which are clearly visible in the news as a function on the sphere , arise from the nonlinearity of the equations and can not be observed in a perturbative evolution . figures [ fig : quad ] and [ fig : dimple ] illustrate the angular behavior of the news at early time ( @xmath192 ) and at late time ( @xmath193 ) , respectively . we plot the real part of the bondi news on a single stereographic patch for the case @xmath194 . the graphs clearly show that at early times the news is pure quadrupole , in agreement with the perturbative regime of the close approximation . the nonlinearity of the problem subsequently introduces higher harmonics in the news . these higher harmonics are clearly visible at later times in the plots of the bondi news as a function on the sphere . the most notable feature is the dimple seen in fig . [ fig : dimple ] on the profile of the news at the pole ( the center of the stereographic patch ) . this feature is partially due to the retardation effect introduced by the angular - dependent redshift of the bondi frames at future null infinity @xmath5 . we have chosen the initial cut of @xmath5 , corresponding to a constant affine time @xmath24 on the horizon , to be a cut of constant bondi time . we then follow the inertial observers at @xmath8 to define a bondi time slicing . figure [ fig : bondi ] , which displays the angular dependence of bondi time @xmath188 for a fixed horizon time @xmath24 , reveals a pronounced increase in bondi time at the pole relative to the equator . we have computed a family of spacetimes exterior to a head - on white hole fission ranging from the close approximation to the nonlinear regime . the results reveal a dramatic time and angular dependence in the waveforms produced in the extreme nonlinear regime . at early times , the results agree with close - approximation perturbative calculations as expected . while the results presented here are for the axisymmetric non - spinning case , the data and evolution codes are not restricted to any symmetry . it will be interesting to see how the results for a head - on collision are modified in the fission of a spinning white hole . reexpressed in terms of the time - reversed scenario of a black hole merger , the boundary conditions for a fission corresponds to no _ outgoing _ radiation in the black hole case . nevertheless , the results pave the way for an application of characteristic codes to calculate the fully nonlinear waveform emitted in a binary black hole collision in the time period from merger to ringdown . waveforms from a black hole merger can be expected to differ from those from a white hole fission , as has been observed in close approximation studies @xcite . the fission process is directly observable at @xmath5 whereas the merger waveform arises indirectly from the preceding collapse of the matter or gravitational wave energy that forms the black holes . this suggests that the fission is a more efficient source of gravitational waves and that the high fractional mass losses computed here can not be attained in a black hole merger . we thank y. zlochower for helpful discussions and careful checking of the numerical code . this research has been partially supported by nsf grants phy 9800731 and phy 9988663 to the university of pittsburgh , and nsf grant phy-0135390 to carnegie mellon university . l. l. thanks pims and cita for support . r. g. thanks the albert einstein institute for hospitality . computer time for this project was provided by the pittsburgh supercomputing center .
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we present a fully nonlinear calculation of the waveform of the gravitational radiation emitted in the fission of a vacuum white hole . at early times , the waveforms agree with close approximation perturbative calculations but they reveal dramatic time and angular dependence in the nonlinear regime .
the results pave the way for a subsequent computation of the radiation emitted after a binary black hole merger .
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among the large number of theoretical models proposed to either solve the hierarchy problem and/or explain dark matter with a new stable particle , the minimal supersymmetric model ( mssm ) remains one of the favourite . supersymmetry not only provides a solution to both these problems but also predicts new physics around the tev scale . the main drawback of the mssm apart from the lack of evidence for supersymmetric particles is the large number of unknown parameters most of which describe the symmetry breaking sector . with the improved sensitivities of dark matter searches in astroparticle experiments @xcite , the precise determination of the dm relic density from cosmology @xcite , the latest results from the tevatron @xcite and the precision measurements , large regions of the parameter space of the supersymmetric models are being probed . this will continue in the near future with a number of direct and indirect detection experiments improving their sensitivities @xcite and most importantly with the lhc starting to take data . the lhc running at the full design energy of 14tev offers good prospects for producing coloured supersymmetric particles lighter than 2 - 3 tev , for discovering one or more higgs scalars @xcite and for measuring the rare processes in the flavour sector , in particular in b - physics @xcite . furthermore some properties of the sparticles , in particular mass differences can be measured precisely in some scenarios @xcite . the first studies that extracted constraints on supersymmetric models worked in general within the context of the mssm embedded in a gut scale model such as the cmssm @xcite . after specifying the fundamental model parameters at the high scale , the renormalisation group equations are used to obtain the weak scale particle spectrum . this approach provides a convenient framework for phenomenological analyses as the number of free parameters is reduced drastically compared to the general mssm ( from o(100 ) to @xmath3 and @xmath4 parameters in the case of the cmssm ) . the drawback is that one is often confined to very specific scenarios , for example in the cmssm the lsp is dominantly bino over most of the parameter space . this has important consequences for the dark matter relic abundance . furthermore it was customary to choose some specific values for some of the mssm or even the sm parameters for a convenient representation of the parameter space in two - dimensions . while the link between specific observables and allowed region of parameter space is easier to grasp in this framework , the allowed parameter space appeared much more restrictive than if all free parameters were allowed to vary . in the last few years efficient methods for exploring multi - dimensional parameter space have been used in particle physics and more specifically for determining the allowed parameter space of the cmssm . this approach showed that the often narrow strips in parameter space obtained when varying only two parameters at a time fattened to large areas @xcite after letting all parameters of the cmssm and the sm vary in the full range . with this efficient parameter space sampling method it becomes possible to relax some theoretical assumptions and consider the full parameter space of the mssm . because the number of experimental constraints on tev scale physics is still rather limited it seems a bit premature to go to the full fledge @xmath5 parameters of the mssm or even to the 19 parameters that characterize the model when assuming no flavour structure and equality of all soft parameters for the first and second generations of sfermions ( for an approach along these lines see @xcite ) . furthermore many parameters , for example those of the first and second generations of squarks , once chosen to be equal to avoid strong flavour - changing neutral current constraints , do not play an important role in the observables selected to fit the model . here we consider a model where input parameters of the mssm are defined at the weak scale and we add some simplifying assumptions : common slepton masses ( @xmath6 ) and common squark masses ( @xmath7 at the weak scale for all three generations and universality of gaugino parameters at the gut scale . this implies the following relation between the gaugino masses at the weak scale , @xmath8 . we furthermore assume that @xmath9 is the only non - zero trilinear coupling . while , as we just argued , the first assumption should not impact much our analysis , the second should certainly be considered as a theoretical bias . this assumption is however well motivated in the context of models defined at the gut scale . most importantly in our approach we keep the higgsino parameter @xmath10 and the gaugino mass @xmath11 as completely independent parameter . the relation between the gaugino and higgsino parameters is what determines the nature of the lsp and plays an important role in determining the lsp - lsp annihilation in the early universe . in that sense our model has many similarities with the non - universal higgs model which also has @xmath10 and @xmath11 as independent parameters @xcite . the observables selected to constrain the model include the relic density of dark matter , @xmath12 , direct searches for higgs and new particles at colliders , searches for rare processes such as the muon anomalous magnetic moment as well as various b - physics observables . note that the dark matter relic abundance is computed within the standard cosmological scenario . the direct detection of dark matter while providing stringent constraint on the model introduces additional unknown parameters both from astrophysics and from strong interactions . we therefore prefer to consider the direct detection rate as an observable to be predicted rather than as a constraint keeping in mind that folding in the astrophysical and hadronic uncertainty could however easily introduce an order of magnitude uncertainty in that prediction @xcite . we find that each individual parameter of the mssm model is only weakly constrained , in particular the parameters of the sfermion sector . the very large allowed parameter space only reflects the still poor sampling of the total parameter space by experiments . the neutralino sector is better constrained with a preferred value for the lsp of a few hundred gev s and a small likelihood for masses above 900gev , similarly charginos above 1.2tev are disfavoured . we also find a lower limit on the pseudo - scalar mass as well as on @xmath13 . furthermore some correlations between parameters of the model are observed , most notably the one between @xmath10 and the gaugino mass . this is because those two parameters determine the higgsino content of the lsp . after having determined the allowed parameter space , we examined the predictions for direct detection as well as for lhc searches both in the higgs and susy sector as well as for b - physics observables . although each type of search can only probe a fraction of the total parameter space we find a good complementarity between the different searches with less than 10% of scenarios leading to no signal . for example large signals for direct detection are expected in the mixed bino / higgsino lsp scenario that are hard to probe at the lhc . the lhc searches in the susy and higgs sector are also complementary and b - observables are specially useful in scenarios with large @xmath13 and a pseudoscalar that is not too heavy . the predictions for susy searches can be different from that expected in the constrained cmssm with in particular a large fraction of models that only have a gluino accessible at lhc , the squarks being too heavy . to ascertain how experiments that will take place in the near future could further constrain the parameter space of the model we consider specific case studies . for example we consider the impact of a signal in @xmath14 at tevatron or of the observation of a signal in direct detection experiments . finally we examine in more details the susy signals at the lhc , analysing the preferred decay chains for models that have either a gluino or a squark within the reach of the lhc . in this analysis we did not include the constraints from indirect detection experiments because the rates predicted feature a strong dependence on additional quantities such as the dark matter profile or the boost factor . the predictions for the rates for @xmath15 will be presented in a separate publication @xcite . the paper is organised as follows . the model and the impact of various constraints are described in section 2 . the method used for the fit is described in section 3 . the results of the global fits are presented in section 4 together with the impact of a selected number of future measurements . the susy signatures are detailed in section 5 . the conclusion contains a summary of our results . we consider the mssm with input parameters defined at the weak scale . we assume minimal flavour violation , equality of the soft masses between sfermion generations and unification of the gaugino mass at the gut scale . the latter leads to @xmath8 at the weak scale ( relaxing this assumption is kept for a further study ) . we allow for only one non - zero trilinear coupling @xmath9 . for the b - squark the mixing which is @xmath16 is driven in general by @xmath17 rather than by the trilinear coupling , this approximation is however not very good in the small sample of models with @xmath18 gev . note also that the higgs mass at high @xmath13 can show some dependence on the sbottom mixing . for first and second generations of squarks the mixing which depends on fermions masses is negligible except for the neutralino - nucleon cross section since the dominant contributions to the scalar cross section are also dependent on fermion masses . however since the squark exchange diagram is usually subdominant as compared to higgs exchange , the neglected contribution of the trilinear coupling falls within the theoretical uncertainties introduced by the hadronic matrix elements @xcite . similarly neglecting the the muon trilinear mixing , @xmath19 , could affect the prediction for @xmath20 but this effect is not large compared with the uncertainties on the value extracted from measurements . the top quark mass @xmath21 is also used as an input although it has a much weaker influence on the results than in the case of gut scale models . for the latter the top quark mass enters the renormalization group evolution and can have a large impact on the supersymmetric spectrum in some regions of the parameter space . in the general mssm the top quark mass mainly influences the light higgs mass . we fix @xmath22 and @xmath23 gev . the free parameters of our mssm model with unified gaugino masses , mssm - ug , are @xmath24 the range examined for each of these parameters is listed in table [ tab : param ] . mssm - ug has a far more restricted set of paramters than the general mssm , still this model will show how the possibilities for susy scenarios open up . the observables that will be used in the fit are listed in table [ tab : constraints ] . we first review the expectations for the role of each observable in constraining the mssm parameter space . .[tab : param ] range of the free mssm - ug parameters . [ cols="^,<,<",options="header " , ] [ tab : squark ] the left - handed quark which couples strongly to the wino and/or higgsino component has a wide variety of decay modes . the frequency of each dominant decay chain are displayed in table [ tab : squark ] for each lsp configuration . for the bino lsp the dominant mode is usually @xmath25 with typical branching fractions around 60% . the chargino will decay either into @xmath26 or @xmath27 when light sleptons are present . the subdominant mode in those scenario is @xmath28 with @xmath29 . the decay chains are similar to those of the cmssm . in some cases the second chargino , a mixed higgsino / wino , is kinematically accessible and the dominant mode will be @xmath30 with subdominant decays into @xmath31 and @xmath32 . @xmath33 will decay preferentially into @xmath34 or in other neutralinos as well as into @xmath35 . the higgs can be produced in either @xmath36 or further in @xmath37 . a fraction of models ( 7.7% ) feature the dominant decay into the lsp @xmath38 . because the squark @xmath39 has a suppressed rate to the bino , this channel is dominant only when other two - body channels are kinematically forbidden . for a mixed lsp ( @xmath40 ) the relative importance of the various decay channels shifts . the decay @xmath38 is dominant in less than 3% of the cases although because of the higgsino component of the lsp this can occur even when heavier neutralinos are kinematically accesssible . by far the most frequent dominant decay is @xmath30 with significant branching fractions in @xmath41 or @xmath42 . the heavier chargino always has two - body decay modes , @xmath43 ( preferably @xmath34 ) or @xmath36 . the @xmath44 and @xmath45 in turn feature mostly 3-body decays . note that decay modes into higgs bosons @xmath36 can involve even the heavy higgs bosons . as usual when light sleptons are present the decay @xmath46 can be dominant . for the higgsino lsp ( @xmath47 ) , the dominant mode is either @xmath48 or @xmath49 with some contributions from @xmath49 and @xmath31 . the @xmath33 channel has similar decay chains as the mixed lsp except that the dominant mode is usually @xmath50 rather than channels involving higgses . the @xmath51 can in a few cases decay via two - body , @xmath52 or @xmath35 , but in most cases it decays via three - body dominantly into @xmath53 . these decays mainly give signatures into jets and missing energy . the @xmath45 produced in squark or neutralino decays will also decay via three body final states . in summary @xmath39 decays dominantly into heavy charginos with further decay chains involving other chargino / neutralino states . decay chains involving slepton production dominater only in 25% of scenarios . finally recall that the elastic scattering cross section also differs significantly depending on the nature of the lsp giving an opportunity to correlate susy signals at lhc with those of direct detection . for the bino lsp , @xmath54 pb while @xmath55 pb for the mixed ( higgsino ) lsp . we do not discuss in detail the case where both squarks and gluinos are below 2tev . the decay chains can be rather complicated with the possibility of producing the gluino in squark decay and vice - versa . the case where the gluino is heavier than the squarks features the same decay chains for the squarks as the case just discussed . increasing the number of free parameters as compared to the cmssm model has opened up the possibilities for supersymmetric scenarios that are compatible with all experimental constraints and this even maintaining the universality of gaugino mass . although the parameter space of the model is still not very well constrained , we found that the most favoured models have a lsp of a few hundred gev with a significant higgsino fraction ( @xmath56 ) . contrary to the cmssm case the higgsino lsp is not fully correlated with a very heavy squark sector although all our scenarios favour squarks above the tev scale . a very light pseudoscalar is also disfavoured with @xmath57 gev , this means that large deviations from the sm in b - physics observables are expected only in a small fraction of allowed scenarios . our favoured scenarios predict few signals at the tevatron , the pseudoscalar higgs as well as the coloured sector are too heavy to be accessed by direct searches . only very few scenarios have a potentially large enough rate for trilepton searches at the tevatron . the complementarity between future experiments to probe this class of models was emphasized . even though susy or heavy higgs signals are not guaranteed at lhc , the majority of allowed models predict at least one signal either at the lhc ( including the flavour sector ) or in future direct detection experiment . furthermore the light higgs is expected to be around 120gev with sm - like couplings . we have also explored the various dominant decay chains for gluinos and squarks that could be produced at lhc in the mssm - ug as well as for the heavy neutralinos appearing in the decays of these coloured sparticles . we found that for models with gluinos accessible at lhc , a significant fraction of the heavy neutralinos produced decayed dominantly into a gauge or higgs boson . furthermore states which decayed into sleptons are rarely dominant . we also showed how the preferred squarks decay channels are determined to a large extent by the neutralino composition . whether one can exploit these decay chains to determine some properties of the sparticles remains to be seen . in our analysis the relic density measurement plays the dominant role in constraining the model , since the relic density computation implicitly assumes a standard cosmological scenario , relaxing this requirement affects significantly the allowed parameter space of the model . finally we comment on the difference between our results and other recent analyses done within the framework of the mssm with 24 parameters , either using a mcmc likelihood approach or applying @xmath58 constraints on each of the observables @xcite . first these studies were done in a more general model than the one we have considered , with in particular no universality condition on the gaugino masses . this means that the lsp can have a significant wino component and therefore is more likely to be at the tev scale as was found in @xcite using linear priors . recall that a tev scale wino annihilates efficiently into gauge bosons pairs . the analysis of @xcite also emphasizes the prior dependence with a generally much lighter spectrum using log priors . this is due mostly to the poorly constrained parameter space @xcite . as in our analysis squarks and sleptons ran over the full range allowed in the scan and the pseudoscalar mass can be very heavy . the analysis of @xcite used a different statistical treatment but most importantly did not require that the neutalino explained all the dm in the universe ( only an upper bound on @xmath12 was imposed ) . this means that a large number of models with small mass splitting between the lsp and the nlsp appeared in the scan calling for a careful study of collider limits . in our approach such models are ruled out since they have @xmath59 . this analysis further emphasized the light susy spectrum in their scans so naturally found preferred lsp mass below the tev scale . we thank j. hamann for many useful discussions on the mcmc method . we acknowledge support from the indo french center fro promotion of advanced scientific research under project number 30004 - 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using a markov chain monte carlo approach , we find the allowed parameter space of a mssm model with seven free parameters . in this
model universality conditions at the gut scale are imposed on the gaugino sector .
we require in particular that the relic density of dark matter saturates the value extracted from cosmological measurements assuming a standard cosmological scenario .
we characterize the parameter space of the model that satisfies experimental constraints and illustrate the complementarity of the lhc searches , b - physics observables and direct dark matter searches for further probing the parameter space of the model .
we also explore the different decay chains expected for the coloured particles that would be produced at lhc . _
_ date : * constraining the mssm with universal gaugino masses and implication for searches at the lhc * + g. blanger@xmath0 , f. boudjema@xmath0 , a. pukhov@xmath1 , r. k. singh@xmath2 + _ 1 ) lapth , univ . de savoie , cnrs , b.p.110 , f-74941 annecy - le - vieux , france + 2 ) skobeltsyn inst . of nuclear physics , moscow state univ . ,
moscow 119992 , russia + 3 ) institut fr theoretische physik und astrophysik , universitt wrzburg , + d-97074 wrzburg , germany _ +
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the phenomenon of soft gamma repeaters ( sgrs ) may allow us in the near future to determine fundamental properties of strongly magnetized , compact stars . already , there exist at least two sources in which quasi - periodic oscillations ( qpos ) have been observed in their x - ray tail , following the initial discovery by @xcite , see @xcite for a recent review . the frequency of many of these oscillations is similar to what one would expect for torsional modes of the solid crust of a compact star . this observation is in support of the proposal that sgrs are magnetars ( compact objects with very strong magnetic fields ) @xcite . during an sgr event , torsional oscillations in the solid crust of the star could be excited @xcite , leading to the observed frequencies in the x - ray tail . however , not all of the observed frequencies fit the above picture . for example , the three lowest observed frequencies for sgr 1806 - 20 are 18 , 26 , 30hz . only one of these could be the fundamental , @xmath6 torsional frequency of the crust , as the first overtone has a much higher frequency . @xcite stressed the importance of crust - core coupling by a global magnetic field and of the existence of an alfvn continuum , while @xcite considered model with simplified geometry , in which alfvn oscillations form a discrete spectrum of normal modes , that could be associated with the observed low - frequency qpos . in @xcite , the existence of a continuum was stressed further and it was shown that the edges or turning points of the continuum can yield long - lived qpos . in addition , numerical simulations showed that drifting qpos within the continuum become amplified near the frequencies of the crustal normal modes . within this model , levin suggested a likely identification of the 18hz qpo in sgr 1806 - 20 with the lowest frequency of the mhd continuum or its first overtone . the above results were obtained in toy models with simplified geometry and newtonian gravity . in this letter , we perform two - dimensional numerical simulations of linearized alfvn oscillations in magnetars . our model improves on the previously considered toy models in various ways : relativistic gravity is assumed , various realistic equations of state ( eos ) are considered and a consistent dipolar magnetic field is constructed . we do not consider the presence of a solid crust , but only examine the response of the ideal magnetofluid to a chosen initial perturbation . spherical stars have generally two type of oscillations , _ spheroidal _ with polar parity and _ toroidal _ with axial parity . the observed qpos in sgr x - ray tails may originate from toroidal oscillations , since these could be excited more easily than poloidal oscillations , because they do not involve density variations . in newtonian theory , there have been several investigations of torsional oscillations in the crust region of neutron stars ( see e.g. , @xcite for reference ) . on the other hand , only few studies have taken general relativity into account @xcite . sgrs produce giant flares with peak luminosities of @xmath7 @xmath8 erg / s , which display a decaying tail for several hundred seconds . up to now , three giant flares have been detected , sgr 0526 - 66 in 1979 , sgr 1900 + 14 in 1998 , and sgr 1806 - 20 in 2004 . the timing analysis of the latter two events revealed several qpos in the decaying tail , whose frequencies are approximately 18 , 26 , 30 , 92 , 150 , 625 , and 1840 hz for sgr 1806 - 20 , and 28 , 53 , 84 , and 155 hz for sgr 1900 + 14 , see @xcite . in @xcite ( hereafter paper i ) , it was suggested that some of the observational data of sgrs could agree with the crustal torsional oscillations , if , e.g. , frequencies lower than 155 hz are identified with the fundamental oscillations of different harmonic index @xmath9 , while higher frequencies are identified with overtones . however , in paper i and above , it will be quite challenging to identify all observed qpo frequencies with only crustal torsional oscillations . for example , it is difficult to explain all of the frequencies of 18 , 26 and 30 hz for sgr 1806 - 20 with crustal models , because the actual spacing of torsional oscillations of the crust is larger than the difference between these two frequencies . similarly , the spacing between the 625hz and a possible 720hz qpo in sgr 1806 - 20 may be too small to be explained by consecutive overtones of crustal torsional oscillations . one can notice , however , that the frequencies of 30 , 92 and 150 hz in sgr 1806 - 20 are in near _ integer ratios_. as we will show below , the numerical results presented in this letter are compatible with this observation , as we find two families of qpos ( corresponding to the edges or turning points of a continuum ) with harmonics at near integer multiples . furthermore , our results are compatible with the ratio of 0.6 between the 18 and 30hz frequencies , if these are identified , as we suggest , with the edges ( or turning points ) of the alfvn continuum . with this identification , we can set an upper limit to the dipole magnetic field of @xmath1 to @xmath10 g . if the drifting qpos of the continuum are amplified at the fundamental frequency of the crust , and the latter is assumed to be the observed 26hz for sgr 1806 - 20 , then our results are compatible with a magnetar mass of about @xmath4 to 1.6@xmath5 and an eos that is very stiff ( if the magnetic field strength is near its upper limit ) or moderately stiff ( for lower values of the magnetic field ) . unless otherwise noted , we adopt units of @xmath11 , where @xmath12 and @xmath13 denote the speed of light and the gravitational constant , respectively , while the metric signature is @xmath14 . the general - relativistic equilibrium stellar model is assumed to be spherically symmetric and static , i.e. a solution of the well - known tov equations for and perfect fluid and metric described the line element @xmath15 we neglect the influence of the magnetic field on the structure of the star , since the magnetic field energy , @xmath16 , is orders of magnitudes smaller than the gravitational binding energy , @xmath17 , for magnetic field strengths considered realistic for magnetars , @xmath18 . for simplicity , we assume that the magnetic field is a pure dipole ( toroidal magnetic fields will be treated elsewhere , see @xcite . details on the numerical method for constructing the magnetic field , as well as representative figure of the magnetic field lines , can be found in paper i. mhd oscillations of the above equilibrium model are described by the linearized equations of motion and the magnetic induction equations , presented in detail in papers i and ii ( we neglect perturbations in the spacetime metric , as these couple weakly to toroidal modes in a spherically symmetric background ) . the perturbative equations presented in papers i and ii can readily be converted from an eigenvalue problem to the form of a two - dimensional time - evolution problem , by defining a displacement @xmath19 due to the toroidal motion ( the coefficient of shear viscosity is set to @xmath20 , as we neglect the presence of a solid crust in the present work ) . the contravariant _ coordinate component _ of the perturbed four - velocity , @xmath21 , is then related to time derivative of @xmath22 through @xmath23 the two - dimensional evolution equation for @xmath19 is @xmath24 where @xmath25 , @xmath26 , @xmath27 , @xmath28 , @xmath29 , and @xmath30 are functions of @xmath31 and @xmath32 , given by @xmath33 e^{-2(\phi - \lambda ) } , \\ { \cal a}_{20 } & = & \frac{{a_1}^2}{\pi r^4}\cos^2\theta , \\ { \cal a}_{11 } & = & -\frac{a_1 { a_1}'}{\pi r^4}\sin\theta \cos\theta , \\ { \cal a}_{02 } & = & \frac{{{a_1}'}^2}{4\pi r^4}\sin^2\theta , \\ { \cal a}_{10 } & = & \left(\phi ' - \lambda ' \right ) \frac{{a_1}^2}{\pi r^4 } \cos^2\theta + \frac{a_1 { a_1}'}{2\pi r^4 } \sin^2\theta , \\ { \cal a}_{01 } & = & \left[\frac{a_1}{\pi r^4}\left(2\pi j_1 - \frac{a_1}{r^2}\right)e^{2\lambda } + \frac{3{{a_1}'}^2}{4\pi r^4 } \right]\sin\theta\cos\theta,\end{aligned}\ ] ] and where @xmath34 and @xmath35 are the radial components of the electromagnetic four - potential and the four - current , respectively . in the above equations , a prime denotes a partial derivative with respect to the radial coordinate . in our numerical scheme , we employ the 2nd - order , iterative crank - nicholson scheme . a numerical instability , which sets in after many oscillations , was treated by adding a 4th - order kreiss - oliger dissipation term @xcite , shown as @xmath36 in equation ( [ eq2d ] ) above . we experimented with various values of the dissipation coefficient @xmath37 and found the evolution to be stable for values as small as a few times @xmath38 . we verified that in this limit , the solution becomes independent of the strength of the numerical dissipation , as the @xmath39 kreiss - oliger dissipation operator introduces an error of higher order than the 2nd - order iterative crank - nicholson scheme . the numerical grid we use is equidistant , covering only the interior of the star , with ( typically ) 50 radial zones and 40 angular zones ( we also compared our results to simulations with 100x80 points ) . the boundary conditions are : ( a ) @xmath40 at @xmath41 ( regularity ) , ( b ) @xmath42 at @xmath43 ( vanishing traction ) ( c ) @xmath44 at @xmath45 ( axisymmetry ) and ( d ) @xmath40 at @xmath46 ( equatorial plane symmetry of the @xmath3 initial data ) . we obtain the same results if you use a grid that extends to @xmath47 . we have also evolved initial data with @xmath48 , for which the appropriate boundary condition at @xmath46 is @xmath44 . ( magnetic axis ) . three lines in each figure correspond to different radial positions @xmath49 , @xmath50 , and @xmath51 ) . a fundamental ( @xmath52 ) qpo and several overtones ( nearly integer multiples ) are clearly present.,width=207 ] , but at @xmath53 ( magnetic equator ) . a second family of qpos is present . arrows indicate several continuous parts , of which only the first is distinct from the others , which partially overlap.,width=207 ] [ cols="^,^ " , ] , of the lower and upper fundamental alfvn qpo frequencies , obtained for a representative sample of equilibrium models with various eoss and masses . the magnetic field was set to @xmath54 . , width=207 ] we have obtained the lower and upper alfvn qpo frequencies for a representative sample of magnetar models with realistic ( tabulated ) eoss and various masses . these equilibrium models and their detailed properties have already been presented in paper i. in table [ tab : eos1 ] we summarize our numerical results for @xmath54 , for a somewhat smaller sample of models , than the one considered in paper i. as in the case of the polytropic model of sec . [ sec : iii ] , the overtones are nearly integer multiples of the fundamental frequency @xmath55 with an accuracy on the order of 1% or better . in addition , the ratio of the lower to upper qpo frequencies ( also shown in table [ tab : eos1 ] ) roughly agrees with the value of 0.6 for the polytropic model . a quadratic fit in terms of @xmath56 gives : @xmath57,\ ] ] with an accuracy of better than @xmath58 . similarly , a quadratic fit for the ratio of the lower to upper first overtone frequencies yields @xmath59,\ ] ] with a similar accuracy . we find that the frequencies of the alfvn qpos scale _ linearly _ with the strength of the magnetic field ( with an accuracy on the order of 1% or better ) at least for magnetic field strengths on the order of @xmath60 . within the representative sample of the equilibrium models we display in table [ tab : eos1 ] , the frequencies of all lower and upper alfvn qpos ( with @xmath61 ) are given by the following two empirical relations @xmath62 \nonumber \\ & & \times \left(\frac{b}{4\times 10^{15 } { \rm g } } \right ) , \label{eq : empirical - low } \\ f_{u_n } ( { \rm hz})&\simeq & 86.1(n+1)\left[1 - 4.58 \left(\frac{m}{r}\right ) + 6.06 \left(\frac{m}{r}\right)^2\right ] \nonumber \\ & & \times \left(\frac{b}{4\times 10^{15 } { \rm g } } \right ) , \label{eq : empirical - up}\end{aligned}\ ] ] with an accuracy of less than about 4% . the quadratic fits in terms of @xmath56 , for @xmath54 is shown in fig . [ fig : fit ] . we emphasize that in all of the above relations , the ratio @xmath56 is dimensionless , in gravitational units ( @xmath11 ) . to restore units , it has to be replaced by @xmath63 . several of the low - frequency qpos observed in the x - ray tail of sgr 1806 - 20 can readily be identified with the alfvn qpos we compute . in particular , one could identify the 18hz and 30hz observed frequencies with the fundamental lower and upper qpos , correspondingly , while the observed frequencies of 92hz and 150hz would then be integer multiples of the fundamental upper qpo frequency ( three times and five times , correspondingly ) . with this identification , our empirical relations eqs . ( [ eq : empirical - low ] ) and ( [ eq : empirical - up ] ) constrain the magnetic field strength of sgr 1806 - 20 ( if is dominated by a dipolar component ) to be between @xmath64 g and @xmath10 g . furthermore , an identification of the observed frequency of 26hz with the frequency of the fundamental torsional @xmath3 oscillation of the magnetar s crust ( eq . ( 79 ) of paper i ) implies a very stiff equation of state and a mass of about 1.4 to 1.6@xmath5 . for example , for the @xmath65 model constructed with eos l+dh , one obtains the following frequencies : @xmath66hz , @xmath67hz , @xmath68hz , @xmath69hz and @xmath70hz , for @xmath71 g. alternatively , one could also identify the 18hz and 30hz observed frequencies with overtones ( which are also at a near 0.6 ratio ) . in this case , the strength of the magnetic field derived above is only an upper limit and the actual magnetic field may be weaker . then , if one assumes that the observed frequency of 26hz is due to the fundamental @xmath3 crust mode for a weak magnetic field , our numerical data agree best with a 1.4@xmath5 model constructed with an eos of moderate stiffness . for example , for the 1.4@xmath5 model constructed with the apr+dh eos one obtains @xmath72hz , @xmath73hz , @xmath74hz , @xmath75hz and @xmath76hz , for @xmath77 g. we have already verified that our main qpo frequencies agree with frequencies obtained with an independent , fully nonlinear numerical code @xcite , for the same initial model . we caution , however , that we have not yet considered the crust - core interaction , different magnetic field topologies or the coupling to the exterior magnetosphere . these effects have to be taken into account and already @xcite , find that the observed qpos could lead to constraints on the magnetic field topology . to complete the picture , a three - dimensional numerical simulation , that includes a proper coupling of the crust to the mhd interior and to the exterior magnetosphere will be required and our current results provide a good starting point . extensive details of our computations will be presented in @xcite . it is a pleasure to thank yuri levin for stressing the importance of the mhd continuum . we are grateful to pablo crda - duran and toni font for comparing our main frequency results with their fully nonlinear code and to demosthenes kazanas for helpful discussions . we also thank the participants of the nsdn magnetic oscillations workshop in tuebingen for useful interactions . this work was supported by the marie - curie grant mif1-ct-2005 - 021979 , the pythagoras ii program of the greek ministry of education and religious affairs , the eu network ilias and the german foundation for research ( dfg ) via the sfb / tr7 grant .
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we investigate torsional alfvn oscillations of relativistic stars with a global dipole magnetic field , via two - dimensional numerical simulations .
we find that a ) there exist two families of quasi - periodic oscillations ( qpos ) with harmonics at integer multiples of the fundamental frequency , b ) the lower - frequency qpo is related to the region of closed field lines , near the equator , while the higher - frequency qpo is generated near the magnetic axis , c ) the qpos are long - lived , d ) for the chosen form of dipolar magnetic field , the frequency ratio of the lower to upper fundamental qpos is @xmath0 , independent of the equilibrium model or of the strength of the magnetic field , and e ) within a representative sample of equations of state and of various magnetar masses , the alfvn qpo frequencies are given by accurate empirical relations that depend only on the compactness of the star and on the magnetic field strength .
the lower and upper qpos can be interpreted as corresponding to the edges or turning points of an alfvn continuum , according to the model proposed by levin ( 2007 ) .
several of the low - frequency qpos observed in the x - ray tail of sgr 1806 - 20 can readily be identified with the alfvn qpos we compute . in particular , one could identify the 18hz and 30hz observed frequencies with the fundamental lower and upper qpos , correspondingly , while the observed frequencies of 92hz and 150hz are then integer multiples of the fundamental upper qpo frequency ( three times and five times , correspondingly ) . with this identification
, we obtain an upper limit on the strength of magnetic field of sgr 1806 - 20 ( if is dominated by a dipolar component ) between @xmath1 and @xmath2 g .
furthermore , we show that an identification of the observed frequency of 26hz with the frequency of the fundamental torsional @xmath3 oscillation of the magnetar s crust is compatible with a magnetar mass of about @xmath4 to 1.6@xmath5 and an eos that is very stiff ( if the magnetic field strength is near its upper limit ) or moderately stiff ( for lower values of the magnetic field ) .
[ firstpage ] relativity mhd stars : neutron stars : oscillations stars : magnetic fields gamma rays : theory
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anisotropy measurements of the cosmic microwave background ( cmb ) are a very effective tool for testing and constraining models of cosmic structure formation . with the discovery of large angular scale anisotropy by cobe ( smoot et al . , 1992 ) , there has been an increased interest in characterizing anisotropy on degree angular scales . although all cmb measurements have to overcome a long list of systematic effects and foreground contaminants ( wilkinson , 1994 ) , they have the potential to constrain many of the global parameters of the universe and thus discriminate among the plethora of cosmic structure formation models . over the past six years , we have travelled to the south pole three times to perform these degree scale anisotropy measurements . the results from our 1988 - 89 measurements are detailed in meinhold and lubin , 1991 ( sp89a ) and meinhold et al . , 1993 ( sp89b ) while the results from the 1990 - 91 measurements are detailed in gaier et al . , 1992 ( sp91a ) and schuster et al . , 1993 ( sp91b ) . the results from these measurements and the five balloon - borne millimeter - wave anisotropy experiment ( max ) are summarized in lubin , 1994 . in order to obtain additional data and frequency coverage , we returned to the amundsen - scott south pole station during the austral summer 1993 - 94 . the sp94 observations used the advanced cosmic microwave explorer ( acme ) as have all our previous degree scale anisotropy measurements . acme is a one meter off axis gregorian telescope which has been described in detail in sp89b . during the observations , the ellipsoidal secondary oscillated sinusoidally at 8 hz with a peak to peak throw of 3@xmath0 on the sky . the receiver signals were phase synchronously demodulated using a `` square - wave '' lockin amplifier and sampled every 0.5 seconds . the beam profile of the telescope can be approximated as a gaussian beam with a 1 @xmath17 dispersion which varies in frequency as given below . two different total power radiometers were used in these observations . the lower frequency ( ka - band ) receiver is similar to that described in sp91a and incorporates a very low noise , high electron mobility transistor ( hemt ) amplifier ( pospieszalski et al . , 1990 ) cooled to 4 k in a @xmath18he dewar . this receiver operated at four center frequencies ( 27.25 , 29.75 , 32.25 , and 34.75 ghz ) with 2.5 ghz 3 db bandwidths . the band subdivision is used to compensate for gain variations across the full band and to obtain spectral information which can be used to discriminate between the various astrophysical foregrounds . for the ka - band system , the beam dispersion is given by @xmath19 the higher frequency receiver ( q - band ) is described in gundersen et al . , 1994 and also uses a cryogenic hemt amplifier based on a design developed at the national radio astronomy observatory ( nrao ) . this amplifier was built at ucsb with assistance from nrao and uses an alinas / gainas / inp hemt ( pospieszalski et al . , 1994 ) in the first of five amplification stages . the q - band system was multiplexed into 3 equal bands centered at 39.15 , 41.45 , and 43.75 ghz with nominal 3 db bandwidths of 2.3 ghz , and the beam dispersion is given by @xmath20 the hemt amplifiers introduce intrinsic cross correlations between the bands which can be characterized by the correlation coefficient between any two frequencies . the measured correlations were typically 0.25 and 0.50 for the ka and q - band systems , respectively , including atmospheric correlations . the radiometers are calibrated to 10% absolute accuracy and 3% relative accuracy using a combination of cryogenic cold loads , the sky , ambient eccosorb , and the moon . the long term stability of the system was checked daily by inserting an ambient load `` calibrator '' . these calibrations varied by less than 3% over the time scale of an observation and contribute a negligible amount to the final error estimate . two observations were performed between january 9 , 1994 and january 22 , 1994 and collected 261 hours of data . the first observation used the q - band receiver and the second observation used the ka - band receiver . these observations consisted of smooth , constant declination , constant velocity scans of length 20@xmath0 on the sky about a center @xmath2145@xmath22 , @xmath23 - 62@xmath0 . the closest approach to the sun was 60@xmath0 on the sky and the closest approach to the plane of the galaxy corresponds to @xmath24 , @xmath25 . this is a low foreground emission region which overlaps some of the region observed in sp91 . our measurement of the eta carina region showed that our absolute elevation was one degree lower than we expected . the offset has been attributed to sag in the inner frame of the telescope mount and makes a direct comparison between these measurements and the sp91 measurements problematic . the instantaneous right ascension of the beam for any of the 3 observations can be given by @xmath26 where @xmath27 enumerates the scan number , @xmath28 is the sinusoidal chop amplitude , @xmath29 hz is the chop frequency , and @xmath30 mhz is the scan frequency . the instantaneous beam position on the sky is then given by @xmath31 . observations of the moon established the absolute pointing at low elevations and this was confirmed with observations of eta carina at high elevations . the error in absolute pointing is @xmath32 in right ascension and @xmath33 in declination while the error in relative pointing is @xmath34 in right ascension and @xmath34 in declination . if a temperature at position @xmath35 is compared to a temperature at position @xmath36 at the same declination , then the dimensionless window function can be written as @xmath37 ^2}4h_0 ^ 2[(\ell-2r)\alpha _ o]j_0 ^ 2[(\ell-2r)\delta \varphi /2]\cos [ ( \ell -2r)\phi _ { ij}^{kl}]\\ ] ] where @xmath38 @xmath39 is the angular difference ( or lag ) between the temperature measured at bin @xmath40 with channel @xmath41 and the temperature measured at bin @xmath42 with channel @xmath43 . the beam profile function is given by @xmath44 , $ ] @xmath45 is the struve function of 0 index , @xmath46 is the 0@xmath47 order spherical bessel function and @xmath48 is the bin size in radians on the sky . the indices are given by @xmath49 to @xmath50 bins and @xmath51 to @xmath52 for the q - band data and @xmath51 to @xmath53 for the ka - band data . this window function is shown in figure 2 for the different combinations of beamsizes and is a specific example taken from a more general expression in white and srednicki ( 1994 ) . data were rejected for a number of reasons including poor pointing / chopper performance ( 1.2% ) , performance of other observations ( 0.2% ) , calibration sequences ( 0.7% ) , telescope / receiver maintenance ( 9.1% ) , temperature variations of the cold plate and backend electronics ( 0.9% ) , and bad weather ( 12.9% ) . from a total of 261 hours of data , 196 hours were used in the data analysis . the poor weather data were determined in a way similar to sp91b in which the data from a single scan were combined into position bins from which an average and one sigma error bar were calculated . the @xmath54 of the individual scans was then calculated , and if the probability of exceeding @xmath54 for 43 degrees of freedom was less than 0.01 for any channel , then the data from _ all _ channels for that scan were removed . as a cross check , other weather filters similar to sp91a were also implemented with no significant changes in the final data set . as with sp89 and sp91 , an offset and gradient were removed in time over the time scale of a single scan ( 100 secs ) for each of the channels in an observation . the offsets were between 1 and 2 mk depending on the channel . the offset and gradient subtraction are taken into account in the analysis by creating a matrix @xmath55 such that @xmath56 where @xmath57 are the @xmath58 temperature means and @xmath59 are the @xmath60 projected temperatures with @xmath61 @xmath62 the formation of @xmath55 is discussed in bunn et al . 1994 and we have made @xmath55 orthogonal such that @xmath63 . the coadded means and 1 sigma error bars are shown in figure 1 and show statistically significant , correlated signals for each observation . all quoted temperatures have been converted from antenna temperature to thermodynamic temperature and have been corrected for atmospheric absorption . in order to determine the origin of the observed structure , the data were binned in azimuthal and heliocentric coordinate systems . various subset analyses were performed including dividing the data set into four roughly equal quarters in time and dividing the data into four roughly equal quarters depending on the azimuthal position of the telescope beam . none of these analyses suggest that the observed structure is anything but celestial in origin . there are several astrophysical foregrounds which could contaminate these observations . these include diffuse synchrotron and free - free emission from within our own galaxy and extragalactic emission from discrete radio sources . neither the sunyaev - zeldovich effect nor diffuse 20 k dust emission is expected to contribute more than a few @xmath9k signal . if we assume that the 408 mhz map ( haslam et al . ) is a tracer of diffuse , high galactic latitude synchrotron emission , then we calculate the rms differential synchrotron emission to be 1.0 k at 408 mhz for these observations . given the spectral index for diffuse synchrotron is @xmath64 with an antenna temperature given by @xmath65 , we estimate that the diffuse synchrotron contribution to the observed rms to be @xmath66 @xmath9k in the lowest frequency channel and @xmath67 @xmath9k in the highest frequency channel . the small amount of differential emission that exists in this region of the sky at 408 mhz can be correlated with discrete radio sources which have been observed in other source surveys at 408 mhz and are identified in the pkscat90 database ( wright and otrupcek , 1990 ) . unlike diffuse synchrotron emission , discrete radio sources and free - free emission can not be dismissed on amplitude arguments alone . for diffuse free - free emission , there are no all - sky surveys which would allow a direct estimate of the free - free contribution to the observed structure . instead , we have to rely on estimates based on a 10@xmath0 @xmath68 12@xmath0 h@xmath69 map ( reynolds , 1992 ) made at a similar angular scale to predict an average background free - free brightness temperature that can be expressed as t@xmath70 . reynolds data suggests that the variations in the h@xmath69 intensity may be a factor of 2 above the average , such that @xmath71t@xmath72t@xmath73 . from this we calculate the differential brightness temperature due to free - free emission ( upon the closest approach to the galactic plane ) to range from @xmath71t@xmath74 @xmath9k at the lowest frequency to @xmath71t@xmath75 @xmath9k at the highest frequency . there have been many discrete source surveys at lower frequencies which are compiled in the pkscat90 database . the parkes - mit - nrao ( pmn ) survey at 4.85 ghz ( wright et al . , 1994 ) serves as the most sensitive survey with a flux limit of 30 mjy in our observation region and includes all the sources identified in pkscat90 for our region . the sensitivity of the telescope is 90 @xmath9k / jy ( assuming 100% aperture efficiency ) , so a 30 mjy flat spectrum source ( with a flux density s(jy)@xmath76 @xmath77@xmath78 would produce a 3 @xmath9k signal , which is well below the noise of the observations . since the effective solid angle , @xmath79 , of the telescope varies like @xmath80 , a flat spectrum point source with a solid angle @xmath81 and @xmath82 would give @xmath83 , while a thermal point source produces an antenna temperature @xmath84 . if we make the worst case assumption that all the point sources have flat spectra to 45 ghz , then we estimate that they would produce a @xmath71t@xmath85 @xmath86k . since the worst case estimates of contamination from free - free emission and flat spectrum point sources are comparable to the rms level of the observed structure , we can not dismiss or verify these types of foreground contamination without measuring @xmath87 this is addressed in the following likelihood analysis . we use the bayesian method with a uniform prior ( bond et al . 1991 ) in the determination of the root mean square ( rms ) amplitude of the data and to make an estimate of the broad - band power in the cmb power spectrum ( bond , 1994a , and steinhardt , 1994 ) . the experimental two point correlation function is given by @xmath88 where @xmath89 for a spherical harmonic expansion of the radiation temperature given by @xmath90 , @xmath91 is the window function ( eq . 1 ) and @xmath92 @xmath93 ( mather et al . 1994 ) . the rms amplitude is calculated from @xmath94 where @xmath95 and @xmath96 is the average window function at zero lag . following bond , 1994a , the broad band power estimate is given by @xmath97 we consider a scale invariant , @xmath98 `` flat '' radiation power spectrum given by @xmath99 where the constant of proportionality and @xmath100 are determined in the likelihood analysis and @xmath101 is the center frequency of channel k normalized to the lowest center frequency of the observation(s ) . the full covariance matrix is given by @xmath102 where @xmath103 is the data covariance matrix which is diagonal in each of the @xmath104 @xmath105 submatrices . this covariance matrix accounts for all spatial and channel - channel correlations as well as the error for each bin at each frequency . the offset and gradient subtraction are taken into account by creating a projected covariance matrix given by @xmath106 , where @xmath55 is defined in section 4 . the likelihood of the data set is proportional to @xmath107 , where @xmath108 for @xmath60 degrees of freedom . table 1 lists the resulting rms and band power estimates for the individual observations as well as the combined observations . figure 2 shows the band power estimates with @xmath109 in relation to the cobe band power estimate . the q - band broad band power is consistent with standard cdm normalized to cobe and has a spectral index , @xmath100 , which is consistent with the cmb spectrum . the ka - band observation is also consistent with standard cdm normalized to cobe ; however , the spectral index does not rule out discrete radio sources and thus does not afford such a straightforward interpretation . when the ka and q - band data are combined , the broad band power is consistent with the standard cdm model normalized to cobe and the spectral index is consistent with the cmb spectrum . for the standard cdm power spectrum , the amplitude of the combined ka+q observations corresponds to a @xmath110 @xmath9k . the combined ka+q data show a marked improvement on the 1 @xmath6 confidence interval for the best fit @xmath111 for the most probable @xmath112 , spectra with @xmath113 ( such as diffuse synchrotron and free - free emission ) are formally excluded at the 5 @xmath6 level . the sp94 results are most easily compared to the sp91 results and the results of wollack et al . 1993 ( sk93 ) . a reanalysis ( bond , 1994b ) of the combined sp91a and sp91b observations gives @xmath114 which compares to the ka - band sp94 results of @xmath115 . the bond , sp91 analysis assumes that @xmath109 and that there is no channel to channel correlations . the sk93 result gives @xmath116 , for @xmath117 . the sk93 result is consistent with both the sp94 and the combined sp91 results , although one should note the differences between the experiments which are addressed in sk93 . we would like to thank m. pospieszalski , m. balister , and w. lakatosh of nrao - cdl for useful information regarding amplifiers and for supplying the 26 - 36 ghz hemt amplifier . in addition we would like to thank l. nguyen of hughes research labs for providing the inp hemts which we ve incorporated into our 38 - 45 ghz amplifiers . this project would have been impossible without the support of b. sadoulet and the center for particle astrophysics . d. fischer and the whole antarctic support associates staff provided the valuable support and expertise at the amundsen - scott south pole station . we are grateful to n. sugiyama for providing us with the cdm radiation power spectrum and to r. bond , m. srednicki , p. steinhardt , m. white , and l. page for useful discussions regarding data analysis and window functions . we would like to acknowledge the previous contributions of j. schuster . n. figueiredo is partially supported by conselho nacional de desenvolvimento cientifico e tecnologico , brazil . this work was supported by nsf grant opp 92 - 21468 and ast 91 - 20005 . bond , j. r. , et al . 1991 , phys . lett . , 66 , 2179 bond , j. r. 1994a , cita-94 - 5 , preprint . 1994b , ( private communication ) bunn , e , et al . 1994 , apj , 429 , l1 gaier , t. , et al . 1992 , apj , 398 , l1 gorski , k. m. , et al . 1994 , apj , 430 , l89 gundersen , j. o. , et al . 1994 , proc . of the case western cmb workshop haslam , c. g. t. , et al . 1982 , a&as , 47 , 1 lubin , p. m. , 1994 , proc . of iau 168 mather , j. , 1994 , apj , 420 , 439 meinhold , p. , & lubin , p. 1991 , apj , 370 , l11 meinhold , p. , et al . 1993 , apj , 406 , 12 pospieszalski , m. w. , gallego , j. d. , & lakatosh , w. j. 1990 , ieee mtt - s digest 1253 pospieszalski , m. w. , et al . 1994 , proc . of the international microwave symosium , 1345 schuster , j. , 1993 , apj , 412 , l47 smoot , g. f. , et al . 1992 , apj , 396 , l1 steinhardt , p. j. 1994 , proc . of the 1994 snowmass workshop on particle astrophysics and cosmology , eds . e. kolb and r. peccei sugiyama , n. 1994 , private communication white , m. & srednicki , m. 1994 , apj , in press ( astro - ph/9402037 ) wilkinson , d. 1994 , proc . of the 1994 lake louise winter institute wollack , e. j. , et al . 1993 , apj , 419 , l49 wright , a. e. , & otrupcek , r. e. , eds . 1990 , atnf , `` pkscat90-the southern radio database '' ( sydney : atnf )
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we present results from two observations of the cosmic microwave background ( cmb ) performed from the south pole during the 1993 - 1994 austral summer .
each observation employed a 3@xmath0 peak to peak sinusoidal , single difference chop and consisted of a @xmath1 strip on the sky near our sp91 observations .
the first observation used a receiver which operates in 3 bands between 38 and 45 ghz ( q - band ) with a fwhm beam which varies from @xmath2 to @xmath3 .
the second observation overlapped the first observation and used a receiver which operates in 4 bands between 26 and 36 ghz ( ka - band ) with a fwhm beam which varies from @xmath4 to @xmath5 .
the ka - band system has a similar beamsize and frequency coverage as the system used in our sp91 results .
significant correlated structure is observed in all bands for each observation .
the spectrum of the structure is consistent with a cmb spectrum and is formally inconsistent with diffuse synchrotron and free - free emission at the 5 @xmath6 level .
the amplitude of the structure is inconsistent with 20 k interstellar dust ; however , the data do not discriminate against flat or inverted spectrum point sources .
the root mean square amplitude ( @xmath7 ) of the combined ( ka+q ) data is @xmath8 @xmath9k for an average window function which has a peak value of @xmath10 at @xmath11 and drops to @xmath12 of the peak value at @xmath13 and @xmath14 .
a band power estimate of the cmb power spectrum , @xmath15 , gives @xmath16 .
the band power estimates for the individual ka and q - band results are larger than but consistent with the band power estimate of the combined ka - band sp91 results .
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uncertainty relations are a key item of the quantum theory . this is from fundamental reasons , but also regarding practical applications , since phase - number uncertainty relations are the heart of the quantum limits to the precision of signal detection schemes @xcite . typically , uncertainty relations are expressed in terms of variances and are derived directly from the heisenberg form of commutation relations . however , this approach is not always useful . on the one hand , variance may not be a suitable uncertainty measure . this is specially clear regarding periodic phase - angle variables @xcite . on the other hand , the phase may not admit a simple well - behaved operator description suitable to obey a heisenberg form of commutation relations with the number operator @xcite . this has lead to the introduction of alternative uncertainty relations @xcite , some of them involving characteristic functions @xcite . in this regard , a recent work has proposed an uncertainty relation for position and momentum based on characteristic functions , which is derived directly from the weyl form of commutation relations @xcite . in this work we translate this approach to phase - number variables . despite the problems that quantum phase encounters , a very fundamental approach admits without difficulties the weyl form of commutation relations and has well - defined characteristic functions . therefore , the approach in ref . @xcite is a quite interesting formulation particularly suited to phase - angle variables . we also show that this encounters fundamental ambiguities when contrasting different slightly different alternative implementations , as it also holds for other approaches @xcite . let us point out that the weyl form is equivalent to say that every system state experiences a global phase shift after a cyclic transformation in the corresponding phase space . this implies that the quantum structure including uncertainty relations might be traced back to a geometric phase @xcite . let us consider general systems describable in a finite - dimensional space as a spin @xmath0 . this admits very general scenarios , including especially the phase difference between two modes of the electromagnetic field . this is because the total number of photons @xmath1 is compatible with the phase difference and defines finite - dimensional subspaces of dimension @xmath2 , where @xmath1 plays the role of the spin modulus as @xmath3 @xcite . let us focus on a spin component @xmath4 and the canonically conjugate phase @xmath5 . to avoid periodicity problems we focus on the complex exponential of @xmath5 , we shall call @xmath6 , this is @xmath7 . the eigenvectors @xmath8 can be referred to as phase states @xcite , being latexmath:[\[| \tilde{m } \rangle = \frac{1}{\sqrt{2j+1 } } \sum_{m =- j}^j e^ { -i \frac{2 \pi}{2j+1 } m \tilde{m } } @xmath4 , as usual @xmath10 , and @xmath11 . likewise , we may define the exponential of @xmath4 as @xmath12 these exponentials @xmath6 and @xmath13 are quite suited to the weyl form of commutation relation @xcite @xmath14 for any @xmath15 . it is worth noting that the weyl form has a quite interesting meaning when expressed as @xmath16 this represents a cyclic transformation in the form of a closed excursion over a @xmath17 rectangle in the associated phase space for the problem . the result is that every system state acquires a global phase after returning to the starting point . following ref . @xcite we can construct the gram matrix @xmath18 for the following three vectors @xmath19 where @xmath20 is an arbitrary state assumed pure for simplicity and without loss of generality , so that @xmath21 involving the characteristic functions @xmath22 and @xmath23 which is the term invoking the weyl commutator ( [ ws ] ) . after the positive semi - definiteness of @xmath18 we get @xmath24 where @xmath25 . from this point we can follow exactly the same steps in ref . these involve to construct another gram matrix after replacing @xmath26 by @xmath27 and @xmath28 by @xmath29 , adding the two determinants , using eq . ( [ ws ] ) and then following some clever simple algebraic bounds . this leads to @xmath30 with @xmath31 therefore , most of the analysis and results found in ref . @xcite could be translated here . even , the limit of vanishing argument of the characteristic functions may be reproduced in the limit of very large @xmath0 . besides the sums , uncertainty relations can be also formulated as the products of uncertainty estimators . in our case from eq . ( [ urs ] ) we can readily derive a bound for the product of characteristic functions @xmath32 a rather interesting point is that this can lead to conclusions fully opposite to the sum relation ( [ urs ] ) . this is specially so regarding the minimum uncertainty states , as we shall clearly show by some examples below . the smallest value for the bound @xmath33 is obtained for @xmath34 . in such a case , the sum of the two gram matrices commented above leads directly to the uncertainty relation @xmath35 where the @xmath36 term is expressing phase - number correlations that in standard variance - based approaches is expressed by the the anti - commutator . if this correlation term is ignored we get the more plain relations : @xmath37 let us note that these relations might be called _ certainty _ instead of _ uncertainty _ relations since we get upper bounds for characteristic functions , that take their maximum value when there is full certainty about the corresponding variable . the most simple and illustrative example is provided by the case @xmath38 . the most general state is of the form @xmath39 where @xmath40 are the pauli matrices , @xmath41 is the identity , and @xmath42 is a three - dimensional real vector with @xmath43 . we can chose the basis so that @xmath44 and @xmath45 . the only nontrivial uncertainty relation holds for @xmath46 so that @xmath34 , @xmath47 @xmath48 and eqs . ( [ urs ] ) , ( [ urt ] ) , and ( [ urp ] ) become , respectively @xmath49 actually , @xmath50 is a well - known duality relation expressing complementarity @xcite . the minimum uncertainty both for eqs . ( [ urs ] ) and ( [ urt ] ) holds for every pure state @xmath51 with @xmath52 . turning our attention to the alternative product of characteristic functions in eq . ( [ urp ] ) we get that the minimum uncertainty states are those pure states with @xmath52 and @xmath53 . on the other hand , the states with @xmath52 and @xmath54 or @xmath55 are of maximum uncertainty , contrary to the predictions of the sum relations ( [ urs ] ) and ( [ urt ] ) . next we address the case of the number and phase for a single field mode . there is always the possibility of addressing this from the number and phase difference taking a suitable reference state in one of the modes @xcite , but the direct approach has also its advantages . one of them is that it faces the fact that , roughly speaking , there is no phase operator . the exponential of the phase is not unitary , but represented instead by the one - sided unitary susskind - glogower operator @xcite @xmath56 where @xmath57 are the eigenstates of the number operator @xmath58 , and @xmath59 are the phase states @xmath60 where @xmath61 is the orthogonal projector on the subspace with less than @xmath26 photons @xmath62 and @xmath63 is the identity . this does not prevent the existence of a proper probability distribution for the phase in any field state @xmath64 . this can be defined thanks to the phase states ( [ ps ] ) as @xmath65 , that lead to the characteristic function @xmath66 where the last equality holds because for all @xmath26 @xmath67 in spite of the fact that the phase states are not orthogonal . this is to say that the lack of unitarity is equivalent to a description of phase in terms of a positive - operator measure . despite the lack of unitarity of @xmath68 there is also a suitable weyl form of commutation relations @xmath69 since the operator @xmath6 is not unitary , the change of @xmath26 by @xmath27 is not trivial , so in order to follow the procedure in ref . @xcite we have to construct explicitly the two gram matrices . the first one for the vectors @xmath70 is @xmath71 with @xmath72 being in this case @xmath73 @xmath25 , and @xmath74 the second gram matrix corresponds to the change of @xmath5 by @xmath75 and @xmath26 by @xmath27 , so the three vectors are now @xmath76 leading to @xmath77 where eqs . ( [ wpn ] ) and ( [ lu ] ) have been used , being @xmath78 and the same @xmath79 , @xmath80 and @xmath36 in eqs . ( [ phitphi ] ) and ( [ om ] ) . this leads to the determinant @xmath81 for the same @xmath82 above . at this point several routes can be followed . for definiteness from now on we will focus always in the most stringent scenario of @xmath83 . in such a case we readily get from the sum of eqs . ( [ detg+ ] ) and ( [ detg- ] ) the following bound : @xmath84 the lack of unitarity of the exponential of the phase reflects in the presence of the @xmath85 term . thus , whenever this term is absent @xmath86 we recover the same expressions obtained for the spin - like systems . otherwise , this term might be also moved to the right - hand side of the relation meaning that the nonunitarity implies a lower upper bound in accordance with the noisy nature of positive - operator measures . for definiteness , on what follows we will consider the following forms @xmath87 and @xmath88 as well as the product @xmath89 readily simple examples are provided by the eigenstates of @xmath58 and @xmath6 . for the number states @xmath57 we get for all @xmath5 , @xmath26 and @xmath90 that @xmath91 , @xmath92 , and there is no effect of the @xmath85 term . thus these are minimum uncertainty states . note that we have the opposite conclusions regarding the uncertainty product . on the other hand , the phase states @xmath93 do not provide a suitable example since they are not normalizable . instead , we can use their normalized counterparts , that are also eigenstates of @xmath6 @xmath94 . these states can be suitably approached in practice via quadrature squeezed states @xcite . in this case it can be readily seen that @xmath95 and @xmath96 in fig . 1 we have represented the combinations @xmath97 , @xmath98 and @xmath99 in eqs . ( [ up ] ) and ( [ upp ] ) as functions of @xmath100 for @xmath101 and @xmath102 . the minima of these functions represent maximum uncertainty and hold for phase states with very small mean number of photons , i. e. , @xmath103 , and 1.3 , for @xmath97 , @xmath98 , and @xmath99 , respectively . on the other hand , when @xmath104 , this is when @xmath105 , we get @xmath106 , @xmath107 , and @xmath108 , as expected for ideal phase states , becoming minimum uncertainty states . ( solid ) , @xmath98 ( dashed ) , and @xmath99 ( dotted ) as functions of @xmath109 for @xmath101 and @xmath102 for the phase states ( [ pc]).,width=226 ] however when considering the product @xmath110 in eq . ( [ v ] ) again for @xmath101 and @xmath102 it can be easily seen after eq . ( [ pc1 ] ) that when @xmath104 and @xmath111 we get maximum uncertainty @xmath112 , while @xmath110 attains its maximum value ( i. e. , minimum uncertainty ) , @xmath113 , when @xmath114 , this is @xmath115 . thus we see another clear example where maximum and minimum uncertainty states exchange their roles depending on the assessment of joint uncertainty considered . states with gaussian statistics are usually minimum uncertainty states in typical variance - based uncertainty relations . then it is worth examining the case in which the number statistics can be approximated by a gaussian distribution . this will work provided that the distribution is concentrated in large photon numbers and that it is smooth enough so that the number @xmath90 can be treated as a continuous variable . thus let us consider a pure state @xmath116 with @xmath117 , \ ] ] where @xmath118 represents the mean number , @xmath119 is given by the inverse of the number variance @xmath120 , and @xmath121 provides phase - number correlations taking positive as well as negative values . consistently with the above approximations we shall consider @xmath122 as well as @xmath123 this situation includes the glauber coherent states @xmath124 for large enough mean photon numbers @xmath125 with @xmath126 . throughout @xmath127 will be assumed . in these conditions we readily get @xmath128 , \ ] ] and @xmath129 with @xmath130 . the first thing we can notice is that @xmath121 increases phase uncertainty . let us begin with the simplest case @xmath126 . we can focus first on the plain sum relation @xmath97 in eq . ( [ up ] ) , which is plotted in fig . 2 in solid line as a function of @xmath119 . we can see that @xmath131 is just a function of @xmath132 and that minimum uncertainty , this is maximum @xmath97 , holds for @xmath132 tending both to 0 and infinity : this is when the state tends to be phase or number state , respectively , in accordance with the above results . in between we get a maximum uncertainty state , i. e. , minimum @xmath97 , when @xmath133 that correspond to @xmath134 , this is uncertainty equally split between phase and number . similar results are obtained for @xmath98 in the same eq . ( [ up ] ) , as shown in fig . 2 in dashed line . ( solid ) and @xmath98 ( dashed ) as functions of @xmath135 for gaussian states with @xmath126.,width=226 ] on the other hand , the situation is quite the opposite for the certainty product @xmath110 in eq . ( [ v ] ) : we have maximum uncertainty @xmath112 for phase and number states @xmath136 , while we have minimum uncertainty , this is maximum @xmath110 , for @xmath137 . this is just the opposite of the conclusion of the sum of characteristics . for the case @xmath138 in fig . 3 we have plotted the certainty sums @xmath97 and @xmath98 as functions of @xmath121 for @xmath139 and @xmath101 , showing that from @xmath126 increasing @xmath121 increases uncertainty until reaching @xmath140 where a revival of @xmath98 is produced , reaching the same certainty values around @xmath126 . regarding the product @xmath110 we have the same behavior of the case @xmath126 but the minimum uncertainty state holds for @xmath141 . ( solid ) and @xmath98 ( dashed ) as functions of @xmath121 for @xmath101 and gaussian states with @xmath139.the two lines overlap around @xmath126.,width=226 ] looking for states with interesting phase - number relations we may consider the eigenstates of @xmath143 , where @xmath144 is a real parameter @xcite : @xmath145 that has the following solution , for @xmath146 for definiteness , @xmath147 are the corresponding modified bessel functions . it could be interesting to apply the previous approach to these states ( [ i1 ] ) looking for the @xmath144 that lead to minimum uncertainty . easily we obtain : @xmath148 and @xmath149 as before , we focus on the case @xmath101 and @xmath150 . performing the numerical computation , we obtain plots of @xmath97 and @xmath151 in eq . ( [ up ] ) similar to the ones obtained in previous cases , as we can see in fig . the maximum uncertainty states are given by the minimum value of @xmath97 and @xmath151 for a given @xmath26 . in the present case the values of @xmath144 which minimizes these functions are @xmath152 with @xmath153 , and @xmath154 with @xmath155 , respectively . here again we obtain opposite results for the certainty product @xmath110 in eq . ( [ v ] ) . ( solid ) and @xmath98 ( dashed ) as functions of @xmath144 for the states ( [ i1]).,width=226 ] joint uncertainty relations of two observables are often minimized by states with properties somewhat intermediate between the two observables . the simplest case is a readily coherent superposition of number and phase states of the form @xmath156 we focus on the case @xmath157 and @xmath158 with the idea that @xmath159 approaches the ideal phase states ( [ ps ] ) . in such a case it can be seen that the normalization condition is just @xmath160 and we shall consider @xmath101 and @xmath102 . thus , eq . ( [ com ] ) reads @xmath161 minimum uncertainty holds just in the limiting cases @xmath162 and @xmath163 , recovering the cases of phase and number states . on the other hand maximum uncertainty holds for the intermediate state @xmath164 . clearly , the situation is reversed if we consider the product @xmath110 so that maximum and minimum uncertainty are exchanged . we have successfully derived meaningful phase - number uncertainty relations from the weyl form of commutation relations . this can be applied to study phase - number statistical properties of meaningful field states , especially intermediate states that have already demonstrated interesting properties regarding uncertainty relations @xcite . moreover , this can be a suitable tool to explore quantum metrology limits . in typical interferometry @xmath58 is the generator of phase shifts . thus the characteristic function @xmath79 is actually expressing the distinguishability of the probe state before and after a phase shift , which should be naturally related to detection resolution . therefore this uncertainty relations may be connected to optimized signal detection schemes @xcite . we are grateful to ukasz rudnicki for stimulating discussions , ivn lvarez domenech for his selfless help , and demosthenes ellinas for valuable comments . g. d. gratefully thanks a collaboration grant from the spanish ministerio de educacin , cultura y deporte . l. thanks support from project fis2012 - 35583 of spanish ministerio de economa y competitividad and from the comunidad autnoma de madrid research consortium quitemad+ s2013/ice-2801 . e. breitenberger , uncertainty measures and uncertainty relations for angle observables , found . phys . * 15 * , 353 ( 1985 ) ; t. opatrn , mean value and uncertainty of optical phase - a simple mechanical analogy , j. phys . a * 27 * , 7201 ( 1994 ) . j. c. garrison and j. wong , canonically conjugate pairs , uncertainty relations and phase operators , j. math . phys . * 11 * , 2242 ( 1970 ) ; a. galindo , phase and number , lett . phys . * 8 * , 495 ( 1984 ) ; * 9 * , 263 ( 1984 ) . j.m . lvy - leblond , who is afraid of nonhermitian operators ? a quantum description of angle and phase , ann . ( n.y . ) * 101 * , 319 ( 1976 ) ; m. grabowski , spin phase , int . j. theor * 28 * , 1215 ( 1989 ) ; on the phase operator , rep * 29 * , 377 ( 1991 ) ; j. bergou and b .- g . englert , operators of the phase fundamentals , ann . ( n. y. ) * 209 * , 479 ( 1991 ) ; r. lynch , the quantum phase problem : a critical review , phys . * 256 * , 367 ( 1995 ) ; v. peinov , a. luk and j. peina , _ phase in optics _ ( world scientific , singapore , 1998 ) . a. s. holevo , in _ quantum probability and applications to the quantum theory of irreversible processes _ , springer lecture notes in mathematics vol . 1055 , edited by l. accardi , a. frigerio , and v. gorini ( springer , berlin , 1984 ) , p. 153 . a. luk and v. penov , number - phase uncertainty products and minimizing states , phys . a * 45 * , 6710 ( 1992 ) ; a. luk , v. penov , and j. k epelka , special states of the plane rotator relevant to the light field , _ ibid_. * 46 * , 489 ( 1992 ) . a. luis , complementarity and certainty relations for two - dimensional systems , phys . a * 64 * , 012103 ( 2001 ) ; complementarity and duality relations for finite - dimensional systems , * 67 * , 032108 ( 2003 ) . a. luis , effect of fluctuation measures on the uncertainty relations between two observables : different measures lead to opposite conclusions , phys . a * 84 * , 034101 ( 2011 ) ; s. zozor , g.m . bosyk , and m. portesi , on a generalized entropic uncertainty relation in the case of the qubit , j. phys . a * 46 * , 465301 ( 2013 ) ; general entropy - like uncertainty relations in finite dimensions , * 47 * , 495302 ( 2014 ) . a. luis and l. l. snchez - soto , phase difference operator , phys . a * 48 * , 4702 ( 1993 ) ; quantum phase difference , phase measurements and stokes operators , _ progress in optics _ , edited by e. wolf ( elsevier , amsterdam , 2000 ) * 41 * , 421 ( 2000 ) . t. s. santhanam , canonical commutation relation for operators with bounded spectrum , phys . a * 56 * , 345 ( 1976 ) ; i. goldhirsch , phase operator and phase fluctuations of spins , j. phys . a * 13 * , 3479 ( 1980 ) ; a. vourdas , su(2 ) and su(1,1 ) phase states , phys . rev . a * 41 * , 1653 ( 1990 ) ; d. ellinas , phase operators via group contraction , j. math . phys . * 32 * , 135 ( 1991 ) . h. weyl , _ theory of groups and quantum mechanics _ ( dover , new york , 1950 ) ; t. s. santhanam and a. r. tekumalla , quantum mechanics in finite dimensions , found . * 6 * , 583 ( 1976 ) ; d. ellinas , quantum phase angles and su(@xmath165 ) , j. mod . opt . * 38 * , 2393 ( 1991 ) ; su(2 ) action - angle varaibles , arxiv : hep - th/9305153 ; a. vourdas and c. bendjaballah , duality , measurements , and factorization in finite quantum systems , phys . a * 47 * , 3523 ( 1993 ) . g. jaeger , a. shimony , and l. vaidman , two interferometric complementarities , phys . a * 51 * , 54 ( 1995 ) ; b .- englert , fringe visibility and which - way information : an inequality , phys . lett . * 77 * , 2154 ( 1996 ) . m. ban , number - phase quantization in ultra - small tunnel junctions , phys . a * 152 * , 223 ( 1991 ) ; phase state and phase probability distribution , opt . commun . * 94 * , 231 ( 1992 ) ; phase operator in quantum optics , phys . lett . a * 176 * , 47 ( 1993 ) .
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we derive suitable uncertainty relations for characteristics functions of phase and number variables obtained from the weyl form of commutation relations .
this is applied to finite - dimensional spin - like systems , which is the case when describing the phase difference between two field modes , as well as to the phase and number of a single - mode field . some contradictions between the product and sums of characteristic functions are noted .
| 6,732 | 97 |
the solutions of the helmholtz equation for the right isosceles triangle with sidelength , @xmath3 ( chosen for convenience ) are given by @xmath4 @xmath5 . this consists of two terms , each being a product of @xmath6 functions . of course , it can be re - written in a variety of equivalent ways by employing trigonometric identities . with just one term of a product of sine functions , the nodal lines are straight lines and they form a checkerboard pattern . this would be the case also for a product of any other special function . + , ( b ) @xmath7 and ( c ) @xmath8 . all three eigenfunctions belong to the same equivalence class @xmath9 $ ] and the similarity of the nodal pattern is evident as the wavefunction evolves from one state to another within members of the same class.,title="fig:",height=124 ] ( a ) , ( b ) @xmath7 and ( c ) @xmath8 . all three eigenfunctions belong to the same equivalence class @xmath9 $ ] and the similarity of the nodal pattern is evident as the wavefunction evolves from one state to another within members of the same class.,title="fig:",height=124 ] ( b ) , ( b ) @xmath7 and ( c ) @xmath8 . all three eigenfunctions belong to the same equivalence class @xmath9 $ ] and the similarity of the nodal pattern is evident as the wavefunction evolves from one state to another within members of the same class.,title="fig:",height=124 ] ( c ) for instance , the solutions of the helmholtz equation for a circular , elliptical , circular annulus , elliptical annulus , confocal parabolic enclosures are each a product of functions like bessel for circular , mathieu for elliptic and so on @xcite . + eq . ( [ eq : iso ] ) can be rewritten in a way that will be more useful : @xmath10 \nonumber \\ & = & \frac{1}{2 } \re { \rm tr~ } \left[\begin{array}{cc } \{e^{i(mx - ny)}-e^{i(mx+ny)}\ } & 0\\ 0 & \{-e^{i(my - nx)}+e^{i(my+nx)}\ } \end{array}\right ] \nonumber \\ & : = & \frac{1}{2 } \re { \rm tr~ } { \mathcal i}. \end{aligned}\ ] ] all the eigenfunctions can be classified into equivalence classes labelled by @xmath11 @xcite . within each class , it was shown that the number of domains , @xmath12 for one eigenfunction is related to @xmath13 by a difference equation @xcite . we can , in fact , write down the operator ( in the matrix form ) which actually takes us along the ladder of states beginning with @xmath14 , up and down . the matrix is @xmath15.\ ] ] to confirm , we get the eigenfunction @xmath16 as @xmath17 thus , we have generated all the states beginning anywhere ; note that @xmath18 could be any integer as long as we keep the inequality between the two quantum numbers . the eigenfunctions of an equilateral triangle of side length @xmath3 , satisfying the dirichlet boundary conditions , can be written as three terms , each a product of trigonometric functions @xcite . there are two possible solutions - one with cosine and th other with sine functions . first we discuss the function with cosines : @xmath19 this can be re - written as @xmath20 \nonumber \\ & = & \im \frac{1}{2}{\rm tr~}{\mathcal a}\end{aligned}\ ] ] where @xmath21 is @xmath22\end{aligned}\ ] ] the matrix operator for this state is @xmath23\ ] ] similarly for the eigenfunctions written in terms of sine functions , @xmath24 in complex form , it can be re - written as @xmath25\end{aligned}\ ] ] and in matrix form as @xmath26.\ ] ] where @xmath27 is @xmath28\ ] ] the corresponding matrix operator is @xmath23\ ] ] this operator is the same as for the cosine form of the eigenfunctions for equilateral triangle billiard . the eigenfunctions of separable billiards are a single product of special functions - trigonometric for rectangular billiard , bessel and trigonometric functions for circular billiards ( and related annuli ) , mathieu and trigonometric functions for elliptical billiards ( and annuli ) , and parabolic cylinder functions for confocal parabolic billiards . in all these cases , the tower of states can be trivially constructed along the lines described here . this is because the index that classifies states for all separable billiards is ( @xmath29 ) . for the non - separable billiards described here , we have shown in earlier papers that all the states can be classified by ( @xmath30 ) or ( @xmath31 ) . here , we have shown that within a class , all the states can be constructed from the energetically lowest state . we can also make a transformation from an excited state to the lowest state . we hesitate to call this a ` ground state ' as there will be one lowest state for an index , @xmath32 , @xmath33 . the results given here are for billiards with dirichlet boundary conditions . of course , these results are trivially extended to the case of periodic boundary conditions . the raising and lowering operators will remain the same . for twisted boundary conditions , these may be generalized by introducing phases in the matrix representation of raising and lowering operators .
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for planar integrable billiards , the eigenstates can be classified with respect to a quantity determined by the quantum numbers .
given the quantum numbers as @xmath0 , the index which represents a class is @xmath1 for a natural number , @xmath2 .
we show here that the entire tower of states can be generated from an initially given state by application of the operators introduced here .
thus , these operators play the same role for billiards as raising and lowering operators in angular momentum algebra
. quantum billiards are systems where a single particle is confined inside a boundary on which the eigenfunctions vanish @xcite .
one seeks the solutions of the time - independent schrdinger equation , which is the same as the helmholtz equation in the context of general wave phenomena .
the solutions of this problem for an arbirarily shaped enclosure is a very challenging open problem , even when we restrict ourselves to two - dimensional cases @xcite .
there are some very interesting connections between exactly solvable models and random matrix theories , a summary may be seen in @xcite .
the helmholtz operator is separable in certain coordinate systems - for these cases , the solutions can be found @xcite . the non - separable problems for which the classical dynamics is integrable have been recently studied in detail @xcite .
although the solutions of these systems have been known , there remain many questions regarding the nature of nodal curves and domains .
the nodal domains of the eigenfunctions of the systems for which the schrdinger equation is separable , form a checkerboard pattern @xcite .
the number of crossings actually count the number of domains .
moreover , the checkerboard patterns are trivially self - similar . counting the nodal domains of non - separable plane polygonal billiards
is very difficult in general @xcite .
even if we restrict to systems that are classically integrable , the problem poses considerable challenge .
progress on this otherwise intractable problem could be made recently due to the observation that the eigenfunctions could be classified in terms of equivalence classes @xcite .
fig .
[ fig : iso ] shows examples of eigenfunctions belonging to an equivalence class in the right isosceles triangle billiard .
one can not miss the remarkable similarity in each family , they seem genetically related . here we shall present operators that make any other state appear starting from one in a family .
thus , we can construct the tower of states by repeated application of this operator .
this reminds us of the usual raising and lowering operators in quantum mechanics .
we explain in the following sections the construction of raising " and lowering " operators for the right isosceles and equilateral triangle billiard , and summarize with remarks about other systems .
| 1,635 | 695 |
high precision wavelength calibrators for astrophysical spectrographs will be key components of new precision radial velocity ( rv ) observations , including the search for earth - like extra - solar planets ( exoplanets ) @xcite and direct observation of cosmological acceleration @xcite . recent work has demonstrated the potential of octave - spanning femtosecond laser frequency combs @xcite ( astro - combs " ) to serve as wavelength calibrators for astrophysical spectrographs providing rv sensitivity down to 1 cm / s @xcite . exoplanet searches place stringent demands upon such calibrators . for example , the rv amplitude of the reflex motion of a solar - mass star induced by an earth - mass planet in an orbit within the habitable zone is about 10 cm / s . the current state of the art astrophysical spectrograph , harps , has demonstrated stellar rv sensitivity @xmath2 cm / s @xcite , largely limited by its thorium argon lamp calibrator @xcite . these calibrators are limited by their unevenness in line spacing and intensity as well as the slow variation of their line wavelengths with time . an astro - comb provides emission lines with uniform intensity and controllable spacing , which can be referenced to atomic frequency standards and the global positioning system ( gps ) , yielding excellent long - term stability and reproducibility . to date , astro - combs consist of a combination of an octave - spanning femtosecond laser frequency comb ( source comb ) and a fabry - prot cavity ( fpc ) , see fig . [ fig : astro - comb setup ] . spectral lines generated by the source comb are spaced by the pulse repetition rate ( @xmath3 ) , currently @xmath4 ghz , which results in a line spacing too dense to be resolved by broadband astrophysical spectrographs @xcite . the fpc serves as a mode filter with a free spectral range ( fsr ) set to an integer multiple of the repetition rate , fsr=@xmath5 , with @xmath6 , depending on the spectrograph resolution . ideally , the fpc passes only every @xmath7 source comb spectral line , providing thousands of calibration lines well matched to a practical spectrograph s resolution , with fractional frequency uncertainty limited only by the stability of the rf reference used to stabilize the source comb and fpc . this frequency uncertainty can be @xmath8 using commonly available atomic clock technology , which corresponds to @xmath9 khz uncertainty in the optical frequency or 0.3 cm / s in rv precision . however , because the spectrograph fails to resolve neighboring source comb lines , finite suppression of these neighboring lines by the fpc affects the lineshape and potentially the centroid of measured astro - comb lines . for example , in the results presented here , source comb modes neighboring the astro - comb line , with intensities after passing through the fpc that differ by 0.1% of the main astro - comb peak , shift the measured line centroid by 1 mhz , which corresponds to an rv systematic error of 1 m / s . in practice , such systematic rv shifts are inevitable over spectral bandwidths of 1000 due to the dispersive properties of the mirrors of the fpc . although these systematic shifts can be very stable over timescales of years , the correction of such shifts is crucial to high accuracy astrophysical spectroscopy . in this paper , we demonstrate an _ in - situ _ technique to determine the systematic shifts of astro - comb lines due to fpc dispersion , which can be applied at a telescope - based spectrograph to enable wavelength calibration accuracy better than 10 cm / s . by measuring the intensity of astro - comb lines as the fpc length is adjusted , we determine ( i ) the offset of each fpc resonance from the nearest source comb line ; ( ii ) fpc finesse as a function of wavelength ; and ( iii ) the intensity of the astro - comb lines and their neighboring ( normally suppressed ) source comb lines . these parameters can be determined quickly and reliably over the full 1000 bandwidth of the astro - comb with only @xmath10 measurements at slightly different fpc lengths , and can be performed quickly ( @xmath11 hour ) and reliably . the measurement has also been performed with a lower resolution commercial optical spectrum analyzer with consistent results . the astro - comb line characterization technique presented here builds on past work in which femtosecond lasers coupled to swept cavities were used to study both the medium in which the cavity was immersed and the cavity mirrors @xcite . imperfect suppression of unwanted source comb lines , e.g. , due to finite fpc finesse , affects the astro - comb lineshape observed on a spectrograph . the lineshape can be modeled with knowledge of the fpc properties including mirror reflectivity and round trip phase delay . the intensity of a source comb line after the fpc is @xmath12 where @xmath13 is the intensity of the source comb line of optical frequency @xmath14 ; @xmath15 is the resonant transmission of the fpc at optical frequency @xmath14 ; @xmath16 is the finesse of the fpc near frequency @xmath14 ; and @xmath17 is the round trip phase delay . the phase delay may be expressed as @xcite @xmath18 where @xmath19 is the length of the cavity , @xmath20 is the speed of light in vacuum , @xmath21 is the refractive index of the medium inside the cavity ( air , vacuum , etc . ) at optical frequency @xmath14 , and @xmath22 is the frequency - dependent group delay of the mirrors . the first term in parentheses in eq . ( [ eqn : phi_0_f ] ) is the distance between the mirrors expressed in wavelengths ; while the second term , the integral of @xmath22 , is the phase delay of the mirrors and represents the frequency - dependent penetration distance of light into the mirror . maximum transmission occurs when a source comb line is resonant with the fpc , or equivalently @xmath23 , with @xmath24 an integer ( see fig . [ fig : comb_fp]a ) . the fsr , @xmath25 , is the frequency difference between two consecutive fpc resonant frequencies . assuming that @xmath26 varies slowly with frequency , the fsr can be approximated by @xmath27^{-1}.\ ] ] for the astro - comb shown in fig . [ fig : astro - comb setup ] , which we deployed as a wavelength calibrator for the tillinghast reflector echelle astrophysical spectrograph ( tres ) @xcite , typical operational parameters are @xmath28 ghz and @xmath29 , with source comb lines nearest to the astro - comb peak suppressed by @xmath30 db ( see fig . [ fig : comb_fp]b and section [ section : experiment ] ) . this imperfect suppression leads to systematic inaccuracies in astro - comb line centers at the 50 cm / s level , as observed on the tres spectrograph across a 1000 bandwidth . the effects of these systematic shifts can be characterized by determining the fpc finesse ( @xmath16 ) and the frequency difference between astro - comb lines and the fpc resonance over the full spectral width . as described below , compensation can then be applied to determine wavelength solutions ( conversions from spectrograph pixel number to wavelength ) with accuracy at least an order of magnitude below the 50 cm / s level of these systematic shifts . c + we determine @xmath16 and the frequency offset of the fpc resonance from the astro - comb lines by adjusting the length of the fpc and measuring the intensities of astro - comb lines over the full spectral bandwidth . when we change the frequency of one fpc resonance from @xmath31 to @xmath32 the cavity length changes by @xmath33\delta\ ! f \\ & \approx & - \frac{c}{2n_df_d\delta_d}\delta \ ! f,\end{aligned}\ ] ] where @xmath34 and @xmath35 are the fsr of the fpc and the phase delay of the mirrors , respectively , at the frequency of the controlled resonance . the change of the cavity length leads to a change of the round trip phase delay of an astro - comb line @xmath14 by @xmath36 as a result , the intensity of the astro - comb line varies with the change of the frequency of one fpc mode according to eq . ( [ eqn : fp_transmission ] ) , from which we can derive the finesse and the phase errors at all astro - comb line frequencies . the astro - comb employed in our experimental demonstrations is shown schematically in fig . [ fig : astro - comb setup ] . the octave - spanning source comb spectrum ( fig . [ fig : source comb spectrum ] ) is generated by a mode - locked titanium - sapphire femtosecond laser ( octavius , menlosystems , inc . ) with repetition rate @xmath37 ghz . the absolute frequencies of the comb lines can be expressed as @xmath38 , where @xmath39 is an integer and @xmath40 is the carrier - envelope offset frequency . a pin diode detects @xmath3 , which is then stabilized by adjusting the laser cavity length . the @xmath41-@xmath42 self - referencing method is used to produce a signal at @xmath40 on an avalanche photodiode : comb lines around 11400 are frequency doubled in a 1 mm thick lbo ( lithium triborate ) crystal and beat with comb lines around 5700 . @xmath40 is then stabilized by intensity modulation of the 7.6 w , 5320 pump laser . both @xmath3 and @xmath40 are referenced to low - noise radio - frequency synthesizers , which are stabilized to a commercial rubidium frequency reference . for our source comb , typical values of these frequencies are : @xmath43 ghz and @xmath44 ghz , and the resulting linewidth of individual source - comb spectral lines is @xmath11 mhz . ) . the source comb consists of @xmath45 narrow lines ( width @xmath11 mhz ) , equally spaced in frequency ( @xmath46 ghz , not resolved here ) , and spanning more than an octave between 6000 and 12000 .,width=432 ] the source comb beam passes through the fpc , filtering out unwanted comb lines and increasing the transmitted line spacing . two flat mirrors with @xmath47% reflectivity and minimal group delay dispersion ( @xmath48 fs@xmath49 from 7500 to 9000 ) comprise the fpc . due to the low dispersion mirrors , the fsr of the fpc is almost constant within the 1000 bandwidth : typically one in thirty comb lines is resonant with the fpc ( fig . [ fig : comb_fp]a ) . the measured finesse of the fpc is @xmath50 , which is consistent with the theoretical limit estimated from the mirror reflectivity . we stabilize the fpc by locking one transmission resonance to an injected diode laser . the diode laser is modulated at 30 mhz with an electro - optic modulator ( eom ) and injected into the fpc with a polarization orthogonal to the comb light . after the fpc , the diode beam is separated from the comb , demodulated , and used to derive a feedback signal that stabilizes the fpc to the diode laser . the diode laser itself is offset - locked to one of the source comb lines at a wavelength @xmath51 . the frequency of the offset lock ( see fig . [ fig : comb_fp]a ) is adjusted to optimize the astro - comb bandwidth , and compensates for fpc dispersion between the diode laser wavelength and the central astro - comb wavelength due to mirror properties and air in the cavity as discussed in secs . [ s.model ] and [ s.results ] . the diode laser wavelength is close to the rb d1 transition . absorption spectroscopy of the diode laser light using a thermal rubidium vapor cell thus identifies the diode frequency to within one source comb line . the offset lock then determines the absolute frequency of the diode laser with the accuracy of the underlying atomic frequency reference . in its current incarnation as a wavelength calibrator for an astrophysical spectrograph , the astro - comb spectrum is analyzed by the tillinghast reflector echelle spectrograph ( tres ) @xcite , a fiber - fed multi - order echelle spectrograph at the 1.5 m telescope of the whipple observatory at mt . hopkins in arizona . spectral dispersion is provided by an echelle grating with cross - dispersion from a prism operated in double - pass mode . tres covers a spectral bandwidth from 3700 to 9100 with a resolving power of @xmath52 @xcite corresponding to a resolution of @xmath53 ghz at 8000 . astro - comb light is injected into an integrating sphere and then sent to tres on a 100-@xmath54 m multimode fiber . the integrating sphere reduces lineshape fluctuations due to input - dependent illumination of the spectrograph optics . each astro - comb spectral measurement by tres is typically integrated for 60 s and recorded on a cooled ( 100 k ) two dimensional ccd ( see fig . [ fig : comb_fp]b ) . flat - field correction is applied to the astro - comb spectrum in order to remove artifacts caused by variations in the pixel - to - pixel sensitivity of the detector and by distortions in the optical path . the implementation of both the integrating sphere and the flat - field correction are essential for high accuracy astrophysical spectroscopy but is not crucial to the work described in this paper . the separation of the fpc mirrors is set such that astro - comb lines are separated by @xmath55 resolution elements of the spectrograph . in the measurement of the intensity of each astro - comb lines , typically the counts from 18 ccd pixels around the central pixel are binned from the one dimensional extracted spectrograph spectrum to obtain each measurement points in fig . the 18 pixel integrations capture all the intensity from each astro - comb line ( spectrograph resolution @xmath56 pixels ) while avoiding cross - talk between neighboring lines ( line separation @xmath57 pixels ) . we have also varied the binning and found no appreciable change in the result . the length of the fpc was swept ( by changing the diode frequency ) back and forth through the fpc resonance : we took steps of 80 mhz in one direction and then steps in the reverse direction at intermediate positions . different step sizes or schemes have led to consistent results . we also characterized the astro - comb spectrum with a commercial optical spectrum analyzer ( osa , ando aq6315 ) , which has much lower resolution ( @xmath58 ghz ) when operated in broadband ( 1000 ) mode . there are then @xmath59 astro - comb lines in one resolution element . extraction of astro - comb intensities from the osa is significantly simpler than the two dimensional tres spectrograph , and the osa , therefore , provides a useful cross - check . we find that all parameters measured with the osa are consistent with those measured with tres . additionally , the calibration procedure has been performed more than 10 times with both the tres spectrograph and the osa over 10 days . all measurements are found to be consistent over this time period . we find good agreement between tres measurements of the astro - comb spectrum and the model presented above . for example , figure [ fig : data_amp_vs_phase ] shows the measured variation of the peak intensity of one astro - comb line vs diode laser frequency ( and thus cavity length ) , as well as a fit of eq . [ eqn : fp_transmission ] to the data . from the fit , we derive the astro - comb resonant intensity @xmath60 ; the fpc finesse @xmath16 ; and the phase deviation @xmath61 . the phase deviation is the offset of the round trip phase from an integer multiple of @xmath62 at an astro - comb line , and is determined from the detuning of the astro - comb peak at frequency @xmath14 from center of the fpc resonance at nominal diode laser setting @xmath63 . the phase deviation is then given by @xmath64 where @xmath65 is the free spectral range near frequency @xmath14 . the finesse , @xmath66 , of the fpc can be derived from the fit as @xmath67 where @xmath68 is the full width at half maximum in phase of an fpc resonance . the uncertainties in measured fpc resonance centers and widths are approximately 1 mhz , corresponding to phase uncertainties of 0.2 mrad . the variation of these parameters as a function of wavelength provides the information needed to characterize the astro - comb spectrum as a wavelength calibrator . to the data ( solid line ) . uncertainties in the linewidth and in the offset of the peak from zero diode detuning are approximately 1 mhz . , width=4 ] as shown in fig . [ fig : fitted_phase_error ] , the fpc finesse and phase deviation are found to vary slowly across the entire astro - comb spectrum , as measured with tres . the fpc finesse is approximately 180 ( fig . [ fig : fitted_phase_error]a ) and the phase deviation of the fpc relative to astro - comb lines is @xmath69 mrad ( fig . [ fig : fitted_phase_error]b ) . reflections from the back surfaces of the fpc mirrors used in these measurements produce the small , rapid phase variations observed in the figure . despite antireflection - coatings the mirror substrates reflected 0.1% of the incident power . a simple model with realistic parameters reproduces the observed phase variations and allows systematic effects associated with these reflections to be eliminated . in future work , slight wedging ( for example , 0.5@xmath70 ) of the substrates will eliminate these variations . after fitting the phase deviation with a sixth order polynomial , we compared our result with the phase deviation derived from a model of air dispersion @xcite and the mirror phase delay measured with a white light interferometer ( fig . [ fig : gdd ] ) . the two different methods agree well , given systematic limitations to the white light interferometer measurement and resultant effect on the model fit . using these _ in - situ _ measurements of the fpc finesse and the phase deviation , along with the source comb intensity variation and resonant fpc transmission , we can determine the suppression of unwanted source comb lines and the resultant frequency shifts of astro - comb line centers . the variation with optical frequency of source comb line intensity and resonant fpc transmission ( @xmath71 in eq . [ eqn : fp_transmission ] ) is measured by offsetting the diode laser frequency from its nominal value by the source comb repetition rate , and thereby tuning the fpc to resonance with a neighboring ( and normally suppressed ) source comb mode . the maximal transmission of each transmitted source comb line and thus @xmath71 is measured ; and then the procedure is repeated for all normally suppressed source comb modes . we find that the variation of @xmath71 is within our measurement uncertainties ( @xmath72 ) . evaluating eq . [ eqn : fp_transmission ] for a single astro - comb line and all of its neighboring ( suppressed ) source comb lines within one spectrograph resolution element , and determining the net transmitted spectrum weighted by the frequency offset from the central astro - comb frequency , allows us to evaluate the effective line center shift in the astro - comb spectrum . for moderate finesse and @xmath73 , the systematic shift @xmath74 of an astro - comb line of frequency @xmath14 , as measured on a spectrograph , can be approximated as @xmath75 \label{eqn : simpleshift}\ ] ] where the term in parentheses parameterizes the mean suppression of nearest source comb side modes , the first term in square brackets results from the difference in source comb intensity between the upper and lower side modes , and the second term in square brackets results from fpc phase deviation leading to asymmetry of the comb lines relative to the fpc mode . the frequency shift in an astro - comb line centroid due to variations of neighboring source comb line intensities at the 1% level is thus @xmath76 khz or 7 cm / s . note that we do not expect neighboring source comb lines to differ as much as 1% ; also a gaussian distribution of comb - line intensities will average away much of this astro - comb centroid frequency shift . nonetheless , we expect that a double pass configuration through the same fpc @xcite , moderately higher fpc finesse ( @xmath77 ) , a higher repetition rate for the source comb ( @xmath78 ghz ) , or more precise measurements of line to line intensity variations will be required to assure 1 cm / s accuracy in practical astro - comb calibrators . in fig . [ fig : correction ] , we show the shift of the estimated systematic error in the wavelength calibration of tres as a function of wavelength , when using our current astro - comb as the calibration reference , if the effect of nearest suppressed source - comb lines are not included in the fit model . with proper inclusion of such effects , residual uncertainty in the wavelength calibration is at the 1 cm / s level . thus we conclude that while astro - comb line frequency shifts caused by mirror dispersion and resultant wavelength dependence of fpc finesse and phase deviation can be absorbed into the wavelength calibration if reproducibility is all that is desired , at the few cm / s level these corrections must be applied to the spectrograph calibration to achieve accurate stellar radial velocity measurements . in summary , we have demonstrated an _ in - situ _ method to determine systematic shifts of astro - comb spectral lines due to imperfect suppression of source comb lines by the fabry - pert filter cavity ( fpc ) . this method involves measurement of fpc finesse as well as the phase deviation of all astro - comb lines over a bandwidth of 1000 . such measurements can be performed at a telescope with either an astrophysical spectrograph or a commercial optical spectrum analyzer of lower resolution . from the measured phase deviation , the dispersion of the fpc is derived and found to be consistent with that calculated from the intra - cavity air dispersion and the group delay dispersion of the fpc mirrors . applying resultant corrections for shifts in the centroid of astro - comb spectral lines allows us to model the astro - comb spectrum as measured on an astrophysical spectrograph with accuracy to better than 0.1 mhz , i.e. , 10 cm / s for measurement of a stellar radial velocity . such high accuracy is important for many applications of astro - comb wavelength calibrators , such as the search for habitable exoplanets , direct measurement of the expansion of the universe , and searches for a temporal variation of physical constants .
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improved wavelength calibrators for high - resolution astrophysical spectrographs will be essential for precision radial velocity ( rv ) detection of earth - like exoplanets and direct observation of cosmological deceleration .
the astro - comb is a combination of an octave - spanning femtosecond laser frequency comb and a fabry - prot cavity used to achieve calibrator line spacings that can be resolved by an astrophysical spectrograph .
systematic spectral shifts associated with the cavity can be 0.1 - 1 mhz , corresponding to rv errors of 10 - 100 cm / s , due to the dispersive properties of the cavity mirrors over broad spectral widths .
although these systematic shifts are very stable , their correction is crucial to high accuracy astrophysical spectroscopy .
here , we demonstrate an _ in - situ _ technique to determine the systematic shifts of astro - comb lines due to finite fabry - prot cavity dispersion .
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can one recover ( some aspects of ) the initial conditions of the universe from the distribution of galaxies at @xmath5 ? a conventional answer to this question is affirmative , _ provided _ that the effect of a spatial bias is well understood and/or if it does not significantly alter the interpretation of the observed distribution . this consensus underlies the tremendous effort in the past and at present to extract the cosmological implications from the existing and future galaxy redshift surveys . the two - point correlation function @xmath6 is a good example supporting this idea ; on large scales it is trivially related to the primordial spectrum of mass fluctuations , @xmath7 . furthermore the effective power - law index of the two - point correlation function on sufficiently small scales is related to the initial power - law index @xmath8 of @xmath9 as @xmath10 ( e.g. @xcite ; @xcite ; @xcite ) . in other words , the initial conditions of the universe are imprinted in the behavior of galaxies on small scales ( again apart from the effect of bias ) . this is why the phenomenological fitting formulae for the nonlinear power spectrum @xcite turn out to be so successful . this fact , however , seems to be in conflict with the universal density profile proposed by navarro , frenk & white ( 1996,1997 ; hereafter nfw ) for virialized dark matter halos . in their study , nfw selected halos which look to be virialized , and found that the density profiles universally obey the nfw form @xmath11 . it is yet unclear to which degree their results are affected by their selection criterion which is not well - defined . in general , different halos should have experienced different merging histories depending on their environment and mass . thus even if the halos do have a _ universal _ density profile _ statistically _ ( i.e. , after averaging over many realizations ) , it is also natural that individual halo profiles are intrinsically scattered around the universal profile @xcite . definitely this is a matter of semantics to a certain extent ; the most important finding of nfw is that such halo - to - halo variations are surprisingly small . a universal density profile was also reported by @xcite on the basis of high resolution simulations of one cluster - mass halo and four galaxy - mass halos , and they claim that the density profile @xmath12 in the most inner region . in what follows , we will address the following quantitative and specific questions concerning the halo profile , especially its most inner region , using the high - resolution @xmath13-body simulations ; the inner slope of the halo profile is really described by @xmath14 or @xmath15 _ universally _ as nfw and @xcite claimed ? if not , does the slope vary among the different halos ? is there any systematic correlation between the slope and the mass of halos ? in fact , some of the above questions have been partially addressed previously with different approaches and methodologies @xcite . in order to revisit those in a more systematic and unambiguous manner , we have developed a nested grid p@xmath16 m n - body code designed to the current problem so as to ensure the required numerical resolution in the available computer resources . this enables us to simulate 12 realizations of halos in a low - density cold dark matter ( lcdm ) universe with @xmath17 particles in a range of mass @xmath18 . as @xcite and later @xcite demonstrated , the inner profile of dark matter halos is substantially affected by the mass resolution of simulations . to ensure the required resolution ( at least comparable to theirs ) , we adopt the following two - step procedure . a detailed description of the implementation and resolution test will be presented elsewhere . first we select dark matter halos from our previous cosmological p@xmath16 m n - body simulations with @xmath19 particles in a @xmath20 cube @xcite . to be specific , we use one simulation of the lcdm model of @xmath21 , @xmath22 , @xmath23 and @xmath24 according to @xcite . the mass of the individual particle in this simulation is @xmath25 . the candidate halo catalog is created using the friend - of - friend grouping algorithm with the bonding length of 0.2 times the mean particle separation . we choose twelve halos in total from the candidate catalog so that they have mass scales of clusters , groups , and galaxies ( table [ table : halos ] ) . except for the mass range , the selection is random , but we had to exclude about 40% halos of galactic mass from the original candidates since they have a neighboring halo with a much larger mass . we use the multiple mass method to re - simulate them . to minimize the contamination of the coarse particles on the halo properties within the virial radius at @xmath26 , @xmath27 , we trace back the particles within @xmath28 of each halo to their initial conditions at redshift @xmath29 . this is more conservative than that adopted in previous studies , and in fact turned out to be important for galactic mass halos . note that we define @xmath27 such that the spherical overdensity inside is @xmath30 times the critical density , @xmath31 . then we regenerate the initial distribution in the cubic volume enclosing these halo particles with larger number of particles by adding shorter wavelength perturbation to the identical initial fluctuation of the cosmological simulation . next we group fine particles into coarse particles ( consisting of at most 8 fine particles ) within the high - resolution region if they are not expected to enter the central halo region within @xmath28 . as a result , there are typically @xmath32 simulation particles , @xmath33 fine particles and @xmath34 coarse particles for each halo . finally about @xmath17 particles end up within @xmath27 of each halo . note that this number is significantly larger than those of nfw , and comparable to those of @xcite and @xcite . the contamination of the coarse particles , measured by the ratio of the mass of the coarse particles within the virial radius to the total virial mass , is small , about @xmath35 , @xmath36 , and @xmath37 for cluster , group , and galactic halos respectively . we evolve the initial condition for the selected halo generated as above using a new code developed specifically for the present purpose . the code implements the nested - grid refinement feature in the original p@xmath16 m n - body code of @xcite . our code implements a constant gravitational softening length in comoving coordinates , and we change its value at @xmath38 , 3 , 2 , and 1 so that the proper softening length ( about 3 times the plummer softening length ) becomes 0.004 @xmath39 . thus our simulations effectively employ the constant softening length in proper coordinates at @xmath40 . the first refinement is placed to include all fine particles , and the particle - particle ( pp ) short range force is added to compensate for the larger softening of the particle - mesh ( pm ) force . when the cpu time of the pp computation exceeds twice the pm calculation as the clustering develops , a second refinement is placed around the center of the halo with the physical size about @xmath41 of that of the first refinement . the mesh size is fixed to @xmath42 for the parent periodic mesh and for the two isolated refinements . the cpu time for each step is about @xmath43 minutes at the beginning and increases to @xmath44 minutes at the final epoch of the simulation on one vector processor of fujitsu vpp300 ( peak cpu speed of 1.6 gflops ) . a typical run of @xmath45 time steps , which satisfies the stability criteria ( couchman et al . 1995 ) , takes @xmath46 cpu hours to complete . figure [ fig : haloplot ] displays the snapshot of the twelve halos at @xmath26 . clearly all the halos are far from spherically symmetric , and surrounded by many substructures and merging clumps . this is qualitatively similar to that found by @xcite for their high - resolution halos in the @xmath47 cdm model . the corresponding radial density profiles are plotted in figure [ fig : haloprof ] . the halo center is defined as the position of the particle which possesses the minimum potential among the particles within the sphere of radius @xmath39 around the center - of - mass of the fine particles . in spite of the existence of apparent sub - clumps ( fig . [ fig : haloplot ] ) , the spherically averaged profiles are quite smooth and similar to each other as first pointed out by nfw . the inner slope of the profiles , however , is generally steeper than the nfw value , @xmath48 , in agreement with the previous findings of @xcite and @xcite . we have fitted the profiles to @xmath49 with @xmath50 ( similar to that used by moore et al.1999 ; the solid curves ) and @xmath51 ( nfw form ; the dotted curves ) for @xmath52 , where @xmath53 is the radius within which the spherical overdensity is @xmath54 . the resulting concentration parameter @xmath55 , defined as the @xmath56 , is plotted in the left panel of figure [ fig : haloslope ] . this is the most accurate determination of the concentration parameter for the lcdm model . there exists a significant scatter among @xmath55 for similar mass @xcite , and a clear systematic dependence on halo mass ( nfw , @xcite ) . the most important result is that the density profiles of the 4 galactic halos are all well fitted by @xmath50 , but those of the cluster halos are better fitted to the nfw form @xmath51 . this is in contrast with @xcite who concluded that both galactic and cluster halos have the inner density profile @xmath57 , despite that they considered one cluster - mass halo alone . in fact , our current samples can address this question in a more statistical manner . cl1 has significant substructures , and the other three are nearly in equilibrium . interestingly the density profiles of cl2 and cl3 are better fitted to the nfw form , and that of cl4 is in between the two forms . the density profiles of the group halos are in between the galactic and cluster halos , as expected . one is better fitted to the nfw form , whereas the other three follow the @xmath50 form . to examine this more quantitatively , we plot the inner slope fitted to a power - law for @xmath58 as a function of the halo mass in the right panel of figure [ fig : haloslope ] . this figure indicates two important features ; a significant scatter of the inner slope among the halos with similar masses and a clear systematic trend of the steeper profile for the smaller mass . for reference we plot the predictions for the slope , @xmath59 by @xcite and @xmath60 by @xcite , using for @xmath61 the effective power - law index @xmath62 of the linear power spectrum at the corresponding mass scale(table 1 ) . with a completely different methodology , @xcite argue that the slope of the density profile within @xmath63 is in between the above two values . on the basis of the slope mass relation that we discovered , we disagree with their interpretation ; for the galactic halos , the analytical predictions could be brought into agreement with our simulation only if the effective slope were @xmath64 , which is much larger than the actual value @xmath65 on the scale . we would like to emphasize that our results are robust against the numerical resolution for the following reasons . since we have used the same time steps and the same force softening length in terms of @xmath53 , the resolution effect , which is generally expected to make the inner slope of @xmath66 shallower , should influence the result of the galactic halos more than that of cluster halos . in fact this is opposite to what we found in the simulation . furthermore our high resolution simulation results agree very well with those of the lower resolution cosmological simulations ( open triangles ) for the cluster halos on scales larger than their force softening length ( short thin lines at the bottom of figure [ fig : haloprof ] ) . we have also repeated the simulations of several halos employing 8 times less particles and 2 times larger softening length , and made sure that the force softening length @xmath67 ( the vertical dashed lines of figure [ fig : haloprof ] ) is a good indicator for the resolution limit . in this _ letter _ we have presented the results of the largest , systematic study on the dark matter density profiles . this is the first study which simulates a dozen of dark halos with about a million particles in a flat low - density cdm universe . this enables us to address the profile of the halos with unprecedented accuracy and statistical reliability . while qualitative aspects of our results are not inconsistent with those reported by moore et al . ( 1999 ) , our larger sample of halos provides convincing evidence that _ the form _ of the density profiles is not universal ; instead it depends on halo mass . since mass and formation epoch are linked in hierarchical models , the mass dependence may reflect an underlying link to the age of the halo . older galactic halos more closely follow the @xmath50 form while younger cluster halos have shallower inner density profiles fitted better by the nfw form . whether this difference represents secular evolution remains to be investigated in future experiments . our results are not fully expected by the existing analytical work . although the analytical work @xcite concluded that the inner profile should be steeper than @xmath48 , their interpretation and/or predicted mass - dependence are different from our numerical results . this implies that while their arguments may cover some parts of the physical effects , they do not fully account for the intrinsically complicated nonlinear dynamical evolution of non - spherical self - gravitating systems . we also note that the small - scale power which was missed in the original cosmological simulation has been added to the initial fluctuation of the halos . the fact that each halo has approximately the same number of particles means that more ( smaller - scale ) power has been added to the low mass halos than to the high mass ones . it is yet unclear how much effect this numerical systematics would have on the mass dependence of the inner slope found in this paper , and we will investigate this question in future work . in summary , the mass dependence of the inner profile indicates the difficulty in understanding the halo density profile from the cosmological initial conditions in a straightforward manner . even if the density profiles of dark halos are not universal to the extent which nfw claimed , however , they definitely deserve further investigation from both numerical and analytical points of view . we thank j. makino for many stimulating discussions and suggestions , and the referee for a very detailed report which significantly improves the presentation of this paper . y.p.j . gratefully acknowledges support from a jsps ( japan society for the promotion of science ) fellowship . numerical computations were carried out on vpp300/16r and vx/4r at adac ( the astronomical data analysis center ) of the national astronomical observatory , japan , as well as at resceu ( research center for the early universe , university of tokyo ) and kek ( high energy accelerator research organization , japan ) . this research was supported in part by the grant - in - aid by the ministry of education , science , sports and culture of japan ( 07ce2002 ) to resceu , and by the supercomputer project ( no.99 - 52 ) of kek . avila - reese , v. , firmani , c. , klypin , a. , kravtsov , a.v . 1999 , astro - ph/9906260 couchman , h.m.p . , thomas , p.a . , & pearce , f.r . 1995 , apj , 452 , 797 eke , v.r , navarro , j.f . , & frenk , c.s . 1998 , apj , 503 , 569 evans , n.w . & collett , j.l . 1997 , apj , 480 , l103 fukushige , t. , & makino , j. 1997 , apj , 477 , l9 hamilton , a.j.s . , kumar , p. , lu , e. , & matthews , a. 1991 , apj , 374 , l1 hoffman , y. , & shaham , j. 1985 , apj , 297 , 16 jain , b. , mo , h.j . , & white , s.d.m . 1995 , mnras , 276 , 25 jing , y. p. 1998 , apj , 503 , l9 jing , y. p. 1999 , apj , submitted ( astro - ph/9901340 ) jing , y.p . & fang , l.z . 1994,apj , 432 , 438 jing , y.p . & suto , y. 1998 , apj , 494 , l5 kitayama , t. & suto , y. 1997 , apj , 490 , 557 lokas , e.l . 1999 , mnras , in press ( astro - ph/9901185 ) ma , c .- p . 1998 , apj , 508 , l5 moore , b. , governato , f. , quinn , t. , stadel , j. , & lake , g. 1998 , apj , 499 , l5 moore , b. , quinn , t. , governato , f. , stadel , j. , & lake , g. 1999 , mnras , submitted , astro - ph/9903164 navarro , j.f . , frenk , c.s . , & white , s.d.m . 1996 , apj , 462 , 563 navarro , j.f . , frenk , c.s . , & white , s.d.m . 1997 , apj , 490 , 493 nusser , a. , & sheth , r.k . 1999 , mnras , 303 , 685 peacock , j.a . & dodds , s.j . 1996 , mnras , 280 , l19 peebles , p.j.e . 1980 , the large scale structure of the universe ( princeton university press : princeton ) suginohara , t. , suto , y. , bouchet , f.r . , & hernquist , l. 1991 , apjs , 75 , 631 suto , y. 1993 , prog.theor.phys . , 90 , 1173 syer , d. , & white , s.d.m . 1998 , mnras , 293 , 337 cccccccccccccc gx 1 & @xmath68 & 458,440 & 0.269 & @xmath69gx 2 & @xmath70 & 840,244 & 0.356 & @xmath71gx 3 & @xmath72 & 694,211 & 0.330 & @xmath73gx 4 & @xmath74 & 1,029,895 & 0.363 & @xmath71gr 1 & @xmath75 & 772,504 & 0.735 & @xmath76gr 2 & @xmath77 & 907,489 & 0.736 & @xmath76gr 3 & @xmath78 & 831,429 & 0.764 & @xmath79gr 4 & @xmath80 & 901,518 & 0.758 & @xmath79cl 1 & @xmath81 & 522,573 & 1.59 & @xmath82cl 2 & @xmath83 & 839,901 & 1.42 & @xmath84cl 3 & @xmath85 & 664,240 & 1.35 & @xmath86cl 4 & @xmath87 & 898,782 & 1.39 & @xmath88 [ table : halos ]
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we perform a series of high resolution n - body simulations designed to examine the density profiles of dark matter halos .
from 12 simulated halos ranging the mass of @xmath0 ( represented by @xmath1 million particles within the virial radius ) , we find a clear systematic correlation between the halo mass and the slope of the density profile at 1% of the virial radius , in addition to the variations of the slope among halos of the similar mass .
more specifically , the slope is @xmath2 , @xmath3 , and @xmath4 for galaxy , group , and cluster mass halos , respectively . while we confirm the earlier simulation results that the inner slope is steeper than the _ universal _ profile originally proposed by navarro , frenk & white , this mass dependence is inconsistent with the several analytical arguments attempting to link the inner slope with the primordial index of the fluctuation spectrum .
thus we conclude that the dark matter density profiles , especially in the inner region , are not universal .
| 5,503 | 244 |
coleman and weinberg pointed out several years ago [ 1 ] that quantum loops and radiative corrections can play an important role in determining the structure of the vacuum of a quantum field or system of fields and particles . specifically they showed that while a scalar field with mass parameter @xmath0 and classical tree potential is unable to develop a vacuum expectation value ( vev ) and spontaneous symmetry breaking ( ssb ) , such vev and ssb become achievable when we take quantum loops into account . this is particularly so if the scalar field is coupled to a gauge field and dimensional transmutation can occur . the overall conclusion is that an alternative electroweak symmetry breaking ( ewsb ) mechanism exists along side the standard model ewsb mechanism that is based on the classical tree potential : @xmath1 with a non - zero negative mass parameter @xmath2 . the coleman - weinberg alternative model dispenses with the mass parameter @xmath3 in equation ( 1 ) but adds quantum loops or higher order potential terms . this alternative mechanism has become known as radiative electroweak symmetry breaking ( rewsb ) . + the fact is that these two mechanisms do not just differ in what terms appear or do not appear in the scalar field potential , but lead to very different physical perceptions of what causes electroweak symmetry breaking . the standard model based on equation ( 1 ) with its @xmath2 attributes electroweak symmetry breaking to a primordial fundamental scalar field that unlike any existing particle known so far has a negative mass parameter @xmath2 . in contrast , the coleman - weinberg rewsb mechanism would attribute ewsb to quantum dynamics and a quantum origin . such quantum dynamics if favored would widen the scope for our search and understanding of ewsb and the origin of mass . another way of looking at the matter is to say that in the standard model mechanism cewsb , the quantum system starts off with a pre - assigned mass scale @xmath4 compared to the coleman - weinberg model rewsb where the same quantum system is left to fix its own scale of dynamics . the question arises what role the pre - assigned scale parameter @xmath5 plays in the quantum dynamics , and whether we can find any observable features that distinguish the outcome of these two dynamics . + because of the widely differing physical interpretations of the two mechanisms , we consider it worthwhile to examine in some quantitative detail , what observational features and signatures distinguish rewsb on the one hand , from the conventional standard model electroweak symmetry breaking ( cewsb ) on the other hand . al . [ 2 - 6 ] have considered the problem from one perspective . + our approach to the problem is to first set out in section 2 , aspects of electroweak symmetry breaking that do not depend on any explicit choice of the scalar field potential @xmath6 . such aspects will therefore be common to both rewsb and cewsb and we may call these aspects the core of electroweak symmetry breaking . thereafter , we take up equation ( 1 ) in section 3 , with its @xmath7 , as one explicit choice of the scalar potential . then we work out what new relations besides the core ewsb equations , follow from this one choice of @xmath6 . these new relations or features we can call the specific signatures of the standard model cewsb . in section 4 , we make a different choice of the scalar potential , by setting @xmath3 which is the coleman - weinberg rewsb model , with quantum loops added . we work out the new features in relation to the core equations , and call these the signatures of rewsb . we proceed further in sections 5 and 6 , to re - examine the same rewsb using the more modern aspects of the coleman - weinberg model known as the renormalization group improved effective potential [ 7 - 10 ] . further improvement using 2-loop @xmath8 and @xmath9 functions is considered in section 7 . the two sets of signatures , cewsb and rewsb , are compared in section 8 . our final results and conclusions are stated in section 9 . in terms of a scalar field , the core features that break electroweak symmetry even when no explicit scalar potential is specified , are a complex scalar field that may or may not be a fundamental field . this complex scalar field must be a doublet under the electroweak @xmath10 gauge symmetry . the complex doublet scalar field must have a non zero vacuum expectation value we take to be v. given such a field one is able to immediately write or parameterize it in the form : + @xmath11 in which only one component field @xmath12 acquires the non - zero vacuum expectation value written : @xmath13 this form of the scalar field is all that is required to derive many results and features of the ewsb . these several features include various charged and neutral currents as well as couplings of fermions and gauge bosons to the scalar field . the details of these core features can be seen in several places such as [ 11 ] . among these relations that we obtain without our specifying any explicit scalar potential , are the gauge boson masses : @xmath14 @xmath15 where @xmath16 is @xmath17 coupling constant , and @xmath18 is @xmath19 coupling constant . + it turns out also that a definitive numerical value of v = 246 gev can be obtained without recourse to any explicit choice of the scalar potential . one simply combines the v - a current structure of the gauge theory [ 11 ] , with such accessible processes as @xmath5 decays , and obtains expression for v in terms of the fermi constant : + @xmath20 given this value of v and the known masses of the gauge bosons stated in equations ( 4 ) and ( 5 ) , one deduces the values of the two gauge coupling constants at the ewsb scale v : @xmath21 similarly from yukawa couplings of equation ( 2 ) to fermions , we obtain fermion mass relations and couplings : @xmath22 where @xmath23 stands for yukawa coupling constant of a given fermion f , such as the top quark @xmath24 . again these relations do not require any explicit choice or specification of the scalar potential . in particular , the relations hold whether @xmath25 as in rewsb , or @xmath26 ( cewsb ) . + we now note one area where we are not able to obtain any information at all based on equation ( 2 ) alone , without specifying some explicit form of the scalar potential . this area is in the determination of physical higgs particle mass @xmath27 , and the self coupling parameter @xmath28 of the scalar field and its components . this becomes the area where we can focus our search for any distinguishing features between rewsb and cewsb . we will therefore proceed by devising ways to determine these two quantities @xmath27 and @xmath28 , first in a cewsb model and then in a rewsb model . both quantities turn out to be derivable from some explicit form of the scalar potential we now write as @xmath29 where @xmath30 stands for the scalar field ( component ) that has the non - zero vev . our main focus is in the structure and analysis of such two potentials @xmath31 and @xmath32 , and the higgs masses and couplings they predict . the conventional ewsb is based on equation ( 1 ) . this potential defines the ground state of the system @xmath33 and gives a value of the vev of @xmath30 in terms of the parameters of the chosen potential : @xmath34 with @xmath35 upon plugging equations ( 2 ) and ( 10 ) into equation ( 1 ) , we obtain a mass relation for the physical higgs particle h(x ) : it is : @xmath36 more formally the mass of the higgs particle is given in terms of a chosen potential by : @xmath37 equations ( 10 ) and ( 11 ) become new relations which used separately or in combination with the core equations ( 2 ) - ( 8) may lead to specific signatures of the cewsb . thus from equations ( 10 ) and ( 11 ) we get @xmath38 as a relation specific to cewsb model . we can go further to use the core value v = 246 gev and relate @xmath27 specifically to @xmath28 as : @xmath39 if we assume that the scalar field interaction was perturbative + in the dynamics leading to ewsb at scale v , and that @xmath40 , we obtain that : @xmath41 which becomes a signature of cewsb mechanism . on the other hand , if @xmath42 , then cewsb predicts @xmath43 gev . notably however , the cewsb does not give us any specific value for higgs mass nor a value for @xmath28 . we consider next the coleman weinberg rewsb potential and what signatures we can tag onto it . for a @xmath44 scalar field theory , coleman and weinberg obtained the general expression for its effective potential : @xmath45^n\ ] ] where @xmath46 is the classical field that has non - zero vacuum expectation value . @xmath47 are 1pi vertex greens functions to which n external legs @xmath48^n $ ] attach , each carrying zero momentum . when these 1pi functions are expanded perturbatively into feynman loops , the above series can be re - organized into a perturbative loop expansion for the same effective potential : @xmath49 where @xmath50 is the tree potential given by equation ( 1 ) ; @xmath51 is the one - loop potential ; @xmath52 is the 2-loop potential , etc . the above loop potentials require in general to be renormalized . the renormalized effective potential determines the state of ewsb and the higgs mass through the standard equations : @xmath53 @xmath54 @xmath55 for a purely scalar field theory , the above loops involve only virtual scalar particles . when however other fields are present in the system that can couple radiatively to the scalar field , the loop contributions of these other fields to @xmath56 have to be taken into account . in the specific case of ewsb where fermions and gauge bosons are present besides the scalar field , the coleman weinberg potential generalizes to include quarks , leptons and gauge boson loops . these extended loop calculations have been carried out in a number of places [ 7 , 11 ] . we write down some of the results up to 1-loop potential for rewsb . in one form cheng and li write [ 11 ] : @xmath57 where @xmath58 is the renormalization scale ; and c is given by : @xmath59\ ] ] here @xmath60 are vector boson masses ( @xmath61 ; @xmath62 is scalar ( physical higgs ) boson mass , and @xmath63 are the fermion masses , quarks and leptons . + using core equations ( 4)-(8 ) without using any equations from cewsb section 3 , we can rewrite equation ( 21 ) in terms of coupling constants as follows : @xmath64\ ] ] if we neglect contributions from all leptons and quarks except the top quark , we can write the rewsb potential equation ( 20 ) up to 1-loop potential finally as [ 2 ] : @xmath65 ( log\frac{\phi^2}{m^2 } - \frac{25}{6})\ ] ] here @xmath24 is the top quark loop yukawa coupling . the neglect of other fermion couplings follows from equation ( 8) , where a fermion yukawa coupling is proportional to its mass , and the top quark with its dominant mass clearly overshadows all other fermions . regarding the parameter m in the above equation , we note that in the absence of a pre - set mass scale @xmath3 in the rewsb system , the quantum dynamics sets its own scale which we take to be the renormalization scale m , as well as the scale of any ssb in the system . therefore we can re - define our renormalization conditions on the effective potential as ; @xmath66 @xmath67 @xmath68 in place of equation ( 23 ) , we can also write the 1-loop effective potential in the form due to ford et.al . [ 8 ] , and to casas et . al.[9 - 10 ] . @xmath69 we will work with the form ( 23 ) . we plug equation ( 23 ) into equations ( 24 ) - ( 26 ) and obtain relations that embody rewsb signatures : @xmath70 putting c value from equation ( 22 ) we obtain a quadratic equation for @xmath28 : @xmath71 using the core coupling values from equation ( 7 ) , the quadratic equation becomes : @xmath72 this gives two possible values of @xmath28 in terms of top quark yukawa coupling constant @xmath24 , both at the renormalization scale m = @xmath73 : @xmath74 and @xmath75 if we require that both coupling constants are positive : @xmath76 then solution ( 31 ) implies that @xmath77 with @xmath78 at @xmath79 . on the other hand , based on equation ( 32 ) , we find at @xmath80 while for @xmath81 decreases to zero , @xmath82 . this solution would indicate that the scalar dynamics in the ewsb has a coupling constant @xmath28 that lies between zero and a value 0.004456 while the yukawa top quark coupling has a value that lies between zero and 0.502758496 . we may take the mean of each parameter as its value at the ewsb scale in this case of solution ( 32 ) . then analogous to equation ( 7 ) we can write the @xmath81 and @xmath24 values at the ewsb scale for solution ( 32 ) as : @xmath83 in this way , we may claim that solution ( 32 ) represents a situation where @xmath84 at ewsb , while solution ( 31 ) represents a situation where @xmath85 at ewsb scale . these two situations we can also try to distinguish in our signature analysis of rewsb . + in the case of solution ( 31 ) , we can use the known top quark mass of 175 gev along side equation ( 8) to estimate @xmath24 at ewsb scale . we get that @xmath86 giving a corresponding value of @xmath28 from equation ( 31 ) : @xmath87 as seen from equations ( 33 ) and ( 34 ) both solutions ( 31 ) and ( 32 ) are perturbative , if we define perturbativity condition as : @xmath88 the two solutions differ however in a profound way . this is that if the dynamics of ewsb is very weakly perturbative ( both @xmath28 and @xmath89 very small ) as per equations ( 32 ) and ( 33 ) , rewsb predicts that the top quark coupling is relatively much stronger than the scalar self interaction @xmath28 at the ewsb scale . on the other hand , if the ewsb dynamics is more strongly coupled , ( both @xmath28 and @xmath24 large ) , though still perturbative as per equations ( 31 ) and ( 34 ) , the scalar self coupling constant @xmath28 is relatively much larger than the top quark yukawa coupling constant @xmath24 . these signatures of the rewsb can be ascertained experimentally . + we can go further to analyze equations ( 25 ) and ( 23 ) for higgs mass . we obtain : @xmath90 using @xmath91 from equation ( 28 ) , this becomes : @xmath92\ ] ] using core values from equation ( 7 ) we can rewrite equation ( 37 ) as : @xmath93\ ] ] next we use equation ( 31 ) or ( 32 ) to eliminate @xmath28 . we obtain : @xmath94\ ] ] finally using core equations ( 7 ) and ( 8) we obtain a higgs boson mass for the rewsb model : @xmath95\ ] ] taking a top quark mass @xmath96 this evaluates to @xmath97 ( the positive value ) . this appears high . the coleman - weinberg 1-loop effective potential equation ( 23 ) used in the above analysis of rewsb , has the feature that it was defined at one and only one arbitrarily chosen mass scale point m which is the renormalization scale of the otherwise divergent effective potential . this one scale point m had also to be chosen as the ewsb point @xmath98 , with @xmath99 this restricted choice of @xmath100 was necessitated by the need for the log terms in equations ( 23 ) - ( 27 ) to vanish or remain small and perturbative . this feature of the coleman - weinberg potential equation ( 23 ) remains even if we make another arbitrary choice of the renormalization point m. we remain bound to study our effective potential and its signature values , only for field values @xmath30 in the vicinity of an arbitrarily chosen renormalization scale m. in effect , @xmath101 can not be studied generally as a function of higgs field @xmath100 to ascertain how the potential behaves in these other regions of larger or smaller @xmath30 , and to what extent any new features such as new minima or maxima of @xmath102 found in those regions , affect or modify the ewsb features or signatures of @xmath103 found from earlier limited observation at one arbitrary renormalization point m. a way out of the problem is to free @xmath103 of its dependence on the renormalization scale m. that is we make @xmath103 a physical observable , whose field @xmath30 and coupling constant parameters @xmath104 can now vary widely over any desired domains , unaffected by any one arbitrarily chosen renormalization point m. of course this freedom of the parameters of @xmath103 to vary widely still calls for some restraint such that the perturbative character of the overall loop expansion equations ( 15 ) and ( 16 ) can still be guaranteed . both objectives and the automatic summing up of leading logarithms , are achieved by making the entire effective potential equation ( 15 ) or ( 16 ) , satisfy a renormalization group equation(rge ) given by : @xmath105 v_{eff}(m , \lambda , g_i , \phi .. ) = 0\ ] ] here @xmath106 , and @xmath107 are the rg functions ( the beta functions and the anomalous dimension @xmath108 of the higgs field ) . this rge has a standard solution we write in the form : @xmath109 and call the rg improved effective potential . all its parameters @xmath110 vary or run widely , each in a coordinated manner prescribed by its beta or gamma function . the variable @xmath111 parameterizes the arbitrary renormalization scale , while the parameters of @xmath112 run according to the equations : @xmath113 @xmath114 with : @xmath115 also : @xmath116 @xmath117 in the above equations , @xmath118 is some fixed initial renormalization point or reference scale , which we shall choose as @xmath119 , the @xmath120 gauge boson mass . similarly the value of the higgs field @xmath121 at @xmath122 we shall choose as the field with vev @xmath123 where @xmath124 gev . the beta and gamma functions have to be calculated separately to some loop order , and plugged into the above running equations . explicit expressions of these @xmath125 and @xmath126 functions computed to 1-loop and 2-loop orders for the ewsb system , have been tabulated by ford et al . [ 8 ] . we shall rely on their values to compute numerically the parameters @xmath127 defined in equations ( 43)-(47 ) . + in order to solve these integral and differential equations , we require to specify the boundary conditions of the equations . these boundary conditions are the values of the parameters at some chosen scale we specify as @xmath122 which means scale @xmath128gev . our explicit values of these parameters at @xmath122 we take from equations ( 7 ) , ( 33 ) and ( 34 ) . + once we have computed the explicit details of how the parameters vary with @xmath111 over a wide range of values in equations ( 43 ) - ( 47 ) , our aim will be to use the new rg improved effective potential equation ( 42 ) to re - examine the issue of signatures of rewsb . to proceed with the analysis , we need first to expand equation ( 42 ) into perturbative loop series and truncate the series at a convenient loop order chosen here as 1-loop order . thus by analogy with equations ( 16 ) and ( 20 ) , we write our rg improved effective potential equation ( 42 ) as : @xmath129 to 1-loop order we write the rg improved rewsb potential as : @xmath130 ( log\frac{\phi^2(t)}{m^2(t ) } - \frac{25}{6})\ ] ] which replaces equation ( 23 ) . similarly , corresponding to equation ( 27 ) we would now write the rg improved 1-loop potential of ford et . 8 ] and casas et . 9 - 10 ] : @xmath131 we will work with equation ( 49 ) , and comment on equation ( 50 ) later . our analysis of equation ( 49 ) now proceeds in stages as follows . + first we note that by construction the full ( untruncated ) rg improved effective potential equation ( 42 ) or ( 48 ) becomes independent of scale , satisfying : @xmath132 this scale invariance applies also to all order derivatives of @xmath112 with respect to any of its parameters such as @xmath30 . thus we also have the conditions : @xmath133 and : @xmath134 the implication of these equations is that equations ( 24)-(26 ) from which we determine the state of occurrence of ssb and the resulting higgs mass as well as signatures of rewsb ( all arising from inherent dynamics of the system ) , have become independent of any one choice of renormalization scale t or m(t ) at which we evaluate the system . in a way , @xmath112 is now defined simultaneously over a whole range of the renormalization scale from @xmath135 to @xmath136 at any one point of which we can freely compute the dynamical quantities in equations ( 24 ) - ( 26 ) and determine at what points we have ssb and its corresponding mass spectra , unaffected by the value of the renormalization scale t at that point , except that the results obtained at any one t scale or point , differ from the results at another t scale or point by mere change or transformation of scale , but not intrinsic difference of dynamics of the system . + we note however that the above discussion regarding scale invariance of the rg improved effective potential holds strictly only where the potential is not truncated as in equations ( 49 ) and ( 50 ) . where @xmath112 is truncated and its arguments are evaluated with beta and gamma functions that are also computed only to a finite order ( 1-loop , 2-loop , ... beta and gamma functions ) , equations ( 51)-(53 ) cease to hold exact . we correct for this by looking for a restricted stretch of the t scale , call it @xmath137 with @xmath138 , where the truncated functions in equations ( 49 ) - ( 53 ) may still be considered reasonably flat or minimally dependent on t. we work with such flat regions for potentials ( 49)- ( 53 ) , in place of the region @xmath139 of the exact scale invariant quantities . now we analyze equations ( 49 ) , ( 24)-(26 ) . + our first step is to compute the various running parameters of @xmath140 specified in equations ( 43 ) - ( 47 ) . we illustrate our computational procedures by first considering the simpler case of 1-loop beta and gamma functions . later we treat the case of 2-loop beta and gamma functions . to 1-loop beta and gamma functions [ 8 ] , the running parameter equations to compute are : + @xmath141 @xmath142 @xmath143 with @xmath144 from equation ( 7 ) and @xmath145 + also we get values of @xmath146 by numerically integrating the following equations : @xmath147}\ ] ] @xmath148 where @xmath149 ; and @xmath150 here we use as boundary values , the pair of values ( @xmath151 ) from equation ( 34 ) . later we shall consider the other pair of values ( @xmath152 ) of equation ( 33 ) . finally we have for the @xmath153 and @xmath154 : @xmath155dt'\right)\ ] ] with this integral separated into two parts for purposes of numerical computation : @xmath156dt ' = \frac{1}{16 \pi^2}\int_{g_{t0}}^{g_t(t)}[3 g_t^2]dg_t(t ' ) - \frac{1}{16 \pi^2}\int_0^t [ \frac{9}{4 } g_2 ^ 2 + \frac{3}{4 } g_1 ^ 2]dt'\ ] ] we proceeded by assigning only positive values to scale t , and insisting in our numerical computations of equations ( 54 ) - ( 60 ) that we obtain only matching positive values of the coupling constants @xmath157 , and @xmath158 . the results we obtained are shown in * figure 1*. the potential @xmath112 equation ( 49 ) is next plotted against t , to search for a scale region @xmath137 where @xmath112 is reasonably flat , independent of t. from the plot shown in * fig . 2 * we see that equation ( 49 ) is reasonably scale invariant in the region : @xmath159 . this allows us to compute the quantities : @xmath160 and @xmath161 at any point within this flat @xmath137 region , and to run the quantities down to our reference scale @xmath122 . it is in this way that we obtain below , a higgs ( running ) mass at the electroweak scale @xmath122 . the relevant quantities to compute and run in region @xmath162 are : @xmath163 implying a ssb condition any where in region @xmath137 given by : @xmath164 where : @xmath165\ ] ] also : @xmath166 which combined with equation ( 62 ) gives within region @xmath137 , a running higgs mass of : @xmath167 a plot of this @xmath168 is shown in * figure 2 * and yields a higgs ( running ) mass value of @xmath169 gev at the electroweak scale t = @xmath137 = 0 . the corresponding physical higgs mass differs from this by only a small vacuum polarization correction discussed by casas et . al . [ 9,10 ] we will not go into this . + based on * figure 2 * , we can interpret the region @xmath170 not only as a region beyond which the 1-loop truncated rg improved effective potential is not reliable , but also as a cut off point @xmath171 beyond which the standard model ewsb theory is probably not valid . this will lead us to say that radiative electroweak symmetry breaking ( rewsb ) with rg improved effective potential ( at 1-loop beta function ) predicts a higgs mass around 245.6 gev , as well as a cut off energy scale @xmath172 gev or less . we examine next the features of rewsb in the case @xmath174 at electroweak scale , presented by equation ( 33 ) , in place of equation ( 34 ) . we recall that equations ( 33 ) and ( 34 ) provided us a means to discriminate between two scenarios where at ewsb scale , the scalar field coupling @xmath28 is stronger ( equation ( 34 ) ) or weaker ( equation ( 33 ) ) than the top quark yukawa coupling . our calculations in sections 4 and 5 , dealt with the case @xmath175 . here we re - compute the same rg improved effective potential equation ( 49 ) under this alternative solution equation ( 33 ) . specifically we take as our new boundary values , the values given in equation ( 33 ) : @xmath176 and @xmath177 . we then re - compute our running parameters and the effective potential ( 49 ) , together with the running higgs mass . the results and features we find with equation ( 33 ) and the 1-loop beta and gamma function equations ( 43 ) - ( 47 ) are shown in * figure 3 * . they lead us to a running higgs mass of only 6.65gev shown in * figure 4*. since experiments already rule out such low mass higgs boson , we can draw one conclusion that the rewsb model clearly selects only the solution ( 34 ) : @xmath178 at ewsb scale . + working now with only solution ( 34 ) , we consider finally the case of 2-loop beta and gamma function rg improvement of the effective potential equation ( 49 ) . according to kastening [ 12],bando et . al [ 13 ] , one gets optimal improvement of a 1-loop effective potential if we evaluate it using 2-loop beta and gamma functions . we test this out by re - computing our rewsb features and comparing with our 1-loop results of sections 4 and 5 . accordingly , we replace equations ( 54 ) to ( 60 ) by the following 2-loop beta function running parameters : @xmath179 } \ ] ] @xmath180 } \ ] ] @xmath181 } \ ] ] with @xmath144 and @xmath182 as before . + for 2-loop @xmath183 and @xmath158,we compute positive values of the upper integration limits @xmath184 and @xmath158 that match an assigned t value in the equations : @xmath185 and : @xmath186 where : @xmath187 the boundary values we use are from equation ( 34 ) : @xmath188 and @xmath189 also because of the greatly intertwined nature of the 2-loop @xmath190 functions and the running parameters , certain approximations had to be made in which quantities like @xmath191 , in the above integrands , were treated as constants having values corresponding to a chosen t value , or else to their known 1-loop values . + finally for @xmath153 and @xmath154 we have the 2-loop gamma function equation : @xmath192dt'\right)\ ] ] where : @xmath193 the results of computing these 2-loop running parameters , and plugging into equations ( 49 ) , ( 24 ) - ( 26 ) , are shown in * figures 5 and 6 . * some individual running parameters appear greatly modified at the 2-loop level . this is particularly so for @xmath158 and @xmath194 seen from figures 1 and 6 . correspondingly , the 2-loop rg improved effective potential does not exhibit much of a flat @xmath137 region seen in figure 6 compared to 1-loop figure 2 . the value of running higgs mass at ewsb scale : @xmath122 predicted in both cases is however about the same : @xmath195 gev . we can compare these features of our rg improved rewsb with features found in conventional electroweak symmetry breaking calculations . we already showed in section 3 that cewsb ( at tree level ) , does not predict any explicit value of the higgs mass @xmath27 nor the scalar coupling constant @xmath28 . but the conventional electroweak symmetry breaking potential ( with @xmath196 can also have higher order potential terms added to it as in equation ( 16 ) , and treated to a rg group improvement as for our rewsb model considered in sections 5 and 6 . this has been done by a number of authors [ 7 - 17 ] , particularly ford et . al . [ 8 ] and casas et we can look at these rg improved cewsb results and compare them to our rewsb results to see any particular signature features . these authors [ 8 - 10 ] worked with 1-loop effective potential of the form shown in equations ( 27 ) and ( 50 ) to which non - zero scalar field mass term @xmath197 is added from equation ( 1 ) . this rg improved cewsb equation ( 50 ) is then analyzed by the usual equations ( 24 ) - ( 26 ) , combined with the additional requirement of vacuum stability or @xmath198 . al [ 9,10 ] obtained a higgs mass estimate of @xmath199gev , to be compared with our explicit rewsb result of running higgs mass @xmath169 gev . admittedly , many of these cewsb rg improved effective potential analysis , were done at a time the top quark mass was still considered unknown free parameter , in contrast to our treatment . al . [ 8 ] for example stated their result in terms of the top quark mass : @xmath200gev . overall , there appears a need to re - analyze the rg improved cewsb effective potential case for better comparison with rewsb , and to ascertain the role the scalar field mass parameter @xmath26 actually plays . + another comparison we can make is with the recent work of elias et . al.[2 - 6 ] who analyzed the same radiative electroweak symmetry breaking ( rewsb ) , and raised the same issue of the possible experimentally accessible signature differences of rewsb and cewsb . they worked not with the standard coleman - weinberg feynman loop calculated effective potential equations ( 15 ) , ( 23 ) , ( 27 ) , ( 49 ) , ( 50 ) , but took advantage of the fact that the loop calculated effective potentials invariably involved leading logarithms as in equation ( 20 ) , multiplied by a factor c that involves products of ew coupling constants and the scalar field @xmath30 . al . showed that this usual coleman - weinberg rg improved effective potential in the particular case of rewsb can be realized as a direct summation to infinity of leading logarithm terms . putting such leading logarithm parameterized effective potential through the usual equations ( 24 ) - ( 26 ) , elias et . al . obtained a rewsb higgs mass ranging in value from @xmath201 gev . they obtained ssociated scalar field coupling @xmath202 . these results are comparable to our values @xmath203 gev , and @xmath204 . their values for other ew coupling constants are also very comparable , thus while for the top quark , elias et . al . obtained the value @xmath205 at ewsb scale , we obtained in equation ( 34 ) the value @xmath206 . + + we find close agreement between our results based on the standard coleman - weinberg loop calculated , and renormalization group improved , effective potential , and the elias et . al . results based on direct summation of leading logarithm terms . basing on both calculations of the same radiative electroweak symmetry breaking ( rewsb ) , we can re - affirm that rewsb model not only remains a viable mechanism of electroweak symmetry breaking , but is able to give an explicit value of the ( running ) higgs mass . the ( running ) higgs mass predicted by the rewsb mechanism , lies in the range 215 - 250 gev , at electroweak scale . in contrast the cewsb models of ford et . al [ 8 ] , casas et al . [ 9.10 ] and others [ 7,14 - 16 ] , are only able to put a lower bound typically @xmath207gev on the higgs mass . because of this limited information from cewsb , we can not use the higgs mass value as a conclusive signature to distinguish between rewsb from cewsb . + we are left to consider the coupling constants @xmath28 and @xmath24 of the system . our calculations established two points regarding @xmath28 and @xmath24 . we found that of the two possibilities : @xmath208 or @xmath173 at ewsb scale , the @xmath209 is the correct one , while the case @xmath173 that implies a higgs mass @xmath210 is already ruled out by experiment . explicitly , we obtained in our equation ( 34 ) that the ratio @xmath211 . elias et . al . found a lower ratio of 2.15 . the two results are however consistent to the extent @xmath212 + + if we accept the rewsb value of higgs mass of 215 - 250 gev as physical higgs mass holding whether the mechanism of ewsb is by rewsb or by cewsb , we can deduce the corresponding @xmath213 value from equations ( 13 ) and ( 14 ) . we find @xmath214 to be compared with @xmath215 at the same higgs mass of 245.6 gev . we conclude as did elias et . al . [ 2,3 ] , that the rewsb mechanism is associated with a greatly enhanced scalar field coupling @xmath216 . + this can become an important signature difference between rewsb and cewsb that can be verified experimentally . the fact is that based on equations ( 1 ) and ( 2 ) , the individual scalar field components @xmath217 all couple with the same strength , @xmath213 or @xmath218 depending on the model . when this fact is combined with the equivalence theorem [ 18 ] between ewsb nambu - goldstone bosons and linearly polarized gauge bosons , one comes to the conclusion that interactions of @xmath219 can provide experimentally observable coupling constant based signature differentiation between rewsb and cewsb , a fact pointed out by elias et . al . [ 2,3 ] , but needing further study . + the observed enhanced coupling @xmath220 by itself calls for explanation . it suggests as part of our main finding that the scalar field couples less strongly to achieve the same end result of ewsb in a quantum system that has a pre - set mass scale @xmath221 , compared to the same quantum system with no pre - set scale @xmath3 . the same scalar field couples more strongly in the latter case . whether this is a general principle of quantum dynamics is something for further study . + + * references :* + 1 . s. coleman and e. weinberg , phys . d7 ( 1973 ) 1888 . v. elias , r.b . mann , d.g.c.mckeon , and t.g . steele , phys . 91 , 251601 . f. a. chishtie , v. elias , and t. g. steele , intern . a20 ( 2005 ) 6241 . v. elias , r.b . mann , d.g.c . mckeon and t.g . steele , nucl . b678 ( 2004 ) 147 . + 5 . v. elias , r. b. mann , d.g.c . mckeon and t.g . steele , arxiv.hep-ph/0508107 ( 2005 ) . f. a. chistie , v. elias , r.b . mann , d.g.c.mckeon , and t.g . steele , nucl . b 743 ( 2006 ) 1034 . + 7 . m. sher , phy . report 179 ( 1989 ) 273 . c.ford , d.r.t.jones , p.w stephenson and m b einhorn , nucl . phys.b395 ( 1993 ) 17 . casas , j.r . espinosa , m. quiros and a. riotto , nucl . b436 ( 1995 ) 3 ; ( e ) b439 ( 1995 ) 466 . casas , j.r . espinosa , m. quiros , phys . b342 ( 1995 ) 171 . t. p. cheng and l. f. li , gauge theory of elementary particle physics , oxford university press ( 1984 ) . b. kastening : phys . b283 ( 1992 ) 287 . m. bando , t. kugo , m. maekawa and h. nakano , phys . b301 ( 1993 ) 83 ; prog . ( 1993 ) 405 . g. altarelli and g. isidori phys . b 337 ( 1994 ) 141 + 15.m . lindner , m. sher and h.w . zaglauer , phys . lett . b228 ( 1989 ) 139 . vincenzo branchina and hugo faivre , phys . d72 ( 2005 ) 065017 . p. kielanowski , s.r . juarez w , and h.g . solis - rodriguez , phys . d72 ( 2005 ) 096003 ) . + 18 . p. n. maher , l. durand and k. riesseimann , phys . d 48 ( 1993 ) 1061 . + * figure captions : * figure 1 : running parameters of rg improved effective potential at 1-loop @xmath222 and @xmath126 functions and for equation ( 34 ) , @xmath223 . + figure 2 : flatness @xmath137 test and running higgs mass plot yielding a running higgs mass intercept of @xmath169 gev at ew scale t = 0 . + figure 3 : running parameters of rg improved effective potential at 1-loop @xmath222 and @xmath126 functions , with @xmath224 , equation ( 33 ) case . + figure 4 : flatness @xmath137 and running higgs mass plots at 1-loop @xmath222 and @xmath126 functions , and for @xmath224 equation ( 33 ) case . + figure 5 : running parameters of rg improved effective potential at 2-loop @xmath222 and @xmath126 functions and for equation ( 34 ) , @xmath223 . + figure 6 : 2-loop @xmath222 and @xmath126 functions flatness @xmath137 and running higgs mass plots , yielding a running higgs mass intercept of @xmath169 gev at ew scale t = 0 .
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we present a comparative study of radiative electroweak + symmetry breaking and conventional standard model breaking , and pose the question whether experiments can distinguish one breaking mode from the other .
the importance of the problem lies in the fact that the two breaking modes have very different physical interpretations concerning the mechanism of spontaneous electroweak symmetry breaking and the origin of mass . _
keywords : radiative electroweak symmetry breaking _ + * pacs : 12.38.-t , * + e - mail : fndili@uh.edu
| 11,494 | 128 |
with wilson fermions , straightforward calculations of @xmath0 using the 1-loop improved @xmath2 operator fail due to the large mixing with the wrong chirality operators @xcite . since this mixing is an artifact of lattice discretization , one hopes that it can be significantly reduced by improving the action . by comparing results obtained using the wilson and the tadpole improved clover action ( @xmath3 ) on the same quenched gauge lattices ( 170 lattices of size @xmath4 at @xmath5 ) we show that this is indeed the case . [ f : bkw ] shows the wilson and clover data as a function of @xmath6 . for each data set , @xmath0 is written as the sum of two parts @xmath7 the contribution of the diagonal ( the 1-loop tadpole improved @xmath8 ) operator , and the mixing term which is proportional to @xmath9 . the general form , ignoring chiral logarithms and terms proportional to @xmath10 , for @xmath11 is @xcite @xmath12 the coefficients @xmath13 are pure artifacts , therefore their value can be used to quantify improvement . of these @xmath14 is the most serious as it causes @xmath0 to diverge in the chiral limit . the divergence , in the limit @xmath15 , of the diagonal term due to a non - zero @xmath14 is evident in fig . [ f : bkw ] for wilson fermions . this artifact is only partially cancelled by the 1-loop mixing operator . the situation is considerably improved with clover fermions . the corresponding values at @xmath16 mev are @xmath17 whereas @xmath18 . this improvement arises because the two dominant artifacts @xmath19 and @xmath20 are significantly reduced ; @xmath21 versus @xmath22 , and @xmath23 versus @xmath24 . -0.8 cm -0.6 cm [ f : bkw ] as explained in @xcite , the contributions proportional to @xmath13 can be removed completely by studying the momentum dependence of the matrix elements . short of calculating the mixing coefficients non - perturbatively , the way to remove the artifacts in @xmath25 is to extrapolate to @xmath26 . we have done the calculation at @xmath27 only , where our final results are @xmath28 and @xmath29 for wilson and clover formulations respectively . the benchmark value , including @xmath30 extrapolation , is @xmath31 , as obtained by the jlqcd collaboration @xcite . the chiral condensate @xmath32 is not simply related to the trace of the wilson quark propagator @xmath33 . the breaking of chiral symmetry by the @xmath34 term introduces contact terms that need to be subtracted non - perturbatively from @xmath33 @xcite . this has not proven practical . instead , the methods of choice are to either evaluate the right hand side of the continuum ward identity @xmath35 or cast the gell - mann , oakes , renner relation @xmath36 in terms of lattice correlation functions @xcite . these estimates have errors of both @xmath37 and @xmath38 , and at fixed @xmath39 are therefore expected to agree only in the chiral limit . a comparison of the efficacy of the two methods is shown in fig . [ f : xbarx ] . we find that a reliable extrapolation to the chiral limit can be made using a linear fit , and the two methods give consistent results for both wilson and clover fermions . also , the @xmath38 corrections are significantly smaller for clover fermion . -0.8 cm -0.6 cm [ f : xbarx ] in ref . @xcite we presented a detailed analysis of mass - splittings in the baryon octet and decuplet with wilson fermions . we had found a large non - linear dependence on quark mass for the @xmath40 , @xmath41 , and @xmath42 splittings . extrapolation of the data to the physical masses including these non - linearities gave estimates consistent with observed values . on the other hand we had found a surprisingly good linear fit to the decuplet masses , and the splittings were underestimated by @xmath43 . the data with clover fermions show the same qualitative features . as an illustration , we show a comparison of the @xmath44 splitting in fig . [ f : siglam ] . details of the analysis will be published elsewhere @xcite . -0.8 cm -0.6 cm [ f : siglam ] the improvement coefficient for the axial current , @xmath1 , is calculated using the the axial wi @xcite . if the clover coefficient @xmath45 is tuned to its non - perturbative value @xmath46 at @xmath27 @xcite , the sum @xmath47 of quark masses defined by @xmath48^{(12)}(\vec{x},t ) j^{(21)}(0 ) \rangle } { \sum_{\vec{x } } \langle p^{(12)}(\vec{x},t ) j^{(21)}(0 ) \rangle } \label{ca } \end{aligned}\ ] ] should be independent of @xmath49 and the initial pseudoscalar state created by @xmath50 , up to corrections of @xmath51 . we vary the composition of the initial state by using @xmath52 or @xmath53 and by using `` wall '' or `` wuppertal '' smearing functions in the calculation of the quark propagators . the results in fig . [ f : ca ] show a large dependence on the initial state for wilson fermions and almost none already for @xmath3 ! we estimate @xmath54 from this clover data , whereas the alpha collaboration report @xmath55 at @xmath56 @xcite . we are repeating the calculation at @xmath56 to understand this difference . -0.8 cm -0.6 cm [ f : ca ] the explicit breaking of chiral symmetry in wilson - like fermions gives rise to the problem of `` exceptional configurations '' in the quenched theory . the cause is that the wilson @xmath34 term breaks the anti - hermitian property of the massless dirac operator . as a result , zero modes of the dirac operator extend into the physical region @xmath57 . thus , on a given configuration , as the quark mass is lowered and approaches the first of the unphysical modes , one encounters exceptionally large fluctuations in the correlation functions . such configurations dominate the ensemble average and as discussed in @xcite there is no basis for excluding them . tuning @xmath58 reduces the @xmath37 chiral symmetry breaking artifacts as shown above , however , it does not reduce this problem @xcite . we find , by comparing fluctuations in 2-point and 3-point correlation functions between wilson and clover fermions , that the problem , in fact , gets worse . a deeper understanding of the persistence of the zero mode problem even though the chiral behavior is improved is missing . this work was supported by the doe grand challenges award at the advanced computing lab at los alamos , and by the nato collaborative research grant , contract no . 940451 . 9 r. gupta , , ( 1997 ) 4036 . jlqcd collaboration , ( 1998 ) 5271 . m. bochicchio , , ( 1985 ) 331 . d. daniel , , ( 1992 ) 3130 . t. bhattacharya , , ( 1996 ) 6486 . t. bhattacharya , , in preparation . m. lscher _ etal . _ , nuc.phy . * b491 * ( 1997 ) 323 . -1 w. bardeen , , ( 1998 ) 1633 .
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we present evidence for improvement with tadpole improved clover fermions based on an analysis of the chiral behavior of @xmath0 and the quark condensate . also presented
are a comparison of the mass splittings in the baryon octet and decuplet , a calculation of @xmath1 using standard 2-point correlation functions , and the problem of zero modes of the dirac operator .
# 1#1
| 2,134 | 106 |
while sports are often analogized to a wide array of other arenas of human activity particularly war well known story lines and elements of sports are conversely invoked to describe other spheres . each game generates a probablistic , rule - based story @xcite , and the stories of games provide a range of motifs which map onto narratives found across the human experience : dominant , one - sided performances ; back - and - forth struggles ; underdog upsets ; and improbable comebacks . as fans , people enjoy watching suspenseful sporting events unscripted stories and following the fortunes of their favorite players and teams @xcite . despite the inherent story - telling nature of sporting contests and notwithstanding the vast statistical analyses surrounding professional sports including the many observations of and departures from randomness @xcite the ecology of game stories remains a largely unexplored , data - rich area @xcite . we are interested in a number of basic questions such as whether the game stories of a sport form a spectrum or a set of relatively isolated clusters , how well models such as random walks fare in reproducing the specific shapes of real game stories , whether or not these stories are compelling to fans , and how different sports compare in the stories afforded by their various rule sets . here , we focus on australian rules football , a high skills game originating in the mid 1800s . we describe australian rules football in brief and then move on to extracting and evaluating the sport s possible game stories . early on , the game evolved into a winter sport quite distinct from other codes such as soccer or rugby while bearing some similarity to gaelic football . played as state - level competitions for most of the 1900s with the victorian football league ( vfl ) being most prominent , a national competition emerged in the 1980s with the australian football league ( afl ) becoming a formal entity in 1990 . the afl is currently constituted by 18 teams located in five of australia s states . games run over four quarters , each lasting around 30 minutes ( including stoppage time ) , and teams are each comprised of 18 on - field players . games ( or matches ) are played on large ovals typically used for cricket in the summer and of variable size ( generally 135 to 185 meters in length ) . the ball is oblong and may be kicked or handballed ( an action where the ball is punched off one hand with the closed fist of the other ) but not thrown . marking ( cleanly catching a kicked ball ) is a central feature of the game , and the afl is well known for producing many spectacular marks and kicks for goals @xcite . the object of the sport is to kick goals , with the customary standard of highest score wins ( ties are relatively rare but possible ) . scores may be 6 points or 1 point as follows , some minor details aside . each end of the ground has four tall posts . kicking the ball ( untouched ) through the central two posts results in a ` goal ' or 6 points . if the ball is touched or goes through either of the outer two sets of posts , then the score is a ` behind ' or 1 point . final scores are thus a combination of goals ( 6 ) and behinds ( 1 ) and on average tally around 100 per team . poor conditions or poor play may lead to scores below 50 , while scores above 200 are achievable in the case of a ` thrashing ' ( the record high and low scores are 239 and 1 ) . wins are worth 4 points , ties 2 points , and losses 0 . of interest to us here is that the afl provides an excellent test case for extracting and describing the game story space of a professional sport . we downloaded 1,310 afl game scoring progressions from http://www.afltables.com[http://afltables.com ] ( ranging from the 2008 season to midway through the 2014 season ) @xcite . we extracted the scoring dynamics of each game down to second level resolution , with the possible events at each second being ( 1 ) a goal for either team , ( 2 ) a behind for either team , or ( 3 ) no score @xcite . each game thus affords a ` worm ' tracking the score differential between two teams . we will call these worms ` game stories ' and we provide an example in fig . [ fig : sog.example_worm ] . the game story shows that geelong pulled away from hawthorn their great rival over the preceding decade towards the end of a close , back and forth game . each game story provides a rich representation of a game s flow , and , at a glance , quickly indicates key aspects such as largest lead , number of lead changes , momentum swings , and one - sidedness . and game stories evidently allow for a straightforward quantitative comparison between any pair of matches . for the game story ecology we study here , an important aspect of the afl is that rankings ( referred to as the ladder ) , depend first on number of wins ( and ties ) , and then percentage of ` points for ' versus ` points against ' . teams are therefore generally motivated to score as heavily as possible while still factoring in increased potential for injury . we order the paper as follows . in sec . [ sec : sog.basics ] , we first present a series of basic observations about the statistics of afl games . we include an analysis of conditional probabilities for winning as a function of lead size . we show through a general comparison to random walks that afl games are collectively more diffusive than simple random walks leading to a biased random walk null model based on skill differential between teams . we then introduce an ensemble of 100 sets of 1,310 biased random walk game stories which we use throughout the remainder of the paper . in secs . [ sec : sog.gameshapes ] and [ sec : sog.gamemotifs ] , we demonstrate that game stories form a spectrum rather than distinct clusters , and we apply coarse - graining to elucidate game story motifs at two levels of resolution . we then provide a detailed comparison between real game motifs and the smaller taxonomy of motifs generated by our biased random walk null model . we explore the possibility of predicting final game margins in sec . [ sec : sog.prediction ] . we offer closing thoughts and propose further avenues of analysis in sec . [ sec : sog.conclusion ] . while every afl game is officially comprised of four 20 minute quarters of playing time , the inclusion of stoppage time means there is no set quarter or game length , resulting in some minor complications for our analysis . we see an approximate gaussian distribution of game lengths with the average game lasting a little over two hours at 122 minutes , and 96% of games run for around 112 to 132 minutes ( @xmath0 minutes ) . in comparing afl games , we must therefore accommodate different game lengths . a range of possible approaches include dilation , truncation , and extension ( by holding a final score constant ) , and we will explain and argue for the latter in sec . [ sec : sog.gameshapes ] . in post - game discussions , commentators will often focus on the natural chapters of a given sport . for quarter - based games , matches will sometimes be described as ` a game of quarters ' or ` a tale of two halves . ' for the afl , we find that scoring does not , on average , vary greatly as the game progresses from quarter to quarter ( we will however observe interesting quarter - scale motifs later on ) . for our game database , we find there is slightly more scoring done in the second half of the game ( 46.96 versus 44.91 ) , where teams score one more point , on average , in the fourth quarter versus the first quarter ( 23.48 versus 22.22 ) . this minor increase may be due to a heightened sense of the importance of each point as game time begins to run out , the fatiguing of defensive players , or as a consequence of having ` learned an opponent ' @xcite . -72 , -71 to -66 , , -6 to -1 , 1 to 6 , 7 to 12 , , @xmath1 72 . as for most sports , the probability of scoring next increases approximately linearly as a function of current lead size . ] in fig . [ fig : sog.conditional_scoring ] , we show that , as for a number of other sports , the probability of scoring next ( either a goal or behind ) at any point in a game increases linearly as a function of the current lead size ( the national basketball association is a clear exception ) @xcite . this reflects a kind of momentum gain within games , and could be captured by a simple biased model with scoring probability linearly tied to the current lead . other studies have proposed this linearity to be the result of a heterogeneous skill model @xcite , and , as we describe in the following section , we use a modification of such an approach . we next examine the conditional probability of winning given a lead of size @xmath2 at a time point @xmath3 in a game , @xmath4 . we consider four example time points the end of each of the first three quarters and with 10 minutes left in game time and plot the results in fig . [ fig : sog.conditional_winning_quarters ] . we fit a sigmoid curve ( see caption ) to each conditional probability . as expected , we immediately see an increase in winning probability for a fixed lead as the game progresses . these curves could be referenced to give a rough indication of an unfolding game s likely outcome and may be used to generate a range of statistics . as an example , we define likely victory as @xmath5 and find @xmath2 = 32 , 27 , 20 , and 11 are the approximate corresponding lead sizes at the four time points . losing games after holding any of these leads might be viewed as ` snatching defeat from the jaws of victory . ' similarly , if we define close games as those with @xmath6 , we find the corresponding approximate lead sizes to be @xmath7 6 , 5 , 4 , and 2 . these leads could function in the same way as the save statistic in baseball is used , i.e. , to acknowledge when a pitcher performs well enough in a close game to help ensure their team s victory . expanding beyond the afl , such probability thresholds for likely victory or uncertain outcome may be modified to apply to any sport , and could be greatly refined using detailed information such as recent performances , stage of a season , and weather conditions . at the end of the first three quarters * ( a c ) * and with 10 minutes to go in the game * ( d)*. bins are comprised of the aggregate of every 6 points as in fig . [ fig : sog.conditional_scoring ] . the dark blue curve is a sigmoid function of the form @xmath8^{-1 } $ ] where @xmath9 and @xmath10 are fit parameters determined via standard optimization using the python function scipy.optimize.curve_fit ( note that @xmath10 should be 0 by construction . ) as a game progresses , the threshold for likely victory ( winning probability 0.90 , upper red lines ) decreases as expected , as does a threshold for a close game ( probability of 0.60 , lower red line ) . the slope of the sigmoid curve increases as the game time progresses showing the evident greater impact of each point . we note that the missing data in panel * a * is a real feature of the specific 1,310 games in our data set . , scaledwidth=49.0% ] a natural null model for a game story is the classic , possibly biased , random walk @xcite . we consider an ensemble of modified random walks , with each walk ( 1 ) composed of steps of @xmath11 6 and @xmath11 1 , ( 2 ) dictated by a randomly drawn bias , ( 3 ) running for a variable total number of events , and ( 4 ) with variable gaps between events , all informed by real afl game data . for the purpose of exploring motifs later on , we will create 100 sets of 1,310 games . an important and subtle aspect of the null model is the scoring bias , which we will denote by @xmath12 . we take the bias for each game simulation to be a proxy for the skill differential between two opposing teams , as in @xcite , though our approach involves an important adjustment . in @xcite , a symmetric skill bias distribution is generated by taking the relative number of scoring events made by one team in each game . for example , given a match between two teams @xmath13 and @xmath14 , we find the number of scoring events generated by @xmath15 , @xmath16 , and the same for @xmath17 , @xmath18 . we then estimate a posteriori the skill bias between the two teams as : @xmath19 in constructing the distribution of @xmath12 , @xmath20 , we discard information regarding how specific teams perform against each other over seasons and years , and we are thus only able to assign skill bias in a random , memoryless fashion for our simulations . we also note that for games with more than one value of points available for different scoring events ( as in 6 and 1 for australian rules football ) , the winning team may register less scoring events than the losing one . represents a team s relative ability to score against another team and is estimated a posteriori by the fraction of scoring events made by each team eq . ( [ sog.eq:balance ] ) . * a * and * b * : kolmogorov - smirnov test @xmath21 statistic and associated @xmath22-value comparing the observed output skill bias distribution produced by a presumed input skill distribution @xmath23 with that observed for all afl games in our data set , where @xmath23 is gaussian with mean @xmath24 and its standard deviation @xmath25 is the variable of interest . for each value of @xmath25 , we created 1,000 biased random walks with the bias @xmath12 drawn from the corresponding normal distribution . each game s number of events was drawn from a distribution of the number of events in real afl games ( see text ) . plot * b * is an expanded version of the shaded region in * a * with finer sampling . we estimated the best fit to be @xmath26 , and we compare the resulting observed bias distribution with that of @xcite in fig . [ fig : sog.resulting_event_biases ] . ] in @xcite , random walk game stories were then generated directly using @xmath20 . however , for small time scales this is immediately problematic and requires a correction . consider using such an approach on pure random walks . we of course have that @xmath27 by construction , but our estimate of @xmath20 will be a gaussian of width @xmath28 , where we have normalized displacement by time @xmath3 . and while as @xmath29 , our estimate of @xmath20 approaches the correct distribution @xmath30 , we are here dealing with relatively short random walks . indeed , we observe that if we start with pure random walks , run them for , say , 100 steps , estimate the bias distribution , run a new set of random walks with these biases , and keep repeating this process , we obtain an increasingly flat bias distribution . to account for this overestimate of the spread of skill bias , we propose the tuning of an input gaussian distribution of skill biases so as to produce biased random walks whose outcomes best match the observed event biases for real games . we assume that @xmath23 should be centered at @xmath31 . we then draw from an appropriate distribution of number of events per game , and tune the standard deviation of @xmath23 , @xmath25 , to minimize the kolmogorov - smirnov ( ks ) @xmath21 statistic and maximize the @xmath22-value produced from a two - tailed ks test between the resulting distribution of event biases and the underlying , observed distribution for our afl data set . we show the variation of @xmath21 and the @xmath22-value as a function of @xmath25 in fig . [ fig : sog.sigma_fitting ] . we then demonstrate in fig . [ fig : sog.resulting_event_biases ] that the @xmath25-corrected distribution produces an observably better approximation of outcomes than if we used the observed biases approach of @xcite . because the fit for our method in fig . [ fig : sog.resulting_event_biases ] is not exact , a further improvement ( unnecessary here ) would be to allow @xmath23 to be arbitrary rather than assuming a gaussian . with a reasonable estimate of @xmath23 in hand , we create 100 ensembles of 1,310 null games where each game is generated with ( 1 ) one team scoring with probability @xmath12 drawn from the @xmath25-corrected distribution described above ; ( 2 ) individual scores being a goal or behind with probabilities based on the afl data set ( approximately 0.53 and 0.47 ) ; and ( 3 ) a variable number of events per simulation based on : ( a ) game duration drawn from the approximated normal distribution described in sec . [ sec : sog.basics ] , and ( b ) time between events drawn from a chi - squared distribution fit to the inter - event times of real games . given in eq . ( [ sog.eq:balance ] ) , dashed blue curve ) with that produced by two approaches : ( 1 ) we draw @xmath12 from a normal distribution using the best candidate @xmath25 value with mean @xmath32 as determined via fig . [ fig : sog.sigma_fitting ] ( red curve ) , and ( 2 ) we choose @xmath12 from the complete list of observed biases from the afl ( green curve , the replication method of @xcite ) . for the real and the two simulated distributions , both @xmath12 and @xmath33 are included for symmetry . the fitted @xmath25 approach produces a more accurate estimate of the observed biases , particularly for competitive matches ( @xmath12 close to 0.50 ) and one sided affairs . inset : upper half of the distributions plotted on a semi - logarithmic scale ( base 10 ) revealing that the replication method of @xcite also over produces extreme biases , as compared to the afl and our proposed correction using a numerically determined @xmath25 . ] , ( solid blue curve ) . we perform fits in logarithmic space using standard least squares regression ( solid black curve for real games , dashed black for the null model ) . the biased random walks satisfactorily reproduce the observed scaling of variance . it thus appears that afl games stories do not exhibit inherently superdiffusive behavior but rather result from imbalances between opposing teams . ] for a secondary test on the validity of our null model s game stories , we compute the variance @xmath34 of the margin at each event number @xmath35 for both afl games and modified random walks ( for the afl games , we orient each walk according to home and away status , the default ordering in the data set ) . as we show in fig . [ fig : sog.biased_margin_variance ] , we find that both afl games and biased random walks produce game stories with @xmath36 and @xmath37 respectively . collectively , afl games thus have a tendency toward runaway score differentials , and while superdiffusive - like , this superlinear scaling of the variance can be almost entirely accounted for by our incorporation of the skill bias distribution @xmath23 . ) . in all examples , dark gray curves denote the game story . for the shorter game of each pair , horizontal solid blue lines show how we hold the final score constant to equalize lengths of games . ] before moving on to our main focus , the ecology of game stories , we define a straightforward measure of the distance between any pair of games . for any sport , we define a distance measure between two games @xmath38 and @xmath39 as @xmath40 where @xmath41 is the length of the game in seconds , and @xmath42 is the score differential between the competing teams in game @xmath38 at second @xmath3 . we orient game stories so that the team whose score is oriented upwards on the vertical axis wins or ties [ i.e. , @xmath43 . by construction , pairs of games which have a relatively small distance between them will have similar game stories . the normalization factor @xmath44 means the distance remains in the units of points and can be thought of as the average difference between point differentials over the course of the two games . in the case of the afl , due to the fact that games do not run for a standardized time @xmath41 , we extend the game story of the shorter of the pair to match the length of the longer game by holding the final score constant . while not ideal , we observe that the metric performs well in identifying games that are closely related . we investigated several alternatives such as linearly dilating the shorter game , and found no compelling benefits . dilation may be useful in other settings but the distortion of real time is problematic for sports . in fig . [ fig : sog.closest_games ] , we present the ten most similar pairs of games in terms of their stories . these close pairs show the metric performs as it should and that , moreover , proximal games are not dominated by a certain type . figs . [ fig : sog.closest_games]a and [ fig : sog.closest_games]b demonstrate a team overcoming an early stumble , figs . [ fig : sog.closest_games]e and [ fig : sog.closest_games]f showcase the victor repelling an attempted comeback , figs . [ fig : sog.closest_games]q and [ fig : sog.closest_games]r exemplify a see - saw battle with many lead changes , and fig . [ fig : sog.closest_games]s and [ fig : sog.closest_games]t capture blowouts one team taking control early and continuing to dominate the contest . having described and implemented a suitable metric for comparing games and their root story , we seek to group games together with the objective of revealing large scale characteristic motifs . to what extent are well - known game narratives from blowouts to nail - biters to improbable comebacks and potentially less well known story lines featured in our collection of games ? and how does the distribution of real game stories compare with those of our biased random walk null model ? ( we note that in an earlier version of the present paper , we considered pure , unbiased random walks for the null model @xcite . ) ) ; * ( c ) * the same as * ( b ) * but with game indices cycled to make the continuous spectrum of games evident . we include only every 20th game for clarity and note that such shuffling is usually performed for entities on a line rather than a ring . the games at the end of the spectrum are most dissimilar and correspond to runaway victories and comebacks ( see also fig . [ fig : sog.dendro_heat ] ) . ] . a noticeable split is visible between the blowout games ( first six clusters ) and the comeback victories ( last three clusters ) . we plot dendrograms along both the top and left edges of the matrix , and as explained in sec . [ subsec:18motifs ] , the boxed numbers reference the 18 motifs found when the average intra - cluster distance is set to 11 points . these 18 motifs are variously displayed in figs . [ fig : sog.biased18motifs ] and [ fig : sog.realmotifs18ratio - random ] . ] we first compute the pairwise distance between all games in our data set . we then apply a shuffling algorithm to order games on a discretized ring so that similar games are as close to each other as possible . specifically , we minimize the cost @xmath45 where @xmath46 is the shortest distance between @xmath38 and @xmath39 on the ring . at each step of our minimization procedure , we randomly choose a game and determine which swap with another game most reduces @xmath47 . we use @xmath48 by choice and other powers give similar results . in fig . [ fig : sog.minimize_by_shuffling_ring100_005 ] , we show three heat maps for distance @xmath21 with : ( a ) games unsorted ; ( b ) games sorted according to the above minimization procedure ; and ( c ) indices of sorted games cycled to reveal that afl games broadly constitute a continuous spectrum . as we show below , at the ends of the spectrum are the most extreme blow outs , and the strongest comebacks i.e . , one team dominates for the first half and then the tables are flipped in the second half . while little modularity is apparent there are no evident distinct classes of games we may nevertheless perform a kind of coarse - graining via hierarchical clustering to extract a dendrogram of increasingly resolved game motifs . even though we have just shown that the game story ecology forms a continuum , it is important that we stress that the motifs we find should not be interpreted as well separated clusters . adjacent motifs will have similar game stories at their connecting borders . a physical example might be the landscape roughness of equal area regions dividing up a country two connected areas would typically be locally similar along their borders . having identified a continuum , we are simply now addressing the variation within that continuum using a range of scales . as a function of cluster number @xmath49 . red lines mark the first occurrence in which the average of the intra cluster distance of the n motif clusters had a value below 12 , 11 , 10 , 9 , and 8 ( red text beside each line ) points respectively . the next cut for 7 points gives 343 motifs . ] we employ a principled approach to identifying meaningful levels of coarse - graining , leading to families of motifs . as points are the smallest scoring unit in afl games , we use them to mark resolution scales as follows . first , we define @xmath50 , the average distance between games within a given cluster @xmath38 as @xmath51 here @xmath39 and @xmath9 are games placed in cluster @xmath38 , @xmath52 is the number of games in cluster @xmath38 , and @xmath21 is the game distance defined in eq . ( [ eq : sog.gamedist ] ) . at a given depth @xmath53 of the dendrogram , we compute @xmath54 for each of the @xmath55 clusters found , and then average over all clusters to obtain an average intra - cluster distance : @xmath56 we use ward s method of variance to construct a dendrogram @xcite , as shown in fig . [ fig : sog.dendro_heat ] . ward s method aims to minimize the within cluster variance at each level of the hierarchy . at each step , the pairing which results in the minimum increase in the variance is chosen . these increases are measured as a weighted squared distance between cluster centers . we chose ward s method over other linkage techniques based on its tendency to produce clusters of comparable size at each level of the hierarchy . at the most coarse resolution of two categories , we see in fig . [ fig : sog.dendro_heat ] that one sided contests are distinguished from games that remain closer , and repeated analysis using @xmath9-means clustering suggests the same presence of two major clusters . as we are interested in creating a taxonomy of more particular , interpretable game shapes , we opt to make cuts as @xmath57 first falls below an integer number of points , as shown in fig . [ fig : sog.point_breaks ] ( we acknowledge that @xmath57 does not perfectly decrease monotonically ) . as indicated by the red vertical lines , average intra - cluster point differences of 12 , 11 , 10 , 9 , and 8 correspond to 9 , 18 , 30 , 71 , and 157 distinct clusters . our choice , which is tied to a natural game score , has a useful outcome of making the number of clusters approximately double with every single point in average score differential . . for real games , we obtain by comparison 18 and 71 motifs ( vertical red lines in * a * and * b * ) , which exceeds all 100 motif numbers in both cases and indicates afl game stories are more diverse than our null model would suggest . ] first drops below 11 points . in each panel , the main curves are the motifs the average of all game stories ( shown as light gray curves in background ) within each cluster , and we arrange clusters in order of the motif winning margin . all motifs are shown with the same axis limits . numbers of games within each cluster are indicated in the bottom right corner of each panel along with the average number of the nearest biased random walk games ( normalized per 1,310 ) . motif colors correspond to relative abundance of real versus random game ratio @xmath58 as red : @xmath59 ; gray : @xmath60 ; and blue : @xmath61 . see fig . [ fig : sog.realmotifs18ratio - random ] for the same motifs reordered by real game to random ratio . ] in the remainder of section [ sec : sog.gamemotifs ] , we show and explore in some depth the taxonomies provided by 18 and 71 motifs at the 11 and 9 point cutoff scales . we first show that for both cutoffs , the number of motifs produced by the biased random walk null model is typically well below the number observed for the real game . in fig . [ fig : sog.randomwalk_motif_numbers ] , we show histograms of the number of motifs found in the 100 ensembles of 1,310 null model games with the real game motif numbers of 18 and 71 marked by vertical red lines . the number of random walk motifs is variable with both distributions exhibiting reasonable spread , and also in both cases , the maximum number of motifs is below the real game s number of motifs . these observations strongly suggest that afl generates a more diverse set of game story shapes than our random walk null model . we now consider the 18 motif characterization which we display in fig . [ fig : sog.biased18motifs ] by plotting all individual game stories in each cluster ( light gray curves ) and overlaying the average motif game story ( blue / gray / red curves , explained below ) . all game stories are oriented so that the winning team aligns with the positive vertical axis , i.e. , @xmath62 ( in the rare case of a tie , we orient the game story randomly ) . and motifs are ordered by their final margin ( descending ) . in all presentations of motifs that follow , we standardize final margin as the principle index of ordering . we display the final margin index in the top center of each motif panel to ease comparisons when motifs are ordered in other ways ( e.g. , by prevalence in the null model ) . we can now also connect back to the heat map of fig . [ fig : sog.dendro_heat ] where we use the same indices to mark the 18 motifs . in the bottom right corner of each motif panel , we record two counts : ( 1 ) the number of real games belonging to the motif s cluster ; and ( 2 ) the average number of our ensemble of 100 @xmath63 1,310 biased random walk games ( see sec . [ sec : sog.randomwalks ] ) which are closest to the motif according to eq . ( [ eq : sog.gamedist ] ) . for each motif , we compute the ratio of real to random adjacent game stories , @xmath58 , and , as a guide , we color the motifs as * red if @xmath59 ( real game stories are more abundant ) ; * gray if @xmath60 ( counts of real and random game stories are close ) ; and * blue if @xmath61 ( random game stories are more abundant ) . we immediately observe that the number of games falling within each cluster is highly variable , with only 3 in the most extreme blowout motif ( # 1 , fig . [ fig : sog.biased18motifs]a / fig . [ fig : sog.realmotifs18ratio - random]a ) and 169 in a gradual - pulling - away motif ( # 8 , fig . [ fig : sog.biased18motifs]h / fig . [ fig : sog.realmotifs18ratio - random]b ) . but reordered according to decreasing ratio of adjacent real to biased random games , @xmath58 , and with closest biased random walk rather than real game stories plotted underneath in light gray . see the caption for fig . [ fig : sog.biased18motifs ] for more details . ] the average motif game stories in fig . [ fig : sog.biased18motifs ] provide us with the essence of each cluster , and , though they do not represent any one real game , they are helpful for the eye in distinguishing clusters . naturally , by applying further coarse - graining as we do below , we will uncover a richer array of more specialized motifs . looking at figs . [ fig : sog.biased18motifs ] and [ fig : sog.realmotifs18ratio - random ] , we now clearly see a continuum of game shapes ranging from extreme blowouts ( motif # 1 ) to extreme comebacks , both successful ( motif # 17 ) and failed ( motif # 18 ) . we observe that while some motifs have qualitatively similar story lines , a game motif that has a monotonically increasing score differential that ends with a margin of 200 ( # 1 ) is certainly different from one with a final margin of 50 ( # 6 ) . in considering this induced taxonomy of 18 game motifs , we may interpret the following groupings : * # 1#6 , # 8 : one - sided , runaway matches ; * # 9 : losing early on , coming back , and then pulling away ; * # 7 and # 10 : initially even contests with one side eventually breaking away ; * # 11 and # 12 : one team taking an early lead and then holding on for the rest of the game ; * # 13 , # 14 , and # 16 : variations on tight contests ; * # 15 and # 17 : successful comebacks ; * # 18 : failed comebacks . we note that the game stories attached to each motif might not fit these descriptions we are only categorizing motifs . as we move to finer grain taxonomies , the neighborhood around motifs diminishes and the connection between the shapes of motifs will become increasingly congruent with its constituent games . the extreme blowout motif for real games has relatively fewer adjacent random walk game stories ( fig . [ fig : sog.realmotifs18ratio - random]a ) , as do the two successful comeback motifs ( fig . [ fig : sog.realmotifs18ratio - random]c and fig . [ fig : sog.realmotifs18ratio - random]f ) , and games with a lead developed by half time that then remains stable ( fig . [ fig : sog.realmotifs18ratio - random]e ) . a total of 5 motifs show a relatively even balance between real and random ( i.e. , within 10% ) including two of the six motifs with the tightest finishes ( figs . [ fig : sog.realmotifs18ratio - random]h and [ fig : sog.realmotifs18ratio - random]i ) . biased random walks most overproduce games in which an early loss is turned around strongly ( fig . [ fig : sog.realmotifs18ratio - random]q ) or an early lead is maintained ( fig . [ fig : sog.realmotifs18ratio - random]r ) . in terms of game numbers behind motifs , we find a reasonable balance with 603 ( 46.0% ) having @xmath59 ( 7 motifs ) , 430 ( 32.8% ) with @xmath60 ( 5 motifs ) , and 277 ( 21.1% ) with @xmath61 ( 6 motifs ) . depending on the point of view of the fan and again at this level of 18 motifs , we could argue that certain real afl games that feature more often that our null model would suggest are more or less ` interesting ' . for example , we see some dominating wins are relatively more abundant in the real game ( # 1 , # 2 , and # 4 ) . while such games are presumably gratifying for fans of the team handing out the ` pasting ' , they are likely deflating for the supporters of the losing team . and a neutral observer may or may not enjoy the spectacle of a superior team displaying their prowess . real games do exhibit relatively more of the two major comeback motifs ( # 15 and # 17)certainly exciting in nature though less of the failed comebacks ( # 18 ) . increasing our level of resolution corresponding to an average intra - cluster game distance of @xmath64 , we now resolve the afl game story ecology into 71 clusters . we present all 71 motifs in figs . [ fig : sog.biased71motifs - final - margin - order ] and [ fig : sog.biased71motifs - ratio - order ] , ordering by final margin and real - to - random game story ratio @xmath58 respectively ( we will refer to motif number and fig . [ fig : sog.biased71motifs - ratio - order ] so readers may easily connect to the orderings in both figures ) . with a greater number of categories , we naturally see a more even distribution of game stories across motifs with a minimum of 1 ( motif # 1 , fig . [ fig : sog.biased71motifs - ratio - order]ac ) and a maximum of 48 ( motif # 43 , fig . [ fig : sog.biased71motifs - ratio - order]ah ) . as for the coarser 18 motif taxonomy , we again observe a mismatch between real and biased random walk games . for example , motif # 14 ( fig . [ fig : sog.biased71motifs - ratio - order]af ) is an average of 25 real game stories compared with on average 15.13 adjacent biased random walks while motif # 20 ( fig . [ fig : sog.biased71motifs - ratio - order]cs ) has @xmath58=10/22.67 . using our 10% criterion , we see 25 motifs have @xmath59 ( representing 553 games or 42.2% ) , 23 have @xmath60 ( 420 games , 32.0% ) , and the remaining 23 have @xmath61 ( 337 games , 25.7% ) . generally , we again see blowouts are more likely in real games . however , we also find some kinds of comeback motifs are also more prevalent ( @xmath59 ) though not strongly in absolute numbers ; these include the failed comebacks in motifs # 67 ( fig . [ fig : sog.biased71motifs - ratio - order]ad ) and # 71 ( fig . [ fig : sog.biased71motifs - ratio - order]ae ) , and the major comeback in motif # 64 ( fig . [ fig : sog.biased71motifs - ratio - order]ab ) . in fig . [ fig : sog.biasedratio_vs_margin_18_and_71 ] , we give summary plots for the 18 and 71 motif taxonomies with motif final margin as a function of the of the real - to - random ratio @xmath58 . the larger final margins of the blowout games feature on the right of these plots ( @xmath59 ) , and , in moving to the left , we see a gradual tightening of games as shapes become more favorably produced by the random null model ( @xmath61 ) . the continuum of game stories is also reflected in the basic similarity of the two plots in fig . [ fig : sog.biasedratio_vs_margin_18_and_71 ] , made as they are for two different levels of coarse - graining . returning to figs . [ fig : sog.biased71motifs - final - margin - order ] and [ fig : sog.biased71motifs - ratio - order ] , we highlight ten examples in both reinforcements and refinements of motifs seen at the 18 motif level . we frame them as follows ( in order of decreasing @xmath58 and referencing fig . [ fig : sog.biased71motifs - ratio - order ] ) : * fig . [ fig : sog.biased71motifs - ratio - order]ab , # 64 ( @xmath65 ) : the late , great comeback ; * fig . [ fig : sog.biased71motifs - ratio - order]ae , # 71 ( @xmath66 ) : the massive comeback that just falls short ; * fig . [ fig : sog.biased71motifs - ratio - order]aj , # 52 ( @xmath67 ) : comeback over the first half connecting into a blowout in the second ( the winning team may be said to have ` turned the corner ' ) ; * fig . [ fig : sog.biased71motifs - ratio - order]am , motif # 13 ( @xmath68 ) : an exemplar blowout ( and variously a shellacking , thrashing , or hiding ) ; * fig . [ fig : sog.biased71motifs - ratio - order]ax , # 55 ( @xmath69 ) : rope - a - dope ( taking steady losses and then surging late ) ; * fig . [ fig : sog.biased71motifs - ratio - order]bz , # 68 ( @xmath70 ) : hold - slide - hold - surge ; * fig . [ fig : sog.biased71motifs - ratio - order]cd , # 56 ( @xmath71 ) : see - saw battle ; * fig . [ fig : sog.biased71motifs - ratio - order]ck , # 62 ( @xmath72 ) : the tightly fought nail - biter ( or heart stopper ) ; * fig . [ fig : sog.biased71motifs - ratio - order]cp , # 50 ( @xmath73 ) : burn - and - hold ( or the game - manager , or the always dangerous playing not - to - lose ) ; * fig . [ fig : sog.biased71motifs - ratio - order]cq , # 36 ( @xmath74 ) : surge - slide - surge . these motifs may also be grouped according to the number of ` acts ' in the game . motif # 53 ( fig . [ fig : sog.biased71motifs - ratio - order]ao ) , for example , is a three - act story while motifs # 56 ( fig . [ fig : sog.biased71motifs - ratio - order]cd ) and # 68 ( fig . [ fig : sog.biased71motifs - ratio - order]bz ) exhibit four acts . we invite the reader to explore the rest of the motifs in fig . [ fig : sog.biased71motifs - ratio - order ] . for real afl games at the 18 and 71 motif levels , panels * a * and * b * respectively , with linear fits . on the right of each plot , extreme blowout motifs ending in high margins have no or relatively few adjacent random walks . ( red , @xmath59 ) . on the left , game stories are more well represented by random walks ( blue , @xmath61 ) . there is considerable variation however , particularly in the 71 motif case , and we certainly see some close finishes with @xmath75 ( e.g. , the massive comeback , motif # 71 , fig . [ fig : sog.biased71motifs - ratio - order]ae ) . ] can we improve our ability to predict the outcome of a game in progress by knowing how games with similar stories played out in the past ? does the full history of a game help us gain any predictive power over much simpler game state descriptions such as the current time and score differential ? in this last section , we explore prediction as informed by game stories , a natural application . suppose we are in the midst of viewing a new game . we know the game story @xmath76 from the start of the game until the current game time @xmath77 , where @xmath78 is the eventual length of game ( and is another variable which we could potentially predict ) . in part to help with presentation and analysis , we will use minute resolution ( meaning @xmath79 for @xmath80 ) . our goal is to use our database of completed games for which of course we know the eventual outcomes to predict the final margin of our new game , @xmath81 . we create a prediction model with two parameters : ( 1 ) @xmath49 : the desired number of analog games closest to our present game ; and ( 2 ) @xmath82 : the number of minutes going back from the current time for which we will measure the distance between games . for a predictor , we simply average the final margins of the @xmath49 closest analog games to @xmath76 over the interval @xmath83 $ ] . that is , at time @xmath3 , we predict the final margin of @xmath76 , @xmath84 , using @xmath82 minutes of memory and @xmath49 analog games as : @xmath85 where @xmath86 is the set of indices for the @xmath49 games closest to the current game over the time span @xmath83 $ ] , and @xmath87 is the final second of game @xmath38 . ) . we start with a game story @xmath76 ( red curve ) for which we know up until , for this example , 60 minutes ( @xmath88 ) . we find the @xmath89 closest game stories based on matching over the time period 45 to 60 minutes ( memory @xmath90 ) , and show these as gray curves . we indicate the average final score @xmath91 for these analog games with the horizontal blue curve . ] analogs and a memories of @xmath92 ( blue curve ) and @xmath93 ( green curve ) , compared with the naive model of assuming that the current leader will ultimately win ( red curve ) . ] for an example demonstration , in fig . [ fig : sog.prediction_example ] , we attempt to predict the outcome of an example game story given knowledge of its first 60 minutes ( red curve ) and by finding the average final margin of the @xmath94 closest games over the interval 45 to 60 minutes ( @xmath90 , shaded gray region ) . most broadly , we see that our predictor here would correctly call the winning team . at a more detailed level , the average final margin of the analog games slightly underestimates the final margin of the game of interest , and the range of outcomes for the 50 analog games is broad with the final margin spanning from around -40 to 90 points . having defined our prediction method , we now systematically test its performance after 30 , 60 , and 90 minutes have elapsed in a game currently under way . in aiming to find the best combination of memory and analog number , @xmath82 and @xmath49 , we use eq . ( [ eq : sog.prediction ] ) to predict the eventual winner of all 1,310 afl games in our data set at these time points . first , as should be expected , the further a game has progressed , the better our prediction . more interestingly , in fig . [ fig : sog.memory_vs_analogs ] we see that for all three time points , increasing @xmath49 elevates the prediction accuracy , while increasing @xmath82 has little and sometimes the opposite effect , especially for small @xmath49 . the current score differential serves as a stronger indicator of the final outcome than the whole game story shape unfolded so far . the recent change in scores momentum is also informative , but to a far lesser extent than the simple difference in scores at time @xmath3 . based on fig . [ fig : sog.memory_vs_analogs ] , we proceed with @xmath89 analogs and two examples of low memory : @xmath92 and @xmath93 . we compare with the naive model that , at any time @xmath3 , predicts the winner as being the current leader . we see in fig . [ fig : sog.prediction_performance ] that there is essentially no difference in prediction performance between the two methods . thus , memory does not appear to play a necessary role in prediction for afl games . of interest going forward will be the extent to which other sports show the same behavior . for predicting the final score , we also observe that simple linear extrapolation performs well on the entire set of the afl games ( not shown ) . nevertheless , we have thus far found no compelling evidence for using game stories in prediction , nuanced analyses incorporating game stories for afl and other professional sports may nevertheless yield substantive improvements over these simple predictive models @xcite . overall , we find that the sport of australian rules football presents a continuum of game types ranging from dominant blowouts to last minute , major comebacks . consequently , and rather than uncovering an optimal number of game motifs , we instead apply coarse - graining to find a varying number of motifs depending on the degree of resolution desired . we further find that ( 1 ) a biased random walk affords a reasonable null model for afl game stories ; ( 2 ) the scoring bias distribution may be numerically determined so that the null model produces a distribution of final margins which suitably matches that of real games ; ( 3 ) blowout and major comeback motifs are much more strongly represented in the real game whereas tighter games are generally ( but not entirely ) more favorably produced by a random model ; and ( 4 ) afl game motifs are overall more diverse than those of the random version . our analysis of an entire sport through its game story ecology could naturally be applied to other major sports such as american football , association football ( soccer ) , basketball , and baseball . a cross - sport comparison for any of the above analysis would likely be interesting and informative . and at a macro scale , we could also explore the shapes of win - loss progressions of franchises over years @xcite . it is important to reinforce that a priori , we were unclear as to whether there would be distinct clusters of games or a single spectrum , and one might imagine rough theoretical justifications for both . our finding of a spectrum conditions our expectations for other sports , and also provides a stringent , nuanced test for more advanced explanatory mechanisms beyond biased random walks , although we are wary of the potential difficulty involved in establishing a more sophisticated and still defensible mechanism . finally , a potentially valuable future project would be an investigation of the aesthetic quality of both individual games and motifs as rated by fans and neutral individuals @xcite . possible sources of data would be ( 1 ) social media posts tagged as being relevant to a specific game , and ( 2 ) information on game - related betting . would true fans rather see a boring blowout with their team on top than witness a close game @xcite ? is the final margin the main criterion for an interesting game ? to what extent do large momentum swings engage an audience ? such a study could assist in the implementation of new rules and policies within professional sports . d. p. kiley , a. j. reagan , l. mitchell , c. m. danforth , and p. s. dodds . the game story space of professional sports : australian rules fbootball . draft version of the present paper using pure random walk null model . available online at http://arxiv.org/abs/1507.03886v1 . accesssed january 17 , 2016 , 2015 . b. morris . skeptical football : patriots vs. cardinals and an interactive history of the nfl , 2014 . http://fivethirtyeight.com/features/skeptical-football-patriots-vs-cardinals-and-an-interactive-history-of-the-nfl/ , accessed on june 25 , 2015 . j. bryant and a. a. raney . sports on the screen . in d. zillmann and p. vorderer , editors , _ media entertainment : the psychology of its appeal _ , lea s communication series . , pages 153174 . lawrence erlbaum associates publishers , mahwah , nj , us , 2000 .
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sports are spontaneous generators of stories . through skill and chance ,
the script of each game is dynamically written in real time by players acting out possible trajectories allowed by a sport s rules . by properly characterizing a given sport s ecology of ` game stories ' , we are able to capture the sport s capacity for unfolding interesting narratives , in part by contrasting them with random walks . here
, we explore the game story space afforded by a data set of 1,310 australian football league ( afl ) score lines . we find that afl games exhibit a continuous spectrum of stories rather than distinct clusters .
we show how coarse - graining reveals identifiable motifs ranging from last minute comeback wins to one - sided blowouts . through an extensive comparison with biased random walks ,
we show that real afl games deliver a broader array of motifs than null models , and we provide consequent insights into the narrative appeal of real games .
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planetary nebulae ( pns ) have become increasingly important in extragalactic astronomy , for distance determinations via their luminosity function ( lf ) ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) , as kinematic tracers of the dark halos of galaxies @xcite , and as tracers for the distribution and kinematics of the diffuse stellar population in galaxy clusters @xcite . due to their strong narrow line emission at @xmath3 \lambda 5007 $ ] , pns can be easily detected out to distances beyond @xmath4 with narrow - band photometry and slitless spectroscopy @xcite , and to @xmath5 with multi - slit imaging spectroscopy @xcite . moreover , they are observed in elliptical and spiral galaxies , making them an indispensible tool to support distances obtained by other methods ( such as cepheids , surface brightness fluctuations , the tully - fisher relation , sne ia ) , and to measure the kinematics of stellar populations whose surface brightness is too faint for absorption line spectroscopy . for distance determination the planetary nebulae luminosity function ( pnlf ) is normally modeled as having a universal shape that depends only on the absolute bright magnitude cutoff @xmath6 : @xmath7 where @xmath8 is the number of pns with absolute magnitude @xmath9 @xcite . observationally , the cutoff magnitude @xmath6 has a quasi - universal value of @xmath10 with only a weak dependence on host galaxy metallicity expressed by the system s oxygen abundance , and which can be compensated for by a quadratic relation in @xmath11 $ ] @xcite . in practice , the pn magnitudes @xmath12 , after correcting for the interstellar reddening , are fitted to the model pnlf of eq . [ pnlfeqn ] convolved with the photometric error profile , yielding a value of the distance modulus @xcite . the absence of any systematic variations in @xmath6 and the pnlf shape has been verified in galaxies with significant population gradients , and among galaxies of different morphologies within galaxy clusters / groups up to virgo ( * ? ? ? * ; * ? ? ? * and references therein ) . this universality of the pnlf and the cutoff magnitude @xmath6 must be considered surprising , given that the pn luminosity in the @xmath3 \lambda 5007 $ ] line depends on the mass and metallicity of the central star , as well as on the electron gas temperature , optical thickness and dust extinction of the surrounding nebula . indeed , some current semi - analytic simulations of the pnlf seem to be at odds with the observational trends . @xcite indicate small possible dependencies of @xmath6 on the total size of the pn population , on the time elapsed since the last episode of star formation , and on how optically thin the pns are ; concluding , however , that only careful studies would detect such effects in the observed pnlf . in contrast , more recent pnlf simulations by @xcite contradict the observed narrow spread in @xmath6 and predict large variations of several magnitudes depending on a variety of realistic star formation and evolution scenarios . so is the pnlf truly quasi - universal and its cutoff magnitude nearly independent of population age and metallicity ? pns are also important as test particles to study the kinematics and dark matter distribution in the halos of elliptical galaxies . since the pn population is expected to arise from the underlying galactic stellar distribution , their radial velocities can be used as effective kinematic tracers of the mass distribution . however , the required pn sample sizes are many 100s @xcite , or at least 100 or more in conjunction with absorption line spectroscopy , which has limited this application to only a few nearby galaxies @xcite . in recent simulations of disk galaxy mergers involving dark matter , stars , and gas , @xcite predict that the young stars formed in the merger have steeper density profiles and larger radial anisotropy than the old stars from the progenitor galaxies , and they argue that if the pns observed in elliptical galaxies were to correspond to the young population rather than to all stars in the simulations , their velocity dispersion profile would match the measured dispersion profiles of @xcite . so do pns really trace the stars and their kinematics in elliptical galaxies ? different stellar populations may have , and in general would have , different phase - space distributions in the same galaxy potential . the simplest approach for dynamical modelling , taking the pn velocities as a random sampling of the stellar velocities , is however valid only when the pn population properties and their kinematics are uncorrelated . except in special cases this also requires that the pnlf is independent of the stellar population . vice - versa , if there existed differences in the pnlf or the bright cutoff magnitude for different stellar populations , they would best be identified by studying the correlations between pn magnitudes and kinematics or positions of these tracers , in a single galaxy where all pns are at the same distance . in this paper , we report on such a study in the elliptical galaxy ngc 4697 , an excellent target for this purpose because of the large sample of pn velocities known from @xcite . our analysis shows the existence of distinct pn populations which differ in their kinematics , brightnesses , and spatial distributions . this suggests that the answer to both the questions posed above may be no in general , different stellar populations may have slightly different pnlfs , and the observed pn population in elliptical galaxies may not be a fair tracer of their stars . the paper is organised as follows : in [ data ] we review the properties and pn data of this galaxy and discuss the magnitude and velocity completeness of our sample . our statistical analysis of these data is given in [ analysis ] where we demonstrate the inhomogeneity of the sample in the space of velocities , magnitudes , and positions . [ discn],[result ] conclude our work , giving also a brief discussion of its implications . ngc 4697 is a normal , almost edge - on e4 - 5 galaxy located along the virgo southern extension . from the multi - colour ccd photometry of @xcite , the effective radius is @xmath13 , the mean ellipticity is 0.45 , and the pa is constant , consistent with a near - axisymmetric luminosity distribution . isophotal analysis shows that this galaxy has a positive @xmath14 coefficient suggesting a disk - like structure within @xmath15 @xcite . from optical spectroscopy , its dominant stellar population has an age of @xmath16 gyr @xcite , consistent with the red mean b - v=0.91 colour . stellar absorption line kinematics along the major axis ( pa @xmath17 ) of ngc 4697 have been reported by @xcite and @xcite ; these velocity data can be well described by dynamical models based on the luminous mass distribution only . @xcite detected and measured radial velocities of 531 pns extending out to @xmath18 in this galaxy , with observational errors of @xmath19 . via dynamical analysis , they determined a constant mass to light ratio @xmath20 within @xmath21 , which is consistent with a @xmath22 gyr old stellar population with a salpeter mass function and slightly super solar metallicity . x - ray observations with rosat @xcite show only small amounts of hot gas in the halo of this galaxy . using more recent _ chandra _ data , @xcite could resolve most of the x - ray emission into nonuniformly distributed x - ray binary point sources ( xps ) , suggesting that ngc 4697 has lost most of its interstellar gas . though ngc 4697 does not show any signature of recent interactions , @xcite present evidence that the distribution of these x - ray sources is inconsistent with the optical morphology of ngc 4697 , and propose that these sources are mostly high mass x - ray binaries ( hmxbs ) associated with young stellar populations related to fallback of material in tidal tails onto a relaxed merger remnant , or to shock - induced star formation along tidal tails . they estimate typical fallback times of such tidal tails to be much longer than the settling timescale of the remnant and expect similar results for other elliptical galaxies with populations of @xmath23 gyr age . for the work in this paper we use the pn sample presented by @xcite . after removing the possible contaminants and unclear detections , they report unambigous detection of 535 pns . however only 526 out of 535 pns have confirmed measurements of velocity _ and _ magnitude , and we use these in our analysis . in order to determine the pnlf , it is crucial to estimate the magnitude where pn detection incompleteness sets in . detectability of a pn varies with the background galaxy surface brightness ; for a statistically complete sample the surface number density of pns should be directly proportional to it . @xcite show that their pn sample is statistically complete down to @xmath24 magnitudes outside an elliptical region of semi - major axis @xmath25 . in our analysis , we have thus defined two data sets : a _ complete sample _ ( with pns brighter than @xmath26 , outside the central ellipse of semi - major axis @xmath25 ) , and a _ total sample _ ( consisting of all pns with measured magnitude and radial velocity ) . the total number of pns in these data sets is 320 and 526 , respectively . the systemic velocity ( @xmath27 ) of ngc 4697 , obtained by averaging the observed velocity of all 526 pns is @xmath28 , which agrees with the values quoted in the literature ( * ? ? ? * and references therein ) . the on - band filter configuration used to detect and measure velocities of these pns has a peak wavelength of @xmath29 , peak transmission of @xmath30 , equivalent width of @xmath31 , and fwhm of @xmath32 @xcite . the fwhm corresponds to a velocity range of @xmath33 around the systemic velocity of ngc 4697 . thus the filter transmission width is large enough to facilitate observations of pns with all velocities bound to ngc 4697 , irrespective of their magnitude . indeed , even at magnitudes as faint as @xmath34 in the total sample , pns with large velocities @xmath35 are detected . thus the velocity coverage in both samples ( total and complete ) is not biased with respect to the pn magnitudes . the pn magnitudes were measured by @xcite from their undispersed images ; they are accurate to 0.1 and 0.2 mag for @xmath12 brighter and fainter than 26.5 , with systematic effects below 2% . as a further test relevant for the present work , @xcite used the redundancy provided between their e and w fields : plotting magnitude differences between the two measurements ( e and w ) of pn candidates as a function of difference in distance from the center of the ccd , they found a scatter diagram without any evidence of correlation . @xcite estimated the errors in the pn velocities from calibration , image registration , spectrograph deformation and guiding errors to be @xmath36 . the velocities of 165 pns were measured independently in the e and w field of @xcite ; these velocities agree within a standard deviation of @xmath37 . in order to check whether a systematic difference between the velocities of bright and faint pns could be introduced by an asymmetric psf ( a possibility suggested by k. freeman ) , we have superposed the psf s of three groups of 10 of the brightest pns , one selected at random , and two selected among those pns with the highest and lowest radial velocities . in the three cases we estimated the shift of the centroid of the entire psf with respect to the centroid of the upper part . the shifts were smaller than @xmath38 , and in some cases they were in the opposite sense compared to the results discussed below . in this section , we search for stellar population effects in the kinematics of the pns in ngc 4697 , by analysing the total and complete data sets with respect to their three observables : velocity , magnitude and position . for both data sets , we convert the observed pn radial velocities into co - rotating or counter - rotating velocities , as follows . with the galaxy center at the origin of the reference frame , and the x - axis oriented along the major axis ( pa@xmath39 deg ) , the absorption line stellar - kinematic data predict positive line - of - sight mean velocity with respect to the galaxy systemic velocity , at slit positions towards the south - west of the center with x coordinate @xmath40 , and vice versa . we denote this sense of rotation as _ co - rotating _ , and the opposite sense as _ counter - rotating_. by definition , the major axis absorption line data is _ co - rotating_. after subtracting the systemic velocity from the pn radial velocities , we define reduced velocities @xmath41 and denote the pns with @xmath42 @xmath43 ( @xmath44 ) 0 as co - rotating ( counter - rotating ) . by definition , the major axis absorption line data have @xmath45 . the resulting values of @xmath42 are displayed against the observed magnitudes in figure [ mvplot ] . even at magnitudes as faint as @xmath34 in the total sample , pns with large velocities @xmath35 are detected , showing that there is no kinematic bias at faint magnitudes . henceforth , unless stated otherwise , we will always use the complete sample for our analysis . figure [ mvplot ] shows that the complete pn sample appears to exhibit a correlation between magnitudes and kinematics , with faint pns showing more co - rotation than bright pns . we have performed several statistical tests to verify the significance and look for the origin of this correlation . table [ tabpear ] shows the results of pearson s r - test for correlated data . velocities of counter - rotating pns are strongly linearly correlated with their brightness , while those of co - rotating pns are almost independent of their magnitude distribution . further , we divided our sample into 3 equal magnitude bins each of size @xmath46 , hereafter referred to as _ faintest _ , _ intermediate _ , and _ brightest pns _ , and computed the mean reduced velocity and its dispersion in each of these bins along with the significance of their differences . as shown in table [ tabvtftest ] , the _ faint _ and _ bright pn subsamples _ defined through these bins have different mean reduced velocity and dispersions at @xmath47 and @xmath48 confidence , respectively . in figure [ 6bins ] we have plotted the cumulative velocity distribution of pns in the brightest and faintest magnitude bins . there is a visible excess of bright pns with counter rotating velocities . it is particularly evident from this figure that the brightest counter rotating pns display a velocity distribution that differs from the rest of the pns from the complete sample with high confidence . thus it is clear that the observed correlation between the pns kinematics and their magnitudes is compelling , and it arises because the faintest and brightest pns have significantly different velocity distributions . there appears to exist an additional component of bright , counter - rotating pns with respect to the overall sample . .pearson s r - test for linear correlation of pn magnitudes with reduced velocity @xmath49 , for the entire complete sample , the co - rotating , and the counter - rotating subsamples . values of r close to @xmath50 indicate a strong linear correlation ; values close to @xmath51 indicate little or no correlation . @xmath52 is the probability that two uncorrelated variables would give the r - coefficient as large as or larger than the measured one , for a normal distribution of r. small values of @xmath52 imply significant correlation . [ cols="^,^,^",options="header " , ] , orange dashed line ) , faintest pns ( @xmath53 , purple dashed - dotted line ) , and the entire complete sample ( black solid line ) . the entire velocity range is shown in the top panel . in the middle panel the velocities are divided according to their sense of rotation . the bottom panel shows the cumulative distribution ( shown from @xmath54 to @xmath55 values ) , normalised at @xmath56 . the kolmogorov smirnov probabilities show that the brightest counter rotating pns have significantly different velocity distribution from the rest of the pns.,title="fig : " ] , orange dashed line ) , faintest pns ( @xmath53 , purple dashed - dotted line ) , and the entire complete sample ( black solid line ) . the entire velocity range is shown in the top panel . in the middle panel the velocities are divided according to their sense of rotation . the bottom panel shows the cumulative distribution ( shown from @xmath54 to @xmath55 values ) , normalised at @xmath56 . the kolmogorov smirnov probabilities show that the brightest counter rotating pns have significantly different velocity distribution from the rest of the pns.,title="fig : " ] , orange dashed line ) , faintest pns ( @xmath53 , purple dashed - dotted line ) , and the entire complete sample ( black solid line ) . the entire velocity range is shown in the top panel . in the middle panel the velocities are divided according to their sense of rotation . the bottom panel shows the cumulative distribution ( shown from @xmath54 to @xmath55 values ) , normalised at @xmath56 . the kolmogorov smirnov probabilities show that the brightest counter rotating pns have significantly different velocity distribution from the rest of the pns.,title="fig : " ] if these correlations have a physical origin , they should also be manifest in the spatial distribution of these pns . thus we now enquire whether the pn kinematics and magnitudes depend on their spatial location in the galaxy . in figure [ tspace ] we plot the spatial locations of all the 526 pns in this galaxy . the central incompleteness ellipse is also displayed . pns brighter than @xmath57 ( which is @xmath58 magnitude deeper than the brightest pn ) are shown as filled blue squares ( co - rotating ) and filled red triangles ( counter - rotating ) . inside the incompleteness ellipse , the distribution of the bright pns appears to be concentrated around an elliptical annulus . however , we did not find any kinematic evidence ( like a rotation curve signature ) relating these bright pns to the central stellar disk : either they are not physically related to the disk , or the evidence from the data is inconclusive . outside the incompleteness ellipse , the distribution of bright pns does not follow the surface brightness of the host galaxy : there is a significant left - right asymmetry , with more bright pns to the right side of the galaxy minor - axis ( @xmath59 ) than to the left side ( @xmath60 ) . @xcite discuss at length the possible differences in their e ( east ) and w ( west ) fields , and are convinced that the maximum systematic errors in the measured photometry , positions and radial velocities are below @xmath61 , @xmath62 and @xmath63 , respectively . hence we conclude that the left - right asymmetry in number counts of bright pns is not affected by detection uncertainties . subsequently , we carried out several tests to check whether the brightest and faintest , or the co- and counter - rotating pns are distributed differently in the galaxy . it turns out that the radial pn distribution is independent of their sense of rotation . however , the distribution of pn distances from the galaxy mid - plane differs for co- and counter - rotating pns at a confidence level of @xmath64 . are shown by the orange dashed line , the red dash - triple dotted line , and the purple dash - dotted line , respectively . all pns in the complete sample are shown by the solid black line . note the large increase in the distribution of bright pns and the moderate increase in the distribution of intermediate pns , relative to the faint pns , at @xmath65 ( see text for details ) . the ks probability of the faintest and brightest pns being drawn from the same underlying distribution is only @xmath66 , while the intermediate pns are still compatible with the faint pns ( ks probability @xmath67).,title="fig : " ] are shown by the orange dashed line , the red dash - triple dotted line , and the purple dash - dotted line , respectively . all pns in the complete sample are shown by the solid black line . note the large increase in the distribution of bright pns and the moderate increase in the distribution of intermediate pns , relative to the faint pns , at @xmath65 ( see text for details ) . the ks probability of the faintest and brightest pns being drawn from the same underlying distribution is only @xmath66 , while the intermediate pns are still compatible with the faint pns ( ks probability @xmath67).,title="fig : " ] . solid ( dashed ) histograms show the mean velocity ( dispersion ) and their errors in different angular bins . the red ragged lines show running averages of mean velocity . _ top : _ velocities of pns brighter than @xmath68 in the complete sample . bin sizes are chosen so as to have approximately constant number of pns ( @xmath69 ) in each bin . the mean velocity along the major ( minor ) axis is positive ( negative ) on both sides of the center . the running average is also over @xmath70 pn velocities . the velocity dispersion is largest ( smallest ) on the minor ( major ) axis . _ middle : _ velocities of pns with @xmath71 in the same angular bins as in the top panel . the running average is over @xmath72 pns . _ bottom : _ velocities of pns with @xmath73 in the same angular bins as in the top panel . the faint pns show a rotation pattern as for the absorption line data but with a smaller peak velocity , and are consistent with a flat dispersion of @xmath74 . the running average is over @xmath75 pns . the blue dotted line shows a sinusoidal fit . the kinematics of the intermediate brightness pns is intermediate between the faint and bright pns.,title="fig : " ] . solid ( dashed ) histograms show the mean velocity ( dispersion ) and their errors in different angular bins . the red ragged lines show running averages of mean velocity . _ top : _ velocities of pns brighter than @xmath68 in the complete sample . bin sizes are chosen so as to have approximately constant number of pns ( @xmath69 ) in each bin . the mean velocity along the major ( minor ) axis is positive ( negative ) on both sides of the center . the running average is also over @xmath70 pn velocities . the velocity dispersion is largest ( smallest ) on the minor ( major ) axis . _ middle : _ velocities of pns with @xmath71 in the same angular bins as in the top panel . the running average is over @xmath72 pns . _ bottom : _ velocities of pns with @xmath73 in the same angular bins as in the top panel . the faint pns show a rotation pattern as for the absorption line data but with a smaller peak velocity , and are consistent with a flat dispersion of @xmath74 . the running average is over @xmath75 pns . the blue dotted line shows a sinusoidal fit . the kinematics of the intermediate brightness pns is intermediate between the faint and bright pns.,title="fig : " ] . solid ( dashed ) histograms show the mean velocity ( dispersion ) and their errors in different angular bins . the red ragged lines show running averages of mean velocity . _ top : _ velocities of pns brighter than @xmath68 in the complete sample . bin sizes are chosen so as to have approximately constant number of pns ( @xmath69 ) in each bin . the mean velocity along the major ( minor ) axis is positive ( negative ) on both sides of the center . the running average is also over @xmath70 pn velocities . the velocity dispersion is largest ( smallest ) on the minor ( major ) axis . _ middle : _ velocities of pns with @xmath71 in the same angular bins as in the top panel . the running average is over @xmath72 pns . _ bottom : _ velocities of pns with @xmath73 in the same angular bins as in the top panel . the faint pns show a rotation pattern as for the absorption line data but with a smaller peak velocity , and are consistent with a flat dispersion of @xmath74 . the running average is over @xmath75 pns . the blue dotted line shows a sinusoidal fit . the kinematics of the intermediate brightness pns is intermediate between the faint and bright pns.,title="fig : " ] the left - right asymmetry is confirmed by inspecting the azimuthal distribution of the faint and bright pns . in the literature we found a related analysis by @xcite who compared the azimuthal distribution of _ chandra _ x - ray point sources ( xps ) with the optical surface brightness of ngc 4697 . we follow their pa convention , and plot the cumulative angular distributions of the bright , intermediate , and faint pns in our complete sample in figure [ zezang ] . for comparison , the right panel of fig . 2 from @xcite is also shown . the angular distribution of all pns in the complete sample has a shape somewhere in between that of the xps and that of the optical light . the brightest pns are in complete disagreement with either of these distributions ; they seem to be more concentrated in a narrow angular sector between @xmath76 ( see fig . [ tspace ] ) , with only @xmath66 probability that the faint and bright pn subsamples are drawn from the same azimuthal distribution . at the same time , the radial distribution of the faint and bright subsamples are not significantly different ( figure [ radial ] ) . the velocity distribution of the brightest pns is also correlated with their azimuthal distribution . in figure [ vang ] we plot the mean radial velocity and its dispersion in angular sectors containing approximately constant number of pns from the bright subsample , as well as an angular running average of their mean radial velocity . along both sides of the major - axis of the galaxy , the brightest pns have positive velocity , while showing a relatively low velocity dispersion , perhaps due to infall as suggested for the xps by @xcite . along the minor - axis they have a large dispersion with a mean velocity that is negative . this kinematics is compatible with neither the faint pn velocities nor the stellar absorption line data . on the other hand , the fainter pns show a regular azimuthal distribution in the mean line - of - sight radial velocity and velocity dispersion . note that their velocity dispersion is in the mean smaller than that of the bright sample . the kinematics of the intermediate luminosity pns is intermediate between those of the bright and faint subsamples . the intermediate luminosity pns thus contain significant contributions from both of the different populations that dominate the bright and faint bins , respectively . of the major axis for bright and faint / intermediate pns separated at @xmath77 . blue circles show stellar absorption line spectroscopy ( als ) data from @xcite , red crosses are individual pn velocities , and grey diamonds are near neighbour running averages . the faint / intermediate sample is in approximate agreement with the als data , while the bright sample has a velocity consistent with zero within @xmath78 and positive velocities on both sides of the galaxy center . ] furthermore , including the bright pns in deriving kinematics for the whole pn sample introduces signficant contamination effects . in figure [ majkindb ] , we demonstrate this by plotting the mean pn velocities along a @xmath79 wide slit about the major - axis . the pns have been separated into faint / intermediate and bright sub - samples at @xmath77 . inside the @xmath78 , the mean velocities of the faint pns agree with stellar absorption line spectroscopic ( als ) data from @xcite , while the bright pns have much smaller mean velocities , consistent with @xmath80 . in the outer parts , the velocity asymmetry in the bright pns leads to a positive mean velocity on both sides of the galaxy center . both the faint / intermediate sample and the entire pn sample also show some asymmetry : the outer mean velocities on the @xmath81 side reach zero , but not the negative values expected from reflecting the positive values at @xmath82 . this confirms that also the intermediate / faint subsample contains a fraction of pns stemming from the out - of - equilibrium population traced by the bright pns . however , the streaming velocity of the majority of the faint pns does appear to decrease on both sides of the center . similarly contaminated results could be expected for the derived velocity dispersions . several important conclusions can be drawn from these figures . ( i ) the bright pns as defined in section [ subpop ] and fig . [ mvplot ] do not trace the azimuthal distribution of light in ngc 4697 . ( ii ) they do not trace the fainter pns in their azimuthal kinematics ; thus , a large fraction of them must belong to a separate pn subcomponent originating from a separate stellar population . ( iii ) third , they are not in dynamical equilibrium in the gravitational potential of ngc 4697 . ( iv ) because this subpopulation does not trace the stars , including its pn velocities into dynamical analysis of the galaxy will lead to significant errors in the results . probability . _ bottom : _ cumulative magnitude distributions of the main pn sample ( @xmath83 , blue dotted line ) and the extreme counter - rotating sample ( @xmath84 , red dash - dotted line ) defined in the text , along with that of the total sample ( black solid line ) . the first two luminosity functions are different with 99.7% confidence ; thus the pnlf can not be universal.,title="fig : " ] probability . _ bottom : _ cumulative magnitude distributions of the main pn sample ( @xmath83 , blue dotted line ) and the extreme counter - rotating sample ( @xmath84 , red dash - dotted line ) defined in the text , along with that of the total sample ( black solid line ) . the first two luminosity functions are different with 99.7% confidence ; thus the pnlf can not be universal.,title="fig : " ] what can we learn about the luminosity functions of the two kinematic components of the pn system in ngc 4697 ? the cumulative magnitude distributions of the co - rotating and counter - rotating subsamples in fig . [ mvplot ] have only @xmath67 ks probability of stemming from the same distribution ( top panel of figure [ ksmag ] ) . however , the fact that there are both co - rotating and counter - rotating bright pns in the bright sample whose azimuthal distribution is unmixed , shows that counter - rotation is not a _ clean _ discriminator after all for the secondary population of pns in ngc 4697 . thus the luminosity functions of the main and secondary pn populations in this galaxy may be a lot more different than this figure of @xmath67 would suggest . from fig . [ vang ] we estimate that the main population of pns in ngc 4697 has a mean radial velocity @xmath85 , with a dispersion of @xmath86 , so its reduced mean velocity is @xmath87 . thus in the bottom panel of fig . [ ksmag ] we show the cumulative magnitude distribution of the pns in the velocity range @xmath88 and compare it with the cumulative distribution of the pns with @xmath84 . these velocity ranges are dominated by the main and secondary pn populations in the sample , respectively . now the kolmogorov smirnov significance test shows that the magnitude distributions from these sections of fig . [ mvplot ] have only probability @xmath89 of being drawn from the same distribution . this result is strong enough to imply that the pnlf can not be universal the pnlf in ngc 4697 depends on a kinematic selection . in the following , we will use the velocity range @xmath90 as an approximate kinematic selection criterion for the main pn population in ngc 4697 . the luminosity function of the strongly counter - rotating pns in fig . [ mvplot ] a first approximation to the luminosity function of the secondary pn population in ngc 4697 differs from that of the main pn distribution so defined in the sense that it contains more bright pns near the cut - off and fewer faint pns than the main population ( see fig . [ ksmag ] ) . now an important question is : in what proportion do both populations contribute to the brightest pns , and do they have different cutoff magnitudes ? . circled symbols denote counter - rotating pns ; many of these are kinematic outliers . the two grey lines denote the position angle of the minor axis . the distribution of blue diamonds is approximatively uniform in pa except near pa@xmath91 where there are 12 pns ( crossed diamonds ) instead of the expected 2 pns . _ bottom : _ radial velocity magnitude plane . the symbols are as in the top panel . 6/8 of the brightest pns are either kinematic outliers or are found in the overdense angular region . , title="fig : " ] . circled symbols denote counter - rotating pns ; many of these are kinematic outliers . the two grey lines denote the position angle of the minor axis . the distribution of blue diamonds is approximatively uniform in pa except near pa@xmath91 where there are 12 pns ( crossed diamonds ) instead of the expected 2 pns . _ bottom : _ radial velocity magnitude plane . the symbols are as in the top panel . 6/8 of the brightest pns are either kinematic outliers or are found in the overdense angular region . , title="fig : " ] to investigate this , we show in figure [ vla ] the velocities , magnitudes , and position angles of the entire bright pn subsample ( see figs . [ mvplot ] and [ vang ] ) . the top panel shows ( i ) that 13/16 of the bright pns , whose radial velocities differ most from those of the faint population , are counterrotating . this explains the differences between the velocity distributions of co- and counter - rotating pns that first suggested more than one pn population in section [ subpop ] . ( ii ) also , even in the kinematically normal bright pns , there is a large overdensity ( 10/12 ) in the angular range pa@xmath91 . the bottom panel shows in addition that many of the brightest pns are either kinematic outlyers or found in the angular overdensity ( 6/8 ) . clearly , to arrive at a main population of ngc 4697 pns that is in dynamical equlibrium in the gravitational potential , we must remove the angular overdensity . then we are left with 16 pns in the range @xmath92 of a total bright subsample of 42 pns ; however , there is some freedom in the way the angular overdensity is removed . in the following , we explore two assumptions : ( i ) using all 12 pns in the angular overdensity , but weighting each one by @xmath93 , and ( ii ) removing the brightest 10 of the 12 pns in the overdensity . the resulting luminosity functions for both cases are plotted in figure [ clumfun ] ; they differ only slightly . thus in the following we use the kinematic condition @xmath94 together with assumption ( ii ) above to ensure azimuthal uniformity of the bright pns , as an improved selection criterion for pns in the main population in ngc 4697 . we keep all pns in the intermediate luminosity bin with @xmath92 , because fig . [ zezang ] showed that their azimuthal asymmetry is not large . also , we have checked that the mean angular velocity distribution in this magnitude bin after applying the kinematic selection follows approximately the sinusoidal variation of the faint pns , and the velocity disperion is approximately constant . the resulting main population sample is identified by their brightness , distribution and kinematic properties . comparing the luminosity function in figure [ clumfun ] of this main pn population , with the luminosity function of all pns in ngc 4697 , we see that the bright cutoff of the main population is shifted to fainter magnitudes . we note that the bright cutoff could be shifted further to fainter magnitudes if some of the kinematically normal and azimuthally uniform bright pns were also part of the secondary population , which we can not tell from the present data . we can ask now what is the effect , in practice , on the pnlf distance determination . the reduced sample for the main pn population in ngc 4697 has 214 objects . after binning these data into 0.2 mag intervals , we transform the apparent magnitudes @xmath12 into absolute magnitudes , adopting the extinction correction of 0.105 mag @xcite and assuming different distance moduli , and we compare the results with the pnlf simulations of @xcite . this is the same procedure used in @xcite for the pnlf distance determination . the comparison is shown in fig . we conclude that the pnlf distance modulus should be increased slightly from 30.1 ( the earlier determination based on the full sample ) to 30.2 or 30.25 . for comparison , the @xmath95-fit to the same data ( blue line in fig . [ clumfun ] ) gives 30.22 . this correction would bring the pnlf distance modulus in better ( but not perfect ) agreement with the surface brightness fluctuation ( sbf ) distance modulus ( @xmath96 ) reported by @xcite . ) of 214 objects , binned into 0.2 mag intervals . the apparent magnitudes @xmath12 have been transformed into absolute magnitudes @xmath97 by adopting an extinction correction of 0.105 mag and distance moduli indicated in each plot . the three lines are pnlf simulations @xcite . for a distance modulus 30.15 the brightest pns are a bit too weak , therefore the distance must be increased . but 30.35 is clearly excessive . the best fit is for 30.2 or 30.25 . , title="fig : " ] ) of 214 objects , binned into 0.2 mag intervals . the apparent magnitudes @xmath12 have been transformed into absolute magnitudes @xmath97 by adopting an extinction correction of 0.105 mag and distance moduli indicated in each plot . the three lines are pnlf simulations @xcite . for a distance modulus 30.15 the brightest pns are a bit too weak , therefore the distance must be increased . but 30.35 is clearly excessive . the best fit is for 30.2 or 30.25 . , title="fig : " ] bright pns play a significant role in the analysis of the last section . is it possible that the brightest pns in ngc 4697 are contaminated by compact hii regions such as those observed in @xcite ? the recent observations of @xcite have shown that this can not be so . these authors took spectra of 13/42 pns in our bright subsample in ngc 4697 with fors2@vlt ; these have no detectable continuum and the line ratios of metal - rich pns . the same argument also shows that these bright pns can not be background emission line galaxies . moreover , ly alpha emission galaxies would come in at [ oiii ] magnitudes of m@xmath98 ( see fig . 4 of * ? ? ? * ) , while the bright pns in ngc 4697 have m@xmath99 . furthermore , one may wonder whether the bright pns in ngc 4697 might simply be foreground objects which could be closer to us and hence brighter than the true ngc 4697 pns on which they would be superposed . given that ngc 4697 is located in the southern extension of the virgo cluster , known to contain an intracluster population of pns @xcite , this possibility deserves to be considered . however , the following observational facts show that the bright pns in ngc 4679 are not intracluster pns ( icpns ) . ( 1 ) ngc 4697 is in fact closer than the virgo cluster . the pnlf from @xcite places it at @xmath100 ( m - m@xmath101 ) , while the distance modulus from the pnlf of m87 is m - m@xmath102 @xcite . even if we discarded the entire brightest 0.3 mag of the ngc 4697 pns , ngc 4697 would still be at 75% of the distance of m87 . ( 2 ) the velocity dispersion of the bright and unrelaxed pns in ngc 4697 is @xmath103 , but varying azimuthally , while that of the fainter main population is @xmath104 ( table [ tabvtftest ] , figs . [ mvplot ] , [ zezang ] ) . both are much smaller than the velocity dispersion of icpns in virgo @xcite . the radial distribution of the bright population outside the incompleteness ellipse is concentrated towards the galaxy center and ks compatible ( 92% ) with the radial distribution of the fainter pns ( see fig . [ radial ] ) . thus also the bright pns in ngc 4697 are bound to the galaxy . ( 3 ) the surface density of pns with m@xmath99 is 0.58 pns / arcmin@xmath105 , while the mean surface density of virgo icpns is 0.02 pns / arcmin@xmath105 , much smaller @xcite . this last argument also rules out significant contamination by chance superpositions of pns even closer than ngc 4697 . we conclude that the bright pn population in ngc 4697 consists of genuine pns , and that it is dynamically bound to the ngc 4697 system . the irregular angular distribution and kinematics of these pns , by comparison with the fainter main population , show that they must belong to a separate stellar population not yet in dynamical equilibrium with ngc 4697 . pn samples as large as the one for ngc 4697 are still the exception . yet to undertake the analysis described in section [ analysis ] it was crucial to work with a complete sample of some 300 pns . the pn.s project @xcite may lead to complete samples of similar size but as of now the typical sample sizes are @xmath106 @xcite and their magnitude distributions and completeness have not been studied . the only other elliptical galaxy with a comparable ( even larger ) sample is cen a @xcite . a detailed analysis of the distribution of cen a pns in the magnitude velocity plane is still pending , but @xcite analysed the pnlf in cen a as a function of radius , based on narrow - band surveys . they concluded that no population effect on the pnlf bright cut - off could be seen , suggesting that the 0.3 mag difference between their main and outer halo samples was due to filter transmission uncertainties . analysis of the large sample from @xcite will clarify whether the pns in cen a are consistent with one or more subpopulations . this will be particularly interesting because cen a is believed to be the remnant of a galaxy merger , so one might expect pns from both the older stellar populations of the progenitors as well as from the stars formed in the subsequent interaction between them . without large kinematic samples , searches for pn subpopulations must be based on the pn luminosity distributions . the work of jacoby , ciardullo and collaborators cited in the introduction has shown that the pnlf is remarkably uniform . however , there are exceptions : we recall that in the halo of m84 there exists a small population of overluminous pns whose cutoff is @xmath107 mag brighter than that of the main m84 population , but which appear nonetheless bound to the halo of m84 @xcite . they must therefore be intrinsically bright , due to some stellar population difference . in m87 , the only overluminous pns projected onto the galaxy for which velocities have been measured @xcite , have very large relative velocities with respect to m87 . this is most naturally explained if these pns have fallen into the deep potential well of m87 from far out in the cluster ; this would again imply an intrinsic population difference . however , a larger kinematic sample in m87 is required to be certain . the existence of the bright pn subpopulation in ngc 4697 implies some uncertainty in the pnlf cut - off luminosity for this galaxy . depending on whether or not the azimuthally symmetric brightest pns are part of the main population , the cut - off luminosity of the main pn population in ngc 4697 is fainter than that of the whole population by @xmath108 mag ( figs . [ clumfun ] , [ rob ] ) . while this is consistent with the m84 result , is it also consistent with the systematic studies of pnlf distances ? @xcite have compared the pnlf and sbf distances , finding a distribution of residuals with a systematic offset by @xmath109 mag , which they suggested is due to internal extinction effects , and with a fwhm of @xmath110 mag . the offset has in the mean - time been reduced to @xmath111 mag , following a revision of the sbf distance scale by @xcite . the width of this distribution is consistent with their determination of the observational errors in both methods . however , the offset we have determined for ngc 4697 is also consistent with the distribution of residuals in @xcite . we have shown that a large fraction of the bright pns in ngc 4697 belong to a secondary , dynamically young stellar population that is not well - mixed in the gravitational potential of the galaxy . late infall of tidal structures @xcite or a merger with a smaller galaxy some time ago would be natural ways to add such an unmixed stellar component to ngc 4697 . what physical population difference is correlated with this dynamical youth ? @xcite show from their spectroscopic data for 13 bright pns that these have near - solar metallicities . of these 13 bright pns , 6 are inside the incompleteness ellipse , one has no measured velocity , and 6 belong to our secondary population . @xcite also use long - slit spectroscopy to show that the metallicity of the integrated stellar population within one effective radius has solar or higher metallicities . these observations make it unlikely that metallicity is the main factor responsible for the different magnitude distributions of the main and secondary pn populations in ngc 4697 . thus the more likely driver would appear to be an age difference , as suggested by @xcite and as might generally be expected in an accretion event . based on their result that the distribution of x - ray point sources in ngc 4697 does not follow the stellar light , @xcite have argued that this is because these sources were formed several @xmath112 years ago in tidal tails that are now falling back onto the galaxy . note , however , that the integrated light in ngc 4697 shows no evidence of young stars with mean age @xmath113 gyr @xcite , so this younger component could not be luminous enough to contaminate the integrated light to the level measured . also note that the observed increase of extinction in the pn envelope with pn core mass more than compensates for the increase of core luminosity with core mass , for bright pne in local group galaxies @xcite , so that stars with ages below @xmath114 gyr may not reach the [ oiii ] luminosity at the pnlf cutoff . a secondary stellar population younger than @xmath114 gyr is therefore unlikely as well . recently , @xcite have argued that the brightest pns in the pnlf must have core masses of @xmath115 , corresponding to main sequence masses of @xmath116 . they argue further that for such high - mass objects to occur in elliptical galaxies , these early - type galaxies would either have to contain a small , smoothly distributed component of young ( @xmath117 gyr age ) stars , or more likely , that the bright pns in these systems have evolved from blue straggler stars created through binary evolution . their blue straggler model , due to the assumption of a fixed distribution of primary - to - secondary mass ratios for the initial binaries , predicts that older stellar populations produce fewer bright pns per unit luminosity , as is observed , because the number of binary stars in a stellar population that can coalesce to @xmath116 blue stragglers decreases with time . if correct , this blue straggler model could also explain how the secondary population we found in ngc 4697 can contain a large fraction of the brightest pns in this galaxy , provided that the stellar population corresponding to this secondary pn population is appreciably younger than the main stellar population , whose age is @xmath16 gyr from optical spectroscopy @xcite . at the same time , this secondary stellar population must not be so young to violate either the constraints from the optical colours or from the envelope absorption - pn core mass correlation , i.e. , must be older than @xmath118 gyr . we can give an estimate for the effect of such an intermediate age population on the optical colours as follows . the secondary subpopulation traced by the bright and predominantly counterrotating pns may contain @xmath119 of all pns in the complete sample for ngc 4697 . a stellar population as blue as the bulge of m31 has a luminosity - specific pn density per unit @xmath120 up to 5 times higher than the populations characteristic for old elliptical galaxies @xcite . thus the subpopulation corresponding to the secondary pn population in ngc 4697 would be expected to contain @xmath121 of the blue luminosity of ngc 4697 , spread over a large fraction of at least the e image . to detect this we need deep and accurate photometry . the unmixed spatial and velocity distributions of the secondary pn population in ngc 4697 show that that this population is _ dynamically young _ , i.e , has not had time to phase - mix and come to dynamical equilibrium in the gravitational potential of ngc 4697 . it may well be associated with tidal structures that were formed in a merger or accretion event @xmath122 gyr ago , and that have now fallen back onto the galaxy , or be associated with a more recent merger / accretion with a red galaxy such as described in @xcite . in a universe in which structures form hierarchically , such secondary stellar populations might be quite common in ellipitical galaxies , but they would be difficult to see . the present work shows that studying their pn populations is one promising approach of looking for such secondary populations . however , large pn samples are required ; most existing pn studies of early - type galaxies do not have the statistics for such an investigation . moreover , in only a fraction of cases may there be enough asymmetry signal to detect with a few hundred pns . we have analysed the magnitudes , kinematics and positions of a complete sample of 320 pns in the elliptical galaxy ngc 4697 from @xcite . this data set is large enough for drawing statistically significant conclusions , and it does not suffer from detection incompletenesses in either magnitudes or radial velocities . we know of no systematic effects in the data that could explain our results . our main conclusions are : 1 . bright and faint pns in ngc 4697 have significantly different radial velocity distributions . the mean velocities of the faint and bright subsamples ( co - rotating and @xmath80 , respectively ) and their velocity dispersions are different , with 94% and 99.3% confidence . thus the pns in ngc 4697 do not constitute a single population that is a fair tracer of the distribution of all stars . 2 . the luminosity functions of the extreme counter - rotating subsample ( @xmath84 ) and of the main population ( defined by @xmath123 ) are different with 99.7% confidence . the pnlf is therefore not universal . 3 . based on this , we suggest that there exist ( at least ) two pn populations in this galaxy . the secondary pn population in ngc 4697 is prominent in a sub - sample of counter - rotating pns brighter than @xmath0 . the luminosity function of the entire extreme counterrotating sample may be a first approximation to the luminosity function of the secondary population . the spatial distribution of bright pns with @xmath124 is different from that of the faint pns . the bright pns do not follow the azimuthal distribution of the optical light , show a left - right asymmetry , and have a positive mean radial velocity on both sides of the galaxy major axis , but zero velocity and larger dispersion on the minor axis . they are not in dynamical equilibrium in the potential of the galaxy . the fainter population has rotation properties more similar to the absorption line velocities , with azimuthally constant dispersion . 5 . using both their kinematics and angular distribution , we can estimate a lower limit to the statistical fraction of bright pns in the secondary population . based on this we estimate that the bright cutoff of the main population is uncertain by @xmath125 mag . our results have two main implications for the use of pns in extragalactic astronomy . first , for distance determinations with the pnlf , it may be important to understand how uniform the pn populations in the target galaxies are . from our analysis in ngc 4697 we estimate that unrecognized subpopulations of pns in smaller samples than that in ngc 4697 may lead to variations of @xmath125 mag in the bright cutoff . this would correspond to distance errors of some @xmath126 , which , although a minor effect , could be significant in some cases . it will be necessary to verify how frequently such subpopulations occur in elliptical galaxies . we recall that also in the halo of m84 there exists a small population of overluminous pns whose cutoff is @xmath107 mag brighter than that of the main m84 population , but which appear nonetheless bound to the halo of m84 @xcite . the second implication concerns the use of pns as tracers for the angular momentum and gravitational potentials of elliptical galaxies . our analysis has shown that in ngc 4697 the bright pns do not trace the distribution and kinematics of stars and are not in dynamical equilibrium in the gravitational potential of the galaxy . the fainter pns look more regularly distributed but may also contain a fraction of pns that belongs to this out - of - equilibrium population . clearly therefore , mass determinations based on pn kinematics will in future require careful study of the pn samples being used , not only to verify that these pns are in dynamical equilibrium , but also to test for different dynamical components . even if in equilibrium , a younger population of stars may be more flattened or have a steeper fall - off than the main body of the elliptical galaxy . if the pns from this population are indeed somewhat brighter than the main population , one can recognize such differences from their lower velocity dispersion or different radial density profile . however , deep observations and large pn samples will be required . we are grateful to m. arnaboldi , r. ciardullo and r. saglia for helpful discussions , and to k. freeman , g. jacoby , e. peng and a. romanowsky for helpful comments on the manuscript . ns and og thank the swiss nationalfonds for financial support under grant 200020 - 101766 . rhm would like to acknowledge support by the u.s.national science foundation , under grant 0307489 .
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we have analysed the magnitudes , kinematics and positions of a complete sample of 320 pns in the elliptical galaxy ngc 4697 . we show ( i ) that the pns in ngc 4697 do not constitute a single population that is a fair tracer of the distribution of all stars . the radial velocity distributions , mean velocities , and dispersions of bright and faint subsamples differ with high statistical confidence .
( ii ) using the combined data for pns brighter than @xmath0 , we have identified a subpopulation of pns which is azimuthally unmixed and kinematically peculiar , and which thus neither traces the distribution of all stars nor can it be in dynamical equilibrium with the galaxy potential .
( iii ) the planetary nebula luminosity functions ( pnlf ) of two kinematic subsamples in ngc 4697 differ with 99.7% confidence , ruling out a universal pnlf .
we estimate that the inferred secondary pn population introduces an uncertainty in the bright cutoff magnitude of @xmath1 mag for this galaxy .
we argue that this secondary pn distribution may be associated with a younger , @xmath2 gyr old stellar population , perhaps formed in tidal structures that have now fallen back onto the galaxy , as has previously been suggested for the x - ray point sources in this galaxy , or coming from a more recent merger / accretion with a red galaxy .
the use of pns for extragalactic distance determinations is not necessarily compromised , but their use as dynamical tracers of dark halos will require deep observations and careful analysis of large pn samples .
| 15,411 | 442 |
in a series of stunning papers stretching over almost a decade @xcite thouless obtained a closed expression for the bandwidth of the hofstadter spectrum @xcite in the @xmath1 limit . here the integer @xmath2 stands for the denominator of the rational flux @xmath3 of the magnetic field piercing a unit cell of the square lattice ; the numerator @xmath4 is taken to be @xmath5 ( or equivalently @xmath6 ) is understood to be equal to @xmath5 . ] . let us recall that in the commensurate case where the lattice eigenstates @xmath7 are @xmath2-periodic @xmath8 , with @xmath9 $ ] , the schrodinger equation _ m+1+_m-1 + 2(k_y+m)_m = e_m[eq ] reduces to the @xmath10 secular matrix m_p / q(e , k_x , k_y)= 2 ( k_y)-e & 1 & 0 & & 0 & e^-i q k_x + 1 & 2 ( k_y+)-e & 1 & & 0 & 0 + 0 & 1 & ( ) & & 0 & 0 + & & & & & + 0 & 0 & 0 & & ( ) & 1 + e^i q k_x & 0 & 0 & & 1 & 2 ( k_y+(q-1))-e + acting as [ sharp ] m_p / q(e , k_x , k_y).=0 on the @xmath2-dimensional eigenvector @xmath11 . thanks to the identity ( m_p / q(e , k_x , k_y))=(m_p / q(e,0,0))-2 ( -1)^q ( ( q k_x)-1 + ( q k_y)-1 ) , the schrodinger equation @xmath12 rewrites @xcite as [ eigen ] ( m_p / q(e,0,0))=2 ( -1)^q ( ( q k_x)-1 + ( q k_y)-1 ) the polynomial b_p / q(e)=-_j=0^[q2]a_p / q(2j)e^2j materializes in @xmath13 [ sososimple](m_p / q(e,0,0))+4(-1)^q=(-1)^q e^qb_p / q(1/e ) so that eq . ( [ eigen ] ) becomes @xmath14 the @xmath15 s ( with @xmath16 ) are related to the kreft coefficients @xcite : one obtains ( see @xcite ) @xmath17 with building blocks 4 ^ 2()=e^-ik_y(1-e^2 i ( k+1 ) p q)e^ik_y(1-e^-2 i ( k+1 ) p q)=_p / q(k)expressed in terms of _ p / q(k)=e^-ik_y(1-e^2 i ( k+1 ) p q ) and its complex conjugate.how to get an explicit expression for these coefficients is explained in kreft s paper . we focus on the hofstadter spectrum bandwidth defined in terms of the @xmath18 edge - band energies @xmath19 and @xmath20 , @xmath21 solutions of e^q b_p/ q(1/e)= 4e^q b_p/ q(1/e)=- 4 respectively ( see figures 1 and 2 ) . the 3 horizontal red segments are the energy bands ; the 3 red dots are the mid - band energies ; the 6 black dots are the @xmath22 edge - band energies.,title="fig:"][figure1 ] the 4 horizontal red segments are the energy bands ; the 4 red dots are the mid - band energies ; the 8 black dots are the @xmath22 edge - band energies there are two degenerate dots located at the center.,title="fig:"][figure1bis ] if one specifies an ordering for the @xmath19 s and the @xmath20 s e_1(4)e_2(4) e_q(4)e_1(-4)e_2(-4) e_q(-4)the bandwidth is [ band ] ( -1)^q+1_r=1^q ( -1)^r ( e_r(-4)-e_r(4 ) ) . the thouless formula is obtained in the @xmath1 limit as [ thouless ] _ q ( -1)^q+1 q _ r=1^q ( -1)^r ( e_r(-4)-e_r(4))=32_k=0^(-1)^k1(2k+1)^2(see also @xcite ) . we aim to extend this result to the @xmath0-th moment defined as [ nmoment ] ( -1)^q+1_r=1^q ( -1)^r ( e_r^n(-4)-e_r^n(4 ) ) . which is a natural generalization - th moment [ graal ] ( -1)^q+1_r=1^q ( -1)^r ( e_r(-4)-e_r(4))^n , here defined for @xmath0 odd , would be of particular interest . we will come back to this question in the conclusion . ] of ( [ band ] ) : one can think of it as _ -4 ^ 4 _ p / q^(e)e^n-1dewhere @xmath23 is the indicator function with value 1 when @xmath24 and 0 otherwise . trivially ( [ nmoment ] ) vanish when @xmath0 is even we will see later how to give a non trivial meaning to the @xmath0-th moment in this case . therefore we focus on ( [ nmoment ] ) when @xmath0 is odd and , additionnally , when @xmath2 is odd , in which case it simplifies further to [ hard ] -2_r=1^q ( -1)^r e_r^n(4 ) = 2_r=1^q ( -1)^r e_r^n(-4)thanks to the symmetry @xmath25 . as said above , the @xmath19 s are the roots of @xmath26 that is , by the virtue of ( [ sososimple ] ) , those of ( m_p / q(e,0,0))=0 . the key point in the observation of thouless @xcite is that if evaluating the first moment rewritten in ( [ hard ] ) as @xmath27 when @xmath2 is odd seems at first sight untractable , still , * thanks to @xmath13 factorizing as ( m_p / q(e,0,0))=-(m_p / q^++(e))(m_p / q^(e))where m_p / q^++(e)= + e-2 & 2 & 0 & & 0 & 1 + 1 & e-2 ( ) & 1 & & 0 & 0 + 0 & 1 & ( ) & & 0 & 0 + & & & & & + 0 & 0 & 0 & & ( ) & 1 + 1 & 0 & 0 & & 1 & e-2 ( q-12)-1 + + m_p / q^(e)= + e-2 ( ) & 1 & 0 & & 0 & 1 + 1 & e- 2 ( ) & 1 & & 0 & 0 + 0 & 1 & ( ) & & 0 & 0 + & & & & & + 0 & 0 & 0 & & ( ) & 1 + 1 & 0 & 0 & & 1 & e-2 ( q-12)+1 + + are matrices of size @xmath28 and @xmath29 respectively , so that the @xmath19 s split in two packets @xmath30 , @xmath31 the roots of @xmath32 and @xmath33 , @xmath34 those of @xmath35 * and thanks to @xmath36 happening to rewrite as -_r=1^q ( -1)^r e_r(4)=_r=1^q+12 |e_r^++|-_r=1^q-12 |e_r^| ( [ hard ] ) becomes tractable since it reduces to the sum of the absolute values of the roots of two polynomial equations . indeed using @xcite _ -i x^i x ( zz - a-1)dz=4a ( xa),_x4a ( xa)=2|a|and _ x2i_-i x^i x ( zz - a-1 ) dz=_x2i _ -i x^i x(-)dz one gets [ ratio]2(_r=1^q+12 |e_r^++|-_r=1^q-12 |e_r^|)=2i_x_-i x^i x ( z ( m_p / q^(z))(m_p / q^++(z))).making @xcite further algebraic manipulations on the ratio of determinants in ( [ ratio ] ) in particular in terms of particular solutions @xmath37 of ( [ sharp ] ) on the one hand @xmath38 and on the other hand @xmath39 and then for large @xmath2 taking in ( [ eq ] ) the continuous limit lead to , via the change of variable @xmath40 , _ q 2q(_r=1^q+12 integral gives the first moment [ thoulessbis]_q q(_r=1^q ( -1)^r ( e_r(-4)-e_r(4)))=4(^(1)()-^2)which is a rewriting of ( [ thouless ] ) ( @xmath41 is the polygamma function of order 1 ) . to evaluate the @xmath0-th moment one follows the steps above by first noticing that -_r=1^q ( -1)^r e_r^n(4)=_r=1^q+12 ( z^nz - a-_k=0^n-1 a^k z^n-1-k ) dz=4a^n ( xa ) , _ x4a^n ( xa ) = 2|a^n| and [ toto]_x2i_-i x^i x ( z^nz - a-_k=0^n-1 a^k z^n-1-k ) dz=_x2i _ -i x^i x - nz^n-1(+_k=1^n-1a^kk z^k ) dz one gets [ hardbis]2(_r=1^q+12 ( m_p / q^(z))(m_p / q^++(z)))-_k=1^n-1_r=1^q+12(e_r^++)^k-_r=1^q-12(e_r^)^kk z^k)dz . in the rhs of ( [ toto ] ) the polynomial @xmath42 cancels the positive or nul exponents in the expansion around @xmath43 of the logarithm term @xmath44 . likewise , in ( [ hardbis ] ) , the same mechanism takes place for @xmath45 with respect to @xmath46 . additionally , the polynomials can be reduced to their @xmath47 even components . further algebraic manipulations in ( [ hardbis ] ) and , when @xmath2 is large , taking the continous limit , lead to , via the change of variable @xmath40 , [ finalbis]_q2 q^n(_r=1^q+12 |e_r^++|^n-_r=1^q-12 |e_r^|^n)=(8i)^n-132_0^n y^n-1(((3/4+y)^2y(1/4+y)^2)+_k=2 , k^n-1e_kk 4^k y^k ) dy . to go from ( [ hardbis ] ) to ( [ finalbis ] ) one has used that for @xmath47 even , necessarily ) is also strongly supported by numerical simulations . more generally the @xmath47-th moments @xmath48 and @xmath49 can be directly retrieved from the coefficients of @xmath50 and @xmath51 respectively . in particular one finds @xmath52 and @xmath53 ; for @xmath47 odd @xmath54 and @xmath55 ; for @xmath47 even @xmath56 . this last result can easily be understood in terms of the number @xmath57 of closed lattice walks with @xmath47 steps @xcite . ] [ infinity ] _ qq^k(_r=1^q+12(e_r^++)^k-_r=1^q-12(e_r^)^k)=(2)^k |e_k| where the @xmath58 s are the euler numbers . indeed in ( [ finalbis ] ) , as it was the case in ( [ toto],[hardbis ] ) , the polynomial @xmath59 cancels the positive or nul exponents in the expansion around @xmath60 of the logarithm term @xmath61 @xcite . it amounts to a fine tuning at the infinite upper integration limit so that after integration the end result is finite . performing this last integral gives the @xmath0-th moment [ final]_q q^n ( _ r=1^q ( -1)^r ( e_r^n(-4)-e_r^n(4)))=4((-1)^n-1^(n)()-2^n ( 2^n+1 - 1 ) ( n+1 ) n!)which generalizes the thouless formula ( [ thoulessbis ] ) to @xmath0 odd ( @xmath62 is the polygamma function of order @xmath0 ) . as said above the @xmath0-th moment trivially vanishes when @xmath0 is even . in this case , we should rather consider a @xmath0-th moment restricted to the positive or equivalently by symmetry negative half of the spectrum one considers _ -4 ^ 0 _ p / q^(e)e^n-1de= n_0 ^ 4 _ p / q^(e)e^n-1de .. ] . in the @xmath2 odd case it is [ half ] -_r=1^(q-1)/2 ( -1)^r(e_r^n(-4)- e_r^n(4))+e_(q+1)/ 2^n((-1)^q+124)= _ ( q+3)/2^q ( -1)^r(e_r^n(-4)- e_r^n(4))+e_(q+1)/2^n((-1)^q-124 ) . it is still true that _ r=1^q+12 ( e_r^++)^n-_r=1^q-12 ( e_r^)^n=_(q+3)/2^q ( -1)^r(e_r^n(-4)- e_r^n(4))+e_(q+1)/2^n((-1)^q-124 ) where , since @xmath0 is even , absolute values are not needed anymore , a simpler situation . it follows that the right hand side of ( [ final ] ) also gives , when @xmath0 is even , twice the @xmath1 limit of the half spectrum @xmath0-th moment as defined in ( [ half ] ) , up to a factor @xmath63 . one reaches the conclusion that [ simple]&&4((-1)^n-1^(n)()-2^n ( 2^n+1 - 1 ) ( n+1 ) n ! ) + & = & 4^n+1 n ! _ k=0^ + & = & n ! ( ( n+1,)- ( n+1,))yields @xmath63 times the @xmath0-th moment when @xmath0 is odd is odd it is also twice the half spectrum @xmath0-th moment _ -4 ^ 4 _ p / q^(e)e^n-1de=2 n_0 ^ 4 _ p / q^(e)e^n-1de .. ] and twice the half spectrum @xmath0-th moment when @xmath0 is even . numerical simulations do confirm convincingly this result ( eventhough the convergence is slow ) . in the @xmath0 even case one already knows from ( [ infinity ] ) that ( [ simple ] ) simplifies further to 2|e_n|(2)^n from which one gets for the @xmath64-moment scaling n ! 2 ^ 2 n where = _ n , n even=2.54647 . ( [ simple ] ) is certainly a simple and convincing @xmath0-th moment generalization of the thouless bandwidth formula ( [ thouless ] ) . it remains to be proven on more solid grounds for example in the spirit of @xcite . in the definition of the @xmath0-th moment ( [ nmoment ] ) one can view the exponent @xmath0 as a magnifying loop of the thouless first moment . ( [ simple ] ) was obtained for @xmath65 ( or @xmath6 ) : it would certainly be interesting to understand what happens for @xmath66 where numerical simulations indicate a strong @xmath4 dependence when @xmath0 increases , an effect of the @xmath0-zooming inherent to the @xmath0-th moment definition ( [ nmoment ] ) . in the @xmath0 even case , twice the half spectrum @xmath0-th moment ends up being equal to @xmath67 , a result that can be interpretated as if , at the @xmath0-zooming level , they were @xmath68 bands each of length @xmath69 . it would be interesting to see if this euler counting has a meaning in the context of lattice walks @xcite ( twice the euler number @xmath68 counts the number of alternating permutations in @xmath70 ) . finally , returning to the bandwidth @xmath0-th moment defined in ( [ graal ] ) for @xmath0 odd , and focusing again on @xmath2 odd , one can expand [ expansion]_r=1^q ( -1)^r ( e_r(-4)-e_r(4))^n=-2_k=0^(n-1)/2(-1)^k_r=1^q ( -1)^r e_r(-4)^ke_r(4)^n - kwhere the symmetry @xmath25 has again been used . the @xmath71 term @xmath72 is the @xmath0-th moment discussed above and one knows that multiplying it by @xmath63 ensures in the @xmath1 limit a finite scaling . let us also multiply in ( [ expansion ] ) the @xmath73 terms by @xmath63 : one checks numerically that _ r=1^q ( -1)^r e_r(-4)^ke_r(4)^n - k&=&n-2kn_q-2q^n _ r=1^q ( -1)^r e_r(4)^n + & = & n-2kn4((-1)^n-1^(n)()-2^n ( 2^n+1 - 1 ) ( n+1 ) n ! ) . using _ k=0^(n-1)/2(-1)^kn-2kn=0one concludes that in the @xmath1 limit the bandwidth @xmath0-th moment is such that _ qq^n_r=1^q ( -1)^r ( e_r(-4)-e_r(4))^n=0when @xmath0 is odd , a fact which is also supported by numerical simulations is even , the bandwidth @xmath0-th moment now defined as _ r=1^q ( e_r(-4)-e_r(4))^nis such that @xmath74 . ] . clearly , multiplying the sum in ( [ expansion ] ) by @xmath63 is insufficient , a possible manifestation of the fractal structure @xcite of the band spectrum . we leave to further studies the question of finding a right scaling for the bandwidth @xmath0-th moment . so that zoomed back to @xmath75 , half the bandwidth is @xmath76 , and the complete bandwidth is @xmath77 . but @xmath78 , so for the complete bandwidth @xmath79 . the @xmath80 point spectrum trace = 1q_r=1^q e_r^n(s ) taken on the @xmath2 roots @xmath81 of @xmath82 respectively = 1q_r=1^q e_r(-s)^non the @xmath2 roots @xmath83 of @xmath84 . is * when @xmath2 even and @xmath0 even : @xmath85 * when @xmath2 even and @xmath0 odd : = 0 * when @xmath2 odd and @xmath0 even : = tr_s h^n_2p / q = _ k 0 _ s^2k_j = 1^(q-1)/2 a_p / q(2j)^_j * when @xmath2 odd and @xmath0 odd : = _ k 0 _ s^2k+1_j = 1^(q-1)/2 a_p / q(2j)^_j s. o. acknowlegdes interesting discussions with eugne bogomolny and stephan wagner and thank alain comtet for a careful reading of the manuscript . discussions with vincent pasquier are also acknowledged . thouless , `` bandwidths for a quasiperiodic tight - binding model '' , phys . b 28 ( 1983 ) 4272 - 4276 ; `` scaling for the discrete mathieu equation '' , commun . math . ( 1990 ) 187 - 193 ; d.j . thouless and y. tan , `` total bandwidth for the harper equation . corrections to scaling '' , j. phys . a math . gen . 24 ( 1991 ) 4055 - 4066 . hofstadter , `` energy levels and wave functions of bloch electrons in rational and irrational magnetic fields '' , phys . rev . b * 14 * ( 1976 ) 2239 . w. chambers , phys . rev a140 ( 1965 ) , 135143 . c. kreft , `` explicit computation of the discriminant for the harper equation with rational flux '' , sfb 288 preprint no . 89 ( 1993 ) . b. helffer and p. kerdelhu , `` on the total bandwidth for the rational harper s equation '' , comm . ( 1995 ) , no . 2 , 335 - 356 ; y. last , `` zero measure for the almost mathieu operator '' , commun . ( 1994 ) 421 - 432 ; `` spectral theory of sturm - liouville operators on infinite intervals : a review of recent developments '' , in w.o . amrein , a.m. hinz , d.b . pearson `` sturm - liouville theory : past and present '' , 99120 ( 2005 ) birkhauser verlag basel / switzerland . it can be shown explicitly that @xmath86 is the series expansion of @xmath87 as @xmath88 , s. wagner , private communication . s. ouvry , s. wagner and s. wu , `` on the algebraic area of lattice walks and the hofstadter model '' , journal of physics a : mathematical and theoretical 49 ( 2016 ) 495205 . igital library of mathematical functions . http://dlmf.nist.gov/ , release 1.0.10 of 2015 - 08 - 07 ( 2015 ) . p. flajolet , x. gourdon , and p. dumas . mellin transforms and asymptotics : harmonic sums . , 144:358 ( 1995 ) . in ( [ finalbis ] ) we want to evaluate the integral @xmath89 for odd integers @xmath0 . we will see later that @xmath86 is indeed the series expansion of @xmath90 as @xmath88 . so for @xmath91 , the expression in parentheses is of order @xmath92 ( @xmath93 for @xmath75 ) , and it is of order @xmath94 as @xmath88 . we can now apply integration by parts to obtain @xmath95 in the following , @xmath96 denotes the digamma function . we first show that @xmath97 to this end , we use the well - known integral representation @xcite @xmath98 which yields @xmath99 as we wanted to show . integration also yields @xmath100 now we consider the mellin transform of @xmath101 , i.e. @xmath102 the integral converges for @xmath103 . using the same integral representation as before , we find @xmath104 now we use the following integral representation of the hurwitz zeta function @xcite : @xmath105 for @xmath106 and @xmath107 . this gives us @xmath108 a priori only for @xmath107 . however , the integral also converges for @xmath109 , so the identity remains true by analytic continuation . hence we have @xmath110 the functional equations of the gamma function and the hurwitz zeta function yield @xmath111 and @xmath112 so @xmath113 now we use the general property of the mellin transform ( see @xcite ) that subtracting off terms of the asymptotic expansion at either @xmath114 or @xmath115 only changes the fundamental strip of the mellin transform , but not the transform itself . thus we have @xmath116 which is ( [ final ] , [ simple ] ) .
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we generalize thouless bandwidth formula to its @xmath0-th moment . we obtain a closed expression in terms of polygamma , zeta and euler numbers .
( * ) lptms , cnrs - facult des sciences dorsay , universit paris sud , 91405 orsay cedex , france
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a widely studied model for growth of porous films is ballistic deposition ( bd ) @xcite , in which the particles incide perperdicularly to the substrate and aggregate at the first contact with the deposit @xcite . bd was originally proposed to describe sedimentary rock formation @xcite and was extended to model thin film growth and related systems by considering other aggregation mechanisms , non - colimated particle flux , or polydispersivity of particle size @xcite . most works on the ballistic - like models address the scaling features of the outer surface of the deposits , particularly for the connections with kardar - parisi - zhang ( kpz ) @xcite roughening . some works also connect the surface growth dynamics and the bulk properties of the porous deposits @xcite . this is an essential step for proposing models of porous materials , which have a large variety of technological applications , frequently in the form of thin films @xcite . some ballistic - like models are in a class of competitive growth models in which uncorrelated deposition ( ud ) is obtained for a certain value of a parameter and , near that value , a crossovers in kinetic roughening is observed @xcite . in the simplest model , particle aggregation follows the bd ( ud ) rule with probability @xmath7 ( @xmath8 ) . it was already studied numerically @xcite and with scaling approaches @xcite . for small @xmath7 , there is an enhancement of characteristic times of the correlated kinetics ( @xmath9 ) by a factor @xmath10 and of the outer surface roughness by a factor @xmath11 ; for a recent discussion of this topic , see ref . these features extend to other ballistic - like models with crossovers to ud @xcite and are related to the lateral aggregation . in case of surface relaxation after aggregation , the exponents in those relations are larger , corresponding to longer crossovers for small @xmath7 @xcite . a renewed interest in these competitive models was recently observed , with a focus on the limitations of scaling relations or with an emphasis on the properties of porous media . @xcite discussed the deviations from the dominant scaling of surface roughness at low @xmath7 , which is essential for a quantitative characterization of surface properties in a variety of growth conditions . the effect of relaxation after collision of incident and aggregated particles was considered in ref . @xcite , also with a focus on surface properties . refs . @xcite considered the effect of a stickness parameter on the aggregation of the incident particles , which may attach to neighboring particles located below the outer surface . simulations in @xmath1 dimensions produced deposits with porosity ranging from very small values to approximately @xmath12 and suggested non - kpz behavior in one of the models @xcite . the first aim of this paper is to study surface and bulk properties of the model proposed in ref . @xcite combining a systematic analysis of simulation data and a scaling approach for small values of the stickness parameter . from the extrapolation of saturation roughness and relaxation times , we show that the model has kpz exponents in @xmath1 dimensions . comparison of roughness distributions confirms kpz scaling in @xmath2 dimensions , thus ruling out the proposal of non - kpz exponents . in the limit of small stickness parameter @xmath0 , the scaling approach shows that the crossover time and the roughness scale as @xmath4 and @xmath5 , respectively , for all substrate dimensions . these results show a shortened crossover when compared to all previously studied models with an ud component @xcite , which is a consequence of subsurface aggregation . the same approach predicts porosity and pore height scaling as @xmath6 and @xmath5 , respectively . these predictions will be confirmed numerically in @xmath1 and @xmath2 dimensions . the approach can be extended to the model introduced in ref . @xcite , with the same crossover exponents due to the similar subsurface aggregation conditions . the rest of this work is organized as follows . in sec . [ model ] , we present the sticky particle deposition model . in sec . [ roughness ] , we analyze the surface roughness scaling of simulated deposits in @xmath1 dimensions . in sec . [ scaling ] , we present a scaling approach that relates surface properties to the stickness parameter , and confirm those predictions with numerical simulations . in sec . [ porosity ] , we extend the scaling approach to relate the porosity and the average pore height with the stickness parameter , again with support from numerical simulations . in sec . [ spd3d ] , we show that the main results extend to @xmath2 dimensions . in sec . [ conclusion ] , we summarize our results and conclusions . in all models discussed in this paper , square particles of size @xmath13 are sequentially released on randomly chosen columns of a one - dimensional discretized substrate of lateral size @xmath14 and move in a direction perpendicular to the substrate . here , @xmath15 is the number of columns . the time interval for deposition of one layer of atoms ( @xmath15 atoms ) is @xmath16 . thus , at time @xmath17 , the number of deposited layers is @xmath18 . the model proposed in ref . @xcite is hereafter called sticky particle deposition ( spd ) . in each site of the trajectory of the incident particle , it interacts with particles in nearest neighbor ( nn ) sites at the same layer ( same height above the deposit ) and particles in next neareast neighbor ( nnn ) sites at the layer immediately below it . this interaction is represented by a probabilistic rule of aggregation at its current position . the probability of aggregation to each neighbor is @xmath19 where @xmath0 is the stickness parameter , @xmath20 is the distance between the centers of the particles ( @xmath21 for nn , @xmath22 for nnn ) , and @xmath23 is an exponent related to the nature of the interaction . in ref . @xcite , the cases @xmath24 and @xmath25 were respectively called coulomb - type and van der walls - type interactions , with most results being presented for the former . here we will restrict the analysis to the case @xmath24 , in which aggregation to nn and nnn have probabilities @xmath0 and @xmath26 , respectively . fig . [ modelspd ] helps to understand the aggregation rules of the spd model and the differences from other ballistic - like models . we first recall the rules of bd and of the next nearest neighbor bd ( bdnnn ) model @xcite . in bd , aggregation occurs at the first contact with a nn occupied site : particle a at position 2 , particle b at position 6 in fig . [ modelspd ] . in bdnnn , aggregation occurs at the first contact with a nnn occupied site : position 1 for particle a and position 5 for particle b in fig . [ modelspd ] . in both cases , the incident particle interacts only with the particles at the top of each column , which are highlighted in [ modelspd ] . the set of top particles is called the outer surface of the deposit . in the spd model , particle a may aggregate at lattice sites marked with circles labeled 1 to 4 , and particle b may aggregate at lattice sites marked with circles labeled 5 to 7 . first , consider the trajectory of particle a. in position 1 , two aggregation trials are executed due to the interaction with two nnn occupied sites ; the probability of aggregation in each trial is @xmath26 . if it does not aggregate there , it moves to the position labeled 2 , in which three aggregation trials are executed : two for interactions with the nn at the same height ( probability @xmath0 for each one ) and one for interaction with the nnn in the layer below , at the left ( probability @xmath26 ) . if the particle does not aggregate at position 2 , then it moves to position 3 and may aggregate there with probability @xmath0 due to the interaction with the nn at the left . if aggregation does not occur in position 3 , the incident particle will permanently aggregate at position 4 , which is the top of the incidence column . relaxation to neighboring columns is not allowed . now we consider the trajectory of particle b. in position 5 , two aggregation trials are executed , each one with probability @xmath26 ( due to interactions with two occupied nnn sites ) . if the particle does not aggregate there , it moves to position 6 , in which three aggregation trials are executed : two for the interactions with the lateral nn ( probability @xmath0 for each trial ) and one for the interaction with the nnn in the layer below ( probability @xmath26 ) . if no aggregation trial is successful at position 6 , the particle moves to position 7 and aggregates there . in contrast to other ballistic - like models ( e. g. bd and bdnnn ) , the spd model allows subsurface aggregation . in fig . [ modelspd ] , position 3 is an example of subsurface position : it is not allowed in bd , nor in bdnnn , nor in any model of solid - on - solid deposition ( which prescribe aggregation at the top of each column ) . in all cases , note that the interaction of an incident particle with an aggregated one is possible in two steps : the first one when they are nnn ( larger distance ) , the second one when they are nn ( smaller distance ) . it represents two possibilities of aggregation in the ingoing part of the trajectory of the incident particle . if the aggregation trials are not accepted , then the incident particle moves to a lower position . in this situation , this particle is in an outgoing trajectory respectively to those aggregated particles . for this reason , no aggregation trial is executed with a nnn aggregated particle in the layer above the current position of the incident particle . for instance , when particle a is at position 3 ( third layer of the deposit ) , we do not execute aggregation trials with the black nnn sites at the fourth layer in fig . [ modelspd ] . the spd model resembles the model introduced in ref . @xcite and the slippery bd model ( sbd ) proposed in ref . @xcite , both studied in three - dimensional deposits ( the latter with line seeds perpendicular to a flat inactive surface ) . most of our simulation work is in @xmath1 dimensions , similarly to ref . @xcite , but in sec . [ spd3d ] we show that the main results are also valid in three - dimensional samples . for simplicity , in the following sections we consider unit values of the lattice constant and of the time of deposition of a layer : @xmath27 , @xmath28 . the outer surface roughness is defined as @xmath29 } ^{1/2 } , \label{defw}\ ] ] where @xmath30 is the height of the top particle of each column , the overbars indicate spatial averages , and the angular brackets indicate configurational averages . in systems with normal ( in opposition to anomalous ) scaling , the roughness follows family - vicsek ( fv ) scaling @xcite as @xmath31 where @xmath32 is the roughness exponent , @xmath33 is a relaxation time , and @xmath34 is a scaling function . in long times ( @xmath35 ) , @xmath36 , so that @xmath37 saturates as @xmath38 where @xmath39 is a model - dependent constant . the saturation time @xmath33 scales as @xmath40 where @xmath41 is the dynamic exponent and @xmath42 is another model - dependent constant . the roughness for @xmath43 scales as @xmath44 with @xmath45 and another model - dependent constant @xmath46 . [ figroughness]a shows the surface roughness evolution of the spd model for three values of @xmath0 in @xmath47 . ( red solid curve ) , @xmath48 ( green dashed curve ) , and @xmath49 ( blue dotted curve ) . the dashed line has slope @xmath50 of kpz scaling . ( b ) saturation roughness as a function of the lattice size for @xmath51 ( red squares ) and @xmath52 ( green triangles ) . the dashed line has slope @xmath53 of kpz scaling . , width=264 ] for short times , there is a crossover from an initial regime of rapid roughness increase to a second regime in which it increases slower . for small @xmath0 , the first regime is mainly of ud and the slope of the @xmath54 plot is near @xmath53 . for @xmath55 , lateral aggregation is frequent , thus the roughness at short times is larger than that of ud ( e. g. @xmath51 in fig . [ figroughness]a ) . it is difficult to find a pure ud regime in this case and to estimate the crossover time with accuracy . after this transient , the growth regime begins , with apparent power law scaling of @xmath37 [ eq . ( [ defbeta ] ) ] . it is difficult to distinguish the different curves for small @xmath0 in fig . [ figroughness]a ; this will be explained by the scaling approach of sec . [ scaling ] . the slope of those curves are near @xmath50 , suggesting kpz scaling . at long times , there is an increase in the saturation roughness as @xmath0 decreases . [ figroughness]b shows the saturation roughness as a function of lattice size @xmath15 for two values of @xmath0 . they seem to be consistent with the kpz exponent @xmath56 . however , linear fits of those plots give slopes slightly smaller than @xmath57 , similarly to what was found in ref . for this reason , a systematic extrapolation of those results is necessary to decide whether the roughness scaling is kpz or not . we proceed by using the same methods of refs . @xcite , in which roughness scaling of various ballistic - like models was studied . effective roughness exponents are defined as @xmath58}{\ln{2 } } . \label{defalphal}\ ] ] assuming that the saturation roughness has scaling corrections as @xmath59 @xcite , where @xmath60 and @xmath61 are constants , we expect @xmath62 , where @xmath63 is another constant . fig . [ alpha ] shows effective exponents as a function of @xmath64 for @xmath51 and @xmath48 , respectively using @xmath65 and @xmath66 . these values of @xmath67 provide the best linear fits of the @xmath68 data for each stickness parameter . the asymptotic ( @xmath69 ) estimates from those fits are @xmath70 and @xmath71 , respectively . for @xmath48 ( red triangles ) with @xmath66 and @xmath51 ( blue squares ) with @xmath72 . , width=264 ] we estimate the dynamical exponent @xmath41 using the method proposed in ref . for each lattice size @xmath15 , a characteristic time @xmath73 is defined as @xmath74 with @xmath75 . the fv relation ( [ fv ] ) shows that @xmath73 is proportional to @xmath33 for fixed @xmath76 , thus @xmath77 . effective dynamical exponents are defined as @xmath78}{\ln{2 } } . \label{defzl}\ ] ] figs . [ zl]a and [ zl]b show @xmath79 for @xmath51 and @xmath48 , respectively , obtained with @xmath80 . in both cases , the exponents oscillate near @xmath81 , suggesting that finite - size corrections are very small . for ( a ) @xmath48 and ( b ) @xmath51.,width=264 ] the estimates of @xmath32 and @xmath41 are in very good agreement with kpz exponents @xmath82 and @xmath83 , which is a strong numerical evidence that this model is in the kpz class in @xmath1 dimensions . a universal scaling is expected in the spd model because there is no change in its symmetries as the stickness parameter changes . in other words , the corresponding hydrodynamic growth equation may have coefficients dependent on the parameter @xmath0 , but the leading spatial derivatives will be the same @xcite . due to the lateral aggregation and consequent excess growth velocity , kpz scaling is expected for any @xmath3 . previous works on ballistic - like models @xcite have already shown that systematic extrapolation of finite - size or finite - time data are necessary to avoid crossover effects . as highlighted in ref . @xcite , this is a consequence of the large fluctuations in height increments , typical of those models . crossovers and finite - size corrections probably are the reasons for the deviations from kpz scaling observed in ref . @xcite . this may also be inferred by comparison with finite - size bd data from ref . @xcite . ref . @xcite suggests @xmath84 for @xmath85 , while ref . @xcite gives effective exponents @xmath86 for bd in the same range of @xmath15 . moreover , the growth exponents @xmath87 in lattice sizes from @xmath88 to @xmath47 for @xmath85 , shown in ref . @xcite , are very near the corresponding estimates for bd in ref . @xcite ( considering minimum linear correlation coefficient @xmath89 in the growth region ) . for small @xmath0 , lateral aggregation is unprobable , thus most particles aggregate at the top of the column of incidence . at short times , the roughness increases as @xcite @xmath90 after a crossover time @xmath91 , kpz scaling appears . our first step is to relate @xmath91 to @xmath0 , for @xmath92 . a typical configuration of two neighboring columns in ud is illustrated in fig . [ columns ] . it has a height difference @xmath93 because their heights increase without correlations . if @xmath94 is large , then a new particle inciding at the right column in fig . [ columns ] may aggregate at a number of positions of order @xmath94 , as indicated by the circles . this means that the number of aggregation trials is of order @xmath94 and the aggregation probability for each trial is of order @xmath0 . thus , the probability of no lateral aggregation after those trials ( i. e. aggregation at the top of the column ) is @xmath95 . the probability that some lateral aggregation occurs is , consequently , @xmath96 the latter approximation requires @xmath97 , which will be confirmed below . the average time for a lateral aggregation event at a given column is @xmath98 . lateral aggregation immediately creates correlations between the heights of neighboring columns , thus the crossover time is @xmath99 . ( [ wrandom ] ) and ( [ plat ] ) at the crossover ( @xmath100 , @xmath101 ) give @xmath102 thus , height fluctuations at the crossover scale as @xmath103 and the crossover time scales as @xmath104 these results confirm that @xmath105 , as the approximation in eq . ( [ plat ] ) requires . the amplitudes of the saturation roughness [ eq . ( [ fv ] ) ] and of the relaxation time [ eq . ( [ scalingttimes ] ) ] scale as @xmath106 and @xmath91 , respectively . following the exponent convention introduced by horowitz and albano @xcite , we have @xmath107 and @xmath108 these results are valid in any spatial dimension because ud properties are not dimension - dependent . the scaling exponents in eqs . ( [ defdelta ] ) and ( [ defy ] ) differ from those obtained in other competitive models with ballistic - like aggregation with probability @xmath7 and ud with probability @xmath8 ; in those systems , @xmath109 and @xmath110 @xcite . in solid - on - solid models with crossovers from ud to correlated growth , the exponents are also different : @xmath111 and @xmath112 . the shorter crossover of the spd model is due to subsurface aggregation , which provides a large number of opportunities ( of order @xmath106 ) for lateral aggregation of the incident particle ( fig . [ columns ] ) . on the other hand , the relation @xmath113 obtained in other competitive models is also obeyed here because it is solely related to ud scaling ( see e. g. the discussion in ref . @xcite ) . we performed simulations of the spd in @xmath47 and small values of @xmath0 , from @xmath52 to @xmath114 , until the steady states ( roughness saturation ) . the saturation roughness @xmath115 and the characteristic times @xmath73 were calculated following the same lines of sec . [ roughness ] . [ wt]a and [ wt]b show @xmath73 and @xmath115 , respectively , as a function of the stickness parameter @xmath0 . since they were measured for constant @xmath15 , they are expected to scale as the amplitudes @xmath42 [ eq . ( [ scalingttimes ] ) ] and @xmath39 [ eq . ( [ wsat ] ) ] , respectively . fits of the data for @xmath116 give exponents @xmath117 [ eq . ( [ defy ] ) and fig . [ wt]a ] and @xmath118 [ eq . ( [ defdelta ] ) and fig . [ wt]b ] . and ( b ) saturation roughness @xmath115 in size @xmath47 as a function of the stickness parameter . solid lines are fits of the data for @xmath119.,width=264 ] the estimate of @xmath120 is in good agreement with the theoretical prediction of eq . ( [ defy ] ) . however , the estimate of @xmath121 shows a discrepancy of @xmath122 from the theoretical prediction of eq . ( [ defdelta ] ) . note that the fits in figs . [ wt]a and [ wt]b considered @xmath123 , which are not very small values of @xmath0 , thus deviations are expected , particularly in the smaller exponent ( @xmath121 ) . unfortunately , it is very difficult to obtain accurate estimates for smaller values of @xmath0 because relaxation times become very large and roughness fluctuations also increase . using smaller system sizes is also inappropriate because it enhances crossover effects @xcite estimated the crossover times for @xmath124 and obtained @xmath125 , which is significanly different from the theoretical prediction in eq . ( [ tc ] ) . however , measuring reliable crossover times is a difficult task , as explained in sec . [ roughness ] . on the other hand , the same work shows that the saturation time for @xmath47 scales as @xmath126 , which is in good agreement with our estimate . the scaling of the amplitude @xmath46 in eq . ( [ defbeta ] ) can be predicted along the same lines of refs . @xcite for other competitive models : @xmath127 for the spd model in @xmath1 dimensions , we obtain @xmath128 . this very small exponent gives a very slow variation of @xmath46 with the stickness parameter . it explains the small distance between the curves for different values of @xmath0 in fig . [ figroughness]a . in ref . @xcite , a model similar to the spd was introduced . the particles incide vertically and , at each site of its trajectory with a nn occupied site , it may aggregate with probability @xmath7 . otherwise , the particle moves down one site . if no lateral aggregation occurs , the particle aggregates at the top of the column of incidence . in fig . [ modelspd ] , particle a may aggregate to positions labeled 2 , 3 , and 4 . in positions 2 and 3 , aggregation trials have probability @xmath7 . if the particle does not aggregate at one of those points , it moves to position 4 and aggregates there . particle b may aggregate at position 6 with probability @xmath7 , otherwise it moves to position 7 and aggregates there . for small @xmath7 , most lateral aggregation trials are rejected , thus ud dominates . large local height fluctuations appear , similarly to fig . [ columns ] . the increase of the local height difference @xmath94 and the probability of lateral aggregation @xmath129 are given by eqs . ( [ deltah ] ) and ( [ plat ] ) , with @xmath0 replaced by @xmath7 . thus , the same reasoning of sec . [ approach ] leads to the same scaling relations of the spd model with @xmath0 replaced by @xmath7 . in the notation of ref . @xcite , exponents @xmath130 and @xmath131 are predicted by our scaling approach . the numerical estimates of that work , @xmath132 and @xmath133 , differ from those predictions , probably because they were obtained by data collapse methods that do not account for scaling corrections . for @xmath85 , the samples have large porosity @xmath134 . when @xmath0 decreases , @xmath135 decreases because ud creates no holes . [ samplesspd ] shows regions of some samples obtained with small stickness parameters . the porosity decrease is accompanied by the formation of longer pores extended in the vertical direction . this is a consequence of the increase of the height fluctuation @xmath106 before a lateral aggregation event [ eq . ( [ deltahc ] ) ] . ( in lattice units ) of samples grown with stickness parameters ( a ) @xmath52 , ( b ) @xmath136 , ( c ) @xmath137 , and ( d ) @xmath138.,width=264 ] the number of deposited layers necessary to attain a steady state value of @xmath135 is relatively small , typically of the order of @xmath91 in eq . ( [ tc ] ) . this is expected because pores are narrow , even in pure bd , thus porosity depends only on short wavelength height fluctuations , which saturate at short times ( in the absence of scaling anomaly @xcite ) . our scaling approach can be used to predict the dependence of the porosity @xmath135 and the average pore height on the parameter @xmath0 , as follows . during the time interval @xmath91 between two lateral aggregation events , the number of particles deposited at the top of a given column is approximately @xmath91 ( note that we are still using unit lattice constant and unit deposition time of a layer , @xmath27 and @xmath28 ) . the size of a long pore produced by the lateral aggregation is @xmath106 [ eq . ( [ deltahc ] ) ] . consequently , for small @xmath0 , the porosity ( pore volume divided by total volume ) is expected to scale as @xmath139 this is valid in the limit of very small @xmath0 , in which @xmath140 . the small exponent in eq . ( [ pscaling ] ) explains why a large decrease of @xmath0 leads only to mild reduction of porosity . this is remarkably illustrated in fig . [ samplesspd ] , in which @xmath0 varies three orders of magnitude , while the porosity decreases from @xmath141 to @xmath142 , i. e. changes by a factor smaller than @xmath143 . the porosity scaling in the spd also differs from other competitive models involving ballistic - type aggregation . examples are the bidisperse ballistic deposition @xcite and the bd - ud competitive model , in which @xmath144 ( @xmath7 is the probability of the ballistic - like component ) . we simulated the spd in size @xmath47 for small values of @xmath0 in order to measure the porosity between times @xmath145 and @xmath146 . in all cases , @xmath147 is much larger than the crossover time and @xmath148 is much smaller than the relaxation time @xmath33 . [ phi]a shows the porosity as a function of the stickness parameter . the linear fit for @xmath149 gives @xmath150 , in excellent agreement with eq . ( [ pscaling ] ) . although these values of @xmath0 are very small , the corresponding values of @xmath6 and of @xmath135 are not very small . thus , scaling corrections are particularly weak in this case . . ( b ) average pore height as a function of the stickness parameter . dashed lines have slopes @xmath151 ( right ) and @xmath152 ( left).,width=264 ] for very small @xmath0 , the pores are long and isolated , as illustrated in fig . [ samplesspd]d . the average pore height is expected to scale as eq . ( [ deltahc ] ) , because a pore is formed only when a lateral aggregation event occurs . however , for @xmath153 or larger , many pores occupy two or more neighboring columns . this can be observed in figs . [ samplesspd]a , [ samplesspd]b , and [ samplesspd]c . here we define pore height as the vertical distance between the aggregation position and the top of the incidence column in any lateral aggregation event . its average value , @xmath154 , is taken over all lateral aggregation events between @xmath147 and @xmath148 in @xmath155 different samples . for small @xmath135 , pores are isolated , thus @xmath154 is a reliable approximation of the average pore height , and is expected to scale as eq . ( [ deltahc ] ) . for @xmath135 not too small , some pores occupy two or more neighboring columns , and all these columns contribute to @xmath154 ( each one had a lateral aggregation event ) . fig . [ phi]b shows @xmath154 as a function of @xmath0 . the slope of that log - log plot evolves from @xmath151 for @xmath156 to @xmath152 for @xmath157 . the latter is @xmath158 smaller than the theoretically predicted value @xmath159 [ eq . ( [ deltahc ] ) ] , which indicates the presence of large scaling corrections . ref . @xcite measured the porosity of samples with @xmath160 , with results in qualitative agreement with ours . however , the low porosity scaling was not addressed there . @xcite suggests that the porosity scales with @xmath7 ( equivalent to @xmath0 ) and with the lattice size @xmath15 . the latter is expected only as vanishing corrections , since porosity does not depend on long wavelength fluctuations . this explains the small ( effective ) exponents @xmath0 and @xmath13 obtained in that work . on the other hand , ref . @xcite estimates the long - time scaling on @xmath7 with exponent @xmath161 , which is to be compared with the theoretical prediction @xmath50 . the discrepancy is probably related to the use of data collapse methods . the aim of this section is to show that the main features of the spd model in @xmath1 dimensions can be extended to @xmath2 dimensions , namely the kpz roughening of the outer surface and the porosity scaling derived by the superuniversal approach of sec . [ scaling ] . the aggregation rules of the spd model have to be extended in this case . first , nn interactions are considered in two substrate directions , with a total of four nn in the same height . secondly , nnn interactions appear with aggregated particles in the same height ( four neighbors ) and with particles at the level immediately below ( four neighbors ) . roughness scaling of ballistic - like models usually show large corrections @xcite . an alternative to search for the universality class of a given model is the comparison of scaled roughness distributions of relatively small systems because the finite - size corrections of those quantities are much smaller @xcite . we simulated the spd model with @xmath48 in substrates of lateral size @xmath88 up to the steady state ( roughness saturation ) . in this regime , the square roughness @xmath162 of several configurations is measured . @xmath163 is the probability density of the square roughness of a given configuration to lie in the range @xmath164 $ ] . this quantity is expected to scale as @xmath165 where @xmath166 is the rms fluctuation of @xmath167 and @xmath168 is a universal function @xcite . fig . [ dist ] shows the scaled roughness distribution of the spd model and the distribution of the restricted solid - on - solid ( rsos ) model @xcite in substrate size @xmath88 . the latter is a well known representative of the kpz class and its roughness distributions have negligible finite - size effects @xcite . the excellent collapse of the curves in fig . [ dist ] is striking evidence that the spd model also belongs to the kpz class in @xmath2 dimensions . ( squares ) and of the rsos model ( solid curve ) in @xmath2 dimensions , with @xmath88 . , width=264 ] we also simulated the spd model in size @xmath47 for small values of @xmath0 and measured the porosity between times @xmath169 and @xmath170 . we observe that the porosity is larger than in the @xmath1-dimensional samples for the same value of @xmath0 . for instance , for @xmath48 , the porosity exceeds @xmath171 . this is a consequence of the larger number of interactions of the incident particle with nn and nnn in @xmath2 dimensions , which facilitates lateral aggregation . [ phi3d ] shows the porosity as a function of @xmath0 , for low values of that parameter . the linear fit for @xmath149 gives @xmath172 , which is also in excellent agreement with eq . ( [ pscaling ] ) . this supports the extension of the scaling approach of sec . [ scaling ] to @xmath2 dimensions . dimensions . the solid line is a linear fit of the data for @xmath173 . , width=264 ] an important consequence of this scaling approach is to facilitate the design of samples with the desired values of porosity and elongate pores . however , one has to take care with the fluctuations in the value of @xmath135 in the first layers of the deposit , typically produced at @xmath174 . we studied surface and bulk properties of porous deposits produced by a model proposed in ref . @xcite , in substrates with one and two dimensions . the model shows a crossover from uncorrelated to correlated growth for small values of the stickness parameter @xmath0 . in @xmath1 dimensions , a systematic analysis of simulation data for saturation roughness and relaxation times shows that the model belongs to the kpz class . finite - size corrections explain the previous claim of deviations from kpz scaling . in @xmath2 dimensions , kpz roughening is confirmed by comparison of roughness distributions . a scaling approach for small values of @xmath0 is proposed to relate the crossover time and the local height fluctuations with that parameter , respectively giving exponents @xmath175 and @xmath159 . these results are consequence of the ud properties , thus they do not depend on the spatial dimension . numerical results confirm these predictions . the crossover exponents are smaller than those of other competitive models that consider aggregation only at the outer surface @xcite . the same approach predicts the porosity scaling as @xmath6 , which is in good agreement with simulation results in @xmath1 and @xmath2 dimensions . this result is important for using the model to produce porous samples representative of real materials . this may also help to model samples with desired porosity and pore height , particularly for the possibility of controlling the scaling properties by changing the kinetics of subsurface aggregation .
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we study surface and bulk properties of porous films produced by a model in which particles incide perpendicularly to a substrate , interact with deposited neighbors in its trajectory , and aggregate laterally with probability of order @xmath0 at each position .
the model generalizes ballistic - like models by allowing attachment to particles below the outer surface . for small values of @xmath0 , a crossover from uncorrelated deposition ( ud ) to correlated growth
is observed .
simulations are performed in @xmath1 and @xmath2 dimensions .
extrapolation of effective exponents and comparison of roughness distributions confirm kardar - parisi - zhang roughening of the outer surface for @xmath3 .
a scaling approach for small @xmath0 predicts crossover times as @xmath4 and local height fluctuations as @xmath5 at the crossover , independently of substrate dimension .
these relations are different from all previously studied models with crossovers from ud to correlated growth due to subsurface aggregation , which reduces scaling exponents .
the same approach predicts the porosity and average pore height scaling as @xmath6 and @xmath5 , respectively , in good agreement with simulation results in @xmath1 and @xmath2 dimensions
. these results may be useful to modeling samples with desired porosity and long pores .
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planetpack is a software tool that facilitates the detection and characterization of exoplanets from the radial velocity ( rv ) data , as well as basic tasks of long - term dynamical simulations in exoplanetary systems . the detailed description of the numeric algorithms implemented in planetpack is given in the paper @xcite , coming with its initial 1.0 release . after that several updates of the package were released , offering a lot of bug fixes , minor improvements , as well as moderate expansions of the functionality . as of this writing , the current downloadable version of planetpack is 1.8.1 . the current source code , as well as the technical manual , can be downloaded at ` http://sourceforge.net/projects/planetpack ` . here we pre - announce the first major update of the package , planetpack 2.0 , which should be released in the near future . in addition to numerous bug fixes , this update includes a reorganization of the large parts of its architecture , and several new major algorithms . now we briefly describe the main changes . the following new features of the planetpack 2.0 release deserve noticing : 1 . multithreading and parallelized computing , increasing the performance of some computationally heavy algorithms . this was achieved by migrating to the new ansi standard of the c++ language , c++11 . several new models of the doppler noise can be selected by the user , including e.g. the regularized model from @xcite . this regularized model often helps to suppress the non - linearity of the rv curve fit . 3 . the optimized computation algorithm of the so - called keplerian periodogram @xcite , equipped with an efficient analytic method of calculating its significance levels ( baluev 2014 , in prep . ) . 4 . fitting exoplanetary transit lightcurves is now implemented in planetpack . this algorithm can fit just a single transit lightcurve , as well as a series of transits for the same star to generate the transit timing variation ( ttv ) data . these ttv data can be further analysed as well in order to e.g. reveal possible periodic variations indicating the presence of additional ( non - transiting ) planets in the system . the transit lightcurve model is based on the stellar limb darkening model by @xcite . also , the transit fitting can be performed taking into account the red ( correlated ) noise in the photometry data . some results of the planetpack ttv analysis of the photometric data from the exoplanet transit database , ` http://var2.astro.cz/etd/ ` , will be soon presented in a separate work . concerning the evolution of the planetpack code , we plan to further develop the transit and ttv analysis module and to better integrate it with the doppler analysis block . we expect that in a rather near future planetpack should be able to solve such complicated tasks as the simultaneous fitting of the rv , transit , and ttv data for the same star . this integration should also take into account subtle intervenue between the doppler and photometry measurements like the rositter - mclaughlin effect .
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we briefly overview the new features of planetpack2 , the forthcoming update of planetpack , which is a software tool for exoplanets detection and characterization from doppler radial velocity data . among other things
, this major update brings parallelized computing , new advanced models of the doppler noise , handling of the so - called keplerian periodogram , and routines for transits fitting and transit timing variation analysis .
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quantum memory can store information in superposition states of a collection of two - level systems . optical ion trap by laser cooling has been prepared to construct quantum logic gates @xcite . in those systems , negative role played by quantum decoherence @xcite is quite significant . randomization of the quantum states produced by entanglement with environmental modes is inevitable in case of storage or processing of non - orthogonal states and environmental interaction allows leakage of some information to the environment @xcite . since it is practically impossible to disentangle the system from the environment , our main efforts are focussed on minimizing decoherence . in this attempt of decoherence minimization , zeno dynamics plays a very significant role @xcite . quantum zeno effect @xcite is depicted as the complete freezing of the decay dynamics due to frequent measurement . it has been shown previously that very frequent measurement of excited states can suppress the decoherence @xcite . in our understanding decoherence and zeno effect has got intrinsic reciprocal relationship between them . the argument behind this statement is as follows : whenever any disturbance in the form of measurement dominates the time evolution of the state of the system , the system is forced to evolve in a subspace of the total hilbert space @xcite . this subspace is called zeno subspace " . nonselective measurement causes the appearance of these subspaces . facchi et.al @xcite have shown that frequent nonselective measurement splits the total hilbert space into invariant quantum zeno subspaces , between which probability leakage is not possible . but probability is conserved within each subspace . so each of the subspace can be considered as an reduced isolated system . if the system undergoes very strong environmental interaction , due to extreme decoherence , these isolated subspaces may not be sustainable . so we can infer that the zeno effect characterized by a certain time scale ( zeno time ) , gives a kind of lower limit to decoherence , below which the process of decoherence will be uncontrollable . the relation between these two phenomena is reciprocal in the sense that within the zeno subspace , due to it s isolated nature , it precludes environment induced decoherence . exploiting this relation , we will formulate the procedure to compare the respective time scales and come up with a certain transitional temperature , below which asymptotic minimization of state decoherence is possible . + the master equation for the density operator in position representation of a certain quantum system can be given as @xcite [ 1.1 ] = --(-)-(x - x)^2 where the first term on the right hand side is the usual commutator term of the von neumann equation . the second term represents dissipation with @xmath0 as the relaxation rate . the third and last term represent the fluctuations leading to random brownian effects . this term being proportional to @xmath1 , though has little effect on the diagonal peaks of the density matrix , but affects the off - diagonal peaks considerably and causes them to decay . hence the effect of this last term leads to the destruction of quantum coherence . from equation ( [ 1.1 ] ) we can easily get that the decay rate of the off - diagonal peaks of the density matrix [ 1.2 ] = -(x - x)^2 = -_dec ^-1where [ 1.3 ] _ dec= is the time scale on which the quantum coherence disappears and is defined as decoherence time . from the solution of equation ( [ 1.2 ] ) , one can easily get [ 1.4 ] ( x , x,t)=(x , x,0 ) ( -t/_dec ) decoherence visibly supresses the interference between macroscopically different quantum states , which is precisely the very property that distinguishes quantum mechanics from it s classical counterpart from observational perspective . here we will consider tunneling in a bistable potential as a model system to develop the expression for decoherence time . as a physically realistic example we will consider a system of laser cooled trapped ion @xcite , where decoherence appears in the dynamics of hyperfine states . comparison between decoherence time and zeno time for this specific case will lead us to find the transitional temperature over which decoherence will dominate the whole process . + let us first concentrate on the calculation of the relaxation rate @xmath2 in presence of dissipative interaction . in a recent paper @xcite we have estimated the weak value of dwell time for a dissipative spin - half system using the same formalism . the approach that has been used here , was originally developed by caldirola and montaldi @xcite introducing a discrete time parameter ( @xmath3 ) incorporating the properties of environment . the schrdinger difference equation in presence of environment induced dissipation is given by [ 2.01 ] h_i|=i it has been shown @xcite that this equation has retarded nature and so naturally implies the dissipative character of it s solution . the discrete time parameter ( @xmath3 ) appears as some sort of relaxation time , incorporating the environment induced dissipation . to supplement this difference equation , we will show further that the time parameter ( @xmath3 ) can be expressed as a function of the energy eigen - values of the quantum states . now as a consequence of the retarded nature of eqn ( [ 2.01 ] ) , we can see that the ground state will also decay . so to stabilize the ground state , the schrdinger difference equation is scaled as [ 2.1 ] ( h_i - h_0)|=i where @xmath4 and @xmath5 are the hamiltonian for i - th and ground state respectively . @xmath5 is introduced in the equation to stabilize the ground state @xcite . we expand @xmath6 in taylor series to get [ 2.2 ] ( h_i - h_0)|=i setting the trial solution as @xmath7 and solving for @xmath8 , we get [ 2.3 ] = ( 1+i(e_i - e_0)/ ) where @xmath9 and @xmath10 are the eigenvalues for the corresponding hamiltonians . expanding the logarithm upto third order , we find that the time evolution takes the form [ 2.4 ] so from ( [ 2.4 ] ) we find the decay rate as [ 2.5 ] = setting the final hamiltonian as @xmath11 , we also find the unknown time parameter as @xcite [ 2.6 ] = so using the value of the time parameter @xmath3 in equation ( [ 2.5 ] ) , we find the decay constant [ 2.7 ] = this is the decay constant for a quantum system decaying from initial to final energy eigenstate denoted by @xmath4 and @xmath11 respectively . we will substitute this decay ( relaxation ) constant in the expression of decoherence time given by equation ( [ 1.3 ] ) . the interaction parameter was given by @xmath3 , which was substituted by a function of the initial and final hamiltonians considering the time evolution dynamics . but to calculate the spatial shift ( @xmath12 ) , we have to consider the spatial dynamics of the system . here we will focus our attention on an asymmetric double well potential approximated as a two - state system with considering only the ground states of the wells separated by an asymmetry energy @xmath13 . we construct our model on the demonstration of a quantum logic gate prepared by a trapped ion laser cooled to the zero point energy @xcite . in this particular case , the target qubit is spanned by two @xmath14 hyperfine ground states ( @xmath15 states ) of a single @xmath16 ion separated by @xmath17 ghz . we set these two energy levels as the ground states of the two wells of the double well potential separated by the asymmetry energy @xmath18 . the control qubit @xmath19 is spanned by the first two states of trapped ion ( @xmath20 ) , which can be identified by the first two states of each well approximated as harmonic oscillators . these two states are separated by @xmath21 mhz.so the basic four eigenstates are given by @xmath22 . let us now consider a quartic potential of the form @xcite [ 2.8 ] v(x)=m^2 x^2 where @xmath23 and @xmath24 are dimensionless constants . for the particular case of double well , @xmath25 and @xmath26 . and @xmath26,width=264,height=188 ] the two potential minima can be found at [ 2.9 ] x_0=0 x_2=these two minima are separated by the barrier maxima situated at [ 2.10 ] x_1=the potential can be expressed in terms of dimensionless variable @xmath27 as [ 2.11 ] v()=m^2a^2 ^2[^2-a+b ] now the dimensionless form of the hamiltonian is given by [ 2.12 ] k = where @xmath28 is the actual hamiltonian and the parameter @xmath29 . here , the wells are approximated as harmonic oscillators . so we expand the potential given by equation ( [ 2.11 ] ) around the first minima @xmath30 to find [ 2.13 ] v()=^2b^2 the normalized ground state wave function for this approximated potential can be set as [ 2.14 ] ( ) = ( ) ^1/2where @xmath31 . + let us consider the transition from left well to the right well . so here the initial position of the particle is @xmath32 . so @xmath33 . now [ 2.15 ] ll < x^2>=a^2<^2>= a^2 _ -^ ^2e^-^2d + = therefore the average of the square of the spatial shift [ 2.16 ] x^2=<x^2>= by using ( [ 2.7 ] ) and ( [ 2.16 ] ) in equation ( [ 1.3 ] ) , we deduce the decoherence time as [ 2.17 ] _ dec= ( ) now the asymmetry energy ( @xmath13 ) can be expressed as [ 2.18 ] = v(x_0)-v(x_2 ) for a double well potential ( @xmath34 ) , from equation ( [ 2.18])we get [ 2.19 ] = = where the width of the well is given by [ 2.20 ] w = x_2-x_0= therefore the decoherence time of equation ( [ 2.17 ] ) can be expressed as [ 2.21 ] _ dec = we will estimate the numerical value of decoherence time after deriving a certain relation between the decoherence time and zeno time under some approximation . + we have already mentioned that , quantum zeno effect is the slow down of quantum to classical transition due to frequent measurement . by definition this effect is something that slows down the process of decoherence . it is possible to control quantum to classical transition by frequent energy measurement . it has been shown @xcite that as a result of extremely frequent measurement , the system - reservoir coupling is eliminated and thus decoherence can be halted . zeno time is the time scale within which the quantum states are frozen , ie the decay is halted . + in previous works @xcite we have derived the weak value of dwell time for interacting ( via dissipation ) systems , which is given by [ 3.1 ] ll _ w^d= + ( ) where @xmath35 is the measurement time . now in explaining hartman effect " @xcite , dwell time can be interpreted as the lifetime of the decaying state in the barrier region @xcite . if the interval between consecutive measurements ( @xmath36 ) is significantly smaller than the zeno time ( @xmath37 ) , then the dynamics of the decay slows down or even asymptotically halted @xcite . given the condition [ 3.2 ] ^m^z we see that the lifetime ( @xmath38 ) of the decaying system , the measurement time ( @xmath36 ) and the zeno time obey the relation [ 3.3 ] ^z if we consider the interpretation of dwell time as the lifetime of the decaying state , then we can set @xmath39 . now under the assumption [ 3.4 ] ^m using ( [ 3.1 ] ) and ( [ 3.3 ] ) we can get the zeno time @xcite as [ 3.5 ] ^z= referring to the equation ( [ 3.5 ] ) , we find that the approximation ( [ 3.4 ] ) is nothing but [ 3.6 ] ^m ^z which is similar to the approximation ( [ 3.2 ] ) . this condition can be fulfilled , if the initial and final energy states ( with eigenvalues @xmath9 and @xmath40 respectively ) of the decaying system are closely spaced . now comparing equation ( [ 2.21 ] ) and ( [ 3.5 ] ) we get [ 3.7 ] = the preservation of quantum coherence leads us to conclude that zeno time represents a certain lower limit of decoherence time , beyond which the system loses it s quantumness " . the reason behind this statement lies in the definition of zeno time itself . if the measurement is frequent enough that @xmath41 , we observe the asymptotic halting of state decay , hence preserves the coherence . but if the decoherence time is shorter than the zeno time , the state will decay even within that estimated zeno interval . as a result we will not be able to preserve the quantum coherence even by frequent measurement . from equation ( [ 3.7 ] ) we find that imposition of this lower limit to the decoherence time leads us to a certain transitional temperature [ 3.8 ] t_tran= above this temperature the coherence of the system ca nt be preserved even by zeno dynamics . we have calculated this transitional temperature for our model system of trapped be atom . since the periodicity of the optical lattice is generally in the micrometer range and @xmath42 is about @xmath43 joules , we get that the transitional temperature is almost 200 microkelvin for be atom . hence above this temperature , decoherence will dominate the whole scenario . this is certainly an achievable temperature in case of raman cooling . in case of raman cooling for optical ion trap @xcite , the cooling temperature of na atom is found to be around 1 microkelvin temperature , which is 0.42 times of the photon recoil temperature @xmath44 . since the recoil temperature is inversely proportional to the mass of the atom ( @xmath45 ) , for a lighter atom like be , even lower cooling temperature can be achieved . it can not be concluded that below the transitional temperature there will not be any loss of coherence . but at least we can predict from our calculation that decoherence should not dominate the scenario . as the decoherence time becomes larger than zeno time , quantumness " of the system can be preserved within the zeno interval . + let us now calculate the decoherence time for a cooling temperature of 5 microkelvin . consider the transition @xmath46 . in this case the zeno time will be @xmath47 @xcite . subsequently , the decoherence time is found to be [ 3.9 ] _ dec^0^+0 ^ -= substituting the corresponding values of the parameters we get that the decoherence time is about 7 nanoseconds , whereas the zeno time is found to be 0.17 nanoseconds . similarly the timescales for other transitions can also be calculated . robust quantum memories are essential to realize the potential advances in quantum computation . optical ion trap can be realized as a quantum storage device . but it is also essential to protect the information , which can be lost due to environmental * _ interaction _ * in the form of decoherence . so it is very important to control the decohering effect in order to build an effective ion trap quantum computer . in this work , we have dealt with the question that whether and under what condition environment induced decoherence can be minimized . as we have discussed that in our understanding , the intrinsic relation between decoherence and quantum zeno effect can be exploited in this aspect . frequent nonselective measurement forces the system to evolve in the reduced zeno subspaces , which can be considered as some quasi - isolated " system . if the zeno effect ( characterized by it s corresponding timescale ) is strong enough , so that the reduced subspaces remains quasi - isolated even under the influence of environmental interaction , effect of decoherence can be controlled . based on this theoretical understanding , we have calculated a certain transitional temperature , by comparing the decoherence and zeno timescales . it is clear from the above analysis that below this transitional temperature we can increase the decoherence time by controlling the parameters ( @xmath48 ) . hence we can minimize the decohering effect asymptotically , though it can never be eliminated completely . 99 c.monroe et.al . ; phys.rev.lett . * 75 * , 4714(1995 ) . j.i.cirac and p. zoller ; phys.rev.lett . * 74 * , 4091(1995 ) . w.h . zurek ; progress in mathematcal physics ; * 48 * ; 1 ( 2007 ) . charles h. bennett ; physics today * 48 * , 10 , 24 ( 1995 ) l. viola and s. lloyd ; phys . a * 58 * , 2733 ( 1998 ) . franson , b.c . jacobs and t.b . pittman ; phys . a * 70 * , 062302 ( 2004 ) . b.mishra and e.c.g.sudarshan ; j.math.phys . * 18 * , 756(1977 ) . c.b.chiu , e.c.g.sudarshan and b.mishra ; phys.rev.d * 16 * , 520(1977 ) . s.tasaki et.al . ; international journal of quantum chemistry * 98 * , 160(2004 ) . p. facchi and saverio pascazio ; j. phys . . jpn . * 72 * , 30 ( 2003 ) . s.bhattacharya and s.roy ; phys.rev.a * 85*,062119(2012 ) . p. caldirola and m. montaldi ; nuovo cimento soc . b * 53 * , 291 ( 1979 ) . mohsen razavy ; _ quantum theory of tunneling_(2003),world scientific publishing , p 255 . p. facchi et.at ; phys . a * 71 * , 022302 ( 2005 ) . s.bhattacharya and s.roy ; arxiv:1209.0279v2 . t. e. hartman ; j. appl . * 33 * , 3427(1962 ) . h. g. winful ; new j. phys . * 8 * , 101 ( 2006 ) . h. g. winful ; phys . rep . * 436 * , 1 ( 2006 ) . h. g. winful ; phys . . lett . * 91 * , 260401 ( 2003 ) . p. facchi and s. pascazio ; j. phys . a * 41 * , 493001 ( 2008 ) . h. j. lee et.al ; phys.rev.lett . * 76 * , 2658 ( 1996 ) .
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decoherence time has been calculated for an optical ion trap of be atoms in a bistable potential model .
comparison has been made between decoherence time and zeno time for double well potential as a special case .
zeno time is considered as a lower limit of decoherence time for sustainable quantum coherence .
equality of the respective timescales provides a certain transitional temperature , below which decoherence can be asymptotically minimized . *
pacs numbers : * 03.65.xp , 03.65.yz + * keywords : * dissipative quantum system , decoherence , zeno time .
| 5,113 | 164 |
microarray technology is an important tool to monitor gene - expression in bio - medical studies @xcite . a common experimental design is to compare two sets of samples with different phenotypes , e.g. diseased and normal tissue , with the goal of discovering differentially expressed genes @xcite . statistical testing procedures , such as such as the t - test and significance analysis of microarrays @xcite , have been extensively studied and widely used . subsequently , multiple testing corrections are usually applied @xcite . a comprehensive review of such approaches are presented in @xcite . differential expression analysis based on univariate statistical tests has several well - known limitations . first , due to the low sample size , high dimensionality and the noisy nature of microarray data , individual genes may not meet the threshold for statistical significance after a correction for multiple hypotheses testing @xcite . second , the lists of differentially expressed genes discovered from different studies on the same phenotype have little overlap @xcite . these limitations motivated the creation of gene set enrichment analysis gsea @xcite , which discovers collections of genes , for example , known biological pathways @xcite , that show moderate but coordinated differentiation . for example , subramanian and tamayo et al . @xcite report that the p53 hypoxial pathway contains many genes that show moderate differentiation between two lung cancer sample groups with different phenotypes . although the genes in the pathway are not individually significant after multiple hypothesis correction @xcite , the pathway is . for those familiar with gsea and its output , figure [ fig : pnas2005toy ] shows the gsea results for the p53 hypoxial pathway . gsea also has the advantages of better interpretability and better consistency between the results obtained by different studies on the same phenotype @xcite . ackermann and strimmer presented a comprehensive review of different gsea variations in @xcite . unfortunately , gsea and related techniques may be ineffective if many individual genes in a phenotype - related gene set have weak discriminative power . a potential solution to this problem is to search for combinations of genes that are highly differentiating even when individual genes are not . for this approach , the targets are groups of genes that show much stronger discriminative power when combined together @xcite . for example , figure [ fig : gb2005toy1 ] illustrates one type of differentially expressed gene combination discovered in @xcite . the two genes have weak individual differentiation indicated by the overlapping class symbols on both the two axes . in contrast , these two genes are highly discriminative in a joint manner indicated by the different correlation structure in the two - dimensional plot , i.e. they are correlated along the blue and red dashed line respectively in the triangle and circle class . such a joint differentiation may indicate that the interaction of the two genes is associated with the phenotypes even though the two genes , individually , are not . figure [ fig : gb2005toy2 ] illustrates another type of phenotype - associated gene combination discovered in @xcite , usually named differential coexpression @xcite , in which the correlation of the two genes are high in one class but much lower in the other class . as discussed in @xcite , existing multivariate tests such as hotelling s @xmath0 @xcite , dempster s t1 @xcite are not suitable to detect such ` complementary ' gene combinations because they only screen for differences in the multivariate mean vectors , and thus will favor pairs that consist of genes with strong marginal effects by themselves but not the genes like the four in fig . [ fig : gb2005toy12 ] . for clarification , we use differential gene combination search ( denoted as dgcs ) to refer to the multivariate data analyses that are designed to detect the complementarity of different genes , rather than those designed to model the correlation structure of different genes ( such as hotelling s @xmath0 and dempster s t1 test ) . a variety of other dgcs measures for complementary gene combination search are proposed for gene pairs @xcite in addition to the two illustrated in figure [ fig : gb2005toy12 ] . several measures are designed for higher - order gene combination beyond pairs @xcite . these approaches can provide biological insights beyond univariate gene analysis as shown in @xcite . the limitations of gsea and the capabilities of dgcs motivate a gsea approach using gene combinations in which the score of a gene set is based on both the scores of individual genes in the set and the scores from the gene combinations in which these genes participate . unfortunately , gene combination techniques have not been used with the gsea approach in any significant way because of two key challenges . 1 . * finding a technique to reduce the vast number of gene combinations . * there are exponentially more gene combinations than individual genes , i.e. in addition to the @xmath1 univariate genes , there are @xmath2 gene - pairs , @xmath3 gene - triplets , etc . many variations of gsea are based on a ranked list of the @xmath1 individual genes as illustrated in figure [ fig : pnas2005toy ] . including combinations in the ranked list might work for size-2 combinations @xcite , but would not be feasible for handling gene combinations of larger sizes . furthermore , this explosion in the number of gene combinations negatively impacts false discovery rates . thus , by adding so many gene combinations , we run the risk that neither groups of genes nor individual genes will show statistically significant differentiation . * combining results from the heterogeneous measures used to score different size gene combinations . * furthermore , because a gene can be associated with the phenotype either as an univariate variable or together with other genes as a combination , the importance of a gene set should be based on both the univariate gene scores and the gene combination based scores of its set members . however , different measures have a different nature , scale and significance , and thus are not directly comparable ( to be detailed in section [ sec : score2pv ] ) . indeed , differences exist even between gene combinations of the same measure but of different sizes . therefore , the challenge lies in how to design a framework to combine different measures ( a univariate measure plus one or more dgcs measures ) together within the gsea framework . to the best of our knowledge , no existing work has sufficiently addressed these two challenges , although recent work presented in @xcite have made initial efforts at adding gsea capabilities to gene combination search . more specifically , two approaches are proposed in @xcite to help the study of a specific type of size-2 differential combinations as illustrated in [ fig : gb2005toy2 ] . the experiments in these two studies provide some evidence about the benefits of the integration . however , a more general framework is needed that can also handle other types of size-2 differential combinations as illustrated in figure [ fig : gb2005toy1 ] , higher order differential combinations ( e.g. sdc@xcite and the n - statistic @xcite ) , and multiple types of differential combinations . * contributions : * in this paper , we propose a general framework to address the above challenges for the effective integration of dgcs and gsea . specific contributions are as follows : 1 . * a gene - combination - to - gene score summarization procedure ( procedure _ a _ ) that is designed to handle the exponentially increasing number of gene combinations . * first , for a given gene combination measure and a certain @xmath4 , the score of a size-@xmath4 combination is partitioned into @xmath4 equal parts which are assigned to each of the @xmath4 genes in the combination . because each gene can participate in up to @xmath5 size-@xmath4 combinations , each gene will be assigned with a score from each of these combinations . secondly , an aggregation statistic , e.g. , maximum absolute value is used to summarize the different scores for a gene . with such a procedure , scores for all the size-@xmath4 gene combinations are summarized to @xmath1 scores for @xmath1 genes . this procedure can effectively retain the @xmath6 length of the ranked list while handling gene combinations of size-@xmath4 ( @xmath7 ) . * a score - to - pvalue transformation and summarization procedure ( procedure _ b _ ) that is designed to integrate the scores contributed ( in procedure @xmath8 ) from different gene combination measures and from gene combinations of different sizes . * the transformation is based on p - values obtained from scores derived from phenotype permutations . such a transformation enables the comparison of scores from different measures ( either univariate or gene combination measures ) and scores from the gene combinations of different sizes . subsequently , among all the p - values of a gene , the best is used as an integrated score of statistical significance . * integration of the above two procedures with gsea * more specifically , after procedures _ a _ and _ b _ , each gene has a single integrated score . unlike traditional univariate scores , these @xmath1 integrated scores are based on both the univariate statistic and the gene combination measures . for the type of gsea variations that depend on phenotype permutation test , @xmath9 lists of @xmath1 integrative scores are computed , one for the real class labels and the other for the @xmath10 permutations . for the type of gsea variations that are based on gene - set permutation test , only the list of integrated scores for the real class labels are needed . an independent matlab implementation of the proposed framework is available for download , which allows most existing gsea frameworks @xcite to directly utilize the proposed framework to handle gene combinations with almost - zero modification . * experimental results that illustrate the effectiveness of the proposed framework . * we integrated three gene combination measures and the gsea approach presented in @xcite and produced experimental results from four gene expression datasets . these results demonstrate that the integrative framework can discover gene sets that would have been missed without the consideration of gene combinations . this includes statistically significant gene sets with moderate differential gene combinations whose individual genes have very weak discriminative power . thus , a gene combination assisted gsea approach can improve traditional gsea approaches by discovering additional disease - associated gene sets . indeed , the integrative approach also improve traditional dgcs since most gene combinations are not statistically significant by themselves . furthermore , we also show that the biologically relevant gene sets discovered by the integrative framework share several common biological processes and improve the consistency of the results among the three lung cancer data sets . * overview : * the rest of the paper is organized as follow . in section [ sec : measures ] , we describe three gene combination measures used in the following discussion and experiments . in section [ sec : methods ] , we present the technical details of the two procedures of the general integrative framework . experimental design and results are presented in section [ sec : exp ] , followed by conclusions and discussions in section [ sec : discssion ] . in this section , we describe three dgcs measures for use in the following discussion and experiments . let @xmath11 and @xmath12 be two phenotypic classes of samples of size @xmath13 and @xmath14 respectively . for each sample in @xmath8 and @xmath15 , we have the expression value of @xmath1 genes @xmath16 . first , we have the following two measures ( denoted as @xmath17 and @xmath18 ) defined for a pair of genes as presented in @xcite : @xmath19 @xmath20 where @xmath21 and @xmath22 are two genes , and @xmath23 represents the correlation of @xmath21 and @xmath22 over the samples in set @xmath24 . as discussed in @xcite , @xmath17 and @xmath18 can detect the joint differential expression of two genes as illustrated in figure [ fig : gb2005toy1 ] and figure [ fig : gb2005toy2 ] respectively . @xmath17 and @xmath18 are used as two representative measures for gene pairs . other options for gene combination measures for gene pairs have been investigated in @xcite . we use the subspace differential coexpression measure ( denoted as @xmath25 ) proposed in @xcite as the representative for measures for size-@xmath4 gene combinations , where @xmath4 can be any integer ( @xmath7 ) . @xmath26 where @xmath27 is a set of genes such that @xmath28 and @xmath29 . @xmath30 and @xmath31 respectively represent the fraction of samples in @xmath8 and @xmath15 over which the genes in @xmath27 are coexpressed . @xmath25 is a generalization of @xmath18 for detecting the differential coexpression of @xmath4 genes ( @xmath7 ) , i.e. the @xmath4 genes are highly coexpressed over many samples in one class but over far fewer samples in the other . other options for size - k combinations include the _ n - statistic _ @xcite , _ supmaxpair _ @xcite , etc . signal - to - noise ratio ( denoted as @xmath32 ) is used as the representative of traditional univariate statistics as in @xcite . @xmath33 where @xmath34 and @xmath35 are the mean expression of @xmath21 in class @xmath8 and @xmath15 respectively , and @xmath36 and @xmath37 are the standard deviation of the expression of @xmath21 in class @xmath8 and @xmath15 respectively . many other univariate statistics can be found in @xcite . in this paper , these four measures are used as representatives of each category for the illustration of the proposed integrative framework . however , the framework is general enough to handle other measures from each of these categories . in section [ sec : intro ] , we motivated the integration of dgcs with gsea , discussed two challenges associated with this integration , and briefly described two main procedures in the proposed framework . in this section , we present the technical details of the two procedures and their integration with gsea . there are two steps in procedure @xmath8 . in step @xmath38 , for each dgcs measure and each size-@xmath4 gene combination , its score is divided into @xmath4 equal parts and assigned to each of the @xmath4 genes in the combination . in step @xmath39 , the scores assigned to a gene from all the size - k combinations in which the gene participates are summarized into a single score by an aggregation functions such as @xmath40 . note that , for most univariate statistic and dgcs measures which can be either positive or negative ( e.g. the four measures described in section [ sec : measures ] ) , the maximum is taken over the absolute values of the scores , and the sign of the score with the highest absolute value is recorded for later use . other simple statistics such as mean or median , or sophisticated ones such as weighted summation @xcite can also be used . since the focus of this paper is the overall integrative framework , we use @xmath40 for simplicity . we provide a conceptual example of procedure @xmath8 for a gene @xmath41 with a certain dgcs measure @xmath17 . this example considers gene combinations up to size @xmath42 for illustration purpose . the gene is associated with scores assigned from gene combinations of size 2 , 3 and 4 ( denoted as @xmath43 , @xmath44 and @xmath45 respectively ) in which @xmath41 participates . in step @xmath39 , the scores from @xmath43 , @xmath44 and @xmath45 are summarized by three maximum values , respectively . please refer to the appendix section for the illustration of this example . procedure @xmath8 serves as a general approach to summarize the @xmath46 scores of all the size-@xmath4 combinations into @xmath1 scores for the @xmath1 genes . if we want to integrate gsea with one dgcs measure and a specific size-@xmath4 , procedure @xmath8 by itself can enable most existing variations of gsea to search , with almost - zero modification , for statistically significant gene sets with moderate but coordinated gene combinations of size-@xmath4 . such a gsea approach can collectively consider the gene combinations affiliated with a gene set , and may provide better statistical power and better interpretability for dgcs , as will be shown in the experiments . the hypothesis tested when one dgcs measure , say @xmath47 , is integrated with gsea ( by procedure @xmath8 ) is that , whether a gene set includes significantly many genes with highly positive ( or highly negative ) combination - based scores measured by @xmath47 . an extended hypothesis can be whether a gene set includes significantly many genes with highly positive ( or highly negative ) scores , either univariate or combination - based scores measured by different dgcs measures . the biological motivation of this extended hypothesis is that , a gene can be associated with the phenotype either as an univariate variable or together with other genes as a combination . to test this extended hypothesis , we design a second procedure ( _ b _ ) that can integrate the scores of a gene from different measures . before describing the steps in this procedure . we first discuss in detail the challenges of integrating heterogeneous scores from different dgcs measures and combinations of different sizes . 1 . * the different nature of different measures * : different measures are designed to capture different aspects of the discriminative power of a gene or a gene combination between the two phenotypic classes . signal - to - noise ratio ( @xmath32 ) , a univariate gene - level statistic , measures the difference between the means of the expression of a gene in the two classes . in contrast , @xmath18 , a differential coexpression measure for a pair of genes describes the difference of the correlations of a gene - pair in the two classes . thus , for a gene , the score of itself measured by @xmath32 and the score assigned and summarized from the @xmath48 size-2 gene combinations measured by @xmath18 are not directly comparable . similarly , the scores of different dgcs measures can also have a different nature , e.g. @xmath17 and @xmath18 as illustrated in figure [ fig : gb2005toy12 ] . * the different scales of different measures * : different measures also have different ranges of values . for example , the range of @xmath32 , @xmath17 , @xmath18 , and @xmath25 are @xmath49 $ ] , @xmath50 $ ] , @xmath51 $ ] and @xmath52 $ ] respectively . thus , they are not directly comparable . * differences in significance between different measures * : even after we normalize the scores of different measures to a single range , say @xmath52 $ ] , they are still not comparable because the scores of different measures have different statistical significance . for example , a normalized @xmath32 score of @xmath53 may be less significant than a normalized @xmath17 score of 0.5 , if there are many genes with normalized @xmath32 score greater than @xmath53 in the permutation test @xcite , but very few genes with normalized @xmath17 score greater than @xmath54 in the permutation test . note that , such differences in statistical significance also exists between gene combinations of different sizes , even for the same measure . take the subspace differential coexpression measure @xmath25 as an example . a score of @xmath54 for a size-2 combination may not be as significant as a score of @xmath54 for a size-3 combination as discussed in @xcite . to handle the above heterogeneity , we propose a score - to - pvalue transformation and summarization procedure that can enable the comparison and integration of the scores of different measures and combinations of different sizes . there are three major steps in procedure @xmath15 . in procedure @xmath15 ( score - to - pvalue transformation ) for gene @xmath21 and measure @xmath18.,scaledwidth=95.0% ] consider a concrete example . for a gene @xmath21 and a measure @xmath18 , procedure @xmath8 computes a single summarized score . in this step , the original phenotype class labels are permutated say @xmath56 times , and for each permutation , the same procedure @xmath8 is applied , and a corresponding score for @xmath21 and @xmath18 is computed . we denote the score of @xmath21 and @xmath18 summarized with the original label as @xmath57 , where @xmath58 is the gene index , and @xmath59 indicates the measure and @xmath60 means it is the score based on the original label . similarly , we denote the scores computed in each of the permutation as @xmath61 , where @xmath62 . these @xmath63 scores are organized in the table on the left in figure [ fig : tabmerge2measures ] . the @xmath56 scores computed in the @xmath56 permutations can be considered as the null - distribution for gene @xmath21 and measure @xmath18 , and a p - value can be estimated for @xmath57 . specifically , if @xmath57 is positive , the p - value is the ratio of the number of scores that are greater or equal to @xmath57 and the number of scores that are positive . similarly if @xmath57 is negative , the p - value is computed as the ratio of the number of scores which are less or equal to @xmath57 and the number of scores which are negative . note that , such a score - to - pvalue transformation is done for both @xmath57 and each of @xmath61 ( @xmath62 ) , if the gsea approach to be integrated is based on phenotype permutation test @xcite . otherwise , only @xmath57 needs to be transformed to p - value and will be used by the gsea approaches that are based on gene - set permutation @xcite . in this paper , we illustrate the proposed framework using the gsea approach presented by subramanian and tamayo et al . @xcite which is based on phenotype permutation test . essentially , step @xmath55 transforms the heterogeneous scores of a gene measured by different measures into their corresponding significance values , which are comparable to each other although their original values are not . suppose that there are @xmath64 different measures to be integrated , one of which is a univariate statistic , and the others are different dgcs measures for which we consider combinations of sizes up to @xmath65 . after step @xmath55 , each gene has a p - value for the univariate measure and up to @xmath65 p - values for each size of gene combination for each measure . in step @xmath59 , the best transformed p - value ] p - value associated with a gene is selected as the integrated significance . essentially , procedure @xmath15 integrates the scores of different dgcs measures for a gene and the univariate statistic of the gene into a single p - value . such a statistical significance - based integration of heterogeneous scores enables the comparison and thus the ranking of all the @xmath1 genes . however , this ranked list does not maintain the original directionality of the integrated scores of each gene . in particular , most univariate statistics and dgcs measures ( e.g. all the four measures described in section [ sec : measures ] ) can be either positive or negative . such directionality information is lost in step @xmath55 and @xmath59 because the p - value is non - negative . next , we describe a third step to maintain the directionality in the integration . in the simple case , the measures to be integrated capture the same type of differentiation between the two phenotype classes , e.g. @xmath18 and @xmath25 . suppose there are two genes @xmath21 and @xmath22 , whose integrated p - values are transformed respectively from two scores measured by @xmath18 and @xmath25 in step @xmath59 . the signs of these two scores are comparable to each other , because both @xmath18 and @xmath25 capture the change of coexpression of a combination of genes . thus , we simply use the signs of these two scores as the signs associated with the integrated p - values of @xmath21 and @xmath22 . similarly , we associate a sign to all the @xmath1 integrated p - values . and these @xmath1 p - values with associated signs can be used to rank the @xmath1 genes based on their significance as well as their direction of differentiation , i.e. p - values associated with positive signs are ranked with descending significance , and afterwards , p - values associated with negative signs are ranked with increasing significance . in the other case , if the measures to be integrated capture different types of differentiation between the two phenotype classes , the directionality can not be fully maintained . for example , suppose there are two genes @xmath21 and @xmath22 , whose integrated p - values are transformed respectively from two scores measured by @xmath32 and @xmath18 in step @xmath59 . the signs of these two scores are not comparable , because @xmath32 captures the change of mean expression , and @xmath18 captures the change of coexpression of a combination of genes . specifically , up - regulation of @xmath21 can be associated with either high or low coexpression of another gene - combination in which @xmath22 participates . thus , it is not reasonable to follow the same strategy to associate signs to the @xmath1 integrated p - values . if we know the correspondence of the signs of different genes in advance , e.g. the up - regulation of @xmath41 is associated with the low coexpression of @xmath67 and @xmath68 , then the signs can be maintained . however , because it is not realistic to assume such prior knowledge , we propose the following heuristic approach which has proved a workable solution for our initial experiments . specifically , since the focus of step @xmath15 is to integrate different dgcs measures in addition to the univariate statistic @xmath32 , we considered @xmath32 as the base measure . for the integrated p - values that are transformed from scores measured by @xmath32 in step @xmath59 ( say there are @xmath69 of them ) , we use the signs of these @xmath69 @xmath32 scores for the @xmath69 integrated p - values . for the signs of the other @xmath70 genes , we assign positive signs to all of them once and negative signs to all of them a second time . correspondingly , we have two ranked lists similar to the simple case described above . note that , if the directionality of differential measures can be preserved , the power of this approach will be enhanced . to deal with the situation where signs are not comparable , other approaches will be explored . from the above description of procedure @xmath8 and @xmath15 , we know that , if only one dgcs measure is used in the gsea framework , only procedure @xmath8 is needed . if one or multiple dgcs measures are integrated together with the univariate statistic @xmath32 in the gsea framework , procedure @xmath15 is needed in addition . in the first case , the integrative framework outputs a ranked list of @xmath1 scores with associated signs for the original class label , and @xmath56 lists corresponding to the @xmath56 permutation tests . in the second case , we have two sets of @xmath63 lists respectively for the two rounds of maintaining directionality in step @xmath66 in procedure @xmath15 . in either case , the @xmath63 ranked lists along with the appropriate parameter settings and specification of gene sets can be used to run gsea . the only modification to gsea is the elimination of the initial gsea step to generate the scores , simulated and actual , that measure the level of differentiation between genes across different phenotypes . the proposed integrative framework is implemented as a matlab function ( available at http://vk.cs.umn.edu/icg/ ) , independently from the gsea framework to be integrated in this paper @xcite . as summarized by ackermann and strimmer @xcite , hundreds of variations of gsea are being used by different research groups . this independently implemented integrative framework can be easily applied to other variations of gsea . in our experiments , in order to have a fair comparison , we transform the @xmath63 ranked lists into the exact sample distribution as the original lists corresponding to @xmath71gsea . specifically , we only use the ranking information in the @xmath63 integrated ranked lists and map to them to the values in the original lists based on @xmath71gsea . essentially , the values in the ranked list passed to the gsea framework are exactly the same among @xmath71gsea , @xmath72gsea , @xmath73gsea , @xmath74gsea and @xmath75gsea , while the only difference is that the @xmath1 genes have different ranks in the lists . such a mapping ensures that the additionally discovered gene sets are because of the integration of gene - combinations in addition to univariate statistic , rather than simply the different value distributions in the @xmath63 lists . in this section , we present the experimental design and results for the evaluation of the proposed integrative framework . we first provide a brief description of the data sets and parameters used in the experiments . second , we describe and discuss the comparative experiments to study whether the integration of dgcs and gsea ( denoted as dgcs@xmath76gsea ) improves both dgcs and gsea . the two major evaluation criteria are the statistical power to discover ( additional ) significant results , and the consistency of the results across different datasets for the study of the same phenotype classes . the four datasets used in the experiments are described as follows : 1 . three lung cancer datasets respectively denoted as boston @xcite , michigan @xcite and standford @xcite : all the three data sets consist of gene - expression profiles in tumor samples from respectively 62 , 86 and 24 patients with lung adenocarcinomas and provide clinical outcomes ( classified as `` good '' or `` poor '' outcome ) . the two phenotypic classes in these three datasets are denoted as @xmath8 and @xmath24 as in @xcite . 2 . a data set from the nci-60 collection of cancer cell lines for the study of p53 status @xcite ( denoted as @xmath77 data set ) : the mutational status of the p53 gene has been reported for 50 of the nci-60 cell lines , with 17 being classified as normal and 33 as carrying mutations in the gene . the two phenotypic classes in this dataset are denoted as @xmath78 and @xmath79 as in @xcite . all four datasets were downloaded from the gsea website@xcite , and were already preprocessed as described in the supplementary file of @xcite . for all four data sets , we use the gene sets from @xmath43 in msigdb@xmath80 as in @xcite , as well as the same parameters . we consider one univariate statistic ( @xmath32 ) , and three gene - combination measures ( @xmath17 , @xmath18 and @xmath25 ) in our experiments . these four measures are described in section [ sec : measures ] . @xmath17 and @xmath18 are defined only for size-2 combinations . for @xmath25 , we considered gene - combinations of size-@xmath59 and size-@xmath66 for the illustration of concept . .number of with fdr less than @xmath81 discovered from the four data sets by each combination measure [ cols="<,<,<,<,<",options="header " , ] in this section , we study whether the question ( q1 ) of whether integration of dgcs and gsea can improve traditional dgcs . for this comparison , we consider the integration of dgcs and gsea as a gsea - assisted dgcs approach . we first apply the traditional dgcs approaches on the four datasets to find statistically significant gene - combinations . we denote the three dgcs approaches respectively with the names of the three measures , i.e. @xmath17 , @xmath18 and @xmath25 . second , we apply the integrative framework , in which gsea is integrated respectively with the three dgcs measures , to find statistically significant gene sets with moderate but coordinated differential gene - combinations . we denote the three instances of the integrative approach respectively as @xmath82gsea , @xmath83gsea and @xmath84gsea . then , we compare the results of @xmath17 , @xmath18 and @xmath25 , respectively with the results of @xmath82gsea , @xmath83gsea and @xmath84gsea . table 1 lists the number of statistically significant gene combinations discovered respectively by the three measures on each of the four datasets , with an fdr threshold of @xmath81 . table 1 lists the number of statistically significant gene sets discovered by integrating gsea respectively with the three dgcs measures on each of the four datasets , also with the same fdr threshold of @xmath81 . three major observations can be made by comparing the two tables : table 1 shows that , in most cases , traditional dgcs discovers very few ( less than 3 ) statistically significant gene combinations ( although @xmath18 and @xmath25 have @xmath85 and @xmath86 gene - combinations on the boston data set , none of them have fdr lower than @xmath87 ) . in contrast , table 2 shows that the integration of gsea with the three combination measures discover multiple significant gene sets in most of the cases . this difference implies that the discovered statistically significant gene sets include many moderate but coordinated differential gene combinations , even though the combinations are not significant by themselves as shown in table 1 . this comparison demonstrates that traditional dgcs , similar to univariate gene analysis , has limited statistical power , and dgcs@xmath76gsea can increase that power . we further compare dgcs and dgcs@xmath76gsea by studying the consistency of their results on the first three data sets that are all from lung cancer studies , as done in @xcite . for dgcs , @xmath17 discovered @xmath59 genes on michigan but nothing from boston and stanford ; @xmath18 discovered @xmath85 combinations on boston but only 1 and 2 from michigan and stanford , respectively , and there are no common ones between the 645 , 1 , and 2 gene combinations ; @xmath25 discovered @xmath86 genes on boston but only @xmath55 gene on michigan and nothing from stanford , and the 10 and 1 combinations do not overlap . the inconsistent results make the follow - up biological interpretation very difficult . in contrast , when the three dgcs measures are integrated with gsea , several consistent themes can be observed : ( i ) apoptosis related pathways ( marked by @xmath8 in table 2 ) : @xmath82gsea discovered four gene sets on boston , three of which are known to be closely related to cancer and specifically to apoptosis , i.e. _ nfkbpathway _ , _ st - gaq - pathway _ and _ tnf - pathway_. this apoptosis theme is shared by the gene sets discovered by @xmath17-gsea from michigan and stanford , i.e. _ monocyte - ad - pathway _ , _ hivnefpathway _ , _ deathpathway _ and _ caspasepathway_. these apoptosis related pathways are enriched with the lung cancer samples with good outcome , which makes sense biologically and also corresponds to the proliferation theme supported by the gene sets enriched with the samples with poor outcome as reported in @xcite . several other examples of the result consistency , as indicated by other superscripts in table 2 , are in the technical report . this comparison demonstrates that traditional dgcs , like univariate gene analysis , has poor result consistency across the three lung cancer data sets , and dgcs@xmath76gsea can improve its consistency by integrating dgcs measures with gsea . the number of significant gene sets discovered by the three versions of gsea varies , i.e. @xmath82gsea and @xmath83gsea discovered a bit larger number of significant gene sets than @xmath84gsea . however , @xmath84gsea still discovered several gene sets that are not discovered by @xmath82gsea or @xmath83gsea , e.g. one gene set from the michigan data set and three from the stanford data set . this indicates that @xmath82gsea , @xmath83gsea and @xmath84gsea have complementary perspectives , i.e. different combination measures capture different aspects of the difference between the phenotype classes ( recall the two types of combinations in figure [ fig : gb2005toy12 ] ) . this also demonstrates the proposed framework is general enough to integrate any type of dgcs with gsea . in this section , we want to answer the question ( q2 ) of whether the integration of dgcs and gsea can improve traditional gsea . for this comparison , we consider the integration of dgcs and gsea as a dgcs - assisted gsea approach . we design three sets of comparisons . firstly , we compare the traditional univariate - statistic based gsea ( denoted as @xmath71gsea ) with the integrative framework where one gene - combinations measure is used instead of @xmath32 . specifically , we compare the gene sets discovered by @xmath71gsea with the gene sets discovered by @xmath82gsea , @xmath83gsea and @xmath84gsea . then , we compare @xmath71gsea with the integrative framework where one gene - combinations measure is used in addition to @xmath32 , i.e. @xmath72gsea , @xmath73gsea and @xmath74gsea . furthermore , we also study the integration of multiple gene - combinations measure in addition to @xmath32 , e.g. @xmath75gsea . figure [ fig : bigtable12312009half ] displays the statistically significant gene sets discovered with different ( combinations of ) measures respectively from the four datasets . an fdr threshold of @xmath81 is used as in @xcite for comparison purpose . the results presented in @xcite are exactly reproduced , i.e. the gene sets listed in the rows corresponding to @xmath71gsea . in each of these four figures , we consider the traditional univariate - statistic based gsea ( @xmath88gsea ) as the baseline , and compare it with the rows corresponding to @xmath89gsea , @xmath90gsea , @xmath91gsea , @xmath72gsea , @xmath73gsea , @xmath74gsea and @xmath92gsea . from these comparisons , the following observations can be made . gsea ) is considered as the baseline . for the other rows , we only list a gene set if it is only discovered by the integrative approach ( with bolded name ) , or it has a non - trivially decreased fdr when it is discovered by the integrative approach ( with bolded fdr ) . please refer to the appendix section for the complete tables . ] first , we compare the rows corresponding to @xmath89gsea , @xmath90gsea , @xmath91gsea with the rows corresponding to @xmath88gsea . we bolded the additional gene sets that are only discovered by @xmath89gsea , @xmath90gsea , @xmath91gsea . for example , with @xmath71gsea , no statistically significant gene sets have been enriched with class a in the boston data set . in contrast , @xmath82gsea discovered @xmath42 gene sets , three out of which ( discussed in @xmath93 ) are related to apoptosis which is consistent with the results on michigan and stanford . on the michigan dataset , @xmath83gsea discovered a gene set _ beta - alanine - metabolism _ that is not discovered by @xmath71gsea . this gene set is related to the responses of hypoxia , which is consistent with the results on boston and stanford . it is worth noting that , although most studies did not report statistically significant gene sets on the stanford dataset due to the very small sample size , @xmath89gsea , @xmath90gsea , @xmath91gsea respectively discovered 4 , 4 and 3 significant gene sets . these additional gene sets were discovered because the three dgcs measures capture different types of the differentiation between the two phenotype classes , compared to the traditional univariate differential expression - based gsea . second , we compare the rows corresponding to @xmath72gsea , @xmath73gsea , @xmath74gsea with the rows corresponding to @xmath88gsea . we bolded the additional gene sets that are only discovered by the integrative approach . for example , on the boston data set , @xmath32 based gsea discovered 8 gene sets . in addition , @xmath72gsea discovered the _ gene set , and @xmath73gsea discovered the _ p53-signaling _ gene set . both ubiquitin - proteasome pathway and p53-signaling pathway are well - known cancer - related pathways that are also specifically related to lung cancer @xcite . ( additional examples are in the technical report . ) the gene sets that are discovered by dgcs - assisted gsea but not by @xmath94-gsea illustrate the benefits of using dgcs to assist gsea . next , we also observed that integrating multiple dgcs measures can further discover statistically significant gene sets . for illustration purpose , we compare the rows corresponding to @xmath72gsea , @xmath73gsea , @xmath74gsea with the rows corresponding to @xmath75gsea . @xmath75gsea discovers the _ g2pathway _ gene set and the _ gsk3pathway _ gene set , respectively from the boston and the michigan dataset . neither of these two pathways are discovered by @xmath88gsea , @xmath72gsea , @xmath73gsea and @xmath74gsea . the curated gene set _ g2pathway _ contains the genes related to the g2/m transition , which is shown to be regulated by p53 @xcite , a well - known cancer - related gene . the curated gene set _ gsk3pathway _ is the signaling pathway of gsk-3-@xmath95 , which has been shown to be related to different types of cancer@xcite . these two cancer - related pathways are discovered by @xmath75gsea but not by @xmath88gsea , @xmath72gsea , @xmath73gsea and @xmath74gsea . this indicates that different members of these two pathways are differential between the two phenotype groups in different manners , i.e. the differentiation of some genes is captured by @xmath32 , some by @xmath17 , some by @xmath18 and some by @xmath25 . these two pathways can be discovered to be statistically significant only when these measures are used together in the integrative framework . this demonstrates the benefits of the proposed framework for integrating multiple dgcs measures with a univariate measure . it is worth noting that , the gene sets discovered by the integrative framework with multiple measures are not necessarily a superset of those discovered by integrating each individual measure with gsea since , when different dgcs measures are integrated with gsea , the null - hypotheses tested in the gsea framework are correspondingly different . the highlight of the integrative framework is that , additional gene sets can be discovered when different dgcs measures are used to assist the traditional univariate statistic - based gsea . in practice , these different versions of gsea should be used collectively . : even when a gene set is discovered both before and after a dgcs measure is integrated into the framework , we can observe several interesting cases where the fdr of a gene set becomes much lower after the integration . we bolded the fdrs that significantly decreased when they are discovered by the integrative approach . for example , @xmath75-gsea , in which @xmath32 , @xmath17 , @xmath18 and @xmath25 are integrated together , discovers _ p53hypoxialpathway _ with an much lower fdr of @xmath96 , two - order lower than @xmath32-gsea . this example indicates that several members of _ p53hypoxialpathway _ have weak individual differentiation measured by @xmath32 , but have more significant differentiation when they are measured by @xmath25 . this and other similar examples demonstrates the benefits of the proposed framework for integrating multiple dgcs measures . as presented in @xcite , @xmath71gsea discovered @xmath97 and @xmath98 gene sets respectively from the boston and michigan data sets , and 5 of the 8 in boston and 6 of the 11 in michigan are common . the three unmatched gene sets that are discovered in boston but not in michigan are _ glut - down _ , _ leu - down _ and _ cellcyclecheckpoint_. interestingly , the latter two are discovered from both the boston and the michigan data sets by @xmath72gsea . such observations suggest that dgcs - assisted gsea also provides new insights to the consistency between different data sets . because different combinations of measures are used in the integrative framework , additional issues of multiple hypothesis testing arise , even though multiple hypothesis testing has been addressed for each measure via the phenotype permutation test procedure in the gsea framework proposed in @xcite . to investigate this , we designed experiments with 4 of the @xmath99@xmath100 possibilities of integrations , i.e. @xmath72gsea , @xmath73gsea , @xmath74gsea and @xmath75gsea . even using a collective ( meta - level ) multiple hypothesis correction , many discovered gene sets would still be significant . for examples , @xmath75gsea discovers _ p53hypoxialpathway _ from the boston data set with a low fdr of @xmath96 , and @xmath75gsea discovers _ deathpathway _ from the michigan data set with a lower fdr of @xmath101 . we also did additional permutation tests , in which we generate random gene sets with the same sizes as the sets in msigdb @xmath43 , and do the same set of experiments as shown in figure [ fig : bigtable12312009half ] . the fdr values of the random gene sets computed in the integrative framework are mostly insignificant ( higher than @xmath81 ) . in this paper we motivated the integration of differential gene - combination search and gene set enrichment analysis for bi - directional benefits on both them . we proposed a general integrative framework that can handle gene - combinations of different sizes ( @xmath7 ) and different gene - combination measures in addition to an univariate statistic used in traditional gsea . the experimental results demonstrated that , on one hand , gsea - assisted dgcs has better statistical power and result consistency than traditional dgcs . on the other hand , dgcs - assisted gsea can discover additional statistically significant gene sets that are ignored by traditional gsea and further improve the result consistency of the traditional gsea . the proposed framework can be extended in several ways . different variations of gsea will be considered . along these lines , we note that the proposed integrative framework is general enough to integrate most existing variations of gsea approaches summarized in @xcite with minimal amount of modification . also , it should be possible to integrate dgcs and gene - subnetwork discovery . both gsea and gene - subnetwork discovery @xcite can discover collections of genes , either known gene sets @xcite or subnetworks in a molecular network ( e.g. protein interaction network ) , that show moderate but coordinated differentiation . in this paper , we integrate dgcs and gsea as an illustration of the general framework for integrating scores from different gene - combination measures and gene - combinations of different sizes , in addition to the traditional univariate statistic , but the same framework also applies to the integration of dgcs and gene subnetwork discovery . another direction is the use of this framework for the analysis of ( gwas ) snp data , by following the methodology proposed in recently work on pathway / network based analysis of gwas datasets @xcite . finally , it may be possible to use constraints on gene - combinations to improve our framework . in procedure @xmath8 , for each gene - combination measure and an integer @xmath4 , the score of a gene is assigned from all the @xmath5 possible gene - combinations . a further extension of procedure @xmath8 is to only consider the gene combinations , in which the @xmath4 genes appear in a common gene set , e.g. a pathway . such gene - set - based constraints may better control false positive gene combinations and improve the statistical power of the whole integrative framework . due to the space limit , table 2 and figure [ fig : bigtable12312009half ] are both summarized from the four complete tables that are available at http://vk.cs.umn.edu / icg/. specifically , table 2 is a high - level summary of the number of gene sets discovered and the biological processes associated with each of the gene sets . in figure [ fig : bigtable12312009half ] , we listed the complete results for @xmath71gsea ( the baseline ) , while for the other rows , we only list a gene set if it is only discovered by the integrative approach ( with bolded name ) , or it has a non - trivially decreased fdr when it is discovered by the integrative approach ( with bolded fdr ) . g. fang , g. pandey , m. gupta , m. steinbach , and v. kumar . mining low - support discriminative patterns from dense and high - dimensional data . technical report 09 - 011 , department of computer science , university of minnesota , 2009 . k. wang , m. narayanan , h. zhong , m. tompa , e. e. schadt , and j. zhu . meta - analysis of inter - species liver co - expression networks elucidates traits associated with common human diseases . , 5(12):e1000616 , 2009 .
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gene set enrichment analysis ( gsea ) and its variations aim to discover collections of genes that show moderate but coordinated differences in expression .
however , such techniques may be ineffective if many individual genes in a phenotype - related gene set have weak discriminative power .
a potential solution is to search for combinations of genes that are highly differentiating even when individual genes are not .
although such techniques have been developed , these approaches have not been used with gsea to any significant degree because of the large number of potential gene combinations and the heterogeneity of measures that assess the differentiation provided by gene groups of different sizes . to integrate the search for differentiating gene combinations and gsea
, we propose a general framework with two key components : ( a ) a procedure that reduces the number of scores to be handled by gsea to the number of genes by summarizing the scores of the gene combinations involving a particular gene in a single score , and ( b ) a procedure to integrate the heterogeneous scores from combinations of different sizes and from different gene combination measures by mapping the scores to p - values .
experiments on four gene expression data sets demonstrate that the integration of gsea and gene combination search can enhance the power of traditional gsea by discovering gene sets that include genes with weak individual differentiation but strong joint discriminative power .
also , gene sets discovered by the integrative framework share several common biological processes and improve the consistency of the results among three lung cancer data sets .
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the scale size of galaxies is one of the fundamental parameters to elucidate the history of galaxy formation and evolution . the change of size and stellar - mass relations over cosmic time would pose strong constraints on models of galaxy evolution . the observational relations between galaxy size and stellar mass have been studied in the local universe , based on the sloan digital sky survey ( shen et al . 2003 ; bernardi et al . 2011 ) . using rest - frame optical bands , which presumably trace the distribution of stellar mass in galaxies , many studies have investigated galaxy sizes at higher redshift as a function of stellar mass for massive galaxies ( @xmath16 m@xmath7 ) . for example , the relations for galaxies at @xmath17 were studied for late - type galaxies ( barden et al . 2005 ) and for early - type galaxies ( mcintosh et al . 2005 ; trujillo , ferreras , & de la rosa 2011 ) . damjanov et al . ( 2009 ) and mancini et al . ( 2010 ) gave size - mass relations of massive galaxies ( @xmath18 m@xmath7 ) at @xmath19 . williams et al . ( 2010 ) studied the relation with large samples of galaxies at @xmath20 . for higher redshifts to @xmath10 , size - mass relations have been obtained for galaxies with @xmath21 m@xmath7 ( e.g. , franx et al . 2008 ; nagy et al . 2011 ; cassata et al . 2011 ) . many studies have corroborated that massive galaxies at high redshifts were much smaller than local galaxies with comparable mass ( e.g. , daddi et al . 2005 ; trujillo et al . 2006 , 2007 ; toft et al . 2007 ; zirm et al . 2007 ; cimatti et al . 2008 ; buitrago et al . 2008 ; van dokkum et al . 2008 , 2009 , 2010 ; akiyama et al . 2008 ; damjanov et al . 2009 ; carrasco et al . 2010 ; cassata et al . 2010 ; szomoru et al . 2010 ; van der wel 2011 ; cassata et al . 2011 ) . at a fixed stellar mass , spheroidal galaxies were significantly more compact at high redshift and evolved with rapid increase of the effective radius by a factor @xmath224 or even larger from @xmath23 ( e.g. , buitrago et al . 2008 ; carrasco et al . 2010 ) and by a factor @xmath222 from @xmath11 ( e.g. , van der wel et al . 2008 ; trujillo et al . 2011 ) . the finding of compact massive galaxies with a high stellar velocity dispersion also supports their existence ( van dokkum 2009 ; van de sande 2011 ) . it contrasts with the absence of such compact massive galaxies in the local universe , though several candidates have been found at @xmath24 ( stockton et al . 2010 ) and in the local universe ( trujillo et al . 2009 ; valentinuzzi et al . the findings demonstrate that massive galaxies have increased their size dramatically since @xmath25 in a different manner from the evolution of less massive galaxies . however , other studies have reached contradictory conclusions . there is significant disagreement between the results of different studies . barden et al . ( 2005 ) found weak or no evolution in the relation between stellar mass and effective disc size for galaxies with @xmath26 m@xmath7 since @xmath11 . for early - type galaxies at @xmath11 , mcintosh et al . ( 2005 ) showed that luminosity - size and stellar mass size relations evolve in a manner that is consistent with a completely passive evolution of the red early - type galaxy population . it is also shown that not all high - redshift early - type galaxies were compact and underwent dramatic size evolution ( e.g. , toft et al . 2007 ; zirm et al . 2007 ; saracco et al . 2009 , 2011 ; mancini et al . 2010 ; stott et al , 2011 ) . from the study of surface brightness in rest - frame @xmath27 and @xmath28 bands at @xmath29 , ichikawa et al . ( 2010 ) gave another evidence for no conspicuous evolution in galaxy sizes . as many previous studies show , the size evolution of galaxies are still controversial . any systematic errors in the observation or analyses could bias results towards such a significant evolution ( e.g. , mancini et al . 2010 ; hopkins et a. 2009b ; bouwens et al . 2004 ) . the origin of the discrepancy could be ascribed to redshift effects that more distant galaxies look more compact due to the difficulty of measuring envelopes at low surface density . the light from the outer portion of high - redshift galaxies is apt to be hidden in noise for low @xmath30 observations . as the consequence , effective radii and total luminosity ( or stellar mass ) would be underestimated . on the other hand , very deep observations ( e.g. , szomoru et al . 2010 ; cassata et al . 2010 ; law et al . 2012 ) or stacking methods to enhance the faint envelope of galaxies ( e.g. , zirm et al . 2007 ; van dokkum et al . 2008 , 2010 ; van der wel et al . 2008 ) have claimed that it is not the case . how significantly have the sizes of less - massive normal galaxies evolved from the early universe to the current epoch ? in this context , we look into the evolution of stellar - mass and size scaling relations on the basis of half- and 90 percent - light encircles , focusing on less massive galaxies ( @xmath31 m@xmath7 ) at @xmath1 , using a deep @xmath4-selected galaxy catalogue . we infer that the outer radius is more influenced by merging effect than star formation or central activity . in 2 , we describe the catalogue we used , which is among the deepest in the @xmath4 band to date . the depth is crucial for studying galaxies of low - surface brightness or galaxies which are dimmed due to the cosmological expansion at high redshift to measure the radius at faint outskirt of galaxies . the data analysis and the result for the size and stellar - mass relation are detailed in 3 and 4 . the results are discussed in 5 . ichikawa et al . ( 2010 ) studied the evolution of surface brightness of galaxies at @xmath29 in rest - frame @xmath27 and @xmath28 bands with the same data as the present study . we will discuss the consistency of the present result with their study . throughout this paper , we assume @xmath32 , @xmath33 , and @xmath34 km s@xmath35 mpc@xmath35 . we use the ab magnitude system ( oke & gunn 1983 ; fukugita et al . we use the @xmath4-band selected catalogue of the moircs deep survey ( mods ) in the goods north region ( kajisawa et al . 2009 , hereafter k09 ; kajisawa et al . 2011 , k11 ) , which are based on our imaging observations in @xmath36 bands with moircs ( suzuki et al . 2008 ) and archived data . four moircs pointings cover 70 percent of the goods north region ( 103 arcmin@xmath37 , hereafter referred as ` wide ' field ) . one of the four pointings , which includes hdf - n ( williams et al . 1996 ) , is the ultra - deep field of mods ( 28 arcmin@xmath37 , ` deep ' field ) . as the accuracy of background subtraction was highly demanded for the study of faint end of galaxies , the background was scrutinized and carefully subtracted ( see k11 for the details ) . the surface brightness limit was extensively examined in k11 . the typical @xmath38 surface brightness fluctuations in one arcsec diameter were found to be @xmath2227 mag arcsec@xmath39 and @xmath2227.5 mag arcsec@xmath39 in @xmath4 band for the wide and deep fields respectively . these are @xmath40 mag deeper than those of previous studies in @xmath4 band . the depth is crucial for the study of low surface brightness features at the outskirts of galaxies because of the strong dependence of cosmological dimming of surface brightness on redshift . for the total magnitude @xmath41 in @xmath4 band , we use mag_auto obtained by sextractor ( bertin & arnouts 1996 ) . the @xmath42-percent completeness of the catalogue is @xmath43 in the wide field and @xmath2226 mag in the deep field . we exclude fainter galaxies from the present analysis . the fwhms of the final stacked images are 0.46 arcsec for the deep image and 0.530.60 arcsec for the wide image . the numbers of galaxies are 3555 and 6063 , respectively . to obtain the stellar mass of mods samples , k09 performed sed fitting of the multiband photometry ( @xmath44 , 3.6 @xmath45 m , 4.5 @xmath45 m , and 5.8 @xmath45 m ) with population synthesis models . we adopt the results with galaxev templates ( bruzual & charlot 2003 ) and the salpeter initial mass function ( see k09 for more details ) . the stellar masses ( @xmath0 ) are obtained from the best - fit stellar mass - to - luminosity ratio in @xmath4 band and scaled with the @xmath4-band flux . a detailed comparison between different mass estimators is given in k09 . in the present analysis , the near - infrared data ( 3.6@xmath45 m ) of spitzer / irac are available for most of the sample galaxies ( 96 percent ) , so that sed fitting is reasonably reliable for the photometry at rest-@xmath27 ( @xmath46 m ) out to redshift @xmath223 . we assume that the size of the stellar system in galaxies is represented by the size measured on @xmath4-band images , which are the rest - frame optical to near - infrared wavelengths at @xmath29 . the difference of the sizes between optical and infrared light is discussed in 4 . in the present catalogue , 209 galaxies are identified as x - ray sources . among them , 61 sources are massive galaxies emitting hard x - rays with @xmath47 m@xmath7 at @xmath48 ( yamada et al . . one would expect galaxies with agn emission to have smaller half - light radius . considering possible effects on the size and mass estimates , we discard x - ray sources in what follows . as the definition of galaxy size , most previous studies have used the scale length ( @xmath49 ) , obtained by fitting the srsic profile ( srsic 1968 ) to galaxy images . the accuracy of the scale length strongly depends on image quality and depth of observations . for bright galaxies , reliable galaxy sizes can be measured from ground - based data ( trujillo et al . 2006 ; franx et al . 2008 ; williams et al . 2010 ) . on the other hand , it is crucial to take an accurate measure of the profile for small objects near the resolution limit . in fact , konishi et al . ( 2011 ) applied two dimensional fitting for the present sample to obtain srsic indexes with a single component for a morphological study of mods galaxies . however , they found that reliable fitting was successful only for comparatively massive galaxies ( @xmath26 m@xmath7 ) at @xmath11 and very bright galaxies ( @xmath50 ) at higher redshift . the resolution of fwhm @xmath51 arcsec like the present sample would not allow reliable fitting with srsic profile to fainter galaxies . srsic parameters are sometimes degenerated between @xmath52 and @xmath49 , which depends on the surface - brightness limit of the images ( e.g. , mancini et al . 2010 ; stott et al . 2011 ) , so that the reliable fitting would be difficult for faint galaxies . moreover high - redshift galaxies exhibit a wide range of disturbed morphologies ( e.g. , kajisawa & yamada 2001 ) . therefore , the fitting for less massive galaxies at high redshift could also be strongly influenced by their more amorphous features in the shape . in addition , we must take into account that we are observing a mix of galaxy morphologies . as such , we define the half - light radius ( @xmath53 ) as the radius of a circular aperture , which encircles half the @xmath4-band light emitted from galaxies . as we here focus not just on massive ( or bright ) galaxies , but also on less massive galaxies up to @xmath10 , we prefer @xmath54 to @xmath49 for the study of the structural parameters . if a galaxy profile extends to infinity , @xmath49 encloses half the flux in the srsic profile . however , as galaxies are supposed to have an edge at several times the scale length , @xmath49 does not always mean half - light radius . therefore , in general , @xmath53 is smaller than @xmath49 . in the same way , we define 90 percent - light radius @xmath55 . the 90-percent radius will give us more information at outer region of galaxies , where the size would be more influenced by merging effect . @xmath53 and @xmath56 are obtained by sextractor with phot_fluxfrac=0.5 and 0.9 figure [ fig - kvsr50 ] shows @xmath53 versus @xmath41 for all objects in mods . mods catalogue lists many spectroscopically confirmed stars , which are also plotted in fig . [ fig - kvsr50 ] . as few spectroscopic data were available for stars fainter than @xmath57 in the present region , we examined the reliability of @xmath54 using artificial stars . the artefacts were convolved with a moffat point spread function ( @xmath58 and fwhm=0.5 arcsec in the deep field and 0.6 arcsec in the wide field ) , and then buried in images as noisy as those of the deep and wide fields . sextractor was applied to the artefacts in the same manner as for the mods catalogue . the results are plotted in fig . [ fig - kvsr50 ] . the location of stars gives a clear boundary of unresolved galaxies . we define galaxies smaller than @xmath59 in the wide field @xmath60 in the deep field as unresolved galaxies . we first study the reliability of the galaxy size . seeing and background noise strongly affect the observed @xmath54 and @xmath55 , especially for faint or small galaxies . to examine the effect on the size estimate , we generated mock galaxies with a 1/4-law or exponential profile extending to 3 times scale length . the images are then convolved with a moffat point spread function ( @xmath58 and fwhm=0.5 arcsec in the deep field and 0.6 arcsec in the wide field ) , and buried in the simulated noise image . the galaxies were randomly generated with various magnitudes and effective radii or scale lengths , and analyzed with sextractor in the same manner as for the mods galaxies . the results are shown in fig . [ fig - mockr90r50 ] . seeing and noise seriously change the observed radii specifically for galaxies with a 1/4-law profile . the observed @xmath53 is significantly smaller than the intrinsic values in general . the effect is stronger for fainter 1/4-law galaxies , whereas the effect is much smaller for exponential galaxies . the apparent smaller size than the intrinsic size would be due to the fact that the faint and large envelope of 1/4-law galaxies is hidden in background noise . we should be aware of this effect if the observation is not deep enough for early - type galaxies ( see also hopkins et al . 2009b ; mancini et al . the effect will be discussed in more detail in the following section . it is noted that williams et al . ( 2010 ) claimed no systematic effects on @xmath49 for @xmath61 , using the shallower mods images . the radii , @xmath53 and @xmath56 , for small galaxies are affected by smearing due to seeing effect and the effect depends on galaxy morphology and size . therefore , we obtained eq . 1 to correct the size effect by fitting the deviation of the size as shown in fig . [ fig - mockr90r50 ] ( dash line ) . however , it is hard to examine the shapes of the present small or high-@xmath28 galaxies , so that we do not correct the effect which depends on morphology . @xmath62 we adopt @xmath63 and 0.35 arcsec for @xmath53 and @xmath56 , respectively . the size errors for mock galaxies are depicted in fig . [ fig - comparionsize ] as a function of observed magnitude . we should take into account that much smaller values are obtained for the size of 1/4-law galaxies . figure [ fig - comparionmag ] is the difference between intrinsic and observed magnitudes . at the faint limit , observed magnitudes are fainter by @xmath64 mag for 1/4-law galaxies , while the effect is much smaller for exponential galaxies . these results suggest that shallower observations tend to more underestimate size and flux ( therefore , stellar mass ) . we discuss later the influence on the results due to the systematic errors . using photometric ( or spectroscopic , if available ) redshift data , @xmath53 and @xmath56 in arcsec are converted to the physical size in kpc ( @xmath2 and @xmath3 ) . the photo-@xmath28 accuracy of the present sample is reasonably good , @xmath65 ( @xmath66 ) ( k11 ) , so that we expect no serious influence on our result due to the photo-@xmath28 error . we will confirm this later using galaxies with spectroscopic redshifts . the results for @xmath2 and @xmath3 are shown in figs . [ fig - massvsr50 ] and [ fig - massvsr90 ] as a function of the stellar mass of galaxies for different redshift bins . the unresolved galaxies are displayed in another figure ( fig . [ fig - r50reject ] ) for reference . the results of effective radius ( @xmath49 ) for local galaxies by shen et al . ( 2003 ) are also shown in fig . [ fig - massvsr50 ] for comparison . size - mass relations for star - forming galaxies are consistent with that of late - type galaxies by shen et al . ( 2003 ) , although our definition of half - light radius is different from that of shen et al . it should be noted that it would be difficult to discuss the consistency for massive early - type galaxies because of the small number of such galaxies in our sample . since our image quality is not high enough for classifying the galaxies into morphological classes at high redshifts , we divided the samples into quiescent and star - forming galaxy groups using a two - color diagnostic plot of rest - frame @xmath67 and @xmath68 colors ( williams et al . 2009 ) . the adopted selection criteria for quiescent galaxies are as follows : @xmath69 where @xmath67 and @xmath68 in the rest frame were obtained with the sed model fit to galaxies . the offset , @xmath70 , is 0.69 , 0.59 , and 0.49 for @xmath71 , @xmath72 , and @xmath73 , respectively . for @xmath74 galaxies , we applied the offset for @xmath75 . therefore the selection would be less reliable ( see williams et al . 2009 ) , though the number of such galaxies is very small . additional criteria of @xmath76 and @xmath77 are required for quiescent galaxies at all redshifts to prevent contamination from unobscured and dusty star - forming galaxies , respectively . the small differences in the slope and offset of the regression lines in different redshift bins in figs . [ fig - massvsr50 ] and [ fig - massvsr90 ] would imply a universal relation between the stellar mass and size of galaxies , irrelevant to redshift . in this context , we plot the stellar - mass surface density ( smsd ) , which is defined as @xmath78 , in a single redshift bin as a function of stellar mass for all galaxies at @xmath79 ( figs . [ fig - sd50all ] and [ fig - sd90all ] ) . the figures are essentially equivalent to the size - mass distribution . however , we can easily identify very compact or low - surface brightness galaxies in the figures , if they exist . for example , galaxies as compact ( or diffuse ) as 3 times smaller ( larger ) in size than the average should be located at @xmath221.0 dex above ( below ) the average . such low surface brightness galaxies can be recognized at @xmath9 m@xmath7 in fig . [ fig - sd90all ] . in contrast , massive and compact galaxies are rare in the present sample . it is noted that most of massive galaxies with @xmath80 m@xmath7 should be resolved in all redshift bins of the deep field . we conjecture that very compact galaxies coalesce in the unresolved galaxies in fig . [ fig - r50reject ] . since the strong correlations of @xmath2 and @xmath3 with @xmath0 in figs . [ fig - massvsr50 ] and [ fig - massvsr90 ] are suggested , we obtain the least square fit between the size and mass of the galaxies with a linear regression , @xmath81 where @xmath82 is 50 for half light radius or 90 for 90% light radius and @xmath83 is the radius at @xmath84 m@xmath7 . @xmath85 is 10 for all and star - forming galaxies , while @xmath86 for quiescent galaxies because the quiescent galaxies with @xmath87 m@xmath7 were not observed at @xmath88 and their @xmath89 is statistically less robust due to the small numbers of the sample in narrow mass ranges at high redshits . we note that quiescent galaxies are located on average below the regression lines in all redshift bins , though the offset is smaller for @xmath3 than for @xmath2 . the fact suggests that quiescent galaxies are more compact than star - forming galaxies at a give mass . the best - fit slopes ( @xmath90 and @xmath91 ) and offsets ( @xmath92 and @xmath93 ) of the regression analysis with mean errors for all , quiescent galaxies ( qsg ) , and star - galaxies ( sfg ) are summarized in table 1 . the dispersion , @xmath94 , of the linear fit is listed in the sixth and last columns . as the exclusion of unresolved sources could biases the results towards larger radii , we obtained in table 2 the least - squares fit with the unresolved galaxies in fig.[fig - r50reject ] . we depict the evolution of the slope and offset as a function of redshift in fig . [ fig - a_evolution ] . while the slope for the quiescent galaxies are a little steeper than those for star - forming galaxies , the figure indicates that the slopes remain within @xmath95 and offsets do not significantly change from @xmath10 to @xmath12 , irrespective of the sample selection except the quiescent population at @xmath96 where the statistical error is very large . for smsd in figs . [ fig - sd50all ] and [ fig - sd90all ] , we define the regression as @xmath97 where @xmath98 is smsd at @xmath99 m@xmath7 . the results are shown in table 3 . ccrcccrccc & & & @xmath2 & & & & @xmath3 & + redshift & sample & @xmath100 & @xmath90 & @xmath101 & @xmath94 & @xmath100 & @xmath91 & @xmath102 & @xmath94 + & & & & ( kpc ) & & & & ( kpc ) & + @xmath103 & all & 5259 & @xmath104 & @xmath105 & 0.130 & 5236 & @xmath106 & @xmath107 & 0.148 + & qsg & 408 & @xmath108 & @xmath109 & 0.110 & 408 & @xmath110 & @xmath111 & 0.099 + & sfg & 4851 & @xmath112 & @xmath113 & 0.129 & 4828 & @xmath114 & @xmath115 & 0.151 + + @xmath116 & all & 543 & @xmath117 & @xmath118 & 0.133 & 542 & @xmath119 & @xmath120 & 0.168 + & qsg & 28 & @xmath121 & @xmath122 & 0.098 & 28 & @xmath123 & @xmath124 & 0.100 + & sfg & 515 & @xmath125 & @xmath126 & 0.133 & 514 & @xmath127 & @xmath128 & 0.170 + @xmath129 & all & 717 & @xmath130 & @xmath131 & 0.119 & 714 & @xmath132 & @xmath133 & 0.124 + & qsg & 70 & @xmath134 & @xmath135 & 0.126 & 70 & @xmath136 & @xmath137 & 0.107 + & sfg & 647 & @xmath138 & @xmath139 & 0.113 & 644 & @xmath140 & @xmath141 & 0.124 + @xmath142 & all & 1083 & @xmath143 & @xmath144 & 0.123 & 1079 & @xmath145 & @xmath146 & 0.151 + & qsg & 134 & @xmath147 & @xmath148 & 0.104 & 134 & @xmath149 & @xmath150 & 0.095 + & sfg & 949 & @xmath151 & @xmath152 & 0.121 & 945 & @xmath153 & @xmath154 & 0.156 + @xmath155 & all & 888 & @xmath156 & @xmath157 & 0.130 & 885 & @xmath158 & @xmath159 & 0.124 + & qsg & 83 & @xmath160 & @xmath161 & 0.105 & 83 & @xmath162 & @xmath163 & 0.085 + & sfg & 805 & @xmath140 & @xmath164 & 0.128 & 802 & @xmath165 & @xmath166 & 0.125 + @xmath167 & all & 519 & @xmath168 & @xmath169 & 0.127 & 514 & @xmath170 & @xmath171 & 0.161 + & qsg & 32 & @xmath172 & @xmath173 & 0.089 & 32 & @xmath174 & @xmath175 & 0.079 + & sfg & 487 & @xmath127 & @xmath176 & 0.127 & 482 & @xmath177 & @xmath178 & 0.165 + @xmath179 & all & 555 & @xmath180 & @xmath181 & 0.128 & 554 & @xmath182 & @xmath183 & 0.128 + & qsg & 28 & @xmath184 & @xmath185 & 0.121 & 28 & @xmath186 & @xmath187 & 0.094 + & sfg & 527 & @xmath188 & @xmath189 & 0.124 & 526 & @xmath190 & @xmath191 & 0.128 + @xmath192 & all & 700 & @xmath193 & @xmath194 & 0.121 & 696 & @xmath195 & @xmath196 & 0.146 + & qsg & 27 & @xmath197 & @xmath198 & 0.109 & 27 & @xmath199 & @xmath200 & 0.097 + & sfg & 673 & @xmath201 & @xmath202 & 0.119 & 669 & @xmath203 & @xmath204 & 0.147 + @xmath205 & all & 254 & @xmath206 & @xmath207 & 0.130 & 252 & @xmath208 & @xmath209 & 0.166 + & qsg & 6 & @xmath210 & @xmath211 & 0.040 & 6 & @xmath212 & @xmath213 & 0.035 + & sfg & 248 & @xmath214 & @xmath215 & 0.129 & 246 & @xmath216 & @xmath217 & 0.167 + ccrcccrccc & & & @xmath2 & & & & @xmath3 & + redshift & sample & @xmath100 & @xmath90 & @xmath101 & @xmath94 & @xmath100 & @xmath91 & @xmath102 & @xmath94 + & & & & ( kpc ) & & & & ( kpc ) & + @xmath103 & all & 6532 & @xmath218 & @xmath219 & 0.161 & 6502 & @xmath220 & @xmath221 & 0.161 + & qsg & 445 & @xmath222 & @xmath223 & 0.116 & 445 & @xmath224 & @xmath225 & 0.108 + & sfg & 6087 & @xmath226 & @xmath227 & 0.162 & 6057 & @xmath228 & @xmath229 & 0.164 + + @xmath116 & all & 646 & @xmath230 & @xmath231 & 0.156 & 645 & @xmath232 & @xmath233 & 0.181 + & qsg & 31 & @xmath234 & @xmath235 & 0.129 & 31 & @xmath236 & @xmath237 & 0.124 + & sfg & 615 & @xmath238 & @xmath239 & 0.156 & 614 & @xmath232 & @xmath240 & 0.183 + @xmath129 & all & 858 & @xmath241 & @xmath242 & 0.136 & 855 & @xmath243 & @xmath244 & 0.134 + & qsg & 73 & @xmath245 & @xmath246 & 0.127 & 73 & @xmath247 & @xmath248 & 0.111 + & sfg & 785 & @xmath249 & @xmath250 & 0.133 & 782 & @xmath251 & @xmath252 & 0.135 + @xmath142 & all & 1313 & @xmath253 & @xmath254 & 0.155 & 1308 & @xmath255 & @xmath256 & 0.161 + & qsg & 141 & @xmath257 & @xmath258 & 0.102 & 141 & @xmath259 & @xmath260 & 0.094 + & sfg & 1172 & @xmath261 & @xmath262 & 0.157 & 1167 & @xmath263 & @xmath264 & 0.166 + @xmath155 & all & 1082 & @xmath265 & @xmath266 & 0.164 & 1077 & @xmath253 & @xmath267 & 0.146 + & qsg & 88 & @xmath268 & @xmath269 & 0.103 & 88 & @xmath270 & @xmath271 & 0.084 + & sfg & 994 & @xmath272 & @xmath273 & 0.164 & 989 & @xmath274 & @xmath264 & 0.148 + @xmath167 & all & 675 & @xmath275 & @xmath276 & 0.159 & 668 & @xmath277 & @xmath278 & 0.176 + & qsg & 35 & @xmath279 & @xmath280 & 0.111 & 35 & @xmath281 & @xmath282 & 0.091 + & sfg & 640 & @xmath283 & @xmath284 & 0.160 & 633 & @xmath285 & @xmath286 & 0.179 + @xmath179 & all & 685 & @xmath287 & @xmath288 & 0.174 & 684 & @xmath289 & @xmath290 & 0.139 + & qsg & 36 & @xmath291 & @xmath292 & 0.128 & 36 & @xmath293 & @xmath294 & 0.120 + & sfg & 649 & @xmath295 & @xmath296 & 0.171 & 648 & @xmath297 & @xmath298 & 0.138 + @xmath192 & all & 923 & @xmath299 & @xmath300 & 0.152 & 918 & @xmath301 & @xmath302 & 0.156 + & qsg & 32 & @xmath303 & @xmath304 & 0.103 & 32 & @xmath305 & @xmath306 & 0.094 + & sfg & 891 & @xmath307 & @xmath308 & 0.151 & 886 & @xmath309 & @xmath310 & 0.157 + @xmath205 & all & 350 & @xmath311 & @xmath312 & 0.150 & 347 & @xmath313 & @xmath314 & 0.175 + & qsg & 9 & @xmath315 & @xmath316 & 0.097 & 9 & @xmath317 & @xmath318 & 0.109 + & sfg & 341 & @xmath319 & @xmath320 & 0.149 & 338 & @xmath321 & @xmath322 & 0.176 + ccrccrccccc & & & @xmath2 & & & & @xmath3 & + selection@xmath323 & sample & n & @xmath324 & @xmath325 & @xmath94 & n & @xmath326 & @xmath327 & @xmath94 + & & & & ( @xmath328 kpc@xmath39 ) & & & & ( @xmath328 kpc@xmath39 ) & + @xmath329 & all & 5259 & @xmath330 & @xmath331 & 0.26 & 5236 & @xmath332 & @xmath333 & 0.30 + & qsg & 408 & @xmath334 & @xmath335 & 0.22 & 408 & @xmath336 & @xmath337 & 0.20 + & sfg & 4851 & @xmath338 & @xmath339 & 0.26 & 4828 & @xmath340 & @xmath341 & 0.30 + with unresolved galaxies & all & 6532 & @xmath342 & @xmath343 & 0.32 & 6502 & @xmath344 & @xmath345 & 0.32 + & qsg & 445 & @xmath346 & @xmath347 & 0.23 & 445 & @xmath336 & @xmath348 & 0.22 + & sfg & 6087 & @xmath349 & @xmath350 & 0.32 & 6057 & @xmath342 & @xmath351 & 0.33 + @xmath352 & & 3980 & @xmath353 & @xmath354 & 0.25 & 3972 & @xmath355 & @xmath333 & 0.28 + @xmath356 & & 2378 & @xmath357 & @xmath358 & 0.25 & 2378 & @xmath359 & @xmath333 & 0.23 + spectroscopic redshift & & 1610 & @xmath360 & @xmath361 & 0.24 & 1610 & @xmath362 & @xmath363 & 0.22 + the errors in stellar mass mainly originated from sed fitting and photometric errors of the observations . in the course of @xmath364 fitting to obtain the best sed model with various parameters ( e.g. , star - formation time scale , photometric redshift for galaxies with no spectroscopic redshift available , age , extinction , and metallicity ) , the probability distributions of stellar mass can be calculated , where the photometry and photometric - redshift errors are included . the error for the observed smsd is dominated by the error for the stellar mass . see more details for the error estimate in k09 and ichikawa et al . ( 2010 ) the size and magnitude errors in accordance with galaxy magnitude have been examined using mock galaxies ( figs . [ fig - comparionsize ] and [ fig - comparionmag ] ) . if all galaxies have a shape of 1/4-law , the error would be significant . recalling the size error due to the morphology strongly depends on magnitude , we examine the effect by confining the sample to bright galaxies , where the systematic error becomes smaller . we selected the samples with the magnitude limits , @xmath365 ( 23 ) , 25(24 ) , 26(25 ) for the deep ( wide ) fields , then obtained the regression lines again . the results are compared in table 3 . it should be noted that the section does not change the result . one would be concerned about the reliance on photometric redshift estimates for high redshift galaxies ( e.g. , mosleh et al . 2011 ) . in order to examine the error , we selected only the galaxies with spectroscopic redshift available and compared the result with the photo-@xmath28 sample . although the spectroscopic samples are limited to lower redshift , the result does not change our conclusion ( table 3 ) . due to possible color gradients in galaxies , it would be best to use images of the same rest - frame band with comparable @xmath30 for measuring the size of galaxies at all redshifts . nevertheless , we used the radius of galaxies measured on @xmath4-band image , because it is deepest among the images we used and because it is the rest - frame optical or longer band at @xmath29 . if the central region of galaxies are younger ( bluer ) than the outer region , the mass - weighted radius could be larger than the luminosity - weighted radius . if it is older ( redder ) , the result would be vice versa . star forming galaxies sometimes show strong morphological variation between observed wavelengths . bond , gawiser & koekemoer ( 2011 ) reported that this was not generically accompanied by a large difference in half - light radius . barden et al . ( 2005 ) measured the disk scale lengths of local galaxies in various bands . the average size is about 10 percent larger in @xmath27 band than in @xmath4 band . on the other hand , macarthur & courteau ( 2003 ) showed the contrary result that the distribution of disc scale lengths was a decreasing trend with increasing wavelengths ( see also cassata et al . 2010 for early - type galaxies at @xmath366 ) . to investigate the effect of the color gradient , we compare the 90 present - light radius of galaxies measured on the nearest rest - frame @xmath27 band for each redshift bin . the sizes of the present samples were obtained on acs @xmath367 , @xmath28 , and moircs @xmath368 , @xmath369 bands and compared with that of @xmath4 band in fig . [ fig - r90vsacs ] . we used acs images binned to 0.117 arcsec per pixel , keeping the original image resolution . [ fig - r90vsacs ] demonstrates that the sizes of acs in @xmath367 and @xmath28 bands are systematically @xmath370 percent smaller than that in @xmath4 band . the convolution of the acs @xmath367-band image with psf and seeing enhances low surface brightness details . if the acs images are convolved with a gaussian ( fwhm=2 pixel , 0.234 arcsec ) , the difference is decreased to @xmath2215 percent . as the convolution enhances the galaxy edge of low surface brightness , it tends to give a larger galaxy size ( see also fig . 7 of mancini et al . it should be noted that the @xmath368 and @xmath369 images gives larger image size for smaller galaxies . the size strongly depends on the depth and the seeing size . although we have found the universal relation between @xmath0 and smsd ( or size ) , which does not depend strongly on redshift , we have good reason to expect offsets from the relation for some galaxy populations ( e.g. , massive compact galaxies ) in certain mass and redshift ranges . the regression analysis could be more weighted on the numerous less massive populations . therefore , it is likely that there are small ( but not highly significant ) offsets between galaxy populations . the small offsets would account for the size evolution of such populations . using the @xmath0@xmath371 relation obtained for the galaxies at @xmath372 ( tables 1 and 2 ) as a reference , we examined the deviation of the median size of galaxies in the mass and redshift bins from the reference . the evolution of the massive ( @xmath47m@xmath7 ) quiescent galaxies , defined in a @xmath67 and @xmath68 diagram ( williams et al . 2010 ) , is also obtained . ( we use median values to avoid unreasonable contributions from outliers with large deviation . ) we show the results in fig . [ fig - r_evolution ] , where unresolved galaxies are included . in addition , the median values of @xmath15 , which represent a sort of _ compactness _ of galaxies , are depicted in the figure to see the evolution of the compactness as a function of redshift . we note that the result with resolved galaxies are in good agreement with that with unresolved galaxies . using the deepest @xmath4-band image , we obtained scaling relations between stellar - mass ( @xmath0 ) and size ( or equivalently smsd ) as a function of redshift for galaxies at @xmath1 in a wide mass range of @xmath373 m@xmath7 . we defined the radii encircling half- ( @xmath2 ) and 90 percent - light ( @xmath3 ) in a circular aperture . the depth was found to be crucial for proper measure of the size and mass for less massive galaxies dimmed due to cosmic expansion . we conclude that there is no strong evidence for the size evolution at a given mass over the redshift range , irrespective of galaxy populations ( star - forming , quiescent ) . the size - mass relation ( @xmath374 ) have a universal slope ( @xmath375 ) and small offsets ( @xmath8 percent ) . in other words , as galaxies grow in mass through star formation or merging process , their sizes evolve in such a way that galaxies in general move along the scaling relation . the universal relation demonstrates that the stellar mass in galaxies with the same stellar mass was built up on average in a similar manner over cosmic times . the trend is insensitive to the stellar populations . our result is comparable with the @xmath376 found by shen et al . ( 2003 ) for low - mass galaxies ( @xmath377 m@xmath7 ) . on the other hand , it is in contrast to the steeper relation for massive galaxies obtained by previous studies ( e.g. , shen 2003 ; bernardi et al . the scaling relations are much tighter ( @xmath3780.17 dex ) than those ( @xmath3790.5 dex ) of previous studies in the local universe ( e.g. , shen et al . 2003 ) or at high redshifts ( e.g. , franx et al . 2008 ) . the weak growth of the galaxy size , irrelevant to galaxy mass , is in disagreement with the scenario that more massive galaxies rapidly changed their sizes . if massive high-@xmath28 galaxies are several times smaller than local galaxies with comparable mass , they should be located well above the stellar mass vs. smsd relations . several candidates of such compact galaxies with @xmath380 possibly coalesce as unresolved galaxies in figs . [ fig - sd50all ] and [ fig - sd90all ] . however , such galaxies are found to be few . our finding is in contrast with the previous results for quiescent early - type massive galaxies ( e.g. , zirm et al . 2007 ; williams et al . 2010 ) . with regard to smsd , the average smsds in @xmath2 for our sample galaxies with @xmath26 m@xmath7 are found not to evolve with redshift ( log @xmath381 8.70 ) . the average smsd for all galaxies at @xmath382 is @xmath383 , which is very consistent with the result , @xmath384 , for disk galaxies with @xmath26 obtained by barden et al ( 2005 ) , though our sample could be a mixture of disk and elliptical galaxies . however , due to small volume of the present study , the results could be subject to statistical uncertainties . the weak size evolution presented in the present study is reconciled with the surface brightness ( sb ) evolution . ichikawa et al . ( 2010 ) presented a universal linear correlation between the stellar mass and sb in rest - frame @xmath27 and @xmath28 bands for galaxies at @xmath1 , using the same sample as the present study . the correlation has a nearly constant slope , independent of redshift and color of galaxies in the rest-@xmath28 frame . in contrast , sb shows a strong dependence on redshift for a given stellar mass . it evolves as @xmath385 . the increase in the luminosity with redshift is estimated by using the expected luminosity evolution from a single burst at @xmath386 for massive galaxies ( @xmath26 ) and constant star formation for blue samples ( @xmath387 ) for less massive galaxies . the redshift dependence of sb evolution is well explained by the pure luminosity evolution of galaxies out to @xmath10 without the need of the size evolution , which supports the results for young early - type galaxies ( e.g. , saracco et al . 2011 ) and for disk galaxies at @xmath388 ( barden et al . 2005 ) . nevertheless , there are some populations having a small offset from the universal scaling relation . we see more or less evolution in the size and compactness for such populations as a function of redshift in fig . [ fig - r_evolution ] . the figure suggests a moderate increase of 3050 percent for @xmath2 and @xmath3 for less massive galaxies ( @xmath9 m@xmath7 ) from @xmath10 to @xmath11 , while the sizes remains unchanged or slightly decrease towards @xmath12 . for massive galaxies ( @xmath13 m@xmath7 ) , the evolution is @xmath14 percent in @xmath3 from @xmath10 to @xmath12 , though that in @xmath2 is weaker . @xmath3 evolved as @xmath389 , it is noted that trujillo et al . ( 2006 ) , using the ground - based @xmath4-band data comparable with our depth , but in a smaller region ( 6.3 arcmin@xmath37 ) , concluded that there was no evidence for significant evolution for the size - mass relation and that there was small increase ( 29 percent ) since @xmath390 for the most massive bin @xmath391 m@xmath7 . in our sample , the average @xmath2 for the most massive bin of @xmath392@xmath393 m@xmath7 is larger by @xmath394 percent at @xmath12 than at @xmath390 , which is in good agreement with the result of trujillo et al . ( 2006 ) , whereas it is much weaker than those of other studies for galaxies of similar mass ( e.g. , franx et al . 2008 ; buitrago et al . 2008 ; van der wel et al . 2009 ; williams et al . 2010 ) . using new deep wfc3 data , cassata et al . ( 2011 ) and law et al . ( 2012 ) have given new evidences of compact galaxies at high - z . although the data in @xmath369 band is shallower by 1 mag arcsec@xmath39 than our @xmath4-band data , their conclusion would be robust . it would be possible that the compact galaxies are mingled in unresolved galaxies of the present sample or that the difference of the definition for half - right radius ( @xmath395 and @xmath53 ) gives the contradict result . the ratio , @xmath15 , in fig . [ fig - r_evolution ] would give a clue to understanding the evolution of the compactness of galaxies . while middle and low mass galaxies shows no evolution in the compactness , the galaxies in the most massive bin are more compact at higher redshift . recalling weaker evolution in @xmath2 than in @xmath3 , we infer that the compactness evolution is ascribed mainly to the expansion of the outer rim of massive galaxies . in other words , at a given mass , massive galaxies in the local universe are more influenced by mergers or star formation at the outer envelopes than those at high redshifts . it would be worth noting that the galaxies in the massive bin of the deep field are complete in sampling over the present redshift range and their @xmath2 and @xmath3 are well resolved in the present psf . however , the present observation samples a comparatively small volume and therefore , the results could be subject to statistical uncertainties for massive galaxies of low number density due to field variances . the size evolution is often used to advocate the merging processes in the hierarchical paradigm of galaxy formation and evolution . minor dry merger is a plausible mechanism for weak size evolution ( e.g. , guo & white 2008 ; bezanson et al . 2009 ; naab , johansson , & ostriker 2009 ; hopkins et al . 2010 ) . in that the size evolution is stronger in massive galaxies , our findings are qualitatively in agreement with previous studies based on simulations ( e.g. , boylan - kolchin , ma , & quataert 2006 ; hopkins et al . naab et al . ( 2009 ) showed that minor mergers or the accretion of relatively low - mass satellites may be the main driver for the late evolution of sizes of massive early - type galaxies . somerville et al . ( 2008 ) showed based on a cdm model of disc formation with @xmath26 m@xmath7 that the average size of discs at fixed stellar mass was about 50 percent larger than that at @xmath10 . the predicted evolution in the mean size at a fixed stellar mass since @xmath11 is about @xmath396 percent , which is comparable with our observations . the newly accreted small galaxies preferentially populate the outer region of massive galaxies . the small size growth of the present result is plausibly accounted for by minor mergers or the accretion of relatively low - mass satellites ( e.g. , naab et al . 2009 ; hopkins et al . naab et al . ( 2009 ) and fan et al . ( 2010 ) showed the fractional variation of the gravitational radius of the main galaxy after @xmath100 minor - merger events with @xmath374 as @xmath397 where @xmath398 and @xmath399 are final and initial galaxy sizes , @xmath400 is the fractional ratio of merging galaxy to the main galaxy . it takes @xmath221 ( @xmath223 ) minor - merger events with @xmath401 to increase the radius by @xmath402 ( @xmath403 ) percent for massive galaxies since @xmath10 , provided that the merger mass ratio is 1:10 . it would not be unreasonable that massive galaxies experienced such a small number of minor merging since @xmath4043 ( e.g. , bundy et al . 2009 ; lpez - sanjuan et al . 2011 ) . deep observations will be important in constraining the exact amount ( or lack thereof ) and distribution of merging galaxies , and how galaxies built up with redshift . in this context , our finding demonstrates that minor mergers in massive system built up an envelope of lower surface density materials . deeper imaging observations in a wider filed with high spatial resolution and consistent analyses from the local universe to high redshift will give constraints on the compactness and the amount of low - surface brightness material at the outer envelope as a function of redshift to improve our understanding of galaxy formation . this work has been supported in part by a grant - in - aid for scientific research ( 21244012 ) of the ministry of education , culture , sports , science and technology in japan . we thank ramsey lundock for careful reading of the manuscript . mods catalogue has been accomplished by moircs builders . we owe the present study to their dedicated efforts . 99 akiyama m. , minowa y. , kobayashi n. , ohta k. , ando m. , and iwata i. 2008 , apjs . , 175 , 1 barden a. , et al . 2005 , apj , 635 , 959 bernardi m. , roche n. , shankar f. , steth , r. k. , 2011 , mnras , 412 , l6 bertin e. , arnouts s. 1996 , a&as , 117 , 393 bezanson r. , van dokkum p. g. , tal t. , marchesini d. , kriek m. , franx m. , coppi p. , apj . , 697 , 1290 bond n. a. , gawiser e. , koekemoer a. m. , 2011 , apj , 729 , 48 bouwens r. j. , illingworth g. d. , blakeslee j. p. , broadhurst t. j. , franx m. , 2004 , apj , 611 , l1 boylan - 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we present scaling relations between stellar - mass ( @xmath0 ) and the size of galaxies at @xmath1 for half- ( @xmath2 ) and 90 percent - light ( @xmath3 ) radii , using a deep @xmath4-band selected catalogue taken with the subaru telescope and moircs in the goods - north region .
the logarithmic slope @xmath5 is independent of redshift in a wide mass range of @xmath6 m@xmath7 , irrespective of galaxy populations ( star - forming , quiescent ) . the offset change is @xmath8 percent .
provided that optical light in the rest frame traces the stellar mass of galaxies , the universal relation demonstrates that the stellar mass was built up in galaxies over their cosmic histories in a similar manner on average irrelevant to galaxy mass .
the small offset in each stellar mass bin from the universal relation shows weak size evolution at a given mass . there is a moderate increase of 3050 percent for @xmath2 and @xmath3 for less massive galaxies ( @xmath9 m@xmath7 ) from @xmath10 to @xmath11 ,
while the sizes remains unchanged or slightly decrease towards @xmath12 . for massive galaxies ( @xmath13 m@xmath7 ) , the evolution is @xmath14 % increase in @xmath3 from @xmath10 to @xmath12 , though that in @xmath2 is weaker .
the evolution of compactness factor , @xmath15 , which becomes smaller at lower redshift , is suggestive of minor merging effect in the outer envelope of massive galaxies .
[ firstpage ] galaxies : evolution galaxies : fundamental parameters galaxies : high - redshift infrared : galaxies .
| 18,464 | 469 |
the creation and motion of large numbers of crystal lattice dislocations is the most fundamental feature of crystal plasticity . during the last half century , the physical properties of individual dislocations and their interactions with localised obstacles have been studied extensively . on the other hand , the complex collective dynamics of strongly interacting many - dislocation systems is still far from being understood . fortunately , everyday plastic deformation processes very often proceed orders of magnitude slower than the typical relaxation times of the underlying dislocation system . these conditions often permit to study the problem in a quasistatic approximation @xcite . beyond the quasistatic limit , however , much less work has been devoted to studying the dynamics of collective dislocation motions which lead to the formation of metastable configurations , and to transitions between such configurations in driven dislocation systems . however , such collective motions are crucial for understanding rapid dislocation processes which not only occur in shock loading but , in the form of dislocation avalanches , are a generic feature of the dynamics of driven dislocation systems @xcite . the first studies of dynamic relaxation processes in dislocation systems were performed by miguel et al . with the protocol of applying a constant external shear stress to well relaxed dislocation configurations @xcite . the ensuing creep relaxation was numerically shown to follow andrade s law stemming from the underlying intermittent and correlated motion of dislocation structures . the connection between the mesoscopic and macroscopic features of the process was , however , not analysed in detail . another direction was taken by the present authors who conducted systematic studies of the relaxation dynamics of initially random configurations of straight dislocations . this is an important issue since the elastic energy density @xmath0 of a random dislocation system of density @xmath1 is known to diverge with the logarithm of system size @xmath2 , @xmath3 @xcite where @xmath4 is the modulus of the dislocation burgers vector . in a well - relaxed dislocation arrangement , on the other hand , the same quantity scales like @xmath5 , i.e. , the screening length corresponds to the mean dislocation spacing @xcite . as the mean square stress is proportional to the elastic energy density , this screening also removes a logarithmic divergence of the width of the internal stress probability distribution @xcite , and of the x - ray line width @xcite . numerical experience showed that , at least in single slip geometries , the relaxation processes that lead to screened dislocation arrangements exhibit slow , power law characteristics for quantities such as the elastic energy or the average dislocation velocity @xcite . a model was proposed which relates the power - law relaxation dynamics to the gradual extinction of initial dislocation density fluctuations @xcite . the present paper presents a comprehensive numerical investigation which allows to check in detail the model predictions and complements the earlier work by extending the investigation to multiple slip geometries and to dislocation systems of non - zero net burgers vector , and by studying the influence of an external driving stress on the relaxation process . the paper is organised as follows . in the problem is defined and technical details of the simulations are presented . unfolds a scaling model of the relaxation process from a chemical analogy and uses this model to predict the evolution of simulation measurables . then gives a detailed comparison between model predictions and numerical results . the results are discussed and conclusions are drawn in . an auxiliary calculation of the elastic energy of a random dislocation wall is presented in the appendix . consider a system of @xmath6 straight edge dislocations running parallel to the @xmath7 axis of a cartesian coordinate system . let all dislocations have a common burgers vector pointing along the @xmath8 axis ( a so - called single slip geometry ) , @xmath9 , where @xmath10 is the sign of the @xmath11th dislocation . assuming overdamped glide motion with a dislocation velocity @xmath12 that is proportional to the local resolved shear stress , and zero dislocation mobility in the climb direction , the equation of motion of dislocation @xmath11 piercing the @xmath13 plane at @xmath14 can be written as @xmath15 , \qquad \tau_{\mathrm{ind}}(\bi{r } ) = g b \frac{x ( x^{2}-y^{2})}{(x^{2}+y^{2})^{2}},\ ] ] where @xmath16 denotes the dislocation glide mobility , @xmath17 $ ] where @xmath18 is the shear modulus and @xmath19 is poisson s ratio of the embedding isotropic crystal , @xmath20 denotes the resolved shear stress field induced by a positive dislocation located at the origin @xcite , and @xmath21 is a constant externally applied resolved shear stress . it is useful to introduce natural coordinates at this point which will be denoted by an apostrophe ( @xmath22 ) in the following . measuring length in units of the average dislocation dislocation distance @xmath23 ( where @xmath1 denotes the total dislocation density of dislocations including both signs and , in multiple slip geometries , including all slip systems ) , stress @xmath24 in units of @xmath25 , and plastic strain @xmath26 in units of @xmath27 leads to the relations @xmath28 where @xmath29 is the elastic energy difference between two states of the system ( energy per unit dislocation length ) . in natural coordinates takes the form @xmath30 , \cr & \tau'_{\mathrm{ind}}(\bi{r } ' ) = \frac{x ' ( x'^{2}-y'^{2})}{(x'^{2}+y'^{2})^{2 } } = \frac{\cos(\varphi ) \cos(2\varphi)}{r ' } , } \ ] ] where @xmath31 denotes the angle between the @xmath8 axis and @xmath32 . to study dislocation relaxation , a large number of discrete dislocation dynamics simulations have been performed . equations of motion were solved with the 4.5th order runge kutta fehlberg method . periodic boundary conditions were applied to a square simulation area with edges parallel to the slip planes , following the method used in @xcite . to avoid overly small timesteps during the final stages of approach of narrow dipoles ( pairs of dislocations of opposite signs ) , a small number of extremely narrow dipoles were excluded from the solution of and forced to move as if they were isolated from the rest of the system . this is justified as the far - field stresses of the dislocations in a narrow dipole cancel , while their pair interaction diverges when the dislocation dislocation distance approaches zero . as a consequence , the dynamics of narrow dipoles is effectively uncoupled from the rest of the dislocation system . we do not allow for annihilation of narrow dipoles , which is a process that is governed by the atomistics of the dislocation cores . this implies that we consider dislocation spacings to be large in comparison with the dipole annihilation distance , which is believed to be of the order of one nanometer . as typical dislocation densities in highly dislocated crystals are of the order of @xmath33 m@xmath34 , i.e. the average dislocation spacings are of the order of a hundred nanometers , this is not a severe restriction . a first set of simulations was started from random configurations of equal numbers of positive and negative dislocations . the number of simulated dislocations @xmath6 ( which defines the system size @xmath35 since @xmath36 ) varied between @xmath37 and @xmath38 . it is well known that the flow stress of single - glide dislocation systems is around @xmath39 in natural units @xcite . to allow the dislocation systems to reach mechanical equilibrium at the end of the relaxation , we restricted the applied external stresses to levels below the flow stress , using stresses between @xmath40 and @xmath41 natural units . as seen in , individual simulations showed strong avalanche - like activity during relaxation , as previously observed in @xcite . to reveal scaling properties of the relaxation process , the evolution of global parameters such as stored energy , mean absolute dislocation velocity , mean square velocity , and mean strain rate was averaged over @xmath42 to @xmath43 simulations starting from different random initial configurations . this ensemble averaging resulted in smooth ensemble averaged graphs as seen in . in addition to the relaxation of ` neutral ' arrangements of dislocations moving on a single slip system aligned parallel to the edge of a square simulation area , we considered two variants of this basic setting ( these simulations were only performed at zero external stress ) : i ) to elucidate the influence of different implementations of the boundary conditions , we performed single slip simulations with the simulation box oriented at an angle @xmath31 to the slip planes . this does not affect the short - range dislocation dislocation interactions but modifies the stress field created by the periodic images . ii ) to study the influence of a net burgers vector on the relaxation process , we investigated the limiting case of fully polarised dislocation systems ( dislocations of one sign only ) of sizes @xmath44 . to investigate the differences between single and multiple slip geometries , we also performed simulations in which dislocations of multiple slip systems were present . in these simulations we considered sets of equally populated slip systems , each containing edge dislocations with a zero net burgers vector that were initially distributed at random . the methodology of the multiple slip simulations was identical to the single slip simulations described above , with the following differences : i ) for each dislocation in the system , the complete elastic stress tensor was computed , again assuming periodic boundary conditions in a square simulation area and using the method of @xcite . the forces acting on the dislocations were calculated from the stress tensor components using the peach koehler formula @xcite . ii ) in addition to narrow dipoles , which were treated in the same way as described above , pairs of attracting dislocations on intersecting slip planes needed to be treated separately . such dislocation pairs react with each other and form a reaction product ( ` dislocation lock ' ) with burgers vector equal to the net burgers vector of the constituent dislocations . in real crystals , both the mobility of dislocation locks and the stress required for separating them into the constituent dislocations depend on their atomic core structure . for simplicity , we assumed all dislocation locks to be immobile and to possess infinite separation stress . new dislocations joining an existing lock were assumed to annihilate with the constituent dislocation of the opposite burgers vector if such a dislocation was present . if a dislocation lock converted to an ordinary dislocation through such an annihilation event , it was assumed to become mobile again . to avoid overly small timesteps during the final stages of dislocation lock formation or reaction with a new dislocation ( again due to diverging elastic interactions between the constituent dislocations ) the moving constituent dislocations were pinned when their relative distance decreased below a small predefined reaction radius which mimics the core extension of the dislocation lock . for simplicity , the stress field of a lock was calculated as the superposition of the stress fields of the pinned constituent dislocations . although the positions of the constituent dislocations are scattered over a small region of the size of the reaction radius , their net stress field is a good approximation of the stress field of the lock at larger distances . finally , we note that this stress field is of long range character since the net burgers vector of a dislocation lock is not zero . in this section a simple scaling argument is used to establish the asymptotic kinetics of a bimolecular combination reaction . we then adapt the fundamental ideas behind this argument to the relaxation of dislocation systems as described in the previous section . based on the adapted model , predictions are made for the evolution of several physical quantities which can be directly obtained from the simulations described in . these predictions are compared in detail to the numerical results in . consider the direct combination reaction @xmath45 starting from a random , balanced configuration of the two reactants @xmath46 where @xmath47 denotes the spatial average of the concentration field @xmath48 . note that the initial concentrations of molecules @xmath49 and @xmath50 are equal only in an average sense because of the thermally induced random positions of the individual molecules . these concentration fluctuations need to be taken into account when modelling the reaction kinetics . for situations where long - distance transport of particles occurs by free brownian motion , a simple scaling argument which captures the essential physics was given by ovchinnikov @xcite ( which the reader is advised to consult for further details ) . broadly speaking , this involves the subsequent dominance of two consecutive mechanisms directly corresponding to the different length scales inherent in the system : i ) the reaction of those molecules that do not need long distance motion to find a reaction partner , followed by ii ) the long range brownian motion and subsequent reaction of excess reactants remaining as a result of initial concentration fluctuations . as discussed in @xcite , the first stage of reaction is controlled by a bimolecular reaction rate @xmath51 which depends on the short - range interactions between the reaction partners . this stage can be discussed in the classical approximation @xmath52 solving for initial conditions yields @xmath53 because of thermal concentration fluctuations , not all of the molecules @xmath49 and @xmath50 can be consumed during stage 1 . spatial fluctuations of the reactant concentrations lead to local excess of molecules of one type . as these excess molecules can not find a local partner , they need to migrate via long range brownian motion , leading to a second kinetic stage controlled by long - range diffusion . to characterise the concentration fluctuations in question we observe that , on scales larger than the range of the ` contact interactions ' which govern the first stage of the reaction kinetics , the positions of molecules are statistically independent . as a consequence , the initial numbers of molecules @xmath49 or @xmath50 in a sufficiently large volume @xmath54 are poisson distributed . this implies that the mean numbers of excess molecules fulfil the relations @xcite @xmath55 where @xmath56 denotes spatial averaging over a large statistically homogeneous system or , equivalently , ensemble averaging over a large number of statistically independent and equivalent realizations . to further discuss the reaction kinetics during stage 2 , we confine ourselves to the limiting case of an infinite bimolecular reaction rate @xmath57 . this choice does not affect the generality of the discussion , it only affects the moment of the crossover between stage 1 and stage 2 kinetics . we introduce the concentration difference @xmath58 as @xmath59 in a hypothetical initial state before the reaction has been ` switched on ' at @xmath60 , @xmath61 and @xmath62 are statistically independent and @xmath58 has the initial statistical properties @xcite @xmath63 to proceed , we note that for @xmath57 molecules @xmath49 and @xmath50 can not coexist for @xmath64 . therefore , @xmath58 gives a complete characterisation of the concentration map for @xmath65 : for @xmath66 , @xmath67 and @xmath68 and for @xmath69 , @xmath70 and @xmath71 . because there exists no other physical length scale in the system , the size of the regions characterised by @xmath72 and @xmath73 is determined by the diffusion length @xmath74 ( for simplicity , the same diffusion constant @xmath75 is assumed for both kinds of reactants ) . consider now the volume referring to the diffusion length at time @xmath76 , @xmath77 . one can suppose that for lengths larger than @xmath78 the fluctuations of @xmath58 are still not affected by diffusion ; therefore , one can write that @xmath79 in some regions , @xmath80 and in others , @xmath81 ; hence , on average @xmath82 meaning a slower kinetics than in stage 1 . ( for a more detailed derivation see @xcite ) . note that is only applicable within certain time limits @xmath83 $ ] . for instance , for @xmath57 , @xmath84 is equal to the time the diffusion length @xmath78 needs to exceed the average intermolecular distance . @xmath85 is determined by the time needed by @xmath78 to reach the system size @xmath86 , independent of the value of @xmath51 . now we consider the relaxation of initially random dislocation configurations with the same number of positive and negative dislocations ( @xmath87 ) following the equations of motion . although not immediately evident , this relaxation process has strong phenomenological similarities to the kinetics of the chemical reaction described in . to elucidate the analogy , the relaxation process will be envisaged as a gradual screening of the long range elastic stress fields of individual dislocations through the formation of dislocation dislocation correlations @xcite . we first envisage ` neutral ' arrangements where dislocations of both signs are present in equal amounts . in a screened dislocation arrangement , the excess of one sign over the other has been eliminated on scales above a few dislocation spacings . any dislocation arrangement where excess dislocations are completely eliminated can be envisaged as an assembly of dipoles where each dislocation has exactly one partner of opposite sign within a distance of the order of @xmath23 , and this picture will be used in the following argument . hence , we envisage the relaxation process as the gradual formation of a large number of dislocation dipoles consisting of dislocations of opposite signs , i.e. , as a bimolecular reaction process analogous to the chemical reaction . the principal difference between the two processes lies in the dynamics of individual particles : for the dislocation system , dislocation glide motion is driven by dislocation dislocation interactions which scale like @xmath88 , whereas in case of the chemical reaction we are dealing with diffusive brownian motion of the reactants . despite this difference , we may again construct a two - stage model for the dislocation relaxation process : i ) in stage 1 adjacent opposite sign dislocations form dipoles ; ii ) in stage 2 initial fluctuations in the excess dislocation density gradually die out from shorter towards longer length scales as excess dislocations which did not find a dipole partner in stage 1 undergo long range glide motion . the process terminates once the length scale on which fluctuations have been eliminated reaches the system size . to understand the time evolution of the system , we again consider the evolution of the typical length scale @xmath78 below which density fluctuations have already died out . to this end we consider that i ) similar to the chemical case , areas of size @xmath89 typically contain mobile excess dislocations ( which have still not ` reacted ' into dipoles ) with only one or the other sign and that ii ) dislocation dipoles give rise only to short range stress fields with a @xmath90 decay . as the typical distance between opposite sign dislocations which try to find each other is proportional to @xmath91 , the typical driving stress towards dipole formation scales as @xmath92 . ( recall that variables with an apostrophe ( @xmath22 ) are measured in natural units . ) the dislocation velocity is proportional to @xmath93 , and therefore the characteristic time for eliminating excess dislocations on scale @xmath94 scales as @xmath95 . hence we find that @xmath96 incidentally , this result is very similar to the evolution law of @xmath78 for brownian motion , @xmath97 . by supposing that the mobile dislocations inherit the initial concentration fluctuations we find that at time @xmath98 , regions of size @xmath99 contain about @xmath94 excess dislocations of one or the other sign , and @xmath100 dislocations in total ( see the chemical model in ) . thus , the fraction of non - paired dislocations is estimated to decrease in time as @xmath101 following the chemical model in , it is straightforward to predict the time interval @xmath102 $ ] during which the above argument is expected to hold . the start time @xmath103 is characterised by @xmath2 reaching the dislocation dislocation distance @xmath23 ( @xmath104 in natural units ) and the process is finished when @xmath2 reaches the system size @xmath86 . with this leads to @xmath105 in natural coordinates where @xmath6 denotes the total number of dislocations in the system . the fraction of ` non - paired ' dislocations is not a convenient quantity for comparing the scaling model with dislocation dynamics simulations , as the definition of ` dislocation pairs ' in a multipolar dislocation arrangement may be ambiguous . instead , we consider the evolution of the @xmath106th moment @xmath107 of the dislocation velocity and of the excess elastic energy , both of which can be determined from the simulations in a straightforward manner . to obtain scaling estimates for the relaxation of these quantities , we assume that all dislocation dipoles are at rest and only the excess dislocations move . furthermore , we assume that the motion of excess dislocations is not hindered by the dipoles already formed ( this can be rationalised with the short range of the dipole stress field ) . the velocity of excess dislocations scales as @xmath108 , leading to @xmath109 where and have been used . the dynamics of dislocations is assumed to be overdamped . hence , the work that is done by the internal stresses in driving the system is completely dissipated : the amount of dissipated energy exactly matches the reduction in elastic energy . the energy dissipated per unit time by a moving dislocation @xmath11 scales like @xmath110 ( the peach koehler force acting on the dislocation is proportional to the stress ) , and consequently the time evolution of @xmath111 can be expressed as @xmath112 since the motion is overdamped , the dislocation velocity is proportional to the acting stress . in natural units and for @xmath113 $ ] , the ensemble averaged elastic energy thus evolves like @xmath114 where was used to estimate the time evolution of the second velocity moment . the scaling argument in the previous section is based on the formation of dipoles consisting of edge dislocations of opposite signs . at first glance , such an argument seems to be completely inapplicable to systems where only dislocations of the same sign are present . dipole formation is clearly impossible in such systems . instead , a most conspicuous feature in the relaxation of single - sign edge dislocation systems is the formation of walls containing many dislocations of the same sign that are aligned in the direction perpendicular to the slip plane @xcite . accordingly , theoretical arguments have focused on parameters characterising the ` condensation ' of dislocations into walls @xcite . however , even though dislocation wall formation is a most conspicuous feature , wall formation alone can _ not _ produce a screened dislocation arrangement . the authors of @xcite evaluate the driving force for wall formation by assuming periodic spacing of dislocations along a wall and using classical results found e.g. in @xcite . if we follow this line of reasoning and note that the energy of a dislocation in a wall decreases with decreasing dislocation spacing , the minimum - energy structure for the system at hand would be a single system - spanning wall . however , for an initially random dislocation system the @xmath115 positions are independent random variables and it is not easy to see how a periodic arrangement could form in the absence of dislocation climb . we calculate the energy of a random wall in the appendix and show that forming such a wall does not produce any energy reduction with respect to the initial random 2d arrangement . how then can an arrangement of dislocations of the same sign be screened ? the answer was provided by wilkens @xcite who demonstrated that a screened dislocation arrangement can be constructed by eliminating dislocation density fluctuations above a certain scale . to this end , he proposed a construction where the crystal cross section is tiled into a grid of cells of size @xmath116 , and the same number of dislocations is randomly distributed within each cell . this construction , which eliminates all density fluctuations on scales above the cell size , leads to an arrangement where the screening radius coincides with the cell size , @xmath117 . taking the wilkens construction as a well - screened reference state offers a surprising outlook on the relaxation of initially random systems of same - sign dislocations . with respect to this reference state , the initial random arrangement contains density fluctuations on all scales which may be either positive ( @xmath118 , positive excess ) or negative ( @xmath119 , negative excess ) . to achieve screening , dislocations must migrate from regions of positive to regions of negative excess , and this process is governed by the long - range stress fields associated with the presence of ( positive or negative ) excess dislocations . in other words , the kinetics of the process follows from exactly the same scaling argument as used in the previous section : we are dealing with the stress - driven elimination of excess dislocation densities , with the only difference that the excess is now not of positive over negative dislocations , but of the local dislocation density over the average one . if the above argument is correct , the relaxation kinetics of same - sign dislocation systems should be characterised by a slow power - law stage which has the same characteristics as the relaxation of neutral dislocation systems as discussed in the previous sections . we demonstrate in the next section that this is indeed the case . the numerically determined evolution of the elastic energy is displayed in . the zero value of the energy was chosen to correspond to the final relaxed state of the system . as seen on the figure , the evolution of the elastic energy @xmath120 per dislocation can be fitted satisfactorily with the prediction in , @xmath121 . power - law relaxation occurs from times @xmath122 onwards , in good agreement with the model prediction @xmath123 in . the final equilibrium value of the elastic energy is not a priori known but was fitted to the data such as to achieve a maximum extension of the linear scaling regime . unfortunately this precludes determination of the second critical time @xmath124 from these results . the presence or absence of an external stress below the flow stress of the relaxed dislocation system seems to have negligible influence on the evolution of the elastic energy . note that it is possible to collapse the curves for different system sizes @xmath125 by normalising the graphs with @xmath126 , as done in . this observation is consistent with the fact that the elastic energy of the initial random dislocation system is of the order of @xmath127 while the energy of the final relaxed state is of the order of @xmath128 @xcite . for estimating the value of @xmath129 we used that the range of dislocation pair correlations in mechanical equilibrium is of the order of the mean dislocation dislocation distance @xmath23 @xcite . therefore , the elastic energy difference per dislocation between the initial and final states is of the order of @xmath130 despite this relation connecting the initial and final states of the system , the numerical finding that the @xmath131 curves for different system sizes can be collapsed on their entire course by normalising them with @xmath126 is not trivial , as the agreement extends also to the relaxation kinetics and characteristic crossover time @xmath132 . and external stress values @xmath133 . ] the numerically calculated evolution of the mean square velocity @xmath134 is displayed in for zero applied stress and different system sizes . due to the connection between the mean square velocity and the elastic energy of the system , it is not surprising that similar statements apply here as for the evolution of the elastic energy . as seen in the figure , the model prediction @xmath135 in fits the data well from @xmath122 . due to the fact that @xmath136 is proportional to the time derivative of the elastic energy , its graphs are much noisier than those obtained for the energy , preventing again the detection of the supposed upper critical time @xmath124 . as for the energy , size effects can be scaled out with a normalisation factor @xmath137 which is a direct consequence of and . a final analogy to the evolution of the elastic energy is that external stresses have only negligible influence on the evolution of the mean square velocity . for this reason , simulations with non - zero external stresses were omitted from . for different system sizes @xmath6 at zero external stress . ] in the evolution of the mean absolute velocity @xmath138 can be seen for different system sizes . again , a power law time dependence can be observed from @xmath122 although an exponent @xmath139 gives a better fit than the theoretically predicted @xmath140 expected according to equation . one may argue that slowly moving dislocation dipoles play a bigger role in this case , as their small velocities contribute more strongly to @xmath141 than to @xmath142 . therefore , the gradually increasing number of slowly moving dislocations might be responsible for the reduced relaxation exponent . what makes this figure very interesting is the possibility to estimate values of @xmath124 . it was found that @xmath143 gives a good approximation , in line with the model prediction @xmath144 in . it was also observed that normalisation with @xmath145 collapses the graphs referring to different system sizes @xmath125 in the region of small @xmath146 . this is consistent with the relations for the elastic energy and the mean square velocity . finally , as in case of the energy and the mean square velocity , the evolution of the mean absolute velocity is not changed by the presence of external stresses below the macroscopic flow stress . for different system sizes @xmath6 at zero external stress ] another numerically measurable quantity is the plastic strain rate , defined as @xmath147 in the following the evolution of @xmath148 is studied for applied shear stresses @xmath149 below the macroscopic flow stress for the present dislocation geometry . from the data of miguel and co - workers @xcite , this is estimated to be @xmath150 in natural units . as it was demonstrated in , external stresses in this range do not appreciably change the evolution of the elastic energy . we study the relaxation of the strain rate mainly in order to assess , by comparing with the work of miguel et al . , the relevance of different initial conditions on the creep behaviour of dislocation systems . shows the numerically determined evolution of the plastic strain rate @xmath148 for different system sizes and external stress values . as can be seen , the plastic strain rate scales roughly in proportion with the external stress . the relaxation does not follow any discernible power law but is roughly exponential . this is in marked contrast with the findings of miguel et al . @xcite who for well relaxed initial configurations demonstrate an andrade - type power - law decay , @xmath151 . the discrepancy points to the crucial importance of initial conditions for relaxation processes in dislocation systems a factor which is also borne out by the history dependence of creep relaxation processes that was demonstrated by miguel et al . @xcite . normalised with the external stress @xmath133 for different system sizes @xmath6 and external stress values @xmath133 . ] in this section we investigate the influence of different ways of implementing the periodic boundary conditions by tilting the angle between the edges of the simulation box and the trace of the slip planes . simulations with different tilt angles @xmath31 are physically equivalent except for the spatial arrangement of the periodic images of each dislocation ( see ) . this arrangement affects the dislocation dislocation interactions on scales comparable to the simulation box size . also , the interaction energy of each dislocation with its periodic images affects the initial elastic energy of the system , which is smallest for @xmath152 and has a maximum for @xmath153 . ( left ) and @xmath153 ( right ) . the latter configuration has a higher elastic energy as the nearest neighbours of each dislocation are in an energetically unfavourable configuration.,scaledwidth=80.0% ] the influence of simulation box orientation on the relaxation process is illustrated in . the absolute values of the squared velocity ( or equivalently the energy dissipation rate ) are higher for @xmath153 than for @xmath154 . however , both curves differ only by a constant factor ( the ratio of the initial excess energies ) , while the dynamics of the relaxation processes is otherwise identical . in systems with a single slip geometry with the simulation box oriented at different angles @xmath31 to the slip planes . ] for systems of dislocations of the same sign we have numerically evaluated the evolution of the mean square velocity @xmath155 ( or , equivalently , of the energy dissipation rate ) and of the mean absolute velocity @xmath138 . all calculations were performed at zero external stress since any applied stress would induce a sustained drift motion of the dislocation arrangement . results are shown in and together with the fit functions obtained for neutral dislocation arrangements ( see and ) . it is evident that the relaxation of same - sign dislocation systems follows the same scaling laws that have been observed for systems containing equal numbers of dislocations of both signs . this provides strong support for our basic conjecture that relaxation is governed by the stress - driven elimination of excess dislocations in a process that progresses from small to large scales . the processes occurring on short scales , on the other hand , are evidently different for the two systems ( dipole formation vs. formation of walls ) . this is reflected by the fact that the relaxation process in the single - sign dislocation systems shows an initial size dependence which is not present in neutral dislocation systems ( see and for comparison ) . for different system sizes @xmath6 for dislocations of the same sign . ] for different system sizes @xmath6 for dislocations of the same sign . ] compares the relaxation of dislocation systems in single , double and triple slip . as seen in the figure , at long times the relaxation in multiple slip geometries accelerates in comparison with the relaxation in single slip . this is consistent with the idea that in multiple slip geometries relaxation processes proceed through the formation of dislocation locks and the annihilation of mobile dislocations at these locks . the long range stress fields associated with dislocation locks and the removal of dislocations accelerate the relaxation as seen in the figure . one of the main assumptions behind our scaling model , namely that the motion of mobile dislocations is governed mainly by their mutual interaction , no longer holds in multiple slip geometries . therefore , the scaling model can not be applied to these situations . indeed , as seen in , no power law relaxation regime can be detected in the multiple slip simulations . in dislocation systems with single and multiple slip geometries . in the figure , @xmath1 means the total density of dislocations on all slip systems . the last two curves were shifted downwards to help comparison . ] the present paper discusses the relaxation of initially random arrangements of straight , parallel edge dislocations . following a phenomenological analogy with the kinetics of bimolecular reactions @xcite the relaxation process can be divided into three consecutive stages . stage 1 is characterised by rapid rearrangements of neighbouring dislocations , leading to the formation of dipoles , multipoles , and/or short wall segments . stage 2 hosts the gradual extinction of initial fluctuations in the burgers vector density on ever increasing length scales through the long range transport of excess dislocations . this stage gives rise to characteristic power - law relaxation dynamics . the relaxation process terminates in stage 3 when the characteristic fluctuation length reaches the system size . our considerations focus on the power - law relaxation dynamics in stage 2 . in section 3 , we formulated a scaling theory for this process by making the analogy with a bimolecular reaction . to this end , we considered a highly simplified picture where pair interactions between positive and negative excess dislocations lead to long - range dislocation transport resulting in the formation of dislocation dipoles . this led to predictions for the evolution of the elastic energy and the first two moments of the dislocation velocity . these predictions were then compared to ensemble averages of discrete dislocation dynamics simulations , and convincing agreement was found for single slip geometries . for multiple slip geometries , however , the persistent long range stress fields of dislocation locks accelerate the relaxation process , to which the scaling model can no longer be applied . the actual dislocation processes in many - dislocation systems are much more complex than the simplified picture underlying our scaling arguments . instead of long - range transport of excess dislocations and dipole formation , we see complex rearrangements resulting in dislocation dipoles , multipoles , and walls . in addition , it is well known that dislocations have a propensity to form large - scale heterogeneous patterns consisting of dislocation - rich and dislocation depleted regions . in the following we briefly discuss how these complex static and dynamic features fit into the idealised picture we used in the previous sections . we have developed our scaling argument for dipole formation which can be considered a bimolecular reaction between positive and negative dislocations . however , actual dislocation arrangements are much more complex . it is therefore important to emphasise that the core of the argument is the elimination of large - scale fluctuations in the excess burgers vector density , and _ not _ the resulting arrangement of nearby dislocations . for the long - time asymptotics of the relaxation process , which is governed by the elimination of fluctuations on larger and larger scales , the small - scale arrangement of dislocations is virtually irrelevant at least as long as the local features ( dipoles , multipoles , short walls and combinations of all these ) do not give rise to long - range stresses . according to the present argument , a well - screened dislocation arrangement is one in which burgers vector fluctuations have been eliminated on all scales . if we are dealing with a neutral dislocation system ( equal numbers of positive and negative dislocations ) this means that the net burgers vector is zero in each small volume , i.e. , for each dislocation we can find exactly one partner of opposite sign nearby . this motivates the dipolar picture even if the actual dislocation arrangements may be more complex . an alternative mechanism for creating well - screened dislocation arrangements is the formation of system - spanning walls of dislocations of the same sign . periodic arrangement of edge dislocations of one sign into a wall perpendicular to the glide plane removes the logarithmic divergence of the dislocation energy and introduces a screening length that is proportional to the dislocation spacing along the wall @xcite . walls are conspicuous both in simulations and in many experimentally observed dislocation microstructures . at first glance , wall formation mechanism seems to be completely at odds with the mechanism discussed in the present paper : formation of walls of same - sign dislocations increases , rather than reduces , the burgers vector density fluctuations . however , a closer investigation reveals that wall formation by itself is not a screening mechanism at all . forming a wall of randomly spaced dislocations does not reduce the energy in comparison with a random 2d dislocation arrangement ( see appendix ) . instead , the screening effect is contingent on the equal spacing of dislocations , i.e. on suppressing fluctuations of the burgers vector density along the wall direction . if we start from a random dislocation arrangement this is not easy to obtain : either the dislocations must have climb degrees of freedom ( which we do not consider in the present study ) , or dislocation motions that lead to the formation of multiple walls must be correlated over large distances such as to ensure that each wall collects only those dislocations that fit into an evenly spaced pattern . in the latter case , we are again dealing with the suppression of burgers vector density fluctuations on all scales above the wall spacing , and the long time asymptotics of this process is expected to obey our scaling theory . this is confirmed by the simulations , which however also demonstrate that the short - time behaviour is different for neutral dislocation systems where the local arrangement of dislocations is characterised by dipolar and multipolar patterns , and for single - sign dislocation systems where the local arrangement of dislocations is characterised by walls ( compare and ) . our scaling argument considers the stress - driven long - range transport of excess dislocations . the picture underlying the argument is schematically shown in ( top ) : a positive excess dislocation at a is attracted by a negative excess dislocation at b and the two recombine by long - range motion which is not affected by the stress field of the dislocations in between a and b. the figure also indicates that this idealisation may not be feasible when we are dealing with multipolar arrangements rather than isolated narrow dipoles : in that case the mutual interaction of the excess dislocation may be much weaker than their interaction with other dislocations ` on the way ' . as a consequence , recombination is much more likely to occur by a collective rearrangement as shown in ( bottom ) . how does this affect our scaling argument ? the total driving force for the process is the same in both cases . however , a collective rearrangement on scale @xmath2 is likely to involve @xmath156 dislocations [ @xmath157 in ( bottom ) ] . hence the driving force per dislocation is reduced by a factor of the order of @xmath6 . however , the same is true for the characteristic distance that has to be covered by each dislocation ( @xmath2 in the case of direct transport , @xmath158 in the case of collective rearrangement ) . as we assume that the dislocation velocity is proportional to the driving force , it follows that the characteristic time scale for eliminating the excess dislocation is the same in both cases , and our scaling argument remains valid . it is a well known phenomenon that dislocation microstructures forming during plastic deformation form heterogeneous patterns consisting of regions of high and low dislocation density , with characteristic lengths that are large in comparison with the dislocation spacing . on a conceptual level , possible mechanisms underlying this patterning were discussed by nabarro @xcite who pointed out that it may be energetically favourable to ` segregate ' the dislocation microstructure into areas of high and low density : if we are dealing with a well - screened dislocation arrangement , the energy density scales like @xmath159 . in this case it can be easily shown that it is energetically favourable to increase @xmath1 in some regions and decrease it proportionally in others . such ` energetically driven ' dislocation patterning could be a reason for the formation of dislocation - dense and dislocation - depleted regions that is observed in many experiments . while this mechanism is not covered by the present model , it is not at variance with our considerations : in a neutral dislocation arrangement , the formation of dislocation - dense and dislocation - depleted regions might occur without disturbing the burgers vector balance . we note , however , that in our simulations large - scale dislocation patterning is not observed either because it does indeed not occur in single slip , or because the dislocation numbers in our simulations might be too small . in conclusion we discuss the relevance of the processes discussed in the present paper for real - world systems . the relaxation of a random dislocation system has no direct counterpart in real deformation experiments , since it is impossible to ` prepare ' such a random system in the first place . our analysis of the screening of same - sign dislocation systems is , however , of general importance for understanding real dislocation patterns since it demonstrates that wall formation , though a conspicuous feature , can by itself not account for screening . this observation points to the importance of investigating long - range correlations between dislocation positions both within the walls and across different walls , and offers ample scope for future investigations . the investigated processes are of significant importance for discrete dislocation dynamics simulations of plasticity as the slow nature of the relaxation makes it difficult to obtain well - defined and energetically stable initial configurations . our comparison of strain - rate relaxation experiments with the results of miguel et al . demonstrates that the collective behaviour of dislocation systems may depend significantly on initial conditions . the analysis of this dependence is still in its infancy , yet understanding it is indispensable for carrying out dislocation plasticity simulations in a controlled and well - defined manner . financial support of the hungarian scientific research fund ( otka ) under contract no . k 67778 , of the european community s human potential programme under contract nos . mrtn - ct-2003 - 504634 [ sizedepen ] and nmp3-ct-2006 - 017105 [ digimat ] and of nest pathfinder programme trigs under contract nest-2005-path - com-043386 are gratefully acknowledged . we consider a wall of infinite height running along the plane @xmath160 in an isotropic material . edge dislocations of burgers vector @xmath161 are distributed randomly along the wall with average linear density @xmath162 . the geometry corresponds to a plane - strain situation , hence the elastic energy density can be written as @xmath163 . \label{edens}\ ] ] the total energy of the system is obtained by integrating over the system volume , @xmath164 where we have used that , for an infinite system , the second and third terms on the right - hand side do not contribute to the total energy . this can be shown as follows : for plane - strain deformation , the stresses can be written as derivatives of the airy stress function @xmath16 , @xmath165 , @xmath166 and @xmath167 . hence , @xmath168 { \rm d}^2 r = \int_v [ \partial_x\partial_y\chi \partial_x \partial_y \chi - \partial_x^2\chi \partial_y^2 \chi ] { \rm d}^2 r. \label{airy}\ ] ] partially integrating the second term in the integral on the right - hand side with respect to @xmath8 and @xmath115 shows that this integral contributes only surface terms to the total energy . these terms are negligible in the infinite - system limit . the ensemble - averaged stress at any point is given by summing over the stress fields of the individual dislocations in the wall and averaging over the different realizations of the random dislocation positions : @xmath169 where @xmath170 is the @xmath171 component of the stress created at @xmath172 by the @xmath106th dislocation . the elastic energy of the system depends on the averages of products @xmath173 where @xmath174 $ ] . in evaluating these averages we use that the @xmath115 coordinates of the individual dislocations are independent random variables : @xmath175 . \label{indep}\end{aligned}\ ] ] we now make the following observations : * the average stresses @xmath176 and their products @xmath177 depend on the @xmath8 coordinate only . * the average single - dislocation stresses @xmath178 and @xmath179 become zero in the limit @xmath180 , since these stresses are antisymmetric functions of the @xmath115 coordinate . the same is true for the average total stresses @xmath181 and @xmath182 . with these observations and using we can write the system energy as @xmath183 , \label{efinal}\end{aligned}\ ] ] where the second step follows by interchanging the averaging and the integration . @xmath2 is the system size ( tending to infinity ) which , in the absence of any other screening mechanism , delimits the divergence of the dislocation self - energy . for periodic boundary conditions , as used in our simulations , @xmath2 must be understood as the size of the periodic simulation box which in this case defines the screening length for an otherwise uncorrelated dislocation arrangement . @xmath184 is the total number of dislocations in the system . it follows from that the energy per dislocation is equal to the energy of a single unscreened dislocation and , hence , equals the energy in a completely random 2d arrangement . in other words , the arrangement of dislocations of the same sign in a random wall does ( with the possible exception of surface terms that are negligible in the infinite system limit ) not produce any reduction of the total energy . accordingly , the thermodynamic driving force towards forming such a wall is zero . zaiser m , _ statistical modelling of dislocation systems _ , 2001 _ mat . a * 309310 * 30415 groma i , csikor f f and zaiser m , _ spatial correlations and higher - order gradient terms in a continuum description of dislocation dynamics _ , 2003 _ acta mater . _ * 51 * 127181 zaiser m , _ scale invariance in plastic flow of crystalline solids _ , 2006 _ adv . * 55 * 185245 miguel m c , vespignani a , zaiser m and zapperi s , _ dislocation jamming and andrade creep _ , 2002 165501 miguel m c , moretti p , zaiser m and zapperi s , _ statistical dynamics of dislocations in simple models of plastic deformation : phase transitions and related phenomena _ , 2005 _ mat _ a * 400401 * 1918 miguel m c , laurson l and alava m , _ material yielding and irreversible deformation mediated by dislocation motion _ , 2008 _ eur . phys . b * 64 * 44350 wilkens m , _ das spannungsfeld einer anordnung von regellos verteilten versetzungen _ , 1967 _ _ * 15 * 14127 wilkens m , _ das mittlere spannungsquadrat @xmath185 begrenzt regellos verteilter versetzungen in einem zylinderfrmigen krper _ , 1969 _ 11559 zaiser m , miguel m - c and groma i , _ statistical dynamics of dislocation systems : the influence of dislocation dislocation correlations _ , 2001 b * 64 * 224102 zaiser m and seeger a , _ long - range internal stresses , dislocation patterning and work hardening in crystal plasticity _ , 2002 , _ dislocations in solids vol . 11 _ , ed f r n nabarro and m s duesbery ( elsevier ) csikor f f and groma i , _ probability distribution of internal stress in relaxed dislocation systems _ , 2004 b * 70 * 064106 krivoglaz m a 1969 _ theory of x - ray and thermal neutron scattering by real crystals _ ( new york : plenum ) csikor f f , kocsis b , bak b and groma i , _ numerical characterisation of the relaxation of dislocation systems _ , 2005 _ mat . eng . _ a * 400401 * 2147 csikor f f and zaiser m , _ scaling and glassy dynamics in the relaxation of dislocation systems _ , 2006 _ proc . conf . on statistical mechanics of plasticity and related instabilities ( 29 august2 september 2005 bangalore ) _ ed m zaiser _ et al _ ( proceedings of science ) 058 hirth j p and lothe j 1982 _ theory of dislocations _ ( new york : wiley - interscience ) bak b , groma i , gyrgyi g and zimnyi g , _ dislocation patterning : the role of climb in meso - scale simulations _ , 2006 _ comp . * 38 * 228 ovchinnikov a a and zeldovich ya b , _ role of density fluctuations in bimolecular reaction kinetics _ , 1978 _ chem . _ * 28 * 2158 groma i , gyrgyi g and kocsis b , _ debye screening of dislocations _ , 2006 165503 thomson r , koslowski m and lesar r , _ energetics and noise in dislocation patterning _ , 2006 b * 73 * 024104 nabarro f r n , _ complementary models of dislocation patterning _ , 2000 _ phil . mag . _ a * 80 * 75964
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we study the relaxation dynamics of systems of straight , parallel crystal dislocations , starting from initially random and uncorrelated positions of the individual dislocations . a scaling model of the relaxation process
is constructed by considering the gradual extinction of the initial density fluctuations present in the system .
the model is validated by ensemble simulations of the discrete dynamics of dislocations .
convincing agreement is found for systems of edge dislocations in single slip irrespective of the net burgers vector of the dislocation system .
it is also demonstrated that the model does not work in multiple slip geometries .
_ keywords _ : defects ( theory ) , fluctuations ( theory ) , plasticity ( theory )
| 13,132 | 158 |
thermopower , @xmath1 , is an important transport coefficient that offers valuable information about the electronic structure , the scattering processes and the mechanisms of carrier - phonon coupling in a system . in the last few years there has been growing experimental interest in @xmath1 of single wall carbon nanotubes ( swcnts ) . several groups have reported thermopower measurements on bulk swcnt materials ( e.g. , mats , fibers , films ) @xcite and on individual swcnts @xcite . however , only modest progress has been made up to now in understanding the unique features of @xmath1 in these systems . interesting issues concerning the large positive thermopower ( @xmath2 80 @xmath3v / k ) in pristine samples @xcite , the change of sign of @xmath1 upon exposure to oxygen @xcite and the effect of carrier - phonon coupling @xcite on @xmath1 still remain open . @xmath1 consists of two additive contributions which are diffusion , @xmath4 , and phonon - drag , @xmath0 . @xmath4 is due to the carrier diffusion in the presence of a temperature gradient and for degenerate systems varies linearly with @xmath5 according to mott s expression . @xmath0 originates from the interchange of momentum between acoustic phonons and carriers via the carrier - phonon interaction . the first theoretical models for the study of the phonon drag in metals @xcite and semiconductors @xcite were developed half a century ago . more recently , extensive theoretical and experimental work has been carried out on @xmath0 of low - dimensional semiconductor structures @xcite . recent experiments on @xmath1 in p - doped swcnt films and fibers @xcite provided clear evidence for the presence of @xmath0 at @xmath6 k. on the theory level , however , there is still an ongoing discussion about the role of @xmath0 in measured thermopower @xcite . so far , the theoretical studies of @xmath0 are confined to metallic armchair ( 10,10 ) tubes @xcite . however , in perfect metallic tubes with mirror electron - hole symmetry both @xmath4 @xcite and @xmath0 @xcite are expected to be negligibly small compared to the experimental data , due to the competition between the opposite contributions of electrons and holes . we note that the accuracy of the existing theoretical models @xcite for @xmath0 in metallic tubes has been questioned recently by mahan @xcite . also , a recent theoretical work @xcite pointed out that thermopower vanishes in one - dimensional conductors with a linear energy dispersion ( as in the case of metallic tubes ) due to electron - hole symmetry . in this paper we propose a theoretical model for the phonon - drag thermopower in semiconducting swcnts that are characterized by a non - linear energy dispersion . ( a brief discussion on the behavior of @xmath0 in this kind of nanotubes appears in [ ] . ) we suggest that the measured thermopower in doped samples is due to the contribution of degenerate semiconducting nanotubes . in our model @xmath0 originates from carrier - phonon intraband scattering within the first 1d subband . as we discuss below , the dominant contribution to @xmath0 is made by long - wavelength acoustic phonons that backscatter carriers across the fermi surface . in this case the carrier - phonon coupling is much weaker in metallic tubes than in semiconducting tubes @xcite and , consequently , @xmath0 is expected to be substantially larger in the latter ones . we note that upon chemical or electrostatic doping the fermi level can be pushed into the conduction or valence band and the degenerate semiconducting tubes can be considered as one - dimensional metals . therefore the terms metallicand semiconductingrefer only to the different electronic structure in the two types of tubes ( see , for example , ref . there are two equivalent theoretical approaches to the problem of phonon drag@xcite . in the first approach phonons are perturbed in the presence of a weak temperature gradient @xmath7 . non - equilibrium phonons transfer part of their momentum to carriers due to the carrier - phonon coupling . then the phonon - drag contribution to the thermoelectric current @xmath8 is calculated by solving the coupled boltzmann equations for carriers and acoustic phonons @xcite . the phonon - drag thermopower is readily obtained by @xmath9 where @xmath10 is the carrier conductivity . in the second approach carriers are accelerated isothermally in the presence of a weak electric field @xmath11 and impart some of their momentum to phonons due to the carrier - phonon coupling . then the resulting phonon heat current and the phonon - drag contribution to the peltier coefficient is calculated @xcite . this method of evaluating @xmath0 is referred as @xmath12-approach @xcite because it provides a direct estimation of the peltier coefficient . the equivalence of the above two approaches is secured by onsager s symmetry relation . in this paper we follow the second approach which is more general and it can be applied even in systems where carriers do not behave semiclassically @xcite . the paper is organized as follows . in sec.ii we introduce the theoretical model for the calculation of @xmath0 in the semiclassical transport regime . an explicit expression for @xmath0 is derived in sec.iib and in sec.iic we derive a simple approximate expression for @xmath0 for the case of a highly degenerate semiconducting tube . numerical results for @xmath0 as a function of temperature , tube radius and position of fermi level are presented in sec.iii . in the same section we discuss the effect of screening . in sec.iv we compare our theory with available experimental data for acid - doped bulk swcnt samples . we assume that the nanotube is a long indefinitely thin cylinder of radius @xmath13 and length @xmath14 . the nanotube axis is along the @xmath15direction . the carrier wave function is @xcite @xmath16 where , @xmath17 is the space vector , @xmath18 is the carrier wave vector along the axial direction , @xmath19 is the azimuthal angle and @xmath20 labels 1d orbital subbands associated with the carrier confinement along the circumference . we assume , that the fermi level , @xmath21 , is located between the first and the second 1d subbands ( i.e. , only the ground subband is occupied ) . then , the carrier energy is @xmath22 where @xmath23 is the carrier effective mass and @xmath24 denotes the position of the first van hove singularity . in carbon nanotubes phonons also exhibit 1d character . the lattice displacement at a point @xmath17 is @xcite @xmath25 where , @xmath26 is the polarization vector , @xmath27 is the phonon wave vector in the axial direction and @xmath28 denotes the phonon modes associated with phonon confinement along the circumference . due to the conservation of angular momentum only the three low - energy acoustic modes with @xmath29 ( the so - called twisting , stretching and breathing modes ) contribute to the carrier - phonon intraband scattering . the phonon frequencies and polarization vectors have been calculated within the continuum model proposed by suzuura and ando @xcite . the carrier - phonon interaction in carbon nanotubes has been studied in several texts within the tight - binding approximation @xcite or a continuous elastic theory @xcite . here we follow the continuous model of suzuura and ando @xcite according to which the carrier - phonon coupling is described via the acoustic deformation potential @xmath30 where @xmath31 is the deformation potential constant . the deformation potential approximation provides a good description of the carrier interaction with long - wavelength acoustic phonons . the last term in eq . ( [ def ] ) accounts for the nonzero curvature of the nanotube @xcite . the twisting mode does not participate to carrier - phonon scattering via the deformation potential coupling . moreover , in the long - wavelength limit ( @xmath32 ) , which is the regime of our interest , the breathing mode is dispersionless and does not contribute to @xmath0 . thus , in what follows we consider only the stretching mode which is characterized by a linear dispersion @xmath33 where , @xmath34 is the sound velocity . the phonon polarization vector , @xmath35 , for this mode in the limit @xmath32 is @xmath36 where @xmath37 and @xmath38 is poisson s ratio . ignoring the terms proportional to @xmath39 the above expression becomes identical with the one derived by de martino _ et al_. @xcite . we assume a small electric field @xmath40 in the axial direction of the nanotube . the presence of @xmath40 creates a net flux of carriers along the axis of the tube which results in a momentum transfer to phonons through the carrier - phonon coupling . we calculate the resulting phonon heat flux @xmath41 and obtain the phonon - drag contribution to the transport coefficient @xmath42 to get @xmath0 we utilize the onsager s relation @xmath43 where @xmath10 is the carrier conductivity and @xmath5 the absolute temperature . the phonon heat flux is given by @xmath44 where @xmath45 is the phonon group velocity and @xmath46 is the first order perturbation of the phonon distribution function . the perturbation @xmath47 is determined by the steady - state boltzmann equation for phonons in the relaxation time approximation when @xmath48 . namely , @xmath49 where @xmath50 is the phonon relaxation time associated with phonon - phonon collisions and phonon scattering by imperfections . for simplicity we have ignored the dependence of @xmath50 on @xmath27 . @xmath51 is the rate of change of the phonon distribution function @xmath52 due to phonon scattering by carriers . it is written in the standard form @xmath53 where @xmath54 and @xmath55 are the spin and the valley degeneracies , respectively , @xmath56 is the carrier distribution function and @xmath57 are the transition rates at which the carrier in a state @xmath18 is promoted to a state @xmath58 by absorbing ( emitting ) one phonon with wave vector @xmath27 . when the external field @xmath40 is weak eq . ( [ ph - c ] ) is linearized and is solved in terms of @xmath47 . then we get @xmath59 where , @xmath60 is the phonon relaxation time associated with scattering by carriers given by @xmath61,\ ] ] and @xmath62 is the average equilibrium rate of absorption of phonons with wave vector @xmath27 . it is given by @xmath63 where @xmath64 + 1\}^{-1}$ ] ( with @xmath65 ) is the fermi - dirac function and @xmath66 denotes the transition rate in equilibrium . assuming that phonon - phonon scattering and phonon scattering by impurities dominate over the phonon - carrier scattering ( @xmath67 ) , eqs.([boltzmann ] ) and ( [ n1nn ] ) give @xmath68 in the above equation @xmath69 is the first order perturbation of the carrier distribution function . it is worth noting that eq . ( [ n1 ] ) can be regarded as a starting point for the calculation of @xmath0 in all the problems treated within the @xmath12-approach@xcite . now , by substituting the phonon perturbation into ( [ q ] ) we take for the heat flux @xmath70 to determine the perturbation of the carrier distribution function @xmath69 entering eq . ( [ heat ] ) we use the 1d steady - state boltzmann equation @xmath71 where @xmath72 is the carrier charge and the rhs of eq . ( [ boltzmann2 ] ) is the rate of change of the carrier distribution function due to elastic collisions with static imperfections . in the relaxation time approximation this term is written as @xmath73 where @xmath74 is the carrier relaxation time . equation ( [ boltzmann2 ] ) is linearized to give @xmath75 where , @xmath76 is the carrier group velocity . by substituting eq . ( [ f1 ] ) into ( [ heat ] ) and making use of eqs.([m ] ) , ( [ onsager ] ) and ( [ gamma ] ) we finally get @xmath77\nonumber\\ \times f^{0}_{k}(1-f^{0}_{k^{\prime}})p^{a0}_{q}(k , k^{\prime}).\end{aligned}\ ] ] the above expression is equivalent to the expression derived by kubakaddi and butcher@xcite for a quantum wire coupled to 3d phonons . the authors in ref . [ ] followed a different approach than this described here . they followed bailyn s theory @xcite and they calculated the phonon - drag contribution to the thermoelectric current that originates from the carrier scattering with non - equilibrium phonons in the presence of a small temperature gradient across the wire . their calculation was based on the solution of the coupled equations for electrons and phonons . the transition rate @xmath66 is calculated by using fermi s golden rule . the lattice displacement for the stretching mode is written in second quantized form @xmath78 where @xmath79 and @xmath80 are the phonon creation and annihilation operators , respectively , @xmath81 is the nanotube surface area and @xmath82 is the mass density . for the stationary carrier states considered here one easily finds @xmath83 where , @xmath84^{-1}$ ] is the phonon distribution in equilibrium , @xmath85 is the square of the carrier - phonon matrix element for the deformation potential coupling and @xmath86 is the 1d static dielectric function . by utilizing eqs . ( [ def ] ) and ( [ eta ] ) the matrix elements @xmath85 in the limit @xmath32 are written as @xmath87 where @xmath88 . we note that the @xmath27-dependence of @xmath85 is typical for the carrier interaction with longitudinal acoustic phonons via an isotropic deformation potential @xcite . a similar expression to the one we derive here is given in ref . the dielectric function for a 1d gas confined to the surface of the carbon nanotube is calculated within the random phase approximation @xcite . for the carrier wave functions considered here we obtain @xmath89 where @xmath90 and @xmath91 are the modified bessel functions of the first and the second kind , respectively , and @xmath92 is the background dielectric constant . @xmath93 is the standard factor that accounts for finite temperature effects on the static polarization function @xcite @xmath94}.\ ] ] to obtain an explicit expression for @xmath0 we substitute eq . ( [ rate ] ) into ( [ sgs1 ] ) . then the summation over @xmath58 is readily carried out by replacing @xmath58 by @xmath95 as a consequence of the momentum conservation condition imposed by the kronecker symbol @xmath96 . moreover , the summations over @xmath27 and @xmath18 are transformed to the integrals @xmath97 the presence of the @xmath98function in eq . ( [ rate ] ) allows the immediate evaluation of the @xmath18integration . we see by inspection that @xmath99 where , @xmath100 . the minus and the plus signs correspond , respectively , to positive and negative @xmath27 . now , after some algebra , we finally obtain @xmath101 where , @xmath102 is the phonon - mean - free path , @xmath103^{1/2}$ ] is the fermi wave number and @xmath104 is the product of the fermi occupation factors @xmath105\ ] ] with @xmath106 . in deriving eq . ( [ sgf ] ) we have ignored the energy dependence of the carrier relaxation time and in eq . ( [ sgs1 ] ) we have replaced @xmath74 by its value at the fermi level , @xmath107 . this is a good approximation when @xmath108 @xcite . moreover , we have replaced @xmath10 by @xmath109 where @xmath110 is the density of carriers per unit length . interestingly , @xmath0 becomes independent of the carrier relaxation time . at low @xmath5 and assuming that @xmath111 is a small quantity compared to @xmath21 the product @xmath104 is approximated by @xcite @xmath112 with @xmath106 . the @xmath98-function can be written in the following form @xmath113.\ ] ] we see that @xmath114 resonates at @xmath115 for positive @xmath27 . when the expression ( [ appr3 ] ) for @xmath104 is substituted into eq . ( [ sgf ] ) the integration over @xmath27 is carried out straightforwardly by using the condition @xmath115 . we note that @xmath116 and consequently , stretching phonons with @xmath117 make the dominant contribution to @xmath0 . equation ( [ sgf ] ) is now significantly simplified and is written in the convenient approximate form @xmath118 where @xmath119 is given by @xmath120 in the above equations , @xmath121 is the frequency of a stretching phonon with @xmath117 and @xmath122 is an approximate expression for the dielectric function . to obtain @xmath122 we replace @xmath27 by @xmath123 in the denominator and in the arguments of the modified bessel functions @xmath90 and @xmath91 in eq . ( [ dielectric ] ) . the factor @xmath93 is replaced by the average @xmath124 that is given by the expression @xmath125 @xmath124 has been evaluated numerically for several values of @xmath126 and @xmath13 and we find that in the degenerate limit and when @xmath127 ( where @xmath128 is the fermi temperature ) the following expression provides a very good fit @xmath129\ ] ] where , @xmath130 , @xmath131 , @xmath132 and @xmath133 . at low @xmath5 the effect of screening is severe and unity can be neglected in eq.([dielectric ] ) . in this case the @xmath5-dependence of the dielectric function is described by eq.([aver2 ] ) . at temperatures where @xmath134 the dielectric function shows a weak @xmath5-dependence . then @xmath0 follows the law @xmath135 this activated behavior is characteristic in 1d systems where the fermi surface consists of two discrete points @xmath136 @xcite . we note that when screening is ignored and the phonon mean - free path is constant the @xmath5-dependence of @xmath0 given by eq . ( [ appr1 ] ) is similar to what predicted by scarola and mahan @xcite for an armchair ( 10,10 ) metallic swcnt due to interband electron scattering between the two linear bands . however , the absolute magnitude of @xmath0 in a metallic tube is expected to be much lower than that predicted in ref . [ ] due to the competing contributions of electrons and holes to the thermoelectric current . we assume that the free carriers are holes and we examine the dependence of @xmath0 on temperature , the radius of the nanotube and the position of the fermi level with respect to the position of the first van hove singularity . the analysis is the same for the case of electrons with the only difference being the sign of @xmath0 . the values for the material parameters used in the calculations are @xmath137 , @xmath138 ev @xcite , @xmath139 @xcite , @xmath140 @xcite , @xmath141 kgr / m@xmath142 and @xmath143 km / s @xcite . the hole effective mass is taken to be @xmath144 where @xmath145 is the tube radius in nm @xcite . we assume that @xmath146 @xmath147 . against temperature for a swcnt of radius 0.5 nm . results are shown for four values of @xmath148 : 60 mev(dotted line ) , 90 mev ( dashed - dotted line ) , 120 mev ( dashed line ) and 150 mev ( solid line ) when screening is taken into account ( a ) and when screening is ignored ( b ) . the phonon mean - free path is taken to be 1 @xmath3 m . the inset shows the ratio @xmath149 as a function of @xmath150.,height=340 ] in fig . 1a we show the @xmath0 evaluated from eq . ( [ sgf ] ) for a p - type swcnt of radius @xmath151 nm as a function of @xmath5 . the solid , dashed , dashed - dotted and dotted lines correspond , respectively , to @xmath152 , 120 , 90 and 60 mev . we note that at temperatures where the carriers are non - degenerate we have taken into account the thermal broadening effects on @xmath10 . to assure the accuracy of the approximate expression ( [ appr1 ] ) , in the inset of fig . 1 we show the ratio @xmath149 as a function of @xmath150 for @xmath153 mev . @xmath0 and @xmath154 are calculated from eqs . ( [ sgf ] ) and ( [ appr1 ] ) , respectively . calculations of @xmath155 for 90 , 120 and 150 mev also fall on to the same curve . we can see that in the degenerate limit the approximate result agrees very well with the exact expression for @xmath0 . finally , in fig . 1b @xmath0 is calculated in the absence of screening , @xmath156 . it turns out that screening induces a strong suppression of @xmath0 by 1 - 2 orders of magnitude . inspection of eq . ( [ dielectric ] ) shows that screening effects become more severe as @xmath13 decreases . we note that in the absence of screening @xmath0 levels off at high @xmath5 in agreement with previous estimations in metallic swcnts @xcite . however , when screening is introduced @xmath0 shows a quasi - linear @xmath5-dependence at high @xmath5 due to the temperature dependence of the dielectric function . the dielectric function @xmath122 as a function of the inverse temperature for a swcnt with @xmath157 nm is shown in fig.2 in fig.3 we show the dependence of @xmath0 on the fermi level with respect to the position of the first van hove singularity for temperatures @xmath158 k. the shown structure is due to two competing mechanisms which are the suppression of the carrier - phonon scattering and the increase of @xmath159 as @xmath126 increases . the tube radius is 0.5 nm . finally , in fig.4 we present the calculated values of @xmath0 as a function of the nanotube radius . at temperatures higher than 100 k we find that @xmath0 follows a law close to @xmath160 . at lower temperatures @xmath0 shows a weaker dependence on @xmath13 especially at large values of r. so far there is no clear evidence about the phonon - drag effect in isolated swcnts . the most relevant experiments were performed by yu et al . @xcite in an individual swcnt at temperatures above 100 k. the observed thermopower showed a linear @xmath5-dependence which was attributed to the linear diffusion component and a constant phonon - drag component of about 6 @xmath3v / k without , however , excluding the possibility of an additional contact effect . according to our analysis in section iii , the phonon - drag thermopower at relatively high temperatures shows a quasi - linear @xmath5-dependence and this makes difficult the separation of the diffusion and the phonon - drag contributions . nevertheless , when the calculated values for @xmath0 shown in fig.1a are fitted by a linear function of @xmath5 we find that the intercepts vary from 1.4 to 7.4 @xmath3v / k when the position of the fermi level with respect to the first van hove singularity varies from 60 to 150 mev . these values are in agreement with the experimental estimation of @xmath0 in ref . we note that the intercepts depend linearly on the phonon mean - free path and vary approximately as @xmath161 . vavro _ et al_. @xcite and zhou _ et al . _ @xcite have reported thermopower measurements in p - doped bulk swcnt samples in a wide temperature range ( 10 - 200 k ) that show clearly the signature of phonon drag . normally , in bulk samples nanotubes are self - organized into long ropes , which contain a large number of nanotubes ( tens to hundreds ) @xcite , forming a 3d network of complex geometry . thermopower in these nanotube networks exhibits a very similar behavior as this of an individual nanotube described in section iii . we have recently proposed a simple argument based on a model of parallel conductors which suggests that in a network with homogeneous doping and with a narrow distribution of tube diameters the measured thermopower resembles that of an individual tube @xcite . the resistivity measurements in the samples under consideration showed weak coupling between metallic nanotubes @xcite and hence the contribution from metallic tubes to the total conductivity is neglected . we also recall that the contribution of metallic tubes in @xmath1 is expected to be small compared to this of semiconducting tubes . therefore , we can use the theory for isolate semiconducting swcnts developed here to interpret the data in @xcite . in fig.5 the circles are the measured thermopower for a bulk sample prepared by pulsed laser vaporization ( plv ) and doped with hno@xmath162@xcite . the tube radius is @xmath163 nm . at low temperatures ( up to 100 k ) we fit the data for the total thermopower , @xmath1 , by the expression @xmath164 the first term is the approximate expression ( [ appr1 ] ) for @xmath0 and the second term corresponds to the diffusion component @xmath4 . the sample is highly degenerate and at temperatures up to 100 k eq.([appr1 ] ) accurately describes @xmath0 . the @xmath5-dependence introduced by the dielectric function is given by eq . ( [ aver2 ] ) . the values we obtained for the parameters @xmath126 , @xmath165 and @xmath166 are shown in table i. the logarithmic term in @xmath4 secures an excellent fit to the measured thermopower at all temperatures up to 100 k. if this term is neglected the theoretical values for the total thermopower are significantly larger than the experimental ones at high temperatures . we speculate that the @xmath167 term in @xmath4 is due to 2d weak localization ( wl ) effects @xcite . if this speculation is valid we would also expect a signature of wl in the conductivity measurements . we note that the relative change in conductivity should be the same as in @xmath4 but with an opposite sign @xcite . interestingly , we find that at temperatures @xmath168 k the conductivity follows the law @xmath169 where the value of @xmath170 is shown in table i. we see that @xmath166 and @xmath170 agree to each other remarkably well . the origin of the 2d wl in these samples is not well understood . it is likely related to the individual rope although in this case a 1d localization behavior would be expected @xcite . however , the phase coherence length , in the samples we discuss here , is comparable to the diameter of the rope @xcite and the 2d limit might be approached . et al . _ @xcite have also observed a @xmath171 dependence of the conductance for an individual multiwall cnt at 0.1 - 100 k which was attributed to 2d wl . finally , we should remark that wl is expected to have a negligible effect on @xmath0 @xcite . now , by using the values for @xmath126 we obtained from the fitting of the thermopower data at low @xmath5 we calculate @xmath0 in the whole temperature range from 10 to 200 k by using the exact expression ( [ sgf ] ) . the only remaining unknown is the value of the phonon - mean - free path @xmath172 which is determined from the experimental data when the diffusion contribution is subtracted . we find that @xmath173 nm . this value is consistent with the values 0.25 - 0.75 @xmath3 m reported recently for an individual swcnt @xcite . our estimation for the total thermopower is shown as solid line in fig.5 . the dashed and the dashed - dotted lines correspond to the phonon - drag and the diffusion contributions , respectively . by following a similar procedure as this described above we have interpreted the thermopower data for another bulk sample prepared by high pressure decomposition of co ( hipco ) and doped with h@xmath174so@xmath175 @xcite . the tube radius varied from 0.4 - 0.7 nm . conductivity measurements for this sample ( designated as hpr93c ) appear in @xcite . the experimental data for the ratio @xmath176 are shown as squares in fig.6 . the values for the fitting parameters @xmath126 , @xmath165 and @xmath166 are shown in table i. we also present the value of @xmath170 for comparison . in the calculations the tube radius is taken to be the average @xmath177 nm while for the phonon - mean - free path we obtained the value @xmath178 nm . the calculated values for @xmath176 is shown as solid line in fig.6 . .the values for the parameters @xmath165 , @xmath166 and @xmath126 obtained from the fit of the thermopower data @xcite for @xmath179 k by using eq . ( [ fit ] ) . in the last column we show for comparison the values for @xmath170 obtained from the resistivity data @xcite in the range 10 - 100 k. [ cols="^,^,^,^,^,^",options="header " , ] concerning the consistency of the fitting parameters @xmath165 and @xmath126 we should make the following remarks . by using the values for @xmath126 shown in table i and a simple tight binding model for the estimation of the first van hove singularity ( see , for example , ref . [ ] ) the values we get for @xmath21 are in good agreement with those determined from reflectivity and raman measurements @xcite . also , @xmath165 varies inversely with @xmath180 in agreement with mott s expression for @xmath4 . moreover , the values we extract for @xmath126 support recent arguments according to which h@xmath174so@xmath175 is a stronger dopant than hno@xmath162 @xcite . namely , according to our estimation for the fermi wave numbers , the fermi level is shifted by 94 and 155 mev below the top of the valence band for the plv+hno@xmath162 and hipco+h@xmath174so@xmath175 samples , respectively . in order to show clearly the effect of phonon drag in fig.6 we have plotted the ratio @xmath176 as a function of @xmath5 . the circles and the squares are the measured values for the samples plv+hno@xmath162 and hipco+h@xmath174so@xmath175 , respectively . the dashed lines are the theoretical estimates for @xmath0 and the solid lines are the calculated values for the total thermopower . the peaks at @xmath181 are associated to phonon - drag thermopower . the shift between the experimental and the theoretical values for @xmath182 is due to the logarithmic term in @xmath4 . the position of the peak moves towards to higher temperatures as doping increases . this dependence can be understood by maximizing the ratio @xmath183 using eq . ( [ appr1 ] ) . then we get the following dependence @xmath184 it is important to add that the exponential suppression of @xmath0 at low temperatures is unique for 1d systems . in higher dimensions @xmath0 exhibits a power - law @xmath5 dependence at low temperatures @xcite . the observed peak in @xmath176 , which is ascribed to phonon drag , underlies the 1d character of thermopower . this adds another confirmation that thermopower in bulk carbon nanotube - based materials is a property of the individual tube rather than a property of the network . in summary , we have presented a rigorous model for the calculation of the phonon - drag thermopower in degenerately doped semiconducting swcnts . by using the derived model we investigated the dependence of @xmath0 on temperature , tube radius and position of the fermi level . we found that @xmath0 decreases with the increase of the tube radius following approximately a @xmath161 law at high temperatures . in the degenerate limit , we derive a simple expression for @xmath0 which can be used as a probe for the estimation of the free carrier density in doped tubes . according to this expression @xmath0 shows an activated @xmath5 dependence at low temperatures . screening effects of the carrier - phonon coupling reduce the magnitude of @xmath0 severely and result to a quasi - linear @xmath5-dependence of phonon drag at high @xmath5 . finally , we have compared our model with available data in acid - doped bulk samples @xcite and we found a very good agreement in a wide temperature range . the author wishes to thank dr . k. papagelis for extensive and useful discussions and prof . r. fletcher for stimulating remarks . 99 l. grigorian , k.a . williams , s. fang , g.u . sumanasekera , a.l . loper , e.c . dickey , s.j . pennycook and p.c . eklund , phys . lett . * 80 * , 5560 ( 1998 ) ; l. grigorian , g.u . sumanasekera , a.l . loper , s. fang , j.l . allen and p.c . eklund , phys . b * 58 * , r4195 ( 1998 ) . afonin , y.m . galperin and v.l . gurevich , zh . fiz * 87 * , 335 [ sov jetp * 60 * , 194 ] ( 1984 ) ; m.j . kearney and p.n . butcher , j. phys . c : solid state phys . * 21 * , l265 ( 1988 ) ; c. castellani , c. di castro , m. grilli and g. strinati , phys . rev . b * 37 * , 6663 ( 1988 ) . a. miele , r. fletcher , e. zaremba , y. feng , c.t . foxon and j.j . harris , phys . b * 58 * , 13181 ( 1998 ) ; c. rafael , r. fletcher , p.t . coleridge , y. feng and z.r . wasilewski , semicond . * 19 * , 1291 ( 2004 ) .
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a theoretical model for the calculation of the phonon - drag thermopower , @xmath0 , in degenerately doped semiconducting single - wall carbon nanotubes ( swcnts ) is proposed .
detailed calculations of @xmath0 are performed as a function of temperature , tube radius and position of the fermi level .
we derive a simple analytical expression for @xmath0 that can be utilized to determine the free carrier density in doped nanotubes . at low temperatures
@xmath0 shows an activated behavior characteristic of the one - dimensional ( 1d ) character of carriers .
screening effects are taken into account and it is found that they dramatically reduce the magnitude of @xmath0 .
our results are compared with previous published experimental data in bulk p - doped swcnt materials .
excellent agreement is obtained in the temperature range 10 - 200 k for a consistent set of parameters .
this is a striking result in view of the complexity of these systems .
| 9,571 | 243 |
jets are found in accreting systems throughout the visible universe . for stellar - mass accretors , the evolution of such jets occurs on human timescales , and can be probed by resolved monitoring observations . x - ray binaries are one such class of objects , in which two distinct types of jets are observed , with a clear connection between the x - ray state of the source and the observed radio emission . from the flat radio spectra seen in the hard x - ray state , the presence of steady , compact , partially self - absorbed outflows is inferred , which have been directly resolved in two black hole ( bh ) systems ( ( * ? ? ? * dhawan et al . 2000 ) ; ( * ? ? ? * stirling et al . 2001 ) ) . brighter , optically - thin , relativistically - moving jets are associated with high - luminosity , soft x - ray states during outbursts ( ( * ? ? ? * mirabel & rodrguez 1994 ) ) . our current understanding of the duty cycles and disc - jet coupling in black hole x - ray binaries ( bh xrbs ) derives from a compilation of x - ray spectral and timing information , together with radio flux density monitoring and a limited set of high - resolution radio imaging . the current paradigm , or ` unified model ' ( ( * ? ? ? * fender et al . 2004 ) ) suggests that the jet morphology and power correlate well with position in an x - ray hardness - intensity diagram ( hid ) . steady , self - absorbed jets are inferred to exist in the very low luminosity quiescent state and the higher - luminosity hard state . as the x - ray intensity increases in the hard state , so too does the jet power , with the radio and x - ray luminosities following a non - linear correlation , @xmath0 . at about @xmath1@xmath2 , the x - ray spectrum begins to soften , and the jet speed increases as the inner disc radius moves inwards . below a certain x - ray hardness ( the ` jet line ' ) , the core jet switches off and internal shocks develop in the flow , which are observed as bright , relativistically - moving radio ejecta . the source may remain at high luminosity for several weeks , making repeated transitions back and forth across the jet line , before the x - ray luminosity eventually decreases to @xmath3 where the spectrum hardens ( note the hysteresis effect compared to the higher - luminosity hard - to - soft transition ) , the core jet is re - established , and the source fades back into quiescence . of the many different classes of neutron star ( ns ) x - ray binaries , only the low - magnetic field systems have shown evidence for radio emission . these systems are divided by mass accretion rate into two main classes ; the z - sources and the atoll sources , each with distinct x - ray spectral and timing characteristics . the z - sources are consistently accreting at or close to the eddington luminosity , whereas the atolls are accreting at a somewhat lower level . the x - ray spectral and timing properties of atolls show many similarities to black hole systems , with distinct soft ( so - called ` banana ' ) and hard ( ` extreme island ' ) x - ray states , making them the best sources to compare with black hole x - ray binary outbursts . to date , only a handful of atoll sources have been detected in the radio band during simultaneous radio / x - ray observations , showing them to be systematically fainter than the black hole sources at the same eddington - scaled x - ray luminosity ( * ? ? ? * ( migliari & fender 2006 ) ) . however , it appears that a similar correlation between radio and x - ray luminosities holds in the hard - state atoll sources , but with a lower normalization and a steeper power - law index ; @xmath4 . at high x - ray luminosities , close to the eddington limit where the sources show z - type behavior , bright transient ejecta are thought to exist , just as in black hole systems , although there appears to be only mild suppression of radio emission in the atoll sources when they reach a soft x - ray state . a third class of interacting binaries where mass is transferred from a donor to a degenerate compact object has a white dwarf as the accretor . one class of such systems , the dwarf novae ( a type of cataclysmic variable ; cv ) , also have accretion discs , which periodically develop disc instabilities , leading to a sudden increase in the accretion rate and causing short - lived outbursts . during such outbursts , which recur at intervals of several weeks , these systems brighten by several magnitudes in the optical band . despite similarities to xrbs , with accretion onto a compact object and outbursts triggered by disc instabilities , no jets have thus far been directly resolved in cvs . generalizing the hid to a ` disc - fraction luminosity diagram ' ( dfld ) by plotting the optical flux of the system against the fraction of emission arising from the power - law spectral component ( as opposed to disc emission ) shows that outbursts of dwarf novae follow a very similar track to bh and ns systems . extending this analogy with the unified model then suggests that they should show flat - spectrum radio emission in the rise phase of an outburst , and resolved ejecta during the subsequent spectral softening . the former prediction was spectacularly confirmed during an outburst of the dwarf nova ss cyg ( ( * ? ? ? * krding et al . 2008 ) ) . the radio emission was highly variable , peaking at 1.1mjy , and coincident with the optical outburst . during the decay , the radio spectrum was slightly inverted , suggestive of a compact jet , as resolved in bh xrbs . however , the existence of such a jet can only be directly verified with high - resolution imaging . the jet acceleration and collimation probe of transient x - ray binaries ( jacpot xrb ) project aims to probe the similarities and differences in the jet launching process between these three different source classes by conducting intensive monitoring of an outburst of each of these different classes of accreting compact object . time - resolved , high angular resolution monitoring in the radio band using the vlba and ( e)vla , plus wsrt and evn when available , allows us to directly image the evolution of the jets over the course of an outburst . simultaneous multi - wavelength coverage using _ swift _ bat , _ rxte _ pca , and _ maxi _ in the x - ray band , plus optical and infrared monitoring using the two faulkes telescopes , fancam and ctio ( via the smarts consortium ) , then enables us to tie the jet behavior to the corresponding changes in the accretion flow . with such detailed , multi - wavelength monitoring , we aim to directly test the prevailing paradigm for black hole x - ray binary outbursts ( * ? ? ? * ( fender et al . 2004 ) ) and , by comparing the jet behavior across the three different source classes , to ascertain the role played in jet formation by the depth of the gravitational potential well and the presence or absence of a stellar surface and stellar magnetic field . one outburst of each class of accreting compact source has now been observed as part of the jacpot xrb project , and data reduction and analysis are underway . monitoring campaigns on one further black hole and one neutron star system have been approved by nrao , and are awaiting triggering events . , and the crosses mark the position of the central binary . ] we triggered our first black hole observing campaign on the 2009 outburst of the black hole candidate source h1743 - 322 . the x - ray evolution of the outburst has already been analysed in detail by ( * ? ? ? * motta et al . ( 2010 ) ) and ( * ? ? ? * chen et al . ( 2010 ) ) . we monitored the outburst with the vla , atca and vlba , tracking the evolution of the radio emission through the entire outburst . while the vlba observations were hampered by the strong scattering and lack of good calibrators in the direction of the galactic center , we detected both the compact core during the initial rising hard state , and also the launching of ejecta immediately following the transition to the soft intermediate state ( fig . [ fig : h1743_images ] ) . the disc - jet coupling in this system follows the standard pattern outlined by ( * ? ? ? * fender et al . ( 2004 ) ) , as shown in the hardness - intensity diagram ( hid ) in fig . [ fig : h1743_hids ] . we see flat - spectrum radio emission from a compact , unresolved core jet in the hard intermediate state ( hims ) at the beginning of the outburst . this core radio emission then quenches , giving rise to very faint , steep - spectrum emission as the source moves into the soft intermediate state ( sims ) , following which we detect bright , optically - thin ejecta . the radio emission does not completely vanish during the high soft state ( hss ) , but shows a modest increase in flux density once the source returns to the hims , before fading with time during the decay phase in the low hard state ( lhs ) . we draw particular attention to the pre - flare ` quench ' phase seen during the hard to soft transition . originally noted by ( * fender et al . ( 2004 ) ) , this short - lived phase is often missed owing to limited temporal sampling in the radio band during outbursts of black hole x - ray binaries , but is surely key to understanding the nature of the transition from steady compact jets to bright , relativistically - moving ejecta . over the reverse transition , the reactivation of the compact jet occurs at a hardness ratio of @xmath5 , and appears to support the assertion of ( * ? ? ? * fender et al . ( 2009 ) ) that the ` jet - line ' is not vertical . the 2009 november outburst of aql x-1 was observed using the vla , vlba , e - evn and _ rxte _ pca , resulting in the first milliarcsecond - scale vlbi detection of the source . we obtained full spectral coverage of the outburst from the radio through infrared ( fancam , smarts ) , optical ( faulkes ) , ultraviolet ( _ swift _ uvot ) and x - ray bands . modeling of the full spectral energy distribution and its evolution is underway . the radio and x - ray lightcurves , plus the hardness - intensity diagram ( hid ) and color - color diagram ( ccd ) of the outburst are shown in fig . [ fig : aqlx1_lcs ] . while full details of the observations and analysis may be found in ( * miller - jones et al . ( 2010 ) ) , we summarize the most salient points here . while hysteresis in the hid of aql x-1 is well known ( * ( maitra & bailyn 2004 ) ) , there have been few previously - reported radio detections ( * ? ? ? * ( tudose et al . 2009 ) ) . our monitoring campaign provided the most complete radio coverage to date of an entire outburst of aql x-1 , finding the radio emission to be consistent with being activated by transitions from a hard spectral state to a soft state , and also by the reverse transition , just as seen in bh systems . a further similarity with black hole systems was the quenching of the radio emission above a certain x - ray luminosity ( @xmath6% of the eddington luminosity ) while in a soft spectral state . however , in contrast to the bh systems , the vlbi observations combined with radio spectral information showed no evidence for steep - spectrum , optically - thin ejecta after the hard - to - soft state transition . the radio emission was at all times consistent with a compact , partially self - absorbed jet as seen in the hard states of bh systems . this may suggest a fundamental difference in the jet formation mechanism between the two classes of source . ss cyg is the brightest dwarf nova , at a distance of only 166pc ( * ? ? ? * ( harrison et al . 1999 ) ) . it undergoes several outbursts each year , and in 2010 april , the aavso notified us that a new outburst of the source was beginning . we were able to get on source with evla within 24h , and followed up the ensuing detection with a full monitoring campaign using evla , wsrt , vlba , _ rxte _ , _ swift _ and aavso . unfortunately the source is too bright for _ swift _ uvot to provide coverage in the ultraviolet . the x - ray and radio light curves of the outburst are shown in fig . [ fig : sscyg_lcs ] . the outburst proceeded in a very similar fashion to that monitored by ( * ? ? ? * wheatley et al . ( 2003 ) ) , showing a sharp drop in the hard x - ray emission at the peak of the optical outburst , and a subsequent recovery as the optical emission faded . as seen by ( * ? ? ? * krding et al . ( 2008 ) ) , the radio emission rose rapidly at the beginning of the outburst , showing a steep radio spectrum ( @xmath7 , with @xmath8 ) immediately after the peak , suggestive of optically - thin ejecta . during the decay phase later in the outburst , the observed flatter radio spectrum is more consistent with a partially self - absorbed compact jet . the source was detected by the vlba during 5 of the 6 epochs . these detections imply a brightness temperature in excess of @xmath9k . as argued by ( * ? ? ? * krding et al . ( 2008 ) ) , together with the radio spectral constraints this suggests the presence of an unresolved jet , as seen in accreting neutron star and black hole systems . as an aside , the astrometric accuracy of the vlba observations allowed us to fit for the source proper motion over the 10 days of observation . assuming a distance of 166pc ( * ? ? ? * ( harrison et al . 1999 ) ) , we find a proper motion of @xmath10masy@xmath11 in r.a . , and @xmath12masy@xmath11 in dec . , fairly consistent with that given in ucac3 ( * ? ? ? * ( zacharias et al . 2010 ) ) , but significantly different from the proper motion derived by ( * ? ? ? * harrison et al . ( 2000 ) ) , with implications for the accuracy of the hst astrometric measurements . a future revision of the distance could remove the discrepancy between predictions of the disc instability model and the observed outburst luminosity ( * ? ? ? * ( schreiber & lasota 2007 ) ) . we have embarked on an ambitious project to test the disc - jet coupling in three different classes of accreting stellar - mass compact objects , using high - resolution vlbi radio observations coupled with multiwavelength monitoring data . using these data , we aim to ( a ) test the ` unified model ' of ( * ? ? ? * fender et al . ( 2004 ) ) for the disc - jet coupling in black hole x - ray binaries , and ( b ) to determine the role played in jet formation by the depth of the gravitational potential well , the stellar surface and the stellar magnetic field . we have already observed one outburst from each of a black hole , a neutron star and a white dwarf binary system , and while preliminary analysis of the data largely confirms the existing paradigm , some interesting similarities and differences between the evolution of black hole and neutron star outbursts have been found . we are very grateful to the nrao , _ rxte _ and _ swift _ schedulers for their flexibility and prompt responses which have made these observing campaigns feasible . we would also like to thank mickael coriat and stephane corbel for sharing their atca data on h1743322 , and matthew templeton and the worldwide network of aavso observers who triggered and supported our observing campaign on ss cyg .
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relationships between the x - ray and radio behavior of black hole x - ray binaries during outbursts have established a fundamental coupling between the accretion disks and radio jets in these systems .
i begin by reviewing the prevailing paradigm for this disk - jet coupling , also highlighting what we know about similarities and differences with neutron star and white dwarf binaries . until recently
, this paradigm had not been directly tested with dedicated high - angular resolution radio imaging over entire outbursts . moreover , such high - resolution monitoring campaigns had not previously targetted outbursts in which the compact object was either a neutron star or a white dwarf . to address this issue ,
we have embarked on the jet acceleration and collimation probe of transient x - ray binaries ( jacpot xrb ) project , which aims to use high angular resolution observations to compare disk - jet coupling across the stellar mass scale , with the goal of probing the importance of the depth of the gravitational potential well , the stellar surface and the stellar magnetic field , on jet formation .
our team has recently concluded its first monitoring series , including ( e)vla , vlba , x - ray , optical , and near - infrared observations of entire outbursts of the black hole candidate h1743 - 322 , the neutron star system aquila x-1 , and the white dwarf system ss cyg . here
i present preliminary results from this work , largely confirming the current paradigm , but highlighting some intriguing new behavior , and suggesting a possible difference in the jet formation process between neutron star and black hole systems .
| 4,597 | 417 |
the group of about @xmath8 so - called central compact objects ( ccos ) in supernova remnants ( snrs ) are distinguished by their steady surface thermal x - ray flux , lack of surrounding pulsar wind nebula , and non - detection at any other wavelength @xcite . three ccos are known pulsars , with periods in the range @xmath9 s , and spin - down rates that provide an estimate of their surface dipole magnetic field strength , which falls in the range @xmath10 g @xcite , smaller than that of any other young neutron star ( ns ) . this weak magnetic field is evidently the physical basis of the cco class . the homogeneous properties of the approximately seven remaining ccos that have not yet been seen to pulse suggest that they have similar or even weaker @xmath7-fields than the known cco pulsars , and a more uniform surface temperature . that ccos are found in snrs ( of ages @xmath11 yr ) in comparable numbers to other classes of nss implies that they must represent a significant fraction of ns births , probably greater than that of magnetars , for example , as only 45 galactic snrs are known to host magnetars @xcite . the subsequent evolution of ccos is a glaring unknown , their immediate descendants not being evident in any existing survey . ccos should persist as cooling nss , detectable in thermal x - rays , for @xmath12 years according to ns cooling curves @xcite . if some are also radio pulsars , that phase could last for @xmath13 years . while there are not yet enough ccos to know whether they are intrinsically radio - quiet , it is very unlikely that the huge expected population of cco descendants are _ all _ hiding simply due to unfavorable radio beaming . therefore , it is difficult to understand why the region of ( @xmath14 ) space in which ccos are found , between the bulk of the ordinary radio pulsars and the recycled `` millisecond''pulsars in binary systems , is relatively empty . most of the pulsars in this sparse region ( see figure [ fig : ppdot ] ) are thought to be `` mildly recycled , '' having been spun up by accretion from a high - mass companion for a relatively short time before a second sn occurred . defined as having @xmath0 ms and @xmath1 g , mildly recycled pulsars include double ns systems , and single ones thought to be the disrupted recycled pulsars ( drps , @xcite ) ejected when the binary is unbound after the second sn . ( these are in contrast to the millisecond pulsars , which have low - mass companions . ) the drps have characteristic ages @xmath15 of @xmath16 yr . historically , it was thought that hardly any pulsars are born with @xmath17 g , so that all such pulsars must be recycled . but the discovery of young ccos in this region of parameter space invalidates that assumption . just as the @xmath18 yr characteristic age of a cco is meaningless , the possibility that _ any _ low @xmath7-field radio pulsar is much younger than its characteristic age may now be considered . lllrrclrccc j0609@xmath192130 & 06 09 58.89 & @xmath1921 30 02.8 & 56 & @xmath20 & 38.73 & 1.2 & 22 & @xmath21 & @xmath22 & 1 + j1038@xmath190032 & 10 38 26.93 & @xmath1900 32 43.6 & 29 & @xmath23 & 26.59 & 1.2 & 880 & @xmath24 & @xmath25 & 2 + j1320@xmath263512 & 13 20 12.68 & @xmath2635 12 26.0 & 458 & @xmath27 & 16.42 & 0.68 & 310 & @xmath28 & @xmath29 & 3 + j1333@xmath264449 & 13 33 44.83 & @xmath2644 49 26.2 & 346 & @xmath30 & 44.3 & 1.4 & 410 & @xmath31 & @xmath32 & 4 + j1339@xmath264712 & 13 39 56.59 & @xmath2647 12 05.5 & 137 & @xmath33 & 39.9 & 1.2 & 310 & @xmath34 & @xmath35 & 4 + j1355@xmath266206 & 13 55 21.34 & @xmath2662 06 20.1 & 277 & @xmath36 & 547 & 8.3 & 22 & @xmath37 & @xmath38 & 5 + j1548@xmath264821 & 15 48 23.26 & @xmath2648 21 49.7 & 146 & @xmath39 & 126.0 & 4.4 & 360 & @xmath40 & @xmath41 & 5 + j1611@xmath265847 & 16 11 51.31 & @xmath2658 47 42.3 & 355 & @xmath42 & 79.9 & 1.7 & 160 & @xmath43 & @xmath44 & 6 + j1753@xmath261914 & 17 53 35.17 & @xmath2619 14 58 & 63 & @xmath45 & 105.3 & 2.2 & 130 & @xmath46 & @xmath47 & 6 + j1816@xmath265643 & 18 16 36.46 & @xmath2656 43 42.1 & 218 & @xmath48 & 52.4 & 1.6 & 470 & @xmath49 & @xmath50 & 4 + j1821@xmath190155 & 18 21 38.88 & @xmath1901 55 22.0 & 34 & @xmath51 & 51.75 & 1.8 & 24 & @xmath52 & @xmath53 & 7 + b1952@xmath1929 & 19 54 22.55 & @xmath1929 23 17.3 & 427 & @xmath54 & 7.932 & 0.70 & 9 & @xmath55 & @xmath56 & 8 + j2007@xmath192722 & 20 07 15.83 & @xmath1927 22 47.91 & 24 & @xmath57 & 127.0 & 5.4 & 250 & @xmath58 & @xmath59 & 9 + j2235@xmath191506 & 22 35 43.70 & @xmath1915 06 49.1 & 60 & @xmath60 & 18.09 & 1.1 & 630 & @xmath61 & @xmath62 & 10 [ tab : drps ] -@xmath63 diagram , including magnetars ( blue crosses ) , inss ( magenta asterisks ) , ccos ( filled red stars ) and drps ( open blue stars ) . black dots are isolated pulsars and circled dots are pulsars in binaries . ( pulsars in globular clusters are excluded as their period derivatives are not entirely intrinsic . ) dashed lines of constant characteristic age and magnetic field are indicated . ] the majority of ccos may have magnetic fields even weaker than those of the known cco pulsars , and may fall among the drps in ( @xmath14 ) space . once the snr associated with a cco has dissipated , it would be difficult to distinguish an `` orphaned cco '' from a drp by timing alone if some ccos are radio pulsars . thermal x - ray emission , however , would allow a recently orphaned cco to be recognized as such up to @xmath64 yr . thermal emission from the cooling ns is the diagnostic that would distinguish an evolving cco from an old drp , whose negligible rotation - powered x - ray emission , thermal or non - thermal , would be orders of magnitude weaker . in this paper , we report an x - ray search for orphaned ccos from among the population drps , whose timing parameters are expected to be comparable . in section [ sec : observations ] we describe the new and archival observations of the drps . section [ sec : results ] gives the resulting upper limits on their temperatures and luminosities . in section [ sec : discussion ] we discuss the implication of these results for the possible evolutionary tracks of ccos . our targets selected for x - ray observations are the 12 radio pulsars classified as drps by @xcite , plus the recently discovered psr j2007 + 2722 @xcite . these comprise all but one of the isolated pulsars in the galactic disk with magnetic field strength @xmath65 g and spin period @xmath0 ms listed in the atnf catalog ( * ? ? ? * v1.46 ) . their properties are listed in table [ tab : drps ] . the latest drp , psr j1821 + 0155 @xcite , the @xmath66 member of the class , was discovered too recently to be included in our x - ray sample . for ten of these objects not already observed in x - rays , we obtained 3.5 ks observations to search for point - like emission at their known ( subarcsecond ) radio locations . we justified this short observing time based on its ability to detect thermal emission from a cooling ns younger than @xmath67 yr , while thermal or nonthermal emission from a @xmath68 yr old drp would be many orders of magnitude less . detailed calculations of the detection limits on temperature and luminosity from these observations are presented below . we also analyzed 5 ks archival exposures on psr j0609@xmath192130 and psr b1952@xmath1929 , and tabulate our prior results for psr j2007@xmath192722 @xcite . all observations were taken with the advanced ccd imaging spectrometer ( acis , * ? ? ? * ) , operating in timed / faint exposure mode , with the targets placed on the back - illuminated ccd . acis has @xmath69 pixels , comparable to the on - axis point - spread function . the nominal acis pointing uncertainty is a radius of @xmath70 . all data reduction and analysis was performed with the interactive analysis of observation software ( ciao , * ? ? ? * ) version 4.5 , using the calibration database ( caldb ) v4.1.3 . the background rates for these observations showed no evidence of flaring behavior , and the full exposure time was retained for analysis in each case . cccccl j0609@xmath192130 & 12687 & 4.99 & @xmath71 & @xmath72 & @xmath73 + j1038@xmath190032 & 13801 & 3.50 & @xmath74 & @xmath75 & @xmath73 + j1320@xmath263512 & 13797 & 3.42 & @xmath76 & @xmath77 & @xmath78 + j1333@xmath264449 & 13800 & 3.42 & @xmath79 & @xmath80 & @xmath81 + j1339@xmath264712 & 13799 & 3.42 & @xmath71 & @xmath82 & @xmath83 + j1355@xmath266206 & 13806 & 3.42 & @xmath84 & @xmath85 & @xmath86 + j1548@xmath264821 & 13805 & 3.42 & @xmath87 & @xmath88 & @xmath89 + j1611@xmath265847 & 13802 & 3.41 & @xmath90 & @xmath91 & @xmath92 + j1753@xmath261914 & 13803 & 3.42 & @xmath93 & @xmath94 & @xmath95 + j1816@xmath265643 & 13804 & 3.42 & @xmath96 & @xmath97 & @xmath98 + b1952@xmath1929 & 12684 & 4.99 & @xmath99 & @xmath100 & @xmath101 + j2007@xmath192722&6438,7254,8492&94.04 & @xmath87 & @xmath102 & @xmath103 + j2235@xmath191506 & 13798 & 3.42 & @xmath104 & @xmath105 & @xmath106 [ tab : limits ] figure [ fig : images ] presents thumbnail images in the 0.310 kev band centered around the radio coordinates of each drp , excluding psr j2007@xmath192722 reported elsewhere @xcite . in these @xmath107 sub - images , no pixel that is not definitely associated with a significant source contains more than two counts in the @xmath108 kev energy band . examination of each image shows no evidence for a source at the radio location within twice the nominal @xmath109 pointing uncertainty . in fact , no counts are detected in an adopted aperture of radius @xmath110 at the position of any target . this is not unexpected given the mean background rate of @xmath111 counts s@xmath112 pixel@xmath112 , uniform across the 12 observations . for this rate , the mean number of counts in a 3.5 ks observation is 0.073 in a @xmath110 radius circle . there is a @xmath113 probability of detecting no counts in that aperture for a single observation , and only a @xmath114 chance of getting one or more counts in any of 12 observations . in no case are the coordinates of the nearest detected x - ray source consistent with the radio location , the closest being @xmath115 from psr j1816@xmath265643 . with no evidence of any photon at the location of each drp , we calculate an upper limit on the thermal flux from an assumed cooling ns of radius @xmath116 km , to match the radius used to derive the theoretical cooling curves discussed in section [ sec : discussion ] . as photon counts follow the poisson distribution , the probability of having gotten zero photons is 0.0023 when the expected number of photons from a source is six . therefore the @xmath117 confidence ( @xmath118 ) upper limit on the source flux is that which would predict six counts . we determine the blackbody temperature required for a fiducial source to produce six counts plus background in the detector by convolving an absorbed blackbody spectrum through the acis spectral response and computing the total counts in the @xmath108 kev bandpass generated for each observation . the blackbody flux normalization is fixed by the ratio @xmath119 for each target , where @xmath120 is the distance derived using the ne2001 galactic free electron density model of @xcite . an absorbing column density @xmath121 is estimated from the dispersion measure ( dm ) assuming a rule - of - thumb @xmath122 , i.e. , @xmath123 dm @xcite . table [ tab : limits ] presents the upper limits computed in this way on the blackbody temperature and bolometric luminosity of each pulsar , quantities measured at infinity . these generally correspond to @xmath124 in the range @xmath125 k and log @xmath126=31.8 - 32.8 $ ] . the two outliers are psr j1355@xmath266206 and psr j1548@xmath264821 , which are less constrained because of their large distances and @xmath121 . the uncertainties on these upper limits are dominated by systematic errors involving the dm derived distances and column densities . dm distance can have fractional uncertainty of 25% or larger ( e.g. , * ? ? ? the neutral column density estimated using a typical ionized fraction involves another uncertain assumption . furthermore , an error on @xmath127 amplifies the error on the temperature measurement , which comes from the low - energy end of the acis - s instrument response , around @xmath128 kev , where the detector sensitivity falls off rapidly and is poorly calibrated . unfortunately , these effects are difficult to quantify . we repeat , for completeness , that we would not expect to detect any of the drps if they are old , rotation - powered nss with spin - down power @xmath129 . for comparison , we can use the dozen old pulsars whose x - ray detections were compiled by @xcite . these typically have @xmath130 with a scatter of a factor of 10 . the same x - ray efficiency for the drps would produce @xmath131 erg s@xmath112 , which is orders of magnitude below our upper limits . the upper limits on temperature and luminosity of each drp can be compared with standard ( minimal ) ns cooling curves , ( e.g. , * ? ? ? * ) to place a lower limit on its age . these limits depend strongly on uncertain variables such as the critical temperature for superfluid neutron pairing , and the composition of the ns envelope , which is why there can not be a unique age limit for each entry in table [ tab : limits ] . roughly speaking , a luminosity limit of log @xmath132=32.8 $ ] requires an age @xmath133 yr for heavy element envelopes and @xmath134 yr for light elements , while log @xmath132=31.8 $ ] implies that @xmath135 yr ( light ) or @xmath136 yr ( heavy ) . the cooling curves for light and heavy element envelopes cross over in this range of luminosities . the upper limits on temperatures and luminosities for the drps ( with the possible exception of psr j1355@xmath266206 ) are smaller than those of all ccos but one . in no case does a drp overlap in possible age with the snr ages of the known ccos , which are @xmath11 yr . the dozen drps fail to qualify as evolved ccos in the age range that is , roughly speaking , 10 times the ages of the known ccos , where we expect their descendants to be 10 times as numerous . the meaning of these x - ray non - detections of drps for the evolution of ccos depends on the volume sampled by the surveys that discovered both populations , and their relative completeness . both are difficult to evaluate ; however , the volumes appear to be at least comparable . the @xmath2 ccos are found in snrs up to a maximum distance of @xmath137 kpc , and the drps appear to have a similar distribution of distance and galactic coordinates . therefore , the absence of radio pulsar counterparts of orphaned ccos appears to be real , at least in the range of magnetic field strengths which define the drps . @xcite noted that roughly four of the drps so defined could actually be interlopers from the population of normal pulsars , as extrapolated from the statistics of studies such as @xcite . however , as we argued previously , it may not be possible to make such a distinction . in any case , it would not change our conclusions regarding the fate of ccos , that there are no known radio pulsars with @xmath1 g that are their immediate , @xmath138 yr old , descendants , where we would expect to find @xmath139 orphans . another clue to the age of drps should be their distribution of heights @xmath140 above the galactic plane as listed in table [ tab : drps ] . however , as discussed by @xcite , these heights are smaller than one would expect for the average ns kick velocity of @xmath141 km s@xmath112 @xcite , which makes it difficult to use @xmath140 as an indicator of age for drps . at this velocity a ns would travel only 270 pc in @xmath142 yr , implying that x - ray detected orphaned ccos could have a similar @xmath140 height as the drps , which are thought to be much older . since they are old , the small scale height of the drps still requires an explanation . @xcite propose that the first sn in the parent binary was of a different type that would give little or no kick to the system , perhaps an electron - capture sn . a priori , one might not have expected dprs to be orphaned ccos . as it is , there are not enough drps compared to double ns systems according to standard evolutionary models that link them @xcite . any drp that is reassigned to a different population would only exacerbate this shortage . still , the evolutionary fate of ccos remains unknown after this survey . one possible solution is that radio luminosity is a declining function of spin - down power . if so , radio surveys could be grossly incomplete in detecting such low @xmath129 pulsars even though they are on the active side of the radio pulsar death line . there is good evidence that ordinary radio pulsars behave this way , with @xmath143 @xcite , because there is no pileup in the number of pulsars near the death line . however , it is not clear that this effect alone could explain the absence of orphaned ccos , because there are in fact many radio pulsars with lower @xmath129 than the cco pulsars . such an effect may also apply to the seven discovered , radio - quiet isolated neutron stars ( inss , * ? ? ? * ) which , however , have strong magnetic fields @xcite , but are close to the radio pulsar death line . the inss ( fig . [ fig : ppdot ] ) are a good analogy to our problem in that they _ are _ plausibly the descendants of the magnetars , following a fast epoch of magnetic field decay around @xmath144 yr @xcite . it is likely that the inss are kept hot for longer than ccos by their continuing magnetic field decay for up to @xmath64 yr @xcite , which could account for their abundance relative to the elusive orphaned ccos . it may be difficult to detect and/or recognize orphaned ccos if they cool faster than ordinary nss . one effect that can accelerate cooling is an accreted light - element envelope , which has higher heat conductivity than an iron surface @xcite . however , this effect actually makes ccos hotter than bare nss for their first @xmath145 yr , after which their temperatures plummet . therefore , the prediction that cco descendants should be detectable in soft x - rays remains robust . another plausible home for orphaned ccos would be among the radio pulsars with magnetic fields comparable to or higher than those of the cco pulsars . one theory for ccos postulates that they are born with a canonical ns magnetic field of @xmath146 g that was largely buried by fall - back of a small amount of supernova ejecta , @xmath147 , during the hours and days after the explosion . the buried field will diffuse back to the surface on a time scale that is highly dependent on the amount of mass accreted @xcite , after which the ccos will join the bulk of the population of ordinary pulsars . for accretion of @xmath148 , the regrowth of the surface field is largely complete after @xmath149 yr , but if @xmath150 is accreted , then the diffusion time could be millions of years . such a scenario addresses the absence of ccos descendants ; they turn into ordinary pulsars . it also has the advantage of not requiring yet another class of ns to exist that would only exacerbate the apparent excess of pulsars with respect to the galactic core - collapse supernova rate , a problem emphasized by @xcite . furthermore , magnetic field growth has long been considered a reason why measured pulsar braking indices are all less than the dipole value of 3 . in this picture , ccos represent one extreme in the evolution of surface magnetic field , and almost any radio pulsar _ might _ be a former cco . finally , an intrinsically strong crustal magnetic field appears to be necessary to explain the existence of the thermal hot spots that enable us to detect pulsations from ccos in the first place ( see discussion in @xcite ) . for the first @xmath67 yr , rapid field growth can only move a cco vertically upward in the @xmath151 diagram . such movement is difficult to detect directly using ccos , because it would require measuring the braking index or observing the change of the dipole magnetic field spectroscopically , neither of which is likely to be possible if the relevant time scale is @xmath152 yr . however , during their first @xmath145 yr , orphaned ccos in this scenario should still have periods of @xmath153 s and could have magnetic fields in the range @xmath154 g. a search of all 159 isolated radio pulsars in this range for thermal x - ray emission from such `` old '' pulsars would provide a promising avenue for finding orphaned ccos . of these , two are known x - ray sources , the faint ( @xmath155 erg s@xmath112 ) nearby radio pulsars psr b1451@xmath2668 and psr b0950 + 08 . x - rays from these sources are attributed to a combination of heated polar caps and non - thermal ( magnetospheric ) emission @xcite . if further x - ray surveys of radio pulsars fail to find any orphaned ccos , then it will be difficult to escape the conclusion that they are intrinsically radio quiet . following the discovery that ccos have weaker magnetic fields than any other young pulsar , it became apparent that their descendants were not obviously present in radio or x - ray surveys . if their magnetic fields at birth are intrinsic , and do not change with time , then the region around the ccos in the @xmath156 diagram of radio pulsars should be densely populated with all of their descendants , unless they are radio quiet . the fact that this area is quite sparsely populated led us to survey a large fraction of the available radio candidates in x - rays , those which were previously understood to be mildly recycled pulsars . the `` smoking pulsar '' evidence of an orphaned cco should be an x - ray hot ns that could be detected , in a short observation , at an age up to @xmath145 yr , which is much younger than the characteristic ages of the targeted drps but much older than the known ccos . only upper limits on their thermal x - ray luminosities were found , in the range log @xmath3=31.8 - 32.8 $ ] , which implies cooling ages @xmath4 yr . up to the age limits implied by the x - ray non - detections , there should be @xmath157 cco descendants in the volume sampled by radio pulsar surveys . since none have been found among radio pulsars with @xmath1 g , the next step should be to search for young , cooling nss among the radio pulsars with larger @xmath7-fields , comparable to or even larger than that of the cco , with @xmath158 g. an especially interesting possibility is that ccos have intrinsically strong @xmath7-fields that were promptly buried by a small amount of supernova debris , but will grow back to `` normal '' strength in @xmath144 yr . if such descendants of ccos are found in thermal x - rays among the ordinary radio pulsar population , it would help solve problems about their surface thermal patterns in addition to their evolution . otherwise , if the orphaned ccos are truly radio silent for some unknown reason , they could still be found in more sensitive all - sky surveys in soft x - rays , by analog with the ( evidently more luminous ) inss that were discovered this way . we thank dr . reinhard prix for discussion during the proposal phase . this investigation is based on observations obtained with the observatory . financial support was provided by award go2 - 13097x issued by the x - ray observatory center , which is operated by the smithsonian astrophysical observatory for and on behalf of nasa under contract nas8 - 03060 .
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we present a x - ray survey of the disrupted recycled pulsars ( drps ) , isolated radio pulsars with @xmath0 ms and @xmath1 g. these observations were motivated as a search for the immediate descendants of the @xmath2 central compact objects ( ccos ) in supernova remnants , three of which have similar timing and magnetic properties as the drps , but are bright , thermal x - ray sources consistent with minimal neutron star cooling curves . since none of the dprs were detected , there is no evidence that they are `` orphaned '' ccos , neutron stars whose supernova remnants has dissipated .
upper limits on their thermal x - ray luminosities are in the range log @xmath3=31.8 - 32.8 $ ] , which implies cooling ages @xmath4 yr , roughly 10 times the ages of the @xmath5 known ccos in a similar volume of the galaxy .
the order of a hundred cco descendants that could be detected by this method are thus either intrinsically radio quiet , or occupy a different region of ( @xmath6 ) parameter space from the drps .
this motivates a new x - ray search for orphaned ccos among radio pulsars with larger @xmath7-fields , which could verify the theory that their fields are buried by fall - back of supernova ejecta , but quickly regrow to join the normal pulsar population .
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interacting fermionic particles play a central role in the structure of matter and exist over a very broad range of energies , from extremely low temperature trapped atomic fermi gases , where @xmath10 k @xcite , to very high temperature primordial matter , like quark - gluon plasmas , where @xmath11 k @xcite . for all of these systems , the most intriguing physics is related to very strong interactions between fermionic particles , such as the strong coupling between electrons in high-@xmath8 superconductors and the strong interactions between neutrons in neutron matter . current many - body quantum theories face great challenges in solving problems for strongly interacting fermi systems , due to the lack of a small coupling parameter . for example , the critical temperature of a superfluid - normal fluid transition in a strongly interacting fermi gas has been controversial for many years . the critical temperature @xmath12 has been predicted to have values in the range between 0.15 and 0.35 by different theoretical methods @xcite . a complete understanding of the physics of strongly interacting systems can not yet be obtained from a theoretical point of view . there is a pressing need to investigate strongly interacting fermions experimentally . in recent years , based on progress in optical cooling and trapping of fermionic atoms , a clean and controllable strongly interacting fermi system , comprising a degenerate , strongly interacting fermi gas @xcite , is now of interest to the whole physics community . strongly interacting fermi gases are produced near a broad feshbach resonance @xcite , where the zero energy s - wave scattering length @xmath13 is large compared to the interparticle spacing , while the interparticle spacing is large compared to the range of the two - body interaction . in this regime , the system is known as a unitary fermi gas , where the properties are universal and independent of the details of the two - body scattering interaction @xcite . in contrast to other strongly interacting fermi systems , in atomic gases , the interactions , energy , and spin population can be precisely adjusted , enabling a variety of experiments for exploring this model system . intense studies of strongly interacting fermi gases have been implemented over the past several years from a variety of perspectives . some of the first experiments observed the expansion hydrodynamics of the strongly interacting cloud @xcite . evidence for superfluid hydrodynamics was first observed in collective modes @xcite . collective modes were later used to study the @xmath14 equation of state throughout the crossover regime @xcite . recently , measurements of sound velocity have also been used to explore the @xmath14 equation of state @xcite . below a feshbach resonance , fermionic atoms join to form stable molecules and molecular bose - einstein condensates @xcite . fermionic pair condensation has been observed by projection experiments using fast magnetic field sweeps @xcite . above resonance , strongly bound pairs have been probed by radio frequency and optical spectroscopy @xcite . phase separation has been observed in spin polarized samples @xcite . rotating fermi gases have revealed vortex lattices in the superfluid regime @xcite as well as irrotational flow in both the superfluid and normal fluid regimes @xcite . measurement of the thermodynamic properties of a strongly interacting fermi gas was first accomplished by adding a known energy to the gas , and then determining an empirical temperature that was calibrated using a pseudogap theory @xcite . recent model - independent measurements of the energy and entropy @xcite provide a very important piece of the puzzle , because they enable direct and precision tests that distinguish predictions from recent many - body theories , without invoking any specific theoretical model @xcite . one of the major challenges for the experiments in strongly interacting fermi gases is the lack of a precise model - independent thermometry . two widely - used thermometry methods are model - dependent , in that they rely on theoretical models for calibration . the first relies on adiabatic magnetic field sweeps between the molecular bec regime and the strongly interacting regime @xcite . subsequently , the temperature of the strongly interacting gas is estimated from the measured temperature in the bec regime using a theoretical model of the entropy @xcite . the second method , used by our group @xcite , is based on determining an empirical temperature from the cloud profiles that is calibrated by comparing the measured density distribution with a theoretical model for the density profiles . currently two model - independent thermometry methods have been reported for strongly - interacting gases . one is the technique employed by the mit group @xcite , which is only applicable to imbalanced mixtures of spin - up and spin - down atoms . that method is based on fitting the noninteracting edge for the majority spin after phase separation . another model - independent method is demonstrated in ref . @xcite , which is applicable to both balanced and imbalanced mixtures of spin - up and spin - down fermions . the energy @xmath1 and entropy @xmath2 are measured and then parameterized to determine a smooth curve @xmath4 . then the temperature in both the superfluid and normal fluid regime is obtained from the fundamental thermodynamic relation @xmath5 . in this paper , we will describe our model - independent thermodynamic experiments on a strongly interacting fermi gas of @xmath0li , which we have conducted at duke university . first , we will describe our measurements of both the total energy @xmath1 and the total entropy @xmath2 of a trapped strongly - interacting fermi gas tuned near a feshbach resonance . then , we determine the temperature @xmath5 after showing that the @xmath4 data are very well parameterized by using two different power laws that are joined with continuous @xmath1 and @xmath6 at a certain entropy @xmath7 that gives the best fit . to examine the sensitivity of the temperature to the form of the fit function , we employ two different fit functions that allow for a heat capacity jump or for a continuous heat capacity at @xmath7 . we find that the @xmath6 values closely agree for both cases . we find a significant change in the scaling of @xmath1 with @xmath2 above and below @xmath7 , in contrast to the behavior for an ideal fermi gas , where a single power - law well parameterizes @xmath4 over the same energy range . by interpreting @xmath7 as the critical entropy for a superfluid - normal fluid transition in the strongly interacting fermi gas , we estimate the critical energy @xmath15 and critical temperature @xmath8 . both the model - independent @xmath4 data and the estimated critical parameters are compared with several recent many - body theories based on both analytic and quantum monte carlo methods . we also show how parameterizing the @xmath4 data provides experimental temperature calibrations , which helps to unify , in a model - independent way , the results obtained by several groups @xcite . first we relate the endpoint temperatures for adiabatic sweeps of the bias magnetic field between the strongly interacting and ideal noninteracting regimes , as used in the jila experiments to characterize the condensed pair fraction @xcite . this enables the ideal gas temperature observed for the onset of pair condensation @xcite to be related to the critical temperature of the strongly interacting fermi gas . the temperature obtained by parameterizing the strongly interacting gas data also calibrates the empirical temperature based on the cloud profiles , as used in our previous studies of the heat capacity @xcite . these temperature calibrations yield values of @xmath8 close to that estimated from our @xmath4 data . next , we discuss three different methods for determining the universal many - body parameter , @xmath16 @xcite , where @xmath17 is the energy per particle in a uniform strongly interacting fermi gas at @xmath14 in units of the energy per particle of an ideal fermi gas at the same density . first , we describe the measurement of the sound velocity at resonance and its relationship to @xmath16 . then , we determine @xmath16 from the ground state energy @xmath18 of the trapped gas . here , @xmath18 is obtained by extrapolating the @xmath4 data to @xmath9 , as suggested by hu et al . this avoids a systematic error in the sound velocity experiments arising from the unknown finite temperature . finally , to explore the systematic error arising from the measurement of the number of atoms , @xmath16 is determined in a number - independent manner from the ratio of the cloud sizes in the strongly and weakly interacting regimes . all three results are found to be in very good agreement with each other and with recent predictions . finally , we obtain three universal thermodynamic functions from the parameterized @xmath4 data , the energy @xmath19 , heat capacity @xmath20 , and global chemical potential @xmath21 . our experiments begin with an optically - trapped highly degenerate , strongly interacting fermi gas of @xmath0li @xcite . a 50:50 mixture of the two lowest hyperfine states of @xmath0li atoms is confined in an ultrastable co@xmath22 laser trap with a bias magnetic field of 840 g , just above a broad feshbach resonance at @xmath23 g @xcite . at 840 g , the gas is cooled close to the ground state by lowering the trap depth @xmath24 @xcite . then @xmath24 is recompressed to a final trap depth of @xmath25k , which is much larger than the energy per particle of the gas , for the highest energies employed in the experiments . this suppresses evaporation during the time scale of the measurements . the shallow trap yields a low density that suppresses three body loss and heating . the low density also yields a weakly interacting sample when the bias magnetic field is swept to 1200 g , although the scattering length is @xmath26 bohr , as discussed in detail in [ sec : entropy ] . the shape of the trapping potential is that of a gaussian laser beam , with a transverse gaussian profile determined by the spot size and an axial lorentzian profile determined by the rayleigh length . to simplify the calculations of the ideal gas properties in subsequent sections , as well as the theoretical modelling , we take the trap potential to be approximated by a three dimensional gaussian profile , @xmath27 where @xmath28 is the @xmath29 width of trap for each direction . here , we take the zero of energy to be at @xmath30 . when the cold atoms stay in the deepest portion of the optical trap , where @xmath31 , the gaussian potential can be well approximated as a harmonic trap with transverse frequencies @xmath32 , @xmath33 and axial frequency @xmath34 , where @xmath35 here @xmath36 is the @xmath0li mass . at our final trap depth @xmath37 , the measured transverse frequencies are @xmath38 hz and @xmath39 hz . the axial frequency is weakly magnetic field dependent since the total axial frequency has both an optical potential contribution @xmath40 determined by eq . [ eq : trapfreqwelldepth ] and a magnetic potential contribution arising from magnetic field curvature , @xmath41 . the net axial frequency is then @xmath42 . we find @xmath43 hz at 840 g and @xmath44 hz at 1200 g. the total number of atoms is @xmath45 . the corresponding fermi energy @xmath46 and fermi temperature @xmath47 at the trap center for an ideal noninteracting harmonically trapped gas are @xmath48 , where @xmath49 . for our trap conditions , we obtain @xmath50k . using @xmath51 , we can rewrite eq . [ eq : udipole ] as a symmetric effective potential , @xmath52 where @xmath53 is the scaled position vector . here , @xmath54 with @xmath55 . to obtain the anharmonic corrections for the gaussian trap , we expand eq . [ eq : reducedtruepotential ] in a taylor series up to second order in @xmath56 , @xmath57 model - independent energy measurement is based on a virial theorem , which for an ideal gas in a harmonic confining potential @xmath58 yields @xmath59 . since the harmonic potential energy is proportional to the mean square cloud size , measurement of the cloud profile determines the total energy . remarkably , a trapped unitary fermi gas at a broad feshbach resonance obeys the same virial theorem as an ideal gas , although it contains superfluid pairs , noncondensed pairs , and unpaired atoms , all strongly interacting . this has been demonstrated both theoretically and experimentally @xcite . the virial theorem shows that the total energy of the gas at all temperatures can be measured from the cloud profile using @xmath60 where @xmath24 is the trapping potential and @xmath61 is the position vector . . [ eq : energymeas ] can be shown to be valid for any trapping potential @xmath24 and for any spin mixture , without assuming either the local density approximation or harmonic confinement @xcite . using eq . [ eq : truepotentialquartic ] in eq . [ eq : energymeas ] and keeping the lowest order anharmonic corrections , we obtain the energy per atom in terms of the axial mean square size , @xmath62 . \label{eq : energygaussian}\ ] ] here , we have used the local density approximation with a scalar pressure , which ensures that @xmath63 . for the ground state , where the spatial profile is a zero temperature thomas - fermi profile , we have @xmath64 . for energies @xmath65 , where the spatial profile is approximately gaussian , we have @xmath66 . since the anharmonic correction is small at low temperatures where the cloud size is small , we use the gaussian approximation over the whole range of energies explored in our experiments . for the conditions of our experiments , there is no evidence that the local density approximation breaks down for a 50:50 spin mixture . in this case , measurement of the mean square size in any one direction determines the total energy . from eq . [ eq : energygaussian ] , we see that by simply measuring the axial mean square size @xmath67 at 840 g and measuring the axial trap frequency by parametric resonance , we actually measure @xmath68 , the total energy per particle of the strongly interacting fermi gas at 840 g. this determines the total energy per particle in a model - independent way @xcite . the entropy @xmath2 of the strongly interacting gas at 840 g is determined by adiabatically sweeping the bias magnetic field from 840 g to 1200 g , where the gas is weakly interacting @xcite . the entropy @xmath69 of the weakly interacting gas is essentially the entropy of an ideal fermi gas in a harmonic trap , which can be calculated in terms of the mean square axial cloud size @xmath70 measured after the sweep . since the sweep is adiabatic , we have @xmath71 the adiabaticity of the magnetic field sweep is verified by employing a round - trip - sweep : the mean square size of the cloud at 840 g after a round - trip - sweep lasting 2s is found to be within 3% of mean square size of a cloud that remains at 840 g for a hold time of 2s . the nearly unchanged atom number and mean square size proves the sweep does not cause any significant atom loss or heating , which ensures entropy conservation for the sweep . the background heating rate is the same with and without the sweep and increases the mean square size by about 2% over 2s . the mean square size data are corrected by subtracting the increase arising from background heating over the 1 s sweep time @xcite . at 1200 g in our shallow trap , we have @xmath72 , where the fermi wavevector @xmath73 and the s - wave scattering length @xmath74 bohr @xcite . we find that the gas is weakly interacting : for the lowest temperatures attained in our experiments , the gas at 1200 g is a normal fluid that we observe to expand ballistically . we have calculated the ground state mean square size at 1200 g in our gaussian trap , based on a mean - field theory , @xmath75 @xcite , which is close to that of an ideal harmonically trapped gas , @xmath76 . here , @xmath77 is the mean square size corresponding to the fermi energy of an ideal noninteracting fermi gas at magnetic field @xmath78 , which includes the magnetic field dependence of the axial trapping frequency : @xmath79 . we expect that the entropy of the gas at 1200 g is close to that of an ideal gas , except for a mean field shift of the energy . we therefore assume that a reasonable approximation to the entropy is that of an ideal fermi gas , @xmath80 , where @xmath81 is the ground state mean square size of an ideal fermi gas in the gaussian trapping potential of eq . [ eq : reducedtruepotential ] . here , we apply an elementary calculation based on integrating the density of states for the gaussian trap with the entropy per orbital @xmath82 $ ] , where @xmath83 is the ideal fermi gas occupation number at temperature @xmath6 for an orbital of energy @xmath84 . by calculating @xmath85 as a function of the _ difference _ between the finite temperature and ground state mean square cloud sizes , we reduce the error arising from the mean field shift at 1200 g , and ensure that @xmath86 for the ground state . the exact entropy of a weakly interacting gas @xmath69 at 1200 g , @xmath87 , has been calculated using many - body theories @xcite for the gaussian potential of eq . [ eq : reducedtruepotential ] . in the experiments , we determine the value of @xmath88 , where we take @xmath89 , the value measured at our lowest energy at 1200 g by extrapolation to @xmath14 using the sommerfeld expansion for the spatial profile of an ideal gas . this result is close to the theoretical value , @xmath90 . the entropy versus cloud size curve for an ideal noninteracting fermi gas and the exact value for a weakly interacting gas @xmath69 at 1200 g are plotted in fig . [ fig : entropysizeoverlap ] . we find that the entropies @xmath91 and @xmath92 ) , agree within a few percent over most of the energy range we studied , except at the point of lowest measured energy , where they differ by 10% . the results show clearly that the shape of the entropy curve of a weakly interacting fermi gas is nearly identical to that of an ideal gas when the mean field shift of the ground state size is included by referring the mean square cloud size to that of the ground state . so we have to a good approximation , @xmath93 since the corrections to ideal gas behavior are small , the determination of @xmath94 by measuring the axial mean square size @xmath70 relative to the ground state provides an essentially model - independent estimate of the entropy of the strongly interacting gas . sound velocity measurements have been implemented for fermi gases that are nearly in the ground state , from the molecular bec regime to the weakly interacting fermi gas regime @xcite . a sound wave is excited in the sample by using a thin slice of green light that bisects the cigar - shaped cloud . the green light at 532 nm is blue detuned from the 671 nm transition in lithium , creating a knife that locally repels the atoms . the laser knife is pulsed on for 280 @xmath95s , much shorter than typical sound propagation times @xmath96 ms and excites a ripple in the density consisting of low density valleys and high density peaks . after excitation , the density ripple propagates outward along the axial direction @xmath97 . after a variable amount of propagation time , we release the cloud , let it expand , and image destructively . in the strongly interacting regime , we use zero - temperature thomas - fermi profiles for a non - interacting fermi gas to fit the density profiles , and locate the positions of the density valley and peak . by recording the position of the density ripple versus the propagation time , the sound velocity is determined . a detailed discussion of potential sources of systematic error is given by joseph et al . @xcite . for a strongly interacting fermi gas in the unitary limit , the sound velocity @xmath98 at the trap center for the ground state is determined by the fermi velocity of an ideal gas at the trap center , @xmath99 and the universal constant @xmath16 , @xmath100 a precision measurement of the sound speed therefore enables a determination of @xmath16 @xcite . as discussed below , the values of @xmath16 determined from the @xmath4 data and the sound velocity data are in very good agreement . in the experiments , the raw data consists of the measured mean square cloud sizes at 840 g and after an adiabatic sweep of the magnetic field to 1200 g. using this data , we determine both the energy and the entropy of the strongly interacting gas . the data is then compared to several recent predictions . we begin by determining the axial mean square cloud sizes at 840 g and after the adiabatic sweep to 1200 g. since the atom number can vary between different runs by up to 20% , it is important to make the comparison independent of the atom number and trap parameters . for this purpose , the mean square sizes are given in units of @xmath77 , as defined above . the measured mean square sizes are listed in table [ tbl : energyentropy ] . in the experiments , evaporative cooling is used to produce an atom cloud near the ground state . energy is controllably added by releasing the cloud and then recapturing it after a short time @xmath101 as described previously @xcite . for a series of different values of @xmath101 , the energy at 840 g is directly measured from the axial cloud size according to eq . [ eq : energygaussian ] . then the same sequence is repeated , but the cloud size is measured after an adiabatic sweep to 1200 g. in each case , the systematic increase in mean square size arising from background heating rate is determined and subtracted . the total data comprise about 900 individual measurements of the cloud size at 840 g and 900 similar measurements of the cloud size after a sweep to 1200 g. to estimate the measurement error , we split the energy scale at 840 g into bins with a width of @xmath102 . measured data points within the width of the energy bin are used to calculate the average measured values of the cloud sizes and the corresponding standard deviation at both 840 g and 1200 g. ' '' '' & @xmath103&@xmath104&@xmath105&@xmath106&@xmath107&@xmath108 + ' '' '' 1&0.568(4)&0.743(6)&0.548(4)&0.63(8)&0.91(23)&0.97(5 ) + ' '' '' 2&0.612(5)&0.776(13)&0.589(5)&0.99(11)&1.18(22)&1.24(9 ) + ' '' '' 3&0.661(5)&0.803(11)&0.634(5)&1.22(8)&1.36(20)&1.42(7 ) + ' '' '' 4&0.697(9)&0.814(15)&0.667(8)&1.30(10)&1.43(18)&1.49(8 ) + ' '' '' 5&0.74(1)&0.87(4)&0.71(1)&1.6(2)&1.72(18)&1.8(2 ) + ' '' '' 6&0.79(1)&0.89(2)&0.75(1)&1.7(1)&1.79(15)&1.9(1 ) + ' '' '' 7&0.83(1)&0.94(2)&0.79(1)&2.0(1)&2.03(16)&2.1(1 ) + ' '' '' 8&0.89(2)&1.02(2)&0.84(2)&2.3(1)&2.32(18)&2.4(1 ) + ' '' '' 9&0.91(1)&1.02(3)&0.86(1)&2.3(1)&2.31(16)&2.4(1 ) + ' '' '' 10&0.97(1)&1.10(1)&0.91(1)&2.55(4)&2.57(17)&2.64(4 ) + ' '' '' 11&1.01(1)&1.17(2)&0.94(1)&2.74(7)&2.75(19)&2.82(7 ) + ' '' '' 12&1.05(1)&1.18(1)&0.98(1)&2.78(4)&2.80(17)&2.87(4 ) + ' '' '' 13&1.10(1)&1.22(1)&1.03(1)&2.89(2)&2.90(15)&2.97(2 ) + ' '' '' 14&1.25(2)&1.35(5)&1.15(1)&3.21(12)&3.20(14)&3.27(12 ) + ' '' '' 15&1.28(1)&1.39(3)&1.18(1)&3.28(6)&3.28(14)&3.35(6 ) + ' '' '' 16&1.44(2)&1.49(2)&1.31(2)&3.49(4)&3.48(9)&3.55(4 ) + ' '' '' 17&1.53(2)&1.62(6)&1.39(2)&3.74(10)&3.73(11)&3.80(10 ) + ' '' '' 18&1.58(1)&1.63(1)&1.42(1)&3.76(2)&3.74(8)&3.81(2 ) + ' '' '' 19&1.70(2)&1.73(6)&1.52(1)&3.94(9)&3.92(7)&3.99(9 ) + ' '' '' 20&1.83(5)&1.79(2)&1.62(4)&4.03(3)&4.01(1)&4.08(3 ) + ' '' '' 21&1.93(3)&1.96(3)&1.70(2)&4.28(4)&4.26(6)&4.32(4 ) + ' '' '' 22&2.11(5)&2.17(3)&1.83(4)&4.55(3)&4.53(7)&4.59(3 ) + the ratio of the mean square axial cloud size at 1200 g ( measured after the sweep ) to that at 840 g ( measured prior to the sweep ) is plotted in fig . [ fig : sizeratio ] as a function of the energy of a strongly interacting gas at 840 g. the ratio is @xmath109 , since for an adiabatic sweep of the magnetic field from the strongly interacting regime to the weakly interacting regime , the total entropy in the system is conserved but the energy increases : the strongly interacting gas is more attractive than the weakly interacting gas . a similar method was used to measure the potential energy change in a fermi gas of @xmath110k , where the bias magnetic field was adiabatically swept between the strongly interacting regime at the feshbach resonance and a noninteracting regime above resonance @xcite . the resulting potential energy ratios are given as a function of the temperature of the noninteracting gas @xcite . in contrast , by exploiting the virial theorem which holds for the unitary gas , we determine both the energy and entropy of the strongly interacting gas , as described below . [ fig : sizeconvertentropy ] shows the entropy which is obtained from the mean square size at 1200 g @xmath111 as listed in table [ tbl : energyentropy ] . first , we find the mean square size relative to that of the ground state , @xmath112 . we use the measured value @xmath113 for the lowest energy state that we obtained at 1200 g , as determined by extrapolation to @xmath14 using a sommerfeld expansion for the spatial profile of an ideal fermi gas . then we determine the entropy in the noninteracting ideal fermi gas approximation : @xmath114 $ ] , where we have replaced @xmath81 by the ground state value at 1200 g. as discussed above , this method automatically ensures that @xmath9 corresponds to the measured ground state @xmath115 at 1200 g , and compensates for the mean field shift between the measured @xmath116 for a weakly interacting fermi gas and that calculated @xmath117 for an ideal fermi gas in our gaussian trapping potential . as shown in fig . [ fig : entropysizeoverlap ] , the entropy obtained from a more precise many - body calculations is in close agreement with the ideal gas entropy calculated in the ideal gas approximation . the energy is determined from the cloud profiles at 840 g using eq . [ eq : energygaussian ] . finally , we generate the energy - entropy curve for a strongly interacting fermi gas , as shown in fig . [ fig : energyentropy ] . here , the energy @xmath1 measured from the mean square axial cloud size at 840 g is plotted as a function of the entropy @xmath2 measured at 1200 g after an adiabatic sweep of the magnetic field . we note that above @xmath118 ( @xmath119 ) the @xmath4 data ( blue dots ) for the strongly interacting gas appear to merge smoothly to the ideal gas curve ( dashed green ) . for an ideal fermi gas . for this figure , the ideal gas approximation to the entropy is used , @xmath106 of table [ tbl : energyentropy ] . [ fig : energyentropy],width=384 ] in addition to the entropy calculated in the ideal gas approximation , table [ tbl : energyentropy ] also provides a more precise entropy @xmath120 versus the axial mean square cloud size . these results are obtained by hu et al . @xcite using a many - body calculation for @xmath121 at 1200 g in the gaussian trap of eq . [ eq : reducedtruepotential ] . perhaps the most important application of the energy - entropy measurements is to test strong coupling many - body theories and simulations . since the energy and entropy are obtained in absolute units without invoking any specific theoretical model , the data can be used to distinguish recent predictions for a trapped strongly interacting fermi gas . fig . [ fig : comparetheory ] shows how four different predictions compare to the measured energy and entropy data . these include a pseudogap theory @xcite , a combined luttinger - ward - de dominicis - martin ( lw - ddm ) variational formalism @xcite , a t - matrix calculation using a modified nozires and schmitt - rink ( nsr ) approximation @xcite , and a quantum monte carlo simulation @xcite . the most significant deviations appear to occur near the ground state , where the precise determination of the energy seems most difficult . the pseudogap theory predicts a ground state energy that is above the measured value while the prediction of ref . @xcite is somewhat low compared to the measurement . all of the different theories appear to converge at the higher energies . the temperature @xmath6 is determined from the measured @xmath4 data using the fundamental relation , @xmath5 . to implement this method , we need to parameterize the data to obtain a smooth differentiable curve . at low temperatures , one expects the energy to increase from the ground state according to a power law in @xmath6 and a corresponding power law in @xmath2 , i.e. , @xmath122 . for a harmonically trapped ideal fermi gas , we have in the sommerfeld approximation an energy per particle in units of @xmath46 given by @xmath123 . the corresponding entropy per particle in units of @xmath124 is @xmath125 , so that @xmath126 . we attempt to use a single power law to fit the @xmath127 curve for a noninteracting fermi gas in a gaussian trapping potential , with @xmath128 , as in our experiments . the energy and entropy are calculated in the energy range @xmath129 and displayed as dots in fig . [ fig : idealgasfit ] . we find that a single power law @xmath130 fits the curve very well over this energy range . note that the power law exponent is @xmath131 , close to the low temperature value . using the fit function , we can extract the reduced temperature @xmath132 as a function of @xmath85 and compare it to the theoretical reduced temperature @xmath133 at the same @xmath85 . the results are shown as the green dashed line in fig . [ fig : idealgasfit ] . we see that the agreement is quite good except below @xmath134 and above @xmath135 , where the deviation is @xmath136% . to improve the fit and to make a more precise determination of the temperature , we employ a fit function comprising two power laws that are joined at a certain entropy @xmath7 , which gives the best fit . when used to fit the data for the strongly interacting fermi gas , we consider two types of fits that incorporate either a jump in heat capacity or a continuous heat capacity at @xmath7 . in this way , we are able determine the sensitivity of the temperature and critical parameters to the form of the fit function . the two types of fits yield nearly identical temperatures , but different values of @xmath7 and hence of the critical parameters , as discussed below . we take the energy per particle @xmath1 in units of @xmath46 to be given in terms of the entropy per particle in units of @xmath124 in the form @xmath137 we constrain the values of @xmath138 and @xmath3 by demanding that energy and temperature be continuous at the joining point @xmath7 : @xmath139 by construction , the value of @xmath140 does not affect these constraints and is chosen in one of two ways . fixing @xmath141 , the fit incorporates a heat capacity jump at @xmath7 , which arises from the change in the power law exponents at @xmath7 . alternatively , we choose @xmath140 so that the second derivative @xmath142 is continuous at @xmath7 , making the heat capacity continuous . the final fit function has 5 independent parameters @xmath143 , and takes the form @xmath144+e(s - s_c)^2 ; \,\,s\geq s_c . \label{eq : evsslowengtwopower}\end{aligned}\ ] ] here , when @xmath140 is not constrained to be zero , it is given by @xmath145 fig . [ fig : idealgasfit ] shows the improved fit to the calculated energy versus entropy of a noninteracting fermi gas in a gaussian trap for @xmath128 , using eq . [ eq : evsslowengtwopower ] with @xmath146 , since the ideal gas has no heat capacity jump . in this case , both power law exponents @xmath147 and @xmath148 are close to @xmath149 as for the single power law fit . the temperature determined from the fit agrees very closely with the exact temperature , as shown in fig . [ fig : idealgasfit ] ( red solid line ) . in contrast to the noninteracting case , we have found that the energy - entropy data of a strongly interacting fermi gas is not well fit by a single power law function @xcite . however , the two power - law function fits quite well , with a factor of two smaller value of @xmath150 than for the single power - law fit . here , we use @xmath151 , where @xmath152 ( @xmath153 ) is the fitted ( data ) value for the @xmath154 point , and @xmath155 is the corresponding the standard error . motivated by the good fits of the two power - law function to the ideal gas energy versus entropy curve and the good agreement between the fitted and exact temperature , we apply the two power - law fit function to the data for the strongly interacting fermi gas . ) . for comparison , the dot - dashed green curve shows @xmath4 for an ideal fermi gas . for this figure , the ideal gas approximation to the entropy is used , @xmath106 of table [ tbl : energyentropy ] . [ fig : energyentropyfit],width=384 ] fig . [ fig : energyentropyfit ] shows the fit ( red solid curve ) obtained with a heat capacity jump using eq . [ eq : evsslowengtwopower ] with @xmath141 and @xmath156 , @xmath157 , @xmath158 , the ground state energy @xmath159 , and the critical entropy @xmath160 . also shown is the fit ( blue dashed curve ) with continuous heat capacity ( @xmath146 ) and @xmath161 , @xmath162 , @xmath163 , the ground state energy @xmath164 , and the critical entropy @xmath165 . the fit functions for the @xmath4 data for the strongly interacting fermi gas exhibit a significant change in the scaling of @xmath1 with @xmath2 below and above @xmath7 . the dramatic change in the power law exponents for the strongly interacting gas suggests a transition in the thermodynamic properties . the power law exponent is @xmath166 above @xmath7 , comparable to that obtained for the ideal gas , where @xmath167 . the power law exponent below @xmath7 is @xmath168 , which corresponds to the low temperature dependence @xmath169 , close to that obtained in measurements of the heat capacity , where the observed power law was @xmath170 after the model - dependent calibration of the empirical temperature @xcite , see [ sec : ttwiddle ] . if we interpret @xmath7 as the critical entropy for a superfluid - normal fluid transition in the strongly interacting fermi gas , then we can estimate the critical energy @xmath15 and the critical temperature @xmath171 . for the fits of eq . [ eq : evsslowengtwopower ] with a heat capacity jump @xmath172 ) or with continuous heat capacity ( @xmath146 ) , we obtain @xmath173 using the fit parameters in eq . [ eq : criticalparam ] yields critical parameters of the strongly interacting fermi gas , which are summarized in table [ tbl : criticalparam ] . the statistical error estimates are from the fit , and do not include systematic errors arising from the form of the fit function . ' '' '' & @xmath174 & @xmath175 & @xmath176 + ' '' '' expt @xmath4 fit@xmath177 & 2.2(1 ) & 0.83(2 ) & 0.21(1 ) + ' '' '' expt @xmath4 fit@xmath178 & 1.6(3 ) & 0.70(5 ) & 0.185(15 ) + ' '' '' heat capacity experiment@xmath179 & & 0.85 & 0.20 + ' '' '' theory ref . @xcite & & & 0.30 + ' '' '' theory ref . @xcite & & & 0.31 + ' '' '' theory ref . @xcite & & & 0.27 + ' '' '' theory ref . @xcite & & & 0.29 + ' '' '' theory ref . @xcite & 2.15 & 0.82 & 0.27 + ' '' '' theory ref . @xcite & 1.61(5 ) & 0.667(10 ) & 0.214(7 ) + we note that the fit function for @xmath180 previously used in ref . @xcite to determine the temperature was continuous in @xmath2 and @xmath1 , but intentionally ignored the continuous temperature constraint in order to determine the entropy as a power of @xmath181 both above and below the joining energy @xmath15 . as the continuous temperature constraint is a physical requirement , we consider the present estimate of the temperature @xmath6 to be more useful for temperature calibrations and for characterizing the physical properties of the gas than the estimate of ref . @xcite . in contrast to the temperature @xmath6 , the estimate of @xmath8 depends on the value of the joining entropy @xmath7 that optimizes the fit and is more sensitive to the form of fit function than the temperature that is determined from the @xmath1 and @xmath2 data . for the fit function @xmath180 used in ref . @xcite , the temperatures determined by the fit function just above @xmath15 , @xmath182 , and below @xmath15 , @xmath183 , were different . an average of the slopes @xmath184 and @xmath185 was used to estimate the critical temperature . from those fits , the critical energy was found to be @xmath186 , the critical entropy per particle was @xmath187 . the estimated critical temperature obtained from the average was @xmath188 , significantly higher than than the value @xmath189 obtained using eq . [ eq : evsslowengtwopower ] , which incorporates continuous temperature . we are able to substantiate the critical temperature @xmath189 by using our data to experimentally calibrate the temperature scales in two other experiments . in [ sec : onsetpaircond ] , we find that this value is in very good agreement with the estimate we obtain by calibrating the ideal gas temperature observed for the onset of pair condensation . nearly the same transition temperature is obtained in [ sec : ttwiddle ] by using the @xmath4 data to calibrate the empirical transition temperature measured in heat capacity experiments @xcite . table [ tbl : criticalparam ] compares the critical parameters estimated from the power - law fits to the @xmath4 data with the predictions for a trapped unitary fermi gas from several theoretical groups . we note that calculations for a uniform strongly interacting fermi gas at unitarity @xcite yield a lower critical temperature , @xmath190 , than that of the trapped gas , where @xmath191 is the fermi temperature corresponding to the uniform density @xmath192 . extrapolation of the uniform gas critical temperature to that of the trapped gas shows that the results are consistent @xcite . using the parameters from the fits and eq . [ eq : evsslowengtwopower ] , the temperature of the strongly interacting fermi gas , in units of @xmath47 can be determined as a function of the entropy per particle , in units of @xmath124 , @xmath193 here @xmath7 is given in table [ tbl : criticalparam ] from the fits to the @xmath4 data for the strongly interacting gas , eq . [ eq : criticalparam ] gives @xmath8 . [ fig : tempvsentropy ] shows the temperature as a function of entropy according to eq . [ eq : temperaturestrongint ] for fits with a heat capacity jump and for continuous heat capacity . the estimates of the temperature of the strongly interacting fermi gas as a function of the entropy can be used to experimentally calibrate the temperatures measured in other experiments , without invoking any specific theoretical models . the jila group measures the pair condensate fraction in a strongly interacting fermi gas of @xmath110k as a function of the initial temperature @xmath194 in the noninteracting regime above the feshbach resonance @xcite . in these experiments , a downward adiabatic sweep of the bias magnetic field to resonance produces a strongly interacting sample . using our @xmath4 data , we relate the endpoint temperatures for adiabatic sweeps of the bias magnetic field between the ideal and strongly interacting fermi gas regimes . we therefore obtain the critical temperature for the onset of pair condensation in the strongly interacting fermi gas , and find very good agreement with our estimates based on entropy - energy measurement . in addition , we calibrate the empirical temperature based on the cloud profiles , which was employed in our previous measurements of the heat capacity @xcite . we relate the endpoint temperatures for an adiabatic sweep between the strongly interacting and ideal fermi gas regimes . [ eq : temperaturestrongint ] gives the temperature of the strongly interacting gas as a function of entropy , i.e. , @xmath195 . next , we calculate the entropy per particle @xmath196 for an ideal fermi gas in our gaussian trap , in units of @xmath124 , with @xmath197 in units of @xmath47 , as used in [ sec : idealgas ] to determine @xmath198 . for an adiabatic sweep between the strongly interacting and ideal fermi gas regimes , where @xmath199 , the temperature of the strongly interacting gas is related to that for the ideal fermi gas by @xmath200 , \label{eq : adiabactictempcal}\ ] ] which is shown in fig . [ fig : tempsivni ] . + for an adiabatic sweep from the ideal fermi gas regime to the strongly interacting fermi gas regime at low temperature @xmath201 , the reduced temperature of the strongly interacting gas is greater than or equal to that of the ideal gas . this arises because the entropy of the strongly interacting gas scales as a higher power of the temperature than that of the ideal gas . in our present experiments , we could not take data at high enough temperatures to properly characterize the approach of the temperature to the ideal gas regime . above @xmath8 , our @xmath4 data are obtained over a limited range of energies @xmath202 to avoid evaporation in our shallow trap . in this energy range , our data are reasonably well fit by a single power law . however , such a power law fit can not completely describe the higher temperature regime . we expect that the temperatures of the strongly interacting gas and ideal gas must start to merge in the region @xmath203 , where the @xmath4 data for the strongly interacting gas nearly overlaps with the @xmath4 curve for an ideal gas , as shown in fig . [ fig : energyentropy ] . from fig . [ fig : tempvsentropy ] , @xmath204 corresponds to @xmath205 , approximately the place where the calibrations from the two different power law fits ( for @xmath141 and @xmath146 ) begin to differ in fig . [ fig : tempsivni ] . we therefore expect that the single power law fit overestimates the temperature @xmath6 of the strongly interacting gas for @xmath206 , yielding a trend away from ideal gas temperature , in contrast to the expected merging at high temperature . in ref . @xcite , projection experiments measure the ideal fermi gas temperature @xmath194 where pair condensation first appears . in those experiments , @xmath194 is estimated to be @xmath207 @xcite . from the calibration , [ fig : tempsivni ] , we see that for @xmath208 , the corresponding temperature of the strongly interacting gas is @xmath209 for both the red solid and blue dashed curves , which is almost the same as the ideal gas value . the critical temperature of the strongly interacting gas for the onset of pair condensation is then @xmath210 , in very good agreement with the values @xmath211 and @xmath212 that we obtain from the two fits to the @xmath4 measurements . this substantiates the conjecture that the change in the power law behavior observed at @xmath8 in our experiments corresponds to the superfluid transition . in our previous study of the heat capacity , we determined an empirical temperature @xmath213 as a function of the total energy of the gas @xcite . the gas was initially cooled close to the ground state and a known energy was added by a release and recapture method . then a thomas - fermi profile for an ideal fermi gas was fit to the low temperature cloud profiles to determine the fermi radius . holding the fermi radius constant , the best fit to the cloud profiles at higher temperatures determined the effective reduced temperature , which is denoted @xmath214 . the @xmath215 data @xcite was observed to scale as @xmath216 for @xmath217 , while below @xmath218 , the energy was found to scale as @xmath219 . the transition point occurs at an energy @xmath220 , which is close to the value @xmath221 obtained from power - law fit to the @xmath4 data for the fit with a heat capacity jump . assuming that @xmath222 corresponds to the superfluid - normal fluid transition , we can determine the corresponding value of @xmath12 for the strongly interacting gas . to calibrate the empirical temperature we start with @xmath215 . then , as discussed in [ sec : energyvstemp ] , eq . [ eq : evsslowengtwopower ] determines @xmath19 and hence @xmath223from the fits to our @xmath4 data . hence @xmath224 $ ] , where @xmath225 is the reduced temperature of the strongly interacting gas and @xmath226 is the reduced energy . for simplicity , we give the analytic results obtained using the @xmath141 fit to the @xmath4 data , @xmath227 fig . [ fig : tempsivni ] shows the full calibration ( green dashed curve ) . for comparison , the calibration obtained from the pseudogap theory of the cloud profiles gave @xmath228 for @xmath229 , and @xmath230 above @xmath218 . for @xmath222 , we obtain from eq . [ eq : tildetcalib ] @xmath231 ( see fig . [ fig : tempsivni ] ) , in good agreement with the value obtained for the onset of pair condensation and with the values @xmath232 and @xmath233 determined from the fits to the @xmath4 data . measurement of the ground state energy of a unitary fermi gas provides a stringent test of competing many - body theoretical predictions and is therefore of great interest . for a unitary fermi gas of uniform density in a 50 - 50 mixture of two spin states , the ground state energy per particle can be written as @xmath234 where @xmath235 is the local fermi energy corresponding to the density @xmath192 . the ground state energy of the unitary fermi gas differs by a universal factor @xmath236 from that of an ideal fermi gas at the same density . the precise value of @xmath237 has been of particular interest in the context of neutron matter @xcite , and can be measured in unitary fermi gas experiments @xcite . the sound speed at temperatures near the ground state determines @xmath16 according to eq . [ eq : soundres ] . we have made precision measurements of the sound speed in a trapped fermi gas at the feshbach resonance @xcite . at 834 g , we vary the density by a factor of 30 to demonstrate universal scaling and obtain the value @xmath238 . using eq . [ eq : soundres ] then yields @xmath239 . note that the reference fermi velocity @xmath240 depends on the fermi energy of an ideal gas at the trap center and hence on both the trap frequencies and atom number ( as @xmath241 ) , which are carefully measured to minimize systematic errors @xcite . while the energy of the gas as measured from the mean square cloud size was close to the ground state value , the precise temperature of the gas was not determined . the universal parameter @xmath16 also can be determined by measuring the ground state energy @xmath18 of a harmonically trapped unitary fermi gas , which is given by @xmath242 our @xmath4 data enables a new determination of @xmath18 by extrapolating the measured energy @xmath4 to @xmath9 . as pointed out by hui et al . @xcite , this method avoids a systematic error arising when the finite temperature is not determined in the measurements . from both of our fit functions below @xmath7 , we obtain @xmath243 . [ eq : hobeta ] yields @xmath244 . this result is slightly more negative than that obtained in the sound speed experiments , which is reasonable since the sound speed measurements are done at finite temperature . both results are in very good agreement . one possible systematic error in these measurements arises from the determination of the atom number . the measurements of @xmath16 from the sound speed and from the energy - entropy measurements were done in different laboratories . the close agreement is gratifying , considering that the imaging systems that determine the atom number employed @xmath245-polarized light for the sound speed experiments , while the entropy - energy measurements used @xmath246-polarized light , for which the resonant optical cross section is a factor of two smaller than for @xmath245 polarization . to examine the systematic error arising from the atom number determination , we employ a third method to measure @xmath16 based on the measured ratio of the cloud size at 840 g and at 1200 g , which is number independent . the ratio of the ground state mean square sizes for the weakly and strongly interacting gases is predicted to be @xmath247 note that we obtain @xmath248 from a mean field calculation @xcite , in agreement with that obtained using a many - body calculation @xcite . our measurements for the ground state mean square size at 1200 g are accomplished by fitting a sommerfeld expansion of the axial density for an ideal fermi gas to the cloud profile @xcite . the fit determines the fermi radius @xmath249 and reduced temperature @xmath133 , yielding @xmath250 for @xmath14 , close to the predicted value of @xmath251 . the ground state energy @xmath252 at 840 g from the entropy - energy experiments determines the ground state mean square size as @xmath253 . hence , @xmath254 . the corresponding @xmath255 from eq . [ eq : groundstateratio ] . since the mean square sizes are determined from the images and the ratio @xmath256 is number independent , this result shows that the systematic error arising from the number measurement is within the quoted error estimate . we also can determine @xmath16 by directly extrapolating to zero entropy the ratio of the axial mean square size of the weakly interacting fermi gas at 1200 g to that of strongly interacting gas at 840 g. when this is done , we obtain @xmath257 , in very good agreement with the estimates based on the sound speed and ground state energy . finally , we can estimate the correction to the ground state energy arising from the finite scattering length at 840 g , @xmath258 . for the trap conditions in the @xmath4 measurements , @xmath259 , where @xmath260 is the wavevector for an ideal fermi gas at the trap center . to estimate the true unitary ground state energy at @xmath261 , we first determine the leading order @xmath262 correction to the trapped atom density , where @xmath263 is the local fermi wavevector corresponding to the density @xmath192 . the local chemical potential is estimated from ref . @xcite . using the notation of eq . [ eq : reducedtruepotential ] and a harmonic approximation , the corrected density yields @xmath264 $ ] , where @xmath265 is the fermi radius for the unitary gas . according to the virial theorem ( see eq . [ eq : energymeas ] ) , the mean square size and energy of the unitary gas are corrected by the same factor . the unitary ground state energy is then @xmath266 for @xmath267 , we obtain @xmath268 and the value of @xmath269 obtained directly from @xmath270 is shifted to @xmath271 . we also obtain the corrected value of @xmath272 in eq . [ eq : groundstateratio ] and @xmath273 . table [ tbl : beta ] compares the values of @xmath16 obtained in our experiments to several recent predictions . note that the table does not include the finite @xmath274 correction for the @xmath4 measurement at 840 g described above . ' '' '' & @xmath16 + ' '' '' @xmath4 experiment & -0.59(2 ) + ' '' '' sound velocity experiment & -0.565(15 ) + ' '' '' cloud size ratio experiment & -0.61(2 ) + ' '' '' ref . @xcite & -0.58(1 ) + ' '' '' ref . @xcite & @xmath275 + ' '' '' ref . @xcite & -0.599 + ' '' '' ref . @xcite & -0.60(1 ) + ' '' '' ref . @xcite & -0.646(4 ) + using the @xmath4 data for the strongly interacting fermi gas and the temperature determined from the two power - law fits , we estimate several universal functions . first , we determine the dependence of the energy on temperature @xmath19 and the corresponding heat capacity , @xmath20 . then we find the global chemical potential of the trapped gas as a function of the energy @xmath21 . the energy is readily determined as a function of temperature using eq . [ eq : evsslowengtwopower ] for the case where there is a heat capacity jump and @xmath141 , @xmath276;\,\,\,0\leq t\leq t_c\nonumber\\ e_>(t)&=&e_c+{\frac{s_ct_c}{d}}\left[\left(\frac{t}{t_c}\right)^{\frac{d}{d-1}}-1\right ] ; \,\,\,t\geq t_c , \label{eq : energyvstemp}\end{aligned}\ ] ] where the energy ( temperature ) is given in units of @xmath46 ( @xmath47 ) and the critical energy @xmath15 is @xmath277 with @xmath18 the ground state energy . for the case with @xmath146 , where the heat capacity is continuous , we determine the ordered pairs @xmath278 $ ] as a function of @xmath2 and plot @xmath19 . fig . [ fig : energyvstemp ] shows the results using the best fits for both cases . of particular interest is the low temperature power law . for @xmath141 , we obtain @xmath279 and @xmath280 . since @xmath147 is near @xmath281 , the energy relative to the ground state scales approximately as @xmath282 . this is consistent with sound modes dominating the low energy excitations . however , one would expect instead that the free fermions on the edges of the trapped cloud would make an important contribution to the low energy excitations @xcite . over an extended range of @xmath201 , the net entropy arising from the bose and fermi excitations has been predicted to scale as @xmath283 , yielding an energy scaling @xcite as @xmath284 . in this case , one would expect that @xmath285 , i.e. , @xmath286 in eq . [ eq : energyvstemp ] , so that @xmath287 . hence , the low energy power law exponents for the entropy should be between @xmath281 and @xmath288 , which is barely distinguishable for our data . the heat capacity at constant trap depth @xmath289 is readily obtained from eq . [ eq : energyvstemp ] ( where there is a heat capacity jump , since we have constrained @xmath141 in eq . [ eq : evsslowengtwopower ] ) . for this parameterization , @xmath290 where @xmath6 and @xmath8 are given in units of @xmath47 , and @xmath7 is given in units of @xmath124 . for the fit with a continuous heat capacity , we use @xmath195 to find @xmath291 , and plot the ordered pairs @xmath292 $ ] . the heat capacity curves for both cases are shown in fig . [ fig : cvst ] . according to eq . [ eq : heatcapacity ] , a jump in heat capacity occurs at @xmath7 : @xmath293 and @xmath294 differ when the power law exponents @xmath147 and @xmath148 are different . this is a consequence of the simple two power - law structure assumed for the fit function @xmath4 given by eq . [ eq : evsslowengtwopower ] for @xmath141 , and can not be taken as proof of a true heat capacity jump . at present , the precise nature of the behavior near the critical temperature can not be determined from our data , and it remains an open question whether the data exhibits a heat capacity jump or a continuous heat capacity . the global chemical potential @xmath295 is readily determined from the fits to the @xmath4 data for a strongly interacting fermi gas , which obeys universal thermodynamics . the local energy density generally takes the form @xmath296 , where @xmath297 is the local internal energy , which includes the kinetic energy and the interaction energy . here , @xmath192 is the local density , @xmath95 is the local chemical potential , @xmath298 is the pressure and @xmath299 is the total entropy per unit volume . the local chemical potential can be written as @xmath300 , where @xmath24 is the trap potential . in the universal regime , where the local pressure depends only on the local density and temperature , we have @xmath301 , as noted by ho @xcite . hence , @xmath302 . integrating both sides over the trap volume and using @xmath303 , where @xmath1 and @xmath304 ) are the total energy and average potential energy per particle , respectively , we obtain @xmath305 where @xmath2 is the entropy per particle . for simplicity , we assume harmonic confinement and use the virial theorem result , @xmath306 , from eq . [ eq : energymeas ] , which holds in the universal regime . then , eq . [ eq : globalchempot1 ] yields the global chemical potential of a harmonically trapped fermi gas in the universal regime , @xmath307 by using the fit to the measured entropy - energy data to obtain the temperature @xmath5 from eq . [ eq : evsslowengtwopower ] , the global chemical potential of a trapped unitary fermi gas can be calculated from eq . [ eq : globalchempot ] . for @xmath141 , where the heat capacity has a jump , the simple power law fits above and below @xmath15 each yield a different linear dependence of @xmath295 on @xmath1 , @xmath308 where @xmath309 . we plot the chemical potential in fig . [ fig : chemenergy ] . the data points are obtained using eq . [ eq : globalchempot ] with the measured energy @xmath1 and entropy @xmath2 and the temperature determined from the fit to the @xmath4 data , using @xmath141 in eq . [ eq : evsslowengtwopower ] , i.e. , with a heat capacity jump . the solid red curve is given by eq . [ eq : globalchempotvse ] . we note that the low temperature data points in @xmath4 are best fit with the power law @xmath279 , which is close to @xmath281 . according to eq . [ eq : globalchempotvse ] , this produces a slope near zero for @xmath310 . since the power - law fit above @xmath15 gives @xmath311 , the slope according to eq . [ eq : globalchempotvse ] changes from nearly zero for @xmath310 to negative for @xmath312 . note that from eq . [ eq : globalchempot ] , we obtain the slope @xmath313 since the entropy @xmath2 is continuous , we see that a jump in the heat capacity produces a corresponding jump in the slope of @xmath295 versus @xmath1 . for comparison , fig . [ fig : chemenergy ] also shows the chemical potential obtained for @xmath146 in eq . [ eq : evsslowengtwopower ] , where the heat capacity is continuous ( blue dashed curve ) . we have studied the thermodynamic properties of a strongly interacting fermi gas by measuring both the energy and the entropy . the model - independent data obtained in both the superfluid and the normal fluid regimes do not employ any specific theoretical calibrations , and therefore can be used as a benchmark to test the predictions from many - body theories and simulations . parameterizing the energy - entropy data determines the temperature of the strongly interacting fermi gas and also yields estimates of the critical parameters . we use the measured data to calibrate two different temperature scales that were employed in observations of the onset of pair condensation and in heat capacity studies . these calibrations yield critical temperatures in good agreement with the results estimated from our energy - entropy data . our data does not determine whether the heat capacity exhibits a jump or is continuous at the critical temperature . however , for a finite system with nonuniform density , the latter is most likely . considering that there is huge interest in determining the detailed behavior of the superfluid transition in a strongly interacting fermi gas @xcite , more precise determinations of the critical temperature , the heat capacity , and the chemical potential near the critical point , as well as the high temperature behavior and the approach to the ideal gas limit , will be important topics for future research . this research has been supported by the physics divisions of the army research office and the national science foundation , the physics for exploration program of the national aeronautics and space administration , and the chemical sciences , geosciences and biosciences division of the office of basic energy sciences , office of science , u. s. department of energy . we thank willie ong for a careful reading of the manuscript .
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strongly interacting fermi gases provide a clean and controllable laboratory system for modeling strong interparticle interactions between fermions in nature , from high temperature superconductors to neutron matter and quark - gluon plasmas .
model - independent thermodynamic measurements , which do not require theoretical models for calibrations , are very important for exploring this important system experimentally , as they enable direct tests of predictions based on the best current non - perturbative many - body theories . at duke university , we use all - optical methods to produce a strongly interacting fermi gas of spin-1/2-up and spin-1/2-down @xmath0li atoms that is magnetically tuned near a collisional ( feshbach ) resonance .
we conduct a series of measurements on the thermodynamic properties of this unique quantum gas , including the energy @xmath1 , entropy @xmath2 , and sound velocity @xmath3 .
our model - independent measurements of @xmath1 and @xmath2 enable a precision study of the finite temperature thermodynamics .
the @xmath4 data are directly compared to several recent predictions .
the temperature in both the superfluid and normal fluid regime is obtained from the fundamental thermodynamic relation @xmath5 by parameterizing the @xmath4 data using two different power laws that are joined with continuous @xmath1 and @xmath6 at a certain entropy @xmath7 , where the fit is optimized .
we observe a significant change in the scaling of @xmath1 with @xmath2 above and below @xmath7 . taking the fitted value of @xmath7 as an estimate of the critical entropy for a superfluid - normal fluid phase transition in the strongly interacting fermi gas
, we estimate the critical parameters .
our @xmath4 data are also used to experimentally calibrate the endpoint temperatures obtained for adiabatic sweeps of the magnetic field between the ideal and strongly interacting regimes .
this enables the first experimental calibration of the temperature scale used in experiments on fermionic pair condensation , where the ideal fermi gas temperature is measured before sweeping the magnetic field to the strongly interacting regime .
our calibration shows that the ideal gas temperature measured for the onset of pair condensation corresponds closely to the critical temperature @xmath8 estimated in the strongly interacting regime from the fits to our @xmath4 data .
we also calibrate the empirical temperature employed in studies of the heat capacity and obtain nearly the same @xmath8 .
we determine the ground state energy by three different methods , using sound velocity measurements , by extrapolating @xmath4 to @xmath9 and by measuring the ratio of the cloud sizes in the strongly and weakly interacting regimes .
the results are in very good agreement with recent predictions . finally , using universal thermodynamic relations , we estimate the chemical potential and heat capacity of the trapped gas from the @xmath4 data .
pacs numbers : 03.75.ss
| 17,424 | 709 |
recent progress in our understanding of faint galaxy data made possible by the combination of _ hubble space telescope _ ( _ hst _ ) deep imaging and ground - based spectroscopy has dramatically increased our knowledge of the evolution of the stellar birthrate in optically - selected galaxies from the present - epoch up to @xmath9 @xcite , @xcite , @xcite . the explosion in the quantity of information available on the high - redshift universe at optical wavelengths has been complemented by the measurement of the far - ir / sub - mm background by dirbe and firas onboard the _ cobe _ satellite @xcite , @xcite , @xcite , by the detection of distant ultraluminous sub - mm sources with the scuba camera @xcite , @xcite , and by theoretical progress made in understanding how intergalactic gas follows the dynamics dictated by dark matter halos until radiative , hydrodynamic , and star formation processes take over @xcite , @xcite , @xcite . the ir data have revealed the ` optically - hidden ' side of galaxy formation , and shown that a significant fraction of the energy released by stellar nucleosynthesis is re - emitted as thermal radiation by dust @xcite , @xcite . the underlying goal of all these efforts is to understand the growth of cosmic structures , the internal properties of galaxies and their evolution , and ultimately to map the star formation history of the universe from the end of the cosmic ` dark age ' to the present epoch . in this talk i will focus on the galaxy number - apparent magnitude relation and its first moment , the integrated galaxy contribution to the extragalactic background light ( ebl ) . the logarithmic slope of the differential galaxy counts is a remarkably simple cosmological probe of the history of stellar birth in galaxies , as it must drop below 0.4 to yield a finite value for the ebl . the recently released _ hubble deep field - south _ ( hdf - s ) images , together with other existing _ hst _ and ground - based observations , provide a unique dataset to estimate the spectrum and amplitude of the optical ebl from discrete sources . together with the far - ir / sub - mm background , the optical ebl is an indicator of the total luminosity of the universe , as the cumulative emission from young and evolved galactic systems , as well as from active galactic nuclei ( agns ) , is recorded in this background . as such it provides , for a given initial mass function , a quantitative estimate of the baryonic mass that has been processed by stars throughout cosmic history . unless otherwise stated , an einstein - de sitter ( eds ) cosmology ( @xmath10 , @xmath11 ) with @xmath12 will be adopted in this talk . all magnitudes will be given in the ab system . the work presented here has been done in collaboration with l. pozzetti . the hdf - s dataset includes deep near - ir nicmos images and the deepest observation ever made with the stis 50ccd filterless imaging mode . the galaxy sample used here was extracted from version 1 of the hdf - s catalog on ftp://archive . stsci.edu / pub / hdf_south / version1/. at near - ir wavelengths ( in the f110w , f160w , and f222 m bandpasses , corresponding to the @xmath13 , @xmath14 , and @xmath15 filters ) , it consists of 425 objects detected in the @xmath16 image , over a field of @xmath17 . the 50ccd ( corresponding roughly to a @xmath18 filter ) stis catalog consists of 674 objects detected again over a field of the same size . galaxy counts as a function of ab magnitudes . the sources of the data points are given in the text . note the decrease of the logarithmic slope @xmath19 at faint magnitudes . _ right : _ extragalactic background light per magnitude bin , @xmath20 , as a function of @xmath21 ( _ filled circles _ ) , @xmath22 ( _ open circles _ ) , @xmath23 ( _ filled pentagons _ ) , @xmath24 ( _ open squares _ ) , @xmath13 ( _ filled triangles _ ) , @xmath14 ( _ open triangles _ ) , and @xmath15 ( _ filled squares _ ) magnitudes . for clarity , the @xmath25 measurements have been multiplied by a factor of 2 , 6 , 15 , 50 , 150 , and 600 , respectively @xcite . , scaledwidth=90.0% ] figure 1 shows the hdf - n and -s galaxy counts compiled directly from the catalogs , as a function of ab isophotal magnitudes in the @xmath1 bandpasses for all galaxies with signal - to - noise ratio @xmath26 within the band . no correction for detection completeness have been made . a compilation of existing _ hst _ and ground - based data is also shown @xcite , @xcite . all magnitudes have been corrected to the ab system , while the second order colour corrections for the differences in the filter effective wavelengths have not been applied to the ground - based data ( for the typical colours of galaxies in the hdf these corrections are less than 0.1 mag ) . the hdf optical counts agree well with previous surveys , to within @xmath27 in the magnitude range @xmath28 . one should note , however , that different algorithms used for ` growing ' the photometry beyond the outer isophotes of galaxies can significantly change the magnitude of faint galaxies . according to @xcite , roughly 50% of the flux from resolved galaxies with @xmath29 mag lie outside the standard - sized apertures used by photometric packages . an extragalactic sky pedestal created by the overlapping wings of resolved galaxies may also contribute significantly to the sky level , and would be undetectable except by absolute surface photometry @xcite . also , at faint magnitude levels , distant objects which are brighter than the nominal depth of the catalog may be missed due to the @xmath30 dimming factor . all these systematic errors are inherent in _ hst _ faint - galaxy photometry ; as a result , our estimate of the integrated flux from resolved galaxies will typically be too low , and must be strictly considered as a _ lower limit_. the contribution of known galaxies to the optical ebl can be calculated directly by integrating the emitted flux times the differential number counts down to the detection threshold . i have used the compilation of ground - based , _ hst _ , and hdf data shown in figure 1 to compute the integrated flux at @xmath31 m . in all seven bands , the slope of the differential number - magnitude relation is flatter than 0.4 above @xmath32 ( 25 ) at near - ir ( optical ) wavelengths , and this flattening appears to be more pronounced at the shorter wavelengths . the leveling off of the counts is clearly seen in figure 1 , where the function @xmath20 is plotted against apparent magnitude in all bands . while counts having a logarithmic slope @xmath33 continue to add to the ebl at the faintest magnitudes , it appears that the hdf survey has achieved the sensitivity to capture the bulk of the near - ultraviolet , optical , and near - ir extragalactic light from discrete sources . the flattening at faint apparent magnitudes can not be due to the reddening of distant sources as their lyman break gets redshifted into the blue passband , since the fraction of lyman - break galaxies at ( say ) @xmath34 is only of order 10% @xcite . moreover , an absorption - induced loss of sources can not explain the similar change of slope in the @xmath35 and @xmath15 bands . while this suggests that the surface density of optically luminous galaxies is leveling off beyond @xmath36 , one should worry about the possibility of a significant amount of light being missed at faint magnitudes . bands ( _ filled dots _ ) , together with the firas 1255000 @xmath37 m ( _ solid and dashed lines _ ) and dirbe 140 and 240 @xmath37 m ( _ filled squares _ ) detections @xcite , @xcite . the _ empty squares _ show the dirbe points after correction for wim dust emission @xcite . also plotted ( _ filled triangle _ ) is a foca - uv point at 2000 @xcite , and a tentative detection at 3.5 @xmath37 m ( _ empty dot _ ) from _ cobe_/dirbe observations @xcite . the empty pentagons at 3000 , 5500 , and 8000 are recent detections from absolute photometry @xcite . upper limits are from @xcite , the lower limit from @xcite . note that the values obtained by integrating the brightness of resolved galaxies are strict lower limits to the ebl intensity . ] the spectrum of the optical ebl is shown in figure 2 , together with the recent results from _ cobe_. the value derived by integrating the galaxy counts down to very faint magnitude levels ( because of the flattening of the number - magnitude relation most of the contribution to the optical ebl comes from relatively bright galaxies ) implies a lower limit to the ebl intensity in the 0.22.2 @xmath37 m interval of @xmath38 . including the tentative detection at 3.5 @xmath37 m by @xcite would boost @xmath39 to @xmath40 . recent estimates of the optical ebl at 3000 , 5500 , and 8000 from absolute surface photometry by @xcite lie between a factor of two to three higher than the integrated light from galaxy counts . applying this correction factor to the range 30008000 gives an optical ebl intensity in excess of @xmath41 in the interval 0.23.5 @xmath37 cobe_/firas @xcite measurements yield @xmath42 in the 1252000 @xmath37 m range . when combined with the dirbe @xcite , @xcite , @xcite points at 140 and 240 @xmath37 m , one gets a far - ir background intensity of @xmath43 . the residual emission in the 3.5 to 140 @xmath37 m region is poorly known , but it is likely to exceed @xmath44 @xcite . a ` best - guess ' estimate to the total ebl intensity observed today is then @xmath45 in the rest of my talk , i will adopt a reference value for the background light associated with star formation activity over the entire history of the universe of @xmath46 . a direct estimate of the contribution of quasars to the ebl depends on the poorly known bolometric correction and the possible existence of a distant population of dusty agns with strong intrinsic absorption , as invoked in many models for the x - ray background . these type ii qsos , while undetected at optical wavelengths , could contribute significantly to the far - ir background . it is in principle possible to bypass some of the above uncertainties by weighing the local mass density of black holes remnants @xcite . recent dynamical evidence indicates that supermassive black holes reside at the center of most nearby galaxies . the available data ( see fig . 3 ) show a correlation ( but with a large scatter ) between bulge and black hole mass , with @xmath47 as a best - fit @xcite . the mass density in old spheroidal populations today is estimated to be @xmath48 @xcite , implying a mean mass density of quasar remnants today @xmath49 since the observed energy density from all quasars is equal to the emitted energy divided by the average quasar redshift , the total contribution to the ebl from accretion onto black holes is @xmath50 ( @xmath51 ) , where @xmath52 is the efficiency of accreted mass to radiation conversion ( in units of 5% ) . therefore , unless dust - obscured accretion onto supermassive black holes is a very efficient process ( @xmath53 ) , a population of quasars peaking at @xmath54 is expected to make a contribution to the brightness of the night sky not exceeding 1020% @xcite , @xcite . . the symbols denote different galaxy types : _ empty circles _ ( e ) , _ filled squares _ ( s0 ) , _ filled circles _ ( sab ) , and _ empty squares _ ( sbc - scd ) . ] with the help of some simple stellar population synthesis tools we can now set a lower limit to the total stellar mass density that produced the observed sky brightness and constrain the cosmic history of star birth in galaxies . one of the most serious uncertainties in this calculation has always been the lower cutoff ( usually treated as a free parameter ) of the initial mass function ( imf ) . observations of m subdwarfs stars with the _ hst _ have recently shed some light on this issue , showing that the imf in the galactic disk can be represented analytically over the mass range @xmath55 ( here @xmath56 is in solar units ) by @xmath57 ( @xcite , hereafter gbf ) . for @xmath58 this mass distribution agrees well with a salpeter function . observations of normal galactic star - forming regions also show some convergence in the basic form of the imf at intermediate and high masses , a power - law slope that is consistent with the salpeter value @xcite . in the following i will use a ` universal ' imf with the gbf form for @xmath59 , matched to a salpeter slope for @xmath60 ; the mass integral of this function is 0.6 times that obtained extrapolating a salpeter function down to @xmath61 . is quite small for a ` typical ' imf . the use of the gbf mass function at low masses instead of salpeter leaves then the total radiated luminosity of the stellar population virtually unaffected . ] , metallicity @xmath62 ( _ solid line _ ) and @xmath63 ( _ dotted line _ ) , and a gbf@xmath5salpeter imf ( see text for details ) . _ right : _ ebl observed at earth from the instantaneous formation at redshift @xmath64 of a stellar population having the same imf , solar metallicity , and mass density @xmath65 and 0.00075 , as a function of @xmath64 . _ solid curves : _ eds universe with @xmath51 . _ dashed curves : _ @xmath66-dominated universe with @xmath67 , @xmath68 , and @xmath69 . , scaledwidth=90.0% ] as shown in figure 4 , the _ bolometric _ luminosity as a function of age @xmath70 of a simple stellar population ( a single generation of coeval , chemically homogeneous stars having total mass @xmath71 , solar metallicity , and the above imf ) can be well approximated by @xmath72 ( cf @xcite ) . over a timescale of 13 gyr ( the age of the universe for an eds cosmology with @xmath51 ) , about 1.3 mev per stellar baryon are radiated away . this number depends only weakly on the assumed metallicity of stars . in a stellar system with arbitrary star formation rate per comoving cosmological volume , @xmath73 , and formation epoch @xmath74 , the comoving bolometric emissivity at time @xmath75 is given by the convolution integral @xmath76 the total background light observed at earth ( @xmath77 ) is @xmath78 where the factor @xmath79 at the denominator is lost to cosmic expansion when converting from observed to radiated ( comoving ) luminosity density . to set a lower limit to the present - day mass density , @xmath7 , of processed gas @xmath5 stars ( in units of the critical density @xmath80 ) , consider now a scenario where all stars are formed instantaneously at redshift @xmath64 . the background light that would be observed at earth from such an event is shown in figure 4 as a function of @xmath64 for @xmath81 ( corresponding to 9 , 6.5 , and 4 percent of the nucleosynthetic baryon density , @xmath82 @xcite ) , and two different cosmologies . a couple of points are worth noting here : ( 1 ) the time evolution of the luminosity radiated by a simple stellar population makes the dependence of the observed ebl from @xmath64 much shallower than the @xmath83 lost to cosmic expansion , as the energy output from stars is spread over their respective lifetimes ; and ( 2 ) in order to generate an ebl at a level of @xmath6 , one requires @xmath84 ( for an eds universe with @xmath51 ) , hence a mean mass - to - blue light ratio today of @xmath85 for a present - day blue luminosity density of @xmath86 @xcite . as shown in figure 4 , the dependence of these estimates on the cosmological model is rather weak . with the adopted imf , about 30% of this mass will be returned to the interstellar medium in @xmath87 yr after intermediate - mass stars eject their envelopes and massive stars explode as supernovae . this ` return fraction ' , @xmath88 , becomes 50% after about 10 gyr . a visible mass density at the level of the above lower limit , @xmath89 , while able to explain the measured sky brightness , requires all the stars that give origin to the observed light to have formed at @xmath90 , and is , as such , rather implausible . a more realistic scenario appears to be one where the star formation density evolves as @xmath91 this model fits reasonably well all measurements of the uv - continuum and h@xmath92 luminosity densities from the present - epoch to @xmath93 after an extinction correction of @xmath94 mag ( @xmath95 mag ) is applied to the data @xcite , and produce a total ebl of the right magnitude ( @xmath96 ) . since about half of the present - day stars are formed at @xmath97 ( hence their contribution to the ebl is redshifted away ) , the resulting visible mass density is @xmath98 ( @xmath99 ) . note that this estimate ignores the recycling of returned gas into new stars . the observed ebl therefore requires that between 7% and 16% of the nucleosynthetic baryons are today in the forms of stars , processed gas , and their remnants . according to the most recent census of cosmic baryons , the mass density in stars and their remnants observed today is @xmath100 @xcite , corresponding to a mean visible mass - to - blue light ratio of @xmath101 ( @xmath51 ) ( about 70% of this mass is found in old spheroidal populations ) . while this is about a factor of 2.5 smaller than the visible mass density predicted by equation ( [ eq : mplot ] ) , efficient recycling of ejected material into new star formation would tend to reduce the apparent discrepancy in the budget . alternatively , the gas returned by stars may be ejected into the intergalactic medium . with an imf - averaged yield of returned metals of @xmath102,^{-1}$ ] , the stellar yields @xmath103 of @xcite , and a gbf@xmath5salpeter imf . ] the predicted mean metallicity at the present epoch is @xmath104 , in good agreement with the values inferred from cluster abundances @xcite .
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i review the constraints imposed by the observed extragalactic background light ( ebl ) on the history of the stellar birthrate in galaxies . at faint magnitudes ,
the logarithmic slope of the galaxy counts is flatter than @xmath0 in all seven @xmath1 optical bandpasses of the _ hubble deep field - south _ imaging survey .
the integration of the number counts provides a lower limit to the surface brightness of the optical extragalactic sky of @xmath2 , comparable to the intensity of the far - ir background from _ cobe _ data .
if the initial mass function has a salpeter slope with a lower mass cutoff consistent with observations of m subdwarf disk stars , a lower limit of @xmath3 ( at hubble constant @xmath4 ) is derived for the visible ( processed gas @xmath5 stars ) mass density needed to generate an extragalactic background light ( ebl ) at a level of @xmath6 .
the current ` best - guess ' estimate to @xmath7 is @xmath8 , about 16% of the nucleosynthetic baryon density .
the contribution of quasar activity to the observed ebl is unlikely to exceed 20% .
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chemical evolution models ( cem ) @xcite were early developed to try to understand the origin of the radial gradients of abundances , observed in our galaxy ( mwg ) . most numerical models in the literature , including the multiphase model used in this work , explain the existence of this radial gradient by the combined effects of a star formation rate ( sfr ) and an infall of gas which vary with galactocentric radius in the galaxy . a radial decrease of abundances has also been observed in most spiral galaxies @xcite although the shape of the radial distribution changes from galaxy to galaxy . among other global trends it is found that for isolated non - barred spirals the steepness of the radial gradient depends on morphological type , with later types showing steeper gradients @xcite , with other general galaxy properties as surface brightness and neutral and molecular gas fractions also playing a role @xcite . the radial gradient tends to be wiped out however for strongly barred galaxies which show flat abundance distributions @xcite . irregulars galaxies also show uniform abundances throughout @xcite . the abundance gradient pattern seems to show an on - off mode @xcite , being very steep for the latest spiral types and very flat for irregulars . all these considerations become clear when the gradient is measured in dex / kpc , but there are indications that suggest a gradient independent of galaxy type when it is measured in dex / scale length @xcite . in order to analyze the behaviour of the radial distribution of abundances and the value of the radial gradient from a theoretical point of view a large number of models is necessary . historically , cem aiming to reproduce radial abundance gradients have been , however , applied only to the mwg . actually , there is a lack of tools to determine the chemical evolutionary state of a particular galaxy , besides our works applying the multiphase models to spiral galaxies . the recent works by @xcite are valid for galaxies other than the mwg . their calculations use the angular momentum and rotation curves as model inputs keeping the star formation efficiency constant for all galaxies @xcite . this technique may not be flexible enough to validate the models against observational data . in fact , a comparison to see if these models reproduce the observed abundance radial distributions of particular galaxies has not been done . it is always possible to extract some information by using evolutionary synthesis models in comparison with spectro - photometric observations . this method , very useful for the study of elliptical galaxies , does not result equally successful in the case of spiral galaxies due to the difficulty of measuring the spectral indices , except for the bulges @xcite , from which ages and metallicities are obtained . furthermore , even when these measurements are done with confidence @xcite , in order to apply this technique to spiral galaxies , a combination of chemical evolution and evolutionary synthesis models is required to solve the uniqueness problem associated to the first ones and the age - metallicity degeneracy associated to the second ones @xcite . at present , the available options are either to use the classical closed box model or a galactic chemical evolution ( gce ) model . however , the closed box scenario is recognised to be inadequate to describe the evolution of most galaxies and in fact its application in many cases can yield misleading results @xcite . in particular , the fact of assuming that a system has a constant total mass with a monotonically decreasing star formation according to a schmidt law , prevents the reproduction of the observational characteristics of most galaxies . on the other hand , the evolution of a galaxy with present time properties different from the milky way will not necessarily be equal to that predicted by a gce model . realistic chemical evolution models adequate to describe different types of spiral and irregular galaxies are therefore clearly needed . the multiphase model , whose characteristics have been described in @xcite , has been applied and checked against observational constraints not only for the milky way galaxy @xcite , as it is commonly done , but also for a sample of spiral galaxies ( discs and bulges ) of different morphological types and total masses @xcite . the observed radial distributions of gas , oxygen abundances and star formation rate have been reproduced rather successfully and the observed correlations between abundance gradients and galaxy characteristics are also reproduced @xcite . this galaxy sample , which includes the best studied objects , is however small ( only 11 ) and encompasses a restricted range of morphologies and masses . the application of the model can however be extended to a larger sample if an adequate parameter space is defined thus providing the required chemical evolution of different types of galaxies . the model uses as input parameters the collapse time scale to form the disc , which depends on the total mass of the galaxy , and the efficiencies to form molecular clouds and stars which we assume different from galaxy to galaxy . the radial distributions of total mass constitute the fundamental input of the multiphase model . they are easily computed when the rotation curves are available ( moll & mrquez , in preparation ) . if this is not the case , some assumptions are necessary . in this work , we have used the universal rotation curve from ( * ? ? ? * hereafter pss96 ) to calculate a large number of mass radial distributions representing theoretical protogalaxies or initial structures which will evolve to form the observed spiral discs or irregulars . the total mass of each simulated galaxy , besides having its own effect on the galaxy evolution , defines the characteristic collapse time - scale or gas infall rate onto the disc . regarding molecular cloud and star formation efficiencies , which will take values between 0 and 1 , we have chosen 10 different sets of values for each radial distribution of total mass . we have computed how the chemical evolution proceeds for galaxies defined by the different parameter combinations . the final bi - parametric grid consists of 440 models simulating galaxies of 44 different total masses . this work represents an extension of our previous work that can help to understand the general trends observed in spiral and irregular galaxies concerning neutral and molecular gas distributions , abundance radial distributions etc . but , most importantly , by using these models we can predict the time evolution of a galaxy when only present time data are known . in section 2 we summarize the general characteristics of the multiphase chemical evolution model and the strategy of its application to the objects of the grid . the results are presented in section 3 including the time evolution of galaxies and some radial distributions for the present time . in section 4 we give the calibration of the grid models with the mwg and a restricted sample of well studied spiral galaxies and we discuss the model results in a global way . finally , the conclusions of this work are presented in section 5 . the results obtained in this work will be available in electronic form at cds via anonymous ftp to cdsarc.u-strasbg.fr ( 130.79.128.5 ) , via http://cdsweb.u-strasbg.fr/abstract.html , or at http://wwwae.ciemat.es/@xmath0mercedes . the model used in this work is a generalization of that developed for the solar neighborhood in @xcite and later applied to the whole mwg @xcite and other spiral galaxies @xcite . the enriched material proceeds from the restitution by dying stars , considering their nucleosynthesis , their initial mass function ( imf ) and hence the delayed restitution and their final fate , via a quiet evolution , or type i and ii supernova explosions . most recent works support the idea that the imf is practically universal in space and constant in time @xcite , showing only local differences . we have adopted the imf from @xcite , very similar to a scalo s law @xcite and in good agreement with the most recent data from @xcite , as can be seen in fig . [ imf ] . the original model has been modified in order to use metallicity dependent yields . nucleosynthesis yields for massive stars have been taken from @xcite . for low mass and intermediate stars we have used the set of yields from @xcite . for type i supernova explosion releases , the model w7 from @xcite , as revised by @xcite has been taken . in this model each galaxy is described as a two - zone system with a halo and a disk . it is assumed that the halo has a total mass which is initially in gas phase . the total mass , m , and its radial distribution , m(r ) , are calculated from the corresponding rotation curve derived from the universal rotation curve of pss96 . these authors use a homogeneous sample of about 1100 optical and radio rotation curves to estimate their profile and amplitude which are analysed statistically . from this study they obtain an expression for the rotation velocity , v(r ) , as a function of the rotation velocity at the optical radius ( the radius encompassing 83% of the total integrated light ) , @xmath1 , the galaxy radius normalised to the optical one , @xmath2 , and a parameter @xmath3 , which represents the ratio of the galaxy luminosity to that of the mwg , @xmath4 : @xmath5 where @xmath6 these authors also found that @xmath1 in @xmath7 , depends on @xmath8 as : @xmath9^{1/2 } } \label{vopt}\ ] ] the same occurs with the optical radius , which depends on luminosity through the expression : @xmath10 kpc , which we will also use . .galaxy characteristics dependent on the total mass . [ cols=">,<,^,^,^,^,^,^ " , ] in table [ grid_m ] we show the characteristics obtained with these equations for 44 different values of @xmath8 . in column ( 1 ) we give the number of the radial distribution , defined by the value of @xmath8 , given in column ( 2 ) . the optical radius , @xmath11 , and the virial radius , @xmath12 , are given in columns ( 3 ) and ( 4 ) respectively . we have defined a characteristic radius for each galaxy , which we will use as our reference radius , as @xmath13 . this radius , in kpc , is given in column ( 5 ) . column ( 6 ) gives the rotation velocity , in @xmath14 , reached at a radius @xmath15 kpc ( see pss96 for details ) . the total mass of the galaxy , calculated with the classical expression @xmath16 @xcite , in units of 10@xmath17 m@xmath18 , is given in column ( 7 ) , and , finally , the characteristic collapse time scale , in gyr , which will be described below , is given in column ( 8) . each galaxy is divided into concentric cylindrical regions 1 kpc wide . from the corresponding rotation curve we calculate the radial distributions of total mass , m(r ) , with the expression @xmath19 . from these distributions of the total mass , we easily obtain the one corresponding to each of the cylinders , @xmath20 . both distributions @xmath21 and @xmath20 are shown in fig . the total mass includes the dark matter component ( dm ) which , in principle , does not take part in the chemical evolution . however , the dm contribution seems to be negligible for the large massive galaxies , mostly in the regions where the chemical evolution is calculated . according to ( * ? ? ? * ; * ? ? ? * and references therein ) 75% of the spiral galaxies are well fitted without a dark matter halo and the failure to reproduce an other 20% is directly related to the existence of non - axisymmetric structures ( bars or strong spiral arms ) . the gas computed following the previous section collapses to fall onto the equatorial plane forming the disc as a secondary structure . the gas infall from the halo is parametrised by @xmath22 , where @xmath23 is the gas mass of the halo and @xmath24 is the infall rate , that is , the inverse of the collapse time scale @xmath25 . the collapse time - scale for each galaxy depends on its total mass through the expression : @xmath26 @xcite , where @xmath27 is the total mass of the galaxy in 10@xmath28 ( column 7 of table [ grid_m ] ) and @xmath29 is its age , assumed to be 13.2 gyr in all cases . we calculate a characteristic collapse time scale @xmath30 for each galaxy from the ratio of its total mass , @xmath31 , and the mwg , @xmath32 : @xmath33 where @xmath34 is the collapse time scale at the characteristic radial region for the mwg , @xmath35 kpc . this collapse time scale ( @xmath36 gyr ) was determined from the corresponding value for the solar neighborhood , @xmath37 gyr . this value for @xmath38 is very similar to that found in other standard galactic chemical evolution models @xcite and it is constrained by a large number of data , such as the [ o / fe ] _ vs _ [ fe / h ] relation for stars in the halo and the disc , the present infall rate , ( 0.7 m@xmath18pc@xmath39gyr@xmath40 , see * ? ? ? * ) , or the g - dwarf metallicity distribution @xcite , all of them well reproduced in our model with this long scale to form the disc ( see * ? ? ? we would like to emphasize that the characteristic collapse time - scale computed with equation [ tau ] is not a free - fall time ( @xmath41 gyr for the mwg ) , being much longer than that since it has been calculated through the calibration with the solar neighbourhood collapse time scale @xmath38 . the mass which does not fall onto the disc will remain in the halo , and yields a ratio @xmath42 for the baryonic component which is also in agreement with observations . the relative normalization of the halo , thick and thin disc surface mass densities @xcite gives an approximated proportion of 1:22:200 , which implies that the halo surface mass density must be about 1/100 that of the disc component , also in agreement with the ratio obtained by the multiphase model . [ tcoll ] a ) shows the characteristic time - scales _ vs _ the rotation velocity @xmath43 for our models . solid dots represent the values used in our previous models for individual spiral galaxies @xcite . an important consequence of the hypothesis linking the collapse time scale with the total mass , is that low mass galaxies take more time to form their discs , in apparent contradiction with the standard hierarchical picture of galaxy formation . this characteristic is , however , essential to reproduce most observational features of spiral and irregular galaxies ( see also * ? ? ? in fact , recent self - consistent hydrodynamical simulations in the context of a cosmological model @xcite show that a large proportion of massive objects are formed at early times ( high redshift ) while the formation of less massive ones is more extended in time , thus simulating a modern version of the monolithic collapse scenario ( * ? ? ? furthermore , this assumption is supported by various sets of observations @xcite which seem to demonstrate that a large proportion ( @xmath44 % ) of the massive galaxies formed their stars at @xmath45 while a much smaller proportion of the less massive ones have converted their baryon mass into stars at that redshift . it is evident that the collapse time - scale must vary with galactocentric radius . assuming that the total mass surface density follows the surface brightness exponential shape , the required collapse time - scale should also depend exponentially on radius , with a scale length @xmath46 ( where @xmath47 is the corresponding scale length of the surface brightness radial distribution ) . thus , we assume : @xmath48 in principle , we might expect that @xmath49 . however , in @xcite we have shown that such large scale length for the infall rate produces a final radial variation for the elemental abundances in disagreement with the observed distributions in spiral galaxies . ( see * ? ? ? * for a wide discussion about the effect of this parameter on the radial distribution of abundances ) . on the other hand , we must bear in mind that the surface brightness distribution is the final result of the combination of both the collapse and the star formation processes , and therefore the collapse time - scale may have in principle a different dependence on radius than the surface brightness itself . for the sake of simplicity , we assume a scale length @xmath50 , corresponding to half the scale length of the exponential disc @xmath51 given by pss96 . the value of @xmath52 decreases with the mass of the galaxy , in agreement with observations ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * see their fig . 4 ) . the radial dependence of the infall rate is not imposed _ a priori _ in our scenario , but it is consequence of the gravitational law and the total mass distribution in the protogalaxy . the physical meaning is clear : galaxies begin to form their inner regions before the outer ones in a classical inside - out scheme . this halo - disc connection is crucial for the understanding of the evolution of a galaxy from early times , the inside - out scenario being essential to reproduce the radial gradient of abundances @xcite . in fact , in a chemo - dynamical model @xcite , this scenario is produced naturally . the radial variation of the collapse time - scale for each galaxy , calculated with equation [ tau_r ] , is shown in fig . [ tcoll]b ) , where we also draw a solid line at 13 gyr , the assumed age of galaxies . if the collapse time - scale is larger than this value , there is not enough time for all the gas to fall onto the disc : only a small part of it has moved from the halo to the equatorial plane and the disc formation is not yet complete . this could correspond to the situation observed by @xcite , who have found an extended component of hi , different from the cold disc , located in the halo , rotating more slowly than the disc and with radial inward motion . the model computes the time evolution of each population which inhabits the galaxy . in the various regions of the disc or bulge and in the halo , which are treated separately , we allow for different phases of matter aggregation : diffuse gas ( @xmath53 ) , clouds ( @xmath54 , except in the halo ) , low - mass stars ( @xmath55 ) , high - mass stars ( @xmath56 ) , and stellar remnants . ] the value for stellar mass range division is related to nucleosynthesis prescriptions : stars with masses lower than 4 m@xmath18 only produce light elements , and do not contribute to the interstellar medium enrichment . , allows a very easy comparison of our resulting metallicity distribution with the observed one , based on g - dwarf low mass stars . ] the mass in the different phases of each region changes by the following conversion processes , related to the star formation and death : 1 . star formation by the gas - spontaneous fragmentation in the halo 2 . cloud formation in the disc from diffuse gas 3 . star formation in the disc from cloud - cloud collisions 4 . induced star formation in the disc _ via _ massive star - cloud interactions 5 . diffuse gas restitution from these cloud and star formation processes in the halo , devoid of molecular clouds , the star formation follows a schmidt law for the diffuse gas @xmath23 with a power @xmath57 and a proportionality factor @xmath58 . in the disk , stars form in two steps : first , molecular clouds , @xmath59 , form out of the diffuse gas , @xmath60 , also by a schmidt law with @xmath57 and a proportionality factor called @xmath61 . then , cloud - cloud collisions produce stars by a spontaneous process at a rate proportional to a parameter @xmath62 . moreover , a stimulated star formation process , proportional to a parameter @xmath63 , is assumed . an advantage of using our model is that it includes a more realistic star formation than classic chemical galactic evolution ones in which the sfr prescriptions are based on a schmidt law , depending on the total gas surface density . instead , the multiphase model assumes a star formation which takes place in two - steps : first , the formation of molecular clouds ; then the formation of stars . this simulates a power law for the gas density with an exponent @xmath64 and with a threshold gas density as shown by @xcite , and , more importantly , it allows the calculation of the two different gas phases present in the interstellar medium . in fact , the actual process of star formation , born by observations @xcite , is closer to our scenario with stars forming in regions where there are molecular clouds , than to the classical schmidt law which depends only on the total gas density . our assumed sfr implies that some feedback mechanisms are included naturally and are sufficient to simulate the actually observed process of creation of stars from the interstellar medium . the formed massive stars induce the creation of new ones . but , at the same time , these star formation processes also may destroy the diffuse or molecular clouds , thus preventing the total conversion of gas into stars and ejecting more gas once again into the ism point ( v). in particular , massive stars destroy the molecular clouds that surround them , due to the sensitivity of molecular cloud condensation to the uv radiation @xcite . this mechanism restores gas to the ism , thus decreasing the star formation . both regulating process are included in our model , although neither heating or cooling mechanisms for the cloud components are included explicitly in our code . the complete set of equations is given in @xcite for each radial region . we summarize here only those related to the star formation processes in the halo , @xmath65 , and in the disk , @xmath66 : @xmath67 where h and d indicate halo and disk . @xmath58 , @xmath61 , @xmath62 and @xmath63 , besides the previously described @xmath25 are the parameters of the model . in the classical method of application of a chemical evolution model to a given region or galaxy , the input parameters are considered as free and chosen as the best ones in order to reproduce the selected observational constraints of the galaxy . in our models , however , not all these input parameters can be considered as free . the parameter @xmath24 is defined by the total mass radial distribution , as already explained . regarding @xmath58 , @xmath61 , @xmath62 and @xmath63 , we have tried to reduce to a minimum their degree of freedom . their radial dependence may be estimated as ( * ? ? ? * see ) : @xmath68 where g is the universal gravitational constant , @xmath69 and @xmath70 are the halo and disk volume of each radial region , . ] @xmath71 is the average cloud density and @xmath72 is the average mass of massive stars . the proportionality factors are called efficiencies that represent probabilities associated with these processes . in this way , the free parameters @xmath58 , @xmath61 , @xmath62 and @xmath63 , proportionality constants in our star and cloud formation laws , but variable for each radial region , are computed through the efficiencies , which represent the efficiencies of star formation in the halo , @xmath73 , cloud formation , @xmath74 , cloud cloud collision , @xmath75 , and interaction of massive stars with clouds , @xmath76 , in the disc . only a given efficiency for each process must be selected for the whole galaxy , although the original parameters , and its corresponding processes , maintain their radial dependence . the term associated to the induced star formation describes a local process and , as a result , its coefficient @xmath76 is considered independent of the galaxy modelled and the location in it . the term @xmath73 is also assumed constant for all haloes . therefore , for all our 440 models , both efficiencies take the same values already used in our previous model for the mwg . only the other two efficiencies @xmath74 and @xmath75 are allowed to vary for each galaxy , being characteristic for each one of them . within each galaxy @xmath75 and @xmath74 are independent of the position . the fact that galaxies with the same gravitational potential or mass but different morphological type or appearance exist , implies that the evolution of a galaxy does not depend solely on gravitation , even if this may be the most important factor , but also on certain dynamical conditions . these conditions can not be taken into account , obviously , in a simple chemical evolution model , but may change the evolution of a galaxy , ( mostly the star formation rate through temperature variations ) . we may consider them as included in our efficiencies to form molecular clouds and stars , @xmath74 and @xmath75 , which are allowed to change from one galaxy to another . we assume that @xmath74 and @xmath75 vary between 0 and 1 , according with its efficiency meaning . in principle both parameters should be allowed to vary independently from each other , what would increase very much the number of models to be calculated . furthermore , some of those models will not be physically possible . on the other hand , on the basis of our previous works , a trend between both efficiencies seems to exist , increasing or decreasing together for a given galaxy respect to the values appropriate for the mwg model . we have therefore studied the observational data related to molecular cloud and star formation from the available gas in order to check if a correlation between these two processes exists . to this aim we have used the data from @xcite which correspond to a large sample of galaxies and refer to atomic and molecular gas masses and to luminosities in the ir band and in h@xmath77 emission , two well known indicators of the star formation rate . with them , we have tried to establish if some correlation appears among our efficiencies , @xmath74 and @xmath75 . since the modelled star formation rate have two steps to form stars , we will analyse the transformation of diffuse gas in molecular gas and of the conversion of the molecular clouds into stars . from equations ( 10 ) and ( 11 ) , our parameters @xmath74 and @xmath75 , efficiencies of transforming the diffuse gas in molecular clouds and forming stars out of them , can be written as : @xmath78 while from equations ( 7 ) and ( 8) we have : @xmath79 in these expressions the subscript @xmath80 has been omitted , since in all cases we refer to the disc . but the second term is much smaller than the first and therefore we can approximate @xmath81 we may approximate @xmath82 , where @xmath83 is the mean time necessary to transform the diffuse gas in a molecular cloud , in units of @xmath84 : @xmath85 where @xmath86 and @xmath87 are the total mass in diffuse and molecular gas in units of @xmath88 , and sfr is the star formation rate in units of @xmath89 . therefore : @xmath90 sfr is usually estimated from the @xmath91 luminosity @xcite : @xmath92 and the volume of the disc is @xmath93 where r is in kpc and @xmath94 is in @xmath95 . we have all data to estimate these efficiencies , except @xmath96 . recent estimates @xcite for this cloud accumulation time scale give values several times the gravitational contraction scale ( which is @xmath97 1 - 4 myr ) , that is @xmath98 myr , and probably smaller than 50 myr . the probable range is @xmath99 $ ] yr . we have given three possible values : @xmath100 , @xmath101 and @xmath102 yr in order to take into account other possible slower modes @xcite . in fig . [ ratio ] we represent the ratio @xmath103 computed for @xmath104 as a function of galaxy morphological type . single galaxies are represented as @xmath105 while solid dots correspond to average values obtained by binning them into 11 types . a constant ratio for all types is consistent with the data points , shown by the solid line and giving a mean value @xmath106 with @xmath107 . to 10 . ] taking into account the large range of variation of @xmath74 and @xmath108 ( more than 5 orders of magnitude ) for the whole set of data , it is surprising that the @xmath109 ratio takes values around 0.35 for all galaxies . for other values of @xmath96 , this value changes but remains constant . they are shown in the graph as dotted lines at values 0.00 ( @xmath110 myr ) and 0.80 ( @xmath111 myr ) . taking this into account , we have assumed a ratio @xmath109= 0.4 for our computed models . ccc n & @xmath112 & @xmath75 + 1 & 0.95 & 0.88 + 2 & 0.80 & 0.57 + 3 & 0.65 & 0.34 + 4 & 0.45 & 0.14 + 5 & 0.30 & 0.05 + 6 & 0.15 & 1.0e-2 + 7 & 0.075 & 1.5e-3 + 8 & 0.037 & 2.6e-4 + 9 & 0.017 & 3.7e-5 + 10 & 0.007 & 4.0e-6 + we have computed 10 models for each mass radial distribution , allowing @xmath74 to take values between 0 to 1 as given in table [ efi ] . for each one of these efficiencies,@xmath75 has been fixed according to the ratio @xmath113 and their values are also given in the table . each set of efficiencies has been labelled by a number n from 1 to 10 . _ summarizing , only the characteristic collapse time - scale , depending on the total mass , and the set of efficiencies , denoted by number n , are varied from model to model . _ we thus obtain 440 different models which represent all possible combinations of the collapse time - scale with the values of n. the results corresponding to the mass of each region and phase , the star formation rate and the supernova rates , for the 440 computed models are shown in tables [ phases ] , and [ sfrs ] . the elemental abundances for the discs are shown in table [ abundances ] . here we only give , as an example , the results of the model corresponding to radial distribution number 22 and @xmath114 , with a rotation velocity of @xmath97 140 km.s@xmath40 , for the first and last time steps of the evolution . the whole tables with the complete time evolution from 0 to 13 gyr , with a time step of 0.5 gyr , for the whole set of models will be available in electronic form at cds via anonymous ftp to cdsarc.u-strasbg.fr ( 130.79.128.5 ) or via http://cdsweb.u-strasbg.fr/abstract.html , or http:/wwwae.ciemat.es/@xmath115mercedes / grid . rrccccccc time & r & mtot & mdisc & mgas(hi ) & mgas(h@xmath116 ) & mstars(m@xmath117 ) & mstars(m@xmath118 ) & mremnants + ( gyr ) & ( kpc ) & ( 10@xmath28 ) & ( 10@xmath28 ) & ( 10@xmath28 ) & ( 10@xmath28 ) & ( 10@xmath28 ) & ( 10@xmath28 ) & ( 10@xmath28 ) + ... & ... & ... & ... & ... & ... & ... & ... & ... + 0.1 & 14 . & 0.69e+01 & 0.47e04 & 0.47e04 & 0.89e08 & 0.40e17 & 0.31e18 & 0.21e19 + 0.1 & 12 . & 0.69e+01 & 0.22e03 & 0.22e03 & 0.11e06 & 0.65e15 & 0.49e16 & 0.34e17 + 0.1 & 10 . & 0.68e+01 & 0.10e02 & 0.10e02 & 0.14e05 & 0.13e12 & 0.93e14 & 0.65e15 + 0.1 & 8 . & 0.67e+01 & 0.48e02 & 0.47e02 & 0.20e04 & 0.28e10 & 0.21e11 & 0.14e12 + 0.1 & 6 . & 0.66e+01 & 0.22e01 & 0.22e01 & 0.31e03 & 0.75e08 & 0.56e09 & 0.39e10 + 0.1 & 4 . & 0.58e+01 & 0.91e01 & 0.87e01 & 0.47e02 & 0.21e05 & 0.16e06 & 0.11e07 + 0.1 & 2 . & 0.29e+01 & 0.21e+00 & 0.17e+00 & 0.39e01 & 0.23e03 & 0.17e04 & 0.13e05 + ... & ... & ... & ... & ... & ... & ... & ... & ... + 13.2 & 14 . & 0.69e+01 & 0.58e02 & 0.46e02 & 0.13e02 & 0.10e04 & 0.22e07 & 0.70e06 + 13.2 & 12 . & 0.69e+01 & 0.27e01 & 0.16e01 & 0.10e01 & 0.89e03 & 0.16e05 & 0.67e04 + 13.2 & 10 . & 0.68e+01 & 0.13e+00 & 0.54e01 & 0.43e01 & 0.28e01 & 0.31e04 & 0.25e02 + 13.2 & 8 . & 0.67e+01 & 0.57e+00 & 0.14e+00 & 0.11e+00 & 0.29e+00 & 0.21e03 & 0.32e01 + 13.2 & 6 . & 0.66e+01 & 0.23e+01 & 0.26e+00 & 0.19e+00 & 0.16e+01 & 0.79e03 & 0.21e+00 + 13.2 & 4 . & 0.58e+01 & 0.49e+01 & 0.18e+00 & 0.18e+00 & 0.39e+01 & 0.88e03 & 0.65e+00 + 13.2 & 2 . & 0.29e+01 & 0.29e+01 & 0.19e01 & 0.51e01 & 0.23e+01 & 0.95e04 & 0.50e+00 + in table [ phases ] we list in column ( 1 ) the time , in gyr and in column ( 2 ) , the galactocentric distance , in kpc , in column ( 2 ) . columns ( 3 ) to ( 9 ) give the mass , in units of 10@xmath119 , in each region and phase : column ( 3 ) the total mass in each region ; column ( 4 ) the mass of the disc region ; column ( 5 ) the mass in the diffuse gas phase;column ( 6 ) the molecular gas ; column ( 7 ) the mass in low and intermediate mass stars ; column ( 8) the mass in massive stars and , finally , column ( 9 ) the mass in remnants . table [ sfrs ] gives , for each time step in gyr and radial distance in kpc , listed in columns ( 1 ) , and ( 2 ) respectively , the star formation rate , in units of @xmath120 , in the disc and the halo regions in columns ( 3 ) and ( 4 ) ; the supernova rates , for types ia , ib and and ii , for the disc in columns ( 5 ) , ( 6 ) and ( 7 ) and for the halo in columns ( 8),(9 ) and ( 10 ) all of them in units of 100 yr@xmath40 . the abundances in the disc for 14 elements are given in table [ abundances ] for each time step in gyr and radial distance in kpc ( columns ( 1 ) , and ( 2 ) ) .the abundances by mass of : h , d , @xmath121he , @xmath122he , @xmath123c , @xmath124c , n , o , ne , mg , si , s , ca , and fe are given in columns ( 3 ) to ( 16 ) . the time evolution of several models is shown in the next figures . in each one , 4 panels are shown , corresponding to 4 different maximum rotation velocities and/or radial distribution of masses . we have selected the values corresponding to @xmath1250.03 , 0.19 , 1.0 and 2.5 corresponding to galaxies with rotation velocities of 48 , 100 , 200 and 290 @xmath126 , respectively , and representing typical examples of spiral and/or irregular galaxies . for each panel we show the results for the 10 selected values of the efficiencies ( or equivalently 10 rates of evolution ) from @xmath127 , corresponding to the highest efficiency values and hence the most evolved models , to @xmath128 , with the smallest values and thus the least evolved ones . we have computed models for galactocentric radii up to @xmath129 with a step @xmath130 which depends on total galactic , being larger ( up to 4 kpc ) for the most massive modelled galaxies and smaller ( only 1kpc ) for the lowest mass ones . results are shown only for the region located at the galactocentric distance closest to the characteristic radius @xmath131 defined in table [ grid_m ] . [ hit_rc ] shows the time evolution of the diffuse gas density . in all cases , the diffuse gas surface density is seen to increase rapidly at early times and then declines slowly . the first abrupt increase is a consequence of the infall rate of gas from the halo and hence does not depend on the model efficiencies . the increase is faster for more massive galaxies . the later decline is due to gas consumption in the process of molecular gas formation and hence depends on the efficiency @xmath74 , producing lower densities for the galaxies with higher efficiencies . fig.[h2t_rc ] shows the time evolution of the molecular gas phase whose formation is delayed respect to that of the diffuse gas . it is clear that the maximum density is reached later than that corresponding to the diffuse gas density . for instance , for @xmath132 , @xmath133 gyr for @xmath134 and 200 km s@xmath40 , while it is 0.3 gyr for @xmath135 km s@xmath40 and reaches @xmath136 gyr when @xmath137 km s@xmath40 . rrrrrrrrrr time & r & sfr(disc ) & sfr(halo ) & sn - ia ( disc ) & sn - ib ( disc ) & sn - ii ( disc ) & sn - ia ( halo ) & sn - ib ( halo ) & sn - ii(halo ) + ( gyr ) & ( kpc ) & ( m@xmath138 ) & ( m@xmath138 ) & ( @xmath139 ) & ( @xmath139 ) & ( @xmath139 ) & ( @xmath139 ) & ( @xmath139 ) & ( @xmath139 ) + ... & ... & ... & ... & ... & ... & ... & ... & ... & ... + 0.1 & 14 . & 0.26e15 & 0.51e01 & 0.17e17 & 0.14e18 & 0.42e14 & 0.45e02 & 0.11e01 & 0.15e+01 + 0.1 & 12 . & 0.43e13 & 0.52e01 & 0.27e15 & 0.14e16 & 0.69e12 & 0.46e02 & 0.11e01 & 0.16e+01 + 0.1 & 10 . & 0.82e11 & 0.54e01 & 0.52e13 & 0.25e14 & 0.13e09 & 0.47e02 & 0.12e01 & 0.16e+01 + 0.1 & 8 . & 0.18e08 & 0.57e01 & 0.12e10 & 0.55e12 & 0.30e07 & 0.51e02 & 0.12e01 & 0.17e+01 + 0.1 & 6 . & 0.49e06 & 0.63e01 & 0.31e08 & 0.15e09 & 0.80e05 & 0.55e02 & 0.14e01 & 0.19e+01 + 0.1 & 4 . & 0.13e03 & 0.61e01 & 0.89e06 & 0.44e07 & 0.22e02 & 0.55e02 & 0.14e01 & 0.18e+01 + 0.1 & 2 . & 0.14e01 & 0.28e01 & 0.11e03 & 0.60e05 & 0.24e+00 & 0.26e02 & 0.65e02 & 0.84e+00 + ... & ... & ... & ... & ... & ... & ... & ... & ... & ... + 13.2 & 14 . & 0.52e05 & 0.45e01 & 0.13e05 & 0.56e05 & 0.16e03 & 0.18e01 & 0.54e01 & 0.14e+01 + 13.2 & 12 . & 0.38e03 & 0.46e01 & 0.10e03 & 0.42e03 & 0.11e01 & 0.18e01 & 0.54e01 & 0.14e+01 + 13.2 & 10 . & 0.74e02 & 0.46e01 & 0.23e02 & 0.86e02 & 0.22e+00 & 0.18e01 & 0.55e01 & 0.14e+01 + 13.2 & 8 . & 0.50e01 & 0.44e01 & 0.18e01 & 0.59e01 & 0.15e+01 & 0.18e01 & 0.53e01 & 0.13e+01 + 13.2 & 6 . & 0.19e+00 & 0.29e01 & 0.73e01 & 0.22e+00 & 0.56e+01 & 0.13e01 & 0.35e01 & 0.86e+00 + 13.2 & 4 . & 0.21e+00 & 0.25e02 & 0.11e+00 & 0.25e+00 & 0.62e+01 & 0.24e02 & 0.31e02 & 0.74e01 + 13.2 & 2 . & 0.22e01 & 0.17e07 & 0.29e01 & 0.28e01 & 0.68e+00 & 0.14e03 & 0.27e07 & 0.51e06 + fig . [ sfrt_rc ] shows the star formation rate history for the same cases as before . these histories result extraordinary different , even for equal efficiencies . taking into account that the radial region shown is equivalent in all galaxies , it indicates that the primary agent driving the time evolution of the sfr is the galaxy mass through the collapse time scale . on the other hand , the resulting star formation history in most models , is smooth , showing values larger than 10@xmath140 only at the early evolutionary phases of the most massive galaxies with very high efficiencies . to 220 mm landscape table 5 to go here . [ abundances ] the time evolution of the oxygen abundance relative to hydrogen , expressed as 12+log(o / h ) , for the models described above is shown in fig.[oht_rc ] . present time oxygen abundances look very similar for all the galaxies with @xmath141 . the models with low efficiencies , @xmath142 , show very low abundances although always larger than 12+log(o / h)= 7 , about the lowest observed oxygen abundance in galaxies . fig.[ofe_rc ] shows the evolution of [ o / fe ] when [ fe / h ] increases . the usual relation for the mwg can be taken to be represented by the @xmath143 line from panel c ) , showing almost a _ plateau _ for low metallicities ( @xmath144<1.5 $ ] ) and then a decline toward the solar value . the _ plateau _ appears in the models corresponding to massive massive galaxies with high efficiencies @xcite . the less massive galaxies show a slow decline without _ plateau _ for low values of n , and a steeper decrease for higher values . thus , we may expect that [ o / fe ] in the less evolved galaxies will soon reach the solar value @xcite . we now analyse the present time results obtained with this bi - parametric grid of models . in order to do this , we start presenting only the present time radial distributions of gas , oxygen abundances and star formation rate which we will analyse in the following subsections . the radial distribution of gas and elemental abundances provide two basic model constraints since they are easily extracted from observations . in addition , in many cases , the star formation surface density and/or the surface brightness can also be derived . the radial distribution of atomic gas surface density for the present time is shown in fig . we can see that the atomic gas surface density shows a maximum somewhere along the disc , as it is usually observed ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * and see fig.[chequeo ] ) . the value of this maximum depends on n : the models with the highest efficiencies have smaller gas quantities and their maximum densities are around 3 - 4 m@xmath145 . for intermediate efficiencies ( @xmath146 ) , these maximum values rise to @xmath147 m@xmath145 . for all efficiencies the radial distributions are very similar independently of their galactic mass , except for those corresponding to @xmath148 ( @xmath149 ) which show much lower densities , except in the central region . in fact , a characteristic shown by all distributions is the similarity among models of the same efficiencies but different total mass , only scaled by their different optical radius . with the exception of the @xmath150 model , all the others show , for a same n , differences small enough as to simulate a dispersion of the data . for the less evolved theoretical galaxies ( n @xmath151 ) , the neutral gas distribution display a clear dependence on galactic mass . the maximum density values are always large , as expected , due to the small efficiencies to form molecular clouds , which do not allow the consumption of the diffuse gas , but this maximum density increases from @xmath152 for @xmath150 ( @xmath153 ) to @xmath154 for @xmath155 ( @xmath156 ) and up to @xmath157 for @xmath158 ( @xmath159 ) . the consequence of a shorter collapse time - scale for the more massive spirals is clearly seen : the maximum is located at radii further away from the centre due to the exhaustion of the diffuse gas in the inner disc which moves the star formation outside . the smaller the galaxy mass , the closer to the centre is the maximum of the distribution , which resembles an exponential , except for the inner region . in fact , a shift in the maximum appears in each panel . in some cases , however , the low values of the surface gas density are due to the fact that the gas did not have enough time to fall completely onto the equatorial disc . this effect is very clear for the model with @xmath150 which shows a very steep distribution with gas densities lower than 15 @xmath160 . in this case the gas shows a radial distribution with a maximum at the centre . the low density out of this central region is not due to the creation of stars but to the long collapse time - scale which prevents the infall of the sufficient gas to be observed . besides the variations due to the differences in total mass , which correspond to different collapse time - scales , and because we have selected different efficiencies to form stars and molecular clouds , a same total mass may have produced discs in different evolutionary states . thus , a same @xmath161 may result in a disc of 7 - 8 kpc and atomic gas densities around 5 @xmath160 , or a disc of only 5 kpc with a maximum density of 40 @xmath160 in the region of 2 kpc . in the same way , a galaxy with a large value of the total mass , as the one with @xmath162 , may show a high gas mass density and a little disc for a high value of n or , on the contrary , be very evolved and therefore show no gas and a large stellar disc if the value of n is low . the first object ( @xmath163 ) could correspond to the case of low surface brightness galaxies , while the last ones ( @xmath164 ) could be identified as the typical high surface brightness spiral galaxies . an important success of the multiphase models has been the ability to reproduce the radial distributions for the atomic gas and the molecular gas separately , which is possible due to the assumed star formation prescription in two steps , thus allowing the formation of molecular clouds prior to the appearance of stars . another important consequence of this sfr law is that it takes into account feedback mechanisms , even negative . if molecular clouds form before stars , this implies a delay in the time of star formation . the molecular gas shows an evolution similar to that of the diffuse gas , but with a certain delay . this delay allows the maintenance of a radial distribution with an exponential shape , as usually observed , for a longer time , although in some evolved galaxies @xmath165 is also consumed in the most central regions , thus reproducing the so - called central _ hole _ of the molecular gas radial distribution , observed in some galaxies @xcite , the mwg being one of them ( see fig6b in next section ) . this is seen in fig.[h2 ] for efficiencies corresponding to @xmath166 , ( depending on the total mass ) , for which the model lines turn over at the inner disc , which corresponds to the regions located at the border between bulge and disc . thus , the galaxies with the highest efficiencies ( @xmath167 ) show a maximum in their radial distribution of @xmath168 , which is always closer to the centre than that of the atomic gas distribution . galaxies with the lowest efficiencies ( @xmath166 ) show larger surface densities of molecular than atomic gas because the efficiency to form stars from molecular clouds is smaller than the efficiency to form these clouds . therefore , the conversion of diffuse to molecular gas occurs more rapidly that the subsequent formation of stars . the total mass converted into stars forms out the stellar disc in each galaxy . these stellar discs are represented in fig.[profiles ] as the stellar surface density radial distributions . these profiles may be compared with brightness surface radial distributions , after the appropriate conversions . the values of efficiencies have a small influence in the resulting stellar surface density distribution shape : the total mass of stars created is similar for all n @xmath169 , although they are formed at different rates , that is the resulting stellar populations have different mean ages . for the most evolved cases , most stars were created very rapidly , while for the less evolved ones , stars formed later as average , as it is shown in fig . [ sfrt_rc ] for the characteristic radius region . therefore , the radial distributions of surface brightness would result very similar for galaxies of a given galactic total mass , but colors are expected to be different , redder for the galaxies with the highest efficiencies for the formation of stars and molecular clouds . a very interesting result is that the central value of the distribution is practically the same , around 100 m@xmath145 , for all rotation curves and efficiencies , in agreement with freeman s law . only galaxies with the smallest efficiencies or the less massive discs show central densities smaller than this value . we can not compute a surface brightness only with these models , but assuming a ratio @xmath170 for the stellar populations , this implies a surface luminosity density of @xmath171 and scale lengths in agreement with observed generic trends . only in models for n@xmath172 the stellar discs show a different appearance : they look less massive , as corresponding to discs in the process of formation . this implies that the surface brightness is lower than for the other types for a similar characteristic total mass . in any case , we remind that all information related to photometric quantities , and this also applies to the disc scale lengths , which may be obtained from the star formation histories and enrichment relations of models , need the application of evolutionary synthesis models , what is out of the scope of this work . one of the most important results of this grid of models refers to the oxygen abundances , shown in fig . a radial gradient appears for most of models . this is due to the different evolutionary rates along radius : the inner regions evolve more rapidly that the outer ones , thus steepening the radial gradient very soon for most model galaxies . then , the radial gradient flattens for the more massive and/or most evolved ( small n ) galaxies due to the rapid evolution , even in the outer regions , which produces a large quantity of elements , with the oxygen abundance reaching a saturation level . this level is found to be around @xmath173 dex . observations in the inner disc of our galaxy support this statement @xcite . this result was already found in our previous works . moreover , the larger the mass of the galaxy , the faster the effect : a galaxy with @xmath174 km.s@xmath40 has a flat radial gradient for efficiencies corresponding to n@xmath175 1 or 2 , while a galaxy with v@xmath176 km.s@xmath40 , shows a flat distribution for @xmath177 . the less massive galaxies maintain a steeper radial distribution of oxygen for almost all efficiencies , with very similar values of the gradients . nevertheless , for any galaxy mass , if @xmath178 the radial abundances distribution are flat . thus , the less evolved galaxies show no gradient , such it is observed in lsb galaxies @xcite , the intermediate ones show steep gradients , and the most evolved galaxies have , once again , flat abundance radial distributions . the largest values of radial gradients correspond to the intermediate evolutionary type galaxies , with the limiting n varying according to the total mass of the galaxy . the more massive galaxies only show a significant radial gradient if n@xmath179 while the less massive ones have a flat gradient only if @xmath180 with the rest having very pronounced radial gradients even for @xmath181 . for the evolved galaxies , the characteristic efficiencies are high for all the disc , thus producing a high and early star formation in all radial regions . in this case , the oxygen abundance reaches very soon a saturation level , flattening the radial gradient developed at early times of the evolution . the characteristic oxygen abundance , measured at r@xmath182 , is higher for the more massive galaxies and lower for the less massive ones . however , this correlation is not apparent when the central abundance is used , due to the existence of the saturation level in the oxygen abundance , which produces a flattening of the radial gradient in the inner disc , even for galaxies with intermediate efficiencies . actually , this saturation level would correspond to the integrated yield of oxygen for a single stellar population . as oxygen is ejected by the most massive stars of a generation of stars , it appears rapidly in the ism . it is therefore impossible to surpass this level of abundance . in fact , the oxygen abundance radial distribution shows sometimes a bad fit to observations in the central parts of the discs , which give frequently abundances larger than 9.10 dex . this absolute value of the oxygen abundance is not reached in any case by the models . all the computations performed within the multiphase approach reach a maximum @xmath183 dex which no model can exceed . we might think that uncertainties in the calculations of the stellar yields elements , still very dependent on the assumed evolution of stars , are a possible reason for this saturation level appears in theoretical models . however , low and intermediate mass stars , for which yields show large variations among the different groups ( see * ? ? ? * and references therein ) , depending on the assumed hypothesis in the calculation of the stellar evolution in the latest stellar evolutionary phases , do not produce oxygen ; these calculations only affect to n and c abundances . on the other hand , the most important uncertainty for the production of oxygen in massive stars refers to the strength of stellar winds responsible for stellar mass loss . this mass loss implies less production of oxygen and a larger ejection of carbon . in our models we have used woosley & weaver s yields which do no include stellar winds . therefore , from this point of view , the modelled oxygen abundances must be considered upper limits . it should be recalled , however , that all oxygen data yielding values larger than 9.1 dex have been obtained from observations of hii regions where the electronic temperature could not be measured , and hence the oxygen abundances have been derived through empirical calibrations which are very uncertain in the high abundance regime . actually , the shape of the radial distribution of oxygen changes depending on the calibration used ( e.g. * ? ? ? the suspicion that these abundances are overestimated at least by 0.2 dex is very reasonable @xcite . abundance estimates in hii regions historically considered as metal - rich , result to be almost solar once the electronic temperature has been finally measured @xcite . radial distributions of star formation rate surface density show an exponential shape in the outer disc , but a less clear one in the inner regions , where some models show a distribution flatter than that of the molecular gas , and even decreasing toward low values of the star formation rate in the centre . these sfr radial distributions are very similar to those estimated by @xcite from h@xmath77 fluxes , as we show in fig.[chequeo ] for a few galaxies . the star formation rate assumed in our models do not produce bursts of massive stars in the low mass galaxies whatever the efficiencies . only the massive galaxies are able to keep a large quantity of gas in a small region , usually at the centre although sometimes at the inner disc regions . on the contrary , low mass galaxies collapse very slowly , and thus the star formation rate maintains a low level during the whole life of the galaxy . in fact , recent works suggest the same scenario for both low mass and low surface brightness galaxies @xcite in order to take into account the observed data . our resulting abundances and gas fractions for low mass galaxies seem to be in rough agreement with these findings @xcite , although our model results can not still be compared with photometric data , for which many more observational sets exist . the first application of any theoretical model is to check its validity for the mwg . a large set of observational data for the solar neighbourhood and the galactic disc exists and therefore the number of constraints is large compared to the number of free parameters of the computed models . the model for @xmath143 , and total mass distribution number 28 , corresponding to @xmath184 and maximum rotation velocity @xmath185 km s@xmath40 , is the model more representative of the mwg . the results corresponding to this model are shown in figs . [ mwg1 ] and [ mwg2 ] together with the available data . [ mwg1 ] shows the time evolution of the star formation history , panel a ) , and the age - metallicity relation , panel b ) , of the region located at r@xmath1758 kpc from the galactic centre , as compared to data for the solar neighbourhood . the observational trends are reproduced adequately , although the model predicted maximum of the star formation rate appears slightly displaced toward earlier times with respect to observations . [ mwg2 ] shows the present time radial distributions of diffuse and molecular gas , mass of stars , star formation rate and oxygen and nitrogen abundances for the galactic disc . the diffuse gas radial distribution , panel a ) , is well reproduced in shape , although the maximum observed gas surface density is somewhat displaced to outer radii compared with the oldest data . it , however , fits well the most recent data obtained from @xcite for radii r@xmath186 kpc . regarding the molecular gas density , panel b ) , the distribution is quasi - exponential from @xmath187 kpc and decreases at the inner disc regions . taking into account that recent data give low densities at these inner regions , we consider that our model results may be adequate . the stellar mass distribution , panel c ) , is exponential in shape in agreement with the surface brightness distribution , and radial distributions , panels e ) and f ) , reproduce those shown by the most recent data . the only feature which is not well reproduced by the model is the sfr radial distribution shown in panel d ) , which decreases toward the inner disc in apparent discrepancy with observations . the modelled star formation rate distribution has a maximum in @xmath188 kpc , while the observed one increases exponentially toward the galactic centre , or levels off at 3 - 4 kpc . on the other hand , the maximum of the atomic gas density is observed around 10 - 11 kpc , and the molecular gas density has its maximum observed at around 6 kpc . therefore , it results difficult to explain , from an observational point of view , how the star formation rate remains so high at the inner disc ( inside the central 3 - 4 kpc ) , where both gas phases are already consumed . in fact , the recent data from @xcite show a decline for r@xmath189 kpc and a decreasing star formation rate for the inner disc regions is also observed in a large number of galaxies , as we will see in the next section . actually , the mwg radial distribution of the present star formation rate is still a matter of discussion @xcite since the detection of young sources , which are embedded in gas clouds , is difficult and might be affected by selection effects . data from pulsars , supernovae or ob star formation @xcite seem to indicate that the radial distribution of the sfr has a maximum around 5 kpc , decreasing toward both smaller and larger radii . on the other hand @xcite obtained a much flatter distribution from the observed cosmic rays and gamma radiation spectra . since a decrease at the inner disc is not unreasonable from the comparison with other spiral galaxy data , we consider that the model results for the mwg are acceptable . there is only a small sample of galaxies for which large observational data sets , including neutral and molecular gas distributions , exist . in what follows we show a comparison of our model results with the data corresponding to theses galaxies . lccccccccccc galaxy & t & type & d & vel@xmath190 & mass distr . & @xmath30 & r@xmath182 & n & @xmath191 & @xmath192 & @xmath193 + name & & class & ( mpc ) & ( @xmath126 ) & number & ( gyr ) & ( kpc ) & & ( gyr ) & & + ngc 300 & 7 & scd / sd & 1.65 & 85 & 13 & 13.31 & 2.3 & 7 & 13.3 & 0.07 & 0.007 + ngc 598 & 6 & sc / scd & 0.84 & 110 & 21 & 8.13 & 2.9 & 6 & 10.3 & 0.05 & 0.005 + ngc 628 & 5 & sc & 11.4 & 220 & 31 & 3.43 & 7.7 & 5 & 3.28 & 0.25 & 0.01 + ngc 4535 & 5 & & 16.6 & 210 & 30 & 3.59 & 7.4 & 5 & 3.50 & 0.28 & 0.02 + ngc 6946 & 6 & scd & 7 & 180 & 25 & 4.94 & 6.2 & 6 & 4.26 & 0.18 & 0.02 + the characteristics and corresponding input parameters of this galaxy sample are given in table [ sample ] . for each galaxy , column ( 1 ) , the morphological type index is given in column ( 2 ) , while the classical hubble type is given in column ( 3 ) . the adopted distance ( taken from references following table [ data ] , column ( 2 ) is in column ( 4 ) ; the maximum rotation velocity is given in column ( 5 ) . the number of the radial distribution of total mass corresponding to the column ( 1 ) of table [ grid_m] chosen to represent each galaxy is in columns ( 6 ) . the characteristic collapse time and the characteristic radius corresponding to each distribution are given in columns ( 7 ) and ( 8) . the number n chosen as the best one to reproduce the observations is in column ( 9 ) . the last columns , ( 10 ) to ( 12 ) give , for comparison purposes , the collapse time - scale and efficiencies for molecular cloud and star formation used in our previous models @xcite for these same galaxies . the radial distributions of the different quantities : atomic and molecular gas densities , star formation rate and oxygen abundance , for these galaxies are shown in fig . [ chequeo ] together with the corresponding observational data , taken from the references given in table [ data ] . in the first row of panels of fig . [ chequeo ] we can clearly see that the radial distributions of neutral hydrogen are very well reproduced by the models . for each galaxy , the distribution shows a maximum along the disc , as predicted . also , for galaxies with similar total mass , such as ngc 4535 and ngc 6946 , this maximum is higher for later morphological type . for galaxies with the same value of t , but different total mass , such as ngc 6946 and ngc 598 , this maximum does not change its absolute value , but it is located at a different galactocentric distance , closer to the galaxy centre for the less massive one , due to the longer collapse time - scale which reflects on a slower evolution . we would like to point out that the agreement between model results and data is much improved when good quality data ( usually the most recently published ones ) are used . the selection of the best obtained distances improves extraordinarily this agreement , as well . the same applies to the rest of the panels in this figure . the radial distributions of molecular cloud surface density for each galaxy , except ngc 300 for which no data exist , are shown in the second row of fig . [ chequeo ] . the agreement between model results and observations is good for the outer discs which follow a quasi exponential distribution , but is not so good for the inner discs where the molecular hydrogen surface density is observed to continue increasing instead of turn over as predicted by the models . we might artificially decrease the efficiencies @xmath75 in these zones , instead to maintain them constant as we do , and thus recover the observed exponential shape . but in this case oxygen abundances will be smaller than observed in these same regions . on the other hand we should recall that the molecular masses are estimated from the co intensity through a calibration factor which depends on metallicity in a way which would produce smaller molecular gas densities than usually assumed at the inner galactic disc @xcite . taking into account that recent data yield low densities at these inner regions , we consider that our model results may represent adequately the reality . in any case , differences between models and data are larger than in the case of the diffuse gas , which is not unexpected , given the larger uncertainties involved in the derivation of molecular hydrogen masses . the radial distribution of the star formation rate for each galaxy is shown in the third row of fig [ chequeo ] . the agreement between model results and data is very good for the most massive galaxies for which both the maximum of the star formation rate and its location is well reproduced . for the less massive galaxies , the predicted central turnover of the distribution is not observed . in the last row of fig . [ chequeo ] , the oxygen abundance radial distribution for each galaxy , except ngc 4535 for which no data exist , is shown . the oxygen radial gradient is reproduced in all cases and also the observed trend of steeper radial distributions larger radial gradients for the late type galaxies is well reproduced by the models . lccccc galaxy & d & h & h@xmath116 & sfr & [ o / h ] + ngc 300 & tu & rog , puc & ... & deh & pag , chr , deh + ngc 598 & tp & new , cs & hey & hey , ken , rk & ka , vil + ngc 628 & mk & wev & al , nis , ry , ss & lr , mk , ry & br , mcc , zee + ngc 4535 & tp & cay , war & ky , nis , ry & ken , mk & ... + ngc 6946 & we & car & ry , ty , wal & ken , mk , ry & br , mcc + al : @xcite ; br : @xcite ; car : @xcite ; chr : @xcite ; cay : @xcite ; cs : @xcite ; deg : @xcite ; deh : @xcite ; hey : @xcite ; ky : @xcite ; ken : @xcite ; ka : @xcite ; lr : @xcite ; mar : @xcite ; mcc : @xcite ; nn : @xcite ; new : @xcite ; pag : @xcite ; puc : @xcite ; rog : @xcite ; rk : @xcite ; ry : @xcite ; ss : @xcite ; ty : @xcite ; tu : @xcite ; tp : @xcite ; zee : @xcite ; vil : @xcite ; wal : @xcite ; war : @xcite ; wev : @xcite we would like to emphasize that the models shown in this section have not been computed specifically for each of the sample galaxies , as was done in our previous works . the models shown in fig.[chequeo ] were selected among the 10 available ones for their corresponding rotation velocity following table[dis ] of the grid . thus the good agreement found between model results and data demonstrate that our bi - parametric models are able to adequately reproduce real galaxies . the computed star formation rate reproduces the relation obtained by @xcite , when it is represented _ vs _ the total gas surface density as can be seen in fig . [ ken ] . in the two upper panels of the figure ( @xmath194 ) , we have over - plotted the data from @xcite and @xcite which are seen to fall in the locus defined by the models . models with @xmath142 seem to be out of the region where this data lie . however , due to their extremely low efficiencies , we may assume that these models do not simulate normal bright spiral galaxies , but other kind of objects with a low stellar content and high gas fractions more similar to dwarf and lsb galaxies . we have , therefore , taken data from @xcite on these kinds of objects and computed the densities for both quantities assuming an optical radius of 5 kpc for all of them . obviously , a change in this radial dimension would vary the final values of our estimates , but our hypothesis is probably valid within a factor of 2 ( radii less than 10 kpc ) . under this assumption , we see that the points , shown in the third panel of fig.[ken ] , fall in the upper locus defined by the models . if the radius were smaller than assumed , the densities would be even higher , and the points would move in the figure following the direction given by the arrow . a second factor not included in these estimates , is the molecular gas content . there are some works suggesting that the molecular gas amounts to less than 10% in this kind of galaxies while other seems to indicate that the molecular mass may be as large as in the brightest massive spirals . in any case , the inclusion of this factor would move the points to the right . in both cases the data points would populate the region of the diagram occupied by the models . we therefore conclude that our models are able to reproduce the observed trend of sfr _ vs _ total gas surface density for different types of galaxies . the model computed abundance radial gradients for oxygen have been obtained by fitting a least squares straigh line to the oxygen abundances for radii @xmath195 kpc . the chosen radial range tries to eliminate the central region ( where the oxygen abundance distribution flattens ) and the outer region where there are no data . thus the calculated gradients may be compared more precisely with the corresponding observed ones . these radial gradients , measured as dex / kpc , are represented as a function of rotation velocity in fig . the relation between both quantities shows that the radial distributions are steeper for the models with lower rotation velocities if n@xmath196 . the radial gradients for a given rotation velocity are larger in absolute value for increasing n , and tend to zero value for the most massive galaxies and low n. the models with @xmath197 , however , tend to deviate from this function and show lower absolute values . that is , for a similar rotation curve , their radial distributions of abundances show a flattening as compared to those corresponding to the models with @xmath198 . the rotation velocity at which the models deviate from the common locus depends on @xmath199 : for @xmath200 it occurs for @xmath201 while for @xmath202 almost all rotation velocity curves produce flat radial gradients of abundances . data from @xcite are over - plotted on this fig . we see that the modelled trend reproduce the observations , although a more profound analysis about this ( and other ) correlation will be performed in the future . in order to use this grid for a given galaxy for which observational data are known , we must first select the radial distribution of mass . this means that we must know the total mass , the maximum rotation velocity or , if none of them is available , the luminosity or the magnitude in the i band . according to this value we may choose the number of the distribution from table [ grid_m ] . then , 10 different models , corresponding to the 10 different efficiencies , are available . the standard procedure in chemical evolution would be to see which of them is able to reproduce with success known data such as elemental abundances or gas densities . if more than one observational constraint is available , some kind of minimum error or maximum probability technique may be used for the purpose of choosing the best model . once this selection is performed we may use the time evolution given by the chosen model to predict the star formation history , the age - metallicity relation , the stellar populations or any other quantity relative to the modelled galaxy . obviously , the largest the observational number of constraints , the smallest the uncertainty in the selection of the best model . the old and well known uniqueness problem of the chemical evolution suggest that more than one model may reproduce the data . actually , we have shown in @xcite that this problem reduces greatly when more than two observational constraints can be used . only 4% of the 500 models computed with with different input parameters could reproduce the observations relative to hi , oxygen abundance and sfr at the same time . therefore we are confident that models may provide the evolutionary history of a given galaxy within a reasonable accuracy . sometimes , however , these observational constraints are not available ( or not with sufficient precision ) to select the adequate model . in that case , some other method to choose the possible evolutionary track is necessary . our best models from previous works give us evidence that the efficiencies to form molecular clouds from atomic gas and stars from molecular gas , seem to depend on the galaxy morphological type . if this were the case , the selection of the best model might be easier . in fact , this dependence was already found in @xcite , where these authors quantified the efficiencies to form molecular clouds and the frequency of cloud - cloud collisions , finding that a variation of 10 in the parameters h and @xmath61 is needed when the hubble type changes from one stage to the next ( sa to sb , etc .. ) . in order to check if a relation between efficiencies and morphological type exists , we have plotted in fig . [ emu1 ] , panel a ) the logarithmic efficiency @xmath74 , computed following the equation ( 19 ) from section 2.3 , as a function of t for the data of @xcite . there @xmath105 symbols correspond to the raw values while solid dots represent the averaged values obtained for 10 bins , one for each t. a clear decreasing correlation appears for @xmath203 . values for @xmath204 fall above the trend but the number of points is small for what we assume that the final increase is probably spurious . based on our previous works , we expect a relation of the form : @xmath205 . therefore , in panel b ) we have plotted the logarithmic efficiency as a function of @xmath206 where a linear correlation is apparent . a least - squares fit gives , for @xmath203 : @xmath207 therefore : @xmath208 on the other hand , taking into account the relationship obtained for the ratio between both efficiencies , @xmath74 and @xmath75 , this latter one may also be expressed by a similar function : @xmath209 where b would be 8 , following our hypothesis from section 2 . the result that efficiencies are related to galaxy morphological type may be useful when a model for a given galaxy must be chosen and the observational data are scarce . in that case , once the total mass distribution has been selected from the rotation curve or the luminosity for the galaxy , an initial choice can be made under the assumption that the galaxy evolution is represented by the model of efficiencies n with @xmath210 . we warn , however , that given the large dispersion of the data around the averaged efficiency values in fig . [ emu1 ] , other values of n are also possible . this finding would imply that the features of spiral galaxies depend at least on two parameters : total mass and morphological type , as some studies about the characteristics of galaxies , as the seminal work by @xcite , show ( see also * ? ? ? * ; * ? ? ? * ; * ? ? ? * among those more related with the chemical evolution of galaxies ) . it is also well known that the radial gradient , measured as dex @xmath211 , observed in spiral galaxies depends on morphological type : late - type galaxies show steeper radial distributions of oxygen abundances than earlier ones , which show in some cases almost no gradient @xcite . at the same time , a correlation seems to exist between oxygen abundance and total mass surface density which , in turn , is related to morphological type . this correlation is stronger when the total mass of the galaxy , including the bulge , instead of the mass of the exponential disc alone , is used @xcite . it is evident from these works that a linear sequence with the mass can not be obtained , and therefore our models , being a bi - parametric grid , would be adequate to find the best model for each galaxy , although the existing relation between the morphological type of galaxies and their total mass does difficult to discriminate which of these features is the origin of the different evolution of galaxies . from the chemical evolution point of view , a recent work ( moll & mrquez , in preparation ) , where the multiphase evolution model has been applied to a large sample ( 67 ) of spiral galaxies , seems to demonstrate that the total mass , through its influence on the collapse time scale , has a larger effect on the evolution of a galaxy and its final radial gradient of abundances than the selected values of the named efficiencies . thus , it seems as if galaxies would evolve mostly due to their total mass with some dispersion around the mean trend depending on hydrodynamical and environmental characteristics , which are taken into account in some way by the efficiencies values . the morphological type would then be the consequence of the different conditions of the intergalactic medium out of which the galaxy formed . we have calculated the chemical evolution of a wide set of theoretical galaxies characterised by their total mass , through the collapse time scale , and their efficiencies to form molecular clouds and massive stars from cloud - to - cloud collisions . with the selection of parameters and inputs described above , we have ran a total of 440 models , with 44 different rotation curves implying 44 values of total mass , characteristic collapse time - scale and disc radius and 10 sets of efficiencies for each one of them , implying 10 evolutionary rates for the star formation and gas consumption in the disc . for each model we have obtained the time evolution of the halo and the disc , and therefore the corresponding radial distributions for the relevant quantities ( masses , abundances , star formation rate , etc ... ) . the star formation history for each radial region ( halo and disc , separately ) and within each one , the mass in each phase of matter : diffuse gas , molecular gas , low - mass stars , massive stars and remnants , has been followed . besides that , we have obtained the abundances of 15 elements : h , d , he3 , he4 , c12 , c13 , n14 , o16 , ne , mg , si , ca , s , fe and neutron - rich nuclei , in all radial regions for both halo and disc . the results of our work can be summarised as follows : 1 . the atomic gas shows a maximum in its radial distribution for all galaxies . this maximum is nearer to the galaxy centre in the low mass or less evolved galaxies than in the more evolved or massive galaxies , for which the maximum is along the disc and moving toward the outer zones . this behavior produces , in some cases , a central _ hole _ in the distribution of the diffuse gas . the diffuse gas radial distribution results to be a very strong constraint for selecting the best model out of the 10 computed ones with different efficiencies , corresponding to the total mass of a given spiral galaxy . 2 . the oxygen abundance reaches a maximum level , as a consequence of a saturation effect which occurs earlier for the massive and more evolved galaxies . the less evolved galaxies do not reach this saturation level , except in the central region , and therefore show a steep radial gradient in their oxygen abundance . the less massive and less evolved galaxies have not yet developed a radial gradient and show flat radial distributions . this simulates an on - off effect : for @xmath200 a radial gradient appears if @xmath212 while at @xmath213 it only appears for @xmath214 . this behavior is in agreement with observations and solves the apparent inconsistency shown by trends showing steep gradients for late type galaxies and flatter ones for the earliest ones , while , at the same time , most irregulars show no gradient at all and very uniform abundances . 3 . the model calculated star formation rates reproduce the observed trend between the surface density of this quantity and the total gas density including , not only the massive normal galaxies data , but also the low surface brightness ones . actually , to our knowledge , there are not other chemical evolution models which compare predicted with observed radial distributions of diffuse and molecular gas , star formation rate and abundances for disc galaxies other than mwg . in this sense our models should be considered as an improvement over the standard ones . a study of the possible correlations among galaxy properties and different features obtained from the complete set of results is now possible , since a statistically significant number of theoretical models is available . this requires a deeper analysis which is out of the scope of this paper and will be done in the next future . we thank an anonymous referee for many useful comments and suggestions that have greatly improved this paper . this work has been partially funded by the spanish ministerio de ciencia y tecnologa through project aya-2000 - 093 . this work has made use of the nasa astrophysics data system , and the nasa / ipac extragalactic database ( ned ) , which is operated by the jet propulsion laboratory , caltech , under contract with the national aeronautics and space administration . braun , j. m. , 2001 , dwarf galaxies and their environment , proceedings of the bonn / bochum - graduiertenkolleg international conference , germany , 23 - 27 january 2001 , eds . : de boer k.s . , dettmar r .- j . , klein u. , shaker verlag , p. 5 meyer , m. r. , adams , f. c. , hillebrandt , l. a. , carpenter , j. m. , & larson , r. b. , 2000 , in protostars & planets iv , eds . manning , v. , boss , a. p. , & russell , s. s. ( tucson : the university of arizona press ) , p.121 rrrrrrrrrrrrrrrr time & r & h & d & @xmath121he & @xmath122he & @xmath123c & @xmath124c & n & o & ne & mg & si & s & ca & fe + ... & ... & ... & ... & ... & ... & ... & ... & ... & ... & ... & ... & ... & ... & ... & ... + 0.1 & 14 . & 0.770 & 0.70e04 & 0.10e04 & 0.230 & 0.11e06 & 0.21e09 & 0.13e07 & 0.92e06 & 0.23e06 & 0.32e07 & 0.55e07 & 0.26e07 & 0.37e08 & 0.42e07 + 0.1 & 12 . & 0.770 & 0.70e04 & 0.10e04 & 0.230 & 0.12e06 & 0.22e09 & 0.13e07 & 0.95e06 & 0.23e06 & 0.34e07 & 0.57e07 & 0.27e07 & 0.38e08 & 0.44e07 + 0.1 & 10 . & 0.770 & 0.70e04 & 0.10e04 & 0.230 & 0.13e06 & 0.23e09 & 0.14e07 & 0.10e05 & 0.25e06 & 0.35e07 & 0.60e07 & 0.28e07 & 0.40e08 & 0.46e07 + 0.1 & 8 . & 0.770 & 0.70e04 & 0.10e04 & 0.230 & 0.14e06 & 0.25e09 & 0.15e07 & 0.11e05 & 0.27e06 & 0.38e07 & 0.65e07 & 0.31e07 & 0.43e08 & 0.50e07 + 0.1 & 6 . & 0.770 & 0.70e04 & 0.10e04 & 0.230 & 0.15e06 & 0.28e09 & 0.17e07 & 0.12e05 & 0.30e06 & 0.43e07 & 0.73e07 & 0.35e07 & 0.49e08 & 0.56e07 + 0.1 & 4 . & 0.770 & 0.70e04 & 0.10e04 & 0.230 & 0.18e06 & 0.32e09 & 0.19e07 & 0.14e05 & 0.35e06 & 0.50e07 & 0.85e07 & 0.40e07 & 0.57e08 & 0.65e07 + 0.1 & 2 . & 0.770 & 0.70e04 & 0.10e04 & 0.230 & 0.45e06 & 0.46e09 & 0.28e07 & 0.38e05 & 0.95e06 & 0.13e06 & 0.22e06 & 0.10e06 & 0.14e07 & 0.15e06 + ... & ... & ... & ... & ... & ... & ... & ... & ... & ... & ... & ... & ... & ... & ... & ... + 13.2 & 14 . & 0.769 & 0.69e04 & 0.12e04 & 0.231 & 0.59e04 & 0.12e06 & 0.68e05 & 0.15e03 & 0.34e04 & 0.52e05 & 0.10e04 & 0.50e05 & 0.71e06 & 0.24e04 + 13.2 & 12 . & 0.768 & 0.69e04 & 0.13e04 & 0.231 & 0.10e03 & 0.21e06 & 0.96e05 & 0.27e03 & 0.57e04 & 0.84e05 & 0.20e04 & 0.96e05 & 0.13e05 & 0.42e04 + 13.2 & 10 . & 0.764 & 0.65e04 & 0.21e04 & 0.234 & 0.43e03 & 0.98e06 & 0.38e04 & 0.11e02 & 0.21e03 & 0.30e04 & 0.86e04 & 0.40e04 & 0.55e05 & 0.17e03 + 13.2 & 8 . & 0.754 & 0.58e04 & 0.44e04 & 0.240 & 0.11e02 & 0.44e05 & 0.16e03 & 0.28e02 & 0.54e03 & 0.76e04 & 0.22e03 & 0.10e03 & 0.14e04 & 0.44e03 + 13.2 & 6 . & 0.746 & 0.51e04 & 0.73e04 & 0.246 & 0.16e02 & 0.11e04 & 0.37e03 & 0.40e02 & 0.78e03 & 0.11e03 & 0.33e03 & 0.16e03 & 0.21e04 & 0.67e03 + 13.2 & 4 . & 0.736 & 0.43e04 & 0.13e03 & 0.253 & 0.21e02 & 0.22e04 & 0.64e03 & 0.49e02 & 0.98e03 & 0.14e03 & 0.44e03 & 0.21e03 & 0.28e04 & 0.93e03 + 13.2 & 2 . & 0.693 & 0.63e05 & 0.43e03 & 0.287 & 0.37e02 & 0.71e04 & 0.18e02 & 0.81e02 & 0.18e02 & 0.26e03 & 0.83e03 & 0.40e03 & 0.56e04 & 0.20e02 +
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we present a generalization of the multiphase chemical evolution model applied to a wide set of theoretical galaxies with different masses and evolutionary rates .
this generalized set of models has been computed using the so - called _ universal rotation curve _ from @xcite to calculate the radial mass distribution of 44 _ theoretical _ protogalaxies .
this distribution is a fundamental input which , besides its own effect on the galaxy evolution , defines the characteristic collapse time - scale or gas infall rate onto the disc .
we have adopted 10 sets of values , between 0 and 1 , for the molecular cloud and star formation efficiencies , as corresponding to their probability nature , for each one of the radial distributions of total mass .
thus , we have constructed a bi - parametric grid of models , depending on those efficiency sets and on the rotation velocity , whose results are valid in principle for any spiral or irregular galaxy .
the model results provide the time evolution of different regions of the disc and the halo along galactocentric distance , measured by the gas ( atomic and molecular ) and stellar masses , the star formation rate and chemical abundances of 14 elements , for a total of 440 models .
this grid may be used to estimate the evolution of a given galaxy for which only present time information such as radial distributions of elemental abundances , gas densities and/or star formation , which are the usual observational constraints of chemical evolution models is available .
[ firstpage ] galaxies : abundances galaxies : evolution galaxies : spirals galaxies : stellar content
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in the ctf ii drive beam gun , cs - te photocathodes are used to produce a pulse train of 48 electron bunches , each 10ps long and with a charge of up to 10nc @xcite . in ctf , the main limit to lifetime is the available laser power , which requires a minimal quantum efficiency ( qe ) of 1.5% to produce the nominal charge . although cs - te photocathodes are widely used , a complete understanding , especially of their aging process , is still lacking . spectra of the qe against exciting photons may help to understand the phenomenon . according to spicer @xcite , the spectra of the quantum efficiency ( qe ) of semiconductors with respect to the energy of the exciting photons ( @xmath0 ) can be described as : @xmath1 where @xmath2 is the threshold energy for photoemission , c@xmath3 and c@xmath4 are constants . to measure the spectral response of photocathodes , wavelengths from the near uv throughout the visible are necessary . to attain these , an * o*ptical * p*arametrical * o*scillator was built @xcite . a frequency - tripled nd : yag laser pumps a betabarium borate ( bbo ) crystal in a double - pass configuration , as shown in fig.[fig : opo ] . the emerging signal - beam , with wavelengths between 409 nm and 710 nm , is frequency doubled in two bbo crystals . the wavelengths obtained are between 210 nm and 340 nm . the idler - beam delivers wavelengths between 710 nm and @xmath5 nm . the measurements of the spectral response of photocathodes were made in the dc - gun of the photoemission lab at cern @xcite , at a field strength of about 8 mv / m . spectra were taken shortly after the evaporation of the cathode materials onto the copper cathode plug , as well as after use in the ctf ii rf - gun @xcite at fields of typically 100 mv / m . to be able to interpret the spectra in terms of spicer s theory , it was necessary to split the data into 2 groups , one at `` low photon energy '' and one at high photon energy , see fig.[fig : cath87 ] . then , the data can be fitted well with two independent curves , following eq.([eq : spicer ] ) , which give two threshold energies . for a typical fresh cs - te cathode , the high energy threshold is 3.5ev , the low one is 1.7ev , as shown in fig.[fig : cath87 ] , upper curve . this might be a hint that two photo - emissive phases of cs - te on copper exist . several explanations are possible : the copper might migrate into the cs - te , creating energy levels in the band gap ; or possibly not only cs@xmath4te , but also other cs - te compounds might form on the surface and these might give rise to photoemission at low photon energy . a hint to this might be that the ratio of evaporated atoms of each element is not corresponding to cs@xmath4te , see below . after use , we found that not only the complete spectrum shifted towards lower quantum efficiency , but also that the photoemission threshold for high qe increased to 4.1ev , which is shown in fig.[fig : cath87 ] , lower curve . one might expect that the photocathode is poisoned by the residual gas , preventing low - energy electrons from escaping . however , because typical storage lifetimes are of the order of months , the effect must be connected to either the laser light , or the electrical field . we also produced a cs - te cathode on a thin gold film of 100 nm thickness . as shown in fig.[fig : cath120 ] , the shoulder in the low energy response disappeared . it is difficult to fit a curve for the spicer model to the low energy data . the high " photoemission threshold is at 3.5ev . at the moment , this cathode is in use in the ctf ii gun and will be remeasured in the future . in terms of lifetime , this cathode is comparable to the best cs - te cathodes , as it has already operated for 20 days in the rf - gun . as a new material presented first in @xcite , we tested rubidium - telluride . we took spectra of qe before and after use in the ctf ii gun , as for cs - te . remarkably , with this material , there was no shift in the photoemission threshold towards higher energies , but only a global shift in qe , see fig.[fig : rb2te ] . this might be due to the lower affinity of rubidium to the residual gas . detailed investigations are necessary to clarify this . long lifetimes for cs - te cathodes are achieved only when they are held under uhv ( @xmath6 mbar ) . other photocathode materials like k - sb - cs are immunized against gases like oxygen by evaporating thin films of csbr onto them @xcite . therefore , we evaporated a csbr film of 2 nm thickness onto the cs - te . fig.[fig : csbr ] shows the spectrum before the csbr film ( square points ) and after it ( round points ) . the qe at 266 nm dropped from 4.3% to 1.2% . in addition , the photoemission threshold was shifted from 3.9ev to 4.1ev . a long - term storage test showed no significant difference between uncoated and coated cathodes . more investigations will determine the usefulness of these protective layers . in order to increase the sensitivity of the on - line qe measurement during evaporation of the photocathodes , we monitored the process with light at a wavelength of 320 nm . we did not see any significant improvement in sensitivity , notably in the high qe region . film thicknesses are measured during the evaporation process by a quartz oscillator @xcite . typical thicknesses for high quantum efficiencies at @xmath7 nm are 10 nm of tellurium and around 15 nm of cesium . this results in a ratio of the number of atoms of each species of @xmath8 , far from the stoichiometric ratio of 0.5 for cs@xmath4te . it is known that tellurium interacts strongly with copper @xcite , so that not all of the evaporated tellurium is available for a compound with subsequently evaporated cesium . therefore , we used also mo and au as substrate material . however , the ratio between the constituents necessary for optimum qe , did not change significantly . another reason might be that instead of cs@xmath4te , cs@xmath4te@xmath9 is catalytically produced on the surface . this compound , as well as some others , was found to be stable @xcite . lifetime in ctf depends on parameters like maximum field strength on the cathode , vacuum and especially extracted charge . typically , a cathode is removed from the gun , if the qe falls below 1.5% . as shown in fig.[fig : lifetime ] , lifetime does not depend on the initial qe ; a cathode having an initial qe of 15% ( round points ) lasted as long as one with 5% ( triangles ) . as shown in table[tab1 ] , the average current produced in ctf ii is nearly a factor 10000 lower than what is required for the clic drive beam . a test to produce 1mc is under preparation in the photoemission laboratory at cern . the exact reproduction of the clic pulse structure would require the clic laser , which is still in the design stage .comparison of cathode relevant parameter [ cols="^,^,^,^",options="header " , ] [ tab1 ] in a collaboration between rutherford appleton laboratory and cern . a test which is compatible with our current installation is the production of 1ma of dc current , which requires a uv laser power of 300mw at the cathode . for this test , we will illuminate the cathode with pulses of 100ns to 150ns pulse length , at repetition rates between 1khz and 6khz . as table[tab1 ] shows , this is a factor 1000 more average current than in ctf ii , and also demonstrates the basic ability of the cathodes to produce the ctf 3 drive beam ( i=26@xmath10a ) . clic is still a factor 75 away . we are currently searching for ways to produce higher charges as well . measurements of qe against photon energy are routinely made after production and after use of photocathodes . we have demonstrated that both low energy and high energy responses agree well with spicer s theory . a gold buffer layer reduces the low energy response of cs - te cathodes . more work is needed to understand the measurements of the stoichiometric ratio of cs - te . coating with 2 nm csbr significantly decreased the quantum efficiency , without improving the storage lifetime . for the high - charge drive beam of clic , it is still necessary to demonstrate the capabilities of cs - te , for which first tests will be done soon . e. chevallay , j. durand , s. hutchins , g. suberlucq , m. wrgel , photocathodes tested in the dc gun of the cern photoemission laboratory " , nuclear instruments methods in physics research section a , vol . 340 , ( 1994 ) 146 - 156 , cern clic note 203 e. shefer , a. breskin , r. chechik , a. buzulutskov , b.k . singh , m. prager , coated photocathodes for visible photon imaging with gaseous photomultipliers " , nuclear instruments methods in physics research section a , vol.433 , no.1 - 2 , ( 1999 ) 502 - 506
|
for short , high - intensity electron bunches , alkali - tellurides have proved to be a reliable photo - cathode material .
measurements of lifetimes in an rf gun of the clic test facility ii at field strengths greater than 100 mv / m are presented . before and after using them in this gun , the spectral response of
the cs - te and rb - te cathodes were determined with the help of an optical parametric oscillator .
the behaviour of both materials can be described by spicer s 3-step model . whereas during the use the threshold for photo - emission in cs - te
was shifted to higher photon energies , that of rb - te did not change .
our latest investigations on the stoichiometric ratio of the components are shown .
the preparation of the photo - cathodes was monitored with 320 nm wavelength light , with the aim of improving the measurement sensitivity .
the latest results on the protection of cs - te cathode surfaces with csbr against pollution are summarized .
new investigations on high mean current production are presented .
| 2,761 | 291 |
the interest in the feas - based superconductors@xcite is ongoing after six years of extensive research as still no consensus has been achieved concerning the superconducting character and pairing mechanism . lifeas is special amongst the many feas - based superconductors , as superconductivity appears in the parent compound at elevated temperatures without doping or application of pressure . this particularity of lifeas most likely arises from its electronic structure with strongly reduced nesting between electron and hole fermi - surface sheets as it was first deduced from angle - resolved photoemission spectroscopy ( arpes ) @xcite . in the 1111 and 122 families ( named after their stoichiometry ) the fermi nesting conditions are excellent stabilizing a spin density wave ( sdw ) , which has to be suppressed by doping@xcite or the application of pressure@xcite in order to reach the superconducting state . lifeas does not exhibit any structural transition nor a magnetically ordered phase.@xcite theoretical calculations@xcite explain this fact by its poor fermi nesting properties and unusually shallow hole pockets around the @xmath2 point , which is in agreement with arpes experiments.@xcite the flat top of the hole pockets implies a large density of states around the @xmath2 point and in combination with small - momentum scattering vectors within the inner hole pocket this would favor ferromagnetic fluctuations and a triplet pairing mechanism.@xcite the symmetry of the order parameter has been a controversial subject , several reports using arpes , quasiparticle interference ( qpi ) or theoretical approaches favor an @xmath3 wave,@xcite while there is also support for a @xmath4-wave state.@xcite although the calculations in ref . support an @xmath3 wave state driven by collinear antiferromagnetic fluctuations , the authors state that ferromagnetic fluctuations stemming from the small hole pocket at the @xmath2 point may dominate at higher energies and/or at higher temperatures . in our previous work@xcite we have established the energy and temperature dependence of an antiferromagnetic excitation located at an incommensurate position @xmath5 resembling magnetic correlations in electron doped bafe@xmath1as@xmath1 . similar results were obtained by wang et al . @xcite the origin of the magnetic signal has been interpreted as scattering between the electron pockets centered around the @xmath6 point and either the outer@xcite or the inner@xcite hole pockets around the zone center.in this work we present a comprehensive inelastic neutron scattering ( ins ) study using different cold and thermal triple - axis spectrometres and a time - of - flight instrument devoted to extend the characterization of the incommensurate antiferromagnetic fluctuations in single - crystalline lifeas . we present the inelastic scattered neutron intensity in absolute units using two different techniques leading to perfectly agreeing results . the magnetic fluctuations have been investigated up to energy transfers of 80 mev and spin - space anisotropies have been studied by polarized neutrons with longitudinal polarization analysis ( lpa ) . furthermore , we have investigated @xmath7 in a broad @xmath8-@xmath9 range to search for any ferromagnetic fluctuation at elevated temperatures and energy transfers . the same single crystal sample as in ref . has been used for all the experiments presented here . the normalization to an absolute intensity scale has been done with data obtained at the thermal triple - axis spectrometer 1 t ( laboratoire lon brillouin , saclay ) , which was used with a pyrolytic graphite ( pg ) monochromator and a pg analyzer . the final neutron energy was fixed at @xmath10 mev . the in20 spectrometer ( institut laue - langevin , grenoble ) was used with the flatcone multianalyzer in order to record @xmath11-maps with different @xmath12 values at different temperatures and energy transfers . in20 has also been used in the polarized mode using polarizing heusler ( 111 ) crystals as a monochromator and an analyzer . for the lpa a set of helmholtz coils was used to guide and orient the neutron polarization . lpa offers the possibility of distinguishing between nuclear and magnetic scattering and it furthermore allows the separation of the two magnetic components perpendicular to the scattering vector . generally , nuclear scattering is a non - spin - flip ( nsf ) process regardless of the initial neutron polarization state . only magnetic components perpendicular to the scattering vector ( @xmath13 by definition ) are accessible in a neutron experiment . the components perpendicular to the polarization axis ( @xmath14 being in the scattering plane and @xmath15 being the perpendicular axis of the spectrometer ) contribute to the spin - flip ( sf ) channel , while those parallel to the axis of polarization scatter into the nsf channel . the puma spectrometer ( frm - ii , garching ) was used with a pg monochromator and a pg analyzer with a fixed final neutron energy of @xmath10 mev . high energy transfers were measured at the time - of flight spectrometer maps ( rutherford - appleton laboratory , didcot ) . the incident beam energies were @xmath16 and 100 mev with @xmath17 parallel to the @xmath18 axis . the measured intensities were normalized to absolute units by using a vanadium standard ( with 30% error ) . in order to express the dynamic susceptibility of lifeas in absolute units data taken on the time - of - flight spectrometer maps and triple - axis spectrometer data from the 1 t instrument were used yielding perfect agreement . the time - of - flight data can be normalized by comparison with incoherent scattering from a vanadium sample and with the sample mass . this procedure is well - established at the maps instrument and described in large detail in reference.@xcite in contrast the normalization of triple - axis data is more complex as the resolution function and the beam profile are more structured . here we follow the most common way to normalize the magnetic scattering by comparison with phonon measurements on the same sample . this method , furthermore , excludes mistakes arising from impurity phases . the scattering potential of the sample is discussed in terms of the double - differential cross section @xmath19 with @xmath20 the final energy . in any ins experiment this entity is folded with the resolution and transmittance function of the instrument . we use the reslib programs@xcite to quantitatively analyse the scattering intensities . in our experiment a neutron monitor between the monochromator and the sample is used to scale the detector counts into the entity counts per given monitor ( note that this monitor is corrected for higher order contaminations ) . the calculation splits the instrumental effects in the finite gaussian resolution and a transmittance term . the intrinsic double differential cross section is first folded with the gaussian resolution in the four - dimensional space consisting of q - space and energy , see eq . [ eq : ddcs ] . here @xmath21 is the resolution matrix according to the popovici approximation @xcite and @xmath22 the four - dimensional difference vector consisting of the q - space coordinates and energy , see eq . [ eq : delta ] . @xmath23 \right ) } \label{eq : ddcs } \\ { \bf \delta } : = & ( q'_x - q_x , q'_y - q_y , q'_z - q_z,\omega'-\omega ) \label{eq : delta}\end{aligned}\ ] ] in order to calculate the intensity in the detector one has to multiply the folded double cross section with a normalization factor @xmath24 describing amongst others the efficiency of the secondary spectrometer and the resolution function normalization @xmath25 . in contrast to the reslib manual we do not include the @xmath26 factor to @xmath27 but follow the common practice keeping this factor in the double - differential cross section.@xcite @xmath28 for known resolution and transmission functions the study of a predictable signal allows one to determine the scale factor @xmath18 describing amongst others the effective sample size . the transformed double - differential cross section @xmath29 thus contains the intrinsic scattering strength of the system combined with the spectrometer properties . we use the scattering by an acoustic phonon for normalization . the single - phonon cross section is given by eq . [ eq : crosssec ] ( refs . ) where @xmath30 is the bose population factor for neutron energy loss . @xmath31 denotes the dynamical structure factor of the particular phonon mode at this scattering vector , which can be calculated with the help of a lattice - dynamical model , see eq . [ eq : fdyn ] . the @xmath32 function in eq . [ eq : fdyn2 ] is approximated in the calculation by a lorentzian profile with finite half width . the symbols in eqs . [ eq : crosssec]-[eq : fdyn2 ] follow the same convention as in refs . . @xmath33 for an acoustic phonon close to the brillouin - zone center one may further simplify the calculation as all atoms in the primitive cell are parallel polarized with components @xmath34 ( here @xmath35 and @xmath36 denote the individual and total masses , respectively ) . the dynamic structure factor then corresponds to that of the nuclear bragg reflection multiplied by the length of the scattering vector , @xmath37 , the inverse square root of the total mass and by the cosine of the angle between scattering vector and phonon polarization , @xmath38 . the latter factor is close to one in a reasonably chosen scan . @xmath39 } \label{eq : fdyn2}\ ] ] the double differential cross section of the phonon scattering is obtained by subtracting the ( refined ) background from the raw data and then by dividing by the ( refined ) scale factor . the phonon dispersion is described by a simple linear relation , @xmath40 . fitting the phonon cross - section with its intensity prefactors to the raw data using the reslib code yields a scale factor of 13.1(8 ) and a constant background of 2 counts per monitor . the raw data can therefore be converted into an absolute scale that still contains the resolution functions of the instrument , see the right axis of ordinate in fig . [ fig : phonon ] . ( color online ) raw data showing the transversal @xmath41-scan across the ( 220 ) phonon at @xmath42 k an energy transfer of 4.5 mev measured at the 1 t spectrometer . the scattered intensity is given in counts/(monitor 25000 ) on the left ordinate and in absolute cross section values on the right ordinate for the folded cross section @xmath43 . , scaledwidth=48.0% ] in order to evaluate the magnetic signal we start with the autocorrelation of the spin fourier coefficients @xmath44 , here @xmath45 denote the space indices , @xmath46 the neutron gyromagnetic factor , @xmath47 the electron charge , @xmath48 the electron mass , @xmath18 the speed of light , @xmath49 the magnetic form factor at the scattering vector , @xmath50 the land factor and @xmath51 the kronecker symbol . note that the second line of eq . [ eq : ddcsmag ] has the unit of an inverse energy ( ev@xmath52 ) . @xmath53 with the fluctuation dissipation theorem one may transform the cross section to the imaginary part of the generalized dynamic susceptibility , which we assume here to be isotropic in spin space . @xmath54\cdot 2\cdot \chi '' ( { \bf q},\omega ) \nonumber \\ \label{eq : ddcsmag2}\end{aligned}\ ] ] a susceptibility can be given in various units creating considerable confusion but here the unit problem drops out due to the term bohr - magneton , @xmath55 , squared in the denominator . the natural microscopic unit to discus thevsusceptibility is thus @xmath56 per formula unit , which we will use in the following . deducing the absolute scale of the cross section of the magnetic fluctuation is now obtained in the same way as in the phonon case by subtracting the background and by dividing by the scale factor obtained from the phonon fit . however , in order to fit the data and deduce the background a model is needed to describe the generalized susceptibility @xmath57 . we assume a superposition of single relaxor functions ( eq . [ eq : relaxor ] ) in energy with a lorentzian q - dependence centered at the four positions @xmath58=(0.5@xmath59@xmath60,0.5@xmath61@xmath60,0 ) and ( 0.5@xmath59@xmath62,0.5@xmath61@xmath62,0 ) . we take only the in - plane components of @xmath63 into account . @xmath60 ( @xmath62 ) is the incommensurability of 0.057(3 ) r.l.u . [ 0.17(2 ) r.l.u . ] ( see sec . [ sec : tas ] for a detailed description of the two signals ) and the hwhm was refined to @xmath64=0.042(9 ) r.l.u [ @xmath65=0.07(3 ) ] , which yields the best agreement with the experimental data . @xmath66 in a first step , the constant background ( 11 counts per monitor 25000 @xmath67 20 s ) was determined and subtracted from the raw data which was then divided by the scale factor deduced from the phonon fit yielding the transformed double - differential cross section @xmath43 . from this one may obtain a susceptibility folded with the instrument resolution and transmission by dividing by all the intensity prefactors in eq . [ eq : ddcsmag2 ] ; this result is shown in fig . 2 on the right coordinate axis . the intrinsic strength and shape of @xmath57 , however , can only be obtained by fitting the model , eq . [ eq : relaxor ] , to the raw data ( a value of 10.9 mev has been used for @xmath2 obtained by the single - relaxor fit to the data presented in sec . [ sec : results].b ) . thereby we obtain @xmath68=7.4(8 ) and at the outer and inner incommensurate positions , respectively , see fig . [ fig : absfluct ] . ( color online ) incommensurate antiferromagnetic fluctuations at @xmath42 k and at an energy transfer of 5 mev measured at the 1 t spectrometer . the ordinates are given in absolute units of cross section ( left ) or of the generalized susceptibility ( right ) . the inset shows the raw magnetic data before subtraction of the background and the division by the scale factor.,scaledwidth=48.0% ] the incommensurate fluctuation has been reinvestigated at the thermal triple - axis spectrometer puma ( frm - ii , garching ) . [ fig : puma ] shows transverse q - scans across the ( 0.5 0.5 0 ) position at different energy transfer clearly documenting a complex q - shape of the magnetic response . the incommensurate magnetic correlations exhibit at least an asymmetric profile with pronounced shoulders towards larger incommensurability ( compared to the ( 0.5,0.5,@xmath69 ) center ) . therefore , the data have been described with two pairs of symmetrical gaussian functions on a constant background . note that parts of the data are contaminated by phonon scattering towards lower energy transfer for which these data points are not shown . the resulting fit curves [ ( red ) solid lines ] ) show a very good agreement with the raw data , while the dashed ( dash - dotted ) curves indicate the contribution of the signal at @xmath70@xmath71(0.43 0.57 0 ) [ @xmath72@xmath71(0.35 0.65 0 ) ] . angle resolved photoemission spectroscopy ( arpes ) experiments @xcite have revealed the fermi surface to consist of two similarly sized electron - like sheets around the @xmath73 point and hole - like sheets around the @xmath2 point . in ref . the authors have identified the ins signal to be connected to scattering between the outer hole pocket and the electron pockets by using a simple tight - binding fit to the arpes data , while involvement of the inner hole pocket was concluded in ref . . deeper understanding of the nesting signal requires the analysis of the orbital character of the various fermi surface sheets which essentially arise from the @xmath74 @xmath75 , @xmath76 and @xmath77 orbitals.@xcite there seems to be agreement that the outer hole pocket can be identified with @xmath77 orbital character which also contributes to the electron pockets . the @xmath77 states should result in two - dimensional bands , but @xmath75 and @xmath76 contributions yield considerable dispersion along the perpendicular directions and strong @xmath78 modulation of the fermi surfaces.@xcite if one associates the nesting magnetic correlations exclusively with @xmath77 orbitals it appears difficult to understand a split signal but an asymmetry or a shoulder can arise from a peculiar detail of the fermi surface shape that is not sufficiently well understood so far . sr@xmath1ruo@xmath79 is a well studied example with incommensurate magnetic correlations arising from fermi - surface nesting,@xcite and this material also exhibits an asymmetric magnetic response with a shoulder . quite recently four theoretical papers aimed to quantitatively model the variation of the superconducting gap on the fermi surface sheets arriving at contradictory results.@xcite the quantitative description of magnetic excitations by analyzing transitions between states with the same or different orbital character will help to arrive at a better understanding of the electronic structure of lifeas . the peak intensities of @xmath70 and @xmath72 have been followed as a function of energy above and below @xmath80 . as it can be seen in fig . [ fig : escanpuma ] the main signal @xmath70 shows the same dependence as already reported in ref . with a crossover between the scattered intensity below and above @xmath80 at 4.5 mev and an increase of intensity above 7 - 8 mev . on the other hand the intensity at @xmath72 suggests a different behaviour in dependence on the energy transfer . the scattered intensity at 20 k stays above the one at 5 k up to an energy transfer of roughly 7 mev , above which the value @xmath81@xmath82@xmath83 becomes stronger than @xmath81@xmath84@xmath83 . the different energy dependences of the signals at @xmath70 and @xmath72 strengthen the assumption of their independent origin and can be explained due to different gap values on different parts of the fermi surface . ( color online ) energy dependence of the ins scattering at @xmath70=(0.425 0.575 0 ) and @xmath72=(0.35 0.65 0 ) measured at the puma spectrometer . the dashed lines indicate where the scattered intensities of the normal [ ( red ) circles ) ] and superconducting state [ ( blue ) squares ] cross.,scaledwidth=42.0% ] the derived amplitude of the excitation at @xmath70 has been corrected for the monitor and the bose factor yielding the imaginary part of the generalized susceptibility which is shown in fig . [ fig : relaxor ] . the data have been fitted with single - relaxor functions . the data does not allow to state a clear tendency of the critical energy , however , a clear reduction of @xmath85 towards higher temperatures is observable . in addition the incommensurate magnetic correlations become strongly broadened at the temperature of only 100 k where the two peak structure has already changed into a broad plateau . ( color online ) imaginary part of the generalized susceptibility at 5 k , 20 k and 100 k as obtained by the amplitude from the fits to the data shown in fig . [ fig : puma ] and correction for the monitor and the bose factor . the solid lines represent fits by a single relaxor functions @xmath86,scaledwidth=42.0% ] by using the triple - axis spectrometer in20 in combination with the flatcone multianalyzer , planar sections of the reciprocal space can be recorded by simple @xmath87 scans which are afterwards converted into @xmath8-space . as theory predicts that a ferromagnetic instability may dominate at higher temperatures and/or higher energies,@xcite maps of the reciprocal space have been recorded up to 150 k and an energy transfer of 40 mev focusing on the ( 100 ) and ( 110 ) positions . however , our obtained data does not give any hint for ferromagnetic fluctuations in lifeas . the in20 spectrometer has then been used with polarized neutrons whose polarization axis after the scattering process has been analyzed . the observation of the incommensurate signal in the sf channels proves its magnetic origin ( note that the sf background has been subtracted according to the description in ref . ) . the peak intensity at the point @xmath8=(0.43 0.57 0 ) has been measured as a function of the energy transfer for the sf@xmath88 and sf@xmath89 channels ( fig . [ fig : escan ] ) . although only the sf@xmath88 channel has been measured with high statistics a slight spin - space anisotropy of the magnetic fluctuation is visible between 6 and 12 mev , where the out - of - plane fluctuation lies above the in - plane fluctuation similar to observations in electron doped bafe@xmath1as@xmath1 @xcite . however , the spin - space anisotropy in lifeas needs further experimental corroboration by measuring the other channels with better statistics . ( color online ) energy scan of the sf@xmath88 and sf@xmath89 intensities at @xmath8=(0.43 0.57 0 ) measured at the in20 spectrometer showing a local anisotropy of the magnetic fluctuation between 6 and 12 mev.,scaledwidth=42.0% ] in order to reveal eventual weak ferromagnetic fluctuations a @xmath41-scan across @xmath8=(110 ) at @xmath90 k and @xmath91 mev has been carried out . all three sf channels revealed neutron counts similar to the sf background meaning that no significant magnetic scattering is present . due to the limitation of triple - axis spectrometers concerning the incident energy , high energy transfers have to be measured using a time - of - flight spectrometer . however , higher incident energies are at the cost of a loss in resolution . with the @xmath18 axis of the sample aligned along the incident beam one obtains a projection of @xmath7 along this axis after the measurement of a curved 3-dimensional hypersurface in the 4-dimensional manifold of reciprocal space . in the projection the @xmath12 component is an implicit variable which changes with energy tranfer , nevertheless being calculable , i.e. the obtained data is three - dimensional in @xmath92-space . in order to visualize the data the program mslice has been used which offers the possibility of averaging the data along a chosen axis to produce a slice or integrating along two axes to produce a cut . by measuring a standard vanadium sample with known mass the intensity can be normalized to an absolute scale in mb/(sr mev f.u . ) by using the sample mass and molar mass . [ fig : fluctuation](a ) shows a slice of the @xmath93 plane which has been integrated between 10 mev and 30 mev for an incident beam energy of 55 mev . two peaks can be observed around the ( 0.5 0.5 0 ) position . by integrating the data perpendicular to the scan path indicated by the dashed line , one obtains the curve shown in fig . [ fig : fluctuation](b ) clearly revealing the incommensurability of the antiferromagnetic fluctuations . however , also the time - of - flight data indicates an asymmetric shape or an additional signal at larger incommensurability . two pairs of symmetrical gaussian functions on a constant background have been fitted to the data , from which the incommensurabilities @xmath94 and @xmath95 could be extracted . the value of @xmath60 is in good agreement with our previous results.@xcite note that the absolute intensity scale obtained by the renormalization to a vanadium standard ( between 10 and 30 mev ) is in very good agreement with our results shown in sec . [ sec : absolute ] ( fig . [ fig : absfluct ] , 5 mev ) and also with a report on polycrystalline samples.@xcite ( color online ) ( a ) time - of - flight data showing @xmath7 in the @xmath93 plane integrated between 10 mev and 30 mev ( @xmath96=55 mev , @xmath97=10 k. the incommensurate peaks are marked by a white ellipse . the black dashed indicates the cut along the [ h -h 0 ] direction shown in ( b ) . the fit of two pairs of symmetrical gaussian functions after the subtraction of a constant background ( determined from the fit in the inset ) yields the incommensurabilities @xmath94 and @xmath95.,scaledwidth=42.0% ] in order to investigate the magnetic signal at higher energy transfers an incident neutron energy of 100 mev has been used . [ fig : e100 ] shows @xmath93 slices of 10 mev thickness each . the magnetic fluctuation can be observed around the ( 0.5 0.5 0 ) point , but the loss in resolution becomes evident . however , there is a significant signal which can be separated from the background up to an energy transfer of 60 mev . for the slice in fig . [ fig : e100](f ) the signal is reduced to the background . due to the limited @xmath8-resolution in comparison to @xmath96=55 mev the incommensurability could not be investigated at higher energy transfers . in summary we have extended our previous work concerning the characterization of the incommensurate antiferromagnetic fluctuations in lifeas . time - of - flight experiments show that the magnetic signal is observable up to energy transfers of 60 mev , while the incommensurability remains unchanged up to 30 mev ( measurements of higher energy transfers were at the cost of resolution prohibiting a quantitative analysis of the incommensurability ) . longitudinal polarization analysis proved the magnetic origin of the observed signal and an eventual spin - space anisotropy between 8 and 10 mev could be deduced that resembles observation in other feas - based superconductors . the asymmetric shape of the incommensurate peak suggests the presence of two different signals which may correspond to scattering between the outer hole pocket and the inner electron pocket as well as between the outer hole pocket and the outer electron pocket . the different energy dependences of the peak intensities of @xmath70 and @xmath72 support the picture of two independent signals . furthermore , we have converted the intensity of the scattered neutrons into an absolute scale making it possible to compare the strength of the magnetic fluctuations in lifeas with those of related compounds . nearly optimally co - doped bafe@xmath1as@xmath1 yields a maximum value of roughly 7.5 mb/(sr mev fe ) at the resonance feature ( ref . ) . by averaging the peak values of the triple - axis ( fig . [ fig : absfluct ] ) and time - of - flight data ( fig . [ fig : fluctuation ] ) we obtain 0.95 mb/(sr mev fe ) rendering the low - temperature fluctuations in lifeas by a factor 8 weaker than the magnetic resonance in co - doped bafe@xmath1as@xmath1 . this perfectly agrees with our earlier work,@xcite where we estimated the same ratio between the incommensurate fluctuations in lifeas and the commensurate resonance in ba(fe@xmath98co@xmath99)@xmath1as@xmath1 by normalizing the magnetic signal to the respective phonon signal . due to the incommensurability a factor of two is recovered for which the magnetic scattering per fe ion in lifeas is by roughly a factor four weaker than the respective scattering in nearly optimally co - doped bafe@xmath1as@xmath1 which must be reconciled with the fact that the superconducting transition temperature is only little reduced in lifeas . this work was supported by the deutsche forschungsgemeinschaft ( dfg ) through the priority programme spp1458 ( grants no . be1749/13 , bu887/15 - 1 and br2211/1 - 1 ) . thanks the dfg for funding in the emmy noether programme ( project 595/3 - 1 ) .
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we present an inelastic neutron scattering study on single - crystalline lifeas devoted to the characterization of the incommensurate antiferromagnetic fluctuations at @xmath0 .
time - of - flight measurements show the presence of these magnetic fluctuations up to an energy transfer of 60 mev , while polarized neutrons in combination with longitudinal polarization analysis on a triple - axis spectrometer prove the pure magnetic origin of this signal .
the normalization of the magnetic scattering to an absolute scale yields that magnetic fluctuations in lifeas are by a factor eight weaker than the resonance signal in nearly optimally co - doped bafe@xmath1as@xmath1 , although a factor two is recovered due to the split peaks owing to the incommensurability .
the longitudinal polarization analysis indicates weak spin space anisotropy with slightly stronger out - of - plane component between 6 and 12 mev .
furthermore , our data suggest a fine structure of the magnetic signal most likely arising from superposing nesting vectors .
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nowadays the family of iron pnictides is a well - established and important prototype system for unconventional high - temperature superconductivity . starting with the first famous compound @xcite in 2008 , today several different sub - families with a wide structural variety are known . all different groups of iron pnictides share some common physical properties , such as their interesting and sometimes puzzling magnetic behavior . most compounds show a phase transition at low temperatures from a tetragonal to an orthorhombic crystal symmetry which is typically accompanied by the formation of long - range antiferromagnetic order.@xcite it is common believe that the suppression of these phase transitions for example by chemical substitution is crucial for the emergence of unconventional superconductivity.@xcite although it is obvious that an understanding of the magnetic fluctuations in the iron pnictides is mandatory to unveil the physics underlying the superconductivity , this task has proven to be more complex than anticipated.@xcite for example , there was discussion in the literature whether the magnetic moments are better described by an itinerant@xcite or a localized@xcite model and there is up to now no consensus concerning the role of correlation effects@xcite . furthermore , the magnitude of the magnetic moments is difficult to reproduce within density functional theory ( dft ) and it is known to be quite sensitive to computational parameters.@xcite one of the most important experimental tools to get insight into the electronic structure of the iron pnictides is angle - resolved photoemission spectroscopy ( arpes ) . there are numerous publications on this topic , although it was shown that dft calculations have typically problems to reproduce all features of the arpes spectra correctly.@xcite this is often ascribed to strong correlation effects , although this question is still under discussion.@xcite another important difficulty which so far is often ignored is the connection between the magnetic phase of the iron pnictides and the resulting consequences for arpes . this is due to the formation of twinned crystals during the phase transition from tetragonal to orthorhombic and it results in mixed magnetic domains which are orthogonal to each other . macroscopic tools like arpes or transport measurements can so only see the averaged information , while information on the anisotropy is lost.@xcite this is a huge drawback considering a comprehensive study of the electronic structure in the iron pnictides , as it is known that the in - plane anisotropy plays a significant role.@xcite in experiment it is possible to effectively detwin the crystals by applying uniaxial stress during the measurement . this was already done successfully for the 122-prototype in the undoped and in the co - doped case . however , such measurements are connected with several technical difficulties and consequently they are rarely done.@xcite yet , to fully understand the electronic properties of the iron pnictide superconductors in a comprehensive way and to get a deeper insight concerning the influence of the in - plane anisotropy in the magnetic phase such studies are absolutely mandatory . although there is nowadays experimental data on detwinned crystals showing clearly the anisotropy in the fermi surface there is hardly any theoretical work focusing on this problem of magnetic anisotropy in arpes data . in this work this issue is addressed by a comprehensive dft study on the magnetic phase of and on the corresponding arpes spectra . the computational results can be directly compared to the available experimental arpes data on detwinned crystals.@xcite in order to deal with this complex situation the korringa - kohn - rostoker - green function ( kkr - gf ) approach is used , which was already shown to be indeed a very useful and accurate tool to deal with the iron pnictides.@xcite the impact of disorder due to substitution is dealt with by means of the coherent potential approximation ( cpa ) , giving results fully compatible to supercell calculations and more reliable than those based on the virtual crystal approximation ( vca).@xcite all calculations have been performed self - consistently and fully relativistically within the four component dirac formalism , using the munich spr - kkr program package.@xcite the orthorhombic , antiferromagnetic phase of is investigated in its experimentally observed stripe spin state using a full 4-fe unit cell . this implies antiferromagnetic chains along the @xmath1- and @xmath2-axes and ferromagnetic chains along the @xmath3-axis . the lattice parameters where chosen according to experimental x - ray data and the experimental as position @xmath4.@xcite to account for the influence of substitution in a linear interpolation for the lattice parameters with respect to the concentration @xmath0 is used based on available experimental data@xcite and vegard s law@xcite . more details on the procedure can be found in a previous publication.@xcite the treatment of disorder introduced by substitution is dealt with by means of the cpa . the basis set considered for a @xmath5 including @xmath6 , @xmath7 , @xmath8 , @xmath9 and @xmath10 orbitals . for the electronic structure calculations the local density approximation ( lda ) exchange - correlation potential with the parameterization given by vosko , wilk and nusair was applied.@xcite the spectroscopical analysis is based on the fully relativistic one - step model of photoemission in its spin density matrix formulation . for more technical details on these calculations see ref.@xcite . the geometry of the spectroscopy setup was taken from experiment including a tilt of the sample around either the @xmath1 or @xmath3 axis . the incident light hit the sample under a constant polar angle @xmath11 and an azimuthal angle @xmath12 of either @xmath13 or @xmath14 . these geometries are referred to as @xmath15 and @xmath16 , meaning the direction of the incident light is either parallel to the antiferromagnetic or the ferromagnetic in - plane directions . the corresponding electrons were collected with an angle @xmath17 of @xmath18 or @xmath19 and a varying angle @xmath20 between @xmath21 and @xmath22 . this geometry is in line to the experimental setup . if not indicated otherwise , an as - terminated surface was chosen . however , the question of surface termination will be discussed in more detail in the following . to describe the anisotropy of the iron pnictides in arpes calculations reasonably well one needs first to ensure that the spin - dependent potentials from the self - consistent field ( scf ) calculations are accurate enough . obviously , the magnetic ordering plays a significant role concerning the anisotropy of the electronic structure and hence the quality of the theoretical description of the arpes spectra is determined by the quality of the spin - dependent potentials . the most meaningful indication for a proper description of the magnetic state is good agreement with experimental data on the magnetic order . for the iron pnictides this is known to be a non - trivial task as the magnetic moments are often overestimated by dft.@xcite for the undoped mother compound a total magnetic moment of @xmath23 was obtained . experiment reports a total magnetic moment of approximately @xmath24 from neutron diffraction@xcite while mssbauer spectroscopy@xcite and @xmath25sr spectroscopy@xcite coherently give a value of around @xmath26 . hence , the calculated total magnetic moment is found in good agreement with experiment and captures the proper order of magnitude accurately.@xcite more importantly , the cpa allows to evaluate the substitution dependent self - consistent evolution of the magnetic moments with increasing co concentration in . the corresponding results are shown in fig . [ fig_magnmom ] , where the results for spin and orbital magnetic moments are given in an atom - resolved way . the total magnetic moment is calculated as substitutionally averaged sum over all contributions . in agreement with experiment the total magnetic moments shows a nearly linear decay until the long - range magnetic order disappears.@xcite in the calculations the critical co substitution for the disappearance of antiferromagnetic order occurs for @xmath27 , which is in reasonably good agreement with the experimental value @xmath28.@xcite it should be mentioned that the results in fig . [ fig_magnmom ] are slightly improved with respect to experiment in comparison with our previous work@xcite due to the higher @xmath29 expansion used here . however , the trends in the magnetic moments and the resulting conclusions are the same . as the one - step model of photoemission fully accounts for matrix - elements as well as for surface effects the resulting spectra can be directly compared to experimental arpes data . as stressed before , it is extremely difficult to see the magnetic anisotropy correctly in experimental spectra because of the twinning of crystals . here reference is made especially to the work of yi _ et al._@xcite , who did remarkable measurements on detwinned single crystals of and by applying uniaxial stress to the crystals . similar results were obtained for example by kim _ et al._@xcite . in this context it is important to note that the brillouin zone ( bz ) of the magnetic 4-fe spin - density - wave ( sdw ) state is only half the size compared to the bz in the nonmagnetic 2-fe state . for that reason it is most appropriate to use in the following the notation for the 4-fe sdw bz where the information of @xmath30 and @xmath31 from the nonmagnetic bz is down - folded to one @xmath32 point.@xcite in fig . [ fig_cal_afmain ] the fermi surface around the @xmath32 point is shown in the sdw bz as calculated from the spin - dependent potentials for a photon energy of @xmath33ev . the overlay of black points corresponds to the experimentally measured bz , reproduced from the work of yi _ et al._@xcite . as can be seen , the agreement of the calculated fermi surface and the experimental data is remarkably good . characteristic are the bright intensity spots along the @xmath34-direction ( i.e. along @xmath1 ) , corresponding directly to the antiferromagnetic order along the @xmath1-axis and the bigger pedal - like structures along the @xmath35-direction ( i.e. along @xmath3 ) which corresponds to the ferromagnetic order along the @xmath3-axis . ( 12,10.0 ) ( 2.20,1.2 ) in the 4-fe sdw bz for a photon energy of @xmath33ev . the overlay of black points is a reconstruction of the sdw bz from experimental arpes data , reproduced from the work of yi _ et al._@xcite.,title="fig : " ] ( 1.2,1.35)(0,1.95)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 1.85,0.9)(1.95,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 0.2,4.0 ) ( 5.2,-0.1)@xmath36 $ ] ( 10.7,1.2 ) in the 4-fe sdw bz for a photon energy of @xmath33ev . the overlay of black points is a reconstruction of the sdw bz from experimental arpes data , reproduced from the work of yi _ et al._@xcite.,title="fig : " ] ( 11.0,9.5)(0,0)max ( 11.0,1.0)(0,0)min it should be noted that in fig . [ fig_cal_afmain ] the intensity over two different light polarizations was averaged , namely for the direction of the incident light either parallel to the antiferromagnetic @xmath1-axis @xmath15 or parallel to the ferromagnetic @xmath3-axis @xmath16 . all features of the electronic structure are visible for both polarizations of light . however , the intensity patterns vary notably with the polarization due to matrix element effects , indicating strong multiorbital character , just as seen in experiment.@xcite if not indicated otherwise this averaging will be applied in the following . for comparison the two contributions to the total fermi surface for @xmath33ev are shown polarization - resolved in fig . [ fig_cal_afpol ] , for the incident light direction being either parallel to the @xmath3-axis @xmath16(fig . [ fig_cal_afpol ] ( a ) ) or parallel to the @xmath1-axis @xmath15(fig . [ fig_cal_afpol ] ( b ) ) . ( 5.9,8 ) ( 1.10,1.2)-axis ( a ) or parallel to the antiferromagnetic @xmath1-axis ( b ) . , title="fig : " ] ( 0.2,1.35)(0,1.3)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 0.75,0.9)(1.3,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.6,3.0 ) ( 2.7,-0.1)@xmath36 $ ] ( 0.65,7.0)(a ) ( 2.75,7.0)@xmath16 ( 7.3,8 ) ( 1.20,1.2)-axis ( a ) or parallel to the antiferromagnetic @xmath1-axis ( b ) . , title="fig : " ] ( 0.85,0.9)(1.3,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 2.8,-0.1)@xmath36 $ ] ( 0.65,6.95)(b ) ( 2.75,6.95)@xmath15 ( 7.0,1.2)-axis ( a ) or parallel to the antiferromagnetic @xmath1-axis ( b ) . , title="fig : " ] ( 7.3,6.95)(0,0)max ( 7.3,1.0)(0,0)min it can be seen that for @xmath16 the intensity of the bright spots along the @xmath1-axis is significantly enhanced while for @xmath15 the intensity around the inner circle of @xmath32 is enhanced . this polarization dependence is again in full agreement with the experimental findings.@xcite at this point it was shown that the detwinned , antiferromagnetic fermi surface obtained by the calculations agrees very well with experiment . one may also ask how the fermi surface of a twinned , antiferromagnetic crystal should look like and how does it differ from the nonmagnetic case . therefore , the calculated fermi surfaces for both cases are shown in fig . [ fig_cal_twin ] and compared with experimental arpes data@xcite of twinned crystals at @xmath37k ( a ) and @xmath38k ( b ) . please note that the transition from a paramagnetic to an antiferromagnetic state occurs at around @xmath39k , accordingly the experimental data shown as overlay of black points corresponds to the nonmagnetic and the twinned antiferromagnetic state , respectively . the representation of the twinned fermi surface is based on a superposition of spectra obtained independently for antiferromagnetic states rotated by @xmath19 against each other which is supposed to be a good approximation for twinned crystals , see for example the work of tanatar _ et al._@xcite . ( 5.9,8 ) ( 1.10,1.2 ) against each other . the overlay of black points is reproduced from the experimental arpes data of yi _ et al._@xcite , title="fig : " ] ( 0.2,1.35)(0,1.3)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 0.75,0.9)(1.3,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.6,3.0 ) ( 2.7,-0.1)@xmath36 $ ] ( 0.65,7.0)(a ) ( 2.1,7.0)nonmagnetic ( 7.3,8 ) ( 1.20,1.2 ) against each other . the overlay of black points is reproduced from the experimental arpes data of yi _ et al._@xcite , title="fig : " ] ( 0.85,0.9)(1.3,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 2.8,-0.1)@xmath36 $ ] ( 0.65,6.95)(b ) ( 2.75,6.95)twinned ( 7.0,1.2 ) against each other . the overlay of black points is reproduced from the experimental arpes data of yi _ et al._@xcite , title="fig : " ] ( 7.3,6.95)(0,0)max ( 7.3,1.0)(0,0)min comparing the nonmagnetic and the twinned antiferromagnetic state with each other it is obvious that there is a significant difference in the shape of the fermi surface which is due to the underlying change of the electronic structure during the magnetic phase transition . however , the twinned fermi surface is in principle isotropic along @xmath34 and @xmath35 due to the fact that the in - plane anisotropy cancels almost completely for two magnetic domains that are rotated by @xmath19 against each other . this means that although some influence of the magnetic ordering can be seen for twinned crystals , it is not possible to deduce information about the important in - plane anisotropy from the corresponding spectra . this stresses again the importance of arpes measurements and calculations on detwinned crystals to investigate the magnetic structure correctly . to summarize , the agreement of the calculations with the experimental data is altogether quite well for the nonmagnetic as well as for the twinned magnetic state . ( 5.9,8.8 ) ( 1.1,1.2 ) dispersion as seen by arpes along the both in - plane real space axes @xmath1 ( a ) and @xmath3 ( b ) . the black lines mark the photon energies where the alternation of @xmath40 and @xmath41 can be seen along @xmath42 . notably , the vertical intensity stripes at @xmath43 in ( a ) seem almost independent on @xmath42 , indicating some connection to a surface related phenomenon . , title="fig : " ] ( 0.4,1.95)(0,1.55)[l]2.5,3.0,3.5,4.0 ( 0.88,0.9)(1.28,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.5,3.7 ) ( 2.8,-0.1)@xmath44 $ ] ( 0.65,8.3)(a ) ( 2.85,8.2)@xmath1-axis ( 7.0,8.8 ) ( 1.1,1.2 ) dispersion as seen by arpes along the both in - plane real space axes @xmath1 ( a ) and @xmath3 ( b ) . the black lines mark the photon energies where the alternation of @xmath40 and @xmath41 can be seen along @xmath42 . notably , the vertical intensity stripes at @xmath43 in ( a ) seem almost independent on @xmath42 , indicating some connection to a surface related phenomenon . , title="fig : " ] ( 0.88,0.9)(1.28,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 2.8,-0.1)@xmath45 $ ] ( 0.65,8.3)(b ) ( 2.85,8.2)@xmath3-axis ( 6.9,1.2 ) dispersion as seen by arpes along the both in - plane real space axes @xmath1 ( a ) and @xmath3 ( b ) . the black lines mark the photon energies where the alternation of @xmath40 and @xmath41 can be seen along @xmath42 . notably , the vertical intensity stripes at @xmath43 in ( a ) seem almost independent on @xmath42 , indicating some connection to a surface related phenomenon . , title="fig : " ] ( 7.25,8.2)(0,0)max ( 7.25,1.0)(0,0)min going back to the original study of detwinned antiferromagnetic crystals the @xmath42 dispersion is shown along the @xmath1- and @xmath3-axes in fig . [ fig_cal_kz ] . the difference between @xmath40 and @xmath41 manifests itself mainly by the alternating intensity distributions . we find @xmath40 for photon energies of @xmath4622ev and @xmath47ev respectively , while @xmath41 can be found at @xmath48 . this is in good agreement with literature which reports @xmath41 at @xmath49ev and @xmath40 at @xmath50ev.@xcite the anisotropic features , namely the bright spots along @xmath34 and the pedals along @xmath35 seem quite independent on @xmath42 , which agrees with the experimental reports on the detwinned crystals.@xcite for further discussion the fermi surfaces for @xmath51ev and @xmath47ev respectively are shown in fig . [ fig_cal_3448 ] . the important aspect to note is that the anisotropic features are preserved independent on @xmath42 , meaning they are preserved for @xmath40 as well as for @xmath41 . however , the most striking anisotropy between the @xmath1- and @xmath3-directions seen in the @xmath42 dispersion are the almost vertical intensity lines along the @xmath1-axis for @xmath43 . they are surprisingly robust concerning the @xmath42 dispersion , already indicating possible surface related phenomena , as will be discussed later in more detail . ( 5.9,8 ) ( 1.10,1.2)ev and @xmath47ev , corresponding to either @xmath41 ( a ) or to @xmath40 ( b ) . it can be seen that the topology of interest , namely the anisotropic features are principally independent on @xmath42.,title="fig : " ] ( 0.2,1.35)(0,1.3)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 0.75,0.9)(1.3,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.6,3.0 ) ( 2.7,-0.1)@xmath36 $ ] ( 0.65,7.0)(a ) ( 2.45,7.0)@xmath51ev ( 7.3,8 ) ( 1.20,1.2)ev and @xmath47ev , corresponding to either @xmath41 ( a ) or to @xmath40 ( b ) . it can be seen that the topology of interest , namely the anisotropic features are principally independent on @xmath42.,title="fig : " ] ( 0.85,0.9)(1.3,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 2.8,-0.1)@xmath36 $ ] ( 0.65,6.95)(b ) ( 2.45,6.95)@xmath47ev ( 7.0,1.2)ev and @xmath47ev , corresponding to either @xmath41 ( a ) or to @xmath40 ( b ) . it can be seen that the topology of interest , namely the anisotropic features are principally independent on @xmath42.,title="fig : " ] ( 7.3,6.95)(0,0)max ( 7.3,1.0)(0,0)min ( 6.8,9.5 ) ( 1.1,1.2 ) ( 0.2,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.4,3.5 ) ( -0.50,8.9)@xmath52 ( 1.55,8.3)(a ) ( 3.25,8.3)@xmath1-axis ( 6.8,8.8 ) ( 1.1,1.2 ) ( 0.12,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 1.55,8.3)(b ) ( 2.7,8.3)nonmagnetic ( 7.8,8.8 ) ( 1.1,1.2 ) ( 0.12,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.55,8.3)(c ) ( 3.25,8.3)@xmath3-axis ( 7.3,1.2 ) ( 7.6,8.2)(0,0)max ( 7.6,1.0)(0,0)min + ( 6.8,9.1 ) ( 1.1,1.2 ) ( 0.2,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.4,3.5 ) ( 3.2,-0.1)@xmath44 $ ] ( -0.50,8.9)@xmath53 ( 1.55,8.3)(d ) ( 3.25,8.3)@xmath1-axis ( 6.8,8.8 ) ( 1.1,1.2 ) ( 0.12,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath45 $ ] ( 1.55,8.3)(e ) ( 2.7,8.3)nonmagnetic ( 7.8,8.8 ) ( 1.1,1.2 ) ( 0.12,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath45 $ ] ( 1.55,8.3)(f ) ( 3.25,8.3)@xmath3-axis ( 7.3,1.2 ) ( 7.6,8.2)(0,0)max ( 7.6,1.0)(0,0)min to complete the study of the in - plane anisotropy in the undoped compound the spin - dependent bands are investigated polarization - dependent along the two in - plane directions @xmath1 and @xmath3 for @xmath33ev and for comparison the isotropic bands of the nonmagnetic case . anisotropies due to the orthorhombic lattice distortion for the nonmagnetic case are very small and have no significant influence , as shown also in earlier work.@xcite the nonmagnetic bands for the polarizations @xmath16 and @xmath15 are shown in fig . [ fig_cal_bs00 ] ( b ) and ( e ) , respectively . already for the nonmagnetic case it becomes obvious , that more information can be deduced for light with a polarization parallel to the ferromagnetic chains . for a perpendicular light polarization the intensity for some bands decrease so strongly that they practically seem to vanish . this is however not due to a vanishing of the bands but only due to the strong intensity variation , i.e. matrix element effects , as already mentioned before and as seen in experiment . for the spin - polarized band structure with antiferromagnetic order along the @xmath1-axis the corresponding cases for @xmath16 and @xmath15 are shown in fig . [ fig_cal_bs00 ] ( a ) and ( d ) , respectively . the green solid lines are guides to the eye which emphasize the two important anisotropic bands in these spectra . first of all , there is some significant reorientation of the bands compared to the nonmagnetic case . most striking is the appearance of a steep double - u shaped band which was not visible in the nonmagnetic case . this new appearance is most likely due to down - folding of the brillouin zone when going from the 2-fe cell to a magnetic 4-fe cell . the second important band is a pure hole pocket which is compared to the nonmagnetic case shifted to higher binding energies . it should be noted that the intensity of this band is extremely polarization dependent . it is the dominating band for @xmath16 while it is barely visible for @xmath16 . comparing with the polarization dependent fermi surface in fig . [ fig_cal_afpol ] it is obvious that this band is also part of the bright intensity spots in the fermi surface along the @xmath1-direction and very characteristic for the anisotropy . it is also noteworthy that these two significant bands cross each other exactly at the fermi level . this crossing is also reported in experiment as can be seen from the black points in fig . [ fig_cal_bs00 ] ( a ) , which are reproduced from the experimental arpes data of yi _ et al._@xcite . thus , the experimental arpes data could be again well reproduced by the calculations . the situation for the magnetic bands with ferromagnetic order along the @xmath3-axis as shown in fig . [ fig_cal_bs00 ] ( c ) and ( f ) is in many aspects similar to that for the bands along the @xmath1-direction . one can identify two prominent anisotropic bands , one with a double - u like shape which has a higher intensity for a light polarization of @xmath15 , while the other band marked with the solid green line is the dominating one for light @xmath16 . the important difference is that these bands do not touch each other as they are significantly shifted away in binding energy . note that also no crossing is reported in experiment.@xcite why these steep bands with the double - u shape can not be seen in experiment for the @xmath3-direction gets also clear : the responsible band is simply completely shifted above the fermi level . this observation can in principle be compared to the band splitting in ferromagnets . note that along the @xmath3-axis there is ferromagnetic coupling while along the @xmath1-axis the magnetic order is antiferromagnetic . thus , for one sees along the ferromagnetic chains a band splitting of approximately @xmath54ev for a magnetic moment around @xmath55 . this is comparable for example to ni which shows a band splitting of approximately @xmath56ev for a moment of approximately @xmath57.@xcite consequently , for decreasing magnetic moments upon alloying one expects a reduced band splitting together with a continuous matching of the anisotropic bands . to investigate this issue in further detail one has to look at the evolution of the arpes band structure for increasing co substitution on the fe position which goes in hand with the reduction of the magnetic moments . ( 6.6,9.3 ) ( 1.1,1.2 ) ( 0.2,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.4,3.5 ) ( 3.2,-0.1)@xmath44 $ ] ( -0.50,8.9)*_a_-axis * ( 1.55,8.3)(a ) ( 3.0,8.3)@xmath58 ( 6.6,8.8 ) ( 1.1,1.2 ) ( 0.25,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath36 $ ] ( 1.55,8.3)(b ) ( 3.0,8.3)@xmath59 ( 6.6,8.8 ) ( 1.1,1.2 ) ( 0.25,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath36 $ ] ( 1.55,8.3)(c ) ( 3.0,8.3)@xmath60 ( 7.0,8.8 ) ( 1.1,1.2 ) ( 0.25,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath36 $ ] ( 1.55,8.3)(d ) ( 3.0,8.3)@xmath61 ( 7.0,1.2 ) ( 7.3,8.2)(0,0)max ( 7.3,1.0)(0,0)min + ( 6.6,9.3 ) ( 1.1,1.2 ) ( 0.2,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.4,3.5 ) ( 3.2,-0.1)@xmath45 $ ] ( -0.50,8.9)*_b_-axis * ( 1.55,8.3)(e ) ( 3.0,8.3)@xmath58 ( 6.6,8.8 ) ( 1.1,1.2 ) ( 0.25,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath45 $ ] ( 1.55,8.3)(f ) ( 3.0,8.3)@xmath59 ( 6.6,8.8 ) ( 1.1,1.2 ) ( 0.25,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath45 $ ] ( 1.55,8.3)(g ) ( 3.0,8.3)@xmath60 ( 7.0,8.8 ) ( 1.1,1.2 ) ( 0.25,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath45 $ ] ( 1.55,8.3)(h ) ( 3.0,8.3)@xmath61 ( 7.0,1.2 ) ( 7.3,8.2)(0,0)max ( 7.3,1.0)(0,0)min substitution of fe with co in is one of the common ways to induce superconductivity in by electron doping . the substitution does consequently diminish the strength of the antiferromagnetic coupling within this compound until the long - range magnetic order collapses and superconductivity emerges.@xcite as the strength of the magnetic order decreases with co doping , in experiment as well as in the calculations , it can be assumed that the strong in - plane anisotropy does also decrease . the breakdown of the long - range antiferromagnetic order in fig . [ fig_magnmom ] appears for a somewhat higher co concentrations than in experiment . thus , the breakdown of the anisotropy is expected at higher doping levels . this is true for a concentration of @xmath61 in which is the first substitution level where the magnetic order did completely vanish in the self - consistent calculation . considering the evolution of the magnetic moments in fig . [ fig_magnmom ] one can see that the initial magnetic moment decreased to approximately @xmath62 , @xmath63 and @xmath64 of the original value for co substitutions of @xmath58 , @xmath65 and @xmath66 respectively . to investigate the impact of alloying in detail , the arpes band structure for these concentrations including the already nonmagnetic @xmath61 for both directions @xmath1 and @xmath3 is presented in fig . [ fig_cal_bsco ] . the calculations are performed for @xmath33ev comparable to fig . [ fig_cal_bs00 ] but they are only shown for a light polarization of @xmath16 because is was already clarified that the anisotropic bands can be best seen with this specific polarization . the black dashed lines in fig . [ fig_cal_bsco ] are shown for comparison and they correspond to the band position of the anisotropic bands in the undoped compound with the highest anisotropy , seen in fig . [ fig_cal_bs00 ] ( a ) and ( c ) . the green solid lines are guides to the eye to identify more easily the corresponding anisotropic bands for the specific co concentrations . the difference between black dashed lines and green solid lines is thus the change of the original anisotropy with increasing co substitution . for the case of the nonmagnetic with @xmath61 shown in fig . [ fig_cal_bsco ] ( d ) and ( h ) the anisotropy has completely vanished and the band structures coincide with each other . this could be expected from experiment and it is reproduced in the calculations . it should be noted again at this point , that the crystal lattice is still orthorhombic , however , the lattice anisotropy is indeed too weak to be visible in the band structure.@xcite some other interesting findings can be deduced from the evolution of the band structure upon co substitution . first of all the intensity of the double - u shaped band decreases continuously , however , it only completely disappears after the collapse of the long - range antiferromagnetic order . the change in anisotropy for the antiferromagnetic order along the @xmath1-axis is mostly characterized with the consequent shift of the hole - pocket to lower binding energies . this is also experimentally reported for a decrease in the magnetic coupling strength , either induced through co doping or increasing temperature.@xcite concerning this situation for the ferromagnetic order along the @xmath3-axis the most prominent feature is the shift of the double - u shaped band to lower binding energies . what can be seen in fig . [ fig_cal_bs00 ] ( e ) to ( h ) is that the energy difference of these two main anisotropic bands does strongly and continuously decrease . this is in agreement with the assumption of a smaller band splitting for decreasing ferromagnetic coupling strength . in summary one can say that for the antiferromagnetic order along the @xmath1-axis mostly the hole - pocket changes while the double - u shaped band stays more or less constant . for the ferromagnetic order along the @xmath3-axis it is the other way round . the double - u shaped band undergoes the strongest change while the other band stays more or less unchanged in energy and shape . the final result is the same in both cases , a matching of the bands and a consequent isotropic in - plane band structure . this detailed analysis allows to follow the change from the strong in - plane anisotropy of the undoped compound to the isotropic behavior in the co substituted system in a continuous way with direct correspondence to arpes . thus , this approach based on kkr - cpa proves its advantages for investigating the iron pnictide superconductors at regions of interest which are difficult to evaluate by means of other band structure methods . c|c ( 6.2,9.3 ) ( 1.1,1.2 ) ( 0.2,1.15)(0,1.48)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 2.10,0.9)(1.45,0)[l]-0.2 , 0.0 , 0.2 ( -0.5,2.8 ) ( 3.0,-0.0)@xmath44 $ ] ( 4.50,8.6)*as - terminated * ( 2.5,7.5)(a ) ( 5.5,7.5)fermi surface ( 7.0,1.2 ) ( 7.1,7.2)(0,0)max ( 7.1,1.0)(0,0)min ( 7.5,8.8 ) ( 1.1,1.2 ) ( 2.10,0.9)(1.45,0)[l]-0.2 , 0.0 , 0.2 ( 3.0,-0.0)@xmath36 $ ] ( 7.0,1.2 ) ( 7.1,7.2)(0,0)max ( 7.1,1.0)(0,0)min & ( 6.2,9.3 ) ( 1.1,1.2 ) ( 0.2,1.15)(0,1.48)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 2.10,0.9)(1.45,0)[l]-0.2 , 0.0 , 0.2 ( 3.0,-0.0)@xmath44 $ ] ( 4.50,8.6)*ba - terminated * ( 2.5,7.5)(c ) ( 5.5,7.5)fermi surface ( 7.0,1.2 ) ( 7.1,7.2)(0,0)max ( 7.1,1.0)(0,0)min ( 7.5,8.8 ) ( 1.1,1.2 ) ( 2.10,0.9)(1.45,0)[l]-0.2 , 0.0 , 0.2 ( 3.0,-0.0)@xmath36 $ ] ( 7.0,1.2 ) ( 7.1,7.2)(0,0)max ( 7.1,1.0)(0,0)min + ( 6.55,9.3 ) ( 1.1,1.2 ) ( 0.2,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 0.9,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.4,3.5 ) ( 3.0,-0.0)@xmath44 $ ] ( 2.0,8.5)(b ) ( 3.0,8.5)band structure along @xmath1-axis ( 7.05,1.2 ) ( 7.25,8.2)(0,0)max ( 7.25,1.1)(0,0)min ( 7.4,8.8 ) ( 1.1,1.2 ) ( 0.9,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.0,-0.0)@xmath36 $ ] ( 7.0,1.2 ) ( 7.25,8.2)(0,0)max ( 7.22,1.0)(0,0)min & ( 6.55,9.3 ) ( 1.1,1.2 ) ( 0.2,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 0.9,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.0,-0.0)@xmath44 $ ] ( 2.0,8.5)(d ) ( 3.0,8.5)band structure along @xmath1-axis ( 7.05,1.2 ) ( 7.25,8.2)(0,0)max ( 7.25,1.1)(0,0)min ( 7.4,8.8 ) ( 1.1,1.2 ) ( 0.9,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.0,-0.0)@xmath36 $ ] ( 7.0,1.2 ) ( 7.25,8.2)(0,0)max ( 7.22,1.0)(0,0)min using the one - step model of photoemission one can identify different surface states and can thus clarify the origin of surface bands . the reason for the occurrence of surface - states has long been developed in multiple scattering theory , which is the underlying basis of the spr - kkr method.@xcite the so - called determinant condition uses the reflection matrices of the bulk crystal @xmath67 and of the surface barrier potential @xmath68 , which connects the inner potential of the bulk crystal with the vacuum level . the appearance of a surface state is given by the following condition : @xmath69 for better visualization we plot @xmath70 in the following . if this expression is bigger than approximately @xmath71 we speak about a surface state . for values between @xmath72 and @xmath71 the state is defined as a so - called surface resonance . for values below one has bulk states . more details can be found in the overview by braun and donath.@xcite the application of this determinant approach is demonstrated in fig . [ fig_cal_det ] . here one can see the fermi surfaces and the band structures along the @xmath1-axis for an as - terminated and a ba - terminated surface , respectively , together with the corresponding plot of @xmath70 on the right hand side of each picture . the bands are shown for @xmath33ev and they are averaged over the two light polarizations @xmath16 and @xmath15 in order to make all relevant contributions equally visible . it should be noted , that the determinant condition itself and without a high intensity in the corresponding band structure plot is only an indication for a surface state or a surface resonance . only if a high intensity in the @xmath70 plot coincides with a band in the band structure one can associate this band with a clear surface character . for example has the bright octagon shape of the determinant plot of the fermi surface in fig . [ fig_cal_det ] ( a ) not a corresponding counterpart in the band structure plot . the two high intensity spots along the @xmath1-axis which are equally visible in the fermi surface as well as in the determinant condition are in clear contrast to this behavior . thus , this feature has a surface related origin , more specifically a surface state as the intensity of @xmath70 is in the order of @xmath73 . this surface state can be also identified in the band structure along the @xmath1-axis as shown in fig . [ fig_cal_det ] ( b ) where a strong intensity in the determinant plot coincides with the steep bands that cut the fermi level and which are part of the already discussed double - u shape . consequently , these bands can be identified as surface states . this is in accordance with the earlier findings for the @xmath42-dispersion in fig . [ fig_cal_kz ] ( a ) where the vertical intensities at @xmath43 were independent on @xmath42 , indicating a connection to a surface related phenomenon . another verification for the surface related origin of these bands is shown in fig . [ fig_cal_det ] ( c ) and ( d ) , where the corresponding fermi surface and band structure are shown for a ba - terminated surface . as already indicated before all other calculations presented in this paper are under the assumption of an as - terminated surface . obviously , the assumed surface termination has also an influence on surface related phenomena and surface states might be shifted significantly in energy . indeed , the surface states discussed for the as - terminated surface have completely vanished in the ba - terminated case . the corresponding high intensities are missing in the @xmath70 plots and in the band structure of fig . [ fig_cal_det ] ( d ) the characteristic steep bands from the as - terminated surface have also vanished . no intensities in the determinant plot coincide with features in the band structure and thus surface effects have been removed by the ba termination . overall , the fermi surface and the band structure have undergone significant changes for the altered surface termination . the characteristic anisotropic features of the fermi surface in fig . [ fig_cal_det ] ( a ) are hardly visible in the ba - terminated case in fig . [ fig_cal_det ] ( c ) . it seems like the ba layer on top acts as some kind of damping layer which reduces the intensity and blurs the electronic states which are clearly visible in an as - terminated surface . in particular one has to note that the agreement with experimental arpes data is significantly better for an as - terminated surface compared to the ba - terminated one . especially the steep bands along the @xmath1-axis are seen in experiment@xcite and they could be successfully identified as surface states which are only visible for an as - terminated surface . this result can be used for conclusions on the most likely surface termination of . interestingly , the surface termination in this material is still not clear and under debate , although several experimental measurements and theoretical calculations exist.@xcite according to first principle calculations only three possibilities for the surface termination exist , namely a fully as - terminated or a fully ba - terminated surface as well as an as surface covered with half of the stoichiometric ba atoms.@xcite there are experimental scanning tunneling microscopy ( stm ) and low - energy electron - diffraction ( leed ) measurements which indicate a ba - terminated surface@xcite . however , there are also experimental stm + leed data which clearly favor an as - terminated surface@xcite . the arpes calculations clearly favor an as - terminated surface as it was shown that agreement with experiment is considerably better compared to the ba - terminated one . this can not rule out the possibility of some partial covering with few ba adatoms but one can state that every ba atoms on top muffles the electronic structure seen in experiment and that this structure is due to an as - terminated compound . so one would expect a more or less clean as - terminated surface as most probable surface termination for . additional covering with some ba atoms might be possible but also might have some degrading influence on the quality of the arpes measurement . the munich spr - kkr package was used for self - consistent and fully relativistic calculations of orthorhombic in its experimentally observed stripe antiferromagnetic ground - state for @xmath74 up to @xmath61 . the substitutional disorder induced by co on fe positions was dealt with on a cpa level which was earlier shown to be fully equal to a comprehensive supercell calculation.@xcite magnetic moments of @xmath23 for undoped were reproduced and additionally a reasonable magnetic behavior for increasing co substitution with a continuous decrease of the magnetic moments until a collapse of the antiferromagnetic order at 15% co concentration was reached . this is in good agreement with experimental behavior.@xcite concerning arpes most experimental data available is actually insufficient to talk about in - plane anisotropy due to twinning effects during the phase transition from the nonmagnetic tetragonal to the antiferromagnetic orthorhombic phase . a complicated detwinning process , typically with uniaxial stress on the single crystal , is necessary to gain anisotropic data of the electronic structure.@xcite referring to the available experimental data it was possible to reproduce the electronic structure of in very good agreement with experiment . the fermi surface shows all important anisotropic features , namely some bright spots of intensity along the antiferromagnetic order along the @xmath1-axis and more blurred pedals along the ferromagnetic order along the @xmath3-axis . also in agreement with experiment a strong dependence on the polarization of light was found , been either parallel to the ferromagnetic or to the antiferromagnetic order , indicating the strong multiorbital character . for comparison the fermi surface of the nonmagnetic phase as well as a hypothetical fermi surface for a twinned arpes measurement as a superposition of two antiferromagnetic cells rotated by @xmath19 to each other was shown . both were again in agreement with experiment . in addition to the anisotropic @xmath42 dispersion some focus was put on the anisotropic band structure along the @xmath1- and @xmath3-axes and it was compared to the nonmagnetic case . one could identify the important anisotropic bands and these could be interpreted in terms of band splitting for the ferromagnetic chains along the @xmath3-axis , principally comparable to typical ferromagnetic band splitting as observed for example in ni . in addition the evolution of these anisotropic bands for small steps of @xmath0 in was presented until the breakdown of long - range antiferromagnetic order . the decreasing band splitting and a continuous matching of the anisotropic bands could be reproduced in great detail and consistent with experimental findings . finally the so - called determinant condition @xmath70 was used to evaluate possible surface states of the band structure . it was possible to identify steep bands along the @xmath1-axis as surface states . these are at least partially responsible for the characteristic bright intensity spots in the electronic structure along the @xmath1-axis and can also be seen in experiment . interestingly , these surface states are only visible near the fermi level for an as - terminated surface . it was shown that a ba - terminated top - layer acts as some kind of damping which moves the surface states far away and blurs the electronic structure . significantly better agreement with experimental data is found for an as - terminated surface . this leads to the conclusion that an as - terminated surface would be most likely , an issue that is in fact experimentally not convincingly clarified.@xcite some ba adatoms might be still possible but one would expect a negative influence on the quality of the measurements . to conclude , this publication was successful in reproducing the strong in - plane anisotropy of and its behavior under substitution in very good agreement with experiment using arpes calculations . these calculations allow even predictions on possible surface terminations . we acknowledge the financial support from the deutsche forschungsgemeinschaft dfg ( projects for 1346 ) and from the bundesministerium fr bildung und forschung bmbf ( project 05k13wma ) . we further thank for the support from centem plus ( l01402 ) . j. fink , s. thirupathaiah , r. ovsyannikov , h. a. drr , r. follath , y. huang , s. de jong , m. s. golden , y .- z . zhang , h. o. jeschke , r. valent , c. felser , s. dastjani farahani , m. rotter , and d. johrendt , phys . b * 79 * , 155118 ( 2009 ) . l. x. yang , y. zhang , h. w. ou , j. f. zhao , d. w. shen , b. zhou , j. wei , f. chen , m. xu , c. he , y. chen , z. d. wang , x. f. wang , t. wu , g. wu , x. h. chen , m. arita , k. shimada , m. taniguchi , z. y. lu , t. xiang , and d. l. feng , phys . lett . * 102 * , 107002 ( 2009 ) . h. gretarsson , a. lupascu , j. kim , d. casa , t. gog , w. wu , s. r. julian , z. j. xu , j. s. wen , g. d. gu , r. h. yuan , z. g. chen , n .- l . wang , s. khim , k. h. kim , m. ishikado , i. jarrige , s. shamoto , j .- h . chu , i. r. fisher , and y .- j . kim , phys . b * 84 * , 100509 ( 2011 ) . p. vilmercati , a. fedorov , f. bondino , f. offi , g. panaccione , p. lacovig , l. simonelli , m. a. mcguire , a. s. m. sefat , d. mandrus , b. c. sales , t. egami , w. ku , and n. mannella , phys . b * 85 * , 220503 ( 2012 ) . v. b. zabolotnyy , d. s. inosov , d. v. evtushinsky , a. koitzsch , a. a. kordyuk , g. l. sun , j. t. park , d. haug , v. hinkov , a. v. boris , c. t. lin , m. knupfer , a. n. yaresko , b. buchner , a. varykhalov , r. follath , and s. v. borisenko , nature * 457 * , 569 ( 2009 ) . d. v. evtushinsky , d. s. inosov , v. b. zabolotnyy , a. koitzsch , m. knupfer , b. bchner , m. s. viazovska , g. l. sun , v. hinkov , a. v. boris , c. t. lin , b. keimer , a. varykhalov , a. a. kordyuk , and s. v. borisenko , phys . b * 79 * , 054517 ( 2009 ) . d. v. evtushinsky , d. s. inosov , v. b. zabolotnyy , m. s. viazovska , r. khasanov , a. amato , h .- h . klauss , h. luetkens , c. niedermayer , g. l. sun , v. hinkov , c. t. lin , a. varykhalov , a. koitzsch , m. knupfer , b. bchner , a. a. kordyuk , and s. v. borisenko , new journal of physics * 11 * , 055069 ( 2009 ) . d. v. evtushinsky , v. b. zabolotnyy , t. k. kim , a. a. kordyuk , a. n. yaresko , j. maletz , s. aswartham , s. wurmehl , a. v. boris , d. l. sun , c. t. lin , b. shen , h. h. wen , a. varykhalov , r. follath , b. bchner , and s. v. borisenko , phys . b * 89 * , 064514 ( 2014 ) . m. yi , d. lu , j .- h . chu , j. g. analytis , a. p. sorini , a. f. kemper , b. moritz , s .- k . mo , r. g. moore , m. hashimoto , w .- s . lee , z. hussain , t. p. devereaux , i. r. fisher , and z .- x . shen , proc . ac . sci . us . * 108 * , 6878 ( 2011 ) . a. a. aczel , e. baggio - 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by means of one - step model calculations the strong in - plane anisotropy seen in angle - resolved photoemission of the well - known iron pnictide prototype compounds and in their low - temperature antiferromagnetic phases is investigated .
the fully - relativistic calculations are based on the korringa - kohn - rostoker - green function approach combined with the coherent potential approximation alloy theory to account for the disorder induced by co substitution on fe sites in a reliable way .
the results of the calculations can be compared directly to experimental spectra of detwinned single crystals .
one finds very good agreement with experiment and can reveal all features of the electronic structure contributing to the in - plane anisotropy .
in particular the local density approximation can capture most of the correlation effects for the investigated system without the need for more advanced techniques .
in addition , the evolution of the anisotropy for increasing co concentration @xmath0 in can be tracked almost continuously .
the results are also used to discuss surface effects and it is possible to identify clear signatures to conclude about different types of surface termination .
| 16,632 | 269 |
when one is dealing with classical field theories on a spacetime , the metric may appear as a given background field or it may be a genuine dynamic field satisfying the einstein equations . the latter theories are often generally covariant , with the spacetime diffeomorphism group as symmetry group , but the former often are considered to have only the isometry group of the metric as a symmetry group . however , @xcite ( see also @xcite ) indicated how theories with a background metric can be parametrized , that is , considered as theories that are fully covariant , if one introduces the diffeomorphisms themselves as dynamic fields . the goal of this paper is to develop this idea in the context of multisymplectic classical field theory and to make connections with stress - energy - momentum ( `` sem '' ) tensors . as we shall see , the multimomenta conjugate to these new covariance fields form , to borrow a phrase from elasticity theory , the piola kirchhoff version of the sem tensor , and their euler lagrange equations are vacuously satisfied by virtue of the fact that the sem tensor is covariantly conserved . thus these fields have no physical content ; they serve only to provide an efficient way of parametrizing a field theory . nonetheless , the resulting generally covariant field theory has several attractive features , chief among which is that it is fully dynamic all fields satisfy euler lagrange equations . structurally , such theories are much simpler to analyze than ones with absolute objects or noncovariant elements . we emphasize that the results of this paper are for those field theories whose lagrangians are built from dynamic matter or other fields and a non - dynamic background metric . one of our motivations was to find a way to treat background fields and dynamic fields in a unified way in the context of the adjoint formalism . many of the ideas are applicable to a wider range of field theories , as @xcite already indicates , but in this paper we confine ourselves to this important class . the general case is presented in @xcite along with a more detailed discussion of parametrization theory and related topics . suppose that we have a metric field theory in which the metric is an absolute object in the sense of @xcite . for instance , one might consider a dynamic electromagnetic field propagating on a schwarzschild spacetime . such a theory is not generally covariant , because the spacetime is fixed , and not all fields are on an equal footing , as the electromagnetic field is dynamic while the gravitational field is not . a somewhat different example is provided by nordstr@xmath0 m s theory of gravity ( see 17.6 of @xcite ) , which is set against a minkowskian background . in this section we explain how to take such a system and construct from it an equivalent field theory that achieves the following goals : ( i ) : : the new field theory is generally covariant , and ( ii ) : : all fields in the new field theory are dynamic . this `` covariance construction '' is an extension and refinement of the parametrization procedure introduced by @xcite . [ [ setup . ] ] setup . + + + + + + as usual for a first order classical field theory , we start with a bundle @xmath1 whose sections , denoted @xmath2 , are the fields under consideration . the dimension of @xmath3 is taken to be @xmath4 , and we suppose that @xmath3 is oriented . let @xmath5 be a lagrangian density for this field theory , where @xmath6 is the first jet bundle of @xmath7 and @xmath8 is the space of top forms on @xmath3 . loosely following the notation of @xcite or @xcite , we write coordinates for @xmath9 as @xmath10 . in addition , in coordinates , we shall write @xmath11 evaluated on the first jet prolongation of a section @xmath2 , the lagrangian becomes a function of @xmath12 ; we shall abbreviate this when convenient and simply write @xmath13 . we assume that the fields @xmath2 are dynamic . [ [ example . ] ] example . + + + + + + + + we will intersperse the example of electromagnetism throughout the paper to illustrate our results . then @xmath14 is the cotangent bundle of 4-dimensional spacetime @xmath3 , sections of which are electromagnetic potentials @xmath15 . the corresponding lagrangian is written below . @xmath16 [ [ a - first - attempt - at - general - covariance . ] ] a first attempt at general covariance . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + suppose that the spacetime @xmath3 comes equipped with a fixed , background metric @xmath17 . the obvious first step in attaining general covariance is to allow @xmath17 to vary ; thus the metric will now be regarded as a genuine _ field _ @xmath18 on @xmath3 . ( when the metric is regarded as variable , we denote it by @xmath18 , and when we want to revert to its fixed value we use @xmath17 . ) so we are led to view the lagrangian density as a map @xmath19 where @xmath20 is the bundle whose sections are lorentz metrics on @xmath3 . we correspondingly write @xmath21 ; the semicolon is used to separate the dynamic from the nondynamic fields . ( we emphasize that @xmath18 being variable does not mean that it is dynamic ; we discuss this point momentarily . ) notice that we have tacitly assumed that the dependence of @xmath22 on the metric is pointwise that is , we have non - derivative coupling . ( the more general case of derivative coupling will be considered in 5 . in any event , we remark that derivatively - coupled theories are considered by many to be pathological . ) [ [ example.-1 ] ] example . + + + + + + + + the electromagnetic lagrangian density @xmath23 is @xmath24 where @xmath25 @xmath16 next , assume that the given lagrangian density @xmath22 has the following ( eminently reasonable ) covariance property for a diffeomorphism @xmath26 : @xmath27 where we assume that a way to lift the spacetime diffeomorphism @xmath28 to a bundle automorphism @xmath29 of @xmath14 has been chosen . [ [ example.-2 ] ] example . + + + + + + + + for the electromagnetic 1-form potential @xmath15 , we take the lift to be push - forward on the fiber , which makes it obvious that holds in this case . when condition holds , we say that the theory is generally covariant , i.e. , the lagrangian density is @xmath30-equivariant . thus we have accomplished objective ( i ) . however , the reader may well remark that this was ` too easy , ' and would be quite right . the problem is that it is not clear how , or even _ if _ , @xmath18 can now be made dynamic . certainly , @xmath18 can not be taken to be variational unless one adds a source term to the lagrangian density for @xmath18 , for otherwise @xmath31 as the metric non - derivatively couples to the other fields . but what should this source term be ? if @xmath18 is gravity , we could use the hilbert lagrangian , but otherwise this is unclear . [ [ the - covariance - field . ] ] the covariance field . + + + + + + + + + + + + + + + + + + + + + the solution to our problem requires more subtlety . we will sidestep both the issues of making @xmath17 variable , and then making @xmath18 dynamic , in one fell swoop as follows . we introduce an entirely new field , the `` covariance field '' into the theory . it will ` soak up ' the arbitrariness in @xmath18 , and will be dynamic . in this way we are able to generate a new generally covariant field theory , physically equivalent to the original one , in which all fields are dynamic . here is the construction . the key idea is to introduce a copy @xmath32 of spacetime into the fiber of the configuration bundle . consider ( oriented ) diffeomorphisms @xmath33 , thought of as sections of the bundle @xmath34 . we regard the diffeomorphisms @xmath35 as new fields , and correspondingly replace the configuration bundle by @xmath36 . next , modify @xmath22 to get the new lagrangian @xmath37 defined on @xmath38 : @xmath39 thus , we obtain a modified field theory with the underlying bundle @xmath40 . the general set up is shown in the figure below . -40pt the general set up for the introduction of covariance fields . let coordinates on @xmath41 be denoted @xmath42 and the associated jet coordinates be denoted @xmath43 . then , writing @xmath44 and similarly for @xmath45 , in coordinates equation reads @xmath46 where from the definition of pull - back @xmath47 we obtain @xmath48 from one verifies that the euler lagrange equations for the fields @xmath49 remain unchanged . [ [ example.-3 ] ] example . + + + + + + + + for the electromagnetic field , our construction produces @xmath50 where @xmath51 is the jacobian of @xmath35 and @xmath52 @xmath16 we pause to point out the salient features of our construction . first , the fixed metric @xmath17 on spacetime is no longer regarded as living on @xmath3 , but rather on the copy @xmath41 of @xmath3 in the fiber of the configuration bundle @xmath40 . so @xmath17 is no longer considered to be a field it has been demoted to a mere geometric object on the fiber @xmath41 . second , the variable metric @xmath18 on @xmath3 is identified with @xmath53 , and thus acquires its variability from that of @xmath35 . so @xmath18 as well is no longer a field per se , but simply an abbreviation for the quantity @xmath53 . finally , we gain a field @xmath35 which we allow to be dynamic ; in the next subsection we will see that this imposes no restrictions on the theory at all . the first key observation is that the modified theory is indeed generally covariant . to this end , recall that , as was explained earlier , given @xmath54 , there is assumed to be a lift @xmath55 . for the trivial bundle @xmath56 , we define @xmath57 the lagrangian density @xmath58 is @xmath59-equivariant , that is , @xmath60 this is an easy consequence of the definitions and , and the covariance assumption . indeed @xmath61 & = { \mathcal{l } } \left ( j^1{\mspace{-1.5mu } } ( \sigma_y(\phi))\ , ; ( \sigma^{-1 } ) ^*(\eta^ * g)\right ) \\[1.5ex ] & = \sigma_*\!\left(\mathcal{l}(j^1 { \mspace{-1.5mu}}\phi)\ , ; ( \eta ^ * g))\right)\\[1.5ex ] & = \sigma_*\!\left(\widetilde { \mathcal{l } } ( j^1 { \mspace{-1.5mu}}\phi , j^1 { \mspace{-1.5mu}}\eta ) \right).\end{aligned}\ ] ] -24pt because of this property , we call @xmath35 the covariance field . [ [ example.-4 ] ] example . + + + + + + + + from it is clear that the modified electromagnetic theory is generally covariant . next we will show something remarkable : the euler lagrange equation for the covariance field @xmath35 is vacuous . this is the main reason that , in the present context , we can introduce @xmath35 as a dynamic field with impunity , namely , its euler lagrange equation does not add any new information to , or impose any restrictions upon , the system . since , as we mentioned earlier , the euler lagrange equations for the fields @xmath49 remain unaltered , we see that _ the parametrized system is physically equivalent to the original system . _ first we compute the multimomenta conjugate to the field @xmath35 for the parametrized field theory with lagrangian @xmath37 . recall that in multisymplectic field theory , the multimomenta conjugate to the multivelocities @xmath62 are defined by @xmath63 using the chain rule together with the relations and , we find that @xmath64 recall from @xcite that , as we have assumed that @xmath18 is the only nondynamic field , and does not derivatively couple to the other fields , the sem tensor density for the _ original _ system with lagrangian @xmath65 and metric @xmath18 is given by the hilbert formula : @xmath66 from we conclude that the multimomenta conjugate to to the covariance field @xmath35 are given by the piola - kirchhoff sem tensor density : @xmath67 this is a familiar object in elasticity theory . observe that @xmath68 is a two - point tensor density : it has one leg ( @xmath69 ) in the spacetime @xmath41 in the fiber analogous to the spatial representation in elasticity theory , and the other leg ( @xmath70 ) in the spacetime @xmath3 in the base analogous to the material representation . now we compute the euler lagrange equations for the @xmath71 . these are : @xmath72 for @xmath73 . expanding the derivatives via the chain rule and using the same type of calculation as in the derivation of to write the equations in terms of @xmath65 rather than @xmath74 , the preceding equation becomes @xmath75 replacing @xmath76 by ( half of ) @xmath77 , and differentiating using the product rule , we obtain @xmath78 for @xmath79 . multiplying by the inverse matrix @xmath80 one gets @xmath81 for @xmath82 . and now , we multiply by @xmath83 , the inverse matrix of the jacobian @xmath84 @xmath85 for @xmath86 . taking into account the symmetry @xmath87 , the preceding equation becomes @xmath88 & - \ 2 \left(\frac{\partial \mathfrak t^{\mu \rho}}{\partial x^\mu}+\mathfrak t^{\mu \nu}\eta ^b{}_{,\mu \nu } \kappa^\rho{}_b \right ) = 0.\end{aligned}\ ] ] recalling the expression of the christoffel symbols of the metric @xmath17 , namely , @xmath89 we obtain @xmath90 finally , recall how the christoffel symbols @xmath91 for @xmath17 and the symbols @xmath92 for @xmath93 are related : @xmath94 using this in gives @xmath95 for @xmath86 , which is exactly the vanishing of the covariant divergence of the tensor _ density _ @xmath77 . thus , we have proven the following basic result . the euler lagrange equations for the covariance field @xmath35 are that the covariant divergence of the sem tensor density @xmath96 is zero . it is known from proposition 5 in @xcite that the sem tensor is covariantly conserved when the metric @xmath18 is the _ only _ nondynamic field . thus , in our context , the equation @xmath97 is an identity , whence the euler lagrange equations for the covariance field @xmath35 are vacuously satisfied . consequently the covariance field has no physical import . we are free to suppose @xmath35 is dynamic , and so we have accomplished goal ( ii ) : we have constructed a new field theory in which _ all _ fields are dynamic . it is interesting to compare the sem tensors for the original and parametrized systems . in @xcite the sem tensor density @xmath98 is defined in terms of fluxes of the multimomentum map @xmath99 associated to the action of the spacetime diffeomorphism group . we rapidly recount some of the basic ideas . consider the lift of an infinitesimal diffeomorphism @xmath100 to @xmath14 ; it can be expressed @xmath101 where we suppose that @xmath102 for some coefficients @xmath103 . the largest value of @xmath104 for which one of the top coefficients @xmath105 is nonzero is the differential index of the field theory . we assume henceforth that the index @xmath106the most common and important case ( e.g. , when the fields are all tensor fields ) . in this context , theorem 1 along with remark 4 of @xcite shows that the sem tensor density @xmath107 for a lagrangian density @xmath108 is uniquely determined by @xmath109 for all vector fields @xmath110 on @xmath3 with compact support and all hypersurfaces @xmath111 , where @xmath112 is the inclusion . the multimomentum map @xmath113 gives , roughly speaking , the flow of momentum and energy through spacetime ; according to the quoted theorem , the fluxes of this flow across hypersurfaces are realized via the sem tensor density . manipulation of ( see formula ( 3.12 ) of @xcite ) shows that @xmath107 is given by @xmath114 where the summation extends over _ all _ fields @xmath115 . we apply this to the newly parametrized theory . note that if the index of the original theory is @xmath106 , then that for the parametrized theory will be also . as well from we see that the lift of @xmath116 to @xmath56 is trivial : @xmath117 that is , there are no terms in the @xmath118 directions in @xmath119 . thus the corresponding coefficients @xmath120 all vanish . the sem tensor for @xmath37 therefore reduces to @xmath121 on the other hand , @xmath122 and @xmath123 so that we can write @xmath124 but the first four terms on the rhs of this equation comprise the sem tensor density of the original theory since the @xmath125 do not derivatively couple to the @xmath49 ( cf . ( 4.4 ) in @xcite ) . thus the sem tensor densities of the original and parametrized systems are related according to : @xmath126 but then @xmath127 on shell by the hilbert formula . therefore , we explicitly see that the sem tensor density for the fully covariant , fully dynamic modified theory vanishes . one can also obtain this result directly by applying the generalized hilbert formula ( 3.13 ) in @xcite to the parametrized theory , since it is fully dynamic . [ [ example.-5 ] ] example . + + + + + + + + in the case of electromagnetism , one may compute directly from that @xmath127 . one could also compute from that @xmath128 -32pt @xmath16 here we briefly consider the situation , although perhaps exotic , when the metric derivatively couples to the other fields . for simplicity , however , we suppose the theory remains first order . so the lagrangian density is taken to be a map @xmath129 as before , modify @xmath22 to get the new lagrangian @xmath37 defined on @xmath130 : @xmath131 ( since @xmath132 depends upon the first derivatives of @xmath35 , @xmath133 will depend upon its second derivatives . thus , we obtain a modified _ second _ order field theory with the underlying bundle @xmath40 . ) the discussion proceeds as in the above , with only obvious changes . in particular , if @xmath134 is diff@xmath135-covariant , then so is @xmath136 . as a simple illustration of a derivatively coupled theory , consider a vector meson with mass @xmath137 . then @xmath14 is the tangent bundle of spacetime , and its sections @xmath138 are klein gordon vector fields . the lagrangian density is the map @xmath139 defined by @xmath140 where the semicolon denotes the covariant derivative with respect to @xmath18 . our construction produces @xmath141 \nonumber \\[1.5ex ] & \times \big [ \phi^{{\mspace{1.5mu}}\rho}{}_{,\nu } + \big(\eta^h{}_{,\nu\xi } + \eta^p{}_{,\nu } { \mspace{1.5mu}}\eta^q{}_{,\xi } { \mspace{1.5mu}}\gamma ^h _ { pq } \big ) \kappa^\rho{}_h{\mspace{1.5mu}}\phi^\xi\big ] \nonumber \\[1.5ex ] & - m^2 \phi^\sigma \phi^{{\mspace{1.5mu}}\rho } \bigg)\sqrt{-g}\ , ( \det \eta _ * ) \ , d^{{\mspace{1.5mu}}4}{\mspace{-1.5mu}}x\end{aligned}\ ] ] where @xmath51 is the jacobian of @xmath35 and we have made use of . now we turn to the euler lagrange equations for the @xmath71 which , since @xmath136 is second order in the @xmath71 , are : @xmath142 for @xmath73 . the calculation of the lhs is similar to the previous one , but slightly more complicated . in any event , we find that @xmath35 satisfies the euler lagrange equations @xmath143 @xmath97 , where now by the hilbert formula @xmath144.\ ] ] thus for ( first order ) derivative couplings the covariance field remains vacuously dynamic . it is likely this will remain true for derivative couplings of arbitrary order , but we have not verified this as yet . we dedicate this paper to darryl holm on his 60@xmath145 birthday . we thank him for his interest in the ideas in this paper and for his many inspiring works over the years . mjg and jem thank the national science foundation for its occasional support of work of this sort . mcl was partially supported by dgsic ( spain ) under grant mtm2007 - 60017 .
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we give an exposition of the parametrization method of @xcite in the context of the multisymplectic approach to field theory , as presented in @xcite .
the purpose of the formalism developed herein is to make any classical field theory , containing a metric as a sole background field , generally covariant ( that is , _ parametrized _ , with the spacetime diffeomorphism group as a symmetry group ) as well as fully dynamic .
this is accomplished by introducing certain `` covariance fields '' as genuine dynamic fields .
as we shall see , the multimomenta conjugate to these new fields form the piola
kirchhoff version of the stress - energy - momentum tensor field , and their euler
lagrange equations are vacuously satisfied .
thus , these fields have no additional physical content ; they serve only to provide an efficient means of parametrizing the theory .
our results are illustrated with two examples , namely an electromagnetic field and a klein gordon vector field , both on a background spacetime .
| 6,120 | 262 |
swimming microorganisms are ubiquitous in nature , and have long been known to play important roles in marine life ecosystems , animal reproduction , and infectious diseases . in these processes , cell motility is crucial.@xcite at the small scales relevant to swimming cells , inertial forces are negligible , and locomotion is constrained by purcell s `` scallop '' theorem stating that any body deformation reversible in time yields zero net motion.@xcite fluid - based cellular motility relies therefore on non - time reversible deformation , for instance by propagating waves along cilia or flagella.@xcite among the various types of locomotion seen in nature , one commonly observed for bacteria is that of helical propulsion , where a flagellum ( or a bundle of flagella ) rotates as a helix , inducing forward propulsion . a typical example of an organism employing helical propulsion is the bacterium _ escherichia coli _ coli_).@xcite this bacterium alternates `` run '' and `` tumble '' periods : in the former , flagella are synchronized in a coherent bundle and propel the cell forward , whereas in the latter flagella are disorganized , changing the cell orientation and subsequent swimming direction . during run periods , when _ e. coli _ cells are isolated in a bulk flow , they swim in straight ( noisy ) lines . however , cell locomotion is strongly affected by nearby boundaries . swimming microorganisms often evolve in confined environments , be it by solid boundaries , free surfaces , or liquid interfaces . in some cases , confinement results from channel boundaries , for example along the mammalian female reproductive tract.@xcite surfaces can also be a key element in the microorganism function , as in the case of surface associated infection or biofilm formation.@xcite since such problems are dominated by viscous dissipation , long - range hydrodynamic interactions have been argued to play important roles , resulting in a significant alteration of the locomotion of microorganisms.@xcite over the past years , intensive theoretical , numerical and experimental work has helped uncover the kinematics and dynamics modifications of swimming properties by boundaries.@xcite for bacteria employing helical propulsion ( such as _ e. coli _ ) , two different effects induced by boundaries have been discovered and quantified . these organisms swim in the forward direction ( cell body forward ) and are being propelled from the back . they thus push on the surrounding fluid forward and backward , and such swimmers are referred to as `` pushers '' . in the presence of a nearby solid wall , _ e. coli _ tends to aggregate close to walls.@xcite this is in fact observed for any kind of pusher , not necessarily one exploiting helical propulsion.@xcite a second property , observed solely for helical swimmers , is a circular motion of the cells in a plane parallel to the surface . this was accounted for both experimentally and theoretically in the case of a solid wall @xcite and a free surface.@xcite notably , the circular motion occurs in an opposite direction in the presence of a solid wall ( clockwise , cw , when viewed from inside the fluid ) or a free surface ( counterclockwise , ccw , see fig . [ fig:1 ] ) . this change in rotation direction is qualitatively similar to the drag increase or decrease observed for the motion of a colloidal particle near a rigid wall and a free surface.@xcite indeed , a solid wall and a free surface induce opposite effects , no - slip for a rigid boundary vs. free slip in the case of a free interface . , and counter - clockwise motion at a free surface ( right panel ) [ reprinted figure with permission from r. di leonardo , d. dellarciprete , l. angelani , and v. iebba , phys . rev . 106 , 038101 ( 2011 ) . copyright ( 2011 ) by the american physical society ] . ] past experimental results have been explained theoretically considering newtonian fluids and perfect interfaces , meaning either a no - slip wall or a shear - free surface . theoretical models do predict a single circular direction , cw in the presence of a solid wall vs. ccw in the presence of a free surface , and are consistent with the results illustrated in fig . [ fig:1 ] . however , recent experiments on _ e. coli _ swimming near glass plates and free surfaces show that the distinction in the direction of the circular motion is not straightforward , and both cw and ccw rotations are observed under seemingly similar experimental conditions.@xcite in the initial study of lemelle _ et al . _ ( 2010),@xcite only cw motion was observed above a glass plate , but both cw and ccw at a free surface , suggesting that particles and surfactants could alter the free slip boundary condition . this hypothesis was further investigated by changing the concentration of a particular polymer that can aggregate at a free surface.@xcite the authors confirmed this qualitative change of behavior , observing a clear dependence on the polymer concentration of the fraction of cells undergoing ccw motion . a similar change in rotation direction was recently highlighted experimentally at a solid wall , when the solution contains polymers.@xcite using a special surface treatment , the polymer concentration at the solid wall was modified , generating possible slip , and resulting in ccw motion . these recent experiments demonstrate that the presence of polymers or surfactants could have a dramatic effect on motility of nearby cells . in this paper we present a modeling approach to quantify the dynamics of swimming bacteria near complex interfaces . when polymers are present in the solution , their concentration close to surfaces is reduced due to higher shear and confinement.@xcite this wall depletion results in the formation of a thin fluid layer of lower viscosity at the wall , thereby modifying significantly the no - slip condition . on scales larger than this thin layer , the equivalent behavior at the wall is an apparent partial slip , characterized by its slip length @xmath0 ranging from @xmath1 nm to 10 @xmath2m.@xcite similarly , a liquid interface covered with surfactants acts as a thin two - dimensional fluid layer separating the liquid phases . this layer has its own rheological properties , and modifies the stress and velocity jumps between the two fluids.@xcite as a consequence , the presence of surfactants can affect significantly the boundary conditions and resulting flow.@xcite in the present work , we address the role of altered boundary conditions on swimming microorganisms , focusing on interface - induced reorientation , attraction vs. repulsion by the surface , and the impact on circular motion . using an analytical framework based on multipole expansions for describing the hydrodynamic interactions between a swimming microorganism and an interface , we show how complex interfaces affect hydrodynamic interactions , providing possible explanations to past experimental results . whereas interface alignment and attraction are seen to be universal properties , the direction of the circular motion turns out to strongly depends on the properties of the fluid , on the bacterium shape and , in some cases , the distance to the interface . in sec . [ sec : modeling ] we present the modeling approach used throughout the paper . we first introduce the different interfaces considered , and then our solution method quantifying the leading - order effect of hydrodynamic singularities . in sec . [ sec : stokesletclosetoaboundary ] we recall some existing results for the flow generated by a point force near boundaries and derive in particular the solution in the case of a surfactant - covered interface . sec . [ sec : resultstruetrue ] is devoted to the main results of the paper , quantifying the impact of complex boundary conditions on swimming bacteria , first on reorientation and attraction , and then on circular motion . we finally conclude in sec . [ sec : ccl ] while some of the technical details are given in appendices [ sec : appb]-[sec : typicaldifferentialequationsandsolutions ] . throughout the paper , we use the word interface to refer to any boundary separating two phases . we denote phase 1 the fluid where the swimming bacterium is located , while the second phase can either be a solid , a liquid , or a gas . the presence of a nearby interface affects bacteria locomotion in different ways depending on its flow boundary conditions . in this work , we consider three types of interfaces , sketched in fig . [ fig:2]a : ( i ) a clean interface , i.e. a fluid - fluid interface with no surfactant , ( ii ) a flat surface with a partial slip condition as a model for a polymer depletion layer , and ( iii ) a liquid interface covered with incompressible surfactants . in the case of flow generated by swimming microorganisms , the typical capillary number , scaling the flow - induced stress with surface tension , is very small , typically lower than @xmath3 . as a consequence we will neglect any interfacial deformation induced by the microorganisms , and focus therefore on planar interfaces with normal @xmath4 ( fig . [ fig:2]a ) . the first type of interface we will consider is a clean interface between two fluids . this problem is characterized by the viscosity ratio between the two fluids , @xmath5 where the reference viscosity @xmath6 is the viscosity of the fluid where the bacterium is located . if the viscosity of the second fluid vanishes , @xmath7 , the interface behaves as a free surface , whereas if the second fluid viscosity tends to infinity , we have @xmath8 , and the interface becomes equivalent to a rigid wall . a clean liquid interface imposes impermeability and the continuity of tangential velocities and stresses.@xcite the corresponding boundary conditions written at @xmath9 are @xmath10 where @xmath11 is the symmetric rate of strain tensor , and where we have used the superscript @xmath12 to denote fluid @xmath12 . in the case of a free surface ( @xmath7 ) , these equations reduce to a free slip condition , whereas in the limit of a rigid wall ( @xmath13 ) , the boundary imposes a no - slip condition , @xmath14 . the second type of interfaces that we will consider models a rigid wall when the fluid contains polymers , and therefore is subjected to wall depletion . for simplicity , we focus solely on the modification of the boundary conditions and assume no change in the fluid rheology . the model problem will therefore become that of a bacterium swimming in a newtonian fluid close to a partial - slip boundary , meaning that the wall velocity is proportional to the normal velocity gradient . defining @xmath0 the slip length , the boundary conditions read @xmath15 where @xmath16 is the flow velocity in the plane of the interface . the last type of interface we are interested in characterizing are those covered with surfactants . this type of interface behaves as an impermeable , bi - dimensional fluid layer separating the two phases , and the bulk stress jump in the absence of surface forces is given by @xmath17,\end{aligned}\ ] ] where @xmath18 is the surfactant concentration , @xmath19 the surface velocity , @xmath20 the in - plane gradient operator , and @xmath21 ( resp . @xmath22 ) the bulk ( resp . surface ) stress tensor.@xcite in the present analysis , it is appropriate to focus on the stationary limit on the length scales of swimming bacteria . additionally , we assume that the concentration of surfactant on the surface is large enough so that the relative change in surface tension is of order 1 . in that case , as the capillary number is very small ( typically lower than @xmath3 ) , the surfactant concentration can be considered to be uniform on the surface.@xcite under this assumption , the interface is incompressible , @xmath23 , and the surface stress tensor reads @xmath24 where @xmath25 is the surface tension and @xmath26 the in - plane rate of strain tensor.@xcite the resulting boundary conditions at @xmath9 state continuity of in - plane velocity , no flow perpendicular to the surface , in - plane incompressibility , and balance between shear stresses from the outside flows , surface viscous stresses and marangoni stresses @xmath27 where @xmath28 is the non - dimensional surface viscosity , @xmath29 @xmath30 being a flow length scale , chosen here to be the bacterium distance to the wall.@xcite the parameter @xmath28 is often referred to as boussinesq number , comparing surface to bulk stresses @xcite . note that the condition for incompressibility prevents the limit of a clean interface to be reached by simply applying @xmath31 , and only the case of a no - slip boundary can be recovered in the limit @xmath32 . in order to analyze the effect of an interface on the swimming microorganism , we decompose the flow into two contributions , one being the flow generated by the same organism in the absence of interface , @xmath33 , and a contribution due solely to the interface , @xmath34 , so that @xmath35 . the effect of the interface on the swimming microorganism is then determined using faxn s law , modeling the bacterium shape as that of a prolate ellipsoid swimming along an intrinsic direction @xmath36 . noting @xmath37 the aspect ratio of the entire organism ( i.e. cell body and flagellar bundle ) , the interface - induced velocity , @xmath38 , and rotation rate , @xmath39 , of the microorganism are given by @xmath40,\end{aligned}\ ] ] where @xmath41 is the location of the bacterium @xcite ( fig . [ fig:2]b ) . the fact that the interface effect is evaluated only at the location of the swimmer is a key point here we do not seek to describe the entire flow field in the presence of the aforementioned interfaces , but only the flow at a specific location . the reynolds numbers associated with swimming microorganisms are very small.@xcite in this limit , the flow generated by a swimming microorganism satisfies the linear stokes equations,@xcite which allow for a multipole expansion of any flow.@xcite in the present work , we will represent the flow induced by a swimming microorganism using flow singularities . general flow singularities and notation are presented below while in sec . [ sec : resultstruetrue ] we will focus on the specific singularity model used for microorganisms using helical propulsion . flow singularities are derived from the green s function for stokes flows , @xmath42 , which gives the flow at position @xmath43 generated by a point force @xmath44 located at @xmath41 and oriented along the direction @xmath45 as @xmath46 where @xmath47 , and @xmath48 . this solution is called the stokeslet , and higher - order singularities are then derived from this fundamental singularity.@xcite we introduce here the stokeslet dipole , @xmath49 , and quadrupole , @xmath50 , @xmath51 and the ( potential ) source dipole , @xmath52 , and quadrupole , @xmath53 , @xmath54 where the notation @xmath55 is used to denote a gradient taken with respect to the singularity location , @xmath41 . an useful combination of stokes dipoles is the rotlet,@xcite which is the anti - symmetric part of a stokeslet dipole , and models the flow generated by a point torque @xmath56 = \frac{{{\bm{c}}}\times{{\bm{r}}}}{r^3},\end{aligned}\ ] ] where @xmath57 . the symmetric part of a stokeslet dipole is called a stresslet , @xmath58 $ ] . note that all these singularities are @xmath59-linear functions of their @xmath59 orientation vectors . a singularity oriented along arbitrary directions can thus be expressed as a combination of similar singularities along the different basis vectors . the mathematical method used for solving the problem of a point singularity located close to a boundary with specific conditions is blake s method , presented first for the problem of a stokeslet close to a rigid wall @xcite . the effect of the interface is mathematically equivalent to an additional flow generated by a system of hydrodynamic image singularities located on the other side of the interface . for complex boundary conditions , the image system is a spatial distribution of singularities . the problem can be significantly simplified by guessing part of the image system . noting @xmath60 the image system , the flow is decomposed as @xmath61 where @xmath33 is the bulk solution due to the singularity itself , @xmath62 is a guess in the image system , and @xmath63 and @xmath64 are the unknowns . for example , in the case of a stokeslet close to a free surface , a good guess for @xmath62 is a stokeslet symmetric with respect to the interface , which is enough to enforce both the impermeability condition and that of zero shear stress , resulting in @xmath65 . the major advantage of this decomposition is that the forcing term due to the singularity in the flow equation is solved by the bulk flow term @xmath33 . as a result , @xmath63 and @xmath64 satisfy stokes equations in the absence of forcing terms @xmath66 where @xmath67 is the velocity field and @xmath68 the pressure , @xmath69 being the fluid viscosity . that problem can be more easily solved in fourier space,@xcite using a two - dimensional ( 2d ) fourier transform , defined as @xmath70 } = \frac{1}{2\pi}\int\int{f(x , y , z)e^{ik_1x+ik_2y}\mathrm{d}x\mathrm{d}y}.\end{aligned}\ ] ] some useful fourier transforms are referenced in appendix [ sec : appb ] . the solution of stokes equations in a 2d fourier space is straightforward , and reads for @xmath71 and @xmath72 @xmath73 the unknown coefficients in this solution are determined using the continuity equation@xmath74 together with the relevant boundary conditions . once the solution is obtained for the stokeslet , it is then possible to derive the solution for higher - order singularities from the stokeslet solution . this is achieved by applying the same operator acting on the singularity position , @xmath75 , where @xmath76 is the direction where the derivative is taken . note that this can be done more easily in fourier space , as @xmath77}=ik_1,\quad { \mathcal{f}\left[{{\bm{e}}_\mathrm{y}}.{\nabla_0}\right]}=ik_2 , \quad { \mathcal{f}\left[{{\bm{e}}_\mathrm{z}}.{\nabla_0}\right]}={\frac{\partial } { \partial h}}.\end{aligned}\ ] ] however , it is also possible to directly apply blake s method to any singularity , which can be more convenient if a good guess of @xmath62 is found . as a singularity is a linear function of its orientation vectors , one only needs to solve the problem for the stokeslet along the directions parallel and perpendicular to the interface . the flow generated by higher - order singularities can then be derived from these projections . by symmetry reasons , all orientations in the plane parallel to the interface are equivalent , we therefore choose @xmath78 to be the singularity plane . we review below some solutions of the flow generated by parallel and perpendicular stokeslets close to a fluid boundary , and then use the method described in the previous section for determining the flow in the presence of a surface covered with incompressible surfactant , providing an alternative to the derivation by bawzdziewicz _ ( 1999).@xcite the solution in the case of slip is given in lauga and squires ( 2005)@xcite along the same lines . the case of parallel and perpendicular stokeslets were derived by blake in the case of a solid wall , a free surface , and a fluid - fluid interface.@xcite in the presence of a clean fluid - fluid interface , the image system is made of a finite number of point singularities located at the image position @xmath79 in fluid 2 ( see notation in fig . [ fig:2]b ) . the image system is composed of a stokeslet , a stokes dipole , and a source dipole , whose intensities vary with the viscosity ratio @xmath80 and the distance @xmath30 to the interface . for stokeslets parallel , @xmath81 , and perpendicular to the surface , @xmath82 , the image system is given by @xmath83 where we use @xmath84 in order to simplify the notations , and where the superscript @xmath85 implies that the singularity is located at the image position , @xmath79 . these singularities give the flow in fluid 1 , where the singularity is located . the flow in fluid 2 is given by a second set of singularities , given by @xmath86,\\ & \text{im}^{(2)}\{{{\bm{g_s}}}({{\bm{e}}_\mathrm{z}})\}= \frac{2 h}{1+{\lambda } } \big[{{\bm{g_{sd}}}}({{\bm{e}}_\mathrm{z}},{{\bm{e}}_\mathrm{z}})+h{{\bm{g_{d}}}}({{\bm{e}}_\mathrm{z}})\big].\end{aligned}\ ] ] this second set of images is located at the singularity position @xmath41 , in fluid 1 ( see fig . [ fig:2]b ) . the solutions are therefore obtained fully analytically , and higher - order solutions can be obtained by taking derivatives with respect to the singularity position ( @xmath87 ) . note that since the image strengths are functions of the singularity distance to the interface @xmath30 , taking derivatives of the image system along @xmath4 generates additional singularities.@xcite in the presence of a solid wall , the second set of images vanishes since there is no flow in the solid region . this was the no - slip solution originally presented by blake.@xcite as we are interested in this work only on the effect of the interface on the swimming bacterium , we will focus only on the first set of images , eqs . and , giving the flow field in the region where the swimming microorganism is located . the no - slip and no - shear solutions have been used in the past to explain different swimming behaviors of bacteria observed experimentally , such as wall attraction , alignment , and circular motion.@xcite the particular case of no - slip boundary ( @xmath88 ) is noteworthy , and will be used in the following as a reference solution @xmath89 in the presence of slip or surfactants , the image system is no longer a set of point singularities . the solution to the problem of a stokeslet close to a partial slip boundary was addressed by lauga and squires,@xcite using blake s method . the case of a surfactant - covered interface has been addressed by bawzdziewicz _ et al._,@xcite using a batchelor - type decomposition of the flow . in order to use the same formalism throughout our paper , we derive below the solution of a stokeslet close to a surface covered with incompressible surfactants using blake s method . we consider here the case of a stokeslet perpendicular and parallel to a surfactant - covered interface as defined in sec . [ sec : interfaces ] . without loss of generality , the stokeslet strength is taken to be 1 . for this problem , we take @xmath62 to be the opposite stokeslet located at the image position @xmath79 . this choice yields at @xmath9 for a perpendicular and parallel stokeslet respectively @xmath90({{\bm{e}}_\mathrm{z } } ) = -\frac{h}{4\pi{\eta}_1}\frac{1}{r_h^3}(x{{\bm{e}}_\mathrm{x}}+y{{\bm{e}}_\mathrm{y}}),\quad \left[{\bm{u}}+{\bm{v}}\right ] ( { { \bm{e}}_\mathrm{x}})= -\frac{h}{4\pi{\eta}_1}\frac{x}{r_h^3}{{\bm{e}}_\mathrm{z}},\end{aligned}\ ] ] where @xmath91 . the problem is then solved in fourier space . in the case of a perpendicular stokeslet , the boundary conditions read in fourier space @xmath92 for @xmath93 and 2 , where index 1 ( resp . 2 ) is associated with the @xmath94 ( resp . @xmath95 ) coordinate . this set of equations is satisfied by @xmath96 , which corresponds to a solid wall . we recover therefore a known result : because of surface incompressibility , the flow generated by a stokeslet perpendicular to an interface covered with incompressible surfactants is identical to that in the presence of a solid wall.@xcite in this case , the boundary conditions are given in fourier space by @xmath97 introducing the coefficients @xmath98 , @xmath99 , @xmath100 and @xmath101 from sec . [ sec : solmeth ] , this system can be solved directly , leading to @xmath102 , and @xmath103 it is interesting to note that the coefficients in the no - slip problem read @xmath104 hence , introducing @xmath105e^{-kz}/(8\pi{\eta}_1)$ ] , the flow field @xmath63 can be written as @xmath106 where @xmath107 is the known solution of the problem in the presence of a no - slip boundary , @xmath108 . after an inverse fourier transform of @xmath109 , one can identify two differential equations satisfied by @xmath110 and @xmath111 in real space , which can be written as a single vectorial differential equation on @xmath112 , @xmath113 where @xmath114 is a dipole along @xmath115 of rotlets along @xmath4 , located at @xmath79 . the rotlet dipole involved in this flow corresponds to two vertical counter - rotating point vortices , merging at @xmath79 . note that this singularity can also be written in a way similar to that of ref.@xcite as @xmath116 . the flow in the second fluid satisfies a similar problem , @xmath117 with the rotlet dipole being located at @xmath41 . as a result , the total flow generated by a stokeslet parallel to an interface covered with surfactants can be written as @xmath118 where @xmath119 is the flow generated by a stokeslet along @xmath120 in the presence of a no - slip boundary . we recover therefore the result of ref.@xcite where the flow is the sum of the no - slip contribution and a `` surface - solenoidal '' flow , i.e. a 2d flow decaying in @xmath121 . in order to obtain the exact expression for the flow at the position of the singularity , one still needs to integrate a third order differential equation , eq . . since our goal is not to derive the entire flow but only the effect of the boundary on the swimming microorganism , we can simplify in the following way . the singularity is located at @xmath122 and since the differential equations giving the flow are equations in @xmath121 , it is possible to set @xmath122 before integrating . as a result , the additional flow induced on the singularity is given by @xmath123 , where @xmath124 and @xmath125 the latter equation yields @xmath126 , which could have been anticipated by symmetry . the former equation can be integrated analytically , as detailed in the next section . in this work , we model a swimming bacterium in the far field and thus assume that it is small compared to any flow length scale . in particular , the distance between the cell and the interface , @xmath30 , must remain larger than the bacterium size . this far - field approach allows for a representation of a swimming microorganism as a combination of point singularities , while the organism size and shape play a role only through the aspect ratio @xmath37 involved in faxn s law , eq . . we focus here on microorganisms using helical propulsion , like _ e. coli _ , as illustrated schematically in fig . [ fig:3]a . such micro - swimmers are force- and torque - free , and can be considered as axisymmetric in average . we choose as a convention that the microorganism swims in the @xmath78 plane , along direction @xmath127 . at leading order ( spatial decay @xmath128 ) , the far - field flow generated by a swimming bacterium is well captured by an axisymmetric stokeslet dipole , @xmath129 , as sketched in fig . [ fig:3]b.@xcite this singularity is force- and torque - free , and accounts for the spatial distribution of thrust and drag on a flagellated swimming cell.@xcite this representation is commonly used for describing far - field hydrodynamic interactions , both to quantify collective dynamics @xcite and wall effects.@xcite however , by symmetry , this leading - order singularity can not be responsible for the observed circular rotation along the normal to the interface since @xmath78 is a symmetry plane of the stokes dipole . it is thus necessary to add higher - order singularities decaying as @xmath130 to capture circular swimming . those singularities could be : ( i ) a stokes quadrupole , related to the length asymmetry between the flagella and the body , ( ii ) a potential source dipole accounting for the finite size of the bacterium , and ( iii ) a rotlet dipole capturing the counter - rotation between the flagella and the body . by symmetry , it is straightforward to see that the only singularity that could potentially lead to nonzero rotation along @xmath4 is the rotlet dipole @xmath131 , illustrated in fig . [ fig:3]c . we are interested in three effects induced by a nearby boundary : ( a ) reorientation in the swimming plane , given by rotation rate @xmath132 ; ( b ) attraction or repulsion by the wall , quantified by wall - induced velocity @xmath133 ; and ( c ) circular motion , measured by the out - of - plane rotation rate @xmath134 . reorientation and attraction can both be accounted for by the leading - order stokes dipole , as analyzed in sec . in contrast , the circular swimming which is due to the rotlet dipole is addressed in sec . [ no ] . we consider here the leading order singularity representing a swimming microorganism , a stokes dipole , and focus on two effects . we first address the issue of reorientation in the @xmath78 plane induced by a boundary on a tilted micro - swimmer moving along the direction @xmath127 . with the stable wall - induced orientations derived , @xmath135 , we then address the attractive vs. repulsive nature of the interface . a stokes dipole tilted along the angle @xmath136 is a combination of parallel and perpendicular stokes dipoles as @xmath137 the wall - induced rotation rate is dependent on the orientation of the microorganism and the local strain rate , see eq . . in the following , the strength of the stokes dipole is taken to be 1 , modeling a `` pusher '' swimmer as relevant to any flagellated bacteria moving cell body first ( a puller corresponds to a negative strength ) . in the presence of a clean interface , the image system is the set of point singularities listed above , and the problem can be solved analytically , leading to the wall - induced rate @xmath138 . \label{omcleansurf}\end{aligned}\ ] ] notably , the sign of this rotation rate is given by @xmath139 as the term in the bracket is always positive . as a result , the interface will always tend to align the ( pusher ) stokes dipole in the direction parallel to the interface . this result was previously shown in the case of a solid wall,@xcite and is thus generalized here to any fluid - fluid interface . the stable orientation is therefore a microorganism swimming parallel to the interface , @xmath140 . in that case , the induced velocity along the vertical axis reads @xmath141 we see thus that a swimming pusher will be attracted by a nearby liquid interface for any value of the viscosity ratio . in the limits of a free surface ( @xmath142 ) and a solid wall ( @xmath143 ) , standard results are recovered.@xcite in the presence of a partial slip boundary , the flow due to the boundary is equivalent to that generated by a continuous distribution of singularities along the vertical axis on the other side of the surface.@xcite taking derivatives of higher - order solutions is not straightforward in real space , but is easily carried out in fourier space . it is interesting to note that @xmath144 is a symmetry plane for the first two singularities of the decomposition shown in eq . , namely @xmath145 and @xmath146 . as a result , these terms do not contribute to any rotation rate through vorticity , only the stresslet does . however , all terms contribute to the total rotation rate , as the swimmer s orientation breaks the symmetry , acting on the strain contribution to the rotation rate , eq . . the three singularities necessary for describing a tilted stokes dipole are derived with a particular choice for the different velocity fields @xmath62 , so that @xmath62 corresponds to the solution in the presence of a free surface . this limit is reached in the partial slip model when @xmath0 tends to infinity . for the stresslet @xmath147 , we choose therefore @xmath148 ; the no - slip solution reads @xmath149 following the procedure outlined above we determine the solution in the presence of a partial slip boundary in fourier space @xmath150 similarly we have for the parallel stokes dipole @xmath145 @xcite @xmath151 , \label{eq : sdpar}\end{aligned}\ ] ] with the no - slip solution @xmath152 . for the perpendicular stokes dipole @xmath146 , we have @xmath153.\end{aligned}\ ] ] the rotation rate of the cell along @xmath115 is then computed in fourier space for the three singularities , and the total rotation rate on a tilted stokes dipole close to a partial slip boundary finally reads @xmath154 evaluated at the singularity position , where index sdx stands for parallel stokes dipole , sdz for perpendicular stokes dipole , and ss for the stresslet contribution . the technical difficulty in this problem is the inverse fourier transform of the rotation rate and we refer to appendices [ sec : appa ] and [ sec : typicaldifferentialequationsandsolutions ] for the details related to the inversion problem and solution of the resulting differential equations . the final solution in real space is given analytically by @xmath155-f_3\!\!\left[\frac{h}{{\ell}}\right]-h_4\!\!\left[\frac{h}{{\ell}}\right]\right ) + { \frac{\gamma^2 - 1}{\gamma^2 + 1}}{\sin^2 \!{\theta}}\nonumber\\ & \qquad\qquad \quad \quad + { \frac{\gamma^2 - 1}{\gamma^2 + 1}}\frac{h}{{\ell } } \bigg[-2{\sin^2 \!{\theta}}f_3\!\!\left[\frac{h}{{\ell}}\right ] + ( 3{\cos^2 \!{\theta}}-2{\sin^2 \!{\theta}})h_4\!\!\left[\frac{h}{{\ell}}\right]\nonumber\\ & \qquad \qquad\qquad \qquad \quad+(1 + 3{\sin^2 \!{\theta } } ) \left(f_4\!\!\left[\frac{h}{{\ell}}\right]-\frac 12f_5\!\!\left[\frac{h}{{\ell}}\right]+\frac 12g_4\!\!\left[\frac{h}{{\ell}}\right]-g_5\!\!\left[\frac{h}{{\ell}}\right]\right)\bigg]\bigg\},\end{aligned}\ ] ] where the functions @xmath156 , @xmath157 and @xmath158 derive from the exponential integral function of order @xmath59 , @xmath159 , as @xmath160 with @xmath161 , characterizing the alignment of the bacterium with the boundary , as a function of the orientation angle @xmath162 and for different values of the dimensionless slip length @xmath163 ( 0.1 , 1 , and 10 , thin solid lines ) . bold dashed lines correspond to the limits of vanishing slip length ( no - slip condition ) and infinite slip length ( no shear ) . ( b ) qualitative physical picture for interface alignment and attraction ; streamlines are thin dashed lines , thin arrows ( dark blue online ) show local effects while thick arrows ( light red online ) show global effect on the swimming microorganism . ] the dependence of the rotation rate @xmath164 induced by a partial slip boundary on a tilted stokeslet dipole is shown in fig . [ fig:4]a , as a function of the tilt angle @xmath162 , for different values of the slip length . we see that the slip length , similarly to the viscosity ratio in the case of a clean liquid interface , does not modify qualitatively the reorientation dynamics and alignment rate with the boundary and a pusher micro - swimmer will always tend to align parallel to the boundary . the limit of infinite slip length , @xmath165 in eq . , leads naturally to the free surface limit . furthermore , using the asymptotic properties of the exponential integral functions , we have @xmath166 and @xmath167 when @xmath94 tends to infinity , which can then be used to recover quantitatively the reorientation dynamics in the no - slip limit . we then compute the velocity induced on a micro - swimmer parallel to the boundary , modeled by a parallel stokes dipole , eq . . the integration can be done in a similar way as for the rotation rate , and using integration by parts we get a simple expression for the vertical velocity as @xmath168 a partial slip boundary will always attract a pusher swimmer for any value of the slip length , generalizing therefore the known results for @xmath169 ( no - slip ) and @xmath170 ( no - shear ) . we finally consider the case of a stokes dipole close to a liquid interface covered with incompressible surfactants . from the stokeslet solution presented in sec . [ sec : surfactstokeslet ] , it is possible to derive higher - order solutions . using eq . , the flow generated by a tilted stokeslet dipole is therefore given by @xmath171 with @xmath172 using the same methodology as that described in the previous section and appendix [ sec : appa ] , we get that @xmath173 ^ 0 + { \cos^2 \!{\theta}}~\!\omega_{xx } + { \cos { \theta}}{\sin { \theta}}~\!\omega_{xz}$ ] , where @xmath174 ^ 0 $ ] is the solution in the presence of a no - slip boundary , given by eq . with @xmath13 , and @xmath175 ( resp . @xmath176 ) is the contribution from the flow @xmath177 ( resp . @xmath178 ) . the resulting rotation rate is given by @xmath179 + { \frac{\gamma^2 - 1}{\gamma^2 + 1}}{\cos^2 \!{\theta}}\frac{3}{\beta } \int_1^\infty{\!\!\ ! \int_1^\infty{\frac{f_5(2bst)}{t^3s^4}\mathrm{d}s}\mathrm{d}t}\nonumber\\ & + \frac{1}{2\beta}f_3(2b)\left[1+{\frac{\gamma^2 - 1}{\gamma^2 + 1}}({\sin^2 \!{\theta}}-{\cos^2 \!{\theta}})\right]\bigg\},\end{aligned}\ ] ] where @xmath180 and the functions @xmath156 are defined in eqs . - . for any value of the parameters and orientation angle , the sign of the rotation sign is that of the prefactor @xmath139 . as a result , a surface covered with incompressible surfactants tends to always align a swimming microorganism with the surface , similarly to the no - slip and no - shear cases . furthermore , since the additional velocity field @xmath112 does not have a vertical component , we see immediately that the vertical velocity induced by the interface is the same as that in the presence of a no - slip boundary , and again a surface covered with surfactants will attract a pusher toward the interface . in summary , the leading - order singularity modeling a swimming microorganism allowed us to generalize results known for both a solid wall and a free surface . the interface , be it a clean fluid - fluid interface , a partial slip boundary , or surface covered with surfactants always induces alignment parallel to the nearest surface and attraction toward it . the only common boundary conditions to these three types of interfaces is impermeability , which strongly confines the fluid in the vertical direction . this confinement provides a qualitative argument for explaining alignment and attraction of pushers near any type of boundary , as sketched in fig . [ fig:4]b . when tilted , a pusher will experience a higher vertical fluid force on the part of the cell closer to the interface , inducing a reorientation in the parallel direction . when swimming parallel to the interface , fluid is attracted towards the cell on its side , leading to attraction toward the surface . from a mathematical point of view , impermeability on the surface is enforced by a symmetric singularity as an image in the second fluid . at leading order it is therefore as if there were two symmetric micro - swimmers , which tend to align and attract each other when they are pushers.@xcite we turn now to a higher - order representation of a swimming bacterium in order to account for its circular motion near interfaces . given the results above concerning the leading - order effect of the interfaces , we will assume that the the microorganism is swimming parallel to it . the micro - swimmer is now modeled by a rotlet dipole along the swimming direction , @xmath120 , which can be written as a combination of stokes quadrupoles , @xmath181.\end{aligned}\ ] ] as a result , the flow generated by a rotlet dipole close to a boundary can be obtained from that of the stokeslet , one only needs to consider the parallel stokeslet along @xmath115 rather than @xmath120 . in this section , in order to perform dimensional analysis , we write the strength of the rotlet dipole as @xmath182 ( units of n.m@xmath183 ) . in the case of a clean fluid - fluid interface , we start by considering the problem using dimensional analysis . in a far - field approach , the bacterium geometry plays a role only through the non - dimensional aspect ratio @xmath37 appearing in faxn s law . the dimensional quantities involved in the rotation rate are then the rotlet dipole strength @xmath182 , the two viscosities @xmath6 and @xmath184 , and the distance @xmath30 to the interface . a straightforward dimensional analysis then yields a rotation rate normal to the interface given by @xmath185 where we recall that @xmath186 . the non - dimensional function @xmath187 needs to be determined analytically , but we can already see from eq . that the sign of @xmath188 , and thus the direction of rotation of the circular motion , will be determined by a comparison between the viscosity ratio and the swimmer geometry , and will be independent of the distance to the interface @xmath30 . moreover , within the context of our far - field approach , we observe that this rotation rate decays as @xmath189 . the circular motion is therefore likely to occur very close to the interface . starting from the image system of a stokeslet in the presence of an interface , we derive the image system for a rotlet dipole along @xmath120 , and get @xmath190,\end{aligned}\ ] ] where @xmath191 is a stresslet dipole . recall that the limit of a free surface ( resp . no - slip wall ) is recovered in the limit of vanishing ( resp . infinite ) viscosity ratio.@xcite the structure of the image system in eq . reveals two roles played by the interface : ( i ) a kinematic role , through the impermeability condition , that forces the existence of an opposite rotlet dipole independent of the viscosity ratio , and ( ii ) a viscous component , related to the balance of tangential velocity and stress . interestingly , in the limit of a free surface ( @xmath7 ) , the viscous contribution vanishes , and the only remaining image singularity is the opposite rotlet dipole.@xcite the rotation rate induced by a clean interface is then computed from eq . using the flow from eq . , leading to the result @xmath192 in eq . we can identify the two different contributions which were apparent in the image system : ( i ) a kinematic contribution independent of the viscosity ratio , and ( ii ) a viscous contribution due to the rotlet and stresslet dipoles ( a potential singularity has no vorticity ) , independent of the micro - swimmer geometry . re - writing eq . , we get another form for the non - dimensional function @xmath193 in eq . as @xmath194 one can see from this expression that the sign inversion for the rotation rate , and thus the transition from ccw to cw circles , occurs for @xmath195 ( see fig . [ fig:5 ] ) . as expected from the dimensional analysis , we found a condition involving the swimmer geometry and the viscosity ratio for determining the rotation direction , regardless of the distance to the interface . for spherical swimmers ( @xmath196 ) , the transition is predicted to occur exactly at a viscosity ratio of 1 , but as the body becomes elongated this threshold is significantly modified . we would therefore expect a clear differentiation in rotation depending on the swimmer shape , for an identical viscosity ratio . , as a function of the viscosity ratio @xmath80 divided by the aspect ratio square @xmath197 ( for a fixed value of @xmath198 ) . circular motion is predicted to be cw when @xmath199 and ccw otherwise . ] in the presence of a clean interface between two given fluids , only the bacterium shape determines the rotation direction , and not the distance to the interface . when both fluid have comparable viscosities , the dominant behavior should be that of the free surface limit , a counter - clockwise rotation , as soon as the bacterium is elongated . this is not consistent with experiments , where both cw and ccw rotations where observed at a free surface or a solid wall.@xcite this indicates a more complex role played by the interface , and motivates the next two sections . in the presence of a partial slip boundary , an additional length is introduced in the problem . a dimensional analysis similar to the one carried out above yields @xmath200 and thus the sign of the rotation rate should depend on the distance to the wall , as opposed to the case of a clean fluid interface . in order to analyize the effect of a partial slip boundary , we derive the solution using blake s method directly for a rotlet dipole . choosing @xmath201 , we have in fourier space @xmath202 where @xmath203 . following an analysis similar to that detailed in appendix [ sec : appa ] , we then obtain the rotation rate induced on the micro - swimmer @xmath204 \nonumber\\ & -\frac{h}{{\ell}}\left[(\gamma^2 - 1)g_5\!\!\left(\frac{h}{{\ell}}\right)+(3\gamma^2 + 1)h_5\!\!\left(\frac{h}{{\ell}}\right)\right]\bigg\},\end{aligned}\ ] ] with the functions @xmath156 , @xmath157 , and @xmath158 defined in eqs . - . from eq . , the known limits of a free surface ( @xmath205 ) and solid wall ( @xmath206 ) are easily recovered . , induced by a partial slip wall on a parallel rotlet dipole as a function of the normalized slip length , @xmath163 , for three values of @xmath37 ( 1 , 3 , and 10 ) ; ( b ) critical value of @xmath163 at which the rotation sign changes as a function of @xmath37 . the regions where the circular trajectories are cw and ccw are indicated on the figure . ] we plot in fig . [ fig:6](a ) the perpendicular rotation rate , @xmath188 , as a function of the normalized slip length , @xmath163 , for different values of @xmath37 . for small slip lengths compared to the distance of the cell to the wall , the effect of slip is negligible , and the standard result of a no - slip boundary is recovered ( cw rotation ) . however , when the swimmer gets close enough to the wall , or when the slip length becomes large enough , the rotation rate changes sign and takes that due to a free surface ( ccw rotation ) . since we argued above that cells are always attracted to interfaces , we thus expect ccw rotations to be observable in this case , consistently with the recent experimental results in the presence of slip - inducing polymers.@xcite in fig . [ fig:6](b ) we further plot the dependence of the critical dimensionless slip length at which the rotation changes sign on the aspect ratio of the cell . we find numerically that @xmath163 scales approximatively as @xmath207 at large values of the aspect ratio . in this last section we consider the case of an interface covered with incompressible surfactants . using dimensional analysis , we obtain that the rotation perpendicular to the surface is @xmath208 and therefore the distance to the wall , @xmath30 , also plays a role as it is included in @xmath28 , the non - dimensional surface viscosity , the flow generated by a rotlet dipole close to a surfactant - covered interface can be derived from the stokeslet solution , or computed directly using blake s method . we have @xmath209 where @xmath210 is the additional flow field generated by a stokeslet along @xmath115 ( i.e. eq . with a rotation of @xmath211 along @xmath4 ) . the rotation rate along the vertical axis can then be derived directly , leading to the analytical result @xmath212\bigg\},\end{aligned}\ ] ] with the functions @xmath156 defined in eqs . - . , induced by an interface covered with incompressible surfactants on a parallel rotlet dipole : ( a ) contour values of @xmath188 for a spherical swimmer ( @xmath213 ) and ( b ) for an ellipsoidal swimmer ( @xmath214 ) . the dashed line ( red online ) indicates the location where the rotation rate changes sign . regions of cw and ccw rotations are schematically shown . ] the dependence of this rotation rate with the two non - dimensional viscosities , @xmath215 and @xmath216 , is shown in fig . [ fig:7 ] . for low cell aspect ratio , the sign of the circular rotation is similar to the no - slip case for a wide range of parameters ( cw rotation ) , and thus opposite to the prediction in the case of clean interface . the sign of rotation is then changing at low @xmath28 ( low surfactant concentration ) and low viscosity ratio @xmath217 . furthermore , as the aspect ratio of the cell @xmath37 increases , the region displaying ccw rotation becomes larger , allowing for both rotations to be potentially observed experimentally in cell populations of different sizes or on surfaces with fluctuations in surfactant concentration . comparing with other interfacial models , we get that when @xmath28 tends to infinity , the solid wall limit is recovered . furthermore , and despite the fact that the surfactant model is not supposed to recover exactly the clean - interface limit , we see that the clean - interface threshold for rotation inversion ( @xmath218 ) , corresponds to the order of magnitude of the vertical asymptote of the dashed line in fig . [ fig:7 ] . , at which the rotation changes sign , as a function of @xmath217 for three values of @xmath37 ( a ) , and as a function of @xmath37 for four values of @xmath217 ( b ) . regions of cw and ccw rotations are indicated on the figure . ] we then plot in fig . [ fig:8 ] the critical value of @xmath28 at which there is a change in sign of the rotation rate , as a function of the viscosity ratio @xmath217 for different aspect ratios ( fig . [ fig:8]a ) , and as a function of @xmath37 for different @xmath217 ( fig . [ fig:8]b ) . for large cell aspect ratios , we find that @xmath28 scales as @xmath197 , similarly to the scaling seen for @xmath217 . this can be interpreted by noting that this problem is the same as that in the presence of a clean interface , but for the presence of a third fluid , and thus a third viscosity that needs to be compared with the viscosity where the swimming microorganism is located . hence , the criterion for a change of sign in the rotation direction is similar for both non - dimensional viscosities , @xmath219 and @xmath28 . there is in general a wide range of surface viscosities , depending on multiple parameters , such as the type of surfactants or temperature.@xcite for instance , for a bacterium swimming at a typical distance of 1 @xmath2 m from the interface , @xmath28 can be of order @xmath220 when the interface is a monolayer at a water - air interface.@xcite as a result , cw rotation ( as in the no - slip case ) is more likely to be observed . in contrast , in the case of an amphiphilic bilayer,@xcite the values can be much smaller , @xmath221 . as a result , there is likely a wide range of parameters where both cw and ccw rotation could be observed in a population of cells . recent experimental results on contaminated free surfaces showed that both cw and ccw circular motion could be seen.@xcite by considering the presence of surfactants on the surface , our analysis shows that the high viscosity fluid film at the interface could indeed alter the natural shear - free rotation direction and lead to cw motion . in this paper we have used a far - field hydrodynamic approach to model the surface swimming of bacteria employing helical flagella . the motivation for this work was the discrepancy between theoretical predictions and experimental observations . specifically , theory predicts that near a rigid wall the cells should always display cw motion , whereas recent experiments where polymers were used to induce slip at the wall showed that rotation in the opposite direction was possible . similarly , cells should rotate in a ccw motion at a free surface whereas if surfactants are present experiments show that cw motion is also observed . to develop a model we have represented the helical swimmer as a superposition of hydrodynamic singularities and investigated its hydrodynamic interactions with three types of surfaces : a clean fluid - fluid interface , a rigid wall with a finite slip length , and an interface covered by incompressible surfactants . the leading - order singularity in the flow field of the cell is a stokes dipole ( stresslet ) , characterized by a @xmath128 spatial decay . the interactions between that singularity and all three types of surfaces systematically lead to a reorientation of the swimming cells parallel to , and an attraction by , the surface . circular motion of the cells are due to wall effects on a higher - order singularity , namely a rotlet dipole , which decays spatially as @xmath130 . in that case , the specific boundary conditions at the interface , together with the shape of the cell , play a crucial role in determining the direction of rotation of the cell , and transitions between cw and ccw are predicted to take place in similar experimental setups . our results indicate thus that the recent experimental finding on transition in rotation direction can be understood as the consequence of complex boundary conditions on the nature of hydrodynamic interactions between the swimming cells and the surfaces . the main assumption made in our paper is that we have only considered the leading - order hydrodynamics effects for all influences of the interfaces ( attraction and rotation ) . this is , admittedly , a severe assumption which is expected to break down as soon as the cell is within about one body length from the interface . in order to obtain more quantitative predictions , one would then need to either include the effect of higher - order singularities , or resort to a fully computational approach . the advantage of our approach however is that it allows us to identify the fundamental physical process at play in setting the direction of rotation , and that it is expected to remain valid generically for all cells exploiting helical swimming . our findings could potentially be exploited in a numbers of ways , for example surface swimming could be used as a proxy for determining the rheological properties of the nearby interface or to selectively stir or sort individual cells from bacterial populations . we hope that our study will motivate further work along these directions . the authors thank the department of mechanical and aerospace engineering at the university of california , san diego where this work was initiated . we reference here some useful fourier transforms @xmath222 } = \frac{1}{k}e^{-kz},\quad { \mathcal{f}\left[\frac{1}{r^3}\right ] } = \frac{1}{z}e^{-kz},\quad { \mathcal{f}\left[\frac{x}{r^3}\right ] } = \frac{ik_1}{k}e^{-kz}\end{aligned}\ ] ] where @xmath223 . we present here the details of the derivation for the rotation rate , @xmath132 , induced by a partial slip boundary on a tilted stokes dipole . the method is general , and the same procedure is applied throughout the paper for deriving the rotation rate induced by a complex interface . in fourier space , the rotation rate induced by a flow @xmath224 on a prolate spheroid of aspect ratio @xmath37 , oriented along @xmath36 , reads @xmath225 & = \frac{1}{2 } \begin{pmatrix } -ik_2{\tilde{u}}_z-{\frac{\partial { \tilde{u}}_2}{\partial z}}\\ { \frac{\partial { \tilde{u}}_1}{\partial z}}+ik_1{\tilde{u}}_z\\ ik_2{\tilde{u}}_1-ik_1{\tilde{u}}_2 \end{pmatrix } \nonumber\\ & \quad+ \frac{1}{2}{\frac{\gamma^2 - 1}{\gamma^2 + 1}}\begin{pmatrix } { \cos { \theta}}{\sin { \theta}}[ik_1{\tilde{u}}_2+ik_2{\tilde{u}}_1]-{\sin^2 \!{\theta}}\left[{\frac{\partial { \tilde{u}}_2}{\partial z}}-ik_2{\tilde{u}}_z\right]\\ [ { \sin^2 \!{\theta}}-{\cos^2 \!{\theta}}][{\frac{\partial { \tilde{u}}_1}{\partial z}}-ik_1{\tilde{u}}_z]-2{\cos { \theta}}{\sin { \theta}}\left[ik_1{\tilde{u}}_1+{\frac{\partial { \tilde{u}}_z}{\partial z}}\right]\\ { \cos { \theta}}{\sin { \theta}}[{\frac{\partial { \tilde{u}}_2}{\partial z}}-ik_2{\tilde{u}}_z]-{\cos^2 \!{\theta}}\left[ik_1{\tilde{u}}_2+ik_2{\tilde{u}}_1\right ] \end{pmatrix}. \label{eq : om_gen}\end{aligned}\ ] ] the tilted stokeslet dipole has three contribution , a stresslet , @xmath226 , and two stokeslet dipoles , @xmath145 and @xmath146 . we compute first the stresslet contribution . for a stresslet @xmath226 , we have @xmath227 , where @xmath228 the coefficients of the no - slip solution read @xmath229 we note @xmath230 $ ] and @xmath231 $ ] . from the expression of the rotation rate in eq . , we have @xmath232 . from eq . , we note that @xmath233 , and thus @xmath234 . this decomposition holds in real space,@xcite leading to @xmath235 the resulting differential equation reads @xmath236 we decompose then @xmath237 in two terms , @xmath238 , so that @xmath239 this last equation can be rewritten as @xmath240 the first term , @xmath241 , can be computed directly knowing the no - slip solution . however , the second term is not straightforward , and needs to be determined in fourier space , as well as @xmath242 . we have then @xmath243 e^{-kz } , \\ { \tilde{\omega}}^2 & = -\frac{1}{8\pi{\eta}_1}\frac{{\ell}k_2 ^ 2 e^{-k(z+h)}}{({1+{\ell}k})({1 + 2{\ell}k})}\left[1+{\frac{\gamma^2 - 1}{\gamma^2 + 1}}\left({\sin^2 \!{\theta}}-{\cos^2 \!{\theta}}+ 2{\cos { \theta}}{\sin { \theta}}\frac{ik_1}{k}\right)\right ] , \label{eq : bug}\end{aligned}\ ] ] these expressions can be inverted , as all contributions are known fourier transforms ( see appendix [ sec : appb ] ) . we have therefore @xmath244 , with @xmath245,\\ & \left(1-{\ell}{\frac{\partial } { \partial z}}\right)\left(1 - 2{\ell}{\frac{\partial } { \partial z}}\right)\omega^2 = \frac{{\ell}}{8\pi{\eta}_1 } \left[1+{\frac{\gamma^2 - 1}{\gamma^2 + 1}}({\sin^2 \!{\theta}}-{\cos^2 \!{\theta}})\right]{\frac{\partial } { \partial z}}\left(\frac{1}{{r}^3}-\frac{3y^2}{{r}^5}\right)\nonumber\\ & \qquad \qquad \qquad \qquad \qquad \qquad\quad + \frac{{\ell}}{8\pi{\eta}_1 } { \frac{\gamma^2 - 1}{\gamma^2 + 1}}\sin(2{\theta}){\frac{\partial ^2}{\partial y^2}}\left(\frac{x}{{r}^3}\right),\end{aligned}\ ] ] where @xmath246 , with @xmath124 . since we are looking for the solution at @xmath122 , we need to integrate the following equations @xmath247 , \\ & \left(1 - 2{\ell}{\frac{\mathrm{d } } { \mathrm{d } { z}}}\right)^2\omega_{ss}^{11 } = -\frac{3{\ell}({\sin^2 \!{\theta}}-{\cos^2 \!{\theta}})}{2\pi{\eta}_1 } { \frac{\gamma^2 - 1}{\gamma^2 + 1}}\frac{1}{{z}^4}\left(1-\frac{4h}{{z}}\right),\\ & \left(1-{\ell}{\frac{\mathrm{d } } { \mathrm{d } { z}}}\right)\left(1 - 2{\ell}{\frac{\mathrm{d } } { \mathrm{d } { z}}}\right)\omega_{ss}^2 = -\frac{3{\ell}}{8\pi{\eta}_1 } \left[1+{\frac{\gamma^2 - 1}{\gamma^2 + 1}}({\sin^2 \!{\theta}}-{\cos^2 \!{\theta}})\right ] \frac{1}{{z}^4}\cdot\end{aligned}\ ] ] the final rotation rate due to @xmath63 reads @xmath248 . for the parallel stokes dipole , @xmath145 , we have @xmath249.\end{aligned}\ ] ] the corresponding coefficients in fourier space are given by @xmath250 following the same procedure , we find that the rotation rate , @xmath132 , induced by a parallel stokes dipole @xmath145 is given by @xmath251 where @xmath252 for the perpendicular stokes dipole , @xmath146 , we have @xmath253.\end{aligned}\ ] ] the rotation rate , @xmath132 , induced by a perpendicular stokes dipole @xmath146 is given by @xmath254 , with @xmath255 the differential equations in @xmath256 giving the rotation rate and velocities induced by the nearby boundary are of the three following types @xmath257 where @xmath59 is a positive integer . we keep here the coefficients corresponding to the partial slip case , however similar equations are obtained in the case of a surfactant - covered interface . knowing that the solution must vanish at infinity , the solutions for these equations read @xmath258 where @xmath159 is the exponential integral function of order @xmath59 defined in eq . . 45ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop _ _ ( , , ) link:\doibase 10.1119/1.10903 [ * * , ( ) ] @noop * * ( ) @noop _ _ ( , ) @noop * * , ( ) link:\doibase 10.1146/annurev.mi.49.100195.003431 [ * * , ( ) ] link:\doibase 10.1007/bf02461846 [ * * , ( ) ] link:\doibase 10.1146/annurev.micro.57.030502.091014 [ * * , ( ) ] link:\doibase 10.1073/pnas.1019079108 [ * * , ( ) ] link:\doibase 10.1098/rspa.2009.0520 [ * * , ( ) ] link:\doibase 10.1073/pnas.1202934109 [ * * , ( ) ] link:\doibase 10.1038/1981221a0 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.101.038102 [ * * , ( ) ] link:\doibase 10.1017/s0022112008004953 [ * * , ( ) ] link:\doibase 10.1103/physreve.82.056309 [ * * , ( ) ] link:\doibase 10.1103/physreve.84.041932 [ * * , ( ) ] link:\doibase 10.1016/s0006 - 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flagellated bacteria exploiting helical propulsion are known to swim along circular trajectories near surfaces .
fluid dynamics predicts this circular motion to be clockwise ( cw ) above a rigid surface ( when viewed from inside the fluid ) and counter - clockwise ( ccw ) below a free surface .
recent experimental investigations showed that complex physicochemical processes at the nearby surface could lead to a change in the direction of rotation , both at solid surfaces absorbing slip - inducing polymers and interfaces covered with surfactants .
motivated by these results , we use a far - field hydrodynamic model to predict the kinematics of swimming near three types of interfaces : clean fluid - fluid interface , slipping rigid wall , and a fluid interface covered by incompressible surfactants .
representing the helical swimmer by a superposition of hydrodynamic singularities , we first show that in all cases the surfaces reorient the swimmer parallel to the surface and attract it , both of which are a consequence of the stokes dipole component of the swimmer flow field .
we then show that circular motion is induced by a higher - order singularity , namely a rotlet dipole , and that its rotation direction ( cw vs. ccw ) is strongly affected by the boundary conditions at the interface and the bacteria shape .
our results suggest thus that the hydrodynamics of complex interfaces provide a mechanism to selectively stir bacteria .
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consider a generic factor model @xcite with a binary configurational space , @xmath12 , @xmath13 , which is factorized so that the probability @xmath14 to find the system in the state @xmath15 and the partition function @xmath16 are @xmath17 where @xmath9 labels non - negative and finite factor - functions @xmath18 with @xmath19 and @xmath20 represents a subset of @xmath21 variables . relations between factor functions ( checks ) and elementary discrete variables ( bits ) , expressed as @xmath3 and @xmath4 , can be conveniently represented in terms of the system - specific factor ( tanner ) graph . if @xmath3 we say that the bit and the check are neighbors . any spin ( a - posteriori log - likelihood ) correlation function can be calculated using the partition function , @xmath16 , defined by eq . ( [ p1 ] ) . general expression for the factor functions of an ldpc code is @xmath22 let us now reproduce the derivation of the belief propagation equation based on the bethe free energy variational principle , following closely the description of @xcite . ( see also the appendix of @xcite . ) in this approach trial probability distributions , called beliefs , are introduced both for bits and checks @xmath23 and @xmath24 , respectively , where @xmath25 , @xmath26 . a belief is defined for given configuration of the binary variables over the code . thus , a belief at a bit actually consists of two probabilities , @xmath27 and @xmath28 , and we use a natural notation @xmath29 . there are @xmath30 beliefs defined at a check , @xmath31 being the number of bits connected to the check , and we introduce vector notation @xmath32 where @xmath33 and @xmath12 . beliefs satisfy the following inequality constraints @xmath34 the normalization constraints @xmath35 as well as the consistency ( between bits and checks ) constraints @xmath36 where @xmath37 stands for the set of @xmath38 with @xmath39 , @xmath40 . the bethe free energy is defined as a difference of the bethe self - energy and the bethe entropy , @xmath41 where @xmath42 , @xmath33 and @xmath12 . the entropy term for a bit enters eq . ( [ bethe ] ) with the coefficient @xmath43 to account for the right counting of the number of configurations for a bit : all entries for a bit ( e.g. through the check term ) should give @xmath44 in total . optimal configurations of beliefs are the ones that minimize the bethe free energy ( [ bethe ] ) subject to the constraints ( [ ineq],[norm],[cons ] ) . introducing these constraints into the effective lagrangian through lagrange multiplier terms @xmath45 and looking for the extremum with respect to all possible beliefs leads to @xmath46 , \nonumber\\ & & \!\!\!\!\!\ ! \frac{\delta l}{\delta b_i(\sigma_i ) } = 0 \label{lbi } \\ & & \!\!\rightarrow\quad b_i(\sigma_i)=\exp\left[\frac{1}{q_i-1}\left(\gamma_i+ \sum\limits_{\alpha\ni i}\lambda_{i\alpha}(\sigma_i)\right)-1\right ] . \nonumber\end{aligned}\ ] ] substituting @xmath47 into eq.([lba],[lbi ] ) we arrive at @xmath48 where @xmath49 is used to indicate that we should use the normalization conditions ( [ norm ] ) to guarantee that the beliefs sum up to one . applying the consistency constraint ( [ cons ] ) to eqs . ( [ ba ] ) , making summation over all spins but the given @xmath21 , and also using eq . ( [ bi ] ) we derive the following bp equations @xmath50 the right hand side of eq . ( [ ba0 ] ) rewritten for the ldpc case ( [ factor_ldpc ] ) becomes @xmath51 thus constructing @xmath52 for the ldpc case in two different ways ( correspondent to left and right relations in eq . ( [ ba0 ] ) ) , equating the results and introducing the @xmath53 field @xmath54 one arrives at the following bp equations for the @xmath53 fields : @xmath55 iterative solution of this equation corresponding to eq . ( [ iter ] ) with @xmath11 is just a standard iterative bp ( which can also be called sum - product ) used for the decoding of an ldpc code . a simplified min - sum version of eq . ( [ iter ] ) is @xmath56 \min_{j\neq i}^{j\in\beta } \big| \eta^{(n)}_{j\beta } \big|+ \frac{1}{\delta}\sum\limits_{\beta\ni i}\eta_{i\beta}^{(n ) } , \nonumber\end{aligned}\ ] ] to illustrate the standard bp iterative decoding , given by eqs . ( [ iter],[min - sum ] ) with @xmath57 , we consider the example of the @xmath58 $ ] code of tanner @xcite performing over awgn channel channel characterized by the transition probability for a bit , @xmath59 , where @xmath60 and @xmath61 are the input and output values at a bit and @xmath62 is the snr . launching a fixed codeword into the channel , emulating the channel by means of a standard monte - carlo simulations and then decoding the channel output constitutes our experimental " procedure . we analyze the probability distribution function of the iteration number @xmath0 at which the decoding terminates . the termination probability curve for the standard min - sum , described by eq . ( [ min - sum ] ) with @xmath11 , is shown in fig . [ tc123 ] for @xmath63 . . notice that the probability of termination ( successful decoding ) without any iterations is always finite . few points on the right part of the plot correspond to the case when the decoding was not terminated even at the maximum number of iterations , @xmath64 ( decoding fails to converge to a codeword ) . [ tc123],width=288 ] the result of decoding is also verified at each iteration step for compliance with a codeword : iteration is terminated if a codeword is recovered . this termination strategy can still give an error , although the probability to confuse actual and a distant codewords is much less than the probability not to recover a codeword for many iterations . if one neglects the very low probability of the codewords confusion , then the probability of still having a failure after @xmath0 iterations is equal to the integral / sum over the termination curve from @xmath65 and up . note also that the probability that even infinite number of iterations will not result in a codeword can actually be finite . discussing fig . [ tc123 ] one observes two distinct features of the termination probability curve . first , in all cases the curve reaches its maximum at some relatively small number of iterations . second , each curve crosses over to an algebraic - like decay which gets steeper with the snr increase . the emergence of an algebraically extended tail ( that is a tail which does not decay fast ) is not encouraging , as it suggests that increasing the number of iteration will not bring much of an improvement in the iterative procedure . it also motivates us to look for possibilities of accelerating convergence of the bp algorithm to a minimum of the bethe free energy . note also the wiggling of termination curves for @xmath66 near the crossover point ( see fig . [ tc123 ] ) . it is possibly related to the cycling of the bp dynamics ( and thus the inability of bp to converge ) . the idea is to introduce relaxational dynamics ( damping ) in an auxiliary time , @xmath67 , thus enforcing convergence to a minimum of the bethe free energy . one chooses @xmath29 as the main variational field and considers relaxing variational equations eqs . ( [ lbi ] ) according to @xmath68 while keeping the set of remaining variational equations eqs . ( [ norm],[lba],[cons ] ) intact . here positive parameters @xmath69 have the physical meaning of correlation / relaxation times . performing calculations , that are completely equivalent to the ones described in section [ sec : bethe ] , we arrive at the following modified bp equations @xmath70 we are interested to approach ( find ) a solution of the original bp eq . ( [ bp1 ] ) . one assumes @xmath71 , thus ignoring the second term under @xmath72 in eq . ( [ qeq ] ) . the resulting continuous equation is @xmath73 eq . ( [ iter ] ) represents a simple discretized version of the eq . ( [ bp3 ] ) where the correlation coefficients @xmath69 are chosen to make the coefficient in front of the second term on the left hand side of eq . ( [ bp3 ] ) independent of the bit index , @xmath8 . then the resulting time dependent coefficient can be rescaled to one by an appropriate choice of the temporal unit ; @xmath74 is the uniform discrete time , @xmath7 is positive integer , @xmath75 ; the left hand side ( right hand side ) of eq.([bp3 ] ) is taken at @xmath76 ( @xmath74 ) and the temporal derivative is discretized in a standard retarded way , @xmath77 . this choice of relaxation coefficients and discretization , resulted in eq . ( [ iter ] ) , was taken out of consideration in the final formula for simplicity , realizability at all positive @xmath10 and also its equivalence to the standard iterative bp at @xmath78 . we test the min - sum version ( [ min - sum ] ) of the modified iterative bp with the monte carlo simulations of the @xmath58 $ ] code at few values of snrs . the resulting termination probability curves are shown in fig . [ tc ] for @xmath79 . the simulations show a shift of the probability curve maximum to the right ( towards larger number of iterations ) with the damping parameter decrease however once the maximum is achieved , the decay of the curve at a finite @xmath10 is faster with the number of iterations than in the standard bp case . the decay rate actually increases as @xmath10 decreases . ( a ) , @xmath80 ( b ) , and @xmath81 ( c ) . [ tc],width=288 ] ( a ) , @xmath80 ( b ) , and @xmath81 ( c ) . [ tc],width=288 ] ( a ) , @xmath80 ( b ) , and @xmath81 ( c ) . [ tc],width=288 ] we conclude that at the largest @xmath0 the performance of a modified iterative bp is strictly better . however to optimize the modified iterative bp , thus aiming at better performance than given by the standard iterative bp , one needs to account for the trade - off between decreasing @xmath10 leading to a faster decay of the termination probability curve at the largest @xmath0 , but on the other side it comes with the price in the actual number of iteration necessary to achieve the asymptotic decay regime . the last point is illustrated by fig . [ er ] , where the decoding error probability depends non - monotonically on @xmath10 . one can also see that the modification of bp could improve the decoding performance ; e.g. , at @xmath82 and maximally allowed @xmath83 ( after which the decoding unconditionally stops ) the decoding error probability is reduced by factor of about 40 by choosing @xmath84 ( see the bottom curve at fig . [ er](b ) ) . for @xmath80 ( a ) and @xmath81 ( b ) . different curves correspond to different maximally allowed @xmath0 : starting from @xmath85 ( top curve ) and increasing @xmath0 by factor of @xmath86 with each next lower curve . the points on the right correspond to the standard bp ( @xmath87 ) . [ er],width=288 ] for @xmath80 ( a ) and @xmath81 ( b ) . different curves correspond to different maximally allowed @xmath0 : starting from @xmath85 ( top curve ) and increasing @xmath0 by factor of @xmath86 with each next lower curve . the points on the right correspond to the standard bp ( @xmath87 ) . [ er],width=288 ] we presented a simple extension of the iterative bp which allows ( with proper optimization in the @xmath88 plane ) to guarantee not only an asymptotic convergence of bp to a local minimum of the bethe free energy but also a serious gain in decoding performance at finite @xmath0 . in addition to their own utility , these results should also be useful for systematic improvement of the bp approximation . indeed , as it was recently shown in @xcite solution of the bp equation can be used to express the full partition function ( or a - posteriori log - likehoods calculated within map ) in terms of the so - called loop series , where each term is associated with a generalized loop on the factor graph . this loop calculus / series offers a remarkable opportunity for constructing a sequence of efficient approximate and systematically improvable algorithms . thus we anticipate that the improved iterative bp discussed in the present manuscript will become an important building block in this future approximate algorithm construction . we already mentioned in the introduction that our algorithm can be advantageous over other bp - based algorithms converging to a minimum of the bethe free energy mainly due to its simplicity and tunability . in particular , the concave - convex algorithms of @xcite , as well as related linear programming decoding algorithms @xcite , are formulated in terms of beliefs . on the contrary our modification of the iterative bp can be extensively simplified and stated in terms of the fewer number of @xmath89 fields each associated with an edge of the factor graph rather than with much bigger family of local code - words . thus in the case of a regular ldpc code with @xmath90 checks of the connectivity degree @xmath31 one finds that the number of variables taken at each step of the iterative procedure is @xmath91 and @xmath92 in our iterative scheme and in the concave - convex scheme respectively . having a tunable correlation parameter @xmath93 in the problem is also advantageous as it allows generalizations ( e.g. by turning to a individual bit dependent relaxation rate ) . this flexibility is particularly desirable in the degenerate case with multiple minima of the bethe free energy , as it allows a painless implementation of annealing as well as other more sophisticated relaxation techniques speeding up and/or improving convergence . m. chertkov , v. chernyak , _ loop calculus in statistical physics and information science _ , phys . e * 73 * , 065102(r ) ( 2006 ) ; cond - mat/0601487 . m. chertkov , v. chernyak , _ loop series for discrete statistical models on graphs _ , j. stat . ( 2006 ) p06009 , cond - mat/0603189 . j. feldman , m. wainwright , d.r . karger , _ using linear programming to decode linear codes _ , 2003 conference on information sciences and systems , the john hopkins university , march 12 - 14 , 2003 .
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the decoding of low - density parity - check codes by the belief propagation ( bp ) algorithm is revisited .
we check the iterative algorithm for its convergence to a codeword ( termination ) , we run monte carlo simulations to find the probability distribution function of the termination time , @xmath0 . tested on an example @xmath1 $ ] code , this termination curve shows a maximum and an extended algebraic tail at the highest values of @xmath0 . aiming to reduce the tail of the termination curve we consider a family of iterative algorithms modifying the standard bp by means of a simple relaxation .
the relaxation parameter controls the convergence of the modified bp algorithm to a minimum of the bethe free energy .
the improvement is experimentally demonstrated for additive - white - gaussian - noise channel in some range of the signal - to - noise ratios .
we also discuss the trade - off between the relaxation parameter of the improved iterative scheme and the number of iterations
. low - density parity - check ( ldpc ) codes @xcite are the best linear block error - correction codes known today @xcite .
in addition to being good codes , i.e. capable of decoding without errors in the thermodynamic limit of an infinitely long block length , these codes can also be decoded efficiently .
the main idea of belief propagation ( bp ) decoding is in approximating the actual graphical model , formulated for solving statistical inference maximum likelihood ( ml ) or maximum - a - posteriori ( map ) problems , by a tree - like structure without loops .
being efficient but suboptimal the bp algorithm fails on certain configurations of the channel noise when close to optimal ( but inefficient ) map decoding would be successful .
bp decoding allows a certain duality in interpretation .
first of all , and following the so - called bethe - free energy variational approach @xcite , bp can be understood as a set of equations for beliefs ( bp - equations ) solving a constrained minimization problem . on the other hand ,
a more traditional approach is to interpret bp in terms of an iterative procedure so - called bp iterative algorithm @xcite .
being identical on a tree ( as then bp equations are solved explicitly by iterations from leaves to the tree center ) the two approaches are however distinct for a graphical problem with loops . in case of their convergence ,
bp algorithms find a minimum of the bethe free energy @xcite , however in a general case convergence of the standard iterative bp is not guaranteed .
it is also understood that bp fails to converge primarily due to circling of messages in the process of iterations over the loopy graph . to enforce convergence of the iterative algorithm to a minimum of the bethe free energy some number of modifications of the standard iterative bp
were discussed in recent years .
the tree - based re - parametrization framework of @xcite suggests to limit communication on the loopy graph , cutting some edges in a dynamical fashion so that the undesirable effects of circles are suppressed .
another , so - called concave - convex procedure , introduced in @xcite and generalized in @xcite , suggests to decompose the bethe free energy into concave and convex parts thus splitting the iterations into two sequential sub - steps . noticing that convergence of the standard bp fails mainly due to overshooting of iterations
, we develop in this paper a tunable relaxation ( damping ) that cures the problem .
compared with the aforementioned alternative methods , this approach can be practically more advantageous due to its simplicity and tunability . in its simplest
form our modification of the bp iterative procedure is given by @xmath2 where latin and greek indexes stand for bits and checks and the bit / check relations , e.g. @xmath3 , @xmath4 express the ldpc code considered ; @xmath5 is the channel noise - dependent value of log - likelihoods ; and @xmath6 is the message associated at the @xmath7-th iteration with the edge ( of the respective tanner graph ) connecting @xmath8-th bit and @xmath9-th check .
@xmath10 is a tunable parameter . by choosing a sufficiently small @xmath10 one can guarantee convergence of the iterative procedure to a minimum of the bethe free energy . on the other hand
@xmath11 corresponds exactly to the standard iterative bp . in the sequel
we derive and explain the modified iterative procedure ( [ iter ] ) in detail .
the manuscript is organized as follows .
we introduce the bethe free energy , the bp equation and the standard iterative bp in section [ sec : bethe ] .
performance of standard iterative bp , analyzed with a termination curve , is discussed in section [ sec : term ] .
section [ sec : relax ] describes continuous and sequentially discrete ( iterative ) versions of our relaxation method .
we discuss performance of the modified iterative scheme in section [ sec : perf ] , where bit - error - rate and the termination curve for an ldpc code performed over additive - white - gaussian - noise ( awgn ) channel are discussed for a range of interesting values of the signal - to - noise - ratios ( snr ) .
we also discuss here the trade - off between convergence and number of iterations aiming to find an optimal strategy for selection of the model s parameters .
the last section [ sec : con ] is reserved for conclusions and discussions .
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chemical master equations ( cmes ) are the accepted mathematical description of chemical systems in well - mixed conditions @xcite . these equations provide a mesoscopic description of chemical kinetics , interpolating between the microscopic regime of molecular dynamics and the macroscopic regime of rate equations ( res ) . it has been shown that cmes are exact descriptions for any well - stirred and thermally equilibrated gas - phase chemical system @xcite . more recently it has been rigorously confirmed that their validity extends to chemical reactions in well - stirred dilute solutions @xcite . however , well before these rigorous demonstrations of the microscopic physical basis of the cme , scientists have employed these equations to probe the nature of mesoscopic chemical kinetics and in particular to understand how this may differ from kinetics on macroscopic length scales ( see mcquarrie for a review @xcite of the literature up till 1967 ) . we briefly review the cme formalism . consider a general chemical system consisting of a number @xmath4 of distinct chemical species interacting via @xmath5 elementary chemical reactions of the type : @xmath6 here @xmath7 is an index running from 1 to @xmath5 , @xmath8 denotes chemical species @xmath9 , @xmath10 and @xmath11 are the stoichiometric coefficients and @xmath12 is the macroscopic rate of reaction . if this system is well - mixed then its mesoscopic state is fully determined by the vector of the absolute number of molecules of each species , @xmath13 , where @xmath14 is the number of molecules of the @xmath15species . the cme is then a time - evolution equation for the probability of the system being in a particular mesoscopic state @xcite : @xmath16 where @xmath2 is the volume of the compartment in which the reactions occur and @xmath17 is a step operator when it acts on some function of the absolute number of molecules , it gives back the same function but with @xmath14 replaced by @xmath18 . the chemical reaction details are encapsulated in the stoichiometric matrix @xmath19 and in the microscopic rate functions @xmath20 . the probability that the @xmath21 reaction occurs in the time interval @xmath22 is given by @xmath23 . for elementary reactions , the microscopic rate function takes one of four different forms , depending on the order of the @xmath21 reaction : ( i ) a zeroth - order reaction by which a species is input into a compartment gives @xmath24 ; ( ii ) a first - order unimolecular reaction involving the decay of some species @xmath25 gives @xmath26 ; ( iii ) a second - order bimolecular reaction between two molecules of the same species @xmath25 gives @xmath27 ; ( iii ) a second - order bimolecular reaction between two molecules of different species , @xmath25 and @xmath28 , gives @xmath29 . the re description of the same system is much simpler . denoting the macroscopic concentration of species @xmath9 by @xmath30 , the set of res describing the macroscopic kinetics of the reactive system represented by eq . ( 1 ) are given by : @xmath31 where @xmath32 is the vector of macroscopic concentrations and @xmath33 is the macroscopic rate function of the @xmath21 reaction which has the general mass - action form , @xmath34 . res provide a continuous deterministic `` many molecule '' description of kinetics . this strongly contrasts with the cme description which constitutes a discrete , stochastic , `` any number of molecule '' description that is faithful to the underlying microscopic basis of chemical reactions . unfortunately , one of the main advantages of cmes over their re cousins , their discrete description , is also the source of their computational intractability . differential - difference equations , such as the cme @xcite , do not lend themselves easily to analysis . in contrast , there is a vast body of literature in engineering , mathematics and physics dealing with the analysis and solution of differential and partial differential equations . thus at an early stage , considerable effort was invested in obtaining a partial differential approximation of the cme . in the 1940 s , kramers @xcite and moyal @xcite developed a taylor series expansion of the cme ; by assuming that all terms with derivatives greater than two are negligible , one obtains the chemical fokker - planck equation ( cfpe , @xcite ) , a second - order partial differential equation of the form : @xmath35 as gardiner mentions in his book @xcite , `` this procedure enjoyed wide popularity mainly because of the convenience and simplicity of the result '' and also because `` it is often simpler to use the fokker - planck equation than the master equation . '' a major and important difference between the cme and the cfpe is that @xmath14 is a positive integer for the cme while it is a real number for the cfpe . several authors have questioned the validity of the cfpe approximation . the approximation is obtained by a perfunctory truncation of the taylor expansion and hence it appears to be an uncontrolled and unjustified approximation of the cme . van kampen , in particular , was a leading and influential critic of the cfpe approximation . in the 1960 s and 70 s , he developed a systematic perturbative expansion of the cme in powers of the inverse square root of the system volume @xmath2 ( the system - size expansion ) and used it to show that to lowest order in the expansion , i.e. in the limit of large volumes the macroscopic limit , one obtains a fokker - planck equation which is of a different form than the cfpe @xcite . of particular concern is that van kampen s fokker - planck equation is linear whereas the cfpe is non - linear . note that by non - linear fokker - planck equation here we mean one such that its drift and diffusion coefficients are generally non - linear functions of the molecule numbers @xmath14 ; this convention is adopted since it is in mainstream use , for example see the book by van kampen @xcite . taking into account higher - order terms in the system - size expansion does not lead to the cfpe as well . however , interestingly , in the limit of large volumes , the cfpe does reduce to van kampen s linear fokker - planck equation @xcite . this led van kampen to conclude that any features arising from the non - linear character of the cfpe are spurious and not to be taken seriously @xcite . we note that the limit of large volumes in van kampen s system - size expansion is taken at fixed macroscopic concentrations and hence it corresponds to the limit of large molecule numbers @xcite . hence van kampen s conclusions can be equivalently stated as : the cfpe becomes a legitimate approximation of the cme in the limit of large molecular populations . a few studies at the time @xcite did suggest that the cfpe s validity extended beyond the linear regime . of particular importance is a result of horsthemke and brenig @xcite which motivated the present study . the authors considered a simple dimerization reaction @xmath36 whereby molecules of a monomer species @xmath37 are introduced in a compartment of volume @xmath2 and subsequently they bind to each other to form dimers @xmath38 . assuming stationary conditions , the cme and cfpe are solved exactly . it is shown that the average concentration of monomers and the variance of the fluctuations from the two formalisms agree exactly to order @xmath39 and are respectively equal to @xmath40 and @xmath41 , where @xmath42 is the macroscopic concentration obtained by solving the corresponding re in steady - state conditions . the same example can be found worked in van kampen s book @xcite wherein he shows that the linear noise approximation gives mean and variance equal to @xmath42 and @xmath41 . as we mentioned before , a linearization of the cfpe will lead to the linear - noise approximation and hence from this example we can conclude that the non - linearity of the cfpe is non - spurious since it leads to a more accurate concentration estimate than that which is obtained from the linear - noise approximation . however one could argue that this higher accuracy is only particular to the dimerization example and not a general feature of the cfpe . because of this or other reasons , the results of hortshemke and brenig do not appear to have received the attention they deserved at the time and van kampen s conclusions about the cfpe were accepted , by and large , by the statistical physics community . approximately 40 years later after the inception of the system - size expansion , gillespie revived the question of the validity of the cfpe by deriving it without invoking truncation of the kramers - moyal expansion of the cme @xcite . to be precise , he derived the chemical langevin equation ( cle ) : @xmath43 where @xmath44 are temporally uncorrelated , independent gaussian white noises . this stochastic differential equation is exactly equivalent to the cfpe in the sense that its solution generates exact sample paths of the cfpe , eq . ( 4 ) . essentially he showed that the cfpe approximation is valid provided two conditions are satisfied . a large number of molecules suffices to ensure that both conditions are satisfied however this is not a necessary condition . this suggests that there are regimes in which the particle numbers may not be very large and yet the cfpe may still provide a reasonably good approximation of the cme . however gillespie s derivation does not provide us with a means to estimate the accuracy of the cfpe for general chemical systems . questions regarding the validity and accuracy of the cfpe and cle are more important now than ever before . in the past decade , interest has virtually exploded in realistic stochastic simulations of biochemical reactions inside cells @xcite . the exact method of sampling the trajectories of the cme , the stochastic simulation algorithm @xcite , is computationally expensive and the cme is analytically intractable ; thus approximate methods such as the cfpe and the cle have come to the foreground as an alternative means to obtain numerical and theoretical insight into the functioning of intracellular biochemical networks @xcite . these networks are typically characterized by a large number of bimolecular reactions in which at least one of the species is present in very small molecule numbers @xcite , indeed the precise conditions in which the fidelity of the cfpe remains unclear . hence the question of the accuracy of the cfpe has nowadays become a practical one how much can we trust the conclusions derived from the cfpe or the corresponding cle ? in this article , we derive formulas to estimate the relative error in the cfpe predictions of the mean concentrations and of the variance of the fluctuations about the mean . the results are valid for all monostable chemical reaction networks . as a byproduct of our derivation , we will also clarify the connection between the cfpe and van kampen s system - size expansion , in particular showing that the non - linear character of the cfpe is not completely spurious and that generally cfpe estimates are more accurate than those obtained from the linear fokker - planck equation . the article is organized as follows . in section ii , we use the multivariate system - size expansion to derive expressions for the mean concentrations and for the variance of the fluctuations as predicted by the cme accurate to order @xmath45 . in section iii , we develop the system - size expansion of the cfpe and use it to derive expressions for the mean concentrations and for the variance of the fluctuations accurate to the same order as derived for the cme in section ii . in section iv , we use the results of the previous two sections to derive expressions for the relative error in the predictions of the cfpe . we also compare the predictions of the cfpe and the linear fokker - planck equation . these results are tested on two bimolecular reaction systems dimerization and an enzyme - catalyzed reaction in section v. we conclude by a discussion in section vi . we will now probe the mesoscopic description provided by the cme using the system - size expansion developed by van kampen @xcite . this method allows one to derive expressions for the mean concentrations and for the variance of the fluctuations about these concentrations , as predicted by the cme , accurate to the order of any desired power of the inverse square root of the volume . the only requirement for the expansion to hold is that the steady - state of the chemical system is asymptotically stable . for the applications that we are interested in , namely biochemical reactions in intracellular conditions , the number of molecules can be very small , in some cases just few tens of molecules of a given species per cell . we will derive equations accurate to @xmath45 this accuracy should be more than sufficient for the applications mentioned since terms of lower order , @xmath46 , already imply corrections to the concentrations of the order of a single molecule in the compartment . to our knowledge this is the first time that the system - size expansion has been carried to this order for a general system of @xmath4 interacting chemical species . van kampen has treated a one species example to the same order in his book @xcite while elf and ehrenberg @xcite have derived the multivariate expansion to @xmath47 . the starting point of the system - size expansion is to write the absolute number of molecules of species @xmath9 as : @xmath48 where @xmath30 is the macroscopic concentration of species @xmath9 as determined by the res . this has the effect of transforming all functions of @xmath14 in the cme into functions of @xmath49 . the expansion of the cme proceeds by writing eq . ( 2 ) in terms of the new variables . details of this transformation can be found in @xcite ; here we will simply state the relevant results and use them for our present derivation . the variable change causes the probability distribution of molecular populations , @xmath50 , to be transformed into the probability distribution of fluctuations , @xmath51 , where @xmath52 . the time derivative , the step operator and the microscopic rate function in the cme , read in the new variables : @xmath53 where @xmath54 note that in eq . ( 9 ) the microscopic rate function is expressed in terms of the macroscopic rate function . as we shall shortly see , this is convenient from a calculation point of view since the final expressions for the means and variances will be solely in terms of functions which appear in the res . note that the upper limit of the sum in eq . ( 9 ) is 2 because all reactions involve at most the interaction of two molecules and hence @xmath55 equals zero for @xmath56 . although our analysis is specifically for elementary reactions , one can easily extend the approach to include `` elementary complex '' reactions @xcite . however we shall not pursue this here . substituting eqs . ( 7 - 9 ) in eq . ( 2 ) we get the following new form of the cme : @xmath57 note that terms proportional to @xmath58 do not appear in the expansion of the cme . this is because when one substitutes eqs . ( 7 - 9 ) in eq . ( 2 ) , one equates terms of this order on both sides of the cme which simply gives us back the macroscopic res , eq . ( 3 ) . to proceed further we need the explicit dependence of the right hand side of eq . ( 14 ) on the new variables @xmath49 . this is obtained by substituting eqs . ( 10 - 13 ) in eq . ( 14 ) which leads to : @xmath59 note that in the above equation , we have used the einstein summation convention where all twice repeated indices are understood to be summed over 1 to @xmath4 . the partial derivative @xmath60 denotes @xmath61 . we have also used the following two convenient definitions : @xmath62 from eq . ( 3 ) it follows that @xmath63 and consequently @xmath64 represents the @xmath9-@xmath65 element of the jacobian matrix associated with the res of the system . note that eq . ( 15 ) to order @xmath66 is the linear fokker - planck equation which was mentioned in the introduction . the drift vector is linear in the @xmath67 variables while the diffusion tensor is independent of them . both depend on time via their own dependence on the macroscopic concentrations . this level of approximation is frequently called the linear - noise approximation , a nowadays popular means of estimating the size of the concentration fluctuations about the macroscopic concentrations @xcite . we are interested in the dynamics on mesoscopic length scales and hence we shall consider terms of higher order than @xmath66 in eq . ( 15 ) . we now proceed to construct equations for the moments of the @xmath67 variables . we start by expanding @xmath51 as a series in powers of the inverse square root of the volume : @xmath68 from which it follows that the moments possess an equivalent expansion : @xmath69_j \omega^{-j/2},\ ] ] where @xmath70_j = \int \epsilon_k \epsilon_m ... \epsilon_r \ \pi_j(\vec{\epsilon},t ) d\vec{\epsilon}.\ ] ] the angled brackets denote the statistical average . some subtle points associated with the perturbative expansion in the probability density and with the physical interpretation of @xmath71_j$ ] are discussed in appendix a. the time - evolution equations for the moments are obtained as follows . one starts by substituting eq . ( 18 ) in eq . ( 15 ) , multiplying the resulting equation on both sides by @xmath72 and integrating over @xmath73 . equating terms of order @xmath74 on both sides of the equation gives the time - evolution equation for @xmath71_j$ ] . finally one constructs the time - evolution equation for the moments using eq . ( 19 ) . as mentioned earlier , our aim is to determine the mean concentrations and the variance of the fluctuations about the means and hence we must relate the latter to the moments of the @xmath67 variables above . using eqs . ( 6 ) and ( 19 ) , one can easily verify that the mean concentration of species @xmath9 and the variance of the fluctuations about it , accurate to order @xmath1 are respectively given by : @xmath75_j \omega^{-j/2 } + o(\omega^{-5/2 } ) , \\ \sigma_i^2 & = \biggl \langle \biggl(\frac{n_i}{\omega } \biggr)^2 \biggr \rangle - \biggl \langle \frac{n_i}{\omega } \biggr \rangle^2 = \omega^{-1 } ( \langle \epsilon_i^2 \rangle - \langle \epsilon_i \rangle^2 ) \nonumber \\ & = \omega^{-1 } \biggr ( \sum_{j=0}^{2 } [ \epsilon_i^2 ] _ j \omega^{-j/2 } - \biggl(\sum_{j=0}^{1 } [ \epsilon_i ] _ j \omega^{-j/2 } \biggr)^2 - \omega^{-1 } [ \epsilon_i ] _ 0 [ \epsilon_i ] _ 2 \biggl ) + o(\omega^{-5/2}).\end{aligned}\ ] ] hence it is clear that to determine the mean and variance accurate to order @xmath1 , we shall need to determine the first and second moments of the @xmath67 variables accurate to orders @xmath0 and @xmath39 respectively . we proceed by implementing the calculation recipe outlined just after eq . ( 20 ) to derive equations for the corrections to the second moments accurate to order @xmath39 : @xmath76_0 & = j_r^{w } [ \epsilon_w \epsilon_k ] _ 0 + ( r \leftrightarrow k ) + d_{rk } , \\ \frac{\partial}{\partial t } [ \epsilon_r \epsilon_k ] _ 1 & = j_r^{w } [ \epsilon_w \epsilon_k ] _ 1 + \frac{1}{2 } j_r^{wp } [ \epsilon_w \epsilon_p \epsilon_k ] _ 0 - \frac{1}{2 } j_r^{w(2 ) } \phi_w [ \epsilon_k ] _ 0 \nonumber \\ & + ( r \leftrightarrow k ) + j_{kr}^w [ \epsilon_w ] _ 0 , \\ \frac{\partial}{\partial t } [ \epsilon_r \epsilon_k ] _ 2 & = j_r^{w } [ \epsilon_w \epsilon_k ] _ 2 + \frac{1}{2 } j_r^{wp } [ \epsilon_w \epsilon_p \epsilon_k ] _ 1 - \frac{1}{2 } j_r^{w(2 ) } \phi_w [ \epsilon_k ] _ 1 \nonumber \\ & - \frac{1}{2 } j_r^{w(2 ) } [ \epsilon_w \epsilon_k ] _ 0 + ( r \leftrightarrow k ) + j_{kr}^w [ \epsilon_w ] _ 1 + \frac{1}{2 } j_{rk}^{wm } [ \epsilon_w \epsilon_m ] _ 0 - \frac{1}{2 } j_{rk}^{w(2 ) } \phi_w.\end{aligned}\ ] ] details of the calculations leading to the above equations are illustrated by a step - by - step derivation of eq . ( 25 ) in appendix b. note that the short - hand notation @xmath77 stands for all the expressions of the same form as the ones preceding the notation but with @xmath78 and @xmath79 interchanged . for example in eq . ( 24 ) , @xmath77 stands for @xmath80_1 + \frac{1}{2 } j_k^{wp } [ \epsilon_w \epsilon_p \epsilon_r ] _ 0 - \frac{1}{2 } j_k^{w(2 ) } \phi_w [ \epsilon_r ] _ 0 $ ] . this notation will be used throughout the rest of the article since it enables the equations to be written in a compact way . the equation for @xmath81_0 $ ] , eq . ( 23 ) , is a lyapunov equation which can be solved either analytically ( see for example @xcite ) or else numerically , for example using the built in functions of matlab and mathematica . solution of the equation for @xmath81_1 $ ] , eq . ( 24 ) , requires the solutions of the equations for the first and third moments to order @xmath66 : @xmath82_0 & = j_r^{w } [ \epsilon_w ] _ 0 , \\ \frac{\partial}{\partial t } [ \epsilon_r \epsilon_k \epsilon_l ] _ 0 & = j_l^{w } [ \epsilon_w \epsilon_k \epsilon_r ] _ 0 + ( l \leftrightarrow k ) + ( k \leftrightarrow r ) \nonumber \\ & + d_{rl } [ \epsilon_k ] _ 0 + ( k \leftrightarrow l ) + ( r \leftrightarrow l).\end{aligned}\ ] ] note that in eq . ( 27 ) , @xmath83 stands for two expressions ; the first expression corresponds to the first term on the right hand side of eq . ( 27 ) with @xmath84 and @xmath79 interchanged and the second expression is the first expression just obtained with @xmath79 and @xmath78 interchanged . by a similar reasoning , it follows that @xmath85 in eq . ( 27 ) stands for @xmath86_0 + d_{lk } [ \epsilon_r ] _ 0 $ ] . note that in steady - state conditions , @xmath87_0 = [ \epsilon_r \epsilon_k \epsilon_l ] _ 0 = 0 $ ] and and consequently there is no correction to the second moments to @xmath46 , i.e. , @xmath81_1 = 0 $ ] . solution of the equation for @xmath81_2 $ ] , eq . ( 25 ) , requires the solutions of the corrections to the the first and third moments to order @xmath3 and the second and fourth moments to order @xmath88 : @xmath82_1 & = j_r^{w } [ \epsilon_w ] _ 1 + \frac{1}{2 } j_r^{wp } [ \epsilon_w \epsilon_p ] _ 0 - \frac{1}{2 } j_r^{w(2 ) } \phi_w , \\ \frac{\partial}{\partial t } [ \epsilon_r \epsilon_k \epsilon_l ] _ 1 & = j_l^{w } [ \epsilon_w \epsilon_k \epsilon_r ] _ 1 + \frac{1}{2 } j_l^{wp } [ \epsilon_w \epsilon_p \epsilon_r \epsilon_k ] _ 0 - \frac{1}{2 } j_l^{w(2 ) } \phi_w [ \epsilon_r \epsilon_k ] _ 0 \nonumber \\ & + ( l \leftrightarrow k ) + ( k \leftrightarrow r ) + d_{rl } [ \epsilon_k ] _ 1 + j_{rl}^w [ \epsilon_w \epsilon_k ] _ 0 + ( k \leftrightarrow l ) \nonumber \\ & + ( r \leftrightarrow l ) + d_{rkl } , \\ \frac{\partial}{\partial t } [ \epsilon_r \epsilon_k \epsilon_l \epsilon_m ] _ 0 & = j_r^{w } [ \epsilon_w \epsilon_k \epsilon_l \epsilon_m ] _ 0 + ( r \leftrightarrow m ) + ( m \leftrightarrow k ) + ( k \leftrightarrow l ) + d_{rm } [ \epsilon_k \epsilon_l ] _ 0 \nonumber \\ & + ( m \leftrightarrow l ) + ( l \leftrightarrow k ) + ( r \leftrightarrow m ) + ( m \leftrightarrow l ) + ( k \leftrightarrow m).\end{aligned}\ ] ] the procedure to obtain the second moments to order @xmath39 is now clear . one first solves eq . ( 23 ) to get @xmath81_0 $ ] ; then one solves eqs . ( 26 - 27 ) and substitutes in eq . ( 24 ) to get @xmath81_1 $ ] ; finally one solves eqs . ( 28 - 30 ) and substitutes these , together with the solution of eq . ( 23 ) , in eq . ( 25 ) to get @xmath81_2 $ ] . the first moment equations and the corresponding equations for the mean concentrations can be obtained in an analogous manner as for the second moments . the equations for @xmath87_0 $ ] and @xmath87_1 $ ] have been already derived , eqs . ( 26 ) and ( 28 ) , respectively . the equations for @xmath87_2 $ ] and @xmath87_3 $ ] are given by : @xmath82_2 & = j_r^{w } [ \epsilon_w ] _ 2 + \frac{1}{2 } j_r^{wp } [ \epsilon_w \epsilon_p ] _ 1 - \frac{1}{2 } j_r^{w(2 ) } [ \epsilon_w ] _ 0 , \\ \frac{\partial}{\partial t } [ \epsilon_r ] _ 3 & = j_r^{w } [ \epsilon_w ] _ 3 + \frac{1}{2 } j_r^{wp } [ \epsilon_w \epsilon_p ] _ 2 - \frac{1}{2 } j_r^{w(2 ) } [ \epsilon_w ] _ 1.\end{aligned}\ ] ] the procedure to obtain the first moments to order @xmath0 is now also clear . one first solves eq . ( 26 ) to get @xmath87_0 $ ] ; then one solves eq . ( 23 ) and substitutes in eq . ( 28 ) to obtain @xmath87_1 $ ] ; finally one uses the solutions already obtained when deriving the second moments to solve eqs . ( 31 - 32 ) for @xmath87_2 $ ] and @xmath87_3 $ ] . given the first and second moments accurate to @xmath0 and @xmath39 , one finally determines the mean concentrations and the variance of the fluctuations about them accurate to @xmath1 from eqs . ( 21 - 22 ) . although the procedure of obtaining the latter final expressions is fairly laborious , as we shall see in the next section , in order to obtain the leading order error in the predictions of the cfpe , it will only be necessary for us to solve very few of these equations explicitly . in this section we use the system - size expansion to derive expressions for the mean concentrations and the fluctuations about them , as predicted by the cfpe , accurate to @xmath45 . to the best of our knowledge this is the first time that the expansion has been used on the cfpe although the method is similar in principle to the small - noise expansion of fokker - planck equations as presented by gardiner @xcite . the cfpe is obtained by truncating the kramer s moyal expansion to include at most second - order derivatives : @xmath89 note that the second step above , follows by taylor expanding the step operator . next we perform the system - size expansion on the cfpe , eq . ( 33 ) , i.e. , we make the variable transformation given by eq . ( 6 ) which transforms functions of @xmath14 into functions of the new variables @xmath49 . the probability distribution @xmath50 is transformed into a new one @xmath90 . note that the subscript @xmath91 will be used to distinguish quantities calculated using the cfpe from those previously calculated using the cme . the time derivative on the left hand side of the equation and the microscopic rate function @xmath92 transform as in the case of the cme and are given by eqs . ( 7 ) and ( 9 ) together with the definitions eqs . ( 11 - 13 ) and with @xmath51 replaced by @xmath90 . the operators involving derivatives with respect to absolute particle number transform as follows : @xmath93 where the operators @xmath94 are as defined in eq . hence the cfpe in the new variables reads : @xmath95 note that whereas the transformation given by eq . ( 6 ) on the cme leads to an infinite series in powers of the inverse square root of the volume , eq . ( 14 ) , the same transformation on the cfpe leads to a finite series with the highest order term being of order @xmath0 ( this is only true for elementary reactions ) . the derivation of the equations for the time evolution of the moments of the @xmath67 variables proceeds in an exactly analogous manner as to that presented in detail in section ii . the probability distribution is written as a series in powers of the inverse square root of the volume , @xmath96 and the moments are then generally given by : @xmath97_{f , j } \omega^{-j/2},\ ] ] where @xmath70_{f , j } = \int \epsilon_k \epsilon_m ... \epsilon_r \ \pi_{f , j}(\vec{\epsilon},t ) d\vec{\epsilon}.\ ] ] the equations for the mean concentrations and the variance of the fluctuations about them are given by eqs . ( 21 - 22 ) with the subscript @xmath91 carried throughout . the time evolution equations for the corrections to the moments can be derived as before . although there is some repetition involved , we will state these equations in full so that the differences between them and those derived using the cme are very clear . the equations for the corrections to the second moments accurate to order @xmath39 are : @xmath76_{f,0 } & = j_r^{w } [ \epsilon_w \epsilon_k ] _ { f,0 } + ( r \leftrightarrow k ) + d_{rk } , \\ \frac{\partial}{\partial t } [ \epsilon_r \epsilon_k ] _ { f,1 } & = j_r^{w } [ \epsilon_w \epsilon_k ] _ { f,1 } + \frac{1}{2 } j_r^{wp } [ \epsilon_w \epsilon_p \epsilon_k ] _ { f,0 } - \frac{1}{2 } j_r^{w(2 ) } \phi_w [ \epsilon_k ] _ { f,0 } \nonumber \\ & + ( r \leftrightarrow k ) + j_{kr}^w [ \epsilon_w ] _ { f,0 } , \\ \frac{\partial}{\partial t } [ \epsilon_r \epsilon_k ] _ { f,2 } & = j_r^{w } [ \epsilon_w \epsilon_k ] _ { f,2 } + \frac{1}{2 } j_r^{wp } [ \epsilon_w \epsilon_p \epsilon_k ] _ { f,1 } - \frac{1}{2 } j_r^{w(2 ) } \phi_w [ \epsilon_k ] _ { f,1 } \nonumber \\ & - \frac{1}{2 } j_r^{w(2 ) } [ \epsilon_w \epsilon_k ] _ { f,0 } + ( r \leftrightarrow k ) + j_{kr}^w [ \epsilon_w ] _ { f,1 } + \frac{1}{2 } j_{rk}^{wm } [ \epsilon_w \epsilon_m ] _ { f,0 } - \frac{1}{2 } j_{rk}^{w(2 ) } \phi_w.\end{aligned}\ ] ] note that these are the same as eqs ( 23 - 25 ) but with subscript @xmath91 ; the implicit reason for this is that only terms containing @xmath98 and @xmath99 contribute to the equations for the second moments and all such terms are equally present in eqs . ( 14 ) and ( 36 ) . note also that eqs . ( 23 ) and ( 40 ) lead to the same solution , i.e. , @xmath100_{0 } = [ \epsilon_r \epsilon_k ] _ { f,0}$ ] . the solution of @xmath100_{f,1}$ ] is dependent on the solutions of the time evolution equations for @xmath101_{f,0}$ ] and @xmath102_{f,0}$ ] . the equations for the latter are the same as eqs . ( 26 - 27 ) but with subscript @xmath91 ; this is since eq . ( 14 ) and ( 36 ) are equal to order @xmath66 . it follows that @xmath103_{0 } = [ \epsilon_r ] _ { f,0}$ ] and @xmath102_{0 } = [ \epsilon_r \epsilon_k \epsilon_l]_{f,0}$ ] from which we can conclude using eq . ( 41 ) that @xmath100_{1 } = [ \epsilon_r \epsilon_k ] _ { f,1}$ ] . however , as we now show , generally @xmath100_{2 } \ne [ \epsilon_r \epsilon_k ] _ { f,2}$ ] . the solution of @xmath100_{f,2}$ ] is dependent on the solutions of the time evolution equations for @xmath101_{f,1}$ ] , @xmath102_{f,1}$ ] and @xmath104_{f,0}$ ] which are : @xmath82_{f,1 } & = j_r^{w } [ \epsilon_w ] _ { f,1 } + \frac{1}{2 } j_r^{wp } [ \epsilon_w \epsilon_p ] _ { f,0 } - \frac{1}{2 } j_r^{w(2 ) } \phi_w , \\ \frac{\partial}{\partial t } [ \epsilon_r \epsilon_k \epsilon_l ] _ { f,1 } & = j_l^{w } [ \epsilon_w \epsilon_k \epsilon_r ] _ { f,1 } + \frac{1}{2 } j_l^{wp } [ \epsilon_w \epsilon_p \epsilon_r \epsilon_k ] _ { f,0 } - \frac{1}{2 } j_l^{w(2 ) } \phi_w [ \epsilon_r \epsilon_k ] _ { f,0 } \nonumber \\ & + ( l \leftrightarrow k ) + ( k \leftrightarrow r ) + d_{rl } [ \epsilon_k ] _ { f,1 } + j_{rl}^w [ \epsilon_w \epsilon_k ] _ { f,0 } + ( k \leftrightarrow l ) \nonumber \\ & + ( r \leftrightarrow l ) , \\ \frac{\partial}{\partial t } [ \epsilon_r \epsilon_k \epsilon_l \epsilon_m ] _ { f,0 } & = j_r^{w } [ \epsilon_w \epsilon_k \epsilon_l \epsilon_m ] _ { f,0 } + ( r \leftrightarrow m ) + ( m \leftrightarrow k ) + ( k \leftrightarrow l ) + d_{rm } [ \epsilon_k \epsilon_l ] _ { f,0 } \nonumber \\ & + ( m \leftrightarrow l ) + ( l \leftrightarrow k ) + ( r \leftrightarrow m ) + ( m \leftrightarrow l ) + ( k \leftrightarrow m).\end{aligned}\ ] ] equations ( 43 ) and ( 45 ) have the same form as eqs . ( 28 ) and ( 30 ) respectively . this combined with the fact that the right hand sides of eqs . ( 43 ) and ( 45 ) are functions of @xmath100_{f,0}$ ] and that @xmath100_{0 } = [ \epsilon_r \epsilon_k ] _ { f,0}$ ] , implies that @xmath103_{1 } = [ \epsilon_r]_{f,1}$ ] and @xmath105_{0 } = [ \epsilon_r \epsilon_k \epsilon_l \epsilon_m]_{f,0}$ ] . however note that eq . ( 44 ) has one term missing compared to its counterpart eq . ( 29 ) and hence generally @xmath102_{1 } \ne [ \epsilon_r \epsilon_k \epsilon_l ] _ { f,1}$ ] from which it follows using eq . ( 42 ) that @xmath100_{2 } \ne [ \epsilon_r \epsilon_k ] _ { f,2}$ ] . the only remaining equations to be considered are those paralleling eqs . ( 31 ) and ( 32 ) for which we find : @xmath82_{f,2 } & = j_r^{w } [ \epsilon_w ] _ { f,2 } + \frac{1}{2 } j_r^{wp } [ \epsilon_w \epsilon_p ] _ { f,1 } - \frac{1}{2 } j_r^{w(2 ) } [ \epsilon_w ] _ { f,0 } , \\ \frac{\partial}{\partial t } [ \epsilon_r ] _ { f,3 } & = j_r^{w } [ \epsilon_w ] _ { f,3 } + \frac{1}{2 } j_r^{wp } [ \epsilon_w \epsilon_p ] _ { f,2 } - \frac{1}{2 } j_r^{w(2 ) } [ \epsilon_w ] _ { f,1}.\end{aligned}\ ] ] by similar arguments to the above , these equations imply @xmath103_{2 } = [ \epsilon_r]_{f,2}$ ] and @xmath103_{3 } \ne [ \epsilon_r]_{f,3}$ ] . hence , in summary , we have obtained the following results : 1 . @xmath101_{0 } = [ \epsilon_r]_{f,0}$ ] , @xmath100_{0 } = [ \epsilon_r \epsilon_k ] _ { f,0}$ ] , @xmath102_{0 } = [ \epsilon_r \epsilon_k \epsilon_l]_{f,0}$ ] , @xmath105_{0 } = [ \epsilon_r \epsilon_k \epsilon_l \epsilon_m]_{f,0}$ ] 2 . @xmath103_{1 } = [ \epsilon_r]_{f,1}$ ] , @xmath100_{1 } = [ \epsilon_r \epsilon_k ] _ { f,1}$ ] , @xmath102_{1 } \ne [ \epsilon_r \epsilon_k \epsilon_l ] _ { f,1}$ ] 3 . @xmath103_{2 } = [ \epsilon_r]_{f,2}$ ] , @xmath100_{2 } \ne [ \epsilon_r \epsilon_k ] _ { f,2}$ ] 4 . @xmath103_{3 } \ne [ \epsilon_r]_{f,3}$ ] note that these results are not for the moments but for the corrections to the moments ; the real physical meaning of these results in terms of means and variances will be elucidated in the next section . using eqs . ( 32 ) and ( 47 ) , eqs ( 25 ) and ( 42 ) and eqs . ( 29 ) and ( 44 ) , we can respectively write down simple equations for the differences in the corrections to the first , second and third moments as predicted by the cfpe and the cme : @xmath106 where we have used the convenient definitions : @xmath107_{3 } - [ \epsilon_r]_{f,3 } , \\ \delta_{rk } & = [ \epsilon_r \epsilon_k ] _ { 2 } - [ \epsilon_r \epsilon_k ] _ { f,2 } , \\ \delta_{rkl } & = [ \epsilon_r \epsilon_k \epsilon_l]_{1 } - [ \epsilon_r \epsilon_k \epsilon_l ] _ { f,1}.\end{aligned}\ ] ] in this section we will use the results derived in the last section to obtain formulas for the absolute and relative errors ( to leading order ) in the cfpe predictions of the mean concentrations and the variance of the fluctuations . using these formulas we will be able to deduce the general conditions in which the differences between the cfpe and the cme are minimal . furthermore we will show that the cfpe is generally more accurate than the linear fokker - planck equation of van kampen and that the mean concentrations of the cfpe to order @xmath39 are precisely the same as those obtained from effective mesoscopic rate equations . we will now derive expressions for the leading order term of the absolute and relative errors made by the cfpe in predicting the mean concentrations and the variance of the fluctuations about the mean concentrations . we will also obtain an expression for the leading order term of the absolute error made by the cfpe in predicting the skewness of the probability distribution of the concentrations . the mean concentration predicted by the cme , @xmath108 , is given by eq . ( 21 ) while the mean concentration predicted by the cfpe , @xmath109 is given by the same equation but with the subscript @xmath91 carried throughout . subtracting the two expressions and using the summary of results in section iii together with eq . ( 51 ) we get the absolute error in the cfpe concentration : @xmath110 the relative error follows easily : @xmath111 \biggl \langle \frac{n_i}{\omega } \biggr\rangle^{-1 } = \frac{\delta_i}{\phi_i } \omega^{-2 } + o(\omega^{-5/2}).\ ] ] similarly , using eq . ( 22 ) and using the summary of results in section iii together with eq . ( 52 ) we find the absolute and relative errors in the variance of the fluctuations to respectively be given by : @xmath112 where @xmath113 is the variance in the concentration of species @xmath9 as estimated by the linear - noise approximation , i.e. , @xmath114_{0 } - [ \epsilon_i]_{0}^2)$ ] . hence the recipe for calculating the errors of the cfpe predictions is now complete . one first solves eqs . ( 48 - 50 ) and then substitutes their solution in eq . ( 54 - 57 ) . note that this calculation recipe is valid for all times and not only in steady - state conditions . note also that since the denominator in eq . ( 57 ) is the linear - noise approximation estimate for the variance then the leading relative error term in the variance is proportional to @xmath39 . in contrast the leading relative error term in the mean concentrations , eq . ( 55 ) , is proportional to @xmath1 . hence the cfpe s estimates of mean concentrations are generally expected to be more accurate than those of the variance of the fluctuations about the mean concentrations . finally we obtain the absolute error in the cfpe prediction of the skewness of the probability distribution of the concentration of species @xmath9 . the skewness is defined as : @xmath115 the absolute error in the skewness is then @xmath116 where @xmath117 is the skewness predicted by the cfpe , i.e. , eq . ( 58 ) with subscript @xmath91 throughout . as before , by using using eqs . ( 21 - 22 ) together with the summary of results in section iii and eq . ( 53 ) we get : @xmath118 we can now answer the question : which of the two , cfpe or linear fokker - planck equation , is the most accurate ? we note that the linear fokker - planck equation ( or equivalently the linear - noise approximation ) is obtained by keeping only terms of order @xmath66 in eq . if we do the same on the expansion of the cfpe , i.e. eq . ( 36 ) , then we also get the same linear fokker - planck equation . this equality implies that the cfpe becomes correct for large enough volumes or equivalently for large enough molecular populations . this result was known to van kampen and is discussed in the book by gardiner @xcite . within the linear - noise approximation , one can calculate the two quantities @xmath103_{0}$ ] and @xmath100_{0}$ ] using eqs . ( 26 ) and ( 23 ) respectively . the quantities @xmath103_{m}$ ] and @xmath100_{m}$ ] where @xmath119 are all zero in this approximation since the expansion has only terms to order @xmath66 . now the initial condition for the cme is a delta function centered on the number of molecules as given by the res , i.e. at time @xmath120 , the average number of molecules of the cme and the res agree and hence it follows that @xmath103_{0 } = 0 $ ] initially and for all times @xcite . these results together with eqs . ( 21 ) and ( 22 ) , would seem to imply that within the linear - noise approximation , the mean concentrations are accurate to order @xmath3 while the variance is accurate to order @xmath39 . however by considering terms of higher order than those leading to the linear - noise approximation , one arrives at the conclusion that actually the variance within this approximation is accurate to higher order than @xmath39 . this can be deduced by noting that @xmath103_{0 } = 0 $ ] for all times implies @xmath121_{0 } = [ \epsilon_r \epsilon_k ] _ { 1}= 0 $ ] also for all times . hence it follows from eq . ( 22 ) that the variance in the linear - noise approximation is accurate to order @xmath0 . now from eqs . ( 54 ) and ( 56 ) , it is evident that generally the mean concentration and variance prediction of the cfpe are accurate to at least order @xmath0 . hence the mean concentration prediction of the cfpe is more accurate than that which can be obtained from the linear fokker - planck equation . it is also clear that the higher accuracy comes from taking into account the non - linear character of the cfpe since the @xmath122 term in eq . ( 54 ) is obtained by considering terms in eqs . ( 14 ) and ( 36 ) of higher order than the linear - noise approximation . we can also derive an explicit equation for the mean concentrations predicted by the cfpe accurate to order @xmath39 : @xmath123_{f,0 } \omega^0 + \partial_t [ \epsilon_i]_{f,1 } \omega^{-1/2 } ) + o(\omega^{-3/2 } ) \nonumber \\ & = \partial_t \phi_i + j_i^w \biggl(\biggl \langle \frac{n_i}{\omega } \biggr \rangle_f - \phi_i \biggr ) + \frac{1}{2 } \omega^{-1 } ( j_i^{wp } [ \epsilon_w \epsilon_p]_{f,0 } - j_i^{w(2 ) } \phi_w ) + o(\omega^{-3/2}).\end{aligned}\ ] ] note that the first step proceeds by taking the time derivative of eq . ( 21 ) and the second step follows from using eqs . ( 26 ) and ( 43 ) , bearing in mind that @xmath124_{f,0 } = [ \epsilon_i]_{0}$ ] . hence the computation of the mean concentrations to this order requires only the solution of the res and of the lyapunov equation eq . note that eq . ( 60 ) is exactly the same as the effective rate equations recently derived by grima from the cme ( eq . 60 is the same as eq . ( 22 ) together with eq . ( 24 ) in ref @xcite ) . consider the case where we have @xmath4 species interacting via @xmath5 elementary reactions of the equal - step type , i.e. , in each individual reaction , either @xmath125 molecules of a species are generated or @xmath125 molecules are destroyed or no molecules are generated or destroyed . in such a case , the stoichiometric matrix elements are @xmath126 or @xmath127 , where @xmath125 is a non - zero positive integer . three examples of equal - step reactions are : @xmath128{k_2 } 2x_1 , x_1 \xrightarrow{k4 } \o \nonumber \\ & \o \xrightarrow{k_1 } 2 x_1 \xrightarrow{k_2 } \o \nonumber \\ & \o \xrightleftharpoons[k_2]{k_1 } x_1 , \o \xrightleftharpoons[k_4]{k_3 } x_2 , x_1+x_2 \xrightarrow{k_5 } \o \end{aligned}\ ] ] the first reaction is autocatalytic where @xmath129 is some very abundant species whose number of molecules is considered constant ; this is a one - step , one species reaction scheme . the second reaction involves the burst input of two molecules and their dimerization , a two - step one species reaction scheme . the third reaction involves the production and degradation of two species and their bimolecular interaction ; this is a one - step , two species reaction scheme . for equal - step reactions , one species reaction schemes , the quantity @xmath130 evaluates to zero in steady - state conditions : @xmath131 note that in the last step , use was made of the steady - state condition : @xmath132 . from eqs . ( 48 - 50 ) , we can then deduce that @xmath133 . hence it follows from eqs . ( 54 ) and ( 56 ) that the mean concentrations and the variance of fluctuations predicted by the cfpe for one species , equal - step reactions , are accurate to at least order @xmath1 . this is impressive when one considers that the linear - noise approximation of the cme only leads to estimates accurate to order @xmath3 in the mean and order @xmath0 in the variance . these conclusions lend support to the results of an early investigation of the one species cfpe @xcite . however , this high accuracy of the cfpe is not generally true for multispecies equal - step reactions . for example , for the third reaction scheme in the examples considered above , one finds @xmath134 and @xmath135 . the non - zero values of @xmath136 for some index values implies that the mean and variance predictions of the cfpe in this case are accurate to order @xmath0 . in the previous subsection we have seen how @xmath137 is zero for one species , one - step reaction schemes and how this leads to a particularly high accuracy in the predictions of the cfpe . we now want to find the condition which forces @xmath138 for chemical reactions involving any number of species . consider the case where all reactions are reversible . since each reaction can be paired with its reverse , it follows that the formula for @xmath137 can then be written as : @xmath139,\end{aligned}\ ] ] where the subscripts @xmath140 and @xmath141 indicates quantities evaluated for the forward and backward reactions respectively . the reversibility condition imposes @xmath142 and was used in deriving the last step . furthermore , a system of reversible reactions will always reach chemical equilibrium and in such conditions the system is characterized by detailed balance , i.e. , @xmath143 , the forward and reverse rates of each elementary reversible reaction balance @xcite . hence by eq . ( 63 ) , @xmath138 , in detailed balance conditions , and consequently by eqs . ( 48 - 50 ) and eqs . ( 54 ) and ( 56 ) , the cfpe s predictions of mean and variance are accurate to order @xmath1 . equilibrium conditions always imply detailed balance and hence our results suggest that the size of the differences between the predictions of the cfpe and the cme increase with how far is the system from equilibrium . as a first application of our theory , we will estimate the relative errors in the cfpe predictions for a dimerization reaction . this is the simplest case of a bimolecular reaction mechanism . the main purpose of considering such a reaction is that both its cme and cfpe are exactly solvable and hence it provides us with a direct test of our expressions for the leading order error in the means and the variances as predicted by the cfpe . the set of reactions under study are : @xmath144 monomers , denoted as @xmath37 , are pumped into some compartment at a rate @xmath145 . pairs of monomers react with rate constant @xmath146 to form a dimer molecule , @xmath38 . the concentration of dimers increases with time however the concentration of monomers becomes constant after a short time , i.e. the monomers reach a steady - state . since @xmath38 is not involved in the reaction , the mathematical description is solely in terms of the number of molecules of the monomers for the cme and cfpe and in terms of the monomer concentration for the re . the cme , eq . ( 2 ) , for the dimerization reaction reads : @xmath147 multiplying the equation on both sides by @xmath148 and summing over @xmath149 from @xmath127 to infinity , we get the equivalent generating function equation : @xmath150 where @xmath151 . this partial differential equation is solved in the steady - state with boundary conditions @xmath152 and @xmath153 @xcite leading to : @xmath154 where @xmath155 , @xmath156 and @xmath157 is the modified bessel function of the first kind of order @xmath158 . the mean concentration and variance of the concentration fluctuations about this mean according to the cme are then given by the following expressions : @xmath159 ^ 2 \biggr ) \nonumber \\ & = \frac{\phi_1 ^ 2 [ n_{ode } [ i_1(4 n_{ode})]^2 - n_{ode } [ i_0(4 n_{ode})]^2 + i_0(4 n_{ode } ) i_1(4 n_{ode } ) ] } { n_{ode } [ i_1(4 n_{ode})]^2}.\end{aligned}\ ] ] note that these expressions are obtained within an exact approach and are not approximations as the ones stemming from the system - size expansion of the cme . now we obtain expressions for the mean and variance using the cfpe approach . the cfpe , eq . ( 33 ) , for the dimerization reaction reads : @xmath160 where @xmath161 and @xmath162 . the exact stationary solution of this non - linear second order partial differential equation can be shown to be : @xmath163}{4 k_2 ( n_1 - 1 ) n_1 + k_1 \omega^2 } \biggl ( k_1 + k_2 \int_1^{n_1 } d \eta \exp \biggl [ - \frac{3 k_1 \omega^2 \arctan h(\eta)}{2 \sqrt{k_2 } \sqrt{k_1 \omega^2 - k_2 } } + \eta \biggr ] \biggr),\end{aligned}\ ] ] where @xmath164 . the constants @xmath165 and @xmath166 are to be determined by the boundary conditions and the normalization condition . the boundary conditions of the cfpe are @xmath167 . note that the cfpe unlike the cme does not generally have a natural boundary at @xmath168 since the noise can sometimes drive the system to negative values of @xmath149 @xcite . note that this problem is also implicit in the stationary solution of the linear fokker - planck equation , a gaussian which is non - zero for negative particle numbers @xcite ( see the end of this subsection for a further discussion of boundary conditions ) . the condition at @xmath169 fixes the value of @xmath166 while the condition at @xmath170 is automatically satisfied by the exponential pre - factor . the remaining constant @xmath165 is fixed by the normalization condition . since there is no closed form solution for the integral in eq . ( 71 ) , @xmath165 has to be computed numerically ; once @xmath171 is determined , the mean and variance can be straightforwardly numerically computed as well . the exact relative error in the mean and variance predictions of the cfpe can now be computed . one first fixes the rate constants @xmath145 and @xmath146 and @xmath172 . the normalization constant @xmath165 is found by numerical integration and from the ensuing steady - state probability distribution , one finds the mean , @xmath173 , and variance @xmath174 . the numerical error in the integration is essentially eliminated by performing the integration for a set of decreasing step size values and extrapolating to obtain the integral value at zero step size . using the same values of rate constants and @xmath172 , one uses eqs . ( 68 - 69 ) to compute the mean and variance according to the cme . the exact relative errors in the mean and variance can then be found using @xmath175 and @xmath176 , respectively . the exact absolute values of the relative errors in the cfpe predictions are shown by the red open circles in fig . 1 for parameter values @xmath177 and @xmath178 . note that the relative error in the variance is larger than that in the mean . the errors increase with decreasing steady - state numbers of monomers . even for very small numbers , the errors are quite small . for example for a case in which the res predict 5 monomers in steady - state , the percentage relative errors in the mean and variance predictions of the cfpe are just @xmath179 and @xmath180 respectively . the high accuracy of the cfpe in low particle number conditions is indeed surprising since typically it has only been deemed accurate for systems characterized by large particle numbers . we can now test the accuracy of the theory developed in the previous sections by using it to obtain expressions for the approximate relative errors in the mean and variance and then compare these with the exact values as already obtained above . by inspection of the reaction scheme , eq . ( 64 ) , it can be easily deduced that the stoichiometric matrix is @xmath181 . from the definition of the macroscopic rate function vector ( see introduction ) it also follows that it is equal to @xmath182 . this is all the information needed to calculate the estimates for the relative errors using our theory . the macroscopic concentration and the relevant entries of the d and j matrices evaluated at steady - state are then given by : @xmath183 these are substituted in eqs . ( 48 - 50 ) which are then evaluated at steady - state , leading to : @xmath184 the relative error in the mean concentration to leading order is then given by eq . ( 55 ) : @xmath185 where @xmath186 is the average number of monomers as predicted by the res . to compute the relative error in the variance we need to first estimate the variance to the linear - noise level of approximation . this is done by solving eq . ( 23 ) in steady - state : @xmath187_0 = -\frac{d_{11}}{2 j_1 ^ 1 } = \frac{3 k_1}{8 k_2 \phi}.\ ] ] the variance is then @xmath188_0 $ ] . using the latter and eq . ( 76 ) , it is found that eq . ( 57 ) evaluates to : @xmath189 the theoretical absolute values of the relative errors in the cfpe predictions , as given by eq . ( 78 ) and eq . ( 80 ) , are shown by the solid blue lines in fig . 1 for parameter values @xmath177 and @xmath178 . the theory is generally in very good agreement with the exact solution ; small discrepancies are only apparent in the error for the variance at molecule numbers less than approximately 5 monomers . the comparison has also been done for many other parameters values and as predicted by theory , in all cases , the graphs are the same as shown in fig . 1 . we have also computed the exact errors by solving the cfpe with different boundary conditions . one could argue that constraints should be imposed on the cfpe such that it preserves the natural boundary of the cme at @xmath168 . this can be fulfilled by requiring that the probability current of the cfpe vanishes at @xmath168 @xcite . in such a case the stationary solution of the cfpe has the form of eq . ( 71 ) with @xmath190 and @xmath165 is found by requiring that the solution is normalized on @xmath191 . the exact errors computed with this new solution of the cfpe are practically indistinguishable from the previous solutions shown in fig . 1 except for a small discrepancy at @xmath192 . the excellent agreement of our theoretical solution with both cfpe solutions is simply due to the fact that the probability of @xmath149 taking negative values in eq . ( 71 ) is very small , unless @xmath172 is also very small . as a second application , we consider the catalysis of a substrate species @xmath193 into a product species @xmath194 by an enzyme species via the michaelis - menten mechanism @xcite : @xmath195{k_0}c , \\ & c \xrightarrow{k_2 } e + p,\end{aligned}\ ] ] where @xmath196 denotes the free enzyme , i.e. when it is not bound to substrate , and @xmath197 denotes the substrate - enzyme complex . we will denote substrate , complex and free enzyme as species 1 , 2 and 3 respectively . note that the product species is missing from the kinetic description because it is a byproduct of the reaction and thus not involved in the reactions . the total enzyme concentration is a constant , @xmath198 , since the enzyme is either bound to substrate or unbound . hence we effectively have a two variable system . the reaction system exhibits a steady - state in the concentrations of substrate and complex whenever the inequality @xmath199 is satisfied , i.e. when the rate at which substrate is pumped into the system is less than or equal to the maximum rate at which the enzyme can convert substrate to product . assuming such conditions , our aim is to calculate the relative errors in the mean and variance predictions of the cfpe , i.e. eqs ( 55 ) and ( 57 ) ; to achieve this , we will first need to solve eqs . ( 48)-(50 ) , which we show in detail now . the stoichiometric matrix and the macroscopic rate function vector follow directly from their definitions ( see introduction ) : @xmath200 the rate equations and the @xmath201 and @xmath202 matrices follow by inserting the above in eq . ( 1 ) , eq . ( 16 ) and eq . ( 17 ) to obtain : @xmath203 where @xmath204 , @xmath205 is the michaelis - menten constant and @xmath206 . note that @xmath207 is a measure of enzyme saturation since as the input rate of substrate , @xmath208 , approaches the maximum rate at which the enzyme can catalyze the reaction , @xmath209 , the proportion of enzyme in complex form increases accordingly , as can also be seen from eq . note also that @xmath210 is a measure of how far is the system from equilibrium . this is since if substrate binding would occur at equilibrium , i.e. , @xmath211 , then the relationship between the macroscopic concentrations would be @xmath212 while generally in steady - state conditions , i.e. @xmath213 , the relationship between the macroscopic concentrations is @xmath214 . both @xmath207 and @xmath210 are non - dimensional , positive fractions . substituting eqs . ( 84 - 87 ) in eqs . ( 48 - 50 ) , setting the time derivative to zero and solving the resulting set of simultaneous equations we obtain : @xmath215 where @xmath216 . the leading order term of the relative errors in the mean concentrations of substrate and complex , as predicted by the cfpe , are then given by substituting eq . ( 83 ) together with eq . ( 89 ) in eq . ( 55 ) : @xmath217 note that the relative error in the substrate concentration is always negative , i.e. , the cfpe overestimates the mean substrate concentrations and it increases with the distance from equilibrium , @xmath210 . there is no error in the cfpe estimate for enzyme concentration ( at least to order @xmath1 ) . to calculate the relative errors in the variance using eq . ( 57 ) we first need to compute the variance as estimated by the linear - noise approximation . this is obtained by solving eq . ( 23 ) using eq . ( 84 ) and eq . ( 88 ) : @xmath218 finally substituting the above two equations and eqs . ( 90 - 91 ) in eq . ( 57 ) we obtain the leading order term of the relative errors in the variance of the substrate and complex concentration fluctuations , as predicted by the cfpe : @xmath219 note that both relative errors are always positive implying that the cfpe underestimates the variance . we can now use the formulae given by eq . ( 92 ) , eq . ( 94 ) and eq . ( 95 ) to estimate the relative error of the cfpe when modeling conditions typical of the intracellular environment . a principal characteristic of such an environment is that the number of molecules of some species can be quite small . a detailed protein abundance profiling of the escherichia coli cytosol by ishihama et al @xcite shows that the total number of enzyme molecules per cell approximately varies from a hundred to a few thousands . it is indeed in this limit of small numbers that it is frequently thought that the cfpe and the cle description are not very accurate . we quantitatively test this hypothesis using our formulae . we will first express our error formulae in terms of the average number of molecules of substrate and free enzyme as predicted by the res , i.e. , @xmath220 and @xmath221 . using eqs ( 83 ) we find that : @xmath222 substituting eq . ( 96 ) in eq . ( 92 ) , eq . ( 94 ) and eq . ( 95 ) we get expressions for the errors in terms of @xmath223 , @xmath224 , @xmath207 and @xmath210 . given fixed molecule numbers , @xmath223 and @xmath224 , we can find the maximum error by varying @xmath207 and @xmath210 over their allowed range @xmath225 $ ] . repeating this procedure for various molecule numbers we can obtain simple two dimensional plots of the maximum error . the results for the maximum relative error in the predictions of the variance are shown in fig . the results verify that the predictions of the cfpe become increasingly accurate with increasing molecule numbers . they also show that the error incurred by using the cfpe for cases of small molecule numbers is very small : less than @xmath226 for a few tens of molecule numbers . it is noteworthy that this accuracy is far better than even that hypothesized by proponents of the cfpe @xcite . for example gillespie in his seminal paper on the derivation of the cle @xcite remarks in his conclusion that the cle ( and hence the cfpe ) approximation is probably not a good one when one models a system composed of three time - varying species with total molecular population of 2000 since it appears quite possible that the molecule number of at least one of the species becomes significantly small at some point in time . in contrast our theory seems to predict that the cfpe predictions will still be very accurate even when the molecule numbers are quite low . we have tested these predictions by numerically solving the cle for the michaelis - menten process using the euler - mayurama method to obtain the mean substrate concentrations and the variance of the substrate fluctuations about the means . the same were obtained from stochastic simulation algorithm simulations of the cme . the results are shown in fig . the parameters are chosen to be @xmath227 , @xmath228 , @xmath229 , @xmath230 , @xmath231 and @xmath232 since this gives conditions similar to those mentioned by gillespie above . the re solutions , eqs . ( 83 ) , with the above parameters lead to @xmath233 , @xmath234 and @xmath235 which , given a volume of @xmath232 , would imply @xmath236 , @xmath237 and @xmath238 . the total molecular population of enzyme ( free plus complex form ) is 2500 molecules . each algorithm ( euler - mayurama and stochastic simulation algorithm ) was run 5 times leading to 5 independent estimates @xcite . note that even though the mean number of free enzyme molecules is considerably low , the predictions of the cfpe for both the mean and the variance agree ( within sampling error ) with those of the cme . for comparison we have also plotted the predictions of the linear - noise approximation ( red lines ) and of the mean concentration as predicted by the effective mesoscopic rate equation eq . ( 60 ) ( blue line ) . the results clearly confirm that the cfpe is more accurate than the linear fokker planck equation associated with the linear - noise approximation and that indeed the mean concentrations of the cfpe are in excellent agreement with the effective mesoscopic rate equations derived in ref @xcite . the effective mesoscopic rate equation for the michaelis - menten reaction was first obtained in ref @xcite ( see eq . ( 29 ) in the latter reference ) . for our set of parameters , the theoretical expressions , eqs . ( 92 ) and ( 94 ) , evaluate to @xmath239 and @xmath240 ; these errors are so small that they are clearly masked by the sampling error inherent in the calculation of the mean and the variance from the long - time simulation trajectories . indeed , in agreement with our theory , from fig . 3 one can detect no significant difference between the cfpe and cme predictions . the numerical experiments were performed with various other parameter sets in all cases we could not detect any discrepancy between the cfpe and cme predictions within sampling error . summarizing , in this article we have shown that ( i ) the mean and variance predictions of the cfpe are accurate to order @xmath0 . since those of the linear fokker - planck equation are accurate to order @xmath3 for the mean and @xmath0 for the variance , it is clear that the cfpe is generally more accurate than the linear fokker - planck equation or equivalently the linear - noise approximation . ( ii ) for detailed balance conditions , the predictions of the cfpe are even more accurate , order @xmath1 , i.e. in equilibrium or near equilibrium conditions the cfpe does an excellent job of approximating the cme . ( iii ) accuracy to such high order in inverse powers of the system volume implies that the cfpe estimates should be quite good even for small populations of molecules . our simulations for dimerization and enzyme - catalyzed reactions support these theoretical conclusions , with impressively good agreement down to an average of 5 molecules for the dimerization example . the cfpe s accuracy is indeed surprising given that it arises out of a naive truncation of the kramers - moyal expansion of the cme and that the cfpe can not be obtained from the systematic system - size expansion of the cme . only the linear fokker - planck equation can be derived from the latter expansion by considering terms of order @xmath66 . this equation leads to mean and variance estimates which are accurate to orders @xmath3 and @xmath0 . now if one wants more accurate estimates one needs to consider higher - order terms in the expansion . to get mean concentration estimates to order @xmath39 one needs to consider the term in the system - size expansion proportional to @xmath3 @xcite . to this order , one does not obtain the cfpe , rather one obtains a partial differential equation with a third - order derivative . however , it turns out that the mean calculated from this equation precisely agrees with that calculated from the cfpe to order @xmath39 . if we even wanted to get more accurate means and variance , say both to order @xmath1 , we need to consider terms in the system - size expansion to order @xmath0 . this leads to a partial differential equation for the time evolution of the probability density function with derivatives as high as fifth order . once again this is not the cfpe . however under steady - state conditions obeying detailed balance , the estimates from this high - order differential equation and the cfpe exactly agree to order @xmath1 . hence we have shown that though it is true that the cfpe does not arise out of the system - size expansion , nevertheless its predictions are better than those which can be obtained by considering only the first term of the expansion ( the linear - noise approximation ) as is conventional @xcite . it follows that the non - linear character of the cfpe is not completely spurious as originally suggested by van kampen @xcite . our study is the first one to our knowledge which systematically analyzes the validity of the non - linear multivariate cfpe and which derives approximate expressions for the size of the errors in the cfpe estimates previous studies @xcite have focused on the cfpe for unimolecular reactions and for unimolecular and bimolecular reactions involving one species @xcite . our analysis is based on the system - size expansion and thus has the same limitations , namely that it is only applicable for chemical systems which are `` asymptotically stable in the sense of lyapunov '' . this implies that from our analysis we can not draw any conclusions for bistable systems @xcite . within these constraints , the system - size expansion is a legitimate means of obtaining the moments of the cme accurate to any desired order @xcite . a few authors @xcite have expressed reservations regarding the accuracy of the expansion beyond the linear - noise level , their reasoning stemming from the fact that pawula s theorem @xcite states that a time - evolution equation for a probability density function with higher than second - order derivatives can not describe a stochastic process . however these misgivings are undue the higher - order partial differential equation stemming from the expansion truncated to some order is `` not an exact equation for a markov process that in some way approximates the original process ; rather it is an approximate equation for the exact p. '' @xcite . this statement of van kampen is generally true for any legitimate expansion of the cme , not only the system - size expansion ; for example risken and vollmer @xcite showed that taking into account higher - order derivatives than two in the kramers - moyal expansion of the cme also leads to more accurate solutions than if one just had to use the cfpe . the accuracy of the system - size expansion beyond the linear - noise approximation has also been verified by many recent studies @xcite , putting at rest any small doubts about its general validity . finally , the good agreement of our theoretical expressions for the errors with simulations is a clear indication of the soundness of our system - size expansion based approach . concluding our results offer theoretical and numerical support for gillespie s hypothesis @xcite regarding the validity of the cfpe in both mesoscopic and macroscopic systems . our formulas provide a simple means to estimate the error in the predictions of the cfpe and the associated cle and hence should be of wide applicability to both theoretical and numerical studies of stochastic chemistry . r. g. acknowledges support by sulsa ( scottish universities life science alliance ) . by the normalization condition and the expansion of @xmath51 we have : @xmath241 equating powers of the volume we obtain : @xmath242 an analogous property has been discussed by gardiner in the different though related context of small noise expansions of the fokker - planck equation @xcite . the two properties above are useful in the computation of the integrals needed to arrive to eqs . ( 23 - 32 ) ; for more details see appendix b. it follows from eqs . ( a2-a3 ) that only @xmath243 is a genuine probability density while the higher orders are negative in some regions of the @xmath244 space . from eq . ( 15 ) , we see that to order @xmath66 , the time - evolution of @xmath243 is given by a linear fokker - planck equation : @xmath245 which again verifies that @xmath243 is a probability density . however the time - evolution equations for @xmath246 where @xmath247 , involve derivatives of order larger than two and hence by pawula s theorem @xcite @xmath246 can not be genuine probability density functions . the above arguments also imply that it is not correct to think of @xmath248_j$ ] , where @xmath247 , as genuine statistical moments ; rather they are best considered as placeholders or labels for the associated integrals @xmath249 . in the main text we refer to them as corrections to the moments to order @xmath74 . it is however important to bear in mind that though @xmath248_j$ ] are generally not true statistical moments , their linear superposition via eq . ( 19 ) is a genuine statistical moment . hence it is best to avoid associating any physical meaning to @xmath248_j$ ] and to simply regard them as a means to obtain the desired answer , i.e. , @xmath250 . the time - evolution equations are obtained by substituting eq . ( 18 ) in eq . ( 15 ) , multiplying the resulting equation on both sides by @xmath251 and integrating over @xmath73 . finally we equate terms of order @xmath39 on both sides of the equation to obtain the time - evolution equation for @xmath81_2 $ ] . the right hand side of the resulting equation simplifies by performing integration by parts ; there are 8 integrals which need such evaluation and we treat each one of them below . 1 . @xmath252 d{\vec{\epsilon } } \nonumber \\ & = - j_{i}^w ( [ \epsilon_w \epsilon_k ] _ 2 \delta_{i , r } + [ \epsilon_w \epsilon_r ] _ 2 \delta_{i , k } ) \nonumber \\ & = - j_{r}^w [ \epsilon_w \epsilon_k ] _ 2 - j_{k}^w [ \epsilon_w \epsilon_r ] _ 2.\end{aligned}\ ] ] + note that in eq . ( 15 ) we are summing over all twice repeated indices , which for the above integral are @xmath9 and @xmath253 . use was made of this implicit summation on @xmath9 in the derivation of the last step . @xmath254 d{\vec{\epsilon } } \nonumber \\ & = d_{ip}(\delta_{p , r } \delta_{i , k } + \delta_{p , k } \delta_{i , r } ) \int \pi_2 d{\vec{\epsilon } } = 0 . \end{aligned}\ ] ] + in the last step , we have made use of the fact that @xmath255 , as shown in appendix a. 3 . @xmath256 d{\vec{\epsilon } } \nonumber \\ & = - j_{i}^{wp } ( [ \epsilon_w \epsilon_p \epsilon_r ] _ 1 \delta_{i , k } + [ \epsilon_w \epsilon_p \epsilon_k ] _ 1 \delta_{i , r } ) \nonumber \\ & = - j_{k}^{wp } [ \epsilon_w \epsilon_p \epsilon_r ] _ 1 - j_{r}^{wp } [ \epsilon_w \epsilon_p \epsilon_k ] _ 1.\end{aligned}\ ] ] 4 . @xmath257 d{\vec{\epsilon } } \nonumber \\ & = - j_{i}^{w(2 ) } ( [ \epsilon_r ] _ 1 \delta_{i , k } + [ \epsilon_k ] _ 1 \delta_{i , r } ) \nonumber \\ & = - j_{k}^{w(2 ) } [ \epsilon_r ] _ 1 - j_{r}^{w(2 ) } [ \epsilon_k]_1.\end{aligned}\ ] ] 5 . @xmath258 d{\vec{\epsilon } } \nonumber \\ & = j_{ip}^w [ \epsilon_w ] _ 1 ( \delta_{p , r } \delta_{i , k } + \delta_{p , k } \delta_{i , r } ) \nonumber \\ & = 2 j_{kr}^{w } [ \epsilon_w ] _ 1.\end{aligned}\ ] ] + in obtaining the last step we have used the implicit summation over @xmath9 and @xmath125 and also the symmetrical property , @xmath259 , which follows from the definitions given by eqs . ( 16 - 17 ) . @xmath260 d{\vec{\epsilon } } \nonumber \\ & = - j_{i}^{w(2 ) } ( [ \epsilon_w \epsilon_k ] _ 0 \delta_{i , r } + [ \epsilon_w \epsilon_r ] _ 0 \delta_{i , k } ) \nonumber \\ & = - j_{r}^{w(2 ) } [ \epsilon_w \epsilon_k ] _ 0 - j_{k}^{w(2 ) } [ \epsilon_w \epsilon_r ] _ 0.\end{aligned}\ ] ] 7 . @xmath261 d{\vec{\epsilon } } \nonumber \\ & = j_{ip}^{wm } [ \epsilon_w \epsilon_m ] _ 0 ( \delta_{p , r } \delta_{i , k } + \delta_{p , k } \delta_{i , r } ) \nonumber \\ & = 2 j_{kr}^{wm } [ \epsilon_w \epsilon_m ] _ 0.\end{aligned}\ ] ] + note that in the last step we have used the symmetrical property , @xmath262 , which follows from the definitions given by eqs . ( 16 - 17 ) . 8 . @xmath263 d{\vec{\epsilon } } \nonumber \\ & = j_{ip}^{w(2)}(\delta_{p , r } \delta_{i , k } + \delta_{p , k } \delta_{i , r } ) \int \pi_0 d{\vec{\epsilon } } = 2 j_{kr}^{w(2)}. \end{aligned}\ ] ] + in the last step , we have made use of the fact that @xmath264 , as shown in appendix a and the symmetry property used in the evaluation of the previous integral . ohphs d. t. gillespie , annu . phys . chem . * 58 * , 35 ( 2007 ) d. t. gillespie , physica a * 188 * , 404 ( 1992 ) d. t. gillespie , j. chem * 131 * , 164109 ( 2009 ) d. a. mcquarrie , j. appl . prob . * 4 * , 413 ( 1967 ) n. g. van kampen , _ stochastic processes in physics and chemistry _ ( elsevier , 2007 ) h. kramers , physica * 7 * , 284 ( 1940 ) j. e. moyal , j. r. stat . soc . * 11 * , 151 ( 1949 ) c. w. gardiner , _ handbook of stochastic methods for physics , chemistry and the natural sciences _ ( springer , 2004 ) n. g. van kampen , can . * 39 * , 551 ( 1961 ) n. g. van kampen , adv . * 34 * , 245 ( 1976 ) n. g. van kampen , the diffusion approximation for markov process , pp . 181 - 195 . in : i. lamprecht and a. i. zotin , ed 1982 . _ thermodynamics and kinetics of biological processes_. walter de gruyter and co. , new york t. g. kurtz , j. chem . phys . * 50 * , 460 ( 1969 ) w. horsthemke and l. brenig , zeitschrift fuer physik b * 27 * , 341 ( 1977 ) d. t. gillespie , j. chem . phys . * 113 * , 297 ( 2000 ) t. e. turner , s. schnell and k. burrage , comp . biol . chem . * 28 * , 165 ( 2004 ) r. grima and s. schnell , essays in biochemistry * 45 * , 41 ( 2008 ) j. paulsson , phys . life revs . * 2 * , 157 ( 2005 ) t. c. meng , s. somani and p. dhar , in silico biology * 4 * , 293 ( 2004 ) d. t. gillespie , j. phys . chem . * 81 * , 2340 ( 1977 ) z. hou and h. xin , j. chem . phys . * 119 * , 11508 ( 2003 ) m. l. simpson , c. d. cox and g. s. sayler , j. theor . biol . * 229 * , 383 ( 2004 ) t. xiao , j. ma , z. hou and h. xin , new . j. phys . * 9 * , 403 ( 2007 ) j. wilkie and y. m. wong , chem . phys . * 353 * , 132 ( 2008 ) v. sotiropoulos et al , ieee / acm trans . . bioinform . * 6 * , 470 ( 2009 ) y. ishihama y. et al , bmc genomics * 9 * , 102 ( 2008 ) s. ghaemmaghami et al , nature * 425 * , 737 ( 2003 ) j. elf and m. ehrenberg , genome res . * 13 * , 2475 ( 2003 ) r. grima , j. chem . phys . * 133 * , 1 ( 2010 ) d. t. gillespie , j. chem . phys . * 72 * , 5363 ( 1980 ) j. keizer , _ statistical thermodynamics of nonequilibrium processes _ ( springer , 1987 ) r. m. mazo , j. chem . phys . * 62 * , 4244 ( 1975 ) h. grabert , p. haenggi and i. oppenheim , physica * 117a * , 300 ( 1983 ) h. risken , _ the fokker - planck equation : methods of solution and applications _ ( springer , 1996 ) a. cornish - bowden , _ fundamentals of enzyme kinetics _ ( portland press , 1995 ) the estimates were calculated according to the following procedure . running the cle euler - mayurama algorithm once leads to a set of numbers @xmath265 where @xmath266 is the discretization time step of the algorithm . an estimate of the steady - state average concentration of species @xmath9 as predicted by the cfpe is then given by @xmath267 where @xmath268 is the maximum number of time steps . running the algorithm @xmath4 times leads to @xmath4 slightly different estimates of the mean of the cfpe ; this variation leads to statistical error in the numerical estimation of the mean concentrations and can be reduced by increasing the total number of time steps . the same arguments hold for the estimation of the variance . for the simulations leading to fig . 3 we chose @xmath269 and @xmath270 . smaller @xmath266 and larger @xmath268 did not change the results . estimates of the concentrations and variances predicted by the cme can be obtained by sampling the trajectories of the stochastic simulation algorithm at intervals of time equal to the discretization time step of the cle . the means and variance are then obtained as for the cle . r. grima , bmc sys . biol . * 3 * : 101 ( 2009 ) d. t. gillespie , j. phys . a * 106 * , 5063 ( 2002 ) d. j. higham and r. khanin , open appl . j * 2 * , 59 ( 2008 ) r. f. pawula , phys . rev . * 162 * , 186 ( 1967 ) h. risken and h. d. vollmer , zeitschrift fuer physik b * 35 * , 313 ( 1979 ) r. grima , phys . letts . * 102 * , 218103 ( 2009 ) p. thomas , a.v . straube and r. grima , j. chem . * 133 * , 195101 ( 2010 ) c. cianci , f di patti , d. fanelli and l. barletti , arxiv:1104.5668v1 c. cianci , f di patti , d. fanelli , arxiv:1104.5570v1
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the chemical fokker - planck equation and the corresponding chemical langevin equation are commonly used approximations of the chemical master equation .
these equations are derived from an uncontrolled , second - order truncation of the kramers - moyal expansion of the chemical master equation and hence their accuracy remains to be clarified .
we use the system - size expansion to show that chemical fokker - planck estimates of the mean concentrations and of the variance of the concentration fluctuations about the mean are accurate to order @xmath0 for reaction systems which do not obey detailed balance and at least accurate to order @xmath1 for systems obeying detailed balance , where @xmath2 is the characteristic size of the system .
hence the chemical fokker - planck equation turns out to be more accurate than the linear - noise approximation of the chemical master equation ( the linear fokker - planck equation ) which leads to mean concentration estimates accurate to order @xmath3 and variance estimates accurate to order @xmath0 .
this higher accuracy is particularly conspicuous for chemical systems realized in small volumes such as biochemical reactions inside cells .
a formula is also obtained for the approximate size of the relative errors in the concentration and variance predictions of the chemical fokker - planck equation , where the relative error is defined as the difference between the predictions of the chemical fokker - planck equation and the master equation divided by the prediction of the master equation . for dimerization and enzyme - catalyzed reactions ,
the errors are typically less than few percent even when the steady - state is characterized by merely few tens of molecules .
| 26,928 | 380 |
graphene as the basis of a new generation of electronics@xcite has been the center of much attention in the last years , and devices based on nanostructured graphene have been put forward . the most generic form of nanostructured graphene is graphene nanoribbons ( gnr),@xcite and other structures , such as graphene anti - dot lattices@xcite , can be viewed as networks of them . gnrs are potential candidates for molecular wires with tailored conductance properties . for graphene - based nanostructures the edges and their passivation , as well as defects inside the structure , can play crucial roles for the transport properties.@xcite however , characterization of edge passivation or structural / chemical defects is challenging especially after device fabrication . raman spectroscopy@xcite can give information about defects on large areas of the sample , while tip - enhanced raman spectroscopy ( ters)@xcite have been used in combination with stm on gnrs . however , raman studies involve averages over larger areas ( > 10 nm ) , and does not yield information about the impact of vibrations on transport . in that aspect inelastic electron tunneling spectroscopy ( iets ) serves as a way of performing non - destructive characterization yielding vibrational / phonon fingerprints of a range of defects . in order to interpret iets experiments , theoretical modeling of the inelastic signals in the electronic current due to electron - phonon ( e - ph ) scattering is needed . gnrs have been fabricated using different strategies including lithographic techniques,@xcite chemical synthesis,@xcite epitaxial growth@xcite , and longitudinal unzipping of carbon nanotubes.@xcite furthermore , several groups have succeeded in atomically precise bottom - up fabrication of armchair gnrs ( agnr)@xcite , chiral gnrs,@xcite and agnr hetero - junctions@xcite grown on metal surfaces . experimentally , the vibrational properties have been investigated by raman spectroscopy and the electronic structure has been mapped out by stm , angle - resolved ( two - photon ) photo - emission and high - resolution electron energy loss spectroscopy.@xcite signatures of phonon excitation were observed by stm in the differential conductance spectroscopy performed at the zigzag termini state of agnrs adsorbed on au(111 ) , and these signatures were shown to be sensitive to modifications in the local atomic geometry.@xcite agnrs have also been lifted up from the weakly bonding au(111 ) surface with the tip of a stm enabling measurements of the voltage - dependent conductance in suspended configurations.@xcite from the theoretical side density - functional theory ( dft ) has been used to investigate the stability of structural and chemical reconstructions of gnr edges,@xcite together with the transport and band - gap engineering.@xcite the vibrational properties and phonon band structure have been calculated with empirical potentials@xcite and dft.@xcite in addition , there have been theoretical predictions@xcite of the raman spectrum , in good agreement with experiments.@xcite for a finite agnr the role of zigzag termini states have been studied theoretically , comparing dft to the many - body hubbard model.@xcite inspired by the recent lifting experiments by koch , @xcite we here investigate theoretically the signals of e - ph scattering in the conductance of long gnrs between metal electrodes . our aim is two - fold . first , we want to address the role phonon scattering in the transport characteristics of pristine gnrs . second , we wish to compute detailed iets for different gnrs under varying charge carrier concentrations and explore how different types of realistic defects may modify the iets and thus possibly be directly probed in transport measurements . we focus on the two most generic edge types , namely armchair ( agnr ) and zigzag ( zgnr ) , and pay attention to the effects of spin polarization in the latter case . in actual experiments the substrate or an applied gate potential control the fermi level @xmath0 in the ribbons . to address this variability we scan @xmath0 using a numerically effective scheme enabling fast calculations of the iets.@xcite we find that the agnr generally display two robust iets signals around @xmath1 and @xmath2 mv corresponding to the d- and g - modes of raman spectroscopy and that a dehydrogenated dimer at the edge should further leave a clear defect signal at around @xmath3 mv . for the zgnr we find that the spin polarization breaks the mirror symmetry around the middle of the ribbon resulting in iets signals from a range of modes around the d- and g - mode energies . for both agnr and zngr defects which break the planar symmetry of ribbons allows for contributions to the iets from out - of - plane phonon modes . the paper is organized as follows . first we discuss our atomistic model setup for the density functional and electron transport calculations , and outline the approach for the iets simulations . in sec . iii we present our results for pristine agnr and zgnr and relate their transport properties and iets to the band structures . in sec . iv we turn to the defected systems by considering realistic possibilities of defects in the edge passivation , backbone bonding motifs , and presence of adatoms . finally , a summary and our conclusions are presented in sec . we calculate the electronic and vibrational structure from dft using the academic codes siesta / transiesta.@xcite we employ the generalized gradient approximation ( gga ) for the exchange - correlation functional,@xcite a single - zeta polarized ( szp ) basis set for the carbon and hydrogen atoms , and use a cut - off energy of 400 - 500 ry for the real - space grid . these choices , balancing accuracy and computational cost , provide a good description to investigate trends and general behavior of the substantial number of systems considered in this work . the vibrational degrees of freedom , calculated by diagonalization of the dynamical matrix , and the e - ph couplings are extracted from finite differences as implemented in the inelastica code.@xcite the armchair and zigzag gnrs considered here are shown in fig . [ fig : cleangnr ] . we adopt the usual two - probe setup with the device region ( @xmath4 ) coupled to left ( @xmath5 ) and right ( @xmath6 ) electrodes with all electronic matrix elements expressed in a local basis set . the primitive unit cell of the agnr ( zgnr ) consists of 18 ( 10 ) atoms and in our calculations this unit cell is repeated 10 ( 18 ) times in the transport direction to form the scattering regions illustrated in fig . [ fig : cleangnr](a , e ) . the electrode couplings @xmath7 are included on the two first / last unit cells before folding onto @xmath4 . in our treatment a subset of atoms in @xmath4 is allowed to vibrate . we fix this dynamical region , restricted by the condition that the e - ph couplings are fully included inside @xmath4 , to the 4 and 6 central unit - cells for the agnr and zgnr , respectively . the corresponding e - ph couplings used to calculate the inelastic electron transport are thus expressed in the center 6 unit - cells for the agnr and 8 unit - cells for the zgnr . the convergence of our results with the size of the dynamical region is addressed below . we generally consider nanoribbons that are suspended between two metallic leads . in the case of the lifting experiments,@xcite these would correspond to the metal sample surface and the stm tip . we wish here to focus on the action inside the gnrs and put aside the possible complications due to the detailed electronic structure of the metals , and the metal - gnr interface in particular . to this end we introduce a simple model of the metal electrodes without substantial electronic features : we use semi - infinite gnrs with highly broadened states ( effectively smearing out energy gaps ) . in practice this is done by adding a finite numerical imaginary part @xmath8 to the energy argument in the electrode recursion calculation.@xcite this scheme ensures that the phonon effects originate from the gnrs themselves and not from details of the metal - gnr interface , which is generally unknown in the stm experiments . the electronic band structures for the infinite ribbons , along with the transmission and density of states ( dos ) are shown for @xmath9 ev in fig . [ fig : cleangnr](b , c , d ) and fig . [ fig : cleangnr](f , g , h ) for agnr and zgnr , respectively . we note that the broadened transmission spectrum [ fig . [ fig : cleangnr](d ) ] is quite consistent with the experimentally reported differential conductance curves for agnr.@xcite the electronic states involved in the transport are shown in fig . [ fig : channels ] in terms of the transmission eigenchannels@xcite in the valence and conduction bands of the agnr and zgnr . their spatial symmetry play a significant role for the selection rules involved in the inelastic scattering as discussed later . in principle , the electronic structure should be evaluated at finite bias . however , without a detailed model of the connection to the metal electrodes ( where an important part of the voltage drop will take place ) and for sufficiently long systems ( in which the electric field will be small ) , it is reasonable to use the zero - voltage electronic structure and to simply assume a symmetric voltage drop over the two identical , idealized device - electrode interfaces . more specifically , in the following we consider that the chemical potentials of the electrodes move according to the applied bias voltage @xmath10 and possibly an applied gate voltage @xmath11 ( mimicking actual doping or electrostatic gating that modify the charge carrier concentration in an experimental setup ) according to @xmath12 . for a device strongly coupled to the electrodes , a coupling between the electron current @xmath13 and a phonon mode @xmath14 ideally shows up at zero temperature as a step discontinuity in the differential conductance when the inelastic phonon emission process becomes energetically allowed , that is , when the chemical potential difference exceeds the quantum of vibrational energy , @xmath15 . thus , around the emission threshold the electronic states involved in the scattering process are those at @xmath16 and @xmath17 . the iets signal , conventionally expressed as the ratio between the second and first derivatives of the current with respect to the voltage , @xmath18 is calculated by considering the e - ph coupling as the perturbation on the current , evaluated using the nonequilibrium green s functions ( negf ) . in the so - called lowest order expansion ( loe ) the inelastic part of the differential conductance can be written as,@xcite @xmath19 where summation over the vibration index @xmath14 is assumed . @xmath20 and @xmath21 are the `` universal '' ( system - independent ) functions that depend on the applied bias @xmath10 , phonon energy @xmath22 and the temperature @xmath23 . assuming the electronic and phononic distribution functions are given by the fermi - dirac and bose - einstein distributions , respectively , their analytical expressions can be written as : @xmath24,\nonumber\end{aligned}\ ] ] where @xmath25 is the conductance quantum , @xmath26 is the fermi - dirac function , @xmath27 , and @xmath28 denotes the hilbert transform . the signal amplitudes @xmath29 and @xmath30 of the symmetric and antisymmetric signals in the differential conductance are even and odd in bias , respectively . for a symmetric structure the asymmetric signal vanishes in the wide - band approximation ( loe - wba)@xcite . however , this is not guaranteed in the more general treatment employed here,@xcite where the energy dependence of the electronic structure is explicitly taken into account . the amplitudes @xmath29 and @xmath30 are expressed in terms of electronic structure quantities and e - ph couplings,@xcite @xmath31+{\rm i m } b_\lambda , \label{eq : gamma } \\ \kappa_\lambda = & 2{\rm re } b_\lambda , \label{eq : kappa}\end{aligned}\ ] ] where @xmath32 is defined as @xmath33 . \label{eq : ic4ag}\end{aligned}\ ] ] in the above , @xmath34 denotes the e - ph coupling matrix for mode @xmath14 , @xmath35 the retarded / advanced unperturbed green s functions , and @xmath36 the spectral density matrices for left / right moving states with the time - reversed version @xmath37 . the purely electronic quantities are thus being evaluated at the chemical potentials of the left / right electrodes corresponding to the excitation threshold for each vibration . we compute @xmath34 with the finite - difference scheme of inelastica taking the vacuum energy as a common reference ( in absence of real metal leads to pin the fermi energy).@xcite in the localized atomic basis set of siesta all the above quantities are matrices defined in the electronic space corresponding to region @xmath4 . the second derivatives of the universal functions in eqs . ( [ eq : sym])-([eq : asym ] ) are sharply peaked around the phonon threshold . for this reason the coefficients @xmath38 and @xmath30 can be considered voltage - independent with their values computed exactly at the threshold . due to the computational efficiency of the loe scheme described above we are able to evaluate the iets on a fine grid of gate voltages @xmath11 spanning a large range of relevant values between valence and conduction bands of the gnrs . now we first turn to the iets results of the two pristine ( clean ) ribbons , and in the following section to the impact of selected defects in the iets . as our main system we focus on the agnr systems directly relevant for the lifting experiments.@xcite the results for the zgnr are provided mainly as comparison and to look into the role of chirality and in particular effects rooted in spin polarization , and thus we now discuss these separately . as representative of the agnr class we have investigated a pristine agnr with a width of @xmath39 dimers ( 7-agnr ) corresponding to a c - c edge distance of 7.5 ( see fig . [ fig : cleangnr ] ) . it presents a direct semi - conducting band gap @xmath40 due to the lateral confinement and can be classified as a `` large - gap ribbons '' since @xmath41 is an integer in the relation @xmath42.@xcite we obtain @xmath43 ev at the present level of approximation ( dft - gga and szp basis set ) , as seen from the electronic band structure shown in fig . [ fig : cleangnr](b ) . this value is smaller than those estimated experimentally ( @xmath44 2.3 - 2.6 ev for a flat agnr on au(111 ) @xcite and @xmath45 ev for an agnr suspended between surface and stm - tip@xcite ) due to the underestimation of electron - electron interaction@xcite which plays an more important role in quasi one - dimensional gnrs compared to pristine graphene . dielectric screening from the substrate also influences significantly the actual gap size : a band gap of @xmath46 ev for a 7-agnr was found to be lowered to @xmath47 ev on a hexagonal boron - nitride ( hbn ) substrate using gw calculations,@xcite similar to the lowering calculated for a 7-agnr on au(111).@xcite in general we expect that underestimation of band gaps would mainly amount to a simple scaling the fermi level position within the gap . we first discuss the effect of the finite size of the dynamical region in our treatment . figure [ fig : convclean](a ) shows how the iets signals for the agnr ( at fixed gate voltage @xmath48 v ) vary as a function of the size of the dynamical region , ranging from 1 to 6 unit cells . for easy comparison , the data are normalized by the number of vibrating unit cells . as the signal amplitudes in this representation are roughly constant we conclude that the absolute iets simply scale linearly with the active e - ph coupling region . consequently , the magnitudes in iets may thus provide insight into the active scattering region in actual experiments . further , as we find that both iets amplitude and shape is well converged with 4 vibrating unit cells , we fix the dynamical region to this size in the following analysis . the computed iets signals for the agnr as a function of varying gate voltage are shown in fig . [ fig : ietsclean](a ) as a density plot . specific iets spectra at selected gate voltages are shown in fig . [ fig : ietsclean](c ) for both the intrinsic part ( temperature broadening at @xmath49 k ) as well as that one would observe employing the experimental lock - in technique ( additional broadening due to a modulation voltage of @xmath50 mv ) . we find that for the agnr there are generally two well - defined iets signals appearing around @xmath1 and @xmath2 mev , corresponding to the d- ( ring breathing ) and g- ( e@xmath51 phonon ) modes , respectively , also observed in raman spectroscopy.@xcite . the d - signal also has a shoulder with a local maximum at @xmath52 mev with contributions from several modes . these three distinct features are indicated with vertical lines in fig . [ fig : ietsclean](a , c ) . shifting @xmath0 inside the gap region with a relatively small gate voltage @xmath53 v does not affect the iets appreciably . however , when @xmath0 comes close to the conduction band of the agnr the signal increases by a factor of five and a small peak - dip feature appear similar to the one reported for gated benzene - dithiol molecular contacts.@xcite upon further gating into the conduction band the iets signals undergo a sign reversal ( from peaks to dips ) as the transmission increases beyond approximately @xmath54 for the involved channels.@xcite similar effects are also found by gating into the valence band of the agnr . we can easily identify the most important vibrational mode vectors @xmath55 for the iets from the two amplitudes @xmath56 and @xmath57 given in eqs . ( [ eq : gamma])-([eq : kappa ] ) . these modes can further be analyzed in terms of the phonons in the infinite agnr . to do so we introduce the measure @xmath58 representing the overlap between modes in the finite dynamical cell and the phonon band modes weighted by the size of the iets signal , @xmath59 where @xmath60 is the phonon band mode indexed by @xmath61 , and @xmath62 is the modes in a finite @xmath63 primitive cell long dynamical region index by @xmath14 . the projections @xmath64 are depicted as widths of the phonon bands in fig . [ fig : agnrphononband](a ) , where the red , green and blue colors refer to modes with frequencies in the ranges @xmath65 mev , @xmath66 mev , and @xmath67 mev , respectively . in total four bands contribute to the iets signal corresponding to the four signals seen in the intrinsic part of the iets spectrum in fig . [ fig : ietsclean](c ) . the corresponding @xmath68-point phonon modes inside the primitive cell for the infinite ribbon are shown in fig . [ fig : agnrphononband](b - e ) . we next turn to our results for the pristine zgnr shown in fig . [ fig : cleangnr](e ) . it has a width of @xmath69 zigzag `` chains '' ( 4-zgnr ) corresponding to a c - c edge distance of 7.26 . the breaking of sublattice symmetry for the zgnr and lack of pseudo - phase result in different selection rules for the matrix elements and difference in for example raman signals.@xcite the zgnr generally presents spin - polarized edge states exhibiting a small band gap at the dft level,@xcite in our case @xmath70 ev ( we note that this gap disappears in simpler tight - binding descriptions@xcite or spin - degenerate dft calculations ) . the spin - polarized edge states play the major role for the conduction , see the spin - down eigenchannels visualized in fig . [ fig : channels](e - h ) . since the edge states break the mirror symmetry with respect to the middle of the ribbon , there are fewer symmetry - forbidden inelastic transitions between the scattering states for the zgnr . thus , we expect a wider range of modes to contribute to the iets signal as compared to the agnr case . indeed this is in agreement with the findings shown in fig . [ fig : convclean](b ) and fig . [ fig : ietsclean](b , d ) . the greater number of modes contributing to the iets for the zgnr results in broader signals with similar magnitudes as compared to the iets for agnr . as for the agnr case the iets signal is well converged with a dynamical region consisting of 6 vibrating unit cells [ fig . [ fig : convclean](b ) ] . for zgnrs the ring breathing is forbidden by symmetry , thus the iets is generally characterized by transverse and longitudinal modes . to explore the impact of spin - polarization on the zgnr - iets we compare in fig . [ fig : nospin ] the results from both spin - degenerate and spin - polarized calculations . without gate voltage ( @xmath71 v ) the iets display opposite signs due to the spin - induced gap . only a single peak contributes to the spin - degenerate iets while several peaks contribute to the spin - polarized iets . even if the zgnr is tuned by @xmath72v to become metallic and the two treatments then show the same overall sign in iets , the spin - polarized iets persists to show a much richer structure . this difference suggests that iets could be a way to indirectly observe spin - polarized edge states . projecting the modes contributing to the iets onto the phonon band modes further underlines how several bands with different symmetries contribute to the spin - polarized iets , while only a couple of bands contributes to the spin - degenerate iets , see fig . [ fig : bandzgnr ] . again we use eq . ( [ eq.weightf ] ) for this characterization , where the overlap for @xmath48 v corresponds to the red color and the overlap for @xmath72 v corresponds to the difference between the blue and red color in fig . [ fig : bandzgnr ] , respectively . it is clear that spin - polarization permits more modes to contribute to the iets . in contrast to the spin - degenerate case , where the symmetric electronic states ( with respect to the middle of the ribbon ) only can couple to the symmetric vibration modes , the symmetry lowering of the electronic states by spin - polarization opens up also for scattering also via odd modes . in this section we address the modification and new signals in iets that arise due to various defects in the gnr . regardless of the fabrication method , defects will inevitable occur . for example , if the agnrs are synthesized from a precursor molecule , involving heating and dehydrogenation , as reported by cai @xcite and blankenburg , @xcite there is a chance that the reaction is incomplete and some of the c - c bonds between the precursor molecules do not form . also there is a chance that a part of the final agnr will have dehydrogenated edges or are passivated by two hydrogen atoms . finally , defects may be introduced on purpose by locally dosing a high current from the tip of a stm.@xcite ( color online ) electronic properties of the agnr structures shown in fig . [ fig : xyzagnr ] . the total transmission is shown with black lines . the ratio @xmath73 , where @xmath74 is the transmission originating from the most transmitting eigenchannel is shown with green dashed lines ( this ratio gives a lower bound to the number of contributing eigenchannels ) . the dos for the c atoms in the dynamical region is shown with red lines ( offset by 3 units ) . ] . in fig . [ fig : xyzagnr ] we show the structures of pristine agnr along with 8 different defect configurations which we have considered . these include four defects in the edge passivation as follows : a single edge side with an extra hydrogen atom [ ` 1h - edge ` , fig . [ fig : xyzagnr](b ) ] , two edge sides with each an extra hydrogen atom [ ` 2h - edge ` , fig . [ fig : xyzagnr](c ) ] , one hydrogen replaced by a fluorine atom [ ` 1f - edge ` , fig . [ fig : xyzagnr](d ) ] , and a dehydrogenated edge with 4 hydrogen atoms removed from each side [ ` 8h - free ` , fig . [ fig : xyzagnr](e ) ] . we have also considered defects in the atomic structure in the form of one , two , or four broken c - c bonds [ ` 1c - broken ` , ` 2c - broken ` , ` 4c - broken ` , fig . [ fig : xyzagnr](f)-(h ) ] as well as a cu adatom on the agnr [ ` cu - adatom ` , fig . [ fig : xyzagnr](i ) ] . for all these systems the entire dynamical region was relaxed , i.e. , the parts of the agnrs shown in fig . [ fig : xyzagnr ] . defects may influence the iets signal in two ways . first , a defect can have a direct impact by changing the vibrational degrees of freedom . in order for the change in the vibrational spectrum to give a signal in the iets , the new vibrations must couple to the current , and preferably have frequencies which do not coincide with ones already giving iets signals for the pristine ribbons . second , a defect can substantially change the electronic structure and thereby have an impact on the e - ph couplings associated with the active modes or even the transmission eigenchannels of the pristine ribbons , e.g. , changing a peak in the iets to a dip ( and vice versa ) or enhancing asymmetric contributions via eq . ( [ eq : asym ] ) . the electronic properties of the pristine agnr is shown in fig . [ fig : transagnr](a ) . the carbon dos projected to the device region ( red curve ) reveals a gap as expected from the band structure [ fig . [ fig : cleangnr](b ) ] , which is significantly broadened from the coupling to the metallic electrodes . the two valence and two conduction bands in the considered energy range naturally explain that the total transmission ( black curve ) is bound below a value of 2 . further , the ratio @xmath75 ( green dashed line ) , measuring the minimum number of contributing channels where @xmath74 is the transmission of the most transmitting eigenchannel , shows that both channels play a role for the transport , at least away from the edges of the direct band gap . measurements of shot noise may provide insights into this effective number of conductance eigenchannels.@xcite we can now discuss how the different defects modify the electronic properties . from fig . [ fig : transagnr](b)-(i ) we notice that not all defects change the elastic transmission , and furthermore , a change in elastic transmission needs not be unique for a specific defect . instead , iets may provide a additional fingerprint in the current that can be used to identify the type of defect . figure [ fig : ietsdefectagnr ] shows the computed iets as a function of gate voltage for the 8 different defects . as for the clean structure , the two peaks at @xmath1 and @xmath2 mev corresponding to the d- and g- raman modes are dominant for a range of gate values for all the structures . another feature , which is present in all the systems , is the appearance of several signals close to the band onsets . in the following subsections we discuss in more detail the transport characteristics with the different types of defects in agnrs . considering defects in the edge passivation [ fig . [ fig : xyzagnr](b - e ) ] the gap in the transmission is essentially unchanged [ fig . [ fig : transagnr](b - e ) ] , except for the ` 1h - edge ` structure where a zero - energy resonance appears in the dos and transmission [ fig . [ fig : transagnr](b ) ] . this new peak can be attributed to tunneling via a mid - gap state which appears due to the local breaking of sub - lattice symmetry.@xcite thus , if a h atom is added to the neighboring c atom [ ` 2h - edge ` , fig . [ fig : xyzagnr](c ) ] the peak disappears [ fig . [ fig : transagnr](c ) ] . the addition of one or two h atoms on the same side also results in the closing of one transmission channel between the valence and conduction bands as shown in fig . [ fig : transagnr](b , c ) . concerning the vibrational degrees of freedom , the addition of extra hydrogen to the edge results in new vibrational modes around 330 mev for ` 1h - edge ` and around 343 and 353 mev for ` 2h - edge ` , clearly outside the bulk phonon band ( ranging up to @xmath76 mev ) of pristine agnr.@xcite comparing the iets in fig . [ fig : ietsdefectagnr](a - c ) we find that only ` 1h - edge ` gives a signal which differs significantly from the pristine case . figure [ fig : ietsdefectagnr](k ) shows specific iets for selected gate voltages for ` 1h - edge ` . here , at @xmath77v ( top green curve ) we see how new signals appear at large voltages : for positive bias polarity two signals appear at 330 and 365 mev , respectively , while for negative bias polarity only an asymmetric signal around @xmath78 mev is present . the signal at 330 mev is due to vibrations of the @xmath79 [ fig . [ fig : armmodes](b ) ] , while the signal at 365 mev [ fig . [ fig : armmodes](a ) ] is due to the h atom on the neighboring c atom . further , the amplitude of the signals around 169 and 196 mev is also found to depend on bias polarity . gating onto the zero - energy resonance for ` 1h - edge ` the iets signal [ middle red curve in fig . [ fig : ietsdefectagnr](k ) ] is dominated by large asymmetric signals for low energy vibrations due to the contribution from @xmath30 and eq . ( [ eq : asym ] ) . we note that @xmath30 changes sign with bias polarity for this approximately left - right symmetric structure . this can be seen from the red iets curve in fig . [ fig : ietsdefectagnr](k ) which is roughly an odd function of the bias voltage . in close proximity of the zero - energy resonance a characteristic `` x - shape '' is observed in the gate - dependent iets , while away from it the signals approach that of the pristine agnr [ fig . [ fig : ietsdefectagnr](b ) ] . substituting a h atom with a f atom ( ` 1f - edge ` ) is seen to have virtually no effect in the iets of fig . [ fig : ietsdefectagnr](d ) . this suggests that a significant change in the chemical composition directly involving the @xmath80-electronic system is required in order to obtain a signal although the vibrations are influenced by the heavier passivation . such a significant change in the passivation occurs for instance by removing four h atoms on each side ( ` 8h - free ` ) , giving rise to four very narrow peaks in the dos around the conduction band , [ fig . [ fig : transagnr](e ) ] . these correspond to very localized dangling - bond states on the dehydrogenated dimers and therefore do not show up in the transmission . however , the dehydrogenated edges give rise to localized vibrations outside the range of the pristine vibrational spectrum.@xcite the in - phase vibration of the dehydrogenated c dimers at the armchair edges [ fig . [ fig : armmodes](f ) ] gives rise to an extra iets peak at 244 mev [ fig . [ fig : ietsdefectagnr](l ) ] matching the h - free mode measured by raman.@xcite we find that this signal is robust as it appears in the whole range of gate values . when gating into to the valence band a new signal appears around 43 mev [ @xmath81 v in fig . [ fig : ietsdefectagnr](e ) ] originating from a low energy edge vibration [ fig . [ fig : armmodes](g ) ] . the electronic transmission in gnrs is mediated by the carbon @xmath80 system . thus if a c - c bond fails to be formed during gnr synthesis or if it is broken again at a later stage , a large effect can be expected for the electronic conduction properties . this impact is indeed revealed in fig . [ fig : transagnr](f - h ) . breaking one or two bonds results in the formation of two in - gap states which , broadened by the electrodes , make the gap appear smaller . the iets signals for the ` 1c - broken ` and ` 2c - broken ` in fig . [ fig : ietsdefectagnr](f , g , m ) have the same two signals at @xmath1 and @xmath2 mev as for the clean ribbon . however , the relative amplitudes are interchanged such that the `` d''-peak is now slightly more intense than the `` g''-peak . breaking four c - c bonds [ ` 4c - broken ` , fig . [ fig : xyzagnr](h ) ] , resulting in constrictions of single c - c bonds , totally alter the dos which is now dominated by three sharp peaks as seen in fig . [ fig : transagnr](h ) . the corresponding iets signals are shown in fig . [ fig : ietsdefectagnr](h , n ) . in the proximity of the zero - energy resonance a broad range of signals at low vibrational energies appears ( red curve in panel n ) as well as a characteristic `` x - shape '' in the gate plot ( panel h ) similar to that of ` 1h - edge ` . gating away from the resonance we observe two additional robust iets signals at @xmath82 and @xmath83 mev resulting from vibrations localized at the defect [ fig . [ fig : armmodes](d , e ) ] . transition metals are typically used for growth of graphene or as a substrate for the bottom - up synthesis of gnrs . thus it is of interest to consider the effect of adatoms of this type on gnrs . a cu adatom on graphene adsorbs preferentially in the on - top position.@xcite however , positioning cu such that it breaks the axial symmetry of our agnr , we find that it is most stable in a hollow site at the edge [ ` cu - adatom ` , fig . [ fig : xyzagnr](i ) ] . the dos and transmission in fig . [ fig : transagnr](i ) reveal a n - type doping effect shifting @xmath0 close to the conduction band while leaving the two transmission channels inside the gap relatively intact . for the pristine gnr the e - ph couplings of the out - of - plane vibrations are suppressed due to the symmetry of the @xmath80-orbitals . however , around the onset of the conduction band the iets signals in fig . [ fig : ietsdefectagnr](i , o ) is dominated by large asymmetric signals with significant contributions from out - of - plane phonons . these modes come into play due to breaking of the planar symmetry by the adatom . also note that by gating of @xmath0 within the gap these signatures of the adatom disappear , cf . the lower black curve in fig . [ fig : ietsdefectagnr](o ) . let us next consider a series of defects for the zigzag graphene nanoribbon . in fig . [ fig : structdefectzgnr ] we show the atomic structures of pristine zgnr along with 8 different defect configurations . we consider the following defects in the edge - passivation : a single edge with an extra hydrogen [ ` 1h - edge ` , fig . [ fig : structdefectzgnr](b ) ] , one hydrogen is replaced by either a f atom [ ` 1f - edge ` , fig . [ fig : structdefectzgnr](c ) ] , an oh group [ ` 1oh - edge ` , fig . [ fig : structdefectzgnr](f ) ] , or a @xmath84 group [ ` 1no_2-edge ` , fig . [ fig : structdefectzgnr](g ) ] . we also consider defects in the form of a cu adatom [ ` cu - adatom ` , fig . [ fig : structdefectzgnr](d ) ] or a li adatom [ ` li - adatom ` , fig . [ fig : structdefectzgnr](e ) ] . finally , we also study the effect of a structural defect in form of a 57 reconstruction [ ` r57 ` , fig . [ fig : structdefectzgnr](h ) ] and a substitutional defect where a c atom next to the edge is replaced by a si atom [ ` si - substitute ` , fig . [ fig : structdefectzgnr](i ) ] . for all these systems the entire dynamical region was relaxed , i.e. , the parts of the zgnrs shown in fig . [ fig : structdefectzgnr ] using spin - polarized treatments . the spin degrees of freedom @xmath85 generalizes @xmath86 and @xmath87 [ eqs.- ] corresponding to two independent spin channels , which in general can have quite different amplitudes and even opposite sign . the observable iets would simply be the sum of these two components @xmath88 . similar to the agnr case , the electronic properties in the device region with the different impurity configurations for the zgnr , now spin resolved , are summarized in fig . [ fig : transdefectzgnr ] . the iets of pristine zgnr was already discussed in sec . [ subsec : cleanzgnr ] and below we continue describing the iets fingerprints for the various defects . as commented above , the broader iets signals of pristine zgnr [ figs . [ fig : convclean]-[fig : ietsclean ] ] ( as compared with agnr ) can be understood from the breaking of the axial mirror symmetry and hence fewer symmetry - forbidden inelastic transitions . these broader signals may in general make the detection of defect signatures more difficult . for ` 1h - edge ` [ fig . [ fig : transdefectzgnr](b ) ] the iets resembles that of the pristine zgnr [ fig . [ fig : transdefectzgnr](a ) ] inside the gap . however , gating into the valance band [ black curve in fig . [ fig : transdefectzgnr](k ) ] the edge states start to extend into the middle of the ribbon , partially restoring mirror symmetry , and thus resulting in part of the pristine zgnr signals to disappear . here an extra signal appear due to edge - modes in the frequency range 194 to 199 mev with the most contributing mode at 196 mev as shown in [ fig . [ fig : zigmodes](a ) ] . the resulting iets signal can clearly be seen in the bottom curve in fig . [ fig : ietsdefectzgnr](k ) . as for the agnr substituting a hydrogen with a fluorine atom has a very limited effect on the electronic properties and the iets signal . substituting a hydrogen with an oh group , according to fig . [ fig : transdefectzgnr](f ) and ( o ) , have only a small effect on the spin down electrons , while it shrinks and add additional structure to the gap for the spin up electrons . for the spin up electrons there is a small peak inside the gap which gives rise to a large asymmetric iets signal around @xmath89v in fig . [ fig : transdefectzgnr](o ) lower curve , compared to the pristine case . the most contributing mode to the asymmetric iets signal is shown in fig . [ fig : zigmodes](b ) . however , there is no clear signature of the oh group itself . in the same manner the substitution with a @xmath84 group removes the gap in the electronic properties without leaving any direct fingerprint of the @xmath84 group in the iets signal . as for the agnr we consider the effect of adatoms . for the cu adatom the transport gap shrinks for the spin up electrons while there is an in - gap peak for the spin down electrons , cf . [ fig : transdefectzgnr](m ) . thus , for some gate values the iets signals reflect that the spin down electrons will back scatter while the spin up electrons will be forward scattered , and the observed signal is then the sum of these contributions . for a gate value of @xmath89 v , the iets signal is dominated by spin down electrons . due to the finite width of the in - gap peak , in the spin down transmission , the low frequency phonons ( @xmath90 mev ) give rise to back scattering while the high frequency phonons ( @xmath91 mev ) result in forward scattering . thus , the low and high energy signals have different signs as can be seen from fig . [ fig : ietsdefectzgnr](l ) . interestingly , the low energy signal primarily consists of symmetric contributions from out - of - plane modes [ fig . [ fig : zigmodes](c ) ] . replacing the cu adatom with li , the transmission and dos , shown in fig . [ fig : transdefectzgnr](e , n ) , reveals a spin dependent n - type doping effect , where @xmath0 is shifted the most for spin down . however , no in - gap peak is seen as for cu and the iets show no clear signature of the li atom . the formation of a ` r57 ` reconstruction results in peaks in the dos in the device region , just above @xmath0 for spin up [ fig . [ fig : transdefectzgnr](h ) ] and just below @xmath0 for spin down [ fig . [ fig : transdefectzgnr](q ) ] . the ` r57 ` breaks the symmetry both in the vibrational and electronic structure allowing for iets signals from a wider range of vibrations , resulting in broader peaks , as seen from fig . [ fig : ietsdefectzgnr](h ) and fig . [ fig : ietsdefectzgnr](n ) . one of the contributing modes is localized at the border between the pentagon - ring and middle of the ribbon at @xmath92 [ fig . [ fig : zigmodes](d ) ] . this localized mode yield a relatively small signal compared to the other signals , however , contrary to the other modes the localized mode is not expected to be broadened if the coupling to phonons away from the dynamical region is taken into account . the breaking of symmetry in the electronic structure also give rise to difference signals for the two bias polarities . substituting a carbon with a silicon atom leads to an out - of - plane buckling , see fig . [ fig : structdefectzgnr](i ) . however , both silicon and carbon have an @xmath93 electronic structure , and the electron transmission is basically similar to the pristine . on the other hand , the buckling give rise to low energy peaks in the iets signal originating from the e - ph coupling to the out - of - plane modes [ fig . [ fig : zigmodes](e ) ] . gating close to the band edge of the conduction band gives rise to different sign of the signals at low and high vibrational energies , as seen from the top curve in fig . [ fig : ietsdefectzgnr](o ) . in summary , we have investigated iets signals in symmetrically contacted armchair and zigzag graphene nanoribbons , considering both pristine as well as a selection of defected configurations under varying charge carrier conditions . for the clean agnr inelastic tunneling gives rise to two distinct peaks in the iets spectrum at @xmath1 mv and @xmath2 mv corresponding to the d - and g - modes of raman spectroscopy , respectively . by connecting the iets signals to the phonon band structure , we have clarified how only a single band contributes to the `` g - mode '' while three bands contribute to the broader `` d - mode '' . concerning defects in agnrs we have shown how some leave iets unchanged while others give clear signals . for instance , adding an extra hydrogen atom to a single edge side gives a clear signal for some gate values . this signal can be removed by adding another hydrogen atom to the neighboring edge side because the sub - lattice symmetry is restored . further , exchanging a single hydrogen atom with a fluorine atom in the passivation does not result in any change in both the elastic and inelastic tunneling . however removing 8 hydrogen atoms leaving part of the edge on each side without passivation , gives a clear robust signal throughout the investigated gate values . the signal , due to the vibration of the carbon dimers at the edge , has an energy around @xmath3 mev making it easy to detect since it is outside the vibrational spectrum of the pristine ribbon . breaking of one or two c - c bonds turns out to interchange the relative intensity of the `` g''- and `` d''-peaks . breaking 4 c - c bonds gives rise to signals caused by the defect tilted out of plane . lifting the symmetry of the @xmath80-electrons by adding a cu - adatom allows the out - of - plane modes to contribute . for the zgnr we find relatively broader iets signals especially in the absense of a large gate voltage ( @xmath94 v ) . importantly , this is a consequence of the breaking of the axial mirror symmetry in the ribbon due to the presence of spin - polarized edges . thus , by comparing to spin - degenerate calculations , we suggest that iets can give an indirect proof of spin - polarization in zigzag ribbons . on the other hand , the broader iets features may make it difficult to identify the different defect signals reported in this paper . the presence of a ` r57`-reconstruction also broadens the iets by breaking both the electronic and vibrational symmetry . substituting a carbon atom with a silicon atom makes the ribbon buckle , breaking the planar symmetry , allowing the out - of - plane modes to contribute to the iets . this suggests that iets in principle could be used to gain information of the curvature of gnrs and other graphene - based structures . finally , as an outlook we note that here we presented calculations on long , symmetrically contacted systems where there is a significant overlap with both metallic electrodes . it would be interesting to extend such a study also to the asymmetric situation where a point tunnel contact is made to one end resembling , say , the coupling a stm tip.@xcite
|
inspired by recent experiments where electron transport was measured across graphene nanoribbons ( gnr ) suspended between a metal surface and the tip of a scanning tunneling microscope [ koch _ et al_. , nat .
nanotechnol .
* 7 * , 713 ( 2012 ) ] , we present detailed first - principles simulations of inelastic electron tunneling spectroscopy ( iets ) of long pristine and defected armchair and zigzag nanoribbons under a range of charge carrier conditions . for the armchair ribbons we find two robust iets signals around 169 and 196 mv corresponding to the d- and g - modes of raman spectroscopy as well as additional fingerprints due to various types of defects in the edge passivation . for
the zigzag ribbons we show that the spin state strongly influences the spectrum and thus propose iets as an indirect proof of spin polarization .
| 13,750 | 242 |
the discovery of the acceleration of the universe by the measurements of the luminosity - redshift relation for type ia supervonae @xcite , combined with the observations of the anisotropies in the cosmic microwave background ( cmb ) @xcite , confirmed that about 70% of the density of the universe is made up of an unknown form of energy . this component , usually referred to as dark energy " , is unclumped and smoothly distributed , with an exotic property that it has negative pressure to cause the expansion of the universe to accelerate . the simplest candidate of dark energy is a non - vanishing , positive cosmological constant @xcite . we have to , however , explain why it is non - zero but vanishingly small ( @xmath0 ) : possibly we can resort to some yet unknown fundamental symmetry which makes the cosmological constant vanishingly small but non - zero , or we might invoke an anthropic consideration that the observed small value of the cosmological constant allows life and that is why we observe it @xcite ) different vacua , and some of them may be suitable for some kind of intelligent observers like us . ] . an alternative form of dark energy is a slowly rolling scalar field , called quintessence , which has not yet relaxed at its ground state @xcite . for recent years , there have been considerable developments in the dynamics of the quintessence field . for example , in the context of so - called tracker solutions @xcite , the simplest case arises for a potential of the form @xmath1 , where @xmath2 is positive and @xmath3 is an adjustable constant . by choosing @xmath3 suitably , it is possible to make a transition from an early matter - dominated universe to a later quintessence - dominated universe , free from fine tuning of the initial conditions . another interesting possibility , @xmath4-essence , is to introduce a non - canonical kinetic term for the scalar field @xcite , which makes the evolution of the scalar field dependent on the background equation of state , explaining why dark energy is dominating now . anyway , apart from the details , the situation that a scalar field is slowly rolling down its potential is reminiscent of the primordial inflation @xcite , where a scalar field ( the inflaton ) provides the vacuum - like energy density ( @xmath5 ) necessary for a phase of the accelerated expansion by slowly rolling down its flat potential . hence one may naturally raise a question of how we can couple the early accelerated expansion , the primordial inflation , and the current one together @xcite : is it possible to cause the acceleration we observe recently by the primordial inflation ? in this paper , we are going to discuss this possibility using a simple model based on the hybrid inflation @xcite , which arises naturally in many string - inspired inflation models , in particular in potentials for moduli fields . this paper is organized as follow : in section [ model ] , we present a simple model of dark energy and analyze its dynamics in detail , presenting several conditions which should be satisfied for our model to work properly . unlike the conventional lore that the true minimum of the quintessence potential is presumed to vanish , we find that positive , zero , or negative vacuum energy is possible as we choose a different set of parameters . in section [ discussions ] , we discuss the variation of the equation of state @xmath6 which might be detected in future observations such as hetdex @xcite . also we briefly address the possibility of realizing our model in supersymmetric theories and obstacles to overcome . we briefly summarize this paper in section [ summary ] . in this section , we discuss a simple model of dark energy based on the hybrid model of inflation @xcite . perhaps the simplest way to combine the onset of present acceleration of the universe with the primordial inflation is to directly couple the inflaton with the quintessence field . in this case , however , first we should ensure that inflation lasts for a long enough time ( at least 60 @xmath7-foldings ) without being disturbed by the interaction with the quintessence field . moreover , the quintessence potential must remain extremely flat after the inflaton reaches its minimum and decays to reheat the universe . rather , it is more plausible to couple the quintessence field with some different field which plays no role during the inflaton rolls down its potential and only works to finish inflation : this is what the waterfall field in hybrid inflation model does . hence we can write the effective potential as @xmath8 where we take @xmath9 as the inflaton field , @xmath10 as the waterfall field , and @xmath11 as the quintessence field . as can be seen from the coupled terms above , we will experience two phase transitions , which distinguish different stages of the evolution of the universe . in the following subsections , we will discuss each stage in detail . the effective masses squared of @xmath10 and @xmath11 are @xmath12 and @xmath13 , respectively . hence , for @xmath14 , the only minimum of the effective potential , eq . ( [ potential ] ) , is at @xmath15 . also , with @xmath15 , for @xmath16 , the only minimum of the effective potential is at @xmath17 . thus , at the early stage of the evolution of the universe , @xmath10 and @xmath11 are trapped at 0 while @xmath9 remains much larger than @xmath18 for a long time . this stage lasts until @xmath19 , and at that moment we assume that the vacuum energy density @xmath20 is much larger than the potential energy density of the inflaton field @xmath21 , so that the hubble parameter is given by @xmath22 where @xmath23 is the reduced planck mass . we can additionally assume that the vacuum energy @xmath24 is dominated by @xmath25 , i.e. , @xmath26 so that the subsequent evolutions of the universe after inflation , e.g. , reheating , becomes identical to the usual hybrid model . this immediately gives @xmath27 now , combining eqs . ( [ h1stlast ] ) and ( [ psidomination ] ) , the slow - roll condition for @xmath28 gives @xmath29 note that this gives another bound on @xmath28 as @xmath30 , and the tightest bound on @xmath28 depends on the comparison between the prefactors of this expression and eq . ( [ mbound1 ] ) , i.e. , @xmath31 and @xmath32 . as discussed above , when @xmath9 becomes smaller than @xmath18 , a phase transition with the symmetry breaking for @xmath10 occurs . to finish inflation as soon as @xmath9 reaches the critical value @xmath18 , i.e. , for the so - called waterfall " , we first require that the absolute value of the effective mass squared of @xmath10 be much larger than @xmath33 . during inflation , using the slow - roll equation of motion @xmath34 and eqs . ( [ h1stlast ] ) and ( [ psidomination ] ) , we can find that when the inflaton @xmath9 reaches @xmath18 , it decreases by @xmath35 during the time interval @xmath36 . at that time , the absolute value of the effective mass squared of @xmath10 is given by latexmath:[\[\label{psimass } is much greater than @xmath33 for @xmath38 also , we demand that the time scale for @xmath9 to roll down from @xmath18 to 0 be much shorter than @xmath36 : let us denote @xmath39 as the time @xmath9 takes as it moves from @xmath18 to 0 . then , from @xmath40 where we have used eq . ( [ psimass ] ) and the slow - roll equation of @xmath9 , we have @xmath41 which gives another condition @xmath42 note that the parameters with respect to @xmath9-@xmath10 transition are further constrained from the observed cobe amplitude of density perturbations : we obtain @xcite @xmath43 which looks similar to eqs . ( [ waterfall1 ] ) and ( [ waterfall2 ] ) . combining eqs . ( [ waterfall1 ] ) , ( [ waterfall2 ] ) and ( [ pertconst ] ) , we can extract the valid range of the parameters @xmath28 , @xmath44 , @xmath31 and @xmath45 : e.g. if we take @xmath46 , @xmath47 and @xmath48 , all the conditions are satisfied . after this phase transition , the waterfall field @xmath10 oscillates at the minimum and decays so that the universe is reheated of @xmath11 to @xmath10 , and in turn to the thermal bath , the oscillation of @xmath10 may affect the dynamics of @xmath11 . however , we note that because of the negative sign of their interaction term in eq . ( [ potential ] ) , the pattern of instability is quite different from the usual parametric resonance @xcite even though the interaction is strong enough : we will study this issue in more detail separately . ] . with these conditions , as soon as @xmath9 reaches @xmath18 , inflation ends within a hubble time , @xmath36 . also note that when such a rapid phase transition occurs within observationally interesting range , still we can calculate the power spectrum and the spectral index for the density perturbations @xcite by generalizing the standard perturbative method @xcite : generally , the power spectrum shows a scale dependent oscillations after the phase transition @xcite . while @xmath10 is rolling down to @xmath49 , the effective mass squared of @xmath11 might become negative then another phase transition occurs . for this to happen , we want @xmath50 , unless no instability for @xmath11 would develop . this is equivalent to the condition is settled at @xmath51 . when @xmath10 moves along @xmath52 direction , this is equivalent to the condition @xmath53 this means , combined with eq . ( [ psisigma ] ) , @xmath44 is ( much ) bigger than @xmath54 . we are grateful to andrei linde for pointing out this . ] @xmath55 note that we can obtain the same condition by requiring that the minimum along @xmath11 direction , @xmath56 given by @xmath57 be real , i.e. , @xmath58 once the above condition is satisfied and @xmath11 rolls down the effective potential , after all the fields are settled at @xmath59 where the potential becomes @xmath60 when @xmath61 , eq . ( [ globalmin ] ) is greater than zero and it corresponds to a non - zero , positive vacuum energy . hence no matter @xmath11 evolves quickly or not , we are provided with the source of the current acceleration of the universe in this case , and this acceleration will last forever : the universe will eventually behave as a de sitter space . note that the possible maximum value of @xmath62 , @xmath63 , could be at most as large as the current critical density , @xmath64 . however , an extreme fine tuning is required to match this value : if we restrict our interest to this case only , our model is no better than the conventional @xmath65cdm because the latter is simpler and hence preferable . anyway here we do not try to improve this situation . a study of alleviating this fine tuning problem is very important and interesting , but is outside the scope of the present paper . for the case @xmath66 eq . ( [ globalmin ] ) is exactly zero and no vacuum energy exists . this case corresponds to the usual quintessential inflation @xcite , where the cosmological constant @xmath65 is assumed to be zero due to some unknown symmetry , and the observed dark energy is supplied by e.g. another scalar field , here @xmath11 . to be able to explain the acceleration of the universe , we require that still @xmath11 be rolling down so that the potential is non - zero . that is , @xmath11 should roll extremely slowly . note that although @xmath11 begins to evolve only after @xmath10 reaches @xmath67 , for the most time during @xmath11 rolls down the effective potential to @xmath56 , @xmath10 has already settled at @xmath51 within a time @xmath68 as soon as @xmath69 . thus , we just set @xmath70 throughout the evolution of @xmath11 . under this circumstance , when @xmath11 begins to roll , @xmath71 is given by @xmath72 for @xmath11 to slowly evolve , we require that @xmath73 , the absolute value of the effective mass squared of @xmath11 , be much smaller than @xmath33 , from which we obtain a condition @xmath74 where we have used eq . ( [ v0=0condition ] ) . indeed , the same bound could be found when we also require that @xmath11 roll very slowly so that for @xmath11 to move by an infinitesimal displacement @xmath75 , it takes much longer time than @xmath36 : the effective potential is given by @xmath76 where we have used eq . ( [ v0=0condition ] ) . hence , combining with the slow - roll equation @xmath77 we find @xmath78 i.e. @xmath79 for @xmath80 . requiring @xmath81 gives @xmath82 which is the same as eq . ( [ v0=0con1 ] ) . recent observations find that the vacuum energy , or cosmological constant , is very small and positive , with its value being of order @xmath83 . the case of negative cosmological constant is not observationally justified , which seems to support the claim that the true minimum of the quintessence potential is zero . the vacuum state with negative energy is , nevertheless , an interesting theoretical possibility , especially in string theories where anti de sitter solutions are popular . hence a negative potential may play an important role in cosmology motivated from string theory or m theory , such as cyclic universe model @xcite . we obtain @xmath84 when @xmath85 has a real solution and that this solution is smaller than @xmath56 . if we take the limit @xmath86 , the effective mass squared of @xmath11 is given by @xmath87 imposing the slow - roll condition , this gives @xmath88 the same as eqs . ( [ v0=0con1 ] ) and ( [ v0=0con2 ] ) . since the dark energy observed now is positive definite , @xmath11 should be still evolving and not have crossed @xmath89 : thus we again require that @xmath11 roll down very slowly , from which we obtain the same conditions as the previous section , eqs . ( [ v0=0con1 ] ) and ( [ v0=0con2 ] ) . however , unlike the cases before , the true vacuum energy is negative and after a ( tremendously ) long time the universe will collapse eventually @xcite , no matter how small the magnitude is . generally , for evolving dark energy models the equation of state @xmath6 is not a constant but a slowly evolving function . to provide an acceleration of the universe it is constrained to be smaller than @xmath90 , usually taken to be @xmath91 . recent combined observations of wmap and sdss constrains @xmath6 to be @xmath92 , with uncertainties at the 20% level @xcite . for the first case discussed in the previous section where @xmath61 , this is easily satisfied : @xmath6 is very slowly varying as @xmath11 rolls down its potential , finally becoming exactly @xmath92 at the absolute minimum @xmath62 . when @xmath93 , @xmath6 will be eventually 0 after @xmath11 settles at @xmath56 . for the third case , however , it is not as trivial as the other cases . naively , we may take @xmath94 to be non - zero although very small , then we can write @xmath95 where @xmath96 stands for the kinetic energy , @xmath97 . at first look , it seems that @xmath6 is divergent at @xmath98 and a discontinuity appears and moreover a phantom phase @xmath99 exists after that point . let us more closely see this . the simplest way is to solve the relevant equations numerically : we choose @xmath100 to solve for @xmath101 and @xmath102 . in figure [ fig_w ] , we plot the resulting equation of state @xmath6 where , as is shown , it oscillate . to understand this behavior first let us consider the expansion of the universe with such a negative potential @xcite : the universe keeps expanding until the moment @xmath103 where the scale factor reaches its largest value , and the universe shrinks afterwards . from the friedmann equation describing a flat universe @xmath104 which has no solution with @xmath105 , we can see that to maintain a real value of @xmath71 no matter positive or negative , @xmath106 should remain positive . hence , when @xmath107 is negative , the kinetic energy must compensate this negative potential energy and consequently these energies oscillate out of phase : for example , at the negative bottom of the potential , the field is moving fastest so that we have maximum kinetic energy here so that @xmath6 becomes very large . this is the reason why the equation of state @xmath6 is oscillating , we do nt see such an oscillation . ] and sometimes becomes greater than 1 , the `` stiff '' fluid . we can easily find the relation with respect to the redshift @xmath108 , by making use of the relation @xmath109 : by solving the equations @xmath110 where a prime denotes the derivative with respect to @xmath108 , we can obtain @xmath6 in terms of @xmath108 . a typical relation between @xmath6 and @xmath108 is shown in the right panel of figure [ fig_w ] . supersymmetry is believed to be the most promising candidate related to the fundamental problems in particle physics , such as the hierarchy problem and the gauge coupling unification . since we have not yet observed any supersymmetric partner of known particles , supersymmetry is broken . in the primordial universe where the energy scale is much higher than the present one , however , the rich structure of supersymmetric and supergravity theories should have played an important role . hence it is natural to try to implement inflationary scenario within supersymmetry . there has been an encouraging progress on the hybrid inflation scenario in the context of supersymmetric theories @xcite . for example , consider a simple superpotential @xcite @xmath111 where @xmath112 and @xmath113 are a pair of superfields in non - trivial representations of some gauge group under which @xmath9 is neutral . then , in a globally supersymmetric theory , the effective potential is given by @xmath114 here , the absolute supersymmetric minimum appears at @xmath115 . however , for @xmath116 , @xmath112 and @xmath113 obtain positive masses squared and hence are confined at the origin . by simply adding a mass term @xmath117 which softly breaks supersymmetry , we see that the hybrid inflation scenario is possible . if we take into account radiative corrections instead , the total effective potential including one - loop corrections is @xmath118\ ] ] when @xmath119 , and we can see that still inflation is possible . similarly , we can hope to implement our scenario within supersymmetry . the simplest possibility should be @xmath120-terms since , as we can see from eq . ( [ potential ] ) , the coupled term of @xmath10 and @xmath11 has a negative sign , so that an instability for @xmath11 could be developed as @xmath10 , initially confined at zero , rolls away from the origin . this is easily achieved by assuming that @xmath10 and @xmath11 are oppositely charged under some gauge symmetry . if it is a @xmath121 symmetry , we can write the @xmath120-term contribution as @xmath122 where we assumed that @xmath10 and @xmath11 have charges equal to @xmath92 and @xmath123 respectively , @xmath124 is the gauge coupling , and @xmath125 is a fayet - iliopoulos @xmath120-term . we may also include another gauge symmetry under which @xmath9 and @xmath10 are charged with the same sign , then @xmath120-term is given by @xmath126 where we have introduced another gauge coupling @xmath127 and fayet - iliopoulos term @xmath128 . taking only eqs . ( [ dterm1 ] ) and ( [ dterm2 ] ) into account , the coupled terms are given by @xmath129 we can reproduce the same forms for @xmath130 and @xmath131 terms as eq . ( [ potential ] ) as long as @xmath132 is positive . this sheds some light on the realization of eq . ( [ potential ] ) . however , this is far from a realistic possibility . first of all , we have not considered any @xmath133-term contributions . we do not want to couple @xmath9 and @xmath11 to guarantee their desired behaviors on the flat enough effective potential , but it seems not easy to achieve this through the contributions from @xmath133-term . more fundamentally , in globally supersymmetric theories the scalar potential is either positive or zero , which does not include the interesting case of @xmath84 . also , it is thought that below the planck scale particle physics is described by an effective @xmath134 supergravity theory derived from string theory . hence , a more detailed analysis of our model and associated problems should be addressed in the context of supergravity : for example , the mass of the field should be very small , of order the present hubble parameter @xmath135 , so that it is still rolling toward its true minimum . however , usually scalar fields acquire masses of order the gravitino mass @xmath136 , much heavier than @xmath137 . one way to evade this difficulty is to use pseudo nambu - goldstone bosons problem associated with generic scalar field potentials in supergravity . ] @xcite , e.g. , string / m theory axion , as @xmath11 field @xcite . we will leave such an analysis as a challenging future work . we have investigated a simple dark energy model based on hybrid inflation . the quintessence field @xmath11 is coupled to the waterfall field @xmath10 so that as @xmath10 rolls towards @xmath138 , @xmath11 begins to move along the effective potential provided that eq . ( [ psisigma ] ) is satisfied . a number of bounds on the parameters of the model are found , such as eqs . ( [ psidomination ] ) , ( [ mbound1 ] ) , ( [ mpsibound1 ] ) , ( [ waterfall1 ] ) and ( [ waterfall2 ] ) . an interesting point is that the true minimum of the effective potential , eq . ( [ globalmin ] ) , depends on our choice of the parameters , allowing the vacuum state with positive , negative and zero energy . when it is negative , the universe will eventually collapse , showing an oscillation in the equation of state @xmath6 for dark energy which we can hope to detect in future observations . this model could be realized in supersymmetric theories via @xmath120-term contributions , but including @xmath133-term parts and supergravity effects makes this model not easy to be achieved . we thank kiwoon choi , salman habib , andrei linde and carlos muoz for helpful discussions and suggestions . we are also indebted to the anonymous referee for many important and invaluable comments . jg is grateful to the sf05 cosmology summer workshop , and sk to the summer institute 2005 for hospitalities when this work was in progress . this work was supported in part by the astrophysical research center for the structure and evolution of the cosmos funded by the korea science and engineering foundation and the korean ministry of science , the korea research foundation grant krf pbrg 2002 - 070-c00022 , and the brain korea 21 . p. j. e. peebles and b. ratra , _ astrophys . j. _ * 325 * , l17 ( 1988 ) ; b. ratra and p. j. e. peebles , _ phys . * d 37 * , 3406 ( 1988 ) ; r. r. caldwell , r. dave and p. j. steinhardt , _ phys . _ * 80 * , 1582 ( 1998 ) ` astro - ph/9708069 ` c. armendariz - picon , v. mukhanov and p. j. steinhardt , _ phys . lett . _ * 85 * , 4438 ( 2000 ) ` astro - ph/0004134 ` ; c. armendariz - picon , v. mukhanov and p. j. steinhardt , _ phys . * d 63 * , 103510 ( 2001 ) ` astro - ph/0006373 ` w. h. kinney and a. riotto , _ astrophys . _ * 10 * , 387 ( 1999 ) ` hep - ph/9704388 ` ; p. j. e. peebles and a. vilenkin , _ phys . rev . _ * d 59 * , 063505 ( 1999 ) ` astro - ph/9810509 ` ; m. peloso and f. rosati , _ j. high energy phys . _ * 12 * , 026 ( 1999 ) ` hep - ph/9908271 ` ; a. b. kaganovich , _ phys . * d 63 * , 025022 ( 2001 ) ` hep - th/0007144 ` ; g. huey and j. e. lidsey , _ phys . * b 514 * , 217 ( 2001 ) ` astro - ph/0104006 ` ; a. s. majumdar , _ phys . rev . _ * d 64 * , 083503 ( 2001 ) ` astro - ph/0105518 ` ; k. dimopoulos and j. w. f. valle , _ astropart . * 18 * , 287 ( 2002 ) ` astro - ph/0111417 ` ; n. j. nunes and e. j. copeland , _ phys . _ * d 66 * , 043524 ( 2002 ) ` astro - ph/0204115 ` ; e. i. guendelman and o. katz , _ class . _ * 20 * , 1715 ( 2003 ) ` gr - qc/0211095 ` ; k. dimopoulos , _ phys _ * d 68 * , 123506 ( 2003 ) ` astro - ph/0212264 ` ; m. giovannini , _ phys . rev . _ * d 67 * , 123512 ( 2003 ) ` hep - ph/0301264 ` ; t. biswas and p. jaikumar , _ phys . * d 70 * , 044011 ( 2004 ) ` hep - th/0310172 ` ; m. r. garousi , m. sami and s. tsujikawa , _ phys . _ * d 70 * , 043536 ( 2004 ) ` hep - th/0402075 ` ; m. sami and v. sahni , _ phys . * d 70 * , 083513 ( 2004 ) ` hep - th/0402086 ` ; t. biswas and p. jaikumar , _ j. high energy phys . _ * 08 * , 053 ( 2004 ) ` hep - th/0407063 ` ; a. gonzalez , t. matos and i. quiros , _ phys . _ * d 71 * , 084029 ( 2005 ) ` hep - th/0410069 ` ; s. barshay and g. kreyerhoff , _ mod . lett . _ * a 19 * , 2899 ( 2004 ) ` astro - ph/0410478 ` ; r. rosenfeld and j. a. frieman , _ j. cosmol . astropart . * 09 * , 003 ( 2005 ) ` astro - ph/0504191 ` ; x. zhai and y. zhao , _ nuovo cim . _ * 120 b * , 1007 ( 2005 ) ` gr - qc/0508069 ` ; x. zhai and y. zhao , _ chin . phys . _ * 15 * , 2465 ( 2006 ) ` astro - ph/0511512 ` j. choe , j .- o . gong and e. d. stewart , _ j. cosmol . phys . _ * 07 * , 012 ( 2004 ) ` hep - ph/0405155 ` ; j .- o . gong , _ class . * 21 * , 5555 ( 2004 ) ` gr - qc/0408039 ` ; m. joy , e. d. stewart , j .- o . gong and h .- c . lee , _ j. cosmol . astropart . phys . _ * 04 * , 012 ( 2005 ) ` astro - ph/0501659 ` e. j. copeland , a. r. liddle , d. h. lyth , e. d. stewart and d. wands , _ phys . * d 49 * , 6410 ( 1994 ) ` astro - ph/9401011 ` ; g. lazarides , r. k. schaefer and q. shafi , _ phys . rev . * d 56 * , 1324 ( 1997 ) ` hep - ph/9608256 ` s. barshay and g. kreyerhoff , _ mod . lett . _ * a19 * , 2899 ( 2004 ) ` astro - ph/0410478 ` _ erratum - ibid . _ * a20 * , 373 ( 2005 ) ; r. rosenfeld and j. a. frieman , _ j. cosmol . phys . _ * 09 * , 003 ( 2005 ) ` astro - ph/0504191 `
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the situation that a scalar field provides the source of the accelerated expansion of the universe while rolling down its potential is common in both the simple models of the primordial inflation and the quintessence - based dark energy models .
motivated by this point , we address the possibility of causing the current acceleration via the primordial inflation using a simple model based on hybrid inflation .
we trigger the onset of the motion of the quintessence field via the waterfall field , and find that the fate of the universe depends on the true vacuum energy determined by choosing the parameters .
we also briefly discuss the variation of the equation of state and the possible implementation of our scenario in supersymmetric theories .
kaist - th/2005 - 16 1.5 cm 0.5 cm 1.2 cm
| 8,351 | 173 |
filaments are the most prominent features visible in the galaxy distribution . this finding dates back to a few papers in the seventies and eighties @xcite . subsequent work substantiates this ( e.g. @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite ) and shows the filaments to be statistically significant @xcite . it is now well accepted that galaxies are distributed in an interconnected network of clusters , sheets and filaments encircling voids . this complicated pattern is often referred to as the cosmic web . despite this progress , it still remains a challenge to quantify the cosmic web that is so distinctly visible in galaxy redshift surveys ( eg . sdss dr5 , @xcite ) . statistical measures like the void probability function @xcite , percolation analysis @xcite and the genus curve @xcite each quantifies a different aspect of the cosmic web . the minkowski functionals @xcite are very effective to quantify the shapes of individual structural elements like sheets or filaments . in @xmath5 dimensions there are @xmath10 minkowski functionals , namely the volume , surface area , integrated mean curvature and integrated gaussian curvature . @xcite introduce the shapefinders , essentially ratios of the minkowski functionals , as a very effective shape diagnostic . a @xmath6 dimensional version of shapefinders @xcite has been extensively used to quantify the filamentarity in the galaxy distribution ( @xcite and references therein ) . centered on a galaxy located in the filament.,scaledwidth=40.0% ] though the minkowski functionals and the shapefinders are very effective techniques to quantify the shapes of individual structural elements like sheets or filaments , it is very different when dealing with the cosmic web which is an interconnected network of filaments , sheets and clusters . for example consider a sheet connected to a filament as shown in figure [ fig : exp1 ] . the minkowski functionals are global properties of the entire object the area is the sum of the areas of the sheet and the filament etc . , and the fact that object is actually a combination of two different elements would be lost . it is necessary to quantify the local shape at different points in the object in order to determine that it actually is a combination of a sheet and a filament . in this paper we consider the `` local dimension '' as a means to quantify the local shape of the galaxy distribution at different positions along the cosmic web . we choose a particular galaxy as center and determine @xmath2 the number of other galaxies within a sphere of comoving radius @xmath3 . this is done varying @xmath3 . in the situation where a power law @xmath11 gives a good fit over the length - scales @xmath12 , we identify @xmath0 as the local dimension in the neighbourhood of the center . the values @xmath13 and @xmath5 correspond to a filament , sheet and cluster respectively . it may be noted that the term `` cluster '' here denotes a three dimensional , volume filling structural element and is not to be confused with a `` cluster of galaxies '' . values of @xmath0 other than @xmath14 and @xmath5 are more difficult to interpret . for example , a galaxy distribution that is more diffuse than a filament but does not fill a plane would give a fractional value ( fractal ) in the range @xmath15 . referring to figure [ fig : exp1 ] , we expect @xmath16 and @xmath17 when the center is located in the filament and the sheet respectively . this is provided that the center is well away from the intersection of the filament and the sheet . when the intersection lies within @xmath12 from the center , there will be a change in the slope of @xmath2 when it crosses the intersection . it is not possible to determine a local dimension at the centers where such a situation occurs . we perform this analysis using every galaxy in the sample as a center . in general it will be possible to determine a local dimension for only a fraction of the galaxies . it is expected that with a suitable choice of the @xmath3 range _ ie . _ @xmath18 and @xmath19 , it will be possible to determine the local dimension for a substantial number of the centers . the value of the local dimension at different positions will indicate the location of the filaments , sheets and clusters and reveal how these are woven into the cosmic web . in this _ letter _ we test this idea and demonstrate its utility by applying it to simulations . we have used a particle - mesh ( pm ) n - body code to simulate the @xmath20 dark matter distribution . the simulations have @xmath21 particles on a @xmath21 mesh with grid spacing @xmath22 . the simulations were carried out using a lcdm power spectrum with the parameters @xmath23 . we have identified @xmath24 particles , randomly drawn from the simulation output , as galaxies . these have a mean interparticle separation of @xmath25 , comparable to that in galaxy surveys . this simulated galaxy distribution was carried over to redshift space in the plane parallel approximation . the subsequent analysis to determine the local dimension was carried out using this simulated sample of @xmath24 galaxies . since the resolution of the simulation is about @xmath26 , we ca nt choose @xmath18 to be less than that . the value of @xmath19 is determined by the limited box size . we have chosen the value of @xmath18 and @xmath19 to be @xmath6 and @xmath27 respectively . increasing @xmath19 causes a considerable drop in the number of centers for which the local dimension is defined . the analysis was carried out for @xmath28 different , independent realizations of the dark matter distribution . figure [ fig : exp2 ] shows @xmath2 for three different centers chosen from a particular realization . the @xmath29 error at each data point is @xmath30 due to the poisson fluctuation . for each center we have determined the power law @xmath1 that provides the best fit to the data . the power law fit is accepted if the chi - square per degree of freedom satisfies @xmath31 and the value of @xmath0 is accepted as the local dimension corresponding to the particular center . the power law fit is rejected for larger values of @xmath32 , and the local dimension is undetermined for the particular center . the criteria @xmath31 is chosen as a compromise between a good power law fit and a reasonably large number of centers for which @xmath0 can been determined . the number of centers for which @xmath0 can been determined falls if a more stringent criteria is imposed on @xmath32 . of a center with @xmath9 are shown as points . the galaxy distribution was converted to a set of 1s and 0s on a grid of spacing @xmath33 . connected structures were identified using the friend - of - friend algorithm where adjacent 1s are identified as belonging to the same connected structure . only the two largest structures have been shown , both pass very close to the center and they appear connected in the figure . the @xmath0 values have been painted on these structures with red , blue and green representing @xmath4 and @xmath5 respectively . the @xmath0 value is undetermined for the parts of the structures shown in black . ] the number of centers for which it is possible to determine a local dimension varies between @xmath34 to @xmath35 _ ie . _ less than @xmath36 variation across the @xmath28 different realizations . the @xmath0 values in the interval @xmath37 , @xmath38 and @xmath39 have respectively been binned as @xmath4 and @xmath5 . we have inspected the galaxy distribution in the vicinity of a few centers in order to visually identify the structures corresponding to the respective @xmath0 values . figure [ fig : fil1n ] shows the galaxy distribution within @xmath7 of a center which has local dimension @xmath40 . we expect this center to be located in a filament . we have used the friend - of - friend algorithm with linking length @xmath41 to identify connected patterns in the galaxy distribution . we note that all the galaxies in this region connect up into a single structure if the linking length is doubled to @xmath42 . using @xmath41 yields several disconnected structures of which we show only the largest two . both the structures pass close to the center , and appear connected in the figure . of the two structures , one with two well separated tentacles appears below the center , whereas the other with two nearly connected tentacles appears above it . the filamentary nature of these structures is quite evident . out of the total @xmath43 galaxies in these two structures , @xmath0 is undetermined for @xmath44 . there are @xmath45 galaxies with @xmath46 , all located in a contiguous region near the center , and @xmath47 and @xmath48 galaxies with @xmath49 and @xmath5 respectively spread out along the structure at locations quite different from the center . while it is relatively easy to visually identify the filament corresponding to the centers with @xmath50 , we have not been able to visually identify the sheets and clusters corresponding to the centers with @xmath49 and @xmath5 shown in figure [ fig : fil1n ] . we note that a visual identification of the sheets and clusters corresponding to @xmath51 and @xmath5 respectively has been possible in other fields where the largest structure is predominantly sheetlike or clusterlike . it is quite apparent that the largest structure in figure [ fig : fil1n ] is predominantly filamentary . values in a box of size @xmath52 ^ 3 $ ] , red , blue and green denote @xmath4 and @xmath5 respectively . ] figure [ fig : allshown ] shows the distribution of @xmath0 values over a large region from one of the realizations . the distribution shows distinct patterns , there being regions with size ranging from a few @xmath53 to tens of @xmath54 where the @xmath0 value is constant . the centers with @xmath9 appear to have a more dense and compact distribution compared to the centers with @xmath51 , whereas the centers with @xmath55 appear to have a rather diffuse distribution . figure [ fig : hist1 ] separately quantifies two different aspects of the distribution of @xmath0 values , ( a. ) the fraction of centers with a particular @xmath0 value , and ( b. ) the fraction of volume with a particular @xmath0 value . for this analysis the @xmath0 values were divided into bins of width @xmath56 and the centers for which a @xmath0 value could not be determined were discarded . we find that the fraction of centers and the volume fraction both have very robust statistics with a variation of @xmath57 or less across the @xmath28 different realizations . we first consider the distribution of the fraction of centers with different @xmath0 values . the bin at @xmath58 contains the maximum number of centers , and the two bins at @xmath58 and @xmath6 together contain more than @xmath59 of the centers . this indicates that the maximum number of centers for which @xmath0 can be determined lie in sheets and filaments . assuming that this is representative of the entire matter distribution , we conclude that the bulk of the matter is contained in sheets and filaments , with the sheets dominating . we next consider the the fraction of volume occupied by a particular @xmath0 value . we have estimated this using a mesh with grid spacing @xmath60 . each grid position was assigned the @xmath0 value of the centers within the corresponding grid cell . cells that contain centers with different values of @xmath0 were discarded . it was possible to assign a @xmath0 value to @xmath61 grid cells . we find that the distribution peaks at @xmath62 , the peak is rather broad and more than @xmath59 of the volume for which it was possible to assign a @xmath0 value is occupied by values in the range @xmath6 to @xmath5 . assuming that this result is valid for the entire volume , we conclude that the volume is predominantly filled with sheets and clusters . while the matter is mainly contained in sheets and filaments , the filaments do not occupy a very large fraction of the volume . this leads to a picture where the filaments have relatively higher matter densities as compared to clusters . to test this we separately consider the centers with @xmath0 values @xmath14 and @xmath5 , and for each value we determine the fraction of centers as a function of the density environment . the density was calculated on a @xmath63 grid after smoothing with a gaussian filter of width @xmath64 . we find ( figure [ fig : hist2 ] ) that in comparison to the centers with @xmath9 and @xmath6 , the centers with @xmath55 are more abundant in under - dense regions in preference to the over - dense regions . this is consistent with the picture where filaments and sheets located along the cosmic web contain the bulk of the matter and also most of the high density regions , whereas the centers with @xmath65 are predominantly located in under - dense regions away from the cosmic web , namely the voids . the results presented above are specific to the length - scale range @xmath66 . it is quite possible that the relative abundance of clusters , sheets and filaments would be quite different if the same analysis were carried out over a considerably different range of length - scales , say for example @xmath67 . here we have repeated the entire analysis using a smaller range of length - scales @xmath68 for which the results are also shown in figure [ fig : hist1 ] . we find that the results are qualitatively similar to those obtained using @xmath66 , indicating that our conclusions regarding the relative abundance of clusters , sheets and filaments are quite robust and are not very sensitive to small changes in the range over which @xmath0 is determined . this also indicates that the @xmath0 values determined using length - scales @xmath69 are not severely biased by the presence of other structural elements with @xmath70 of the one on which the center is located . in conclusion we note that the local dimension provides a robust method to quantify the shapes and probe the distribution of the different , interconnected structural elements that make up the cosmic web . we plan to apply this to the sdss and other galaxy surveys in future . p.s . is thankful to biswajit pandey , kanan datta and prasun dutta for useful discussions . p.s . would like to acknowledge senior research fellow of university grants commission ( ugc ) , india . for providing financial support .
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it is now well accepted that the galaxies are distributed in filaments , sheets and clusters all of which form an interconnected network known as the cosmic web .
it is a big challenge to quantify the shapes of the interconnected structural elements that form this network .
tools like the minkowski functionals which use global properties , though well suited for an isolated object like a single sheet or filament , are not suited for an interconnected network of such objects .
we consider the local dimension @xmath0 , defined through @xmath1 , where @xmath2 is the galaxy number count within a sphere of comoving radius @xmath3 centered on a particular galaxy , as a tool to locally quantify the shape in the neigbourhood of different galaxies along the cosmic web .
we expect @xmath4 and @xmath5 for a galaxy located in a filament , sheet and cluster respectively . using lcdm n - body simulations we find that it is possible to determine @xmath0 through a power law fit to @xmath2 across the length - scales @xmath6 to @xmath7 for @xmath8 of the galaxies .
we have visually identified the filaments and sheets corresponding to many of the galaxies with @xmath9 and @xmath6 respectively . in several other situations
the structure responsible for the @xmath0 value could not be visually identified , either due to its being tenuous or due to other dominating structures in the vicinity .
we also show that the global distribution of the @xmath0 values can be used to visualize and interpret how the different structural elements are woven into the cosmic web .
methods : data analysis - galaxies : statistics - large - scale structure of universe
| 3,747 | 422 |
the m31 globular cluster ( gc ) b158 ( a.k.a . bo 158 ) , named following the revised bologna catalogue v3.4 @xcite , contains a bright x - ray source that was discovered by the einstein observatory @xcite , and has shown up in every x - ray observation of the region since ; we call this x - ray source xb158 . xb158 exhibited strong intensity modulation on a 10017@xmath050 s ( @xmath12.78 hr ) period during the 2002 january xmm - newton observation @xcite . @xcite found similar variation in the folded lightcurves from a 1991 june rosat observation and a 2000 june xmm - newton observation ; they found that the amplitude of modulation decreased with increasing source intensity . they found no such modulation in the 2001 june xmm - newton observation , setting a 2@xmath3 upper limit of 10% modulation . assuming that this represents the orbital period , @xcite found that this is probably a neutron star binary with a low mass donor with a separation @xmath510@xmath6 cm ( i.e. a low mass xb - ray binary , lmxb ) . however , analysis of the unfolded 2000 june xmm - newton lightcurve revealed a single deep dip at the end of the observation , with no evidence for dips in the two previous orbital cycles @xcite . furthermore , @xcite analyzed 3 proprietary xmm - newton observations over 2004 july 1719 , finding @xmath1100% dipping for one orbital cycle , and zero evidence for dips in other cycles ; we concluded that the disk was precessing , with dips only visible for some part of the super - orbital cycle . such behavior is associated with the `` superhump '' phenomenon that is observed in accreting binaries where the mass ratio is smaller than @xmath10.3 @xcite . superhumps were first identified in the superoutbursts of the su uma subclass of cataclysmic variables ( accreting white dwarf binaries ) , manifesting as a periodic increase in the optical brightness on a period that is slightly longer than the orbital period @xcite . su umas are a subclass of dwarf novae with orbital periods @xmath72 hr , that exhibit occasional superoutbursts that last @xmath85 times as long as the normal outbursts @xcite . @xcite proposed that these superoutbursts are enhanced by a tidal instability that occurs when the outer disk crosses the 3:1 resonance with the secondary ; the additional tidal torque causes the disk to elongate and precess , and also greatly enhances the loss of angular momentum ( and therefore the accretion rate ) . the disk precession is prograde in the rest frame , and the secondary repeats its motion with respect to the disk on the beat period between the orbital period and the precession period , a few percent longer than the orbital period . the secondary modulates the disk s viscous dissipation on this period , giving rise to the maxima in the optical lightcurve known as superhumps . some short period , persistently bright cvs exhibit permanent superhumps @xcite . cccccccccccc obs & @xmath9 & ra & dec & xrt exp & @xmath10/@xmath11 & counts + + 00032702001 & 0.0 & 00 43 00.81 & + 41 17 00.2 & 2234 & 10.0 & 57 + 00032702002 & 0.9 & 00 43 06.53 & + 41 16 54.6 & 2510 & 9.7 & 41 + 00032702003 & 2.0 & 00 43 03.73 & + 41 16 40.8 & 2299 & 9.5 & 13 + 00032702004 & 3.0 & 00 43 06.35 & + 41 16 51.3 & 2299 & 9.6 & 40 + 00032702005 & 3.1 & 00 43 13.10 & + 41 16 30.4 & 2489 & 9.2 & 43 + 00032702006 & 4.4 & 00 43 03.22 & + 41 13 58.5 & 2572 & 7.0 & 65 + 00032702007 & 5.4 & 00 43 10.44 & + 41 15 10.9 & 2511 & 7.9 & 51 + 00032702008 & 6.4 & 00 43 06.44 & + 41 16 06.4 & 2635 & 8.9 & 57 + 00032702009 & 7.3 & 00 43 11.97 & + 41 16 27.7 & 2476 & 9.1 & 29 + 00032702010 & 8.8 & 00 43 03.76 & + 41 16 15.5 & 2644 & 9.1 & 39 + 00032702011 & 9.5 & 00 43 08.40 & + 41 17 27.6 & 2405 & 10.2 & 53 + 00032702012 & 10.1 & 00 43 07.20 & + 41 16 29.8 & 2502 & 9.2 & 43 + 00032702013 & 11.2 & 00 43 03.63 & + 41 17 36.2 & 2514 & 10.5 & 48 + 00032702014 & 12.4 & 00 43 04.82 & + 41 17 00.3 & 2411 & 9.8 & 38 + 00032702015 & 13.2 & 00 43 07.68 & + 41 15 24.0 & 2490 & 8.2 & 11 + 00032702016 & 14.2 & 00 43 01.12 & + 41 15 11.3 & 2511 & 8.2 & 27 + 00032702017 & 15.1 & 00 43 06.35 & + 41 14 46.6 & 2273 & 7.6 & 54 + 00032702018 & 16.8 & 00 43 02.79 & + 41 17 00.9 & 2314 & 9.9 & 55 + 00032702019 & 17.1 & 00 43 08.59 & + 41 15 26.4 & 1992 & 8.2 & 45 + 00032702020 & 18.1 & 00 43 04.88 & + 41 17 31.5 & 1389 & 10.3 & 13 + 00032702021 & 19.4 & 00 43 03.24 & + 41 17 28.0 & 2690 & 10.3 & 18 + 00032702022 & 20.2 & 00 43 02.68 & + 41 17 26.6 & 2686 & 10.3 & 42 + 00032702023 & 21.1 & 00 43 03.07 & + 41 17 07.0 & 2821 & 10.0 & 65 + 00032702024 & 22.3 & 00 43 04.56 & + 41 16 19.5 & 2659 & 9.2 & 50 + 00032702025 & 23.2 & 00 43 10.49 & + 41 16 54.1 & 2711 & 9.6 & 46 + 00032702026 & 24.3 & 00 43 07.86 & + 41 15 05.1 & 2740 & 7.8 & 35 + 00032702027 & 25.1 & 00 43 06.42 & + 41 16 41.0 & 2424 & 9.5 & 18 + 00032702028 & 26.1 & 00 43 10.04 & + 41 16 54.2 & 2550 & 9.6 & 49 + 00032702029 & 27.2 & 00 43 10.52 & + 41 17 33.9 & 2524 & 10.2 & 59 + 00032702030 & 28.4 & 00 43 06.50 & + 41 17 25.9 & 2331 & 10.2 & 41 + 4u 1916@xmath12053 is a high inclination neutron star lmxb with an x - ray period of 50.00@xmath00.08 min and an optical period 50.458@xmath00.003 min @xcite . it exhibits periodic x - ray intensity dips due to absorption by material in the outer disk ; the amplitude of these dips varies over @xmath10 to @xmath1100% on a @xmath14 day period , the precession period of the disk @xcite . @xcite showed that ns lmxbs with orbital periods shorter than @xmath14.2 hr are likely to exhibit superhumps , and identified 4u 1916 - 053 as a persistent superhumping source . xb158 appears to be somewhat analogous to 4u 1916@xmath12053 @xcite . in @xcite we modeled the 2004 july 17 xmm - newton pn spectrum of xb158 with a blackbody and a power law , finding @xmath13 = 2.0@xmath00.2 kev , the photon index @xmath14 = 2.0@xmath00.3 , @xmath4/dof = 19/19 and the 0.310 kev luminosity was 1.5@xmath00.6@xmath15 erg s@xmath16 ; fitting a single power law emission model yielded a photon index of 0.57@xmath00.09 , which is harder than any black hole spectrum @xcite , meaning that the accretor is a neutron star . we conducted three dimensional smoothed particle hydrodynamical ( sph ) modeling , assuming a 1.4 @xmath17 ns , a 2.78 hr period , a total system mass of 1.8 @xmath17 , and a luminosity @xmath180% of the eddington limit . as a result , we estimated the disk precession period to be 29@xmath01 times the orbital period , i.e. 81@xmath03 hr . in @xcite we examined chandra observations of @xmath130 m31 globular cluster x - ray sources spanning @xmath112 years ( 89 acis and 45 hrc ) , including xb158 . we observed 0.310 kev luminosities @xmath1420@xmath18 erg s@xmath16 , and proposed that this is due to varying accretion rates over the disk precession cycle . since we expected a super - orbital period for xb158 over time - scales of a few days , we obtained a series of 30 swift observations , each with 2.5 ks exposure and spaced @xmath11 day apart . in this work we present our analysis of these swift results , and find evidence for a super - orbital period of @xmath16 days we made 30 swift observations over 2013 february 8 march 9 ( obsids 00327020010032702030 , pi r. barnard ) . xb158 was one of two main targets for this survey , and the pointing was chosen to optimize the results from both targets . however , the actual pointings varied significantly over the 30 observations . a journal of swift observations is provided in table [ journ ] ; for each observation we give the time relative to the first observation , pointing , xrt exposure , and off - axis angle , along with the net source counts . for each observation , we placed circular regions around xb158 and a suitable background region . we obtained the net source counts using the `` counts in regions '' tool in the ds9 image viewer , and estimated the intensity by dividing the net counts by the exposure time . these data were obtained in order to determine the extent to which the varying off - axis angles affected our results . we also created spectra from the same extraction regions , created appropriate ancillary response files using xrtmkarf , and found the appropriate response file using the quzcif tool . none of the spectra were suitable for free spectral fitting , hence we obtained luminosity estimates by assuming a model obtained from previous observations . for chandra acis observations with @xmath2200 net source photons , the mean line - of - sight absorption ( @xmath19 ) was 9@xmath04@xmath20 atom @xmath21 ( @xmath4/dof = 7/11 ) , and the mean power law index ( @xmath14 ) was 0.52@xmath00.03 ( @xmath4/dof = 5/11 ) . this is consistent with the best absorbed power law fit to the 2004 july 17 xmm - newton observation ( @xmath19 = 0.1 , @xmath14 = 0.57@xmath00.09 , @xmath4/dof = 30/24 * ? ? ? as we noted earlier , that xmm - newton spectrum was best described by a blackbody + power law model , but neither the chandra nor swift spectra were sufficient to constrain the two - component emission model . however , we were able to estimate the luminosity for each observation by assuming a fixed emission model , allowing only the normalization to vary . we fitted each spectrum using xspec 12.8.2b , fixing @xmath19 = 9@xmath20 atom @xmath21 and @xmath14 = 0.52 , to find the normalization required to make absorbed model intensity 1.00 count s@xmath16 . we then calculated the unabsorbed flux for this model , allowing us to convert from intensity to flux . multiplying the conversion by the background - subtracted intensity provided by xspec yielded the instrument - corrected , background - subtracted source flux . the luminosity was calculated from the flux assuming a distance of 780 kpc @xcite . we fitted the lightcurve with constant and sinusoidal components using the qdp program provided in heatools , performing a simple search for periodicity . @xcite created a periodogram suitable for unbinned data with a mean of zero that produces exactly equivalent results to such least - squares fitting of sinewaves , but also allows comparison of the best period with other periods . the likelihood that any peak in the periodogram is real is given by the false alarm probability ( @xmath22 ) , where a low value of @xmath22 indicates that the peak is likely to be significant . if the highest frequency to be probed is @xmath23 times higher than the lowest frequency , then the power , @xmath24 , required for a false alarm probability @xmath22 is given by @xmath24 = @xmath25 $ ] ; for small @xmath22 , @xmath26 @xcite . we present our 30 day , 0.310 kev swift xrt lightcurve of xb158 in figure [ swiftlc ] . we see that the 0.310 kev luminosity varied by a factor @xmath15 ; the luminosity dropped from @xmath12.3@xmath15 erg s@xmath16 to @xmath15@xmath18 erg s@xmath16 in @xmath12 days , assuming the mean chandra absorbed power law model . we note that these swift observations are non - contiguous , spacing the 2.5 ks observing time over several hours ; the low intensities ( @xmath10.0050.025 count s@xmath16 ) meant that there was no appreciable variability within each observation . we find no evidence for a dependence of luminosity on off - axis angle ; the luminosities for the observations when xb158 have the largest and smallest off - axis angles have consistent values , while observations at an off - axis angle of 8.2@xmath11 resulted in a factor @xmath15 range in luminosity . the conversion factor for translating 1 count s@xmath16 into 0.310 kev flux ranged over 1.091.29@xmath27 erg @xmath21 count@xmath16 for all observations except the second one , where it was 1.79@xmath27 erg @xmath21 count@xmath16 . this @xmath110% variation about the mean in instrumental correction is clearly not sufficient to account for the factor @xmath15 variation in luminosity . fitting our lightcurve with a constant intensity yielded a best fit luminosity of @xmath11.3@xmath15 erg s@xmath16 , with @xmath4 = 181 for 29 degrees of freedom ( dof ) . however , adding a sinusoidal variation component yielded a much improved fit , with @xmath4/dof = 43/26 ; for this model , the period is 5.65@xmath00.05 days , with an amplitude of 7.1@xmath00.6@xmath18 erg s@xmath16 around a mean luminosity of 1.43@xmath00.04@xmath15 erg s@xmath16 , and a phase of 88.1@xmath00.7 degrees . all uncertainties in this work are quoted at the 1@xmath3 level . this sinusoidal variability yielded @xmath28 = 138 for 30 bins , with 3 extra free parameters ; f - testing showed that the probability for this improvement being due to chance was 3@xmath29 , equivalent to a @xmath25@xmath3 detection . in figure [ swift_per ] we show our swift lightcurve folded on a 5.65 day period and fitted with the best fit sinusoid . we present our lomb - scargle periodogram for the 0.310 kev swift xrt luminosity lightcurve of xb158 in figure [ ls ] , and also indicate the power required for false alarm probabilities @xmath22 = 0.5 , 0.05 , and 0.005 for reference . we tested 30 frequencies , and oversampled each frequency by a factor of 50 ; this resulted in a periodogram covering a wider range of periods than is interesting ( up to @xmath11500 days ) , so we show only part of the periodogram here . the periodogram shows a single strong peak , with the maximum power of 10.4 at a period of 5.60 days , corresponding to @xmath22 = 0.0009 ; the range of periods which are detected at a @xmath23@xmath3 level is 5.45.8 days . while there is a small peak at the 3 day period , @xmath22 = 1 for this peak xb158 is a high inclination x - ray binary associated with the m31 globular cluster b158 . it exhibits deep intensity dips on a 2.78 hr period in some observations but not others , prompting @xcite to suggest that the disk is precessing , caused by the `` superhumping '' phenomenon observed in low mass ratio systems where the disk crosses the 3:1 resonance with the donor star . @xcite predicted a disk precession period of 29@xmath01 times the orbital period , i.e. 81@xmath03 hr , inspiring a month of daily monitoring of the m31 central region by swift . fitting a sinusoid to the lightcurve revealed a 5.65@xmath00.05 day super - orbital period ( 1@xmath3 uncertainties ) ; the 0.310 kev luminosity varied by a factor @xmath15 , which is consistent with the range of luminosities observed in the acis observations of our 13 + year chandra monitoring campaign @xcite . the peak of the lomb - scargle periodogram is at 5.6 days , consistent with that obtained from least squares fitting of the lightcurve with a sinusoid . the suggested super - orbital period is @xmath170% longer than predicted by our 3d sph simulations @xcite . none of the authors of the current paper are experts in sph ; however , j. r. murray stated in a private communication that the longer than expected super - orbital period is likely due to the mass ratio of the donor to the accretor being lower than assumed . the mass of a roche lobe filling main sequence star , @xmath30 , may be approximated as @xmath30 @xmath31 0.11 @xmath32 , where @xmath32 is the orbital period in hours @xcite , and we originally assumed a donor mass of @xmath10.30 @xmath17 . for the accretor , we assumed a 1.4 @xmath33 neutron star . a power law emission model fit to the 2004 july 16 xmm - newton spectrum of b158 yielded a photon index of 0.57@xmath00.09 @xcite , which is considerably harder than any spectrum emitted by a black hole binary . however , some neutron stars have masses @xmath22@xmath17 ( see e.g. * ? ? ? * ; * ? ? ? * ) , and it is possible that xb158 contains a particularly massive neutron star . we noted in @xcite that other xbs such as xb146 exhibited strong luminosity fluctuations between fairly consistent maxima and minima during our chandra monitoring observations , and suggested that this long - term behavior could be indicative of a short period / low mass ratio system . our new findings support this hypothesis . we thank the anonymous referees for suggesting improvements to this work , in particular prompting more rigorous estimation of the super - orbital period . we thank the swift team for making this work possible . this work was funded by swift grant nnx13aj76 g .
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the m31 globular cluster x - ray binary xb158 ( a.k.a . bo 158 ) exhibits intensity dips on a 2.78 hr period in some observations , but not others .
the short period suggests a low mass ratio , and an asymmetric , precessing disk due to additional tidal torques from the donor star since the disk crosses the 3:1 resonance .
previous theoretical 3d smoothed particle hydrodynamical modeling suggested a super - orbital disk precession period 29@xmath01 times the orbital period , i.e. @xmath181@xmath03 hr . we conducted a swift monitoring campaign of 30 observations over @xmath11 month in order to search for evidence of such a super - orbital period . fitting the 0.310 kev swift xrt luminosity lightcurve with a sinusoid yielded a period of 5.65@xmath00.05 days , and a @xmath25@xmath3 improvement in @xmath4 over the best fit constant intensity model .
a lomb - scargle periodogram revealed that periods 5.45.8 days were detected at a @xmath23@xmath3 level , with a peak at 5.6 days .
we consider this strong evidence for a 5.65 day super - orbital period , @xmath170% longer than the predicted period .
the 0.310 kev luminosity varied by a factor @xmath15 , consistent with variations seen in long - term monitoring from chandra .
we conclude that other x - ray binaries exhibiting similar long - term behaviour are likely to also be x - ray binaries with low mass ratios and super - orbital periods .
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factor analysis is one of the most useful tools for modeling common dependence among multivariate outputs . suppose that we observe data @xmath0 that can be decomposed as @xmath1 where @xmath2 are unobservable common factors ; @xmath3 are corresponding factor loadings for variable @xmath4 , and @xmath5 denotes the idiosyncratic component that can not be explained by the static common component . here , @xmath6 and @xmath7 , respectively , denote the dimension and sample size of the data . model ( [ eq1.1 ] ) has broad applications in the statistics literature . for instance , @xmath8 can be expression profiles or blood oxygenation level dependent ( bold ) measurements for the @xmath9th microarray , proteomic or fmri - image , whereas @xmath4 represents a gene or protein or a voxel . see , for example , @xcite . the separations between the common factors and idiosyncratic components are carried out by the low - rank plus sparsity decomposition . see , for example , @xcite . the factor model ( [ eq1.1 ] ) has also been extensively studied in the econometric literature , in which @xmath10 is the vector of economic outputs at time @xmath9 or excessive returns for individual assets on day @xmath9 . the unknown factors and loadings are typically estimated by the principal component analysis ( pca ) and the separations between the common factors and idiosyncratic components are characterized via static pervasiveness assumptions . see , for instance , @xcite among others . in this paper , we consider static factor model , which differs from the dynamic factor model [ @xcite , @xcite ( @xcite ) ] . the dynamic model allows more general infinite dimensional representations . for this type of model , the frequency domain pca [ @xcite ] was applied on the spectral density . the so - called _ dynamic pervasiveness _ condition also plays a crucial role in achieving consistent estimation of the spectral density . accurately estimating the loadings and unobserved factors are very important in statistical applications . in calculating the false - discovery proportion for large - scale hypothesis testing , one needs to adjust accurately the common dependence via subtracting it from the data in ( [ eq1.1 ] ) [ @xcite ] . in financial applications , we would like to understand accurately how each individual stock depends on unobserved common factors in order to appreciate its relative performance and risks . in the aforementioned applications , dimensionality is much higher than sample - size . however , the existing asymptotic analysis shows that the consistent estimation of the parameters in model ( [ eq1.1 ] ) requires a relatively large @xmath7 . in particular , the individual loadings can be estimated no faster than @xmath11 . but large sample sizes are not always available . even with the availability of `` big data , '' heterogeneity and other issues make direct applications of ( [ eq1.1 ] ) with large @xmath7 infeasible . for instance , in financial applications , to pertain the stationarity in model ( [ eq1.1 ] ) with time - invariant loading coefficients , a relatively short time series is often used . to make observed data less serially correlated , monthly returns are frequently used to reduce the serial correlations , yet a monthly data over three consecutive years contain merely 36 observations . to overcome the aforementioned problems , and when relevant covariates are available , it may be helpful to incorporate them into the model . let @xmath12 be a vector of @xmath13-dimensional covariates associated with the @xmath4th variables . in the seminal papers by @xcite and @xcite , the authors studied the following semi - parametric factor model : @xmath14 where loading coefficients in ( [ eq1.1 ] ) are modeled as @xmath15 for some functions @xmath16 . for instance , in health studies , @xmath17 can be individual characteristics ( e.g. , age , weight , clinical and genetic information ) ; in financial applications @xmath17 can be a vector of firm - specific characteristics ( market capitalization , price - earning ratio , etc . ) . the semiparametric model ( [ eq1.2 ] ) , however , can be restrictive in many cases , as it requires that the loading matrix be fully explained by the covariates . a natural relaxation is the following semiparametric model : @xmath18 where @xmath19 is the component of loading coefficient that can not be explained by the covariates @xmath17 . let @xmath20 . we assume that @xmath21 have mean zero , and are independent of @xmath22 and @xmath23 . in other words , we impose the following factor structure : @xmath24 which reduces to model ( [ eq1.2 ] ) when @xmath25 and model ( [ eq1.1 ] ) when @xmath26 . when @xmath17 genuinely explains a part of loading coefficients @xmath27 , the variability of @xmath28 is smaller than that of @xmath27 . hence , the coefficient @xmath19 can be more accurately estimated by using regression model ( [ eq1.3 ] ) , as long as the functions @xmath29 can be accurately estimated . let @xmath30 be the @xmath31 matrix of @xmath32 , @xmath33 be the @xmath34 matrix of @xmath35 , @xmath36 be the @xmath37 matrix of @xmath38 , @xmath39 be the @xmath37 matrix of @xmath19 and @xmath40 be @xmath31 matrix of @xmath5 . then model ( [ eq1.4 ] ) can be written in a more compact matrix form : @xmath41 we treat the loadings @xmath36 and @xmath39 as realizations of random matrices throughout the paper . this model is also closely related to the _ supervised singular value decomposition _ model , recently studied by @xcite . the authors showed that the model is useful in studying the gene expression and single - nucleotide polymorphism ( snp ) data , and proposed an em algorithm for parameter estimation . we propose a projected - pca estimator for both the loading functions and factors . our estimator is constructed by first projecting @xmath30 onto the sieve space spanned by @xmath22 , then applying pca to the projected data or fitted values . due to the approximate orthogonality condition of @xmath42 , @xmath40 and @xmath39 , the projection of @xmath30 is approximately @xmath43 , as the smoothing projection suppresses the noise terms @xmath39 and @xmath40 substantially . therefore , applying pca to the projected data allows us to work directly on the sample covariance of @xmath43 , which is @xmath44 under normalization conditions . this substantially improves the estimation accuracy , and also facilitates the theoretical analysis . in contrast , the traditional pca method for factor analysis [ e.g. , @xcite , @xcite ] is no longer suitable in the current context . moreover , the idea of projected - pca is also potentially applicable to dynamic factor models of @xcite , by first projecting the data onto the covariate space . the asymptotic properties of the proposed estimators are carefully studied . we demonstrate that as long as the projection is genuine , the consistency of the proposed estimator for latent factors and loading matrices requires only @xmath45 , and @xmath7 does not need to grow , which is attractive in the typical high - dimension - low - sample - size ( hdlss ) situations [ e.g. , @xcite ] . in addition , if both @xmath6 and @xmath7 grow simultaneously , then with sufficiently smooth @xmath29 , using the sieve approximation , the rate of convergence for the estimators is much faster than those of the existing results for model ( [ eq1.1 ] ) . typically , the loading functions can be estimated at a convergence rate @xmath46 , and the factor can be estimated at @xmath47 . throughout the paper , @xmath48 and @xmath49 are assumed to be constant and do not grow . let @xmath50 be a @xmath37 matrix of @xmath51 . model ( [ eq1.3 ] ) implies a decomposition of the loading matrix : @xmath52 where @xmath36 and @xmath39 are orthogonal loading components in the sense that @xmath53 . we conduct two specification tests for the hypotheses : @xmath54 the first problem is about testing whether the observed covariates have explaining power on the loadings . if the null hypothesis is rejected , it gives us the theoretical basis to employ the projected - pca , as the projection is now genuine . our empirical study on the asset returns shows that firm market characteristics do have explanatory power on the factor loadings , which lends further support to our projected - pca method . the second tests whether covariates fully explain the loadings . our aforementioned empirical study also shows that model ( [ eq1.2 ] ) used in the financial econometrics literature is inadequate and more generalized model ( [ eq1.5 ] ) is necessary . as claimed earlier , even if @xmath55 does not hold , as long as @xmath56 , the projected - pca can still consistently estimate the factors as @xmath45 , and @xmath7 may or may not grow . our simulated experiments confirm that the estimation accuracy is gained more significantly for small @xmath7 s . this shows one of the benefits of using our projected - pca method over the traditional methods in the literature . in addition , as a further illustration of the benefits of using projected data , we apply the projected - pca to consistently estimate the number of factors , which is similar to those in @xcite and @xcite . different from these authors , our method applies to the projected data , and we demonstrate numerically that this can significantly improve the estimation accuracy . we focus on the case when the observed covariates are time - invariant . when @xmath7 is small , these covariates are approximately locally constant , so this assumption is reasonable in practice . on the other hand , there may exist individual characteristics that are time - variant [ e.g. , see @xcite ] . we expect the conclusions in the current paper to still hold if some smoothness assumptions are added for the time varying components of the covariates . due to the space limit , we provide heuristic discussions on this case in the supplementary material of this paper [ @xcite ] . in addition , note that in the usual factor model , @xmath50 was assumed to be deterministic . in this paper , however , @xmath50 is mainly treated to be stochastic , and potentially depend on a set of covariates . but we would like to emphasize that the results presented in section [ 1541512515 ] under the framework of more general factor models hold regardless of whether @xmath50 is stochastic or deterministic . finally , while some financial applications are presented in this paper , the projected - pca is expected to be useful in broad areas of statistical applications [ e.g. , see @xcite for applications in gene expression data analysis ] . throughout this paper , for a matrix @xmath57 , let @xmath58 and @xmath59 , @xmath60 denote its frobenius , spectral and max- norms . let @xmath61 and @xmath62 denote the minimum and maximum eigenvalues of a square matrix . for a vector @xmath63 , let @xmath64 denote its euclidean norm . the rest of the paper is organized as follows . section [ sec2 ] introduces the new projected - pca method and defines the corresponding estimators for the loadings and factors . sections [ 1541512515 ] and [ s4 ] provide asymptotic analysis of the introduced estimators . section [ sec5 ] introduces new specification tests for the orthogonal decomposition of the semiparametric loadings . section [ sec6 ] concerns about estimating the number of factors . section [ sec7 ] presents numerical results . finally , section [ sec8 ] concludes . all the proofs are given in the and the supplementary material [ @xcite ] . in the high - dimensional factor model , let @xmath50 be the @xmath37 matrix of loadings . then the general model ( [ eq1.1 ] ) can be written as @xmath65 suppose we additionally observe a set of covariates @xmath66 . the basic idea of the projected - pca is to smooth the observations @xmath67 for each given day @xmath9 against its associated covariates . more specifically , let @xmath68 be the fitted value after regressing @xmath67 on @xmath69 for each given @xmath9 . this results in a smooth or projected observation matrix @xmath70 , which will also be denoted by @xmath71 . the projected - pca then estimates the factors and loadings by running the pca based on the projected data @xmath70 . here , we heuristically describe the idea of projected - pca ; rigorous analysis will be carried out afterward . let @xmath72 be a space spanned by @xmath73 , which is orthogonal to the error matrix @xmath40 . let @xmath74 denote the projection matrix onto @xmath72 [ whose formal definition will be given in ( [ eq2.5 ] ) below . at the population level , @xmath74 approximates the conditional expectation operator @xmath75 , which satisfies @xmath76 , then @xmath77 and @xmath78 . hence , analyzing the projected data @xmath79 is an approximately noiseless problem , and the sample covariance has the following approximation : @xmath80 we now argue that @xmath33 and @xmath81 can be recovered from the projected data @xmath70 under some suitable normalization condition . the normalization conditions we impose are @xmath82 under this normalization , using ( [ eq2.1a ] ) , @xmath83 . we conclude that the columns of @xmath84 are approximately @xmath85 times the first @xmath86 eigenvectors of the @xmath87 matrix @xmath88 . therefore , the projected - pca naturally defines a factor estimator @xmath89 using the first @xmath86 principal components of @xmath90 . the projected loading matrix @xmath81 can also be recovered from the projected data @xmath71 in two ( equivalent ) ways . given @xmath33 , from @xmath91 , we see @xmath92 . alternatively , consider the @xmath93 projected sample covariance : @xmath94 where @xmath95 is a remaining term depending on @xmath96 . right multiplying @xmath81 and ignoring @xmath95 , we obtain @xmath97 . hence , the ( normalized ) columns of @xmath81 approximate the first @xmath86 eigenvectors of @xmath98 , the @xmath93 sample covariance matrix based on the projected data . therefore , we can either estimate @xmath81 by @xmath99 given @xmath89 , or by the leading eigenvectors of @xmath98 . in fact , we shall see later that these two estimators are equivalent . if in addition , @xmath100 , that is , the loading matrix belongs to the space @xmath72 , then @xmath50 can also be recovered from the projected data . the above arguments are the fundament of the projected - pca , and provide the rationale of our estimators to be defined in section [ sec2.3 ] . we shall make the above arguments rigorous by showing that the projected error @xmath101 is asymptotically negligible and , therefore , the idiosyncratic error term @xmath40 can be completely removed by the projection step . as one of the useful examples of forming the space @xmath102 and the projection operator , this paper considers model ( [ eq1.4 ] ) , where @xmath17 s and @xmath32 s are the only observable data , and @xmath103 are unknown nonparametric functions . the specific case ( [ eq1.2 ] ) ( with @xmath104 ) was used extensively in the financial studies by @xcite , @xcite and @xcite , with @xmath17 s being the observed `` market characteristic variables . '' we assume @xmath86 to be known for now . in section [ sec6 ] , we will propose a projected - eigenvalue - ratio method to consistently estimate @xmath86 when it is unknown . we assume that @xmath105 does not depend on @xmath9 , which means the loadings represent the cross - sectional heterogeneity only . such a model specification is reasonable since in many applications using factor models , to pertain the stationarity of the time series , the analysis can be conducted within each fixed time window with either a fixed or slowly - growing @xmath7 . through localization in time , it is not stringent to require the loadings be time - invariant . this also shows one of the attractive features of our asymptotic results : under mild conditions , our factor estimates are consistent even if @xmath7 is finite . to nonparametrically estimate @xmath105 without the curse of dimensionality when @xmath17 is multivariate , we assume @xmath29 to be additive : for each @xmath106 , there are @xmath107 nonparametric functions such that @xmath108 each additive component of @xmath109 is estimated by the sieve method . define @xmath110 to be a set of basis functions ( e.g. , b - spline , fourier series , wavelets , polynomial series ) , which spans a dense linear space of the functional space for @xmath111 . then for each @xmath112 , @xmath113 here , @xmath114 are the sieve coefficients of the @xmath115th additive component of @xmath105 , corresponding to the @xmath116th factor loading ; @xmath117 is a `` remaining function '' representing the approximation error ; @xmath118 denotes the number of sieve terms which grows slowly as @xmath45 . the basic assumption for sieve approximation is that @xmath119 as @xmath120 . we take the same basis functions in ( [ eq2.4 ] ) purely for simplicity of notation . define , for each @xmath121 and for each @xmath122 , @xmath123 then we can write @xmath124 let @xmath125 be a @xmath126 matrix of sieve coefficients , @xmath127 be a @xmath128 matrix of basis functions , and @xmath129 be @xmath37 matrix with the @xmath130th element @xmath131 . then the matrix form of ( [ eq2.3 ] ) and ( [ eq2.4 ] ) is @xmath132 substituting this into ( [ eq1.5 ] ) , we write @xmath133 we see that the residual term consists of two parts : the sieve approximation error @xmath134 and the idiosyncratic @xmath40 . furthermore , the random effect assumption on the coefficients @xmath39 makes it also behave like noise , and hence negligible when the projection operator @xmath74 is applied . based on the idea described in section [ sec2.1 ] , we propose a projected - pca method , where @xmath72 is the sieve space spanned by the basis functions of @xmath42 , and @xmath74 is chosen as the projection matrix onto @xmath72 , defined by the @xmath93 projection matrix @xmath135 the estimators of the model parameters in ( [ eq1.5 ] ) are defined as follows . the columns of @xmath136 are defined as the eigenvectors corresponding to the first @xmath86 largest eigenvalues of the @xmath87 matrix @xmath137 , and @xmath138 is the estimator of @xmath36 . the intuition can be readily seen from the discussions in section [ sec2.1 ] , which also provides an alternative formulation of @xmath139 as follows : let @xmath140 be a @xmath141 diagonal matrix consisting of the largest @xmath86 eigenvalues of the @xmath93 matrix @xmath142 . let @xmath143 be a @xmath37 matrix whose columns are the corresponding eigenvectors . according to the relation @xmath144 described in section [ sec2.1 ] , we can also estimate @xmath36 or @xmath81 by @xmath145 we shall show in lemma [ la.1add ] that this is equivalent to ( [ eq2.6 ] ) . therefore , unlike the traditional pca method for usual factor models [ e.g. , @xcite , @xcite ] , the projected - pca takes the principal components of the projected data @xmath71 . the estimator is thus invariant to the rotation - transformations of the sieve bases . the estimation of the loading component @xmath39 that can not be explained by the covariates can be estimated as follows . with the estimated factors @xmath89 , the least - squares estimator of loading matrix is @xmath146 , by using ( [ eq2.1 ] ) and ( [ eq2.2 ] ) . therefore , by ( [ eq1.5 ] ) , a natural estimator of @xmath147 is @xmath148 consider a panel data model with time - varying coefficients as follows : @xmath149 where @xmath17 is a @xmath13-dimensional vector of time - invariant regressors for individual @xmath4 ; @xmath150 denotes the unobservable random time effect ; @xmath5 is the regression error term . the regression coefficient @xmath151 is also assumed to be random and time - varying , but is common across the cross - sectional individuals . the semiparametric factor model admits ( [ eq2.8 ] ) as a special case . note that ( [ eq2.8 ] ) can be rewritten as @xmath152 with @xmath153 unobservable `` factors '' @xmath154 and `` loading '' @xmath155 . the model ( [ eq1.4 ] ) being considered , on the other hand , allows more general nonparametric loading functions . let us first consider the asymptotic performance of the projected - pca in the conventional factor model : @xmath156 in the usual statistical applications for factor analysis , the latent factors are assumed to be serially independent , while in financial applications , the factors are often treated to be weakly dependent time series satisfying strong mixing conditions . we now demonstrate by a simple example that latent factors @xmath33 can be estimated at a faster rate of convergence by projected - pca than the conventional pca and that they can be consistently estimated even when sample size @xmath7 is finite . [ ex3.1 ] to appreciate the intuition , let us consider a specific case in which @xmath157 so that model ( [ eq1.4 ] ) reduces to @xmath158 assume that @xmath159 is so smooth that it is in fact a constant @xmath160 ( otherwise , we can use a local constant approximation ) , where @xmath161 . then the model reduces to @xmath162 the projection in this case is averaging over @xmath4 , which yields @xmath163 where @xmath164 , @xmath165 and @xmath166 denote the averages of their corresponding quantities over @xmath4 . for the identification purpose , suppose @xmath167 , and @xmath168 . ignoring the last two terms , we obtain estimators @xmath169 these estimators are special cases of the projected - pca estimators . to see this , define @xmath170 , and let @xmath171 be a @xmath6-dimensional column vector of ones . take a naive basis @xmath172 ; then the projected data matrix is in fact @xmath173 . consider the @xmath87 matrix @xmath174 , whose largest eigenvalue is @xmath175 . from @xmath176 we have the first eigenvector of @xmath137 equals @xmath177 . hence , the projected - pca estimator of factors is @xmath178 . in addition , the projected - pca estimator of the loading vector @xmath179 is @xmath180 hence , the projected - pca - estimator of @xmath181 equals @xmath182 . these estimators match with ( [ e3.2 ] ) . moreover , since the ignored two terms @xmath183 and @xmath184 are of order @xmath185 , @xmath186 and @xmath187 converge whether or not @xmath7 is large . note that this simple example satisfies all the assumptions to be stated below , and @xmath188 and @xmath189 achieve the same rate of convergence as that of theorem [ th4.1 ] . we shall present more details about this example in appendix g in the supplementary material [ @xcite ] . we now state the conditions and results formally in the more general factor model ( [ eq3.1 ] ) . recall that the projection matrix is defined as @xmath190 the following assumption is the key condition of the projected - pca . [ ass3.1 ] there are positive constants @xmath191 and @xmath192 such that , with probability approaching one ( as @xmath193 ) , @xmath194 since the dimensions of @xmath195 and @xmath50 are , respectively , @xmath196 and @xmath37 , assumption [ ass3.1 ] requires @xmath197 , which is reasonable since we assume @xmath86 , the number of factors , to be fixed throughout the paper . assumption [ ass3.1 ] is similar to the _ pervasive _ condition on the factor loadings [ @xcite ] . in our context , this condition requires the covariates @xmath42 have nonvanishing explaining power on the loading matrix , so that the projection matrix @xmath198 has spiked eigenvalues . note that it rules out the case when @xmath42 is completely unassociated with the loading matrix @xmath50 ( e.g. , when @xmath42 is pure noise ) . one of the typical examples that satisfies this assumption is the semiparametric factor model [ model ( [ eq1.4 ] ) ] . we shall study this specific type of factor model in section [ s4 ] , and prove assumption [ ass3.1 ] in the supplementary material [ @xcite ] . note that @xmath33 and @xmath50 are not separately identified , because for any nonsingular @xmath199 , @xmath200 . therefore , we assume the following . [ ass3.2 ] almost surely , @xmath201 and @xmath198 is a @xmath141 diagonal matrix with distinct entries . this condition corresponds to the pc1 condition of @xcite , which separately identifies the factors and loadings from their product @xmath202 . it is often used in factor analysis for identification , and means that the columns of factors and loadings can be orthogonalized [ also see @xcite ] . [ ass3.3 ] ( i ) there are @xmath203 and @xmath204 so that with probability approaching one ( as @xmath193 ) , @xmath205 \(ii ) @xmath206 . note that @xmath207 and @xmath208 is a vector of dimensionality @xmath209 . thus , condition ( i ) can follow from the strong law of large numbers . for instance , @xmath22 are weakly correlated and in the population level @xmath210 is well - conditioned . in addition , this condition can be satisfied through proper normalizations of commonly used basis functions such as b - splines , wavelets , fourier basis , etc . in the general setup of this paper , we allow @xmath211 s to be cross - sectionally dependent and nonstationary . regularity conditions about weak dependence and stationarity are imposed only on @xmath212 as follows . we impose the strong mixing condition . let @xmath213 and @xmath214 denote the @xmath215-algebras generated by @xmath216 and @xmath217 , respectively . define the mixing coefficient @xmath218 [ ass3.4 ] ( i ) @xmath219 is strictly stationary . in addition , @xmath220 for all @xmath221 ; @xmath222 is independent of @xmath223 . strong mixing : there exist @xmath224 such that for all @xmath225 , @xmath226 weak dependence : there is @xmath227 so that @xmath228 exponential tail : there exist @xmath229 satisfying @xmath230 and @xmath231 , such that for any @xmath232 , @xmath122 and @xmath233 , @xmath234 assumption [ ass3.4 ] is standard , especially condition ( iii ) is commonly imposed for high - dimensional factor analysis [ e.g. , @xcite ] , which requires @xmath235 be weakly dependent both serially and cross - sectionally . it is often satisfied when the covariance matrix @xmath236 is sufficiently sparse under the strong mixing condition . we provide primitive conditions of condition ( iii ) in the supplementary material [ @xcite ] . formally , we have the following theorem : [ th3.1 ] consider the conventional factor model ( [ eq3.1 ] ) with assumptions [ ass3.1][ass3.4 ] . the projected - pca estimators @xmath237 and @xmath238 defined in section [ sec2.3 ] satisfy , as @xmath239 [ @xmath240 may either grow simultaneously with @xmath6 satisfying @xmath241 or stay constant with @xmath242 , @xmath243 to compare with the traditional pca method , the convergence rate for the estimated factors is improved for small @xmath7 . in particular , the projected - pca does not require @xmath244 , and also has a good rate of convergence for the loading matrix up to a projection transformation . hence , we have achieved a finite-@xmath7 consistency , which is particularly interesting in the `` high - dimensional - low - sample - size '' ( hdlss ) context , considered by @xcite . in contrast , the traditional pca method achieves a rate of convergence of @xmath245 for estimating factors , and @xmath246 for estimating loadings . see remarks [ re4.1 ] , [ re4.2 ] below for additional details . let @xmath247 be the @xmath93 covariance matrix of @xmath248 . convergence ( [ eq3.4add ] ) in theorem [ th3.1 ] also describes the relationship between the leading eigenvectors of @xmath98 and those of @xmath249 . to see this , let @xmath250 be the eigenvectors of @xmath249 corresponding to the first @xmath86 eigenvalues . under the _ pervasiveness condition _ , @xmath251 can be approximated by @xmath50 multiplied by a positive definite matrix of transformation [ @xcite ] . in the context of projected - pca , by definition , @xmath252 ; here we recall that @xmath253 is a diagonal matrix consisting of the largest @xmath86 eigenvalues of @xmath98 , and @xmath254 is a @xmath37 matrix whose columns are the corresponding eigenvectors . then ( [ eq3.4add ] ) immediately implies the following corollary , which complements the pca consistency in _ spiked covariance models _ [ e.g. , @xcite and @xcite ] . [ th3.2 ] under the conditions of theorem [ th3.1 ] , there is a @xmath141 positive definite matrix @xmath255 , whose eigenvalues are bounded away from both zero and infinity , so that as @xmath193 [ @xmath240 may either grow simultaneously with @xmath6 satisfying @xmath241 or stay constant with @xmath242 , @xmath256 in the semiparametric factor model , it is assumed that @xmath257 , where @xmath105 is a nonparametric smooth function for the observed covariates , and @xmath19 is the unobserved random loading component that is independent of @xmath17 . hence , the model is written as @xmath258 in the matrix form , @xmath259 and @xmath36 does not vanish ( pervasive condition ; see assumption [ ass4.2 ] below ) . the estimators @xmath237 and @xmath238 are the projected - pca estimators as defined in section [ sec2.3 ] . we now define the estimator of the nonparametric function @xmath29 , @xmath260 . in the matrix form , the projected data has the following sieve approximated representation : @xmath261 where @xmath262 is `` small '' because @xmath39 and @xmath40 are orthogonal to the function space spanned by @xmath42 , and @xmath129 is the sieve approximation error . the sieve coefficient matrix @xmath263 can be estimated by least squares from the projected model ( [ eq4.1 ] ) : ignore @xmath264 , replace @xmath33 with @xmath237 , and solve ( [ eq4.1 ] ) to obtain @xmath265^{-1}\phi ( \bx ) ' \by{\widehat\bf}.\ ] ] we then estimate @xmath29 by @xmath266 where @xmath267 denotes the support of @xmath17 . when @xmath268 , @xmath36 can be understood as the projection of @xmath50 onto the sieve space spanned by @xmath42 . hence , the following assumption is a specific version of assumptions [ ass3.1 ] and [ ass3.2 ] in the current context . [ ass4.1 ] ( i ) almost surely , @xmath201 and @xmath269 is a @xmath141 diagonal matrix with distinct entries . \(ii ) there are two positive constants @xmath191 and @xmath192 so that with probability approaching one ( as @xmath193 ) , @xmath270 in this section , we do not need to assume @xmath271 to be i.i.d . for the estimation purpose . cross - sectional weak dependence as in assumption [ ass4.2](ii ) below would be sufficient . the i.i.d . assumption will be only needed when we consider specification tests in section [ sec5 ] . write @xmath272 , and @xmath273 [ ass4.2 ] ( i ) @xmath274 and @xmath22 is independent of @xmath275 . \(ii ) @xmath276 , @xmath277 and @xmath278 the following set of conditions is concerned about the accuracy of the sieve approximation . [ ass4.3 ] @xmath279 , \(i ) the loading component @xmath280 belongs to a hlder class @xmath281 defined by @xmath282 for some @xmath283 ; \(ii ) the sieve coefficients @xmath284 satisfy for @xmath285 , as @xmath286 , @xmath287 where @xmath288 is the support of the @xmath115th element of @xmath17 , and @xmath118 is the sieve dimension . \(iii ) @xmath289 . condition ( ii ) is satisfied by common basis . for example , when @xmath290 is polynomial basis or b - splines , condition ( ii ) is implied by condition ( i ) [ see , e.g. , @xcite and @xcite ] . [ th4.1 ] suppose @xmath241 . under assumptions [ ass3.3 ] , [ ass3.4 ] , [ ass4.1][ass4.3 ] , as @xmath291 , @xmath7 can be either divergent or bounded , we have that @xmath292 in addition , if @xmath244 simultaneously with @xmath6 and @xmath118 , then @xmath293 the optimal @xmath294 simultaneously minimizes the convergence rates of the factors and nonparametric loading function @xmath29 . it also satisfies the constraint @xmath295 as @xmath296 . with @xmath297 , we have @xmath298 and @xmath299 satisfies @xmath300 some remarks about these rates of convergence compared with those of the conventional factor analysis are in order . [ re4.1]the rates of convergence for factors and nonparametric functions do not require @xmath244 . when @xmath301 , @xmath302 the rates still converge fast when @xmath6 is large , demonstrating the blessing of dimensionality . this is an attractive feature of the projected - pca in the hdlss context , as in many applications , the stationarity of a time series and the time - invariance assumption on the loadings hold only for a short period of time . in contrast , in the usual factor analysis , consistency is granted only when @xmath303 . for example , according to @xcite ( lemma c.1 ) , the regular pca method has the following convergence rate : @xmath304 which is inconsistent when @xmath7 is bounded . [ re4.2]when both @xmath6 and @xmath7 are large , the projected - pca estimates factors as well as the regular pca does , and achieves a faster rate of convergence for the estimated loadings when @xmath19 vanishes . in this case , @xmath305 , the loading matrix is estimated by @xmath306 , and @xmath307 in contrast , the regular pca method as in @xcite yields @xmath308 comparing these rates , we see that when @xmath29 s are sufficiently smooth ( larger @xmath309 ) , the rate of convergence for the estimated loadings is also improved . the loading matrix always has the following orthogonal decomposition : @xmath310 where @xmath39 is interpreted as the loading component that can not be explained by @xmath42 . we consider two types of specification tests : testing @xmath311 , and @xmath312 . the former tests whether the observed covariates have explaining powers on the loadings , while the latter tests whether the covariates fully explain the loadings . the former provides a diagnostic tool as to whether or not to employ the projected - pca ; the latter tests the adequacy of the semiparametric factor models in the literature . testing whether the observed covariates have explaining powers on the factor loadings can be formulated as the following null hypothesis : @xmath314 due to the approximate orthogonality of @xmath42 and @xmath39 , we have @xmath315 . hence , the null hypothesis is approximately equivalent to @xmath316 this motivates a statistic @xmath317 for a consistent loading estimator @xmath318 . normalizing the test statistic by its asymptotic variance leads to the test statistic @xmath319 where the @xmath141 matrix @xmath320 is the weight matrix . the null hypothesis is rejected when @xmath321 is large . the projected - pca estimator is inappropriate under the null hypothesis as the projection is not genuine . we therefore use the least squares estimator @xmath322 , leading to the test statistic @xmath323 here , we take @xmath324 as the traditional pca estimator : the columns of @xmath325 are the first @xmath86 eigenvectors of the @xmath87 data matrix @xmath326 . connor , hagmann and linton ( @xcite ) applied the semiparametric factor model to analyzing financial returns , who assumed that @xmath328 , that is , the loading matrix can be fully explained by the observed covariates . it is therefore natural to test the following null hypothesis of specification : @xmath329 recall that @xmath330 so that @xmath331 . therefore , essentially the specification testing problem is equivalent to testing @xmath332 that is , we are testing whether the loading matrix in the factor model belongs to the space spanned by the observed covariates . a natural test statistic is thus based on the weighted quadratic form @xmath333 for some @xmath334 positive definite weight matrix @xmath335 , where @xmath237 is the projected - pca estimator for factors and @xmath336 . to control the size of the test , we take @xmath337 , where @xmath338 is a diagonal covariance matrix of @xmath339 under @xmath340 , assuming that @xmath341 are uncorrelated . we replace @xmath342 with its consistent estimator : let @xmath343 . define @xmath344 then the operational test statistic is defined to be @xmath345 the null hypothesis is rejected for large values of @xmath346 . for the testing purpose , we assume @xmath347 to be i.i.d . , and let @xmath348 simultaneously . the following assumption regulates the relation between @xmath7 and @xmath6 . [ ass5.1 ] suppose ( i ) @xmath349 are independent and identically distributed ; @xmath350 , and @xmath351 ; @xmath118 and @xmath309 satisfy : @xmath352 , and @xmath353 . condition ( ii ) requires a balance of the dimensionality and the sample size . on one hand , a relatively large sample size is desired [ @xmath354 so that the effect of estimating @xmath342 is negligible asymptotically . on the other hand , as is common in high - dimensional factor analysis , a lower bound of the dimensionality is also required [ condition @xmath350 ] to ensure that the factors are estimated accurately enough . such a required balance is common for high - dimensional factor analysis [ e.g. , @xcite , @xcite ] and in the recent literature for pca [ e.g. , @xcite , @xcite ] . the i.i.d . assumption of covariates @xmath17 in condition ( i ) can be relaxed with further distributional assumptions on @xmath356 ( e.g. , assuming @xmath356 to be gaussian ) . the conditions on @xmath118 in condition ( iii ) is consistent with those of the previous sections . we focus on the case when @xmath357 is gaussian , and show that under @xmath358 , @xmath359 and under @xmath55 @xmath360 whose conditional distributions ( given @xmath33 ) under the null are @xmath361 with degree of freedom , respectively , @xmath362 and @xmath363 . we can derive their standardized limiting distribution as @xmath364 . this is given in the following result . [ th5.1 ] suppose assumptions [ ass3.3 ] , [ ass3.4 ] , [ ass4.2 ] , [ ass5.1 ] hold . then under @xmath358 , @xmath365 where @xmath48 and @xmath49 . in addition , suppose assumptions [ ass4.1 ] and [ ass4.3 ] further hold , @xmath366 is i.i.d . @xmath367 with a diagonal covariance matrix @xmath338 whose elements are bounded away from zero and infinity . then under @xmath55 , @xmath368 in practice , when a relatively small sieve dimension @xmath118 is used , one can instead use the upper @xmath369-quantile of the @xmath370 distribution for @xmath371 . we require @xmath5 be independent across @xmath9 , which ensures that the covariance matrix of the leading term @xmath372 to have a simple form @xmath373 . this assumption can be relaxed to allow for weakly dependent @xmath374 , but many autocovariance terms will be involved in the covariance matrix . one may regularize standard autocovariance matrix estimators such as @xcite and @xcite to account for the high dimensionality . moreover , we assume @xmath338 be diagonal to facilitate estimating @xmath342 , which can also be weakened to allow for a nondiagonal but sparse @xmath338 . regularization methods such as thresholding [ @xcite ] can then be employed , though they are expected to be more technically involved . we now address the problem of estimating @xmath48 when it is unknown . once a consistent estimator of @xmath86 is obtained , all the results achieved carry over to the unknown @xmath86 case using a conditioning argument . , then argue that the results still hold unconditionally as @xmath375 . ] in principle , many consistent estimators of @xmath86 can be employed , for example , @xcite , @xcite , @xcite , @xcite . more recently , @xcite and @xcite proposed to select the largest ratio of the adjacent eigenvalues of @xmath326 , based on the fact that the @xmath86 largest eigenvalues of the sample covariance matrix grow as fast as @xmath6 increases , while the remaining eigenvalues either remain bounded or grow slowly . we extend ahn and horenstein s ( @xcite ) theory in two ways . first , when the loadings depend on the observable characteristics , it is more desirable to work on the projected data @xmath71 . due to the orthogonality condition of @xmath40 and @xmath42 , the projected data matrix is approximately equal to @xmath43 . the projected matrix @xmath376 thus allows us to study the eigenvalues of the principal matrix component @xmath377 , which directly connects with the strengths of those factors . since the nonvanishing eigenvalues of @xmath376 and @xmath378 are the same , we can work directly with the eigenvalues of the matrix @xmath379 . second , we allow @xmath380 . let @xmath381 denote the @xmath116th largest eigenvalue of the projected data matrix @xmath137 . we assume @xmath382 , which naturally holds if the sieve dimension @xmath118 slowly grows . the estimator is defined as @xmath383 the following assumption is similar to that of @xcite . recall that @xmath384 is a @xmath31 matrix of the idiosyncratic components , and @xmath385 denotes the @xmath386 covariance matrix of @xmath339 . [ ass6.1 ] the error matrix @xmath40 can be decomposed as @xmath387 where : the eigenvalues of @xmath338 are bounded away from zero and infinity , @xmath388 is a @xmath7 by @xmath7 positive semidefinite nonstochastic matrix , whose eigenvalues are bounded away from zero and infinity , @xmath389 is a @xmath31 stochastic matrix , where @xmath390 is independent in both @xmath4 and @xmath9 , and @xmath391 are i.i.d . isotropic sub - gaussian vectors , that is , there is @xmath392 , for all @xmath232 , @xmath393 there are @xmath394 , almost surely , @xmath395 this assumption allows the matrix @xmath40 to be both cross - sectionally and serially dependent . the @xmath87 matrix @xmath388 captures the serial dependence across @xmath9 . in the special case of no - serial - dependence , the decomposition ( [ eq5.1 ] ) is satisfied by taking @xmath396 . in addition , we require @xmath339 to be sub - gaussian to apply random matrix theories of @xcite . for instance , when @xmath339 is @xmath397 , for any @xmath398 , @xmath399 , and thus condition ( iii ) is satisfied . finally , the _ almost surely _ condition of ( iv ) seems somewhat strong , but is still satisfied by bounded basis functions ( e.g. , fourier basis ) . we show in the supplementary material [ @xcite ] that when @xmath338 is diagonal ( @xmath5 is cross - sectionally independent ) , both the sub - gaussian assumption and condition ( iv ) can be relaxed . the following theorem is the main result of this section . [ th6.1 ] under assumptions of theorem [ th4.1 ] and assumption [ ass6.1 ] , as @xmath400 , if @xmath118 satisfies @xmath401 and @xmath402 ( @xmath118 may either grow or stay constant ) , we have @xmath403 this section presents numerical results to demonstrate the performance of projected - pca method for estimating loadings and factors using both real data and simulated data . we collected stocks in s&p 500 index constituents from crsp which have complete daily closing prices from year 2005 through 2013 , and their corresponding market capitalization and book value from compustat . there are @xmath404 stocks in our data set , whose daily excess returns were calculated . we considered four characteristics @xmath42 as in @xcite for each stock : size , value , momentum and volatility , which were calculated using the data before a certain data analyzing window so that characteristics are treated known . see @xcite for detailed descriptions of these characteristics . all four characteristics are standardized to have mean zero and unit variance . note that the construction makes their values independent of the current data . we fix the time window to be the first quarter of the year 2006 , which contains @xmath405 observations . given the excess returns @xmath406 and characteristics @xmath17 as the input data and setting @xmath407 , we fit loading functions @xmath408 for @xmath409 using the projected - pca method . the four additive components @xmath280 are fitted using the cubic spline in the r package `` gam '' with sieve dimension @xmath410 . all the four loading functions for each factor are plotted in figure [ fig : gcurves ] . the contribution of each characteristic to each factor is quite nonlinear . , @xmath411 from financial returns of 337 stocks in s&p 500 index . they are taken as the true functions in the simulation studies . in each panel ( fixed @xmath115 ) , the true and estimated curves for @xmath412 are plotted and compared . the solid , dashed and dotted red curves are the true curves corresponding to the first , second and third factors , respectively . the blue curves are their estimates from one simulation of the calibrated model with @xmath413 , @xmath414 . ] we now treat the estimated functions @xmath280 as the true loading functions , and calibrate a model for simulations . the `` true model '' is calibrated as follows : take the estimated @xmath280 from the real data as the true loading functions . for each @xmath6 , generate @xmath366 from @xmath415 where @xmath416 is diagonal and @xmath417 sparse . generate the diagonal elements of @xmath416 from gamma(@xmath418 ) with @xmath419 , @xmath420 ( calibrated from the real data ) , and generate the off - diagonal elements of @xmath417 from @xmath421 with @xmath422 , @xmath423 . then truncate @xmath417 by a threshold of correlation @xmath424 to produce a sparse matrix and make it positive definite by r package `` nearpd . '' generate @xmath425 from the i.i.d . gaussian distribution with mean @xmath426 and standard deviation @xmath427 , calibrated with real data . generate @xmath428 from a stationary var model @xmath429 where @xmath430 . the model parameters are calibrated with the market data and listed in table [ table : calibfactor ] . finally , generate @xmath431 . here @xmath432 is a @xmath433 correlation matrix estimated from the real data . @lccd2.4d2.4c@ & + 0.9076 & 0.0049 & 0.0230 & -0.0371 & -0.1226 & @xmath434 + 0.0049 & 0.8737 & 0.0403 & -0.2339 & 0.1060 & @xmath435 + 0.0230 & 0.0403 & 0.9266 & 0.2803 & 0.0755 & @xmath436 + by projected - pca ( p - pca , red solid ) and traditional pca ( dashed blue ) and @xmath437 , @xmath438 by p - pca over 500 repetitions . left panel : @xmath439 , right panel : @xmath440 . ] and @xmath441 over 500 repetitions , by projected - pca ( p - pca , solid red ) and traditional pca ( dashed blue ) . ] we simulate the data from the calibrated model , and estimate the loadings and factors for @xmath442 and @xmath443 with @xmath6 varying from @xmath444 through @xmath445 . the `` true '' and estimated loading curves are plotted in figure [ fig : gcurves ] to demonstrate the performance of projected - pca . note that the `` true '' loading curves in the simulation are taken from the estimates calibrated using the real data . the estimates based on simulated data capture the shape of the true curve , though we also notice slight biases at boundaries . but in general , projected - pca fits the model well . we also compare our method with the traditional pca method [ e.g. , @xcite ] . the mean values of @xmath446 , @xmath447 , @xmath448 and @xmath449 are plotted in figures [ fig : calibg ] and [ fig : calibf ] where @xmath450 [ see section [ design2 ] for definitions of @xmath451 and @xmath452 . the breakdown error for @xmath453 and @xmath39 are also depicted in figure [ fig : calibg ] . in comparison , projected - pca outperforms pca in estimating both factors and loadings including the nonparametric curves @xmath36 and random noise @xmath39 . the estimation errors for @xmath36 of projected - pca decrease as the dimension increases , which is consistent with our asymptotic theory . and @xmath454 over 500 repetitions . p - pca , pca and sls , respectively , represent projected - pca , regular pca and sieve least squares with known factors : design 2 . here , @xmath328 , so @xmath455 . upper two panels : @xmath6 grows with fixed @xmath7 ; bottom panels : @xmath7 grows with fixed @xmath6 . ] and @xmath456 by projected - pca ( solid red ) and pca ( dashed blue ) : design 2 . upper two panels : @xmath6 grows with fixed @xmath7 ; bottom panels : @xmath7 grows with fixed @xmath6 . ] consider a different design with only one observed covariate and three factors . the three characteristic functions are @xmath457 with the characteristic @xmath458 being standard normal . generate @xmath459 from the stationary var(1 ) model , that is , @xmath460 where @xmath461 . we consider @xmath462 . we simulate the data for @xmath442 or @xmath443 and various @xmath6 ranging from @xmath444 to @xmath445 . to ensure that the true factor and loading satisfy the identifiability conditions , we calculate a transformation matrix @xmath199 such that @xmath463 , @xmath464 is diagonal . let the final true factors and loadings be @xmath465 , @xmath466 . for each @xmath6 , we run the simulation for @xmath445 times . we estimate the loadings and factors using both projected - pca and pc . for projected - pca , as in our theorem , we choose @xmath467 , with @xmath468 and @xmath469 . to estimate the loading matrix , we also compare with a third method : sieve - least - squares ( sls ) , assuming the factors are observable . in this case , the loading matrix is estimated by @xmath470 , where @xmath471 is the true factor matrix of simulated data . the estimation error measured in max and standardized frobenius norms for both loadings and factors are reported in figures [ fig : simpleg ] and [ fig : simplef ] . the plots demonstrate the good performance of projected - pca in estimating both loadings and factors . in particular , it works well when we encounter small @xmath7 but a large @xmath6 . in this design , @xmath328 , so the accuracy of estimating @xmath472 is significantly improved by using the projected - pca . figure [ fig : simplef ] shows that the factors are also better estimated by projected - pca than the traditional one , particularly when @xmath7 is small . it is also clearly seen that when @xmath6 is fixed , the improvement on estimating factors is not significant as @xmath7 grows . this matches with our convergence results for the factor estimator . it is also interesting to compare projected - pca with sls ( sieve least - squares with observed factors ) in estimating the loadings , which corresponds to the cases of unobserved and observed factors . as we see from figure [ fig : simpleg ] , when @xmath6 is small , the projected - pca is not as good as sls . but the two methods behave similarly as @xmath6 increases . this further confirms the theory and intuition that as the dimension becomes larger , the effects of estimating the unknown factors are negligible . we now demonstrate the effectiveness of estimating @xmath86 by the projected - pc s eigenvalue - ratio method . the data are simulated in the same way as in design 2 . @xmath442 or @xmath443 and we took the values of @xmath6 ranging from @xmath444 to @xmath445 . we compare our projected - pca based on the projected data matrix @xmath137 to the eigenvalue - ratio test ( ah ) of @xcite and @xcite , which works on the original data matrix @xmath326 . . p - pca and ah , respectively , represent the methods of projected - pca and @xcite . left panel : mean ; right panel : standard deviation . ] for each pair of @xmath473 , we repeat the simulation for @xmath443 times and report the mean and standard deviation of the estimated number of factors in figure [ fig : estimatek ] . the projected - pca outperforms ah after projection , which significantly reduces the impact of idiosyncratic errors . when @xmath413 , we can recover the number of factors almost all the time , especially for large dimensions ( @xmath474 ) . on the other hand , even when @xmath475 , projected - pca still obtains a closer estimated number of factors . we test the loading specifications on the real data . we used the same data set as in section [ sec7.1 ] , consisting of excess returns from 2005 through 2013 . the tests were conducted based on rolling windows , with the length of windows spanning from 10 days , a month , a quarter and half a year . for each fixed window - length ( @xmath7 ) , we computed the standardized test statistic of @xmath321 and @xmath476 , and plotted them along the rolling windows respectively in figure [ fig : testing ] . in almost all cases , the number of factors is estimated to be one in various combinations of @xmath477 . figure [ fig : testing ] suggests that the semiparametric factor model is strongly supported by the data . judging from the upper panel [ testing @xmath478 , we have very strong evidence of the existence of nonvanishing covariate effect , which demonstrates the dependence of the market beta s on the covariates @xmath42 . in other words , the market beta s can be explained at least partially by the characteristics of assets . the results also provide the theoretical basis for using projected - pca to get more accurate estimation . from 2006/01/03 to 2012/11/30 . the dotted lines are @xmath479 . ] in the bottom panel of figure [ fig : testing ] ( testing @xmath480 ) , we see for a majority of periods , the null hypothesis is rejected . in other words , the characteristics of assets can not fully explain the market beta as intuitively expected , and model ( [ eq1.2 ] ) in the literature is inadequate . however , fully nonparametric loadings could be possible in certain time range mostly before financial crisis . during 20082010 , the market s behavior had much more complexities , which causes more rejections of the null hypothesis . the null hypothesis @xmath328 is accepted more often since 2012 . we also notice that larger @xmath7 tends to yield larger statistics in both tests , as the evidence against the null hypothesis is stronger with larger @xmath7 . after all , the semiparametric model being considered provides flexible ways of modeling equity markets and understanding the nonparametric loading curves . this paper proposes and studies a high - dimensional factor model with nonparametric loading functions that depend on a few observed covariate variables . this model is motivated by the fact that observed variables can explain partially the factor loadings . we propose a projected - pca to estimate the unknown factors , loadings , and number of factors . after projecting the response variable onto the sieve space spanned by the covariates , the projected - pca yields a significant improvement on the rates of convergence than the regular methods . in particular , consistency can be achieved without a diverging sample size , as long as the dimensionality grows . this demonstrates that the proposed method is useful in the typical hdlss situations . in addition , we propose new specification tests for the orthogonal decomposition of the loadings , which fill the gap of the testing literature for semiparametric factor models . our empirical findings show that firm characteristics can explain partially the factor loadings , which provide theoretical basis for employing projected - pca method . on the other hand , our empirical study also shows that the firm characteristics can not fully explain the factor loadings so that the proposed generalized factor model is more appropriate . throughout the proofs , @xmath45 and @xmath7 may either grow simultaneously with @xmath6 or stay constant . for two matrices @xmath481 with fixed dimensions , and a sequence @xmath482 , by writing @xmath483 , we mean @xmath484 . in the regular factor model @xmath485 , let @xmath486 denote a @xmath141 diagonal matrix of the first @xmath86 eigenvalues of @xmath487 . then by definition , @xmath488 . let @xmath489 . then @xmath490 where @xmath491 still by the equality ( [ ea.1add ] ) , @xmath501 . hence , this step is achieved by bounding @xmath502 for @xmath503 . note that in this step , we shall not apply a simple inequality @xmath504 , which is too crude . instead , with the help of the result @xmath505 achieved in step 1 , sharper upper bounds for @xmath502 can be achieved . we do so in lemma b.2 in the supplementary material [ @xcite ] . consider the singular value decomposition : @xmath508 , where @xmath509 is a @xmath93 orthogonal matrix , whose columns are the eigenvectors of @xmath98 ; @xmath510 is a @xmath511 matrix whose columns are the eigenvectors of @xmath512 ; @xmath513 is a @xmath31 rectangular diagonal matrix , with diagonal entries as the square roots of the nonzero eigenvalues of @xmath98 . in addition , by definition , @xmath253 is a @xmath141 diagonal matrix consisting of the largest @xmath86 eigenvalues of @xmath98 ; @xmath254 is a @xmath37 matrix whose columns are the corresponding eigenvectors . the columns of @xmath514 are the eigenvectors of @xmath88 , corresponding to the first @xmath86 eigenvalues . by assumption [ ass3.3 ] , @xmath528 , @xmath529 hence , @xmath530 by lemma b.1 in the supplementary material [ @xcite ] , @xmath531 . similarly , @xmath532 using the inequality that for the @xmath116th eigenvalue , @xmath533 , we have @xmath534 , for @xmath260 . hence , it suffices to prove that the first @xmath86 eigenvalues of @xmath320 are bounded away from both zero and infinity , which are also the first @xmath535 eigenvalues of @xmath536 . this holds under the theorem s assumption ( assumption [ ass3.1 ] ) . thus , @xmath537 , which also implies @xmath523 . fan , j. , liao , y. and mincheva , m. ( 2013 ) . large covariance estimation by thresholding principal orthogonal complements ( with discussion ) . _ journal of the royal statistical society , series b _ * 75 * 603680 .
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this paper introduces a projected principal component analysis ( projected - pca ) , which employs principal component analysis to the projected ( smoothed ) data matrix onto a given linear space spanned by covariates .
when it applies to high - dimensional factor analysis , the projection removes noise components .
we show that the unobserved latent factors can be more accurately estimated than the conventional pca if the projection is genuine , or more precisely , when the factor loading matrices are related to the projected linear space .
when the dimensionality is large , the factors can be estimated accurately even when the sample size is finite .
we propose a flexible semiparametric factor model , which decomposes the factor loading matrix into the component that can be explained by subject - specific covariates and the orthogonal residual component .
the covariates effects on the factor loadings are further modeled by the additive model via sieve approximations . by using the newly proposed projected - pca ,
the rates of convergence of the smooth factor loading matrices are obtained , which are much faster than those of the conventional factor analysis .
the convergence is achieved even when the sample size is finite and is particularly appealing in the high - dimension - low - sample - size situation .
this leads us to developing nonparametric tests on whether observed covariates have explaining powers on the loadings and whether they fully explain the loadings .
the proposed method is illustrated by both simulated data and the returns of the components of the s&p 500 index .
./style / arxiv - general.cfg ,
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