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near - infrared , long baseline interferometry is sensitive to the distribution of dust around the nearest young stars on scales of the order of 1 au , and provides a powerful probe of models of disks and envelopes of such stars .
the herbig ae - be stars are pre - main sequence , emission line objects that are the intermediate mass ( @xmath3 ) counterparts of t tauri stars ( hillenbrand _ et al . _ 1992 ) .
we also observed the fu orionis object v1057 cyg , expected to have a strong disk signature due to the high accretion rate of such objects .
while the evolutionary status of the fu orionis objects remains unclear , they are believed to be t tauri stars undergoing an episode of greatly increased disk accretion , involving a brightening of @xmath4 magnitudes .
v1057 cyg , whose outburst began in 1969 - 70 , is the only fu orionis object for which a pre - outburst spectrum is available , confirming its pre - main sequence nature ( grasdalen 1973 ) . until now ,
only one fu orionis object , fu orionis itself , has been resolved by long baseline optical interferometry ( malbet _ et al . _
1998 ) , and v1057 cyg was chosen for study as the next - brightest such object accessible to pti .
we selected a sample of 5 sources from the thesis of millan - gabet , chosen to satisfy the observing limitations of pti , and to avoid known binaries ( with the exception of mwc 147 , whose companion is too faint to affect the current measurements ) .
details of the instrument are described in colavita _
et al . _
table i describes our final sample .
0.8 cm llllllll + * name * & * alternate * & * ra ( @xmath5 ) * & * dec ( @xmath5 ) * & * @xmath6 * & * @xmath7 * & * spec * & d , pc + & & & & & & + + hbc 330 & v594 cas & 00 43 @xmath8 & + 61 54 40.100 & 9.9 & 5.7 & b8e&650 + hd 259431 & mwc 147 & 06 33 @xmath9 & @xmath10 19 19.984 & 8.7 & 5.7 & b6pe&800 + mwc 297 & nz ser & 18 27 @xmath11 & @xmath12 49 52 & 9.5 & 3.1 & o9e&450 + hd 179218 & mwc 614 & 19 11
@xmath13 & + 15 47 15.630 & 7.4 & 5.9 & b9&240 + hd 190073 & v1295 aql & 20 03
@xmath14 & + 05 44 16.676 & 7.8 & 5.8 & a2pe&280 + hbc 300 & v1057 cyg & 20 58 @xmath15 & + 44 15 28.4 & 11.6 & 5.7 & &575 + observations of each source were interweaved with nearby calibrator stars , chosen to exclude known binaries and variable stars .
system visibility was determined based upon observations of the calibrators and models of the calibrator ( e.g. size based upon multiwavelength photometry ) .
the measured raw source visibilities were then divided by the system visibility .
the resulting calibrated visibilities @xmath16 are presented in table ii .
our reported visibilities are a wideband average produced synthetically from five narrowband channels . as a consistency check
, sources were calibrated first relative to one calibrator , then relative to another , and the results compared to avoid problems with unknown binarity .
the stellar contribution to @xmath16 is subsequently removed , assuming the observed spatial distribution of emission on the sky is the sum of an unresolved point source of known flux , and an extended circumstellar contribution . for the herbig stars ,
mst estimated the fractions of the infrared emission due to the star and due to circumstellar emission at k. in table ii we list the fraction @xmath17 of emission due to circumstellar matter , while that of the star is @xmath18 .
for v1057 cyg , we will assume all the infrared emission is circumstellar .
table ii also gives @xmath19 for the circumstellar contribution , where @xmath20 . because our program stars all have large infrared excesses , the corrections for stellar light are generally small .
upper limits to the visibility squared were determined for sources lacking fringes , based upon the sensitivity of the detection algorithm and measuring the system visibility with a nearby calibrator .
figures 1 - 2 show some of the measured individual visibilities @xmath16 for our resolved sources .
0.8 cm lcccccc + * source * & * baseline*&*@xmath16 * & @xmath17&@xmath21 + & & & & + + v594 cas & nw & @xmath22 & @xmath23&@xmath24 + mwc 147 & nw & @xmath25 & @xmath26&@xmath27 + mwc 147 & ns & @xmath28 & @xmath26&@xmath29 + v1057 cyg & nw & @xmath30 & @xmath31 & @xmath32 + mwc 297 & nw , ns & @xmath33 & @xmath34&@xmath35 + mwc 614 & nw , ns & @xmath33 & @xmath36&@xmath37 + v1295 aql & ns & @xmath33 & @xmath38&@xmath37 + fringes were obtained for a total of four sources , although for one of these , mwc 297 , there are insufficient data to produce a calibrated measurement .
thus , we treat mwc 297 as an upper limit .
based upon the observed circumstellar visibilities @xmath21 , table iii gives approximate source sizes based upon a circular gaussian and a uniform disk model : @xmath39 here @xmath40 , @xmath41 the projected baseline , @xmath42 is the fwhm in radians , @xmath43 is the uniform disk diameter in radians , and @xmath44 is a bessel function .
the baseline lengths are 110 m in ns , and 85 m in nw .
error bars include uncertainties in our measurements and in the stellar and circumstellar fluxes , but not in the distance . 0.8 cm lccccc + * source * & * baseline * & & + & & * ( mas ) * & * ( au ) * & * ( mas ) * & * ( au ) * + + v594 cas & nw & @xmath45&@xmath46&@xmath47&@xmath48 + mwc 147 & nw & @xmath49&@xmath50&@xmath51&@xmath52 + mwc 147 & ns & @xmath53&@xmath54&@xmath55&@xmath56 + v1057 cyg & nw & @xmath57&@xmath58&@xmath59&@xmath60 + mwc 297 & nw & @xmath61&@xmath62&@xmath63&@xmath64 + mwc 614 & nw & @xmath65&@xmath66&@xmath67&@xmath68 + v1295 aql & ns & @xmath69&@xmath70&@xmath71&@xmath72 +
for our observations with the largest range of hour angles and projected baseline orientation , v1057 cygni is consistent with a circularly symmetric source .
as an fu ori type object , there is little doubt that its infrared excess comes from a circumstellar disk and not a spherical distribution of dust .
more modeling is necessary to put limits on the possible orientation of the disk .
our measurements of mwc 147 in the ns baseline are consistent with those of akeson et al . 2000 .
however , the new measurement in the nw baseline is inconsisitent with that of the ns baseline if the source is indeed a circularly - symmetric distribution on the sky .
because the baselines have differring orientations , the difference can be accounted for by an asymmetric source distribution , such as a tilted disk .
we wish to confirm the new nw measurement and perform further modeling of this source .
we have resolved three young stellar objects at milli - arc second scales , two for the first time ( the herbig be star v594 cas , and the fu orionis star v1057 cyg ) . presumably we are sampling the distribution of warm dust close to the stars
. however , these data alone are insufficient to fully constrain the sources , and other explanations besides circumstellar disks ( e.g. a binary companion ) are possible .
no significant variation of the visibility is seen as a function of hour angle on the sky , suggesting a symmetric distribution on the sky .
however , for mwc 147 , the measurements in two different baselines suggest an asymmetric distribution , such as a tilted disk .
this is consistent with recent measurements by eisner _
et al . _ ( 2003 ) for a similar sample of herbig stars , three of which appear to have disks with significant inclinations .
this work has made use of software produced by the michelson science center at the california institute of technology . f.p.w .
is grateful to the observatoire de la cte dazur for a poincar fellowship , and to the nsf international researchers fellows program for financial support .
0.4truecm akeson , r.l . ,
ciardi , d.r . ,
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apj , 543 , 313 + millan - gabet , r. , schloerb , f.p . , & traub , w.a . 2001 , apj , 546 , 358 + | we present observations of a sample of herbig aebe stars , as well as the fu orionis object v1057 cygni .
our k - band ( @xmath0 ) observations from the palomar testbed interferometer ( pti ) used baselines of 110 m and 85 m , resulting in fringe spacings of @xmath1 and @xmath2 , respectively .
fringes were obtained for the first time on v1057 cygni as well as v594 cas .
additional measurements were made of mwc147 , while upper limits to visibility - squared are obtained for mwc297 , hd190073 , and mwc614 .
these measurements are sensitive to the distribution of warm , circumstellar dust in these sources .
if the circumstellar infrared emission comes from warm dust in a disk , the inclination of the disk to the line of sight implies that the observed interferometric visibilities should depend upon hour angle .
surprisingly , the observations of millan - gabet , schloerb , & traub 2001 ( hereafter mst ) did not show significant variation with hour angle .
however , limited sampling of angular frequencies on the sky was possible with the iota interferometer , motivating us to study a subset of their objects to further constrain these systems . |
the analyses reported in this talk were performed using either a sample of @xmath9 @xmath7 events or a sample of @xmath10 @xmath8 events collected with the upgraded beijing spectrometer ( besii ) detector @xcite at the beijing electron - positron collider ( bepc ) .
a new structure , denoted as @xmath0 and with mass @xmath11 gev/@xmath12 and width @xmath13 mev/@xmath12 , was observed by the babar experiment in the @xmath14 initial - state radiation process @xcite .
this observation stimulated some theoretical speculation that this @xmath15 state may be an @xmath16-quark version of the @xmath17 since both of them are produced in @xmath18 annihilation and exhibit similar decay patterns @xcite .
here we report the observation of the @xmath0 in the decays of @xmath19 , with @xmath20 , @xmath21 , @xmath22 .
a four - constraint energy - momentum conservation kinematic fit is performed to the @xmath23 hypothesis for the selected four charged tracks and two photons .
@xmath24 candidates are defined as @xmath25-pairs with @xmath26 gev/@xmath12 , a @xmath6 signal is defined as @xmath27 gev/@xmath12 , and in the @xmath28 invariant mass spectrum , candidate @xmath29 mesons are defined by @xmath30 gev/@xmath12 .
the @xmath31 invariant mass spectrum for the selected events is shown in fig .
[ draft - fit ] , where a clear enhancement is seen around 2.18 gev/@xmath12 .
fit with a breit - wigner and a polynomial background yields @xmath32 signal events and the statistical significance is found to be @xmath33 for the signal .
the mass of the structure is determined to be @xmath34 gev/@xmath12 , the width is @xmath35
gev/@xmath12 , and the product branching ratio is @xmath36 .
the mass and width are consistent with babar s results .
invariant mass distribution of the data ( points with error bars ) and the fit ( solid curve ) with a breit - wigner function and polynomial background ; the dashed curve indicates the background function.,scaledwidth=40.0% ]
structures in the @xmath38 invariant - mass spectrum have been observed by several experiments both in the reaction @xmath39
@xcite and in radiative @xmath7 decays @xcite .
the @xmath2 was first observed by the mark - iii collaboration in @xmath7 radiative decays @xmath40 .
a fit to the @xmath38 invariant - mass spectrum gave a mass of 2.22 gev/@xmath12 and a width of 150 mev/@xmath12 @xcite .
an angular analysis of the structure found it to be consistent with a @xmath41 assignment .
it was subsequently observed by the dm2 collaboration , also in @xmath42 decays @xcite .
we present results from a high statistics study of @xmath43 in the @xmath44 final state , with the @xmath45 missing and reconstructed with a one - constraint kinematic fit .
after kinematic fit , we require both the @xmath46 and @xmath47 invariant masses lie within the @xmath6 mass region ( @xmath48 mev/@xmath12 and @xmath49 mev/@xmath12 ) .
the @xmath38 invariant mass distribution is shown in fig .
[ dalitz ] .
there are a total of 508 events with a prominent structure around 2.24 gev/@xmath12 .
invariant mass distribution for @xmath50 candidate events .
the dashed histogram is the phase space invariant mass distribution , and the dotted curve indicates how the acceptance varies with the @xmath38 invariant mass.,scaledwidth=40.0% ] a partial wave analysis of the events with @xmath51 2.7 gev/@xmath12 was performed .
the two - body decay amplitudes in the sequential decay process @xmath52 , @xmath53 and @xmath54 are constructed using the covariant helicity coupling amplitude method .
the intermediate resonance @xmath55 is described with the normal breit - wigner propagator @xmath56 , where @xmath16 is the @xmath38 invariant mass - squared and @xmath57 and @xmath58 are the resonance s mass and width . when @xmath59 , @xmath60 is fitted with both the @xmath38 and @xmath61 systems in a @xmath62-wave , which corresponds to a pseudoscalar @xmath55 state , the fit gives @xmath63 events with mass @xmath64 gev/@xmath12 , width @xmath65 gev/@xmath12 , and a statistical significance larger than @xmath66 , and a product branching fraction of : @xmath67 .
the presence of a signal around 2.24 gev/@xmath12 and its pseudoscalar character are confirmed , and the mass , width , and branching fraction are in good agreement with previous experiments .
a pseudoscalar gluonium candidate , the so - called @xmath68 , was observed in @xmath69 annihilation in 1967 @xcite and in @xmath7 radiative decays in the 1980 s @xcite .
the study of the decays @xmath70 \{@xmath5 , @xmath6}@xmath71 is a useful tool in the investigation of quark and possible gluonium content of the states around 1.44 gev/@xmath72 .
here we investigate the possible structure in the @xmath71 final state in @xmath7 hadronic decays at around @xmath73 gev/@xmath72 . in this analysis , @xmath5 mesons
are observed in the @xmath74 decay , @xmath6 mesons in the @xmath75 decay , and other mesons are detected in the decays : @xmath76 , @xmath77 .
@xmath71 could be @xmath78 or @xmath79 .
figures [ fig : w - x1440-recoiling ] and [ fig : x1440-phikksp ] show the @xmath80 and @xmath81 invariant mass spectra after @xmath5 selection ( @xmath82 gev / c@xmath83 ) or @xmath6 signal selection ( @xmath84 gev/@xmath72 ) .
clear @xmath4 signal is observed recoiling against the @xmath5 , and there is no significant signal recoiling against a @xmath6 . the @xmath80 invariant mass distribution in @xmath85 ( fig .
[ fig : w - x1440-recoiling](b ) ) is fitted with a bw function convoluted with a gaussian mass resolution function ( @xmath86 mev/@xmath72 ) to represent the @xmath4 signal and a third - order polynomial background function .
the mass and width obtained from the fit are @xmath87 mev/@xmath72 and @xmath88 mev/@xmath72 , and the fit yields @xmath89 events . using the efficiency of @xmath90 determined from a uniform phase space mc simulation ,
we obtain the branching fraction to be @xmath91 , where the first error is statistical and the second one systematic . for @xmath92 mode , by fitting the @xmath81 mass spectrum in fig . [
fig : w - x1440-recoiling](c ) with same functions , we obtain the mass and width of @xmath93 mev/@xmath72 and @xmath94 mev/@xmath72 , and the number of events from the fit is @xmath95 .
the efficiency is determined to be @xmath96 from a phase space mc simulation , and the branching fraction is @xmath97 , in good agreement with the isospin symmetry expectation from @xmath85 mode . the distribution of @xmath98 and @xmath99 invariant mass spectra recoiling against the @xmath6 signal are shown in fig .
[ fig : x1440-phikksp ] , and there is no evidence for @xmath4 . the upper limits on the branching fractions at the @xmath100 c.l .
are @xmath101 and @xmath102 . in conclusion , the mass and width of the @xmath4
are measured , which are in agreement with previous measurements ; the branching fractions we measured are also in agreement with the dm2 and mark - iii results .
the significant signal in @xmath103 mode and the missing signal in @xmath104 mode may indicate the @xmath105 component in the @xmath4 is not significant .
besides conventional meson and baryon states , qcd also predicts a rich spectrum of glueballs , hybrids , and multi - quark states in the 1.0 to 2.5 @xmath107 mass region .
therefore , searches for the evidence of these exotic states play an important role in testing qcd .
the radiative decays of @xmath106 to hadrons are expected to contribute about 1% to the total @xmath106 decay width @xcite .
however , the measured channels only sum up to about 0.05% @xcite .
we measured the decays of @xmath106 into @xmath108 , @xmath109 , @xmath110 , @xmath111 , @xmath112 , @xmath113 , @xmath114 , @xmath115 , @xmath116 , and @xmath117 , with the invariant mass of the hadrons ( @xmath118 ) less than 2.9 @xmath107 for each decay mode @xcite .
the differential branching fractions are shown in fig .
[ difbr ] .
the branching fractions below @xmath119 @xmath120 are given in table [ tot - nev ] , which sum up to @xmath121 of the total @xmath106 decay width .
we also analyzed @xmath122 and @xmath123 modes to study the resonances in @xmath124 and @xmath125 invariant mass spectrum .
significant signals for @xmath126 and @xmath127 were observed , but the low statistics prevent us from drawing solid conclusion on the other resonances @xcite .
differential branching fractions for @xmath128 , @xmath129 , @xmath111 , and @xmath110 . here
@xmath118 is the invariant mass of the hadrons . for each point , the smaller longitudinal error is the statistical error , while the bigger one is the total error . ,
scaledwidth=48.0% ] .[tot - nev ] branching fractions for @xmath130 with @xmath119 @xmath107 , where the upper limits are determined at the 90% c.l . [ cols="<,<",options="header " , ]
using the 58 m @xmath131 and 14 m @xmath106 events samples taken with the besii detector at the bepc storage ring , bes experiment provided many interesting results in charmonium decays , including the observation of the @xmath0 , @xmath2 , @xmath4 , and many @xmath106 radiative decays .
these results shed light on the understanding of strong interaction sector of the standard model .
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d * 77 * , 032005 ( 2008 ) . | we present recent results from the bes experiment on the observation of the @xmath0 in @xmath1 , study of @xmath2 in @xmath3 , and the production of @xmath4 recoiling against an @xmath5 or a @xmath6 in @xmath7 hadronic decays .
the observation of @xmath8 radiative decays is also presented . |
nmt would like thank hitesh changlani , cyrus umrigar , adam holmes , bryan ogorman , bryan clark , and jonathan moussa for useful discussions .
this work was supported through the scientific discovery through advanced computing ( scidac ) program funded by the u.s .
department of energy , office of science , advanced scientific computing research and basic energy sciences .
we used the extreme science and engineering discovery environment ( xsede ) , which is supported by the national science foundation grant no .
oci-1053575 and resources of the oak ridge leadership computing facility ( olcf ) at the oak ridge national laboratory , which is supported by the office of science of the u.s .
department of energy under contract no .
de - ac0500or22725 .
59ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ]
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\doibase http://dx.doi.org/10.1063/1.4869192 [ * * , ( ) , http://dx.doi.org/10.1063/1.4869192 ] link:\doibase 10.1103/physrev.183.23 [ * * , ( ) ]
link:\doibase 10.1002/qua.560070515 [ * * , ( ) ] link:\doibase 10.1007/bf02394557 [ * * , ( ) ] link:\doibase 10.1080/00268977800100581 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.1679199 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0301-0104(83)85011-3 [ * * , ( ) ] link:\doibase 10.1002/jcc.540080105 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.461037 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.460537 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.465368 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.1615956 [ * * , ( ) ] link:\doibase 10.1103/physrevc.79.064324 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.3600351 [ * * , ( ) , http://dx.doi.org/10.1063/1.3600351 ] \doibase http://dx.doi.org/10.1063/1.4905528 [ * * , ( ) , http://dx.doi.org/10.1063/1.4905528 ] link:\doibase 10.1021/acs.jctc.5b01099 [ * * , ( ) ] , \doibase http://dx.doi.org/10.1063/1.4766327 [ * * , ( ) , http://dx.doi.org/10.1063/1.4766327 ] link:\doibase 10.1139/cjc-2013 - 0017 [ * * , ( ) ] , @noop _ _ ( , ) @noop ( ) , @noop ( ) , link:\doibase 10.1137/s0895479898334605 [ * * , ( ) ] , link:\doibase 10.1002/wcms.93 [ * * , ( ) ] link:\doibase 10.1002/wcms.81 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.463096 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.1804498 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.4905124 [ * * , ( ) , http://dx.doi.org/10.1063/1.4905124 ] link:\doibase 10.1021/ct5004252 [ * * , ( ) ] , \doibase http://dx.doi.org/10.1063/1.1829045 [ * * , ( ) , http://dx.doi.org/10.1063/1.1829045 ] link:\doibase 10.1021/cr200137a [ * * , ( ) ] , @noop * * , ( ) \doibase http://dx.doi.org/10.1063/1.4927594 [ * * , ( ) , http://dx.doi.org/10.1063/1.4927594 ] \doibase http://dx.doi.org/10.1063/1.4932595 [ * * , ( ) , http://dx.doi.org/10.1063/1.4932595 ] | development of exponentially scaling methods has seen great progress in tackling larger systems than previously thought possible .
one such technique , full configuration interaction quantum monte carlo , is a useful algorithm that allows exact diagonalization through stochastically sampling determinants .
the method derives its utility from the information in the matrix elements of the hamiltonian , along with a stochastic projected wave function , to find the important parts of hilbert space .
however , the stochastic representation of the wave function is not required to search hilbert space efficiently , and here we describe a highly efficient deterministic method to achieve chemical accuracy for a wide range of systems , including the difficult cr@xmath0 dimer . we demonstrate that such calculations for systems like cr@xmath0 can be performed in just a few cpu hours . in addition
our method also allows efficient calculation of excited state energies , for which we illustrate with benchmark results for the excited states of c@xmath0 .
_ introduction : _ the scope of traditional approaches to full configuration interaction ( fci ) has been limited to simple diatomic molecules @xcite , and there has been little progress in diagonalizing spaces much larger than a billion determinants in recent times @xcite . however , recent progress in alternative approaches to fci problems has increased the scope of fci beyond simple diatomic molecules .
two techniques in particular have been important in this progress , full configuration quantum monte carlo ( fciqmc ) @xcite , and density matrix renormalization group ( dmrg ) @xcite .
both algorithms provide unique advantages , with dmrg being the definitive method for systems in which one can identify degrees of freedom with low levels of entanglement @xcite , and fciqmc showing promise for molecules and extended systems in two or more dimensions @xcite .
the success of dmrg and fciqmc in quantum chemistry is highlighted by their recent applications to unprecedented large - size determinant spaces while also achieving chemical accuracy @xcite .
the fciqmc method is a useful technique with a few limitations which include biased sampling and comparatively computationally expensive simulations .
the biased sampling is a result of the initiator approximation @xcite , generally employed in fciqmc calculations which additionally limits the space in which determinants can be sampled .
the initiator approximation can cause errors in fciqmc calculations that can be unexpectedly large @xcite .
the need for this approximation is related to the monte carlo sampling and not necessarily related to power of the technique which we suggest is finding important determinants . in this letter we suggest an alternative to stochastic sampling in favor of a completely deterministic version of the fciqmc technique that efficiently samples the determinant space .
we denote this deterministic fciqmc algorithm by adaptive sampling ci ( asci ) .
our approach should be contrasted to other techniques for finding energetically important determinant subspaces .
the majority of traditional ci methods encode relevant physical degrees of freedom based on excitation levels from a reference determinant @xcite .
these excitation - based methods can also suffer from inaccuracy as they miss important parts of determinant space . within the ci framework
several promising ways to circumvent this problem have been suggested that focus on selected ci approaches where one selects relevant determinants based on different criteria @xcite . after presenting the asci method ,
we establish a connection between fciqmc and various selected ci approaches .
we then apply the asci method to the cr@xmath0 dimer , a classic hard problem for many computational electronic structure methods @xcite .
finally , we demonstrate that excited states are straightforward to calculate with asci .
we note that the calculation of excited states within stochastic fciqmc is possible , but require specialized techniques @xcite , or stochastic orthogonalization between walker sets , which is quite different from the method described here . _ a path to a deterministic algorithm _ : in the initial development of fciqmc , one of the original improvements on the method was to take part of the projection step and make it deterministic @xcite . in this work ,
we go further and develop a completely deterministic algorithm .
our approach here is to find important determinants in the same manner as fciqmc , i.e. to sample determinants based on the absolute value of the ground state wave function amplitudes .
the fciqmc technique was originally presented as a projector method in imaginary time and we use this approach to motivate our method .
we start by expanding a wave function in the space of determinants , @xmath1 and the propagator in imaginary time , @xmath2 which has an asymptotic solution of a stationary state with @xmath3 . in fciqmc ,
the parameter @xmath4 is a free parameter that controls the population of walkers . here
we will consider @xmath4 to be the ground state energy or our best approximation thereof .
the power of fciqmc is that it ignores the unimportant parts of determinant space , finds important determinants , and samples them according to their amplitudes . for a stationary state we can solve for the individual coefficients as @xmath5 .
the rhs of this equation captures all aspects of the fciqmc algorithm .
the @xmath6 in the numerator corresponds to the spawning step ( transition moves between determinants ) , the sign of @xmath6 in the numerator and specifically summing over positive and negative terms corresponds to the annihilation step ( cancelation of positive and negative walkers ) , and the denominator corresponds to the death / cloning step ( adding or removing walkers from the simulation ) .
the key to turning fciqmc into a non - stochastic algorithm is to remove the stochastic sampling and replace it with a deterministic ranking of important determinants . in both approaches
the hamiltonian matrix elements and a wave function are needed . in fciqmc
the wave function is represented by the distribution of walkers at any given step , whereas for our deterministic algorithm , we use an approximate wave function at each iteration as follows , @xmath7 in this equation , we have labeled the coefficients of the initial wave function as @xmath8 , and the output coefficients as @xmath9 .
thus for any good approximation to the ground state wave function we can use eq .
[ trialwfs ] to determine the importance of determinants in a much larger space that what is initially included in @xmath8 .
this @xmath9 is essentially a first - order perturbation estimate for ci coefficients in the epstein - nesbet perturbation theory @xcite . _
the deterministic algorithm _ : the technique described in this section can be considered a variant of cipsi ( configuration interaction by perturbation with multiconfigurational zeroth - order wave functions selected by iterative process ) @xcite , with a modified search procedure . for the largest systems considered here ,
only a modest amount of memory is needed , and all parts except the diagonalization step can be trivially parallelized .
the algorithm is defined by two determinant subspaces : a core space of size _ cdets _ and a target space of size _
tdets_. the _ core space _ determines the number of terms @xmath10 we include in the sum in eq .
[ trialwfs ] . this is to say we select @xmath10 determinants with the largest c@xmath11 , and consider only those to be non - zero in the sum of equation eq .
[ trialwfs ] .
the space to be searched is the set of single and double excitations of the core set of determinants . since our objective is to find the determinants with the largest amplitudes , we in general only need to search determinants connected to those with large amplitudes .
we illustrate the use of this approximation with numerical tests in the next section .
the _ target space _ contains the top _ tdet _
determinants , as determined from the ranking , and is the rank of the matrix diagonalized in each iteration .
initalize : set size of core ( _ cdets _ ) and target space ( _ tdets _ ) . set the starting wave function to the hartree fock wave function ( c@xmath12 = 1 , c@xmath13 ; e = e@xmath14 ) .
\(1 ) evaluate the perturbed wave function amplitudes over al the single and double substitutions from the core space . @xmath15
\(2 ) for the core determinants of the current wave function \{c@xmath16}@xmath17 , and the current perturbed wave function \{a@xmath16}@xmath18 , select the _
tdet _ largest absolute values to define the new target space .
\(3 ) form and diagonalize @xmath19 in the target space .
\(4 ) the lowest eigenvalue is the new energy @xmath4 .
the largest _ cdet _ amplitudes , by magnitude , define the new core space . if the energy is not converged , return to step ( 1 ) .
the determinants found at the end of the simulation will in general be the most important determinants for the ground state wave function . unlike fciqmc
this technique has no population bias , initiator bias , and no sign problem .
the technique described here provides an inherently variational energy .
however , it is possible to extend the accuracy of the technique with perturbation theory , which comes at the expense of the energy no longer being variational @xcite .
_ discussion _ : when performing an asci calculation the first few steps involve the exploration of higher order excitations . if the starting wave function is a single determinant , then each step increases the maximum number of excitations that have been explored by 2 .
the coefficients @xmath20 , as calculated in step ( 1 ) , can span all single and double excitations from the current wave function , and is generally very large .
truncating the coefficients that are calculated for @xmath20 in the main part of the self - consistent loop allows us to consider applying our algorithm to large systems . to maintain size consistency , the value of _ tdets _ and _ cdets _ will have to grow with system size . for the systems considered in this work , it was possible to converge the value of _ cdets _ , as the energy with respect to this parameter can be extrapolated by running with a few different values .
apart from the diagonalization step , the most computationally expensive task is forming the @xmath20 matrix and finding its largest values .
this can be split among many processors , with the only non - trivial communication occurring during the final aggregation of the final values .
further improvements are possible by generating the natural orbitals after an initial run and using them to recalculate the electron integrals . the natural orbitals can be generated at any point in the simulation from the current best wave function .
the natural orbitals are generally thought to produce highly compact representations of a wave function @xcite . for a difficult system like cr@xmath0
, we find the use of natural orbitals to be crucial to obtain accurate energies .
before we present our results we consider numerical tests for our approximation of the matrix @xmath20 . in general , the coefficients of the ground state wave function will span many orders of magnitude , and searching over the determinants with the smallest amplitude coefficients is inefficient . using _
cdets _ as a parameter to limit the search to only the important determinants is a well controlled approximation that can be converged .
this approach has similarities to the fciqmc initiator approximation @xcite .
we demonstrate the accuracy of this approximation by considering cn in an sto-3 g basis ( 6240 determinants ) . we show that we can find the most important determinants in hilbert space without having to perform a diagonalization over a larger determinant space .
table [ tab3 ] provides a comparison of the most important determinants found with different values of _ cdets _ and _ tdets _ to those obtained from a full diagonalization over the entire space . for a simulation of 100 core determinants and a target space of 200 determinants ( 100/200 ) we found 182 of the top 200 determinants ( 91% ) .
the remaining 9% of missed determinants were found to be close in amplitude to the determinants that replaced them in the target space .
similar result can be seen for for all the simulations presented .
these results suggest that for some simulations that the search algorithm is nt highly dependent on the size of the core space and the small percentage of determinants that are missed by the algorithm are replaced by determinants that are similar in importance .
thus we argue that extremely high accuracy is not needed in determining the ranking order .
this approximation is even further reduced when perturbation corrections are used , which can correct for any important determinants that were missed .
.test of the cn dimer using the full search algorithm .
the energies are in units of ha .
the columns with top 200 ; is the number of determinants we found that agree with the top 200 determinants , by amplitude , of the exact answer . likewise for top 400 and top 800.
the total fci space is 6240 determinants [ tab3 ] [ cols="<,^,>,>,>,>",options="header " , ] for c@xmath0 simulations , the convergence of energies to chemical accuracy was easily achieved , and neither the use of natural orbitals nor perturbation corrections are needed . in comparison with the exact results
@xcite , we were able to achieve an accuracy of 1 mha using a diagonalization no larger than 200,000 determinants .
the total computer time for a simulation of this size was less than 2 cpu hours .
results are presented in table [ c2tab ] , using different values of _ tdets _ and _
cdets_. _ cr@xmath0 with the sv basis set _ : fig .
[ cr2ene ] shows the convergence of our results for cr@xmath0 . in order to make a comparison with previous studies @xcite ,
cr@xmath0 calculations were carried out with the sv basis set @xcite at 1.5 with 24 active electrons in 30 orbitals and a frozen core .
a compact representation of the wave function in this system is dependent on having a good set of orbitals . as part of our algorithm
, we run a preliminary calculation with a target space of 100,000 determinants , after which we calculated the natural orbitals .
the resulting natural orbitals were used to recalculate the integrals for the production run .
the total energy for our most accurate simulation converged to within 16 mha of the predicted full ci basis set energy @xcite .
energy as a function of tdet size , with the largest target space going up to 1 million determinants .
the plotted energies have been shifted by 2086 ha .
the pt correction line represents our result with the added perturbation theory correction , which brings our final energy to within 1 mha of the predicted exact result .
our best energy without the perturbation correction is -2086.40388 ( ha ) . with the perturbation correction it is -2086.4203 ( ha ) .
the dmrg benchmark is -2086.420948 ( ha ) @xcite . ]
a perturbation theory analysis @xcite was performed bringing the final energy within 1 mha of the predicted exact result .
the timing for the largest simulation ( _ _ tdets__=10@xmath21 ) , including the initial run for calculating the natural orbitals but not including the perturbation correction , requires approximately 7 cpu hours when run on a single core of a 2.40 ghz intel xeon processor .
while this is a relatively fast calculation , we expect further improvements will speed up the simulation significantly . to demonstrate the distribution of the determinants located by asci we plot a histogram of excitations from the dominant determinants for cr@xmath0 in fig .
[ crdist ] .
the ratio between different excitations sectors does not change much in increasing _ tdet _
values from 10@xmath22 to 10@xmath21 . for this range ,
the quadruple excitations generally make up half the wave function , while higher excitations above the quadruples make up roughly 30% of the wave function . by excitations from the dominant determinant .
the x - axis is excitation level from the dominant determinant , and the y - axis is the fraction of determinants .
plots ( a ) , ( b ) , ( c ) , ( d ) have tdets = 100k , 400k , 800k , and 1 million respectively .
simulations with different tdet spaces ( for the ones shown here ) have roughly the same fractional importance in the different excitation levels .
thus even for our smallest tdet simulations , large determinant excitations are important .
, title="fig : " ] by excitations from the dominant determinant .
the x - axis is excitation level from the dominant determinant , and the y - axis is the fraction of determinants .
plots ( a ) , ( b ) , ( c ) , ( d ) have tdets = 100k , 400k , 800k , and 1 million respectively .
simulations with different tdet spaces ( for the ones shown here ) have roughly the same fractional importance in the different excitation levels .
thus even for our smallest tdet simulations , large determinant excitations are important .
, title="fig : " ] by excitations from the dominant determinant .
the x - axis is excitation level from the dominant determinant , and the y - axis is the fraction of determinants .
plots ( a ) , ( b ) , ( c ) , ( d ) have tdets = 100k , 400k , 800k , and 1 million respectively .
simulations with different tdet spaces ( for the ones shown here ) have roughly the same fractional importance in the different excitation levels .
thus even for our smallest tdet simulations , large determinant excitations are important .
, title="fig : " ] by excitations from the dominant determinant .
the x - axis is excitation level from the dominant determinant , and the y - axis is the fraction of determinants .
plots ( a ) , ( b ) , ( c ) , ( d ) have tdets = 100k , 400k , 800k , and 1 million respectively .
simulations with different tdet spaces ( for the ones shown here ) have roughly the same fractional importance in the different excitation levels .
thus even for our smallest tdet simulations , large determinant excitations are important .
, title="fig : " ] _ excited states _ : we are also able to calculate excited states with our technique , as they are obtained automatically within the diagonalization procedure . for a 6 - 31 g * basis of c@xmath0 , we compare against previous fci simulations for the first two excited states @xcite . for a simulation at a distance of 1.25 with cdets=@xmath23 and tdets=800000
, we have the following energies for the ground state and the first two excited states ( @xmath24,@xmath25,@xmath26 ) in units of ha .
the exact results are ( @xmath27,@xmath28,@xmath29 ) @xcite .
thus , although the accuracy of the excited states is not as good as the ground state , it is still straightforward to achieve chemical accuracy ( which is generally defined as 0.0016 ha ) .
we can improve the accuracy of excited states by noting that any eigenstate can be used in eq .
[ trialwfs ] .
thus we can find and rank determinants by their importance to individual excited states .
the excited state optimization can be done simultaneously with the ground state method , or in a state - by - state bootstrap method . for the simultaneous optimization algorithm
, we determine a set of important determinants to retain for each excited state .
there is a separate search step for each excited state , but one diagonalization step that combines all retained determinants .
in contrast , the bootstrap method would converge each excited state one by one , where at each step determinants would be added in specifically for the targeted excited state .
the use of natural orbitals averaged over various excited states may also be used to improve the algorithm @xcite .
a detailed study of the targeted excited state technique will be presented elsewhere .
_ connection to selected ci _ : as mentioned earlier , the asci method may also be considered a variant of selected ci techniques @xcite in which there is considerable current interest @xcite .
our method employs first - order perturbation coefficients for selecting determinants , and is thus closely resembles the ci method , cipsi .
another related technique is the @xmath30+sd - ci @xcite method which uses a one - step energy criteria , and a one - step approach together with our eq .
[ trialwfs ] in order to find important determinants . despite these similarities ,
none of these algorithms have been pushed to achieve chemical accuracy for hard systems , and do not appear to have been benchmarked against fciqmc or dmrg .
the largest @xmath30+sd - ci calculations included roughly 50,000 determinants and attained 13 mha accuracy for c@xmath0 6 - 31 g * ( comparable to our results in table [ c2tab ] ) .
as shown in this work , it is easy to go more than an order of magnitude in accuracy using our iterative scheme , without significantly increasing the computational effort .
the largest selected ci techniques we are aware of have been extended up to 4 million determinants @xcite , but for systems in which no benchmarks exist . for the future development of asci and other selected ci techniques , it is important to consider how such methods are different from standard ci methods .
the difference is largely due to the construction of the hamiltonian .
selected ci techniques need unique data structures in order to construct the hamiltonian efficiently @xcite .
a previous study demonstrated that much larger scale simulations , than what we presented here , is possible for selected ci techniques @xcite .
we are currently considering various data structures used previously @xcite and new structures , to determine the best way to scale up our simulations .
_ conclusions _
: we have shown that the underlying dynamics of fciqmc can be used to generate a deterministic algorithm that can be efficiently used to calculate both ground and excited states of chemical systems .
we have applied this technique to a known difficult problem in electronic structure theory , the cr@xmath0 molecule , and shown that chemical accuracy can be achieved with the cpu power available on any modern computer .
our results suggest that the asci method ( and selected ci methods in general ) should be considered as a state of the art ci method in both accuracy and efficiency
. it will be interesting to determine where asci stands in relationship to dmrg and fciqmc , as all these methods have different strengths .
certainly the use of asci and fciqmc is currently important since dmrg and post - dmrg methods are not yet well suited for simulations in two and three dimensions .
the asci method also distinguishes itself from fciqmc in that excited states and other properties , such as the 2-rdm , are inherently easy to calculate @xcite . |
j. a. noble is a royal commission for the exhibition of 1851 research fellow
* corresponding author : s. coussan , stephane.coussan@univ-amu.fr | in the quest to understand the formation of the building blocks of life , amorphous solid water ( asw ) is one of the most widely studied molecular systems . indeed , asw is ubiquitous in the cold interstellar medium ( ism ) , where asw - coated dust grains provide a catalytic surface for solid phase chemistry , and is believed to be present in the earth s atmosphere at high altitudes .
it has been shown that the ice surface adsorbs small molecules such as co , n@xmath0 , or ch@xmath1 , most likely at oh groups dangling from the surface .
our study presents completely new insights concerning the behaviour of asw upon selective infrared ( ir ) irradiation of its dangling modes .
when irradiated , these surface h@xmath0o molecules reorganise , predominantly forming a stabilised monomer - like water mode on the ice surface .
we show that we systematically provoke `` hole - burning '' effects ( or net loss of oscillators ) at the wavelength of irradiation and reproduce the same absorbed water monomer on the asw surface .
our study suggests that all dangling modes share one common channel of vibrational relaxation ; the ice remains amorphous but with a reduced range of binding sites , and thus an altered catalytic capacity .
+ asw is a molecular system which has long provoked interest due , in part , to its role in the formation of molecules key to the origins of life@xcite .
asw has long been known to accrete small molecules such as co , h@xmath0o , n@xmath0 , or ch@xmath1@xcite , initiating chemical and photochemical surface reactivity@xcite . in the ism ,
water in the form of asw is the most abundant solid phase molecular species@xcite .
the production of molecules , from the most simple , h@xmath0@xcite , to the more complex ch@xmath2oh@xcite , and even precursors to the simplest amino acid , glycine@xcite , is catalysed by the asw surface@xcite ; both the outer surface and surfaces within its porous structure are involved .
the selective ir irradiation of crystalline ice@xcite and water clusters@xcite has already been studied . in the former case , the desorption of h@xmath0o molecules , and in the latter , the dissociation of clusters , was stimulated .
we are interested in the behaviour of asw upon selective ir irradiation and have studied the irradiation of the four surface modes of this ice , assigned in the literature@xcite and illustrated in figure [ fig : cartoon ] .
theoretical calculations , supported by experimental studies , suggest that water molecules in the dh mode are bi- or tri - coordinated , presenting one free oh bond dangling at the surface ; do molecules present a free oxygen electronic doublet ; and s4 molecules have a tetrahedral structure at the surface , which is distorted compared to the tetrahedra of bulk asw . ) , do ( 3549 @xmath3 ) , and s4 ( 3503 @xmath3 ) are illustrated on a representative asw sample.,scaledwidth=50.0% ] in these experiments we prepared a pure asw sample ( figure [ fig : cartoon ] ) as follows : deionised water was subjected to multiple freeze - pump - thaw cycles under vacuum to remove dissolved gases .
mixtures of purified h@xmath0o and helium ( air liquide , @xmath4 99.9999 % ) gas were prepared in a stainless steel dosing line with base pressure 10@xmath5 mbar in ratios of @xmath6 .
ices were produced by depositing the gas mixture directly onto a gold - plated copper surface held at 50 k ( to avoid trapping of the vector gas or nitrogen ) then cooled to 3.7 k ( the cooling typically takes around five minutes due to the high cryogenic power of 0.5 w at 4 k ) .
the cooled surface is located in a high vacuum chamber with a base pressure of 10@xmath7 mbar at 3.7 k. ir spectra were recorded in reflection mode using a bruker 66/s ftir spectrometer equipped with a mct detector ( 4000 800 @xmath3 ) .
full details of the experimental setup are given in coussan _
_ et al.__@xcite .
the ices grown in our study were characterised as purely amorphous in nature due to the position of the bulk oh stretch and the characteristic dangling modes at 3720 and 3698 @xmath3 ( see figure [ fig : cartoon])@xcite .
some previous studies using a helium carrier gas have reported the formation of ice nanocrystals or clusters@xcite , but this can be ruled out due to the absence of an absorption feature centred at 3692 @xmath3 . this band was never observed in any deposition of a h@xmath0o : he mixture during the work detailed here .
after deposition , ices were selectively irradiated using a tunable ir opo laserspec ( 1.5 4 @xmath8 m ) , pumped at 10 hz by a pulsed nd : yag quantel brilliant b laser ( 1064 nm , pulse duration 6 ns ) .
the average laser power is @xmath9 35 mw in the @xmath10 domain , except in the range 3520 3500 @xmath3 where it is @xmath9 10 mw , with a fwhm @xmath4 1.5 @xmath3 .
each irradiation was performed for one hour to ensure saturation of the effects . , ( b ) 3698 @xmath3 , ( c ) 3549 @xmath3 , and ( d ) 3503 @xmath3 .
the spectra are coloured as follows : before irradiation , blue ; after irradiation , green ; difference spectrum , red .
the insets magnify the irradiated range.,scaledwidth=50.0% ] figure [ fig : irr ] shows the results of irradiations carried out on the four dangling mode bands .
the irradiations provoke permanent `` hole - burnings '' in the irradiated band which are slightly shifted ( by up to 3 @xmath3 ) compared to the irradiation frequency .
the `` hole - burning '' is clearest for the irradiation of dh at 3698 @xmath3 but is also easily visible for the irradiations at 3720 and 3549 @xmath3 .
interestingly , upon each irradiation , we observe the growth of a new band centred at @xmath11 3725 @xmath3 , with fwhm @xmath9 5 @xmath3 .
both the shifts in frequency upon irradiation , and the narrowness of the `` hole - burning '' effects and the newly created bands at 3725 @xmath3 illustrate the inhomogeneity of the bands ; each band contains a distribution of oscillators , but only one class of oscillator isomerises upon irradiation at a given frequency , producing one new oscillator class .
after irradiation at 3698 @xmath3 , a second , smaller peak at 3638 @xmath3 is also observed ( see figure [ fig : depot]a ) . unirradiated asw samples and their ir spectra were stable and remained unchanged over the timescale of an irradiation study .
the common feature of all the irradiations in figure [ fig : irr ] is the growth of the 3725 @xmath3 band .
what is the source of this new band , previously unidentified in asw spectra ?
considering energetics only , irradiating between 3720 and 3503 @xmath3 could potentially break one or two hydrogen bonds , as the average weak h - bond strength is around 1800 @xmath3 .
is the band , therefore , due to h@xmath0o surface molecules in a different conformation than those in dh , do and s4 modes , or is it due to water molecules which have desorbed then re - adsorbed at the surface ?
the latter response can be immediately discarded based on the results of molecular adsorption studies which have shown a red - shift of the oh dangling frequency upon adsorption of multiple molecular species at the dangling modes@xcite .
for example , in the case of nitrogen adsorption onto asw , manca _ _ et al.__@xcite observed a red shift of 22 @xmath3 of the dh mode@xcite . in our experiments , the new band at 3725
@xmath3 is blue - shifted with respect to the dh modes .
moreover , the dynamic vacuum of 10@xmath7 mbar rapidly evacuates any desorbing molecules , such that the residual pressure is too low to allow redeposition .
thermal effects are discounted based upon annealing of asw samples , which provoked a global decrease of the dangling bonds and no production of narrow peaks at 3725 @xmath3 , in agreement with previous studies@xcite . upon irradiation ,
the newly produced band is narrow , indicating a single , homogeneous vibrational mode .
thus , the 3725 @xmath3 band is clearly not due to a perturbed dh mode , but it rather has dangling oh character , and we propose that it is samples a bi - coordinated h@xmath0o molecule , with two dangling oh and two co - ordinated electron pairs ( d2h ) . this structure explains the blue shift of the peak with respect to the dh mode at 3720 @xmath3 as , because neither of the two oh oscillators is directly hydrogen bonded , the free oh oscillators are less perturbed than the surface dh modes . ,
reproduced from figure [ fig : irr ] ; ( b ) an asw sample plus background - deposited water ; ( c ) spectrum b plus background - deposited nitrogen ; ( d ) an asw sample plus background - deposited nitrogen ; ( e ) spectrum d plus background - deposited h@xmath0o : n@xmath0 mixture.,scaledwidth=50.0% ] we performed further experiments to verify the source of the two new peaks at 3725 and 3638 @xmath3 , as illustrated in figure [ fig : depot ] .
the background deposition of water on a pure asw sample does not result in a new band , but rather the global growth of the dh and bulk oh bands ( figure [ fig : depot]b ) .
however , subsequent background deposition of pure nitrogen , a molecule present as a low - level source of pollution in the chamber , provokes the appearance of a band at 3726 @xmath3 ( figure [ fig : depot]c ) , revealed thanks to the magnifying effect of nitrogen@xcite .
the background deposition of n@xmath0 upon a new asw sample does not produce a clear peak ( figure [ fig : depot]d ) , but the background deposition of a h@xmath0o : n@xmath0 ( 1:10 ) mixture provokes the appearance of two narrow bands at 3725 @xmath3 and 3638 @xmath3 ( figure [ fig : depot ] ) , with a peak area ratio of approximately 10:1 , compared to 6:1 seen after irradiation at 3698 @xmath3 . when this deposition spectrum is compared to the spectrum of a gas - phase water monomer ( see table [ table : frequencies ] ) , the two peaks show red shifts of nearly 200 @xmath3 with respect to the @xmath12 modes of the monomer ( one always compares gas - phase experimental results with those of a `` classical '' theoretical calculation , as classical calculations are carried out for isolated species .
it also allows us to approximate the perturbation induced by the medium ) .
these red shifts are almost precisely those observed in the solid phase for water in a nitrogen matrix , where the @xmath13 ratio is 8:1 , and a h@xmath0o - n@xmath0 complex.@xcite considering our experimental results and the literature , the band at 3725 @xmath3 is positively attributed to a water monomer interacting with the surface via its two electronic doublets ; its large intensity compared to the other dh bands is explained by the magnifying effect of nitrogen , as extensively investigated by hujo _ _ et al.__@xcite .
we suggest that nitrogen molecules , present as a low - level pollutant in the chamber , serendipitously complex the water molecule , as illustrated in figure [ fig : depot ] , stabilising the molecule , preventing any further adsorption , and magnifying the oh stretching bands
. however , the nitrogen must only be present as a trace pollutant on the surface , as no shifting of the dh peak positions of the asw is seen , either during cooling of the water ice sample from 50 k to 3.7 k , or during the irradiation period .
if nitrogen were present at multilayer concentrations , we would expect to see a red shift@xcite of up to 22 @xmath3 .
the @xmath14 peak at 3638 @xmath3 is not observed after irradiation of the other three surface modes , but this is likely due to its low intensity compared to that of @xmath15 .
.comparison between calculated and observed @xmath15 and @xmath14 water monomer frequencies .
frequencies are given in @xmath3 . [ cols="^,^,^,^,^,^ " , ] one explanation for the observed `` hole - burning '' is that amorphous ice is unable to relax all of the vibrational energy injected at the surface through bulk relaxation channels .
as a result , some fraction of this energy is accumulated at the surface and in the immediate sub - layers , where it induces reconstruction of the surface .
we observe saturation of the `` hole - burning '' events within the timescale of the irradiations performed , suggesting that the rearrangement of surface molecules is not an efficient relaxation channel , but is a minority effect compared to the main relaxation channels via the bulk ice .
thus , the production of the 3725 @xmath3 band is not the major result of irradiation , but is due to the inability of the ice to fully dissipate the injected energy .
it has previously been suggested that asw is a disordered material which has no long - range organisation@xcite , which could help to explain the lack of efficiency in the bulk relaxation channels . although it is possible that some h@xmath0o molecules desorb upon irradiation , as in the studies of focsa _ _ et al.__@xcite , we consider this effect to be minor because no increase in pressure is observed in the chamber and the energy injected is not enough to break more than two h - bonds .
it is curious that the absorption band at 3725 @xmath3 is produced upon irradiation of all dangling bonds .
the irradiation effects at 3720 and 3698 @xmath3 , in particular , are very similar , except that the `` hole - burning '' of the doubly - coordinated dh is less pronounced than that of the triply - coordinated dh .
such molecular rearrangement requires only a reorientation of the water molecule and thus is not `` energy consuming '' compared to the energy injected into the system .
these results also provide evidence of a local ordering to the asw structure . upon irradiation of the dh modes ,
both the `` hole - burning '' in absorption bands and the newly produced monomer band at 3725 @xmath3 are relatively narrow ( fwhm @xmath9 5 @xmath3 ) .
this suggests that each surface molecule is surrounded by a locally ordered oscillator network , resulting reproducibly in the production of one oscillator class upon selective irradiation .
if we consider irradiation of the s4 band , centred at 3503 @xmath3 , we were unable to observe a definitive `` hole - burning '' event , likely due to the width of the band .
however , we observed an increase at 3725 @xmath3 , as for the dh modes .
it is unlikely that the tetra - coordinated s4 molecules are themselves ejected from within the surface layer , as this would be highly energetically unfavourable .
however , as we see an increase at 3725 @xmath3 , it is likely that the relaxation channel involves the breaking of h - bonds , with two breaks being enough to `` transform '' a s4 molecule into a monomer - like water molecule .
the case of do is likely intermediate between those of dh and s4 . during each of these irradiations
, we see no interconversion between the modes , suggesting that there is only a single `` surface '' channel for the release of excess vibrational energy . in this work we have provided new insights into the bonding and structure of the surface molecules in amorphous solid water .
the ensemble of our results show that surface modes , in particular the dh dangling bonds , are sensitive to photo - induced rearrangement due to competition between surface reorganisation and the main relaxation channels in the bulk water ice . rather than desorbing from the surface , molecules embedded in the surface layer
become loosely associated with the surface in the form of monomer - like structures interacting through their two free electron pairs .
the fortuitous presence of nitrogen in the chamber both promoted the magnification of the oh stretching mode of the monomer - like molecule and stabilised it on the ice surface .
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several groups have worked for the past two decades on the generation of reliable models of low - mass stars , but it was nt until the late 1990s that they arrived to realistic models of these objects .
the models of the group led by baraffe & chabrier are at present the most widely used ones , since they can reproduce very well many of the observational properties of low - mass stars .
for example , the mass - magnitude and the mass - luminosity relations of these stars are very nicely reproduced by the baraffe et al .
( 1998 ) models . those models , however , still have some problems reproducing the effective temperature scale and the mass - radius relation of these stars . in the case of the @xmath3 scale ,
baraffe et al . (
1998 ) find that at temperatures below @xmath2 3700k the models predict bluer v
i colors than the ones observed .
a possible reason provided by the authors for this mismatch is a missing source of opacity in the optical that causes the stars to be fainter in v than what the models predict . for the mass radius relation
, the models underestimate the radii of the stars by at least 10 % .
this conclusion is based on the observational results from eclipsing binaries with errorbars of 3 % or less ( see figure 1 ) .
the problem may be that the `` standard models '' do not include the effect of magnetic fields .
mullan & macdonald ( 2001 ) find that low - mass star models have larger radii and smaller @xmath3 when magnetic fields are taken into account .
magnetic fields are generally enhanced by stellar rotation , and in close binaries ( where we are measuring systematically larger radii ) the stars are spun up by orbital synchronization .
with the current observational techniques , double - lined detached eclipsing binaries are the only objects where we can measure simultaneously the mass and the radius of stars with error bars of less than 23 % . the technique is a well established one : the radial velocity ( rv ) curves of the binaries provide the masses as a function of the orbital inclination of the system . from their light curves ( lcs ) one
can then measure the orbital inclination of the system and the radius of each star . also , by measuring the lcs at different wavelengths one can estimate the effective temperature of the stars .
we have searched to date five photometry databases ( see companion paper in this proceedings by shaw & lpez - morales ) .
the result of that search are 41 new detached eclipsing binaries with masses below 1@xmath0 . after identifying the binaries from the lcs in those databases ,
we need to conduct follow - up observational campaigns to measure the optical and infrared light curves of the systems and their radial velocity curves .
this is an extensive project that requires of large amounts of telescope time .
currently we have been awarded time in the facilities listed in table 1 .
our final goal is to obtain full , high quality lcs and rv curves to be able to determine the masses and the radii of the stars in those binaries with errors smaller than 3% .
we have completed to date the optical ( vri ) light curves and radial velocity curves of three binaries : gu boo ( lpez - morales & ribas 2005 ) , rxj0239.1 ( torres et al .
, in prep ) , and nsvs01031772 ( hereafter nsvs0103 ; lpez - morales et al . , submitted ) .
near - ir light curves are also available for rxj0239.1 .
table 2 summarizes the masses , radii , and temperatures derived for the components of each binary .
the two stars in gu boo are almost identical to the stars in yy gem .
the stars in the other two binaries , with masses between 0.5 and 0.55 @xmath0 and 0.7 and 0.73 @xmath0 respectively , fill - in two current gaps in the mass - radius relation .
figure 1 shows the mass - radius relation of stars below 1@xmath0 .
the lines represent the predictions of different models , using 0.35 gyr isochrones and a metallicity z = 0.02 .
the open circles correspond to the previously known binaries cm dra ( lacy 1977 ; metcalfe et al .
1996 ) , cu cnc ( delfosse et al . 1999 ; ribas 2003 ) , tres - her0 - 07621 ( creevey et al .
2005 ) , and yy gem ( leung & schneider 1978 ; torres & ribas 2002 ) .
the filled squares show the location in this diagram of the components of gu boo , rxj0239.1 , and nsvs0103 . except for tres - her0 - 07621 ,
all the other stars show a clear trend towards larger radii than what the models predict .
all the stars in binaries are at least 10% larger than what any of the models predict .
figure 2 shows the mass
log(@xmath3 ) relation for gu boo , rxj0239.1 , and nsvs0103 ( open circles ) , yy gem ( filled circle ) , and cu cnc ( open triangles ) .
the top figure corresponds to a metallicity of z=0.01 , the bottom figure is for a metallicity of z=0.02 .
the age of both sets of isochrones is 0.35 gyrs .
the bottom figure ( z=0.02 ) agrees with the trend observed by baraffe et al .
( 1998 ) , where they find that below 37003800k the effective temperatures predicted by the models are larger than the ones observed in low - mass stars .
we present in this paper the first results of an extensive observing campaign primarily aimed at providing an accurate empirical m - r relation for low - mass stars .
our targets are low - mass eclipsing binaries , from where precise stellar masses and radii can be derived .
these systems also provide an estimation of the @xmath3 of the stars .
our current sample contains 41 new binaries with masses between 0.35 and 1.0@xmath0 . here
we present the parameters of the first three of those binaries , gu boo , rxj0239.1 , and nsvs0103 , which provide six new valuable data points .
the addition of those new data points to the mass radius relation diagram ( see figure 1 ) strengthens the trend already suggested by the other binaries ( cm dra , cu cnc , and yy gem ) .
that is , the models underestimate the radii of low - mass stars by at least 10 % .
this is at least the case for the components of binaries .
the few available measurements from single stars present a much larger scatter and larger error bars , preventing the identification of a clear trend , if any in fact exists .
a mismatch is also apparent when the @xmath3 of the binaries are compared to the predictions by the models ( see figure 2 ) . in this case the temperatures are lower than the prediction by the models below @xmath2 37003800k , as already noticed by baraffe et al .
( 1998 ) .
observations begin to reveal what appear to be two different populations of low - mass stars : non - active stars , whose parameters would be succesfully reproduced by the `` standard models '' ( i.e. those not including magnetic fields ) , and active stars , where the magnetic fields play an important role .
+ + + * acknowledgments*. we are grateful to the following institutions for providing support for this work : carnegie institution of washington , nasa astrobiology institute , southeastern association for research in astronomy , university of georgia at athens , instituto de astrofsica de canarias , national science fundation , and the university of north carolina at chapel hill .
we also thank our collaborators jerome a. orosz ( sdsu , usa ) , ignasi ribas ( ieec , spain ) , maria jess arvalo and carlos lzaro ( iac , spain ) , and guillermo torres ( harvard cfa , usa ) for their time and efforts towards this project
. m. l - m . acknowledges research and travel support from the carnegie institution of washington through a carnegie fellowship .
lll technique&&facility + photometry:&optical:&0.9 m @xmath12 telescope ( @xmath13 , usa ) + & & 1.0 m @xmath14 consortium telescope ( @xmath15 , chile ) + & & 2.5 m dupont telescope ( @xmath16 , chile ) + & near ir:&1.5 m carlos snchez telescope ( @xmath17 , spain ) + & & 2.5 m dupont telescope ( @xmath16 , chile ) + spectroscopy:&&2.5 m dupont telescope ( @xmath16 , chile ) + & & 4.0 m mayall telescope ( @xmath13 , usa ) + & & 6.5 m magellan telescopes ( @xmath16 , chile ) + lcccccc binary&m1 ( msun)&m2 ( msun)&r1 ( rsun)&r2 ( rsun)&teff1 ( k)&teff2 ( k ) + @xmath18&0.610 @xmath19 0.007&0.599 @xmath19 0.006&0.623 @xmath19 0.016&0.626 @xmath19 0.020&3920 @xmath19 130&3810 @xmath19 130 + @xmath20&0.730 @xmath19 0.009&0.693 @xmath19 0.006&0.741 @xmath19 0.004&0.703 @xmath19 0.002&4645 @xmath19 20&4275 @xmath19 15 + @xmath21&0.530 @xmath19 0.014&0.514 @xmath19 0.013&0.559 @xmath19 0.014&0.518 @xmath19 0.013&3750 @xmath19 150&3600 @xmath19 150 + | full tests to stellar models below 1@xmath0 have been hindered until now by the scarce number of precise measurements of the stars most fundamental parameters : their masses and radii . with the current observational techniques , the required precision to distinguish between different models ( errors @xmath1 2 - 3 % )
can only be achieved using detached eclipsing binaries where 1 ) both stars are similar in mass , i.e. q = m1/m2 @xmath2 1.0 , and 2 ) each star is a main sequence object below 1@xmath0 . until 2003 only three
such binaries had been found and analyzed in detail .
two new systems were published in 2005 ( creevey et al . ; lpez - morales & ribas ) , almost doubling the previous number of data points .
here we present preliminary results for 3 new low - mass detached eclipsing binaries .
these are the first studied systems from our sample of 41 new binaries ( shaw & lpez - morales , this proceedings ) .
we also provide an updated comparison between the mass radius and the mass@xmath3 relations predicted by the models and the observational data from detached eclipsing binaries .
we define low - mass stars as main sequence stars with masses between 1@xmath0 and the hydrogen burning limit ( 0.070.08@xmath0 ) .
low - mass stars are small cool objects , with radii between 1.0 and 0.1@xmath4 and effective surface temperatures between 6000 and 2500 k. they are also faint as their luminosities between 1 and @xmath5@xmath6 reveal .
they are the most abundant stars in the galaxy , where least 7 of every 10 stars are low - mass main sequence stars .
these objects play a role on studies of baryonic dark matter ( it is thought that low - mass stars , brown dwarfs , and stellar remnants are the main contributors to the baryonic dark matter in the universe ) , on dynamical studies of galaxies and star clusters , and on the detailed characterization of the stars in the solar neighborhood , where intensive searches for earth - like planets around low - mass stars are currently underway .
low - mass stars are also a subject of interest in other fields in physics , given the complicated physical processes that are taking place in these stars .
@xmath7 molecules , tio , @xmath7o , co , and cn become stable in the atmosphere of low - mass stars at temperatures below 50004000 k. below 2800 k , even more complex molecular compounds , such as @xmath8 , @xmath9 , and @xmath10 , also become stable . therefore any model trying to reproduce how the radiation is transmitted from the interior of the stars through their atmospheres
needs to take into account the effect of all these molecules and compounds .
the interior physics of low - mass stars is also challenging .
the interior of these stars is essentially a plasma of ionized h and he under partially or fully degenerate conditions , so one can not model them using ideal equations of state ( eoss ) . instead , we need more elaborate eoss , that include coulomb interactions between electrons and ions , ionization pressure effects , the effect of the electric fields generated by the ions , and so on . |
let @xmath1 be positive integers .
we denote by @xmath2 the @xmath0-dimensional rectangle of sides @xmath3 , that is , @xmath4 .
a @xmath0-dimensional rectangle @xmath5 is said to be _ tiled _ with _ bricks _
( i.e. , small @xmath0-dimensional rectangles ) @xmath6 if @xmath5 can be filled entirely with copies of @xmath7 , @xmath8 ( rotations allowed ) .
it is known @xcite that rectangle @xmath9 can be tiled with @xmath10 if and only if @xmath11 divides @xmath12 or @xmath13 , @xmath14 divides @xmath12 or @xmath13 and if @xmath15 divides one side of @xmath5 then the other side can be expressed as a nonnegative integer combination of @xmath11 and @xmath14 . in 1995 , fricke @xcite gave the following characterization when @xmath16 ( see also @xcite for a @xmath0-dimensional generalization with @xmath17 ) .
[ kler]@xcite let @xmath18 be positive integers with @xmath19 .
then , @xmath20 can be tiled with @xmath21 and @xmath22 if and only if either @xmath12 and @xmath13 are both multiple of @xmath23 or @xmath12 and @xmath13 are both multiple of @xmath24 or one of the numbers @xmath25 is a multiple of both @xmath23 and @xmath24 and the other can be expressed as a nonnegative integer combination of @xmath23 and @xmath24 .
let us consider the following natural question .
[ qq ] does there exist a function @xmath26 such that if @xmath27 then @xmath20 can be tiled with @xmath10 and @xmath28 for some positive integers @xmath29 and @xmath30 ?
an algebraic result due to barnes @xcite seems to show the existence of such @xmath31 .
however , barnesmethod does not give an explicit lower bound for @xmath31 .
the special case when @xmath32 and @xmath33 was posed in the 1991 william mowell putnam examination ( problem b-3 ) . in this case , klosinski _ et . al . _
@xcite gave a lower bound of @xmath31 .
their method was based on knowledge of the _
frobenius number_. the _ frobenius number _ , denoted by @xmath34 , of a set of relatively prime positive integers @xmath35 , is defined as the largest integer that is not representable as a nonnegative integer combination of @xmath35 .
it is well known that @xmath36 however , to find @xmath34 , for general @xmath0 , is a difficult problem from the computational point of view ; we refer the reader to @xcite for a detailed discussion on the frobenius number .
klosinski _ et . al .
_ used equation ( [ frob2 ] ) , with particular integers @xmath37 and @xmath38 , to show that @xmath20 can be tiled with @xmath39 and @xmath40 if @xmath41 .
.3 cm in this paper , we will use the frobeniuis number in a more general way to show that a @xmath0-dimensional rectangle @xmath5 can be tiled with some set of bricks if the sides of @xmath5 are larger than a certain function ( see theorem [ maint ] ) .
we use then theorem [ maint ] to obtain the following result .
[ cor1 ] let @xmath42 be integers with @xmath43 , @xmath44 and @xmath45 .
then , @xmath20 can be tiled with @xmath46 and @xmath47 if @xmath48 in the case when @xmath49 and @xmath50 , corollary [ cor1 ] implies that @xmath20 can be tiled with @xmath39 and @xmath40 if @xmath51 , improving the lower bound given in @xcite .
we remark that this lower bound is not optimal . in @xcite ,
narayan and schwenk showed that , in this particular case , it is enough to have @xmath52 .
however , their tiling constructions allow rotations of both bricks ( and tilings with more complicated patterns ) which is not the case of corollary [ cor1 ] . we shall also use theorem [ maint ] to prove the following result concerning tilings of squares .
[ cor2 ] let @xmath53 be prime integers .
then , @xmath54 can be tiled with @xmath55 if @xmath56 we finally improve the lower bound given in theorem [ cor2 ] in some special cases .
[ cor3 ] let @xmath57 be an odd integer with @xmath58 and let @xmath59 be a positive integer .
then , @xmath60 can be tiled with @xmath61 and @xmath62 if @xmath63 .
moreover , @xmath60 can be tiled with @xmath61 and @xmath64 if and only if @xmath65 and with @xmath61 and @xmath66 if and only if @xmath67 . a collection of some unpublished work , due to d.a .
klarner , in relation with theorem [ cor3 ] can be found in @xcite .
we need to introduce some notation and definitions .
let @xmath68 where @xmath69 are positive integers .
we will write @xmath70 instead of @xmath71 and @xmath72 instead of @xmath73 .
let @xmath74 be a positive integer for each @xmath75 and each @xmath76 .
let @xmath77 , @xmath78 .
we define the set we denote by @xmath82 the rectangle obtained from @xmath83 by sticking together @xmath84 copies of @xmath85 along the @xmath86-axis , that is , @xmath87 .
finally , we denote by @xmath88 the @xmath89-dimensional rectangle obtained from @xmath83 by setting @xmath90 , that is , @xmath91 . _ proof .
_ we shall use induction on @xmath0 .
for @xmath100 we have that @xmath101 and thus @xmath102 . by definition of the frobenius number ,
any integer @xmath103 is of the form @xmath104 where @xmath105 are nonnegative integers .
thus , the 1-dimensional rectangle @xmath106 ( that is , the interval @xmath107 $ ] ) can be tiled by sticking together @xmath108 ( that is , the interval @xmath109 $ ] ) and @xmath110 ( that is , the interval @xmath111 $ ] ) .
.3 cm we suppose that it is true for @xmath112 and let @xmath74 be a positive integer for each @xmath113 and each @xmath114 with @xmath94 for any @xmath115 , @xmath116 and let @xmath117 , @xmath114 and @xmath118 for all @xmath119 . indeed ,
if we consider the rectagle @xmath120 embedded in @xmath127 with @xmath128 then by replacing each brick @xmath129 used in the tiling of @xmath120 by @xmath130 we obtain a tiling of @xmath125 with bricks @xmath126 .
.3 cm now , since @xmath131 then @xmath132 where each @xmath133 is a nonnegative integer . by the above claim
, @xmath134 can be tiled with bricks @xmath135 for each @xmath136 .
thus , @xmath137 can be tiled with @xmath138 by sticking together bricks @xmath139 along the @xmath140-axis .
\(b ) by induction on @xmath166 . for @xmath167
we have @xmath168 since @xmath169 and @xmath170 for all @xmath171 .
we suppose that it is true for @xmath172 and assume that @xmath173 . by equation ( [ eerr ] )
we have _ proof of theorem [ cor2 ] .
_ let @xmath53 be prime integers , @xmath93 .
we consider theorem [ maint ] , where , for each @xmath76 , we let @xmath184 for all @xmath99 .
then , @xmath54 can be tiled with @xmath55 if @xmath185 [ llls ] let @xmath191 and @xmath84 be positive integers with @xmath192 and such that @xmath193 for some integers @xmath194 and @xmath195 for some integers @xmath196 .
then , @xmath197 and @xmath198 can be tiled with @xmath199 and @xmath200 for any integer @xmath201 .
_ proof of theorem [ cor3 ] .
_ by theorem [ kler ] , we have that @xmath60 can be tiled with @xmath210 and @xmath211 if @xmath212 or @xmath213 .
so , we only need to show that @xmath60 can be tiled with @xmath214 and @xmath62 for any odd integer @xmath63 with @xmath215 or @xmath216 .
we have two cases .
.3 cm case a ) @xmath217 .
let @xmath218 for any integer @xmath201 . since @xmath219 then there exist nonnegative integers @xmath220 and @xmath221 such that @xmath222 .
since @xmath223 then , by proposition [ llls ] , @xmath224 can be tiled with @xmath214 and @xmath62 for any @xmath225 and @xmath226 . now , since @xmath227 for some integer @xmath201
then for @xmath228 we have that @xmath229 and by proposition [ llls ] ( with @xmath230 ) , we have that @xmath231 can be tiled with @xmath214 and @xmath62 for any odd integer @xmath232 with @xmath233 .
.3 cm case b ) @xmath234 .
let @xmath235 for any integer @xmath201 . since @xmath219 then there exist nonnegative integers @xmath220 and @xmath221 such that @xmath222 .
since @xmath223 then , by proposition [ llls ] , @xmath224 can be tiled with @xmath214 and @xmath62 for any @xmath236 and @xmath237 .
now , since @xmath238 for some integer @xmath201 then for @xmath228 we have that @xmath229 and by proposition [ llls ] ( with @xmath230 ) , we have that @xmath231 can be tiled with @xmath214 and @xmath62 for any odd integer @xmath232 with @xmath239 .
.3 cm let us set @xmath240 .
it is clear that @xmath241 and @xmath66 can not be tiled with @xmath214 and @xmath64 . by the above cases
, we have that @xmath60 can be tiled with @xmath214 and @xmath64 if @xmath242 and , by theorem [ kler ] , @xmath60 can be tiled with @xmath210 and @xmath211 if @xmath212 or @xmath213 .
this leave us the cases when @xmath243 and @xmath244 . if @xmath245 is trivial .
@xmath246 can be tiled with @xmath214 and @xmath64 since , by tha above case b , the result is true for any odd integer @xmath247 and @xmath226 .
finally , @xmath248 can be tiled as it is illustrated in figure [ fig2 ] .
let us set @xmath249 .
it is clear that @xmath241 , @xmath64 and @xmath246 can not be tiled with @xmath214 and @xmath66 . by the above cases ,
we have that @xmath60 can be tiled with @xmath214 and @xmath66 if @xmath250 and , by theorem [ kler ] , @xmath60 can be tiled with @xmath210 and @xmath211 if @xmath212 or @xmath213 .
this leave us the cases when @xmath251 and @xmath252 . if @xmath253 is trivial . @xmath248 and
@xmath254 both can be tiled since , by the above case a , the result is true for any odd integer @xmath255 with @xmath233 .
finally , @xmath256 can be tiled as it is illustrated in figure [ fig3 ] . | in this paper , we give some sufficient conditions for a @xmath0-dimensional rectangle to be tiled with a set of bricks .
these conditions are obtained by using the so - called frobenius number . |
the confirmation of the temporal variation of the fundamental constants would be the first indication of the universe expansion influence on the micro physics @xcite .
shlyakhter was the first who showed that the variation of the fundamental constants could lead to measurable consequences on the sm isotops concentrations in the ancient reactor waste @xcite .
later damur and dyson @xcite for zones 2 and 5 and also fujii @xcite for zone 10 of reactor oklo made more realistic analysis of the possible shift of fundamental constants during the last @xmath6 years based on the isotope concentrations in the rock samples of oklo core . in this investigation
the idealized maxwell spectrum of neutrons in the core was used .
the efforts to take into account more realistic spectrum of neutrons in the core were made in works @xcite .
new severe constraints on the variation of the fine structure constant have been obtained from reactor oklo analysis in work @xcite : @xmath7 we investigate here how these constraints confine the parameter of bsbm model @xcite of varying @xmath0 .
this theory combines bekenstein extension of electrodynamics @xcite with varying alpha to include gravitational effects of new scalar field @xmath8 .
it respects covariance , gauge invariance , causality and has only two free parameters : the fraction of electromagnetic energy @xmath4 in the total energy of matter including dark matter as well as the dimensional parameter @xmath3 which is having sense of characteristic length . as a result of our analysis
we get the constraints on the combination of the parameters of bsbm model .
bsbm theory @xcite is the extension of the bekenstein @xcite theory to include dynamics of the gravitational field .
total action of this theory has a form : @xmath9 where @xmath10 and @xmath11 .
a parameter @xmath12 here is definite as @xmath13 where dimensional parameter @xmath3 is having sense of characteristic length .
fine structure constant expressed via @xmath8 with the equation : @xmath14 .
varying @xmath8 we get the following equation : @xmath15 for pure radiation @xmath16 , so @xmath8 remains constant during radiation domination epoch . only in matter domination epoch changes in @xmath0
take place .
the only contribution to variation of @xmath8 come mainly from pure electrostatic or magnetostatic energy .
it is convenient to work in the following parameter : @xmath17 and according to @xcite @xmath18 and @xmath19 .
varying the metric tensor and using friedmann metric we get the following friedmann equation : @xmath20,\ ] ] and the equation for @xmath8 takes form : @xmath21 where @xmath22
. we have also energy conservation equations : @xmath23 which have solutions : @xmath24 , and @xmath25 let use critical density : @xmath26 and use also the fractions of all densities relative to critical : @xmath27 .
index @xmath28 will denote the present values of these fractions .
we use the ordinary values for these fractions at present : @xmath29 , @xmath30 , and @xmath31 is determined from the condition that the universe is flat
. then the friedmann equation takes form : @xmath32,\ ] ] and equation for @xmath8 : @xmath33 here constant @xmath34 is equal to @xmath35 . for negative @xmath4
this constant is positive and has the following dependence on the ratio of characteristic and plank lengthes : @xmath36 the result of the numerical integration of these equations is presented of fig.1 for the variation of different components of energy density with red shift @xmath37 , and on fig.2 for the variation of fine structure constant @xmath0 .
here we use the notation : @xmath38 .
we took the value of the characteristic length @xmath3 equal to @xmath39 during this analysis , and assigned the following value for the the parameter @xmath4 : @xmath40 where @xmath41 - is the fraction of energy density in the universe due to ordinary baryonic matter .
the initial values of the scalar field @xmath8 for the second order differential equation ( 6 ) : the value of the scalar field @xmath8 and its derivative during the radiation epoch was taken in such a manner that the present value of the fine structure constant coincide with experiment , and it appeared that the initial value of the @xmath42 during the radiation domination epoch could be assigned a rather arbitrary value because the result of integration influenced rather weakly by this choice .
- dash - dot line.,width=453 ] experimental results for keck telescope @xcite , closed circles - experimental results from vlt telescope ( data were taken from work @xcite ) , red circle at @xmath43 - oklo result.,width=453 ] as it is followed from figure [ dens ] , the scalar field @xmath8 influence rather weakly on the variation of the different components of the energy density with red shift .
the total variation of alpha during the whole history of the universe is about @xmath44 ( as is followed from figure [ alpha ] ) which is not contradict big bang and radiation recombination constraints @xcite . on
the other side the oklo analysis predict about zero result for @xmath45 with the experimental error which could be seen in figure [ alpha ] ) if we increase the scale of figure [ alpha ] one hundred times .
we investigate the constraints on the parameters of bsbm model followed from oklo analysis in the next section .
in analysis of oklo data @xcite we obtained the following constraints on the variation of the fine structure constant @xmath46 during the past @xmath47 years .
the age of the reactor @xmath48 years corresponds to red shift parameter @xmath43 .
we use here also previous constraints obtained in @xcite : @xmath49 and in @xcite : @xmath50 all these constraints are shown on figure [ oklo ] . to provide the solution of the equations ( 5 ) and ( 6 ) which does nt contradict the result of work @xcite ( see figure [ oklo ] )
, we have to set rather severe constraints on the combinations of the parameters of bsbm model .
they have to satisfy the following inequality : @xmath51 for realistic value @xmath52 to fulfill this inequality we have to demand that : @xmath53
a theoretical framework under very general assumptions was worked out by bekenstein to admit the variation of the fine structure constant .
a characteristic length @xmath3 enters into it .
an experimental constraint rules out @xmath0 variability of any kind if it is in clear conflict with predictions of the framework for @xmath3 no shorter than the fundamental length @xmath39 ( @xcite ) . as a result of oklo analysis
we get @xmath54 the oklo geophysical constraints strongly rule out all @xmath0 variability . in this analysis
we have considered only the variation of electromagnetic fine structure constant @xmath0 .
if other fundamental constants also varies the picture would be more complicated as well as the analysis of the oklo phenomenon and the analysis of the cosmological variation of @xmath0 . to do such analysis in our opinion
would be too early because till now we havent had any convincing manifestations of the cosmological variations of the other fundamental constants @xcite .
the author would like to express his gratitude to s. karshenboim and m.s .
yudkevich for useful discussions and critical remarks .
this work was partly supported by the rscf grant ( project 14 - 22 - 00281 ) . | new severe constraints on the variation of the fine structure constant have been obtained from reactor oklo analysis in our previous work .
we investigate here how these constraints confine the parameter of bsbm model of varying @xmath0 . integrating the coupled system of equations from the big bang up to the present time and taking into account the oklo limits
we have obtained the following margin on the combination of the parameters of bsbm model : @xmath1 where @xmath2 cm is a plank length and @xmath3 is the characteristic length of the bsbm model .
the natural value of the parameter @xmath4 - the fraction of electromagnetic energy in matter - is about @xmath5 . as a result
it is followed from our analysis that the characteristic length @xmath3 of bsbm theory should be considerably smaller than the plank length to fulfill the oklo constraints on @xmath0 variation . |
at energies below the electroweak scale the weak interactions are described by local four - fermi operators multiplied by effective coupling constants , the wilson coefficients . the formal framework to achieve this is the operator product expansion ( ope ) which allows one to separate the calculation of a physical amplitude into two distinct parts : the short distance ( perturbative ) calculation of the wilson coefficients and the long distance ( generally non - perturbative ) calculation of the hadronic matrix elements of the operators @xmath11 .
we calculate on the lattice @xmath0 and @xmath1 .
this allows us to calculate the low energy constants in chiral perturbation @xcite which , after incorporating the non - perturbative renormalization factors are then translated into @xmath12 matrix elements .
the cp - pacs collaboration has also presented a very similar calculation at this meeting @xcite .
we have used the wilson gauge action , quenched , at @xmath7 on a @xmath13 lattice which corresponds to an inverse lattice spacing @xmath14 . the domain wall fermion height @xmath8 and fifth dimension @xmath15 give a residual symmetry breaking @xmath16 @xcite ; 400 configurations separated by 10000 heat - bath sweeps were used in this analysis .
@xmath0 matrix elements were calculated in the @xmath17 limit for 5 light quark masses @xmath18 .
since the @xmath1 matrix elements vanish in the @xmath19 limit these matrix elements were calculated with non - degenerate quark propagators for 10 mass combinations subject to the constraint @xmath20 .
we have also calculated the so called eye diagrams with an active charm quark for @xmath21 ( the physical charm quark is around 0.5 ) . however ,
the analysis for charm - in is still in progress ; in this presentation we concentrate on the case with 3-active flavors wherein charm is integrated out assuming it is very heavy .
the calculation took about 4 months on 800 gflops ( peak ) .
quark propagators were calculated using the conjugate gradient method with a stopping residual of @xmath22 with periodic and anti - periodic boundary conditions which amounts to doubling the lattice size in time direction .
the two wall source propagators at @xmath23 and @xmath24 were fixed to coulomb gauge . for eye diagrams we employed random wall sources
spread over time slices @xmath25 with 2 hits per configuration .
dividing the three - point correlation functions by the wall - wall pseudoscalar - pseudoscalar correlation function yields the desired matrix elements up to a factor of @xmath26 which is determined from a covariant fit to the wall - point two - point function in the range @xmath27 for each mass .
since our results unambiguously show that re@xmath4 and re@xmath5 come essentially from the current - current operators ( recall these have the largest wilson coefficients ) we will concentrate on these operators from now on .
quenched chiral perturbation theory predicts @xmath28 % \end{equation } % \end{small}\ ] ] we find a quenched chiral logarithm coefficient @xmath29 which has a negligible contribution in our matrix element calculation . unlike the quenched chiral logarithms , the conventional logarithms coming from quenched chiral perturbation theory induce large corrections to the @xmath3 @xmath0 matrix element as can be seen in figure [ fig : o2_ktopi_3_2 ] .
we fit these amplitudes to @xcite @xmath30 + b_2^{(27,1 ) } m_m^4 % \end{equation}\ ] ] where @xmath31 , @xmath32 , @xmath33 .
the conventional chiral logarithm @xmath34 is almost linear over the mass range we have used so the fitting routine can not distinguish this term from the linear term if we leave the coefficient of the logarithm as a free parameter . since the large coefficient -6 of the logarithm makes the contribution of this term comparable to the contribution of the linear term omitting this term would change @xmath35 by almost a factor of two .
the quenched chiral log contribution is very small .
= -0.3 in -0.2 in @xmath2 @xmath0 matrix elements mix with @xmath36 with a power divergent coefficient @xmath37 .
we define a subtracted matrix element @xmath38 by @xmath39 where @xmath40 is obtained from a linear fit to @xmath41 . for an explanation of this subtraction of the power divergence
we refer the reader to @xcite .
the quenched chiral perturbation theory corrections to @xmath42 our data is consistent with a linear fit @xmath43 with the slope @xmath44 determining the low energy constants @xmath45 and the intercept @xmath46 arising from residual chiral symmetry breaking .
-0.1 in = -0.3 in -0.2 in
we use chiral perturbation theory to compute the lattice @xmath49 matrix elements . using non - perturbative z factors
we obtain the continuum matrix elements which are then multiplied by wilson coefficients to yield the physical amplitudes .
we present an extrapolation to the kaon mass scale to lowest order in chiral perturbation theory and a second extrapolation which includes one loop logarithmic effects .
we multiply the pseudoscalar masses by @xmath50 so that for @xmath51 the chiral perturbation theory extrapolation is increasingly accurate but we need the extrapolation at @xmath52 , the physical point . in figure
[ fig : re_a0_re_a2 ] we present re@xmath47 and re@xmath48 as a function of the parameter @xmath50 .
the chiral logarithm correction for re@xmath47 is large ( about @xmath53 ) .
in addition one expects a large correction ( not included here ) coming from the tree level @xmath54 terms necessary to cancel the dependence on the chiral perturbation theory scale @xmath55 .
-0.1 in = -0.3 in -0.2 in if the z factors and the wilson coefficients were calculated to all orders in perturbation theory the physical amplitudes that we calculate would not depend on the scale @xmath56 where the transition between the lattice and the continuum operators is made .
to a good approximation this is what we find , even though at @xmath57 one expects non - perturbative effects in the z factors and at @xmath58 the discretization errors may be large . in figure
[ fig : delta_i_1_2_mu_dep ] we present the ratio re@xmath47/re@xmath48 , the so called @xmath59 rule which shows a large enhancement in the @xmath60 channel in accord with experiment ( note , the chiral logarithm corrections largely cancel in the ratio so a large enhancement is seen for both extrapolation choices ) .
the residual scale dependence in the physical amplitudes is slight ( see figure [ fig : delta_i_1_2_mu_dep ] ) .
= -0.3 in -0.2 in
in conclusion , re@xmath47 , re@xmath48 and especially the ratio re@xmath47/re@xmath48 were found reasonably close to the experimental values .
we see this as an important success of the lattice method .
however there were a number of major approximations in our calculation , the hardest to quantify is the use of quenched qcd .
also the chiral logarithms in quenched @xmath0 , @xmath61 are not known and we have included only the logarithmic portion of the next - to - leading - order , 1-loop corrections in @xmath12 extrapolations . 6 c. bernard , _ et .
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b * 152 * ( 1985 ) 226 . | we have used domain wall fermions to calculate @xmath0 and @xmath1 matrix elements which can be used to study the @xmath2 rule for k decays in the standard model .
nonlinearities in the @xmath3 matrix elements due to chiral logarithms are explored and the subtractions needed for the @xmath2 matrix elements are discussed .
using renormalization factors calculated using non - perturbative renormalization then yields values for real @xmath4 and @xmath5 .
we present the details of our quenched @xmath6 , @xmath7 , @xmath8 simulation , where a previous calculation showed that the finite @xmath9 chiral symmetry breaking effects are small ( @xmath10 ) . |
liquid helium system has a total hamiltonian as @xmath0 where @xmath1 is the mass of a helium atom , @xmath2 and @xmath3 respectively signify the creation and annihilation operators .
we examine the general form of the total energy via the unitary transformation @xmath4 diagonalizing the hamiltonian @xmath5 .
all eigenstates are described as @xmath6 where @xmath7 denotes the vacuum state .
new creation and annihilation operators are defined as @xmath8 which indicate the creation and annihilation operators of a quasi - particle .
we designate this quasi - particle as a `` dressed boson '' .
the dressed boson number operator is defined as @xmath9 the total number conservation and the total momentum conservation are expressed as @xmath10 that is to say , the total number of helium atoms is equal to the total number of dressed bosons and the total momentum of helium atoms is equal to the total momentum of dressed bosons .
the total energy of the system is a sum of the kinetic energy @xmath11 of the center of mass and galilean invariant terms @xmath12 : @xmath13 where @xmath14 is the total mass of liquid helium .
@xmath15 where galilean invariant terms are described only by relative momenta of dressed bosons : ( galilean invariant terms)= @xmath16 + substitution of eq.([eq : c ] ) into eq.([eq : b ] ) yields @xmath17 where we abbreviate higher terms because three - particle collision is a rare case for diluteness of liquid helium .
the single excitation state has a distribution of @xmath18 and therefore its total energy is derived from eq.([eq : d ] ) as follows : @xmath19 where we have used @xmath20 and the spherical symmetric property of the function @xmath21 .
therein , the latent heat at zero kelvin is equal to @xmath22 . accordingly
the elementary excitation energy at zero kelvin is given by @xmath23 .
this relation engenders a function form of the nonlinear term as @xcite @xmath24
the energy of one dressed boson is an increase value of the total energy when one dressed boson is added to the system . accordingly
the dressed boson energy is defined as @xmath25 .
the calculation result for the derivative of eq .
( [ eq : d ] ) shows @xmath26 where we have used @xmath27 .
the distribution function is determined as @xmath28 we can obtain approximate solutions of the coupled equations of ( [ eq : g ] ) and ( [ eq : h ] ) via the iteration method @xcite .
we adopt the landau distribution function as the zero - th order distribution : @xmath29 the j - th order solutions are derived from the ( j-1)-th distribution function as follows : @xmath30 this j - th energy form produces the j - th distribution function : @xmath31 therein the excitation energy from the bose - einstein condensate of dressed bosons is expressed as @xmath32 we can evaluate the second order solutions @xmath33 and @xmath34 via the iteration processes from the zero - th order distribution .
using the second order excitation energy @xmath33 and the distribution function @xmath34 , we can calculate the second order approximation values of specific heat as follows @xcite ; + @xmath35 + + the evaluated results are shown in fig.[f : 1 ] and fig.[f : 2 ] .
figure [ f : 1 ] and [ f : 2 ] indicate the second order results of specific heat via the nonlinear theory .
the curves express the calculated values .
the dots with red indicate experimental data @xcite . as shown in fig.[f : 1 ] and fig.[f : 2 ] , the theoretical values of the second order are in good agreement with the experimental data for @xmath36 .
it is noteworthy that the present calculation uses the experimental values of excitation energy only for the temperature 1.1 k. of course the iteration method is insufficient in close vicinity of the @xmath37 transition temperature .
we have discussed origin of the logarithmic divergence at the @xmath37 point in the previous paper @xcite . it is clarified that the logarithmic divergence is caused by the nonlinear mechanism of the total energy .
the calculation results are shown in fig .
[ f : 3 ] .
the large dots colored with blue indicate the experimental data @xcite , and the small dots with red are measured by lipa et al @xcite .
bd theory indicates the results in the reference of @xcite .
bcy theory indicates the results in the reference of @xcite .
the curve of the nonlinear theory is the results of the reference @xcite ] thus the nonlinear theory has well explained the temperature dependence of the specific heat of superfluid helium for all temperature region . accordingly
the nonlinear mechanism of total energy is important for understanding the properties of liquid helium .
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sasaki s 1994 _ physica _ b * 194 * 497 . | the specific heat of liquid helium was calculated theoretically in the landau theory @xcite .
the results deviate from experimental data in the temperature region of 1.3 - 2.1 k. many theorists subsequently improved the results of the landau theory by applying temperature dependence of the elementary excitation energy @xcite , @xcite . as well known , many - body system has a total energy of galilean covariant form .
therefore , the total energy of liquid helium has a nonlinear form for the number distribution function .
the function form can be determined using the excitation energy at zero temperature and the latent heat per helium atom at zero temperature .
the nonlinear form produces new temperature dependence for the excitation energy from bose condensate .
we evaluate the specific heat using iteration method .
the calculation results of the second iteration show good agreement with the experimental data in the temperature region of 0 - 2.1 k , where we have only used the elementary excitation energy at 1.1 k. |
global - scale magnetic fields and cycles of magnetic activity in sun - like stars are generated by the interplay of rotation and convection . at rotation rates greater than that of the current sun , such as when our sun was younger , observations tend to show increased magnetic activity indicating a strong global dynamo may be operating ( pizzolato et al .
2003 ) . here
we explore large - scale dynamo action in sun - like stars rotating at three times the current solar rate , or @xmath0 , with a rotational period of 9.32 days . as shown by helioseismology ,
the solar interior is in a state of prominent differential rotation in the convection zone ( roughly the outer 30% by radius ) whereas the radiative interior is in uniform rotation .
a prominent shear layer , or tachocline , is evident at the interface between the convective and radiative regions .
motivated by these observations , a number of theoretical models have been proposed for the solar dynamo .
the current paradigms for large - scale solar dynamo action favor a scenario in which the generation sites of toroidal and poloidal fields are spatially separated ( e.g. , charbonneau 2005 ) .
poloidal fields generated by cyclonic turbulence within the bulk of the convection zone , or by breakup of active regions , are pumped downward to the tachocline of rotational shear at its base .
the differential rotation there stretches such poloidal fields into strong toroidal structures , which may succumb to magnetic buoyancy instabilities and rise upward to pierce the photosphere as curved structures that form the observed active regions .
similiar dynamo processes are believed to be active in sun - like stars rotating several times faster than the current sun . here
we explore a variation to this paradigm by excluding the tachocline and the photosphere from our simulated domain , which extends from @xmath1 to @xmath2 , in order to see if magnetic cycles can be realized in the bulk of the convection zone itself .
radial velocity in global mollweide projection at @xmath3 with fast , narrow downflows in dark tones and broad , slow upflows in light tones .
differential rotation profile , with lines of constant angular velocity @xmath5 largely along cylinders , as expected for rapidly rotating systems .
some deviation toward conical contours is seen at low latitudes .
magnetic wreaths tend to form in the regions of strong shear near the equator . ] using massively - parallel supercomputers , we solve the nonlinear anelastic mhd equations in rotating 3-d spherical shells using the anelastic spherical harmonic ( ash ) code ( brun et al .
the anelastic approximation filters out fast - moving sound and magneto - acoustic waves , allowing us to follow the decidedly subsonic flows in the solar convection zone with overturning times of days to months . in large - eddy simulation ( les ) such as those using ash , the effects of small , unresolved scales on larger scales must be parameterized using a turbulence closure model .
previous ash simulations of convective dynamos in sun - like stars rotating at @xmath0 have yielded large - scale wreaths of strong toroidal magnetic field in the bulk of their convection zones ( brown et al 2010 ) .
these wreaths persist for decades of simulation time , remarkably coexisting with the strongly turbulent flows . here
we explore the effects of decreased levels of diffusion on these wreaths in two simulations , labeled case b and case s. case b uses an eddy viscosity that varies with depth as the square root of the mean density .
case s uses the dynamic smagorinsky model of germano et al .
( 1991 ) , which is based on the assumption of self - similarity in the inertial range of the velocity spectra .
case s has 50 times less diffusion on average than case b. figure 1a shows the radial velocity field for case s near the top of the convection zone with columnar cells at low latitudes and smaller - scale helical convection at higher latitudes .
figure 1b shows the differential rotation profile for case s with roughly 20% ( 250 nhz ) contrast in rotation rate between the equator and poles .
the radial velocity patterns and differential rotation for case b are qualitatively similar to figure 1 . longitudinal magnetic field @xmath6 for case b at @xmath7 in mollweide projection , showing two strong but patchy magnetic wreaths of opposite polarity with peak field strengths of 38 kg . @xmath4 time - latitude plot of @xmath6 averaged over longitude @xmath8 at the same depth over 15 years in case b , with strong negative - polarity wreaths shown in dark tones and strong positive - polarity wreaths shown in light tones , clearly indicating cyclic behavior and reversals in magnetic polarity .
the most remarkable feature of case b is a cyclic variation in the toroidal wreaths of magnetic field . with significantly less diffusion than the simulation of brown et al .
( 2010 ) that produced persistent wreaths with no reversals , case b creates strong toroidal bands of magnetic field as shown in figure 2 with peak field strengths of about 38 kg .
these wreaths of magnetic field vary strongly with time in both polarity and amplitude .
figure 2a shows @xmath6 in the lower convection zone when there are strong wreaths of opposite polarity in each hemisphere with significant longitudinal variation , which we term patchy wreaths .
if we average over longitude , figure 2b shows a time - latitude map of the @xmath9 in the lower convection zone .
the simulation clearly goes through reversals in the magnetic polarity of the wreaths in each hemisphere . at times
the hemispheres are out of phase with each other , occasionally yielding wreaths of the same polarity in both hemispheres .
such behavior might be termed irregular magnetic activity cycles . from case
s , 3-d volume visualization of magnetic field lines in the core of a wreath - segment with the inner and outer simulation boundaries shown as lined surfaces .
view is looking at low latitudes along the rotation axis .
@xmath4 radial location of the top of a buoyant loop as a function of time .
magnetic field strength at the top of the loop is indicated at representative times .
time corresponding to @xmath10 is indicated by circular plotting symbol at day 13.7 . ]
as we move to even less diffusive simulations , case s shows additional features in the strong toroidal wreaths , most notably buoyant loops of magnetic field .
the wreaths are again patchy in longitude and roughly cyclic in time .
the peak magnetic field strength rises to about 45 kg inside the wreaths .
these strong fields combine with the very low levels of diffusion to allow regions of very strong field to coherently move upward without changing the magnetic topology via reconnection or simply diffusing away the strong fields .
such magnetic loops rise due to a combination of magnetic buoyancy and advection by convective giant cells that span the layer .
figure 3a shows a magnetic loop near its maximum size , extending from @xmath11 to @xmath12 .
examination reveals that there is a significant amount of twist present in the loops and that there is a significant deflection poleward as they rise .
the radial location of the top of one buoyant loop as a function of time is shown in figure 3b .
initially the buoyancy of the wreath due to evacuation of fluid from magnetic pressure dominates over the advective force of the convective upflows , but within 6 days advection becomes dominant . after about 10 days
the magnetic tension force begins to balance the advection , causing the top of the loop to stall near @xmath13 .
these simulations suggest that stars rotating slightly faster than the current sun may produce dynamos capable of cycles of magnetic activity and buoyant magnetic structures in the bulk of their convective envelopes despite the absence of a tachocline of shear .
this both challenges and informs the interface dynamo paradigm for sun - like stars .
the essential questions are what drives the magnetic reversals in these simulations and what are the conditions necessary to generate buoyant magnetic loops that can survive transit through the convection zone .
+ this work is supported by nasa heliophysics theory program grants nng05g124 g and nnx08ai57 g and major supercomputing support through nsf teragrid resources .
the presentation of this paper in iau symposium 273 was aided by nsf grants atm 0548260 and ast 0968672 , and nasa grant 09-lwstrt09 - 0039 .
browning is supported by the jeffrey l. bishop fellowship at cita .
brown , b.p . , browning , m.k . , brun , a.s . , miesch , m.s . ,
& toomre , j. , 2010 , `` persistent magnetic wreaths in a rapidly rotating sun '' _ astrophys .
j. _ * 711 * 424 | observations of sun - like stars rotating faster than our current sun tend to exhibit increased magnetic activity as well as magnetic cycles spanning multiple years . using global simulations in spherical shells to study the coupling of large - scale convection , rotation , and magnetism in a younger sun ,
we have probed effects of rotation on stellar dynamos and the nature of magnetic cycles .
major 3-d mhd simulations carried out at three times the current solar rotation rate reveal hydromagnetic dynamo action that yields wreaths of strong toroidal magnetic field at low latitudes , often with opposite polarity in the two hemispheres .
our recent simulations have explored behavior in systems with considerably lower diffusivities , achieved with sub - grid scale models including a dynamic smagorinsky treatment of unresolved turbulence .
the lower diffusion promotes the generation of magnetic wreaths that undergo prominent temporal variations in field strength , exhibiting global magnetic cycles that involve polarity reversals . in our least diffusive simulation
, we find that magnetic buoyancy coupled with advection by convective giant cells can lead to the rise of coherent loops of magnetic field toward the top of the simulated domain . |
this work was carried out under the visiting researcher@xmath31s program of kyokugen at osaka university and was supported in part by a grand - in - aid for scientific research from the ministry of education , science , sports and culture .
m.h . would like to thank professor k. nonoyama of konan women@xmath31s junior college for information about the synthesis of na@xmath0[cu(pba)]@xmath26h@xmath0o and professor s. yamamoto of okayama university for fruitful discussions .
thanks are also due to the chemical analysis units in riken . 99 f. d. m. haldane : phys . rev .
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* 20 * ( 1976 ) 127 . | we report the results of magnetic measurements on a powder sample of nicu(pba)(d@xmath0o)@xmath1@xmath22d@xmath0o ( pba=1,3-propylenebis(oxamato ) ) which is one of the prototypical examples of an @xmath3=1/2 and 1 ferrimagnetic chain .
susceptibility(@xmath4 ) shows a monotonous increase with decreasing temperature ( t ) and reaches a maximum at about 7 k. in the plot of @xmath4@xmath5 versus @xmath6 , the experimental data exhibit a broad minimum and are fit to the @xmath4@xmath5 curve calculated for the ferrimagnetic heisenberg chain composed of @xmath3=1/2 and 1 . from this fit , we have evaluated the nearest - neighbor exchange constant @xmath7=121 k , the g - values of ni@xmath8 and cu@xmath8 , @xmath9=2.22 and @xmath10=2.09 , respectively .
applied external field dependence of @xmath4@xmath5 at low temperatures is reproduced fairly well by the calculation for the same ferrimagnetic model .
extensive studies of one - dimensional systems were prompted by haldane s theoretical work @xcite in 1983 after the initial wave of studies @xcite in the late 1960s and early 1970s .
recently , quantum spin systems with singlet ground states , namely haldane systems @xcite ( linear chain heisenberg antiferromagnets with integer spin values ) , inorganic spin - peierls systems @xcite and even - leg spin ladder systems , @xcite have been studied extensively . in particular , cuprate systems have attracted much attention because of the relation to high @xmath11 superconductors . in regard to the one - dimensional systems with magnetic ground states , an @xmath3=1/2 and 1 ferrimagnetic chain has been theoretically investigated recently , @xcite in addition to some pioneering theoretical works @xcite published in the 1980s . from the low dimensionality and small spin values in this system , we expect a kind of quantum effect .
theoretical studies show some remarkable features as follows : ( 1 ) between two low - lying , gapless and gapped excitation branches , the gapped branch lies higher than that deduced from a conventional spin wave theory . from reliable calculations , @xcite
the gap ( @xmath12/@xmath13 ) has been evaluated to be 1.767@xmath140.003 where @xmath13 is the nearest - neighbor exchange constant .
the definition of the hamiltonian will be shown later .
( 2 ) the spin correlation length between sublattice moments is extremely short .
the length is below unit cell length and can not be evaluated with numerical accuracy .
( 3 ) the full magnetization curve up to saturated magnetization is calculated and is obviously different from that for a classical ferrimagnet .
@xcite on the other hand , although some candidates for the ferrimagnetic heisenberg chain composed of spin 1/2 and 1 exist in real bimetallic substances , @xcite only preliminary magnetic measurements and comparisons with numerical calculations were made .
@xcite thus , we investigate precisely the magnetic properties of an alternating ni and cu chain compound nicu(pba)(d@xmath0o)@xmath1@xmath22d@xmath0o ( pba=1,3-propylenebis(oxamato ) ) . in these measurements , we use a deutrated sample because it is of superior quality to a hydrated one , although reason for this remain unclear and we plan to perform neutron scattering measurements on this deutrated sample .
the format used in this letter is as follows : in the next section , we discuss the synthesis and crystal structure of nicu(pba)(d@xmath0o)@xmath1@xmath22d@xmath0o .
we then report the results of magnetic measurements and of the comparison with numerical calculations for @xmath3=1/2 and 1 ferrimagnetic heisenberg chain .
finally , we show the field dependence of @xmath4 times @xmath6 and compare the experimental data with calculated ones for the same model .
powder samples of nicu(pba)(d@xmath0o)@xmath1@xmath22d@xmath0o were synthesized according to the procedure reported in ref . 13 .
na@xmath0[cu(pba)]@xmath26h@xmath0o was prepared from cuso@xmath15 , naoh and 1,3-trimethylenebis(oxamido ) @xcite which was previously synthesized from ethyl oxamate and 1,3-propanediamine .
then , the title compound was obtained by slow diffusion of aqueous solutions ( d@xmath0o @xmath1699.8% ) of na@xmath0[cu(pba)]@xmath26h@xmath0o and ni(clo@xmath15)@xmath0@xmath26h@xmath0o in a u - tube .
chemical analysis showed a slight deviation of h content from the ratio in the ideal deutrated sample , but the molecular weight of this sample was only about 1% smaller than that of nicu(pba)(d@xmath0o)@xmath1@xmath22d@xmath0o .
crystal structure of nicu(pba)(h@xmath0o)@xmath1@xmath22h@xmath0o has not been analyzed , but that of a similar compound mncu(pba)(h@xmath0o)@xmath1@xmath22h@xmath0o where mn replaces ni , has been analyzed . @xcite powder x - ray diffraction patterns of these compounds show that these belong to the same space group .
thus , nicu(pba)(h@xmath0o)@xmath1@xmath22h@xmath0o crystallizes in the orthorhombic system and belongs to the @xmath17 space group .
@xcite as shown in fig . 1 , the structure consists of ordered bimetallic chains along the @xmath18 axis with octahedral ni@xmath8 and square - pyramidal cu@xmath8 ions bridged by oxamato groups . at the apical positions of ni and cu , water molecules are bound .
magnetic measurements were carried out with a squid magnetometer ( quantum design s mpms - xl7l ) at kyokugen in osaka university .
we show in fig .
2 the dc magnetic susceptibility @xmath4(=@xmath19/@xmath20 where @xmath19 and @xmath20 represent magnetization of the sample and the external magnetic field , respectively ) of a powder sample of nicu(pba)(d@xmath0o)@xmath1@xmath22d@xmath0o .
the susceptibility of nicu(pba)(d@xmath0o)@xmath1@xmath22d@xmath0o increases monotonously with decreasing temperature until about 7 k , at which the susceptibility reaches a maximum . below 7 k
, the long - range order probably occurs due to the interchain couplings .
figure 3 shows @xmath4 times @xmath6 of nicu(pba)(d@xmath0o)@xmath1@xmath22d@xmath0o as a function of temperature .
this plot is familiar to chemists but not to physicists .
therefore , we explain this plot in some details . if a magnetic system is paramagnetic , @xmath4@xmath5 is constant over the whole temperature range . if a magnetic system has a dominant ferromagnetic ( antiferromagnetic ) interaction , @xmath4@xmath5 increases ( decreases ) when the temperature is decreased . in fig . 3
, @xmath4@xmath5 decreases when decreasing the temperature from 300 k , implying that antiferromagnetic coupling exists between the nearest neighbor spins , and reaches a rounded minimum at about 70 k. then , @xmath4@xmath5 increases and reaches a maximum at about 10 k , and hereafter , when the temperature is decreased further , it decreases rapidly .
the increase in @xmath4@xmath5 below 70 k implies that this ferrimagnetic system behaves like a ferromagnetic chain at low temperatures .
interchain ( antiferromagnetic ) couplings probably give rise to the steep decrease of @xmath4@xmath5 below 10 k. we compare the experimental data with numerical calculations ( exact diagonalization method up to five unit cells ( ten sites ) ) for the @xmath3=1/2 and 1 ferrimagnetic heisenberg chain .
the hamiltonian of this system in a magnetic field is defined by @xmath21}\\ -g_{s } \mu_{\rm b}h \sum_{i=1}^{l } { { \bf s_{i}}}-g_{s } \mu_{\rm b}h \sum_{i=1}^{l } { { \bf s_{i}}},\ ] ] where * s * and * s * are the @xmath3=1 and @xmath3=1/2 spin operators , respectively , and @xmath22 and @xmath23 the g - values of the @xmath3=1 and @xmath3=1/2 magnetic moments , respectively , and @xmath24 the bohr magneton and @xmath25 the external magnetic field . here , the periodic boundary condition is imposed , so that * s*@xmath26=*s*@xmath27 . the solid line in fig .
3 shows the result of the best fit to the experimental data between 30 k and 150 k. good agreement between experimental and calculated results is achieved between 30 k and 150 k. slight deviation at high temperatures may arise from the error enhancement of @xmath4@xmath5 at high temperatures or the omission of the single ion anisotropy term which exists in ni(@xmath3=1 ) compounds . from this fit
, we obtain the exchange constant @xmath7=121 k , @xmath22(=@xmath9)=2.22 and @xmath23(=@xmath10)=2.09 .
next , we show the magnetic field dependence of @xmath4@xmath5 in fig . 4 .
experimental data of @xmath4@xmath5 at 0.1 t ( open squares ) , 1 t ( open triangles ) and 7 t ( open circles ) are plotted in the upper panel .
experimental data at 0.1 t and 1 t have a similar tendency , but @xmath4@xmath5 at 7 t at low temperatures deviates significantly from the others .
this behavior is reproduced in the calculation shown in the lower panel . here
, the designated @xmath20/@xmath13 figures represent those of @xmath28/@xmath13 with @xmath29=2.0 and @xmath7=121 k , and magnitudes of @xmath4@xmath5 are calculated for the above hamiltonian using @xmath22=2.22 and @xmath23=2.09 .
the low temperature behavior of @xmath4@xmath5 at 7 t implies that the n@xmath30el order tends to be fixed and the ferromagnetic fluctuation of magnetic moments is suppressed by the external field
. in conclusion , magnetic properties of alternating ni and cu chain compound nicu(pba)(d@xmath0o)@xmath1@xmath22d@xmath0o were investigated by magnetic susceptibility measurements . from comparison with a numerical calculation for the ferrimagnetic heisenberg chain composed of @xmath3=1/2 and 1
, we have obtained the values of the exchange constant and the @xmath29-values of ni and cu .
field dependence of @xmath4@xmath5 at low temperatures has been reproduced by similar calculations in magnetic fields . |
photon dominated regions are predominantly molecular and atomic regions where the physical and chemical processes are dominated by uv radiation ( cf . hollenbach & tielens , 1997 , 1999 ) .
the molecular clouds in the vicinity of the newly formed stars is heated by the fuv photons in the energy range from about 6 to 13.6ev .
these clouds cool through the atomic and molecular spectral lines , such as [ cii]158@xmath0 , [ oi]63,146@xmath0 , [ ci]609,370@xmath0 and the milli - metric and sub - mm co rotational lines .
plane parallel and spherical models of pdrs have been constructed to understand the physical and chemical characteristics of these regions ( e.g. , kaufman et al .
, 1999 ; kster et al . , 1994 ; le bourlot et al . ,
1993 ; sternberg & dalgarno 1995 ; strzer et al . , 1996 ; tielens & hollenbach 1985 ) .
however there are other important factors which affect the uv absorption and scattering as well as the basic heating and cooling processes in pdrs .
several galaxies such as dwarf galaxies , irregular galaxies , the large and small magellenic clouds have low metallicity ( cf .
wilson 1995 ) .
a radial gradient in metallicity of molecular clouds is found in the milky way and several other nearby galaxies ( e.g. , arimoto et al . , 1996 ) .
these low metallicity systems have much higher [ cii]/co and [ ci]/co line ratios as compared to the galactic value ( e.g. bolatto et al .
, 2000 ; madden 2000 ) .
this suggests that the effects of metallicity should be considered while interpreting the molecular and atomic spectral line observations of these sources . since the important surface coolant of the pdrs , the [ cii]158@xmath1 m emission ,
is strongly affected by the metallicity factor , we study the effects of metallicity in pdrs .
in addition , low metallicity pdr models would also help us to understand the star forming regions in dwarf galaxies which resemble the primordial galaxies .
additionally , observations of edge - on pdr have suggested that the molecular clouds are clumpy , and the uv radiation can penetrate deep inside the clouds ( cf .
stutzki et al . ,
1988 ; boiss 1990 ) .
this suggests that a simple single component model may not be sufficient to explain the observed features .
we also include a mass spectrum of clumps to understand the cooling lines of pdrs from low metallicity galaxies .
low metallicity systems have a lower , dust to gas ratio and heavy elemental abundances as compared to the galactic ism .
this reduction in the amount of dust grains affects , the absorption of uv radiation , heating of gas by photo electric emission from dust , formation of h@xmath2 on the dust grains and the cooling of the gas through atomic and molecular lines .
in addition , the chemistry is also affected by the reduction in dust and heavy elements ( van dishoeck & black 1988 ; lequeux et al . , 1994 ) .
we use the self - consistent spherical pdr model of strzer et al .
( 1996 ) to study the metallicity effects .
we scale the dust dependent parameters and the abundance of heavy elements with the metallicity factor , z , in our pdr calculations .
the variation of the temperature at the surface of the pdr clumps with the incident uv field is plotted in figure [ pdrtemp ] , for a clump of mass , m=1m@xmath3 and density , n=10@xmath4 .
the uv field , @xmath5 , is expressed in units of mean uv field of draine ( 1978 ) .
it is seen from the figure that at high uv fields the temperature is proportional to metallicity , whereas at low uv fields there is no significant change in the surface temperature .
this correlation can be understood analytically by balancing the dominant heating and cooling processes . in the case of pdrs exposed to high uv fields grain photo - electric emission ( pe ) dominates the heating .
the photo electric heating rate given by bakes & tielens ( 1994 ) is @xmath6 where z is the metallicity factor .
the cooling is dominated by fine structure [ oi ] emission , gas - grain collisions and fine - structure [ cii ] emission .
the cooling rate can be written as @xmath7 where @xmath8 is the escape probability , @xmath9 is the transition probability and @xmath10 number of atoms at level @xmath11 and @xmath12 is the corresponding frequency of radiation .
following hollenbach & mckee ( 1979 ) , for the population of the first level of the oi atom , the cooling rate of [ oi]63@xmath1 m can be written as , @xmath13 although eqns .
[ peheat ] and [ oicool ] show a similar dependence with metallicity , the grain heating rate has an additional dependence on metallicity through the charge state of the grains .
the charge state of the grains is expressed as @xmath14 which depends on the availability of electrons for recombination .
the main source of electrons at high uv fields is the ionisation of ci . since at the surface almost all of the ci is ionised , the electron density @xmath15 . with this assumption the heating rate , @xmath16 decreases with decreasing @xmath17 whereas the cooling rate
, @xmath18 remains constant .
the equilibrium temperature obtained by solving eqns .
[ peheat ] and [ oicool ] decreases as the metallicity decreases ( cf .
figure [ equitemp ] ) as seen in our pdr calculations .
in our clumpy model , the molecular cloud is modelled as being composed of many spherical clumps of mass spectrum of the form , @xmath19 we use @xmath20 ( e.g. kramer et al . , 1998 ) .
we assume that the turbulent velocity dispersion of the cloud is larger than the intrinsic line width of each clump .
thus the clumps do not interact radiatively and the total intensity of a spectral line from the cloud can be written as , @xmath21 the beam filling factor of each clump is @xmath22 ) where @xmath23 is the solid angle of the clump of mass @xmath24 and @xmath25 is the beam solid angle .
the fraction of the clumps within the beam is given by @xmath26 . by scaling the mass with @xmath27 ( @xmath28 ) @xmath29 @xmath30 where @xmath31 is the beam filling factor , @xmath32 and @xmath33 .
although the total intensity depends on the scaling constant and the beam filling factors , the line ratio between any two spectral lines depends only on @xmath34 and @xmath35 .
the observations of [ cii ] and [ ci ] emission from low metal galaxies have been modelled by bolatto et al .
( 1999 ) , assuming that the size of the cii region scales inversely with metallicity .
it is also assumed that the size of the ci region remains between two limiting scenarios of , an inverse dependence and no variation with metallicity .
our spherical pdr model calculations based on the model of strzer et al .
( 1996 ) show that the size of the cii layer is indeed inversely proportional to the metallicity factor z. however the size of the ci region shows very weak dependence at low z and a roughly inverse dependence at high z for a typical spherical clump of density , @xmath36 .
the intensity ratio @xmath37158\mu m}$]/@xmath38 , observed in many nearby low metal galaxies , show a power law dependence with metallicity .
this dependence has been predicted using a semi - analytical clumpy model by bolatto et al .
we use our clumpy model explained in section [ clumpymodel ] to study the metallicity dependence of this line ratio .
the observed ratios of nearby galaxies can be well represented by a clumpy model of density @xmath39 , exposed to a uv field of @xmath40 as shown in figure [ clumpyplot ] .
the higher observed ratio for the 30 doradus region can be explained by a similar clumpy model , but exposed to an uv field of @xmath41 .
these results compare well with the results shown by the semi - analytical model of bolatto et al .
this trend suggests that at low metallicities cii is a tracer of molecular hydrogen rather than co. however the observed variation of [ ci]/co line ratio with metallicity is steeper than the model prediction .
this is most likely due to large [ ci ] line intensities predicted by the pdr models and requires further investigation .
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, hollenbach d. 1985 , apj , 291 , 722 van dishoeck , e. f. & black , j. h. 1988 , apj , 334 , 771 wilson c. d. 1995 , apj,448 , l97 | several galaxies , such as dwarfs and irregulars as well as outer galactic clouds have low metallicity . at low metallicities
a reduction in the amount of dust and heavy elements plays a significant role on the chemistry as well as the heating and cooling of the gas in the molecular regions , called as photon dominated regions ( pdrs ) .
we present here the effects of reduced metallicity in pdrs and study the important pdr cooling lines ( [ cii ] , [ ci ] and co ) . moreover many observational evidences suggest that molecular clouds are clumpy .
we model the molecular emission from galaxies incorporating a mass spectrum of clumps .
we also compare our results with the semi - analytical results obtained by bolatto et al .
( 1999 ) . |
in constructing a higher dimensional quantum field theory , the regularization and the continuum limit are two important keys . particularly , the problem of the limit associates with hard difficulty in the higher dimensional case .
statistical mechanics usually insists that critical behaviors of the phase transition in the theory are equivalent to those of a mean field theory , which has only a trivial fixed point .
many pioneering works on lattice gauge theories were trying to overcome the difficulty @xcite , although the continuum limit is not taken strictly . where is the continuum limit ? by using well - known 4-dimensional theories such as qed and qcd ,
is it possible to construct the continuum limit by the related critical behavior near the critical point ?
our purpose is to construct a @xmath0-dimensional pure yang - mills theory by arranging a number of 4-dimensional yang - mills theories with appropriate couplings .
a @xmath0-dimensional lattice space is decomposed into @xmath1 layers with 4 dimensions like as fig.[fig : layer ] . originally , fu and nielsen investigated the system to show dynamical dimensional reduction by using the characteristic vacuum which confines for extra dimensional directions and deconfines for 4 dimensions called a layer phase @xcite .
we do not assume that the layer phase exists , but use only the decomposition . from now , we focus on a 5-dimensional theory to study the possibility of the construction for higher dimensional field theories explicitly but an extension to general dimensions
shall be mentioned in the final section .
our starting action for @xmath2 gauge group is written as @xmath3 + \frac{\beta_{5}}{2}\sum_{p_{5}}\left[2-\tr u_{p_{5}}\right ] , \label{lact}\ ] ] where @xmath4 is proportional to a usual 4-dimensional coupling constant inside a layer and @xmath5 is a coupling constant between neighboring layers .
@xmath6 implies a plaquette inside a layer and @xmath7 does a plaquette between neighboring layers .
to display the phase diagram explicitly , two order parameters are introduced ; ( 1 ) creutz ratio ( @xmath8 ) for 4-dimensional wilson loops and ( 2 ) 5-dimensional polyakov loop ( @xmath9 ) .
since a theory with @xmath10 is equivalent to a pure 4-dimensional yang - mills theory which is one phase , we expect a phase structure such as fig.[fig : phase1 ] .
we have calculated numerically these parameters for the system and obtained the numerical phase diagram in fig.[fig : phase2 ] . in the figure , along line i we can see both 4-dimensional and 5-dimensional deconfinement transitions in the large @xmath5 . by wider calculations along lines
ii , we recognize the phase transition is of 1st order near @xmath11 .
the results of lines iii and iv suggest that both 4-dimensional and 5-dimensional deconfinement transitions in the small @xmath5 are of 2nd or weakly 1st order , because we can not see any hysteresis loop in the coupling region .
this critical point in the @xmath12 is noted as @xmath13 .
our surprising remark is that the diagram fig.[fig : phase2 ] is not quantitatively changed for various @xmath14 and its stable property may help us to take the limit @xmath15 .
our simple consideration finds our way to existence of a 4-dimensional continuum system with finite inter - layer coupling(@xmath5 ) . near the critical point ( @xmath16 ) , the inverse ( @xmath17 ) of correlation length for polyakov loop
is written as @xmath18 where the value of @xmath19 is approximately 0.6 and @xmath20 implies the critical exponent which is 0.5 in a mean field theory .
we call the picture that many gauge fields on different layers interacts each other with finite coupling as multi - layer world .
it is noted that link variables with 5-th direction behave as bi - fundamental fields with a finite coupling .
can we construct a 5-dimensional space not an internal space ?
a straightforward way uses excitation masses @xmath21 between layers corresponding to kaluza - klein(k - k ) modes , @xmath22 \right .
\nonumber \\ & + & \langle l_5\rangle^2 \left.\left [ \begin{array}{ccccc } 1 & & & & -1\\ & 0 & & 0 & \\ & & \ddots & & \\ & 0 & & 0 & \\
-1 & & & & 1 \\
\end{array } \right ] \right\ } .
\label{kkmass}\end{aligned}\ ] ] for a lower excited mode with label @xmath23 , the simple formula of the mass is obtained as @xmath24 in order to remain finite masses of k - k modes , we must keep @xmath25 finite for large @xmath26 and @xmath27 with small @xmath28 from eq.([kkm ] ) . to go through a 5-space from our 4-dimensional multi - layer world
, we need to balance inter - layer dynamics and inside - layer one , @xmath29 from eq.([defcalr ] ) , three parameters @xmath30 tunings are necessary ( see fig.[fig : env ] ) . comparing @xmath31 with eq.([kkm ] )
, a lattice spacing @xmath32 along 5th dimension can be defined by @xmath33 .
we summarize main three steps ; @xmath34 to find a second order phase transition in the meaning of 4-dimensional statistical mechanics .
@xmath35 to take a 4-dimensional continuum limit ( multi - layer world ) .
@xmath36 to compare a 4-dimensional scale with an extra dimensional scale , i.e. confinement and kaluza - klein modes . for detailed analysis ,
see ref .
@xcite .
finally , the following problems remain ; @xmath37 further detailed study for large @xmath26 and @xmath27 .
@xmath38 to estimate contribution of bi - fundamental field for @xmath8 .
@xmath39 to recover the rotational symmetry relating to anisotropy between 4-scale and 5-scale .
@xmath40 6 or higher - dimensional
extension of these decomposition is straightforward but their phase diagram analysis is not so easy . 9 m. creutz , phys .
43 ( 1979 ) 533 .
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y. k. fu and h.b .
nielsen , nucl .
b236 ( 1984 ) 167 . m. murata and h. so , hep - lat/0306003 . | a higher dimensional lattice space can be decomposed into a number of four - dimensional lattices called as layers . the higher dimensional gauge theory on the lattice
can be interpreted as four - dimensional gauge theories on the multi - layer with interactions between neighboring layers .
we propose the new possibility to realize the continuum limit of a five - dimensional theory based on the property of the phase diagram . |
in [ k ] khovanov introduced a homology theory for links in @xmath0 that was a categorification of the jones polynomial . in [ aps ] asaeda , przytycki and sikora extended this theory to links embedded in i - bundles .
their homology theory incorporated some of the topology of the i - bundle into their invariant .
turner and turaev showed in [ t ] that the homology from [ aps ] could be recreated using embedded surfaces as elements of the chain groups instead of decorated diagrams . in this paper
we accomplish that in the case of i - bundles over orientable surfaces .
section 2 contains definitions and explains the skein relations on surfaces that are used .
section 3 defines the grading on the chain groups and which surfaces generate the chain groups .
the boundary operator is defined in section 4 and it is also shown that it is well - defined with respect to the relations . in section 5
it is proved that the boundary operator squared is equal to zero , and thus the boundary operator together with the chain groups form a chain complex . finally , in section 6 it is shown that the homology produced from the chain complex coincides with the homology from [ aps ]
let @xmath1 be a surface properly embedded in a 3-manifold @xmath2 . a boundary circle of @xmath1 is said to be * inessential *
if it bounds a disk in @xmath2 , otherwise it is said to be * essential*. if @xmath1 is an oriented surface and @xmath3 is an oriented boundary component of @xmath1 then the orientation of @xmath1 is * compatible * with the orientation of @xmath3 if the boundary orientation of @xmath3 from @xmath1 agrees with the orientation of @xmath3 .
two oriented boundary curves of an orientable connected surface are * compatible * if both curves are compatible with the same orientation on the surface .
if @xmath1 is a connected unoriented orientable surface and @xmath3 is an oriented boundary component of @xmath1 then there is exactly one orientation for the other boundary curves to be oriented compatibly with @xmath3 . [ cols="^,^,^,^,^ " , ] note that under @xmath4 the associated state is nt affected , thus for example if t @xmath5 tt by changing a smoothing before applying @xmath4 , then after applying @xmath4 the boundary circles behave the same way , and an inessential boundary circle turns into two inessential boundary circles by placing a bridge .
the following 21 items show what @xmath6 is in each of the cases when the boundary circles are affected as in the previous table . 1
. note @xmath7 = . after a bridge
is placed there are two trivial boundary curves in the top .
this has euler characteristic equal to 0 , and thus it is a compressible annulus .
compress the annulus to get two disks , each with a dot .
2 . @xmath7 = .
when a bridge is placed there are two non - trivial boundary components in the top .
this is an incompressible annulus with a dot , so it is trivial in the quotient .
@xmath8 = . after a bridge
is placed there are two trivial boundary curves in the top .
this is a compressible annulus .
compress the annulus to get disk with dot , disk + disk , disk with dot .
4 . @xmath8 = . after a bridge
is placed there are two non - trivial boundary curves in the top .
this is an incompressible annulus , so have unoriented annulus = average of oriented annuli .
@xmath9 = . after a bridge
is placed there is a non - trivial boundary curve in the top and a trivial boundary curve in the top .
compress the neck that is near the trivial boundary curve to get an annulus , oriented same way as the original annulus and a disk with a dot .
@xmath9 = . after a bridge
is placed there are two non - trivial boundary curves on the top .
one can only compress and separate boundary curves if we have at least 4 non - trivial and we only have three , so we have a surface that is trivial in the quotient by lemma [ pairofpants ] 7
. refer to 5 8 .
refer to 6 9 .
@xmath10 = . after a bridge
is placed there is one trivial boundary component .
now we have two dots on the same component , so it is trivial in the quotient . 10 .
@xmath11 = . after a bridge
is placed there is one trivial component .
these two disks combined to make a disk with a dot . 11 .
@xmath12 = . after a bridge
is placed there is one trivial boundary component .
this leaves us with a disk .
@xmath13 = . placing
a bridge would result in a trivial boundary component in the top .
thus the original boundary components must have been parallel .
therefore the bridge falls into the category of ( eo ) since they are oriented the same way .
thus the result is trivial in the quotient
@xmath13 = . placing a bridge results in one non - trivial boundary curve on the top .
thus we have an incompressible pair of pants which is trivial in the quotient .
@xmath14 = . after placing a bridge
there is one trivial boundary component .
thus the original non - trivial curves were homotopic .
compress upon the disk that is present near the trivial curve on top .
this results in a disk on top with a dot and an annulus on the bottom + disk on top with an annulus with a dot on the bottom which is equivalent to just having a disk with a dot in the quotient .
@xmath14 = . after a bridge
is placed there is one non - trivial boundary component . as in 13
, we have an incompressible pair of pants which is trivial in the quotient .
refer to 12 17 .
refer to 13 18 .
@xmath15 = . after a bridge
is placed there is one non - trivial boundary curve on the top .
note bridging to a disk does nt change the annulus , except it adds a dot , which makes the foam trivial in the quotient .
refer to 18 20 .
@xmath16 = . after a bridge
is placed there is a non - trivial boundary component on top . absorbing a disk does nt change annulus , so we get the same annulus with the same orientation .
refer to 20 by examining the list and the table , we can see that @xmath17 in each case .
thus note @xmath18 thus @xmath4 is a chain map , as desired . given a link diagram @xmath19 , @xmath20 @xmath21 by @xmath22 .
let @xmath23 , @xmath24 , @xmath25 be the indices coming from the [ aps ] theory and let @xmath26 be an enhanced kauffman state from [ aps ] . then note clearly @xmath27 since the smoothings stay the same under @xmath4 .
we also have , @xmath24 ( @xmath28 ) @xmath29 + 2 ( # positive trivial circles @xmath30 # negative trivial circles ) @xmath31 similarly , @xmath24 ( @xmath30 ) @xmath32 + 2 ( # positive trivial circles @xmath30 # negative trivial circles ) @xmath33 also ,
@xmath24 ( @xmath280 ) @xmath34 + 2 ( # positive trivial circles @xmath30 # negative trivial circles ) @xmath35an annulus with bottom boundary curve oriented in the positive direction ) finally , @xmath24 ( @xmath300 ) @xmath36 + 2 ( # positive trivial circles @xmath30 # negative trivial circles ) @xmath37an annulus with bottom boundary curve oriented in the negative direction ) then note all non - trivial circles that are present in a smoothing of the diagram appear in the bottom of the foam , so the @xmath25-grading is also preserved under @xmath4
. the proof will proceed by induction on the number of crossings in the diagram .
assume @xmath19 has zero crossings . therefore the boundary maps are all the zero map .
thus the chain groups are also the homology groups .
note @xmath4 is an isomorphism on the chain groups since it takes generators to generators , so it is also an isomorphism on homology in this case .
let @xmath19 be a diagram in @xmath38 , with @xmath39 crossings and inductively assume @xmath22 is an isomorphism for all diagrams with less than @xmath39 crossings .
note we have a relation between the short exact sequences coming from the two theories .
the diagram commutes since @xmath40 and @xmath41 are defined identically and the same is true for @xmath42 and @xmath43 .
0 & & |c_i+1,j+1,s(d _ ) & ^| & |c_i , j , s(d_p ) & ^| & |c_i-1,j-1,s(d_0 ) & & 0 + & & _ & & _ & & _ & & + 0 & & c_i+1,j+1,s(d _ ) & ^ & c_i , j , s(d_p ) & ^ & c_i-1,j-1,s(d_0 ) & & 0 + this induces the long exact sequence : & |h_i+1,j-1,s(d_0 ) & ^ & |h_i+1,j+1,s(d _ ) & ^| _ * & |h_i , j , s(d_p ) & ^| _ * & |h_i-1,j-1,s(d_0 ) & ^ & |h_i-1,j+1,s(d _ ) & & + & & & _ _ * & & _ _ * & & _ _ * & & _ _ * + & h_i+1,j-1,s(d_0 ) & ^ & h_i+1,j+1,s(d _ ) & ^ _ * & h_i , j , s(d_p ) & ^ _ * & h_i-1,j-1,s(d_0 ) & ^ & h_i-1,j+1,s(d _ ) & &
+ all @xmath22 , except the middle one , are isomorphisms by the inductive assumption .
also , the diagram commutes since @xmath4 is a chain map .
note by the five lemma the middle @xmath4 is an isomorphism .
thus by induction given a link diagram @xmath19 , @xmath20 @xmath21 by @xmath22 .
since asaeda , przytycki and sikora proved invariance for the @xmath44 homology and @xmath45 by the previous theorem we obtain , @xmath46 is an invariant under reidemeister moves 2 and 3 and a reidemeister 1 move shifts the indices in a predictable way .
* asaeda , marta m. , jozef h. przytycki and adam s. sikora , * categorification of the kauffman bracket skein module of i - bundles over surfaces * , algebr .
4 ( 2004 ) 1177 - 1210 * bar - natan , dror , * khovanov s homology for tangles and cobordisms * , geom .
9(2005 ) 1443 - 1499 *
bar - natan , dror , * on khovanov s categorification of the jones polynomial * , algebr .
. 2 ( 2002 ) 337 - 370 mr1917056 * khovanov , m. , * a categorification of the jones polynomial * , duke math . j. 101 ( 2000 ) 359 - 426 mr1740682 * manturov , v. , * additional gradings in khovanov homology * , arxiv:0710.3741v1 [ math.gt ] * turaev , vladimir and paul turner , * unoriented topological quantum field theory and link homology * , arxiv : math/0506229v2 * viro , oleg , * remarks on definition of khovanov homology * , arxiv : math/0202199v1 [ math.gt ] | this paper introduces a homology theory for links in i - bundles over an orientable surface .
the theory is unique in that the elements of the chain groups are surfaces instead of diagrams .
it is then shown this theory yields the same results as the homology theory constructed by asaeda , przytycki and sikora . |
the experiments were performed in ultrahigh vacuum ( base pressure : @xmath42 mbar ) with a radio - frequency low - temperature scanning tunneling microscope @xcite operated at 5k .
it utilizes a sharp tungsten tip ( electrochemically etched and thermally deoxidized above 1070k ) as , both , imaging probe as well as movable ground - electrode against the flat sample .
the latter is a ag(111 ) single - crystal prepared by repeated cycles of ar@xmath43 ion sputtering ( 600ev ) and thermal annealing at 720k .
the ag(111 ) sample is biased from independent rf- and dc - voltage sources for applying ac and dc @xmath0-fields of @xmath44@xmath45v / m at its surface .
the rf - circuit and electronics are described elsewhere @xcite .
after cooling the sample to 5k , the stm chamber was flooded for 1 min with ar gas at a pressure of @xmath46 mbar yielding an ar coverage of @xmath47monolayers on ag(111 ) .
after preparation by this procedure the ag(111 ) surface is covered by 2d - islands of ar with compact shapes and typical sizes ranging from 30 to 100 nm .
we kindly acknowledge financial support of the project i958 by the austrian science fund ( fwf ) .
s.m . , r.k . and g.s .
designed the experiments .
, s.t . , and s.w .- b . conducted the experiments .
g.s . , s.t .
, s.w .- b . and s.m .
analyzed the data .
, r.k . and g.s .
wrote the manuscript .
s.m . and r.k . planned and supervised the project .
all authors discussed the manuscript .
the authors declare no competing financial interests .
correspondence and requests for materials should be addressed to stefan.muellegger@jku.at .
27ifxundefined [ 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 * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , ) @noop * * , ( ) link:\doibase 10.1002/9780470034590.emrstm0584 [ _ _ ] ( , ) @noop * * , ( ) @noop * * , ( ) @noop _ _ , edited by and , , vol .
( , ) pp . @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ , ed .
( , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) | we present here a powerful method providing simultaneous atomic spatial and nanosecond temporal resolution for investigating dynamics and structure on the atomic scale , in general .
we reveal the dynamic reorganization of surface ( ad)atoms induced by radio frequency alternating charging and decharging of a metal .
our method utilizes taylor - made nano - fabricated two - dimensional islands of physisorbed argon atoms , acting as motion sensors , probed by a radio frequency low - temperature scanning tunneling microscope .
charge oscillations forced by voltages and currents alternating at radio frequency ( rf ) play a dominant role in practically all electronic devices used in our daily life .
such charge oscillations are known to excite plasmons in the quasi - two - dimensional skin layer at the surface of metals @xcite .
in addition , due to electrostriction @xcite , a periodically modulated electric ( @xmath0 ) field gives rise to forced small - amplitude mechanical oscillations of the atomic lattice , @xmath0-fields of @xmath1v / m typically generating strain of 0.1 % @xcite .
such mechanical oscillations are utilized as surface acoustic waves in piezoelectric biosensors @xcite and frequency filters for telecommunication @xcite ; they enable the tuning of electronic and magnetic properties of materials @xcite as well as the operation of artificial muscles for robots and artificial limbs @xcite ; ultrasonic irradiation facilitates to control spin
lattice relaxation times and peak widths in nuclear magnetic resonance @xcite . for measuring strain ,
macroscopic techniques such as x - ray diffraction @xcite or cantilever beam techniques @xcite are well established .
detection of picometer - scale mechanical amplitudes of the surface atomic lattice has been achieved up to 1ghz by an rf - modified scanning tunneling microscope @xcite .
near - field microwave microscopy has demonstrated the electrodynamical response of the material on length scales far shorter than the free - space wavelength of the microwave @xcite .
however , a direct real - space detection and imaging of the surface atomic lattice and its dynamics has remained elusive , to date , due to the lack of simultaneous spatial and temporal resolution of the detection method applied .
we have developed a method that circumvents these difficulties .
here we investigate with nanometer spatial- and nanosecond temporal resolution the impact of radio - frequency alternating electric charging and de - charging of a metal on its atomic surface structure .
in particular , we reveal the dynamic reorganization of surface ( ad)atoms with a time constant of 147 ns induced by applying a 2-ns - periodic rf - voltage ( 530 mhz ) to the metal surface . as origin , we identify the charge - density oscillations in the metal surface skin layer . for detecting such dynamic processes
we utilize nanometer - sized motion sensors ( fig .
[ fig : sensor ] ) consisting of nano - structured two - dimensional ( 2d ) islands of physisorbed noble - gas atoms @xcite .
our motion sensors are shown herein to structurally transform on oscillating substrates like an rf - biased metal surface .
their structural transformations are shown to be powerful analytical probes for characterizing the underlying physical processes of the excitation at the nano - scale .
( a ) nanometer - sized motion sensor : nano - fabricated 2d - island of ar on ag(111 ) imaged by stm at 5k ( @xmath2nm@xmath3 , @xmath4-scale : 200pm , @xmath5v , 70pa ) ; arrows mark lateral channels fabricated by cutting - out from the island by dc - stm manipulation .
( b ) same 2d - island as in ( a ) before the nano - fabrication step ; left inset : magnified view ( @xmath6nm@xmath3 ) revealing atomic resolution on the ar 2d - island as well as single ar vacancies ( labeled 1 ) ; right inset : atomic - resolution image of the ag(111 ) substrate ( @xmath7nm@xmath3 , 1.2 na , @xmath8mv).,width=321 ] figure [ fig : sensor]a shows exemplarily a typical motion sensor imaged by stm at 5k .
it is based on a 2d - island of ar on ag(111 ) that exhibits a strongly non - equilibrium shape obtained after cutting lateral channels ( marked by arrows ) out of the 2d - island .
this is achieved by means of dc - stm manipulation @xcite for the controlled removal of ar atoms at the channels ( see supplementary figures1 ) .
we demonstrate below that these channels are suitable probes for detecting atomic - scale motional dynamics of surface ( ad)atoms . for comparison , fig .
[ fig : sensor]b displays the 2d - island of ( a ) before the nano - structuring step , i.e. exhibiting its natural compact equilibrium shape .
the left inset of fig.[fig : sensor]b displays a magnified view revealing the regular hexagonal ar atomic lattice of the 2d - islands with ar - ar distance of 0.39 nm , in agreement with previous studies @xcite . a single ar vacancy is labeled 1 . for comparison ,
the atomically resolved ag(111 ) lattice ( @xmath9 nm ) is shown in the right inset of fig .
[ fig : sensor]b . similar to a monolayer of ar on ag(111 ) @xcite , the equilibrium - shape islands are stable for > 12h during continuous imaging by dc - stm at sample bias voltage of @xmath5 to @xmath10v and tunneling current of 50200 pa .
more importantly here , also the motion sensors , i.e. the non - equilibrium 2d - islands ( fig .
[ fig : sensor]a ) , are found to be longterm stable against dc - imaging by stm at 5k ( see supplementary figures2 ) .
response of motion sensor to cw rf - excitation .
( a ) sensor island before rf - excitation imaged by stm ( @xmath2 nm@xmath3 , @xmath5v , 70pa ) .
( b)(f ) same sensor island as in ( a ) after successive cw rf - excitation ( 530mhz , @xmath11db ) for accumulating on - time , @xmath12 ( see labels ) .
( g , h ) dependence of the magnitude of sensor response on @xmath12 ; line : numerical fit , function given in red .
( h ) dependence on rf power @xmath13 ; line : numerical linear fit.,width=317 ] for investigating the impact of rf alternating electric charging and de - charging of the metal on its atomic surface structure , we have connected the ag(111 ) sample to the output of an rf generator in parallel to the dc sample - voltage source .
we set a fixed generator frequency to avoid possible effects of frequency - dependent damping of the rf circuitry and to guarantee a constant rf voltage amplitude at the sample surface for all experiments presented herein .
the frequency was @xmath14mhz , which means that the microwave in the ag sample kept at 5k is confined to a surface skin layer @xcite of thickness @xmath15 nm ( resistivity of ag at 5k is @xmath16 @xmath17 m @xcite and permeability @xmath18vs / am ) . in a first step , we have applied a continuous - wave ( cw ) rf - voltage to the sample for varying time spans @xmath12 . before and after each excitation , the motion sensor was imaged by dc - stm .
intriguingly , our motion sensors respond to the cw excitation with characteristic structural changes as evidenced in fig.[fig : response ] .
excitation for @xmath19min causes a gradual closing of the channels .
this is clearly seen by comparing the images of the motion sensors before ( a ) and after ( b ) the rf excitation .
moreover , the large island merges with the isolated small one on the right .
notice the defects labeled a and b being unaffected by the rf excitation .
the total size of the sensor ( area ) has remained approximately constant , indicating that no significant amount of ar atoms is added or subtracted during @xmath12 .
obviously , channel closing proceeds via directed diffusion ( displacement ) of ar atoms across the ag surface .
the perimeter - to - area ratio of the sensor island decreases monotonically , evidencing the non - random nature of the underlying process .
we emphasize that the sensors do not respond to dc tunneling . repeating the 1min cw excitation leads to a further closing of the channels ( fig .
[ fig : response]c ) , which seems to come to rest after a third excitation ( fig .
[ fig : response]d ) .
complete closure of the channels , however , is achieved after applying an additional 5-min ( fig .
[ fig : response]e ) and 10-min ( fig . [ fig : response]f ) cw excitation .
finally , the motion sensor adopts a compact equilibrium - like shape similar to original ar 2d - islands . by determining the total number of ar atoms ( area ) displaced during the on - time of the rf - voltage
, we have quantified the size of the sensor response .
starting from zero , with increasing @xmath12 the response increases and finally saturates at very long times ( fig .
[ fig : response]g ) . within the experimental range of our method
, we have found a nearly linear dependence of the sensor response on the microwave power ( fig .
[ fig : response]h ) .
it exhibits a low - power threshold of @xmath203dbm generator output power , corresponding to an rf - voltage amplitude at the sample of only a few mv zero - to - peak , considering the damping of our rf circuitry @xcite . in the experiments described so far ,
the stm tip was positioned over the sensor 2d - islands in tunnel contact ( typically @xmath5v , 100pa ) during the rf - excitation . to minimize stm - tip effects
, we carefully checked the tip state during the experiments and have repeated them with different tips ( tip formings ) .
it has turned out , however , that the position of the stm tip during rf excitation is irrelevant for the sensor response .
figure[fig : tip]a shows sensor islands with eight fabricated channels in both horizontal and vertical direction , marked by arrows .
after rf - excitation for @xmath21min all of them have responded ( fig.[fig : tip]b ) , although during the excitation the stm tip was placed over the pristine substrate several tens of nanometers away from the sensors ( tip position marked by cross ) .
apparently , the sensor response is based on a `` non - local '' mechanism , i.e. independent of the close - up range of the tip apex , and isotropic in the surface plane ( see fig.[fig : tip]b ) .
we have confirmed the sensor response up to a surface area of @xmath22nm@xmath3 by manual piezo control ( limited by the scan range of our lt - stm instrument ) .
sensor response is independent of stm tip position .
( a , b ) stm images of six sensor islands , marked by arrows , recorded before ( a ) and after ( b ) cw rf - excitation ( 530mhz , @xmath23min ) ; cross marks tunnel position of stm tip during excitation ( + 0.4v ) ; dashed lines mark radial distance from tip position .
( c - f ) difference images of sensor response obtained by subtracting stm images before and after rf - excitation ( @xmath23min ) at different rf - power levels and tunnel conditions ( see labels ) ; red ( blue ) color marks positive ( negative ) response , i.e. accumulation ( removal ) of ar atoms ; unchanged sensor area is plotted in yellow.,width=317 ] even more intriguing are experiments performed at non - tunneling conditions , where the stm tip was perpendicularly retracted by @xmath24200 nm away from the ag(111 ) surface for suppressing electron tunneling ; the local dc @xmath0-field between the stm tip apex and the sample is decreased by a factor of @xmath24200 . for better clarity the respective results , figs.[fig : tip]c f ,
are displayed as difference images , where red ( blue ) color marks sensor area where ar atoms have been accumulated ( removed ) by the rf - excitation .
the sensor response at non - tunneling conditions , fig.[fig : tip]c , is clearly revealed by the channel closing after @xmath23min ( rf power level was 4db above @xmath25 ) .
it is almost indistinguishable from the response at tunneling conditions ( fig.[fig : tip]d ) .
this finding clearly evidences that sensor response is independent of tunneling electrons as well as the magnitude of the dc @xmath0-field between tip and sample .
notice that edge diffusion is observed at the outermost ar atomic row of the sensor 2d - islands in all our experiments independent of rf excitation ( for details see supplementary information ) . repeating the experiments at a decreased power level of 3db below @xmath25 results in zero response ( no channel closing ) at both non - tunneling ( fig.[fig : tip]e ) and tunneling conditions ( fig.[fig : tip]f ) .
the respective decrease of power corresponds to a 50%-decrease of the rf @xmath0-field amplitude .
this result is indeed surprising : there is no response in ( f ) , although it has at least 100 times larger @xmath0-field compared to ( c ) .
obviously , sensor response depends on the rf - power level , but is uncorrelated with the strength of the rf @xmath0-field between stm tip and junction .
our findings therefore contradict a simple @xmath0-field effect .
notice that our argumentation holds independent of the precise value of the rf - voltage amplitude at the tunneling junction , which is not precisely known .
the results of the non - tunneling experiments clearly rule out ( local ) joule heating at the tunnel junction , which relies on the flow of electric current @xcite , as origin of the sensor response .
heating of the sample by microwave radiation is ruled out , because the sensor response happens in the ( reactive ) near - field of the sample surface , where emission of radiant energy is known to be negligible .
heating of the sample by absorption of electromagnetic energy in the cabling and sample crystal is ruled out because we observe negligible warming of the sample ( e.g. , only @xmath26k upon 5min of cw rf - excitation ) .
the presented experiments suggest that 2d - islands of ar on ag(111 ) act as motion sensors responsive to the dynamic reorganization of surface ( ad)atoms . the reorganization is caused by additional elementary diffusion processes of the sensor s atoms induced by the rf alternating electric charging and de - charging of the metal skin layer upon applying an rf - voltage to the sample .
the additional diffusion is absent at dc - voltage conditions at 5k , where only edge - diffusion is observed ( i.e. diffusion along the same atomic row starting off from a kink site ) .
edge diffusion alone can not explain the response , because channel filling requires ar atoms to move out of an edge row to form a new edge , one row in front of the old one .
we estimate the respective energy barrier of this additional diffusion step to be on the order of @xmath27mev , based on our observation that 60min of heating the sensors to 10k ( @xmath28mev ) causes a similarly large sensor response as 5min of 530 mhz rf - excitation with power of 4db above @xmath25 at 5k ( @xmath29mev ) .
still , fundamental questions remain : what causes the additional diffusion process ?
what is the role of the ag substrate atoms ? in the following we discuss the underlying physical mechanism .
sensor response to pulsed rf - excitation .
( a ) schematics illustrating the periodicity @xmath30 of periodic rf pulses .
( b e ) difference images of sensor response to rf excitation ( 530mhz , @xmath31db ) at non - tunneling conditions ; red ( blue ) color marks accumulation ( removal ) of ar atoms ; unchanged sensor area is plotted in yellow ; stm tip is positioned 500 nm away from the island center ; ( b ) after cw - excitation for 1min ; ( c)-(e ) after excitation by pulse train of @xmath32 periodic 50 ns - pulses ( equivalent of 1 min total rf on - time ) with different periodicity of 200 , 600 , and 1000 ns .
( f ) dependence of the magnitude of response on @xmath30 ; red line : numerical fit @xmath33.,width=317 ] to gain further insight , we have investigated the sensor response to pulsed rf - excitation .
we have obtained practically the same results for pulsed experiments at tunneling and non - tunneling conditions ; for brevity we show and discuss herein only the results at non - tunneling conditions .
we have applied pulse trains consisting of periodic 50ns - pulses of frequency 530mhz with different values of pulse period varying between @xmath34 and 1000ns ( fig.[fig : puls]a ) .
each pulse train contained the same total number of @xmath32 pulses , equivalent of a total on - time of 1min of the rf - excitation . compared to cw - excitation for 1min ( fig.[fig : puls]b ) , pulsed excitation yields a significantly smaller response ( fig.[fig : puls]c ) ; increasing the pulse period decreases the response further ( figs.[fig : puls]d , e ) .
this finding indicates that the dynamic processes underlying the sensor response exhibit a time constant @xmath35 with a value similar to the period of pulses applied . a quantitative evaluation of the @xmath30-dependence of the response is displayed in fig.[fig : puls]f .
numerical fitting an exponential function yields a value of @xmath36ns for the time constant of the sensor response .
this value is about six orders of magnitude larger than typical decay times of excited ( surface ) plasmons @xcite , surface - state electrons @xcite , and ( surface ) phonons @xcite on ag(111 ) .
it seems more likely that @xmath35 belongs to the ( collective ) mechanical excitations of ar atoms involved in the restructuring of the sensors .
similar decay times were observed for the collective vibrations of ensembles of small physisorbed molecules on au(111 ) @xcite .
our experimental results obviously rule out several processes as origin of the sensor response : tunnel current , the electric field between stm tip and sample , local joule heating , radiative heating and heating by absorption of electromagnetic energy in the cabling and/or sample crystal .
these findings indicate that sensor response is caused by processes in the ag sample induced by the charging and decharging at radio frequency . in other words ,
we identify the periodic - in - time deviation from charge neutrality in the skin layer of the sample as origin of the observed sensor response .
since the dynamics of electrons and atomic lattice are known to be closely related to each other on time scales of @xmath37ns , studied herein , a manifold of different effects is expected to occur simultaneously , contributing to sensor response : ( i ) electrostriction mechanically strains the surface atomic lattice @xcite affecting the bonding geometry ; reversible mechanical strain as large as 0.15% has been observed in charged nanoporous pt samples @xcite , corresponding to mechanical stress of about 1gpa .
hence , the application of an rf voltage is expected to enforce mechanical vibrations of the surface atomic lattice at the same frequency ( here 530mhz ) with amplitude @xmath38 , contributing to the measured dc tunnel current via the well - known relation @xmath39 .
notice , that the rf - induced change of sample surface height @xmath38 may be misinterpreted as `` rf - induced contribution to dc bias voltage '' , `` apparent shift of work function '' , or `` apparent smearing of @xmath40 '' ( at const-@xmath41 and const-@xmath4 conditions ) .
( ii ) excitation of acoustic surface plasmons ( asps ) , known to exist on ag(111 ) @xcite , is expected to influence adsorbate dynamics ( here : sensor response ) , since the decay of asps can generate ( surface ) phonons @xcite .
our sensors promise to facilitate future experimental studies on the rf - excitation of asps .
( iii ) electric polarization of the surface atoms induces repulsive electrostatic forces between neighboring atoms or even affects the van der waals - london dispersion forces of ar - ar as well as ar - ag atoms @xcite , facilitating enhanced ar diffusion . in conclusion
, we have demonstrated that nano - fabricated monolayer 2d - islands of ar on ag(111 ) at 5k act as experimental probes for detecting surface ( ad)atom mechanical motion and dynamic processes .
to showcase the strength of our method , we reveal a dynamic reorganization of surface ( ad)atoms at the ar / ag(111 ) interface , caused by radio - frequency charge - density oscillations and related electric fields induced by an external rf voltage .
our experimental method is expected to enable quantitative characterization of atomic structural dynamics with unprecedented detail , in general .
this is relevant , in particular , for the study of weakly bound systems with nanometer spatial- and nanosecond temporal resolution , including monolayer solids , surfaces , ( hetero)interfaces and 2d nanostructures . |
the spatial power spectrum of the hi 21 cm intensity in the small magellanic cloud was obtained by stanimirovic ( 1999 ) .
interestingly , it is a power law over scales as large as that of the smc itself .
similar power laws have been observed by crovisier & dickey ( 1983 ) and by green ( 1993 ) in the galaxy .
the outstanding feature in the case of the smc is the large scale of the observed correlations .
the power laws signal underlying long range correlations in what looks like a field of random fluctuations of the intensity . for an optically thin medium along the line of sight , the 21 cm intensity is proportional to the column density
. therefore , the fluctuations in 21 cm intensity , represent fluctuations in density .
a natural interpretation of the observed power spectra is that the underlying correlations in density fluctuations are due to a turbulence in which velocity fluctuations , that are coupled to density fluctuations , give rise to the observed power laws .
the turbulence interpretation was suggested by goldman ( 2000 ) and stanimirovic & lazarian ( 2001 ) .
goldman ( 2000 ) suggested that this large scale turbulence was generated by instabilities in the bulk flows that resulted from the tidal interaction during the last close passage of the large magellanic cloud ( lmc ) @xmath0 ago ( gardiner & noguchi 1996 ) .
however , since the observations catch a snapshot of the intensity field and since the turbulence timescales are very long ( @xmath1gyr ) one can not rule out the possibility of a _
static _ correlated density field that reflects initial conditions . in the present paper
we propose a test to decide between these two alternatives .
staveley - smith ( 1997 ) observed 501 hi super shells in the smc .
the proposed test relies on the fact that the timescale and age of the turbulence ( if indeed there ) are typically @xmath2 orders of magnitude larger than the lifetimes of the super shells .
therefore , they have formed in the turbulent gas and their observed radial velocities should reflect the turbulent velocity field in the gas in which they where formed .
we wish to look at them as markers registering the ambient gas velocity .
if the radial velocity field exhibits spatial correlations consistent with the those of the turbulence , assumed as responsible for the 21 cm intensity spectra , it will strengthen the case for dynamical turbulence as the source of the hi intensity power spectrum .
we use the data of the 501 super shells reported in table 1 of staveley - smith ( 1997 ) . for each super shell
, the residual radial velocity was found by subtracting from the observed velocity the large scale best fit , up to a shear .
@xmath3 with @xmath4 where @xmath5 the coordinates of each shell @xmath6 are in units of pc and were obtained from the angular coordinates by adapting a distance of 60 kpc to the smc .
the velocities are in units of km / s .
the subtracted large scale velocity field is composed of a mean velocity and a shear .
the magnitude of the shear is consistent with values obtained by gardiner & noguchi ( 1996 ) .
we have computed the second order structure function and the autocorrelation for the residual velocity field along lines parallel to the coordinate axes .
interpolation was used to fit the discrete data along the lines to a continuous function .
the different lines yielded similar results . for simplicity
, homogeneous and isotropic velocity field is assumed . in this case , the structure function and the autocorrelation depend only on the distance between the two points , @xmath7 .
the structure function is @xmath8 similarly , the autocorrelation function is @xmath9 the angular brackets denote ensemble averaging .
assuming ergodicity , in addition to homogeneity and isotropy , ensemble averaging equals space averaging .
as stated above , we use averages over lines so that @xmath10 where @xmath11 is the length of the line . similarly , @xmath12 the results of a typical computation are presented in figures 1 - 2 .
figure 1 shows the structure function @xmath13 . for very small values of @xmath14 @xmath15 , for larger values of @xmath14
it varies as @xmath16 and then it saturates .
the index @xmath17 characterizes the inertial range of the turbulent velocity spectral function : @xmath18 . in kolmogorov turbulence characterizing incompressible fluid @xmath19 . in the case of turbulence in compressible medium @xmath20
this was also the value deduced by goldman ( 2000 ) on the basis of the 21 cm intensity power spectrum .
these two power lows are presented in figure 1 .
the precision of the data is not enough to decide between them , even though the @xmath20 line seems to follow better the slope of the computed structure function .
as function of scale in pc
. the thin lines have slopes @xmath21 and @xmath22 . the upper line has a slope of @xmath21 . ]
the autocorrelation function is shown in figure 2 .
it behaves as an autocorrelation function of a turbulent velocity rather than uncorrelated velocity fluctuations .
as function of scale in pc . ]
figure 3 presents the turbulence spectral function @xmath23 computed from the autocorrelation function .
the curve is noisy but a power law range is clear . also here
the turbulence spectral functions with @xmath19 and @xmath20 are plotted .
the two slopes are compatible with the computed spectral function , although @xmath20 seems preferable .
the wavenumber range shown corresponds to spatial scales between @xmath24 , which is in this case the length of the line @xmath11 , and @xmath25 .
higher wavenumbers correspond to spatial scales that are smaller than the average radius of the shells , and therefore the computed turbulence spectrum is not valid for these scales .
the results of the present work strengthen the case for the turbulence interpretation of the 21 cm power spectra of the smc .
the residual radial velocities of the super shells exhibit statistical spatial correlations expected from turbulence .
the turbulence spectrum and structure function are consistent with a kolmogorov spectrum , @xmath26 , and with that of incompressible turbulence , @xmath20 .
the latter seems preferable .
it equals the value deduced by goldman ( 2000 ) from the hi intensity fluctuations .
stanimirovic , s. , stavely - smith , l. , dickey , j. m. , sault , r. j. , and snowden , s. l. 1999 , mnras , 302 , 417 staveley - smith , l. , sault , r. j. , hatzidimitriou , d. , kesteven , m. j , and mcconnell , d. , 1997 , mnras , 289 , 225 stanimirovic , s. , and lazarian , a. 2001 , apj , 551 , l53 | the spatial power spectrum of the hi 21 cm intensity in the small magellanic cloud ( stanimirovic 1999 ) is a power law over scales as large as those of the smc itself .
it was interpreted as due to turbulence by goldman ( 2000 ) and by stanimirovic & lazarian ( 2001 ) .
the question is whether the power spectrum is indeed the result of a dynamical turbulence or is merely the result of a structured static density . in the turbulence interpretation of goldman ( 2000 )
the turbulence was generated by the tidal effects of the last close passage of the lmc about 0.2 gyr ago .
the turbulence time - scale was estimated by goldman to be 0.4 gyr , so the turbulence has not decayed yet .
staveley - smith ( 1997 ) observed in the smc about five hundreds of hi super shells . their age is more than an order of magnitude smaller than the turbulence age .
therefore , if the turbulence explanation holds , their observed radial velocities should reflect the turbulence in the gas in which they formed . in the present work we analyze the observed radial velocities of the super shells .
we find that the velocities indeed manifest the statistical spatial correlations expected from turbulence .
the turbulence spectrum is consistent with that obtained by goldman(2000 ) . |
lm , sl and ac acknowledge the italian miur for financial support .
pu wishes to thank a.j . dean and j.b .
stephen for useful scientific discussions . | using recent polarimetric observations of the crab nebula in the hard x - ray band by integral , we show that the absence of vacuum birefringence effects constrains @xmath0 lorentz violation in qed to the level @xmath1 at @xmath2 cl , tightening by more than three orders of magnitude previous constraints .
we show that planned x - ray polarimeters have the potential to probe @xmath3 by detecting polarization in active galaxies at red - shift @xmath4 .
experimental constraints on the parameters quantifying lorentz invariance violation ( lv ) are of fundamental importance .
because the lowest order corrections predicted in the photon dispersion relation imply the vacuum is birefringent , observations of polarized photons from distant astronomical sources provide very promising tests . in this letter
we exploit the recently discovered linear polarization of hard x - rays from the crab nebula ( cn ) @xcite .
these observations show a remarkably high degree of linear polarization ( @xmath5 ) and very close alignment of the polarization vector with both the optical polarization vector and the projection on the sky of the spin axis of the central neutron star . the high degree of polarization together with the lack of detectable rotation of the polarization vector of these @xmath6kev photons whilst propagating over the intervening @xmath7 cm enables us to tighten existing constraints by three orders of magnitude .
recent years have witnessed a growing interest in the possible high energy violations of local lorentz invariance as well as a flourishing of observational tests .
indeed , specific hints of lv arose from various approaches to quantum gravity @xcite .
however , most tests require a well established theoretical framework to calculate reaction rates and describe the particle dynamics . here , we work within the framework of effective field theory with non - renormalizable , mass dimension 5 lv operators ( see @xcite and references therein ) restricted to qed , for which the most general dispersion relations for photons and electrons are @xmath8 where ( [ eq : disp_rel_phot ] ) refers to photons and ( [ eq : disp_rel_ferm ] ) to fermions . @xcite . ]
we assume @xmath9 to be comparable to the planck mass @xmath10 .
the constants @xmath11 and @xmath12 indicate the strength of the lv .
the @xmath13 signs denote right and left circular polarization in ( [ eq : disp_rel_phot ] ) , and positive and negative helicity states of the fermion in ( [ eq : disp_rel_ferm ] ) .
equation ( [ eq : disp_rel_phot ] ) implies that the direction of polarization rotates during propagation due to the different velocities of the right- and left - handed circular polarizations , @xmath14 .
this effect is known as vacuum birefringence ( vb ) .
although it may seem hopeless to search directly for effects suppressed by the planck energy scale , even tiny corrections can be magnified to measurable ones when dealing with high energies , long distances of signal propagation or peculiar reactions ( see , e.g. , @xcite ) .
recently @xmath15 have been constrained to have a magnitude less than @xmath16 at 95% confidence level ( cl ) by a detailed analysis of the synchrotron component of the cn broadband spectrum @xcite , while the constraint @xmath17 has been obtained by @xcite considering the absence of vb effects during the propagation of optical / uv polarized light from gamma - ray bursts ( grb ) , were claimed in @xcite from grb 021206 observations @xcite ; however the result was later contested @xcite . ] .
there are also preliminary indications , based on an analysis of the photon fraction in ultra - high - energy cosmic rays , that these coefficients might be less than @xmath18 , though nothing conclusive can be claimed yet @xcite . in this work
we tighten the current constraints on @xmath19 suppressed lv by about three orders of magnitude for photons , by considering the limits on vb effects implied by the recently detected @xcite polarized hard x - rays from the cn .
firstly , we set such constraints following the arguments by @xcite , an approach robust against systematic uncertainties related to astrophysical modeling .
we then infer tighter limits that exploit and rely on modeling of the crab nebula and pulsar . finally , we consider the constraints which future x - ray polarization measurements of extragalactic objects , e.g. active galactic nuclei ( agn ) will allow .
this is of particular interest in the light of current experimental efforts to build x - ray polarimeters @xcite . during propagation over a distance @xmath20 ,
the ( cosmological ) distance is given by @xmath21 , which includes a @xmath22 factor in the integrand to take into account the red - shift acting on the photon energies .
as usual , @xmath23 is the present value of the hubble parameter , while @xmath24 and @xmath25 represent the density fractions of cosmological constant and matter in the universe , respectively . ] , the polarization vector of a linearly polarized plane wave with momentum @xmath26 rotates through an angle @xcite , @xmath27 observations of polarized light from a distant source can constrain @xmath28 in two ways , depending on the amount of available information on both the observational and the theoretical ( i.e. source modeling ) side : 1 . since detectors have a finite energy bandwidth , eq .
( [ eq : theta ] ) is never probed in real situations .
rather , if some net amount of polarization is measured in the band @xmath29 , an order - of - magnitude constraint arises from the fact that if the angle of polarization rotation ( [ eq : theta ] ) were to differ by more than @xmath30 over this band , the detected polarization would fluctuate sufficiently for the net signal polarization to be suppressed @xcite . from ( [ eq : theta ] ) , this constraint is @xmath31 this just requires that any intrinsic polarization ( at source ) is not completely washed out during signal propagation .
it thus relies on the mere detection of a polarized signal , without considering the observed polarization degree .
a more refined limit can be obtained by calculating the maximum observable polarization degree , given the maximum intrinsic value @xcite : @xmath32 where @xmath33 is the maximum intrinsic degree of polarization , @xmath34 is defined in eq .
( [ eq : theta ] ) and the average is weighted over the source spectrum and instrumental efficiency , represented by the normalized weight function @xmath35 @xcite .
conservatively , one can set @xmath36 , but a lower value can sometimes be justified on the basis of source modeling .
using , one can then cast a constraint by requiring @xmath37 to exceed the observed value .
2 . suppose that polarized light measured in a certain energy band has a position angle @xmath38 with respect to a fixed direction .
at fixed energy , the polarization vector rotates by the angle ( [ eq : theta ] ) ; if the position angle is measured by averaging over a certain energy range , the final net rotation @xmath39 is given by the superposition of the polarization vectors of all the photons in that range : @xmath40 where @xmath34 is given by ( [ eq : theta ] ) . if the position angle at emission @xmath41 in the same energy band is known from a model of the emitting source , a constraint can be set by imposing @xmath42 although this limit is tighter than that obtained from the previous methods , it clearly hinges on assumptions about the nature of the source , which may introduce significant uncertainties . in the case of the crab nebula ,
a @xmath43% degree of linear polarization in the @xmath44 band has recently been measured by the integral mission @xcite .
this measurement uses all photons within the spi instrument energy band .
however the convolution of the instrumental sensitivity to polarization with the detected number counts as a function of energy , @xmath35 , is maximized and approximately constant within a narrower energy band ( 150 to 300 kev ) and falls steeply outside this range @xcite .
for this reason we shall , conservatively , assume that most polarized photons are concentrated in this band .
given @xmath45 , @xmath46 and @xmath47 , eq .
( [ eq : decrease_pol ] ) leads to the order - of - magnitude estimate @xmath48 .
a more accurate limit follows from ( [ eq : pol ] ) . in the case of the cn
there is a robust understanding that photons in the range of interest are produced via the synchrotron proces , for which the maximum degree of intrinsic linear polarization is about @xmath49 ( see e.g. @xcite ) .
figure [ fig : casea ] illustrates the dependence of @xmath50 on @xmath11 for the distance of the cn and for @xmath51 . the requirement @xmath52 ( taking account of a @xmath2 offset from the best fit value @xmath53 ) leads to the constraint ( at 99% cl ) @xmath54 on @xmath11 for the distance of the cn and photons in the 150300 kev range , for a constant @xmath35 . ]
it is interesting to notice that x - ray polarization measurements of the cn already available in 1978 @xcite , set a constraint @xmath55 , only one order of magnitude less stringent than that reported in @xcite .
constraint ( [ eq : constraint - degree ] ) can be tightened by exploiting the current astrophysical understanding of the source .
the cn is a cloud of relativistic particles and fields powered by a rapidly rotating , strongly magnetized neutron star . both the _ hubble space telescope _ and the _ chandra _ x - ray satellite have imaged the system , revealing a jet and torus that clearly identify the neutron star rotation axis @xcite . the projection of this axis on the sky lies at a position angle of @xmath56 ( measured from north in anti - clockwise ) .
the neutron star itself emits pulsed radiation at its rotation frequency of 30 hz . in the optical band
these pulses are superimposed on a fainter steady component with a linear polarization degree of 30% and direction precisely aligned with that of the rotation axis @xcite . the direction of polarization measured by integral - spi in the @xmath57-rays is @xmath58 ( @xmath59 error ) from the north , thus also closely aligned with the jet direction and remarkably consistent with the optical observations .
this compelling ( theoretical and observational ) evidence allows us to use eq .
( [ eq : constraint - caseb ] ) . conservatively assuming @xmath60 ( i.e. @xmath2 from @xmath41 , 99% cl ) , this translates into the limit @xmath61 and @xmath62 for a @xmath63 deviation ( 95% cl ) .
figure [ fig : caseb ] shows @xmath64 as function of @xmath11 .
the left hand panel reports the global dependence ( the spikes correspond to rotations by @xmath65 ) , while the right hand panel focuses on the interesting range of values . ) rules out the possibility that the polarization angle is close to the expected one after rotating by some multiple of @xmath66 ( the polarization angle is defined on the interval @xmath67 $ ] ) . ]
the constraints presented in ( [ eq : constraint - degree ] ) and ( [ eq : constraint - serious - crab ] ) are remarkably strong .
although based on a cumulative effect , they are achieved using a local ( galactic ) object .
the reason lies , on the one hand , in the quadratic dependence of @xmath34 on the photon energy , in constrast with the linear gain given by distance ( see e.g. eq .
( [ eq : theta ] ) ) . on the other hand ,
the robust theoretical understanding of the cn has enabled us to strengthen the constraints significantly .
further improvements on lv constraints via birefringenge are expected thanks to the forthcoming high - energy polarimeters , such as xeus @xcite , pogolite @xcite , polar - x @xcite and gamma ray imager @xcite which will provide an unprecedented sensitivity , sufficient to detect polarized light at a few % levels also in extragalactic sources .
the lv limits will be optimized by balancing between source distance and observational energy range depending on the detector sensitivity .
this is illustrated in fig .
[ fig : plot ] , where the strength of the possible constraints ( cast with the first , most general method described above ) is plotted versus the distance of sources ( in red - shift @xmath68 ) and for different energy bands ( medium x- and @xmath57-rays ) .
remarkably , constraints of order @xmath69 could be placed if some polarized distant sources ( @xmath70 ) will be observed by such instruments at 1 mev . |
in recent years , higher - order operators have become the object of intense study in the search for possible effects of lorentz invariance violation @xcite .
these planck - mass suppressed higher - order operators allows to describe new physics beyond those obtainable from renormalizable operators , that is , operators with mass dimension four or less @xcite .
for example , the higher - order effective theory may involve additional degrees of freedom associated to ultra - high energies which do not converge perturbatively to the normal ones when taking the limit of the dimensionless parameters in the effective terms to zero .
lee and wick studied these exotic modes in the context of negative metric theories @xcite and in spite of the ghost states that appear , they showed that unitarity can be preserved by demanding all stable particles to be positive norm states @xcite . here
we check perturbative unitarity in a qed consisting of higher - order myers and pospelov photons @xcite and standard fermions .
the myers - pospelov lagrangian density for photons is given by @xmath0 where @xmath1 is a four - vector defining a preferred reference frame , @xmath2 is the planck mass and @xmath3 is a dimensionless parameter
. we can always select a real basis of four - vectors @xmath4 to be orthonormal and to satisfy the properties described in ref . . in analogy with the left and right handed polarizations of usual electrodynamics
we can switch to a basis of complex four - vectors @xmath5 and define the orthogonal projectors @xmath6 as @xmath7 where @xmath8 . to derive the dispersion relation
we can expand the gauge field in term of this complex basis and replace in the equations of motion to arrive at @xmath9 in agreement with the work in ref . .
here we check perturbative unitarity in the process of electron - positron scattering @xmath10 . for this
we use the optical theorem which relates the imaginary part of the forward scattering amplitude @xmath11 with the total cross section as @xmath12 where the sum runs over all intermediate physical states . considering the qed extension model
the amplitudes that contribute to the @xmath13-matrix are the direct amplitude @xmath14 and the exchange amplitude @xmath15 where @xmath16 , @xmath17 and @xmath18 , @xmath19 and where @xmath20 is the usual fermionic normalization constant .
let us start with the left hand side of the unitarity condition ( [ unitarity ] ) .
a similar calculation has been given in the minimal sector of the standard - model extension , see ref . .
to simplify we will consider the lightlike case where we have a ghost state with frequencies @xmath21 and two photons with frequencies @xmath22 , see ref . , and the propagator @xmath23 where and we have included the @xmath24 prescription .
we are interested in the imaginary part of the forward scattering amplitude , therefore let us set @xmath25 and @xmath26 .
moreover , we can see that the direct process does not contribute since the virtual photon can never be on shell for non - zero external momenta , hence @xmath27=0 $ ] .
let us find the contribution of the exchange process and substitute the propagator ( [ propagator ] ) in ( [ amplitud2 ] ) @xmath28 because only the poles can contribute to the imaginary part and due to energy conservation encoded in @xmath29 , we have that only the positive poles of the virtual photon have a chance to contribute .
we can discard the ghost contribution since its energy @xmath30 lies beyond the region of validity of the effective theory .
that is , the external fermions will always satisfy the condition @xmath31 .
hence , we have @xmath32\nonumber \\&= & - e^2 \int d k^0 \int \frac{d^3 \vec k}{(2\pi)^3}\delta^4(p_1+p_2-k ) v^{\mu } v^ { * \nu } \sum_{\lambda}\frac { p^{\lambda}_{\mu \nu } \delta ( k_0-\omega_1^{\lambda } ) } { 2g\lambda ( k_0-\omega_0^{\lambda } ) ( k_0- \omega_2^{\lambda } ) } , \nonumber \\ & = & e^2\int \frac{d^3 k}{(2\pi)^3 } \delta^4(p_1+p_2-k ) v^{\mu } v^ { * \nu } \sum_{\lambda } \frac { \varepsilon_{\mu}^{\lambda } \varepsilon_{\nu}^{*\lambda } } { 2g\lambda ( \omega_1^{\lambda}-\omega_0^{\lambda } ) ( \omega_1^{\lambda}-\omega_2^{\lambda } ) } , \nonumber \\ & = & \int \frac{d^3 k}{(2\pi)^3 } \delta^2(p_1+p_2-k ) \sum_{\lambda } \left| \mathcal m_{\lambda } \right|^2,\end{aligned}\ ] ] where we have used the notation @xmath33 for the physical process @xmath34 and we have introduced the normalization constant @xmath35 . finally we have @xmath36= \int \frac{d^3 k}{(2\pi)^3 } \delta^2(p_1+p_2-k ) \left| \mathcal m_{\rm phys } \right|^2,\end{aligned}\ ] ] and therefore the unitarity condition is satisfied in this scattering process .
with an explicit calculation we have verified that the unitarity condition in the process of electron - positron scattering at tree level order is satisfied .
a next step is to verify unitarity to order @xmath37 that will require to analyze more diagrams .
some of them contain loops where the ghosts can appear off - shell , thus , introducing an extra difficulty .
checking the unitarity condition to these order will give us a robust support in order to make physical predictions in the theory .
this work was supported by the direccin de investigacin de la universidad del bo - bo grant 123809 3/r and fapei .
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b * 848 * , 90 ( 2011 ) . | the unitarity of a lorentz - invariance violating qed model with higher - order myers and pospelov photons coupled to standard fermions is studied .
as expected , we find ghost states associated to the higher - order terms that may lead to the loss of unitarity .
an explicit calculation to check perturbative unitarity in the process of electron - positron scattering is performed and it is found to be possible to be preserved . |
the accreted stellar halo ( ash ) of a galaxy represents a record of the accretion history of the galaxy itself .
its assembly is determined by a large number of free parameters , including the structural properties of each accreted satellite ( virial mass , concentration , stellar content , morphology ) , the orbital properties of each accretion event ( energy and angular momentum at infall ) , the structural properties of the host itself during accretion .
this implies a significant degree of stochasticity , as shown by the observed halo - to - halo scatter ( e.g. , van dokkum et al .
2014 ) and by the dichotomy between the ` broken ' and sharply declining density profile of the stellar halo of the milky way ( e.g. , deason et al .
2013 ) and the more extended halo of andromeda , whose density profile is well described by a single power - law ( e.g. , gilbert et al .
2012 , ibata et al .
in amorisco ( 2015 ) i have isolated the main ingredients that shape the contribution of each accreted satellite to the ash .
i adopted a simplified approach and assumed that the contributing dwarfs are dark matter dominated ( as expected for the case of an @xmath1 host ) and ignored the gravitational influence of the host s disk .
combined with the halo mass - concentration relation ( e.g. , gao et al .
2008 , ludlow et al .
2014 ) , this reduces the structural properties of each minor merger to two dimensionless parameters .
two additional parameters characterise the orbital properties of the satellite at accretion ( e.g. , benson 2005 , jiang et al .
the locus of this parameter space that is relevant to a @xmath0cdm cosmology is explored with a library of minor merger n - body simulations , in which stars are assigned to the most bound 5@xmath2 of the satellite s particles , using a standard particle tagging technique ( e.g. , bullock & johnston 2005 , cooper et al .
this study shows that dynamical friction is a major player in shaping stellar deposition , allowing only the most massive ( and/or concentrated ) accreted satellites to deposit their stars in the innermost regions of the host .
orbital radialisation by dynamical friction causes the stellar populations deposited by such most massive accretion events ( virial satellite - to - host mass ratio at accretion @xmath3 ) to bear little memory of the details of the orbital properties of the progenitor at infall .
galaxies sharing a final virial mass of @xmath4 . panels
b ) and c ) : an example for the procedure of assembly of the accreted stellar halo using the combination of a given accretion history and of a library of minor merger simulations ; each contribution to the halo is color - coded by accretion redshift .
panel c ) : the spherically averaged density profiles of the 150 accreted stellar haloes corresponding to the accretion histories of panel a ) , color - coding refers to the local mass - weighed virial satellite - to - host mass .
panel e ) : the correlation between the total accreted stellar mass in the halo and the number of main accretion events ( see text for details ) . ]
i use the library just described to assemble 500 ashs , for galaxies that share a virial mass of @xmath5 .
i use monte carlo generated accretion histories ( fakhouri et al .
2010 ) , 150 of which are displayed in _ panel a ) _ of fig . 1 , color - coded by the total accreted stellar mass .
satellite stellar masses are assigned based on a redshift - independent abundance matching relation ( garrison - kimmel et al .
2014 , 0.3 dex scatter ) .
_ panels b ) _ and _ c ) _ exemplify the assembly of an individual ash : each accreted and disrupted satellite ( color - coded by its accretion redshift ) is associated to the spherically averaged density profile of the stars it deposits in the halo .
these are retrieved from the library using the relevant time - interval between accretion and @xmath6 , and are re - scaled to physical quantities according to the dimensional properties of the merger at hand .
_ panel d ) _ displays the spherically averaged density profiles of 150 ashs built in this manner . at each radius , the halo - to - halo scatter approaches 2 dex , and increases at @xmath7 , together with an increasing amount of not fully phase - mixed substructure from recent accretion events .
although with a significant scatter , on average , ashs share a logarithmic density slope @xmath8 within 20 kpc , and become steeper with radius , as shown by the dashed guiding lines .
the details of this steepening are highly variable : some profiles have marked and sharp breaks , others ` roll ' gently towards steeper and steeper slopes , other remain comparatively shallower .
the radii where such transitions take place are equally variable . the color - coding in _
panel d ) _ indicates the local mass - weighted satellite - to - host virial mass ratio . on average , the innermost regions of the ash
are contributed by satellites that have larger virial mass ratios at accretion .
this gradient has been observed in cosmological hydrodynamical simulations ( rodriguez - gomez et al .
2016 ) and i conclude is a direct consequence of dynamical friction ( e.g. , amorisco 2015 ) .
color - coding in _
panel d ) _ reveals that the local mean virial mass ratio also correlates positively with the local density .
_ panel e ) _ confirms this link by showing a scatter plot of the total accreted stellar mass of the ash against the ` number of main accretion events ' @xmath9 ( i.e. the ratio between the total accreted stellar mass of the ash and the mean stellar mass of the contributing satellites , mass - weighted by stellar mass itself ) . the most massive ashs result from the accretion of just one / two particularly massive satellites , which dominate the ex - situ mass .
although this technique represents a highly simplified approach , it allows for an efficient exploration of the significant stochasticity of ashs .
physical ingredients that are neglected here ( the host s stellar disk , the morphologies of the accreted dwarfs , any post - accretion star formation etc . ) will result in even increased variability .
detailed analysis of a large sample of ashs concentrating on the systematic correlations that connect density profile and accretion history is the subject of a forthcoming work ( amorisco 2016 ) . | i use a library of controlled minor merger n - body simulations , a particle tagging technique and monte carlo generated @xmath0cdm accretion histories to study the highly stochastic process of stellar deposition onto the accreted stellar halos ( ashs ) of @xmath1 galaxies .
i explore the main physical mechanisms that drive the connection between the accretion history and the density profile of the ash .
i find that : i ) through dynamical friction , more massive satellites are more effective at delivering their stars deeper into the host ; ii ) as a consequence , ashs feature a negative gradient between radius and the local mass - weighed virial satellite - to - host mass ratio ; iii ) in @xmath1 galaxies , most ashs feature a density profile that steepens towards sharper logarithmic slopes at increasing radii , though with significant halo - to - halo scatter ; iv ) the ashs with the largest total ex - situ mass are such because of the chance accretion of a small number of massive satellites ( rather than of a large number of low - mass ones ) . |
" . ] here we summarize which permutations yield equivalent and nonequivalent metastable structures . (
i ) the cyclic variants of any permutations give equivalent structures because of the periodicity of the crystal .
( ii ) pairs of permutations exchanged by @xmath68 ( e.g. , @xmath69 " and @xmath70 " ) yield pairs of structures exchanged by the mirror plane @xmath71 ( plane perpendicular to the units , see fig .
s[fig : permutation - op ] ) .
( iii ) pairs of permutations exchanged by inversion ( e.g. , @xmath69 " and @xmath72 " ) yields structures interchanged by global inversion ( @xmath73 ) .
all the permutations related by these three operations yield equivalent structures in that they give equal values of formation enthalpy .
note that the mirror reflection with respect to plane @xmath74 ( plane parallel to the units , see fig .
s[fig : permutation - op ] ) is represented by the combination of the former operations .
we carried out the first - principles structure optimization for the structures summarized in table [ tab : struct ] , which resulted in the values of the formation enthalpy depicted in fig .
4 ( triangle ) .
note that this list exhausts all the possible structures for @xmath75 , in that any unlisted permutations can be related to either of the listed ones with the operations summarized in fig .
s[fig : permutation - op ] .
the first - principles structure optimization has also been done for other compounds for reference .
the squares in fig .
4 correspond to the body - centered monoclinic ( without spatial modulation ) and @xmath20-po sulfur ( ref . ) , @xmath76- and @xmath77-hs ( ref . ) , @xmath78-h@xmath67s@xmath16 ( ref . ) , @xmath17- and @xmath15-h@xmath14s ( ref . ) , @xmath8- and @xmath9-h@xmath16s ( ref . ) , and @xmath79-h ( ref . ) . here
we summarize the reaction(s ) that can occur in the compressed h@xmath14s and intermediate phases . as discussed in the main text ,
the pressure - induced stabilization of the hs and magnli " phases should stimulate a reaction @xmath18 ( @xmath19)@xmath55@xmath20+hs [ fig .
s[fig : transforms - summary](a ) ] .
there is also another possible reaction forming h@xmath16s slabs @xmath18 ( @xmath19)+h@xmath1@xmath552@xmath20 [ fig .
s[fig : transforms - summary](b ) ] .
the values of formation enthalpy suggests that this reaction is also stimulated upon compression [ figs . 4 and s[fig : transforms - summary](c ) ] .
although the latter reaction requires excess hydrogen atoms , it may also have been relevant in the previous experiments : observation of phases of elemental sulfur @xcite indicates h@xmath14 or any other h - rich phases as their decomposition counterparts .
( @xmath19)@xmath55@xmath20+hs and ( b ) reaction @xmath18 ( @xmath19)+h@xmath1@xmath552@xmath20 , where small and large balls represent hydrogen and sulfur atoms , respectively .
( c ) schematic picture of the possible transformation paths ( see also fig . 5 ) . ] | we theoretically give an infinite number of metastable crystal structures for the superconducting sulfur hydride h@xmath0s under pressure . previously predicted crystalline phases of h@xmath1s and h@xmath2s
have been thought to have important roles for the experimentally observed low and high @xmath3 , respectively .
the newly found structures are long - period modulated crystals where slab - like h@xmath1s and h@xmath2s regions intergrow in a microscopic scale . the extremely small formation enthalpy for the h@xmath1s h@xmath2s boundary indicated with the first - principles calculations suggests possible alloying of these phases through formation of local h@xmath2s regions .
the modulated structures and gradual alloying transformations between them explain peculiar pressure dependence of @xmath3 in sulfur hydride observed experimentally , as well as could they prevail in the experimental samples under various compression schemes .
sulfur hydride has recently been found to become superconductor at extremely high pressure around 200 k @xcite .
this is the first achievement of superconducting transition temperature ( @xmath3 ) exceeding the nitrogen boiling point among the conventional phonon mediated superconductors @xcite , as well as it has broken the long - standing record of 160 k in mercury - cuprate @xcite .
a remarkable feature observed in this superconducting phenomenon is pressure and annealing - scheme dependences of @xmath3 @xcite .
( i ) when the pressure ( @xmath4@xmath5100 gpa ) is applied to the h@xmath1s sample at room temperatures and afterwards cooled down , the observed @xmath3s amount to over 150 k , with 203k being the maximum value ( open circle in fig .
[ fig : crystals ] ) .
( ii ) by pressurizing at temperature around 100 k , on the other hand , the observed @xmath3 remains low and next rapidly increases ( open square in fig .
[ fig : crystals ] ) .
although this behavior suggests a variety of structural phases and their peculiar properties , efficient experimental observations have been obstructed by the extremely high pressure . instead , first - principles calculations have provided insights for the superconducting phases .
it is now established that some of the observed values of @xmath3 with the low- and high-@xmath6 schemes are well reproduced @xcite with crystal structures predicted for compositions of h@xmath1s @xcite , as well as h@xmath7s@xmath1 @xcite and h@xmath2s @xcite ( see fig .
[ fig : crystals ] ) .
however , consistent understanding on the observed behavior still remains unprecedented .
in particular , no clear explanation on the rapid increase of @xmath3 with the low-@xmath6 scheme has been given , despite accumulated first - principles proposals of candidate structures with various composition of h@xmath0s @xcite .
( open circle , square @xcite and star with error bar @xcite ) compared with the first - principles calculations ( solid symbols ) .
the latter data are taken from ref .
( pentagon ) , ref .
( rhombus ) , ref .
( inverted triangle ) , ref .
( circle and square ) , ref .
( double circle ) , ref .
( triangle ) , ref .
( hexagon ) and ref .
( circle with x - mark ) .
the data points in the shaded areas have been obtained from the corresponding predicted crystal structures @xcite , respectively .
note that @xmath8-h@xmath2s is a trigonally distorted variant of @xmath9-h@xmath2s . as discussed later
, we attribute the pressure dependence of @xmath3 ( open square ) to the magnli " phases . the values from ref .
( refs . )
includes the plasmon effect ( phonon anharmonic effect ) . the vertical bars connecting
the rhombi indicate the @xmath3 variation with the empirical coulomb parameter @xmath10 @xcite . ]
the current theoretical attempts have focused on the possible understanding with a minimal number of distinct structural phases .
rather , we provide a different view : not only distinct phases but also their mixture have vital roles .
specifically , we find an infinite number of metastable crystal structures having compositions h@xmath0s@xmath11 with 2/3@xmath12@xmath13@xmath123/4 , which have not been reported in the first - principles structure - search studies .
they can be understood as long - period modulated crystals formed by stacking the h@xmath1s and h@xmath2s slab - like structures , which are reminiscent of the magnli phases in transition - metal oxides @xcite .
all these structures are thermodynamically as stable as the complete separation to h@xmath1s and h@xmath2s phases .
this suggests that the microscopic _ intergrowth _ of the h@xmath1s and h@xmath2s regions requires little activation enthalpy and therefore occurs ubiquitously in the experimental situation , forming , as it were , slab - alloy phase .
we also show that the experimentally observed @xmath3-pressure curve is reproduced from these alloy - like phases .
-h@xmath14s and ( b ) ( 1 0 0 ) view of @xmath15-h@xmath1s , accompanied by their schematic pictures .
the unit cells are depicted in thin lines . in the schematic pictures ,
the bond depicted by panel ( d ) is indicated by thick dashed lines .
( c ) the fundamental unit structure , interlaced square nets " and its schematic picture . in the latter ,
sulfur and hydrogen atoms are located at the corners and the centers of the edges , respectively .
( d ) bonding structure in panels ( a ) and ( b ) , where sulfur and hydrogen atoms are depicted in yellow and red , respectively . ]
we begin with a discussion of the hidden similarity between theoretically predicted relevant structures for h@xmath14s and h@xmath16s . @xmath17-h@xmath1s and @xmath15-h@xmath1s ( fig . [
fig : crystals ] ) are known to give low @xmath3 values @xcite , which agree relatively well with the experimental values with the low-@xmath6 annealing for the low - pressure regime .
@xmath9-h@xmath2s ( fig .
[ fig : crystals ] ) has been thought to give the @xmath3 with the high-@xmath6 annealing cases @xcite , and its presence is recently confirmed with the x - ray diffraction measurements @xcite .
seen from the ( 1 1 0 ) direction [ ( 1 0 0 ) direction ] in the setup of ref . , one can notice that @xmath17- and @xmath15-h@xmath14s can be decomposed into the common unit structures [ figs .
[ fig : relations](a)(b ) ] : two h@xmath14s square nets interlaced with each other [ fig .
[ fig : relations](c ) ] .
the units of this shape are bound with each other so that sulfur atoms adopt local fcc - like stacking [ fig .
[ fig : relations](d ) ] @xcite .
the @xmath17 and @xmath15 structures can now be distinguished by the configuration of the inter - unit bonding . the uniform ( alternating ) bonding orientation yields the @xmath17 ( @xmath15 ) structure .
an important thing is that the @xmath9-h@xmath16s structure can also be formed with the present unit by binding them so that the vertical edges are shared [ fig .
[ fig : unit - bond](c ) ] . " and ( e ) its relaxed counterpart ( see the main text ) .
the @xmath9-h@xmath16s regions are shaded in red . ] the unit - based perspective for h@xmath14s and h@xmath16s is indeed helpful for us to construct a group of metastable structures . in a wide variety of materials such as metal oxides ,
multiple crystalline phases with slightly different stoichiometries emerge depending on the bonding between the unit complexes @xcite .
one of such examples is the magnli phases in molybdenum- @xcite , tungsten- @xcite , titanium- @xcite , and vanadium - oxides @xcite , where two - dimensional defects ( or _ crystallographic shear _
@xcite ) of edge- or face - sharing bonding between metal - oxide complexes are periodically formed . on the analogy to them ,
here we consider possible crystalline phases formed by the structural h@xmath14s unit .
we first define the three types of the inter - unit bonding ( fig [ fig : unit - bond ] ( a)(c ) ) : bonding @xmath18 and @xmath19 correspond to the two bonding orientations seen in @xmath15-h@xmath14s , respectively , whereas bonding @xmath20 corresponds to the bond sharing the edge . with this definition , by arranging the units side by side and next assigning any type of bonding for every neighboring pair of units , we can generate a crystal structure formed by the units . in this sense , every circular permutation composed of desired numbers of @xmath18 , @xmath19 and @xmath20 yields a corresponding long - period modulated crystal structure .
any structures generated in this way have intermediate composition h@xmath21s@xmath11 ( @xmath22 ) , whose unit cells are composed of h@xmath23s@xmath24 with @xmath25 and @xmath26 being the period of the permutation and the number of @xmath20 , respectively .
the
@xmath27 " structure is exemplified in fig .
[ fig : unit - bond](d ) , whose unit - cell formula is h@xmath28s@xmath29 . as the number of @xmath20 increases , h@xmath16s - like structural regions grow [ fig .
[ fig : unit - bond ] ( d)(e ) ] .
note that @xmath17-h@xmath14s , @xmath15-h@xmath14s , and @xmath9-h@xmath16s are generated by permutations
@xmath18 " ( also @xmath19 " ; see supplemental materials @xcite ) ,
@xmath30 " , and @xmath20 " , respectively .
denotes the number of atoms in the unit cell .
the structures corresponding to the points are summarized in supplemental materials @xcite .
the inset panels focus the region between @xmath31 ( h@xmath14s ) and @xmath32 ( h@xmath16s ) , where the formation enthalpies are measured with respect to the values for the phase separation of h@xmath14s and h@xmath16s .
bold lines indicate @xmath3350 kelvin . ] using the structures thus generated from the permutations of period 2 , 3 , and 4 as inputs , we carried out the first - principles structure optimization for various pressures .
our calculations were done using the first - principles code package quantum espresso @xcite with the generalized - gradient approximation for the exchange - correlation potential @xcite .
the unit - cell compositions of the resulting structures are h@xmath34s@xmath16 , h@xmath35s@xmath36 , h@xmath37s@xmath7,h@xmath38s@xmath29 , h@xmath39s@xmath7 , h@xmath28s@xmath29 , h@xmath40s@xmath41 , and h@xmath42s@xmath43 , respectively .
detailed conditions of the calculations are summarized in supplemental materials @xcite .
we have found that the formation enthalpy @xmath44(h@xmath0s@xmath11 ) for all the resulting structures satisfy @xmath45 ( fig .
[ fig : enthalpy ] ) .
we have also confirmed that all these structures retain the bonding characteristics in their initial structures [ fig .
[ fig : unit - bond](e ) ] .
namely , every different permutation gives different optimum metastable structures where well - defined bcc - h@xmath16s slab regions emerge .
although we do not examine the @xmath25@xmath46@xmath47 cases here , an infinite number of metastable structures are expected to be obtained in this way . remarkably , the values of formation enthalpy relative to the complete decomposition into h@xmath14s and h@xmath16s are wholly within 50 k per atom [ insets of fig . [
fig : enthalpy](a)(d ) ] .
this dependence is not an indication of trivial phase separation into h@xmath14s and h@xmath16s because the two regions _ intergrow _ in a microscopic scale in the respective structures ( see fig .
[ fig : unit - bond ] ( e ) ) .
a more appropriate interpretation is that the formation enthalpy for the h@xmath14s
h@xmath16s boundary is extremely small in these structures .
let us argue the relevance of the newly found intermediate phases in the experimental situation .
we here have to bear in mind that the intermediate structures are _ within _ the convex hull of the formation enthalpy in the present pressure regime , apart from the narrow range around 110 gpa ( fig .
[ fig : enthalpy ] and ref . ) .
this means that in the compressed h@xmath14s system , these phases do not grow as the decomposition residues of the emergence of h@xmath16s phase , if annealed with enough time and heat .
however , the decomposition paths otherwise depend on the enthalpy barrier from the pristine h@xmath14s structures .
the transition between two metastable structures with slightly different @xmath13 occurs through transformations of a small fraction of @xmath48-type inter - unit bonds to @xmath20 .
this sporadic character of the bonding transformation and the small formation enthalpy for the h@xmath14s
h@xmath16s boundary revealed above suggest that such transition requires little activation enthalpy .
the intermediate crystalline phases should thus be observable if h@xmath0s systems exhibit either of the @xmath17- or @xmath15-h@xmath1s region and next further compressed at low temperature .
the possible emergence of the intermediate magnli - type phases draws a consistent explanation on the puzzling pressure dependence of @xmath3 observed experimentally . in the present magnli " phases , @xmath3 is expected to increase as the h@xmath2s regions grow . indeed , by selecting plausible phases and calculating their superconducting @xmath3s from the first principles @xcite , we successfully reproduced the experimentally observed @xmath3 behavior ( fig.[fig : crystals ] ) : 41 k ( 130gpa ) , 80 k ( 180gpa ) , 107 k ( 190gpa ) and 121 k ( 200gpa ) . here , the increase of @xmath3 is due to enhancement of electron - phonon coupling ( see supplemental materials @xcite ) .
the succession of the magnli " phases with increasing fraction of the h@xmath2s region is hence the probable origin of the pressure dependence of @xmath3 observed with the low-@xmath6 compression scheme , filling the unsolved gap between the theory and experiments .
although we have selected the @xmath49 " , @xmath50 " , , @xmath51 " , and
@xmath18(@xmath20@xmath529 ) " phases for 130 , 180 , 190 and 200 gpa , respectively , we do not conclude these specific phases were emerging in the experimental situation in ref . but
were more various magnli phases .
for example , there has been a recent first - principles calculation that h@xmath7s@xmath1 , which is actually equivalent to the @xmath50 " phase , also yields a good agreement for @xmath53@xmath12@xmath54 gpa ( circle with x - mark in fig .
[ fig : crystals ] ; ref . ) .
( @xmath19 ) @xmath55 @xmath20+hs " .
the shaded area in the left plot represents the multiple intermediate structures , whereas that in the right indicate the h@xmath2s regions .
] s taken from ref .
, where the horizontal bars indicate regions where we can find similarity to simulated patterns for some magnli phases [ see panel ( c ) ] .
( bottom ) the simulated patterns for the reference structures .
( b ) representative simulated patterns exhibiting multiple diffraction peaks .
( c ) patterns obtained from a selected group of structures which are ordered so that the fraction of @xmath20-bond gradually varies . ]
we can also find a plausible reaction path consistent with the experiment .
the calculated formation enthalpies for hs and the magnli " phases decrease with compression compared with those for h@xmath1s [ see fig .
[ fig : enthalpy ] ( a)(c ) ] .
this pressure - induced phenomenon would stimulate a local reactions forming the h@xmath2s slab : @xmath18 ( @xmath19)@xmath55@xmath20+hs ( fig .
[ fig : transform - scheme ] ; see also supplemental materials @xcite ) .
this supports the present scenario ; the increase of the h@xmath2s region by compression .
the multiplicity of the metastable phases deduces an interesting speculation : arbitrary values of @xmath3 between those of the low- and high-@xmath3 phases in ref .
are observable .
the local h@xmath2s - slab formation can also be stimulated by annealing . by carefully controlling the annealing conditions
, one would observe various values and their temporal evolution even at fixed pressures .
here we note a possible characteristic effect in the present h@xmath0s magnli phases for future studies . in the titanium - oxide magnli phase
, it is known that the local electron - lattice coupling is enhanced by the two - dimensional defects to form bipolarons ( or charge order ) @xcite .
hydrogen atoms are in principle subject to this instability because of their multivalent character .
although we have not found traces of such polaronic phases ( see supplemental materials @xcite ) , in view of the strong electron - phonon coupling , proximity effects of such phases may affect the transport and superconducting properties @xcite . to facilitate experimental exploration of the h@xmath21s magnli phases , we computed theoretical x - ray diffraction patterns at 150gpa using rietan package ( ref . , see supplemental materials@xcite ) .
we focus on two distinct groups of structures .
when the fraction of the @xmath20 bond is small and both @xmath18 and @xmath19 bonds are present , multiple minor peaks are generally seen around the diffraction angle of the dominant peak in @xmath8-h@xmath2s [ ( 110 ) peak ; @xmath56@xmath57@xmath58@xmath59 in fig .
[ fig : xray](a ) ] , as exemplified in fig .
[ fig : xray](b ) .
this behavior is due to subtle distortion of the sulfur lattice and apparently difficult to understand with a simple combination of the patterns for the pristine @xmath8-h@xmath2s , @xmath17-h@xmath1s and @xmath15-h@xmath1s phases .
experimental noise " may actually be contributed to by the diffraction peaks from those phases .
we also show the simulated patterns when the @xmath20 bonds are gradually introduced in either @xmath17- or @xmath15-h@xmath1s phase in fig .
[ fig : xray](c ) .
we observe gradual evolutions between the end points .
remarkably , the intensity of the dominant peak is well retained while peripheral features such as sub - peaks , peak shift and tail are found . in the previous diffraction experiments @xcite , presence of the h@xmath2s phase has been indicated from the ( 110 ) peak , though it was accompanied by subtle sub - peaks depending on the experimental protocol .
the magnli phases of this group may provide a unified understanding on such structures . to demonstrate this , we plotted an experimental pattern quoted from ref . in fig .
[ fig : xray](a ) . the sub - peaks and tails around
@xmath60@xmath61@xmath62@xmath63@xmath64 well resembles to , for example , those for @xmath65 and @xmath66 , as indicated in panels ( a ) and ( c ) .
we additionally assert that not only crystalline magnli phases but also their nanoscale stripes can emerge in an ordered and random fashion , similarly to the cases of the microsyntactic intergrowth @xcite observed in silicon carbide @xcite and metallic oxides including the magnli materials @xcite .
such structures will appear in reality an alloy - like phase formed by the slabs of the metallic h@xmath14s and h@xmath16s .
this alloy phase is a compound analog of the classic superconducting alloys formed by elemental metals @xcite , but quite different from them .
the ingredients of the former alloy h@xmath1s and h@xmath2s slabs can develop in the common h@xmath1s crystalline phases and therefore , even in the pristine sample , their alloying occurs in the self - contained manner through the structural and stoichiometric transformations .
this alloying obviously smears the experimental diffraction pattern , which could appear to be amorphous - like behavior . in summary
, we have provided an interpretation based on the common structural unit for the known crystal structures of the low- and high-@xmath3 h@xmath14s and h@xmath16s and thereby found metastable magnli - type structures formed by their layered microsyntactic intergrowth .
the experimentally observed pressure dependence of @xmath3 is reasonably explained with pressure - induced gradual transformations through such alloy phases .
the present finding gives a new insight into the high-@xmath3 superconducting phenomena in sulfur hydride that the microscopic mixture of the phases can be ubiquitous .
_ -acknowledgment .
_ this work was supported by mext element strategy initiative to form core research center in japan and jsps kakenhi grant numbers 15k20940 ( r. ak ) and 15h03696 ( r. ar ) from japan society for the promotion of science ( jsps ) .
we thank yinwei li and mari einaga for sharing structure data for h@xmath67s@xmath16 in ref . and
x - ray diffraction data in ref . , respectively .
we also thank mikhail eremets , yoshihiro iwasa , hui - hai zhao , and yuta tanaka for enlightening comments .
the crystal structures were depicted using vesta @xcite .
the superconducting properties were calculated at the supercomputer center at the institute for solid state physics in the university of tokyo .
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it is well known that a potential difference @xmath0 is observed on a segment @xmath1 ( with a resistance @xmath2 ) of an asymmetric conventional metal loop @xmath3 ( with a resistance @xmath4 ) when a circular current @xmath5 is induced by the faraday s voltage @xmath6 in this loop . on the other hand
the magnetization measurements give evidence a circular direct current observed in semiconductor [ 1 ] normal metal [ 2 ] and normal state of superconductor [ 3 ] nano - structures in a constant magnetic field , i.e. without the faraday s voltage @xmath7 .
the observed periodical change of the magnetization with magnetic field at the period corresponding to the flux quantum for single electron @xmath8 or pair @xmath9 gives unambiguous evidence that this equilibrium quantum phenomenon , as well as flux quantization in superconductor [ 4 ] , is a consequence of the persistent current @xmath10 existing because of the quantization of the velocity circulation @xmath11 but in contrast to the flux quantization observed as far back as 1961 [ 5 ] the experimental results [ 1 - 3 ] give evidence of the persistent current along the loop with non - zero resistance @xmath12 .
the persistent current at @xmath12 was predicted as far bag as 1970 both in normal state @xmath13 of superconductor [ 6 ] and in non - superconductor mesoscopic structures [ 7 ] .
it was written in [ 7 ] and the later theoretical works [ 8,9 ] have corroborated that the persistent current can be observed at electron scattering ( at a finite mean free path @xmath14 ) , i.e. at non - zero dissipation .
thus , the persistent current can be observed at non - zero dissipation like conventional circular current .
nevertheless most experts are fully confident that a potential difference @xmath15 can not be observed on a segment @xmath1 when the persistent current @xmath10 is observed along the asymmetric mesoscopic loop with non - homogeneous dissipation @xmath16 along its circumference @xmath3 .
the observation [ 10 ] of the quantum oscillation of the dc voltage @xmath17 on a system of aluminum loops in the temperature region corresponding to the superconducting resistive transition , i.e. at @xmath12 , call this confidence in question . the superconducting resistive transition of the nano - structure containing 1050 asymmetric aluminum loops with diameter @xmath18 written at the measuring current with different values @xmath19 .
the inset shows the quantum oscillation of the dc voltage @xmath20 induced by the external as current with the frequency @xmath21 and the amplitude @xmath22 at the temperature @xmath23 corresponding to superconducting state of this nano - structure . ]
the persistent current observed in normal state of superconductor and non - superconductor ( semiconductor and normal metal ) has seminar nature and the theorists demonstrate this likeness .
kulik made the theory of the persistent current in non - superconductor nano - structure [ 7 ] just after the work [ 6 ] on this phenomenon in normal state of superconductor and in twenty years f. von oppen and e. k. riedel have calculated the flux - periodic persistent current in mesoscopic superconducting rings close to @xmath24 [ 11 ] after the calculation of the disorder - averaged persistent current for a non - superconductor mesoscopic ring [ 9 ] .
the persistent current can be observed in a loop when the wave function of electron or superconducting condensate is closed now and again in this loop .
therefore the persistent current can have an appreciable value only if the mean free path @xmath14 is not smaller than the loop length @xmath3 [ 8,9 ] . in the superconducting state
the mean free path of pairs is infinite @xmath25 and the persistent current has a value @xmath26 much large then in a non - superconductor loop @xmath27 [ 8,9 ] .
although the fermi velocity exceeds the pair velocity @xmath28 determined by the relation ( 1 ) the pair number @xmath29 in any real loop is so great at @xmath30 that @xmath31 . because of the large @xmath32 value
the quantum oscillation of the dc voltage @xmath20 with high amplitude can be observed at @xmath30 , fig.1 .
but because of zero resistance @xmath33 an external ac current with the amplitude @xmath34 exceeding the superconducting critical current @xmath35 should be applied at @xmath30 [ 12 ] .
the little - parks oscillations of the resistance @xmath36 reduced to the one in the normal state @xmath37 measured on two nano - structures containing aluminum loops with diameter @xmath18 ( the upper curve ) and @xmath38 ( the lower curve ) demonstrate the increase of the amplitude of the superconducting transition shift @xmath39 in magnetic field with the loop decrease . ]
such switching between quantum states with different connectivity of the wave function can induce a potential difference @xmath40 on segment of an asymmetric loop [ 14,15 ] .
it is expected that its value in the normal state @xmath13 may be larger than in non - superconductor loop since the @xmath32 value in the first case [ 3 ] is larger than in the second one [ 1,2 ] .
the persistent current @xmath41 increases with the loop length @xmath42 decrease .
but at a too small loop @xmath43 the switching between states with different connectivity of the wave function becomes impossible [ 14 ] because of the critical temperature shift @xmath44 [ 16 ] . here
@xmath45 is the superconductor coherence length at @xmath46 and @xmath47 is the width of the superconducting transition .
our measurements have corroborated the @xmath48 amplitude increase with the @xmath49 loop decrease , fig.2 .
we have found that @xmath50 at the diameter of our aluminum loop @xmath51 .
we intend to present results of the @xmath17 measurements on nano - structures with great number of such loops , fig.3 .
it may be these results will answer on the question on a possibility to observe the like phenomenon in semiconductor and normal metal loops .
an electron micrograph of the nano - structure containing 1080 asymmetric aluminum loops with diameter @xmath38 . ]
this work has been supported by grant of the program `` quantum nanostructures '' of the presidium of ras , grant 08 - 02 - 99042-r - ofi of the russian foundation of basic research and grant `` quantum bit on base of micro- and nano - structures with metal conductivity '' of the program `` technology basis of new computing methods '' of itcs department of ras .
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b _ * 37 * , 6050 ( 1988 ) | the observations of the dc voltage proportional to the persistent current on system of asymmetric superconductor loops at a non - zero resistance raise a question on a nature of this quantum phenomenon and its possibility in semiconductor and normal metal mesoscopic loops . |
the discovery of features in the x - ray spectra of the thermal emission spectra of the isolated neutron star 1e 1207 @xcite and three other isolated neutron stars has revived the interest in studies of medium-@xmath1 elements in strong magnetic fields .
the reason is that the observed features could be due to atomic transitions in elements that are fusion products of the progenitor star .
however , to calculate synthetic spectra for model atmospheres , and thus to be in a position to draw reliable conclusions from observed spectra to the elemental composition of the atmosphere and the distribution of elements on different ionization stages , accurate atomic data for these elements at very strong magnetic fields ( @xmath2 to @xmath3 t ) are indispensible . while the atomic properties of hydrogen and , partly , helium at such field strengths have been clarified in the literature over the last 25 years ( for a detailed list of references see , e. g. , ref . ) , for elements with nuclear charges @xmath4 only fragmentary atomic data exist with an accuracy necessary for the calculations of synthetic spectra .
we have tackled @xcite the problem by adapting the diffusion monte - carlo method ( dqmc ) @xcite to the case of neutron star magnetic fields .
this method has the advantage that ground - state energies can be determined practically free from approximations .
the basic idea of dqmc is to identify the ground state wave function @xmath5 ( @xmath6 of an @xmath7-body hamiltonian @xmath8 with a _ particle density _ whose correct distribution is found by following the random walk of many test particles ( `` walkers '' ) in imaginary time in 3@xmath7-dimensional configuration space . to reduce fluctuations
one works with a density distribution @xmath9 , where @xmath10 is a given guiding function used for importance sampling .
the density distribution @xmath11 obeys a drift - diffusion equation in imaginary time . because the importance - sampled green s function is an exponential operator
, one can expand it in terms of a euclidean path integral .
for sufficiently small time steps one can write down accurate approximations to the green s function , and sample it with diffusion monte - carlo @xcite .
the choice of the guiding function is crucial for the success of the dqmc procedure .
we take the guiding function @xmath12 as a slater determinant of single - particle orbitals each of which is a product of a landau state in the lowest level with a given magnetic quantum number and an unkown longitudinal wave function ( `` adiabatic approximation '' @xcite ) .
the different longitudinal wave functions are obtained selfconsistently by an iterative solution of the hartree - fock equations using b - splines on finite elements . to incorporate correlation effects
it is common to multiply the guiding function by a jastrow factor , @xmath13 .
we adopt the form @xmath14 where @xmath15 is the magnetic field strength in atomic units ( @xmath16 , @xmath17 t ) .
this leads to modifications of the adiabatic approximation guiding functions only at small distances , of the order of the larmor radius .
as a representative example , fig .
[ fig1 ] shows for the ground state of neutral iron ( @xmath18 ) at @xmath19 t the typical flow of a diffusion quantum monte carlo simulation .
ions can be treated without additional complication in the same way @xcite .
the figure depicts the energy offset @xmath20 , the block energy @xmath21 and the averaged block energy @xmath22 as a function of the number of blocks performed .
the complete simulation goes through three stages . during the first 100 blocks , a variational quantum monte carlo calculation ( vqmc )
is performed . since the adiabatic approximation guiding wave function is augmented by the jastrow factor ,
the vqmc calculation already lowers the energy in comparison with the initial adiabatic approximation result .
this stage is followed , in the next 300 blocks , by a fixed - phase diffusion quantum monte carlo ( fpdqmc ) simulation .
it is seen that the onset of the simulation leads to a considerable drop in the energy .
finally , in the last 300 blocks a released - phase diffusion quantum monte carlo ( rpdqmc ) simulation is carried out , which still slightly lowers the averaged block energy , by roughly 0.1 per cent .
the dashed vertical lines in fig .
[ fig1 ] indicate the blocks where dynamical equilibrium of the walkers is reached .
the relatively small difference between the fixed - phase and the released - phase results indicates that the phase of the adiabatic approximation wave function already well reproduces the phase of the ground state wave function .
the small fluctuations of the individual block energies @xmath21 evident in fig .
[ fig1 ] are characteristic of diffusion quantum monte carlo simulations .
it is also seen , however , that the averaged block energies @xmath23 quickly converge to constant values in all three stages of the simulation .
our final rpdqmc result for the energy is @xmath24 kev and lies well below the density functional ( df ) value .
the standard deviation of the block energies at the end of the simulation in this case is @xmath25 kev .
( ragged curve ) and the averaged block energy @xmath22 ( smooth curve ) in the dqmc simulation for the ground state energy of neutral iron ( @xmath18 ) at @xmath19 t as a function of the number of blocks . in each block , 200 time steps @xmath26a.u . were performed .
( hffem ( top horizontal line ) : energy value in adiabatic approximation ; df ( second horizontal line from top ) : density functional result of ref ; mcph@xmath27 ( third horizontal line from top ) : result of ref . . ) , width=432 ] @xmath28ref . .
@xmath29ref . .
[ table3 ] table [ table3 ] lists the results for all elements from helium to iron at the magnetic field strength @xmath30
t. the table contains in the first three columns the results of the three stages of the simulation and in the fourth column the energy values in adiabatic approximation calculated with our own hartree - fock finite - element method hffem .
literature values obtained by ivanov and schmelcher @xcite ( 2dhf ) , by mori and hailey @xcite ( mcph@xmath27 , multi - configurational perturbative hybrid hartree - hartree - fock ) and the results of density functional calculations @xcite ( df ) are given in the remaining columns .
the numbers in brackets attached to the hffem , 2dhf , mcph@xmath27 and df results designate the number of electrons occupying an excited hydrogen - like single - particle longitudinal state .
it can be seen that already the fixed - phase results lie slightly below the values that were obtained using the 2dhf method .
the comparison with the results of the mcph@xmath27 method shows that our rpdqmc energy values generally lie below those results , but there are also exceptions where our results lie above the mcph@xmath27 energies .
this may be due to the fact that the hybrid method is not self - consistent , since it evaluates the exchange energy in first - order perturbation theory in a basis of hartree states and it does not include the back - reaction of the excited landau states whose admixtures are taken into account perturbatively on the effective interaction potentials . therefore the method need not necessarily produce an upper bound on the energy .
the comparison with the results of the df calculations shows that these yield lower ground state energies at small nuclear charge numbers than our rpdqmc results , while for large @xmath1 the reverse is the case .
the df results listed in table [ table3 ] differ in the choice of the exchange functional .
given this restriction , it can not be ensured that the df calculations in all cases produce an upper bound on the ground state energy in magnetic fields as do the ab - initio methods used in this work or in a ref . .
we have extendend the released - phase diffusion monte carlo method to the calculation of the ground state energies of atoms and ions from helium to iron neutron star magnetic field strengths by using adiabatic approximation wave functions as guiding wave functions @xcite .
however , for matching observed thermal spectra from isolated neutron stars , wavelength information , and thus energies of excited states , are requisite .
jones et al . @xcite have shown a way how to calculate excited states of small atoms in strong magnetic fields using the correlation function monte carlo method @xcite .
the challenge remains to transfer their method to the dqmc simulations presented in this paper , and to calculate excited states of large atoms in intense fields .
this work was supported by deutsche forschungsgemeinschaft within the sfb 382 `` methods and algorithms for simulating physical processes on high - performance computers '' at the universities of tbingen and stuttgart . | the diffusion quantum monte carlo method is extended to solve the old theoretical physics problem of many - electron atoms and ions in intense magnetic fields .
the feature of our approach is the use of adiabatic approximation wave functions augmented by a jastrow factor as guiding functions to initialize the quantum monte carlo prodecure .
we calculate the ground state energies of atoms and ions with nuclear charges from @xmath0 for magnetic field strengths relevant for neutron stars . |
ngc3801 is a nearby e / s0 galaxy at a distance of @xmath147.9 mpc , with the body of the galaxy being crossed by two main dust features ( heckman et al .
1986 ; verdoes kleijn et al .
a warped dust lane lies along the optical minor axis while patchy dust filaments are seen on the eastern and western halves of the galaxy . at brightness levels
@xmath2@xmath123@xmath324 mag arcsec@xmath4 , the galaxy shows a hysteresis loop like structure while at even fainter levels , a boxy isophotal structure is seen ( heckman et al .
it hosts a small radio galaxy with an angular size of @xmath150 arcsec ( 11 kpc ) , whose jet axis is almost orthogonal to the rotation axis of the stellar component or orthogonal to the minor - axis dust lane ( heckman et al .
millimetre - wave observations have helped identify a radio core and clumps of co(1@xmath30 ) emission suggesting a r@xmath12 kpc circum - nuclear rotating gas disk orthogonal to stellar rotation and perpendicular to the radio jet ( das et al .
_ chandra _ observations reveal shock - heated shells of hot gas surrounding the radio lobes ( croston , kraft , & hardcastle 2007 ) .
hi observations with the arecibo telescope show gas in both emission and absorption , but higher resolution observations are required to determine its distribution and kinematics ( heckman et al . 1983 ) .
we present the first ever imaging study in hi , dust , uv and pa@xmath5 emission .
figure 1 shows the radio continuum image superimposed on an optical dss blue - band image in the left panel and the total intensity hi image superimposed on the 8.0 @xmath0 m dust / pah emission image from the _ spitzer _ on the right panel .
the dust emission shows a prominent linear feature nearly orthogonal to the jet in the central region ( r@xmath12 kpc ) .
we found that both the 8.0 @xmath0 m dust / pah emission and uv emission from _ galex _ show similar @xmath130 kpc wide s - shaped structure , representing young massive star formation in it .
our hi - emission study with the vla shows emission blobs on the eastern ( mostly red - shifted ) and western ( blue - shifted ) sides , roughly coinciding with the tails of the s - shaped structure ( figure 1 , right panel ) .
these hi results suggest a rotating gas disk ( v@xmath6@xmath1280 km s@xmath7 ) , with velocities nearly twice than that of the stars ( cf .
heckman et al . 1985 ) .
in addition , broad , faint , blue - shifted absorption wing and an hi absorption clump associated with the shocked shell around the eastern lobe are seen , possibly due to jet - driven outflow . due to its similarity with kinematically decoupled cores and other properties , we propose that a merger between a gas - rich spiral galaxy and an elliptical galaxy has triggered its agn activity and has shaped its stellar , gaseous and radio - jet structures .
detailed stellar population synthesis studies to understand its time evolution are in progress .
croston , j. h. , kraft , r. p. , & hardcastle , m. j. 2007 , apj , 660 , 191 das m. et al .
2005 , apj , 629 , 757 heckman , t. m. , balick , b. , van breugel , w. j. m. , & miley , g. k. 1983 , aj , 88 , 583 heckman , t. m. , illingworth , g. d. , miley , g. k. , & van breugel , w. j. m. 1985 , apj , 299 , 41 heckman , t. m. et al .
1986 , apj , 311 , 526 verdoes kleijn , g. a. , baum , s. a. , de zeeuw , p. t. , & odea , c. p. 1999 , aj , 118 , 2592 | we present preliminary results from a multi - wavelength study of a merger candidate , ngc3801 , hosting a young fr i radio galaxy , with a z - shaped structure .
analysing archival data from the vla , we find two hi emission blobs on either side of the host galaxy , suggesting a 30 kpc sized rotating gas disk aligned with stellar rotation , but rotating significantly faster than the stars .
broad , faint , blue - shifted absorption wing and an hi absorption clump associated with the shocked shell around the eastern lobe are also seen , possibly due to an jet - driven outflow . while 8.0 @xmath0 m dust and pah emission , from _
spitzer _ and near and far uv emission from _ galex _ is seen on a large scale in an s - shape , partially coinciding with the hi emission blobs , it reveals a @xmath12 kpc radius ring - like , dusty , starforming structure in the nuclear region , orthogonal to the radio jet axis .
its similarities with kinematically decoupled core galaxies and other evidences have been argued for a merger origin of this young , bent jet radio galaxy . |
non - photonic single electron data @xcite , which present an indirect probe of heavy quark energy loss , have significantly challenged the underlying assumptions of jet tomography theory .
a much larger suppression of electrons than predicted @xcite was observed in the @xmath2 gev region . ``
these data falsify the assumption that heavy quark quenching is dominated by [ pqcd based ] radiative energy loss when the bulk [ weakly coupled ] qcd matter parton density is constrained by the observed dn / dy @xmath3 1000 rapidity density of produced hadrons . ''
@xcite whdg @xcite revisited the assumption that pqcd collisional energy loss is negligible compared to radiative energy loss @xcite . as argued there , and references therein , `` the elastic component of the energy loss can not be neglected when considering pqcd jet quenching . '' as shown in whdg and elsewhere @xcite , the computationally expensive integrations over the geometry of the qgp can not be reduced to a simple ` average length ' prescription . indeed , this computation time is essential to produce radiative + collisional energy loss calculations consistent with the pion data .
there are large theoretical uncertainties in the whdg results @xcite . very significant to the electron prediction
is the uncertainty in the charm and bottom cross - sections .
there are also theoretical uncertainties in the energy loss mechanisms . here
, two aspects of the collisional energy loss will be examined with the aim of improving the energy loss model .
similar to radiative energy loss , the fluctuations of collisional energy loss around the mean affect the quenching of the quark spectra .
collisional fluctuations are often modelled in a fokker - planck formalism , characterized by two numbers or functions : drag and diffusion .
whdg implemented an approximation to this scheme applicable for small energy loss by giving the collisional loss a gaussian width around the mean , with @xmath4 , where @xmath5 is the mean energy loss given by a leading log calculation .
the drag - diffusion method is essentially a continuum approximation to a discrete process . a high energy jet traversing
the qgp will undergo only a small number of collisions . in the gyulassy - wang model ,
the expected mean free path of a quark is @xmath6fm , so there is a very significant surface region in which the fluctuations will differ greatly from those given by the continuum approximation .
it is therefore necessary to look at the fluctuations per collision and in the number of collisions . a simple model to investigate this is to model the medium as _ initially _ static objects which will then recoil upon collision , model the interaction between jet and medium using the full htl medium modified propagator .
this gives the probability of longitudinal momentum loss : @xmath7 \nonumber \\
c_l = 2+\frac{1}{e}(\omega + \vec{v}.\vec{q})(2 - \frac{\omega}{m})\,,\ , c_t = \left ( \frac{-\omega}{m}\right)\left ( v^2 - ( \vec{v}.\hat{\vec{q}})^2 \right)\end{aligned}\ ] ] this single collision distribution is then poisson convoluted to give the distribution for a finite number of expected collisions : @xmath8 the mass of the medium particle is tuned to give an average energy loss similar to that of the bt and tg leading log calculations ( @xmath9gev - although here we are not interested in the average energy loss per se ) . in fig .
[ fig : pofeps ] , the probabiliy of fractional energy loss in one collision is shown , similar to a @xmath10 bjorken collisional style model , with screening at small t - values ( shown in the right pane of fig .
[ fig : pofeps ] ) .
figure [ fig : collfluct ] illustrates the distributions in energy loss for a finite number of collisions for bottom and light quark jets .
the results for charm quarks are qualitatively similar to those for light quarks . for a large number of collisions ( eg average number of collisions @xmath11 , l@xmath12fm ) ,
the distributions are roughly symmetric and somewhat similar to the simple whdg gaussian .
this is expected from the central limit theorem .
the @xmath13 values extracted from these distributions are similar , with @xmath14 and the gaussian approximation only differing by @xmath15 .
surprisingly , a similar result for the @xmath13 values is found for @xmath16 collisions for bottom quarks .
the large change arrives for light quarks . for both @xmath17 collisions ,
the gaussian approximation gives a very different distribution for the fluctuations and a very different @xmath13 value .
the gaussian approximation overpredicts the @xmath13 suppression by @xmath18 , which is around a 30% effect for @xmath19 collisions .
this can not be neglected .
a full treatment of the finite number of collisions will reduce the quenching due to elastic energy loss compared to the treatment in whdg .
this conclusion is also applicable to other uses of fokker - planck / langevin formalisms that use a continuum description of the collisional process .
the @xmath13 predictions for bottom quarks are likely only marginally affected , those for light quarks most affected .
in @xcite , the change of the fixed qcd coupling @xmath20 from 0.3 to 0.4 was seen to significantly change the @xmath13 precitions from the whdg model .
there has been much recent work on the effect of a running coupling on the collisional energy loss @xcite ( ie @xmath21 ) . here
, we revisit the collisional energy loss in a similar manner to @xcite , looking at a simple bjorken - style estimate @xcite .
bjorken s estimate for the collisional energy loss is : @xmath22 in @xcite , the running coupling version for very high jet energies is given as : @xmath23 although this neglects the finite energy kinematic bound on the jet . adding in this bound to this calculation gives @xmath24 which is similar in structure to the original fixed coupling estimate .
a numerical comparison of equations [ eqnpesh1],[eqnpesh2],[eqnpesh3 ] is shown in fig .
[ fig : runalpha ] . for reasonable temperatures ,
@xmath25gev , all results are of a similar order of magnitude . for reasonable energies , no qualitatively new behavior is seen ( although , as found in @xcite , the @xmath26 behavior is new , but this affects much higher energies than those of interest at rhic or even lhc ) . when the kinematic bounds are taken into account ,
the result for the average energy loss including running coupling is often larger than the fixed @xmath1 result used in @xcite .
however , the numerical result is very sensitive to the input parameters used , illustrated in the middle and right panes of fig .
[ fig : runalpha ] for changing the prescription for @xmath27 from that used in @xcite to that from @xcite .
it has been argued previously `` that radiative and elastic average energy losses for heavy quarks were in fact comparable over a very wide kinematic range''@xcite , and even `` e @xmath28 10 gev light and charm quark jets have elastic energy losses smaller but of the same order of magnitude as the inelastic losses''@xcite .
hence , collisional energy loss can not be neglected when considering jet quenching of high energy jets in the qgp at either rhic or lhc .
a simple model combining collisional and radiative energy losses significantly reduces the discrepancy between the predictions and data .
two possible improvements to the whdg model have been examined here .
the inclusion of a finite number of collisions is seen to reduce the effect of the collisional energy loss on the quenching of gluons , light and charm quarks , but not to significantly affect the bottom quark @xmath13 .
opposite to this effect , including a running qcd coupling increases the energy loss by up to a factor of @xmath29 .
the combination of these two affects , along with other large uncertainties in the prediction for electron @xmath13 such as the ratio of charm to bottom total cross - sections , hints at the possibility that both the pion and electron @xmath13s may both be within range of purely perturbative calculations .
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j. d. bjorken , fermilab - pub-82 - 059-thy | with the qgp opacity computed perturbatively and with the global entropy constraints imposed by the observed @xmath0 , radiative energy loss alone can not account for the observed suppression of single non - photonic electrons .
collisional energy loss is comparable in magnitude to radiative loss for both light and heavy jets .
two aspects that significantly affect the collisional energy loss are examined : the role of fluctuations , and the effect of introducing a running qcd coupling as opposed to the fixed @xmath1 used previously . |
simulations of the ism in a shearing box domain have shown that turbulence driven by sne leads to an amplification of the mean magnetic field . using the test - field method ( schrinner et al .
@xcite ) , we derived transport coefficients relating the mean electromotive force to the mean magnetic field ( gressel @xcite ) . with these we were able to reproduce the time behaviour seen in the simulations . under conditions found in our own galaxy , and assuming a constant circular velocity , a rotation rate @xmath0 was required for the dynamo to work .
in order to further define the turbulence properties as a function of the star formation rate , rotation and gas density , we analysed a comprehensive set of direct simulations . taking these as an input , we here compute global mean - field maps for a set of different model galaxies .
measuring test - field coefficients for a wide set of direct simulations ( gressel et al .
@xcite , @xcite ) led to the following scaling relations for the relevant diagonal term in the @xmath1 tensor , @xmath2 for the ( downward ) turbulent pumping described by the antisymmetric part of the @xmath1 tensor @xmath3 for the turbulent diffusivity @xmath4 and for the mean vertical outflow velocity @xmath5 the relations were derived for sf rates , @xmath6 , varying from one tenth up to the galactic value @xmath7 , angular velocities between @xmath8 and @xmath9 and midplane densities from @xmath10 up to @xmath11 . from the simulations , we moreover found a vertical gradient of the turbulent velocity @xmath12 independent of the star formation rate , density and angular velocity .
we approximate the vertical profiles for the @xmath1 tensor by a @xmath13 curve with a scale height of @xmath14 .
the value of @xmath15 is chosen to be constant for @xmath16 and linearly growing with a slope of one third outside this range . for simplicity , we assume a constant scale height within our models . we also neglect the anisotropic part of the turbulent diffusivity , which seems to be of minor importance for the current models .
the rotation curve is modelled with a brandt law @xmath17 further we modify the vertical wind above @xmath18 by adding a radial outward velocity of the same size as @xmath19 .
the wind velocities reach values of 100 - 200 km / s at z=4kpc , which is an order of magnitude higher than in the models of moss et al . @xcite . with these input parameters ,
we solve the induction equation @xmath20 in a cylindrical domain with @xmath21 , and of vertical extent @xmath22 . defining @xmath23 and @xmath24
, we obtain a dynamo number @xmath25 .
the pitch angle , @xmath26 , can be estimated by @xmath27 , scaling as @xmath28 .
these estimates show that stronger sf reduces the dynamo number and increases the pitch angle .
it is known that the stationary quadrupole solution of the @xmath29 dynamo exists only in a finite range of the dynamo number . because the final strength of the regular field also depends on the saturation process ,
this estimate does , however , not provide a prediction for the final field strength in dependence of the star formation .
nevertheless , this behaviour still opens the possibility for radially extended regular magnetic fields .
this is because , in an exponential disc , sf decays much faster with radius than the angular velocity , and hence the dynamo number may be nearly constant over a large radial range .
+ applying a kenicutt schmidt - type law , @xmath30 , we can specify our galaxy model by a radial density profile , which we leave constant up to @xmath31 and then exponentially decay with a scale length of @xmath32 , as seen in fig .
[ fig : models ] . for the nonlinear back - reaction
, we use a classical , however anisotropic , @xmath1 quenching . while the usual quenching of only the diagonal terms in the @xmath1 tensor
would lead to solutions with a small pitch angle , independently quenching the pumping term can also saturate the dynamo by the increasing field advection from the wind . in this case , the pitch angle of the kinematic mode can be preserved ( elstner et al .
the models rely on a crude approximation of the vertical profiles for the turbulent transport coefficients , which still leave some uncertainty in the absolute numbers given in table 1 .
nevertheless , the general trend agrees well with the predictions form the local dynamo number analysis .
the pitch angle , measured from the magnetic vectors ( see fig . [
fig : polmap ] ) of the polarisation map at @xmath31 , increases slightly with the star formation rate as predicted by the ratio @xmath33 above .
the growth times of the order of @xmath34 tend to increase with the star formation rate , but there are exceptions ( cf . a1 and a2 of table 1 ) .
no large - scale magnetic field amplification was observed for @xmath35 in model b3 , and in the weakly differentialy rotating model a5 . yet , strong starbursts are usually not long - lasting events and therefore the dynamo may still operate on longer time scales .
the final field strength is not strongly dependent on the sf rate , and only the toroidal field is influenced by the difference in turbulent diffusivity .
the inverse dependence of the dynamo action on the sf activity is mainly due to an enhanced turbulent diffusion .
this does not necessarily increase the turbulent velocity but may equally change the correlation time .
in fact , the preference of magnetic arms _ between _ the optical arms may be based on this very property of ism turbulence ( cf .
rohde et al .
@xcite ) .
* sf rate and rotation determine the dynamo calling for adequate galaxy evolution models .
* low star formation rates favour the dynamo , explaining coherent inter - arm fields .
* strong sf may suppress large - scale dynamo action ( no vertical fields in the centre ) .
* explaining the radio - fir relation will require a different type of amplification mechanism at least for the small - scale field .
elstner d. , gressel o. , rdiger g. , 2009 , iaus 259 , 467 gressel o. , elstner d. , ziegler u. , rdiger g. , 2008 , a&a 486 , l35 gressel o. , 2009 , phd thesis , university of potsdam ( astro - ph:1001.5187 ) gressel o. , elstner d. , rdiger g. , 2011 , iaus 274 , 348 moss d. , sokoloff d. , beck r. , krause m. , 2010 , a&a 512 , a61 rohde r. , elstner d. , beck r. , 1999 , a&a 350 , 423 schrinner m. , rdler k .- h . , schmitt d. et al . , 2005 , an 326 , 245 | magnetic field amplification by a fast dynamo is seen in local box simulations of sn - driven ism turbulence , where the self - consistent emergence of large - scale fields agrees very well with its mean - field description .
we accordingly derive scaling laws of the turbulent transport coefficients in dependence of the sn rate , density and rotation .
these provide the input for global simulations of regular magnetic fields in galaxies within a mean - field mhd framework . using a kennicutt - schmidt relation between the star formation ( sf ) rate and midplane density
, we can reduce the number of free parameters in our global models .
we consequently present dynamo models for different rotation curves and radial density distributions . |
semiclassical general relativity predicts violations of the point - wise energy conditions associated with negative energy phenomena such as warp drives , traversable wormholes@xcite , and even _ _ time machines__@xcite .
however , the same theory can be used to constrain the magnitude and duration of negative energy pulses .
two of these restrictions are given by the quantum inequalities ( qis ) and the quantum interest conjecture ( qic ) .
the qis@xcite impose a lower bound on the expectation value ( in a quantum state @xmath0 ) of the renormalized stress - energy tensor along a timelike geodesic , @xmath1 weighted by a non - negative and normalized test function @xmath2 .
the initial bounds on @xmath3 depended on the modified bessel functions , and were obtained using a lorentzian test function@xcite . eventually flanagan found a more general bound in ( 1 + 1)-dimensional minkowski space for a massless scalar field@xcite , which does not depend on the specific choice of the test function , @xmath4 similarly , fewster and eveson obtained bounds in ( 1 + 1 ) and ( 3 + 1 ) dimensional flat space@xcite .
although their two dimensional inequality is slightly weaker than ( [ flanagan ] ) , their result for ( 3 + 1 ) dimensions is the most general and optimum bound for a massless scalar field , @xmath5''(t)\right)^2\,dt.\ ] ] both inequalities , ( [ flanagan ] ) and ( [ few&eve ] ) , can be written as a more general statement for @xmath6-dimensional spacetime@xcite , @xmath7 with @xmath8 . here
@xmath9 is the derivative operator , and the set of constants @xmath10 are given by @xmath11 the test function is now normalized such that @xmath12 .
it is easy to check that we recover equations ( [ flanagan ] ) and ( [ few&eve ] ) from ( [ qi ] ) , by setting @xmath13 and @xmath14 , respectively .
furthermore , by integrating by parts , the qis become a statement regarding the lack of negative eigenvalues for a one - dimensional pseudo - hamiltonian , @xmath15 where @xmath16 here @xmath17 is effectively a _ potential _ for a quantum mechanical system .
therefore it is possible to use the point of view of one - dimensional quantum mechanics ( after some technical considerations@xcite ) to reduce the qis to a much _ simpler _ framework .
once the quantum mechanical viewpoint is adopted , it is more convenient to also use its notation
. then the _ operator _
@xmath18 can be written as @xmath19 , where @xmath20 and @xmath21 are operators in the usual hilbert space of square - integrable functions .
this technical construction allows us to rewrite the eigenvalue problem in coordinates , as an ordinary differential equation for the eigenfunctions @xmath22 ( the test functions @xmath23 used before ) ; the _ multiharmonic _ time - independent schrdinger equation ( sde ) , @xmath24 where we have set @xmath25 to simplify the algebra . again ,
if @xmath13 we recover the ( 1 + 1)-dimensional case in the form of the time - independent sde , @xmath26 fortunately in this case , there is a theorem by simon@xcite , which guarantees the existence of a negative eigenvalue for the sde , if @xmath27 with this theorem , the qic can be related to the qis in ( 1 + 1 ) dimensions .
it is also possible to reformulate the qic for a more general set of energy pulses@xcite , unlike the original formulation , which is restricted to @xmath28-function pulses@xcite . to see this clearly ,
let us split the potential as @xmath29 its positive part minus its negative part .
then in order to guarantee positive eigenvalues , the potential must violate ( [ simon ] ) .
that is , @xmath30 since here the potential @xmath21 represents the expectation value of the renormalized stress - energy tensor , it is clear that to fulfill the qis the net energy density must always be positive , i.e. , its positive part must always overcompensate the negative one .
this is a simplified version of the original formulation of the qic@xcite .
the ( 3 + 1)-dimensional case can be recovered from the multiharmonic sde by choosing @xmath14 .
this yields the _ biharmonic _ sde , @xmath31 however , we now need to generalize simon s theorem for ( [ bsde ] ) .
first , via a variational argument , the lowest eigenvalue @xmath32 of ( [ bsde ] ) satisfies @xmath33 \d x,\ ] ] assuming all the test functions are normalized .
secondly , let us choose the following test function @xmath34 we then enforce the normalization @xmath35 to obtain @xmath36 and @xmath37 note that the kinetic term of ( [ eigenval ] ) , @xmath38'')^2\d x}{\sigma^4},\ ] ] diverges .
it contains a term proportional to @xmath28-function square , which arises from differentiating twice the absolute value . nevertheless , by expanding the normalized functions @xmath39 into a power series , @xmath40 we can get rid of the troublesome term by properly setting the expansion coefficient @xmath41 to zero . moreover , to make ( [ kinetic ] ) converge at zero , we also need @xmath42 . the rest of the coefficients can be freely chosen .
then we have , from ( [ eigenval ] ) , ( [ test ] ) and ( [ gexpan ] ) @xmath43 with @xmath44'')^2\d x$ ] . now choosing a sufficiently large @xmath45 , it is clear that @xmath46 implies a negative eigenvalue for the differential equation .
lastly , it is possible to collect more information from ( [ expand ] ) if we set @xmath47 .
then the next two terms become relevant . and
since the sign and magnitude of @xmath48 and @xmath49 are arbitrary , either @xmath50 or @xmath51 is a sufficient condition to guarantee the absence of a bound state .
differentiating twice the last expression with respect to @xmath52 , we have @xmath53 and finally @xmath54 .
that is , if @xmath47 , a necessary condition for the lack of a bound state is that @xmath54 .
this proves the extension of simon s theorem for the biharmonic sde , and it also proves the qic in ( 3 + 1)-dimensional minkowski space . to clarify the proof of the qic , we must recover the notation of semiclassical general relativity .
then the version of the qic we just proved states that the qis imply , either @xmath55 everywhere along the world line , _ or _
@xmath56 which is slightly stronger than the awec . once again , splitting the energy density into its positive part minus its negative part , @xmath57 the positive energy density must overcompensate the negative part elsewhere along the world line .
by proving the variant of simon s theorem for the biharmonic sde , we were able to reformulate the qic in ( 3 + 1)-dimensional flat spacetime similarly to the ( 1 + 1)-dimensional case studied in references and .
in flat spacetime , an energy pulse which satisfies the qis ( and the qic proved above ) must also fulfill an awec - like inequality .
although several technical aspects have not been mentioned here , a more detailed proof of the qic in flat spacetime can be found in reference .
this research was supported by the marsden fund administered by the royal society of new zealand . + ga was additionally supported by victoria university of wellington .
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d * 79 * , 065004 ( 2009 ) [ arxiv:0808.1931 [ gr - qc ] ] . | the _ quantum inequalities _ , and the closely related _ quantum interest conjecture _ , impose restrictions on the distribution of the energy density measured by any time - like observer , potentially preventing the existence of exotic phenomena such as _ alcubierre warp - drives _ or _ traversable wormholes_. it has already been proved that both assertions can be reduced to statements concerning the existence or non - existence of bound states of a certain 1-dimensional quantum mechanical hamiltonian . using this approach
, we will informally review a simple variational proof of one version of the quantum interest conjecture in ( 3 + 1)-dimensional minkowski space . 2 |
luc moreau is a postdoctoral fellow of the fund for scientific research - flanders . work done while a recipient of an honorary fellowship of the belgian american educational foundation , while visiting the princeton university mechanical and aerospace engineering department .
this paper presents research results of the belgian programme on inter - university poles of attraction , initiated by the belgian state , prime minister s office for science , technology and culture .
the scientific responsibility rests with its authors . | some biological systems operate at the critical point between stability and instability and this requires a fine - tuning of parameters .
we bring together two examples from the literature that illustrate this : neural integration in the nervous system and hair cell oscillations in the auditory system . in both examples
the question arises as to how the required fine - tuning may be achieved and maintained in a robust and reliable way .
we study this question using tools from nonlinear and adaptive control theory .
we illustrate our approach on a simple model which captures some of the essential features of neural integration . as a result
, we propose a large class of feedback adaptation rules that may be responsible for the experimentally observed robustness of neural integration .
we mention extensions of our approach to the case of hair cell oscillations in the ear .
persistent neural activity is prevalent throughout the nervous system .
numerous experiments have demonstrated that persistent neural activity is correlated with short - term memory .
a prominent example concerns the oculomotor system
see @xcite for a review and experimental facts .
the brain moves the eyes with quick saccadic movements . between saccades , it keeps the eyes still by generating a continuous and constant contraction of the eye muscles ; thus requiring a constant level of neural activity in the motor neurons controlling the eye muscles .
this constant neural activity level serves as a short - term memory for the desired eye position . during a saccade
, a brief burst of neural activity in premotor command neurons induces a persistent change in the neural activity of the motor neurons , via a mechanism equivalent to integration in the sense of calculus .
neural activity of an individual neuron , however , has a natural tendency to decay with a relaxation time of the order of milliseconds .
therefore the question arises as to how a transient stimulus can cause persistent changes in neural activity . according to a long - standing hypothesis , persistent neural activity
is maintained by synaptic feedback loops .
positive feedback can oppose the tendency of a pattern of neural activity to decay .
if the feedback is weak , then the natural tendency to decay dominates and neural activity decays . as the feedback strength is increased , the neural dynamics undergo a bifurcation and become unstable .
when the feedback is tuned to exactly balance the decay , then neural activity neither increases nor decreases but persists without change .
this , however , requires a fine - tuning of the synaptic feedback strength and the question arises as to how a biological system can achieve and maintain this fine - tuning @xcite . some gradient descent and function approximation algorithms performing this fine - tuning
have been proposed @xcite and a feedback learning mechanism based on differential anti - hebbian synaptic plasticity has been studied in @xcite .
nevertheless , it is still unclear how the required fine - tuning is physiologically feasible . for this reason , a different model for neural integration based upon bistability
has recently been proposed in @xcite . in the present paper
, we do not follow the line of research based upon bistability .
instead , we pursue the hypothesis of precisely tuned synaptic feedback .
the present paper proposes an adaptation mechanism that may be responsible for the fine - tuning of neural integrators and that may explain the experimentally observed robustness of neural integrators with respect to perturbations .
before we present this adaptation mechanism in detail , we first discuss a similar phenomenon in the auditory system . in order to detect the sounds of the outside world , hair cells in the cochlea operate as nanosensors which transform acoustic stimuli into electric signals .
in @xcite these hair cells are described as active systems capable of generating spontaneous oscillations .
ions such as @xmath0 are believed to contribute to the hair cell s tendency to self - oscillate . for low concentrations of the ions ,
damping forces dominate and the hair cell oscillations are damped .
as the concentration increases the system undergoes a hopf bifurcation , the dynamics become unstable , and the hair cells exhibit spontaneous oscillations . in @xcite
the hair cells are postulated to operate near the critical point , where the activity of the ions exactly compensates for the damping effects . as before , this requires a fine - tuning of parameters ( the ion concentrations ) and again the question arises as to how this fine - tuning can be achieved and maintained . in @xcite
a feedback mechanism has been proposed which could be responsible for maintaining this fine - tuning .
it thus seems that operating in the vicinity of a bifurcation is a recurrent theme in biology . and
the question as to how proximity to the bifurcation point may be achieved and maintained in a noisy environment may be of considerable , general interest .
we view the two presented examples as special instances of the following general problem .
consider a forced dynamical system , described by a differential equation @xmath1 .
the right - hand side of this equation depends on a parameter @xmath2 and the unforced dynamics @xmath3 are assumed to exhibit a bifurcation when @xmath2 equals a critical value @xmath4 .
the problem consists of finding a feedback adaptation rule for the parameter @xmath2 which guarantees proximity to the bifurcation point ; that is , which steers @xmath2 toward its critical value @xmath4 .
this adaptation law may depend on @xmath2 and @xmath5 but should be independent of @xmath4 , since this critical value is not known precisely .
this abstract formulation captures common features of both biological examples and suggests some unexpected links with the literature .
questions very similar to the present one have been studied extensively in the literature on adaptive control @xcite and stabilization @xcite ; and the general problem is closely related to extremum seeking @xcite , and to instability detection @xcite , where an operating parameter is adapted on - line in order to experimentally locate bifurcations .
although the above general formulation is convenient , there is little hope that a complete and satisfactory theory can be developed that applies to all possible instances of the problem . simplifying assumptions
make it more tractable . in this letter
, we study in detail what is probably the most simple but nontrivial instance of the general problem .
we consider the one - dimensional system @xmath6 which captures some of the essential features of neural integration and is in fact closely related to the autapse model from @xcite . with this interpretation
, @xmath5 is a strictly positive variable representing neural activity in the integrator network and @xmath7 represents the signal generated by the premotor command neurons .
the term @xmath8 corresponds to the natural decay of neural activity and @xmath9 represents a positive , synaptic feedback loop .
of course , when studying neural integration , questions can be investigated at varying levels of detail .
it is clear that a simple model as ( [ e : ni ] ) has several limitations .
because of its one - dimensional nature , the present model is , for example , unable of reproducing the distributed nature of persistent activity patterns observed in the brain .
nevertheless , eq . ( [ e : ni ] ) captures a key feature of neural integration : when the feedback is tuned to exactly balance the decay , eq .
( [ e : ni ] ) behaves as an integrator and produces persistent neural activity .
eq .
( [ e : ni ] ) is therefore a valuable model when studying fine - tuning of neural integrator networks @xcite .
we are interested in the fine - tuning of eq .
( [ e : ni ] ) and study this question using tools from nonlinear and adaptive control theory .
first , we ignore the presence of the input @xmath7 and consider the simpler equation @xmath10 we present a large class of feedback adaptation laws for ( [ e : pa ] ) which steer @xmath2 to its critical value @xmath4 ; thus enabling the automatic self - tuning of parameters and the spontaneous generation of persistent neural activity .
we consider adaptation laws could come from synaptic plasticity . in particular , the term @xmath11 might be related to types of synaptic plasticity that depend on the temporal ordering of presynaptic and postsynaptic spiking , as in @xcite .
] of the form @xmath12 we show that , under three very mild conditions , this adaptation rule guarantees convergence to the bifurcation point for ( [ e : pa ] ) .
the first condition requires that @xmath13 is a strictly increasing function .
this means that the term @xmath14 in ( [ e : adaptation ] ) acts as a negative feedback . as a consequence , if the neural activity @xmath5 were constant in ( [ e : adaptation ] ) , then the synaptic feedback gain @xmath2 would naturally relax to a rest value depending on @xmath5 via the equation @xmath15 .
the second condition states that there exists @xmath16 such that @xmath17 .
this condition implies that , if the neural activity would be constant and equal to @xmath16 in ( [ e : adaptation ] ) , then the synaptic feedback gain @xmath2 would naturally relax to its critical , desired value @xmath4 .
of course there is no guarantee that the neural activity would be equal to , or even converge to , this special value @xmath16 . instead
, the level of neural activity is governed by eq .
( [ e : pa ] ) . therefore , in order for the adaptation law ( [ e : adaptation ] ) to work , we need to impose a last condition , that @xmath18 is a decreasing function .
this means that the level of neural activity negatively regulates the synaptic feedback strength .
we now show that , under these three conditions , the feedback adaptation law ( [ e : adaptation ] ) indeed tunes the synaptic feedback gain @xmath2 to exactly balance the natural decay rate @xmath4 .
we begin with noticing that the combined system of equations ( [ e : pa])([e : adaptation ] ) has a unique rest point .
this equilibrium is determined by setting the right - hand sides of ( [ e : pa])([e : adaptation ] ) equal to zero , yielding @xmath19 and @xmath20 .
although the precise value of @xmath4 is unknown , if we are able to prove that all trajectories of ( [ e : pa])([e : adaptation ] ) converge to this ( unknown ) fixed point , then it follows that @xmath2 indeed converges to its desired , critical value @xmath4 . in order to prove this
, we introduce a coordinate transformation @xmath21 and @xmath22 .
this transforms ( [ e : pa])([e : adaptation ] ) into @xmath23 and @xmath24 . in these new coordinates ,
the dynamics take the form of a nonlinear mass - spring - damper system ( with unit mass , nonlinear spring characteristic @xmath25 and nonlinear damping function @xmath26 ) .
it follows from physical energy considerations that this system exhibits damped oscillations @xcite .
this shows that all trajectories of ( [ e : pa])([e : adaptation ] ) indeed converge to the unique fixed point , where @xmath20 .
the above coordinate transformation reveals a subtle relationship between self - tuning of bifurcations and the internal model principle ( `` imp '' ) from robust control theory ( see @xcite for a discussion of the imp from a systems biology perspective ) .
this relation is made explicit by the equation @xmath23 , which represents an integrator and corresponds to integral action studied in robust control theory .
one regards the constant @xmath4 as an unknown perturbation acting on the system .
the imp implies that , in order to track this constant perturbation , the system dynamics should contain integral action .
the integral action is generated by the biological system itself , and not by the feedback adaptation law .
we have so far ignored the presence of the signal @xmath7 .
we showed that the adaptation law ( [ e : adaptation ] ) tunes the synaptic feedback gain to exactly compensate for the natural decay rate , resulting in the spontaneous generation of persistent neural activity . at these equilibrium conditions ,
the action potential firing rate equals @xmath16 , which is related to @xmath4 by @xmath17 . in the next paragraphs
, we take into account the effect of the input @xmath7 . in this case
, the value @xmath16 will play the role of a parameter that influences the accuracy with which the feedback adaptation law guarantees proximity to the bifurcation point .
the signal @xmath7 will in general result in a time - varying action potential firing rate @xmath27 .
the mechanism with which this happens , is determined by the neural integrator equation ( [ e : ni ] ) and the adaptation law ( [ e : adaptation ] ) . for the purpose of analysis ,
we make two simplifying assumptions , both of which seem to be natural and physically relevant for neural integration .
first , we assume that , over any sufficiently large time interval @xmath28 $ ] , the time spent by @xmath27 in any interval @xmath29 $ ] is approximately independent of @xmath30 . in more mathematical terms , we assume the existence of a function @xmath31 such that for every test function @xmath32 , the time average @xmath33 converges to @xmath34 as @xmath35 , uniformly with respect to @xmath30 .
secondly , we assume that the adaptation law acts on a much slower time scale than the time variations in @xmath27 . under these assumptions ,
the effect of the action potential firing rate @xmath27 on the adaptation law ( [ e : adaptation ] ) may be approximated by the average effect @xmath36 .
it is now clear when the adaptation law guarantees proximity to the bifurcation point : if the compatibility condition @xmath37 is satisfied , then time scale separation arguments suggest that @xmath2 will converge approximately to @xmath4 and the neural integrator will approximately behave as a perfect integrator .
the compatibility condition may by interpreted as follows @xcite . when the premotor command signal @xmath7 has zero time - average and the adaptation law acts on a slow time scale , then eq .
( [ e : ni ] ) behaves as a good integrator and the firing rate @xmath27 equals the time - integral of @xmath7 plus an integration constant .
the compatibility condition ensures that this integration constant is compatible with the desired range for the firing rate @xmath27 .
we illustrate this result on a particular example representative for saccadic eye movements .
we consider the case of periodic saccadic eye movements asking for an action potential firing rate in the motor neurons alternating between @xmath38 and @xmath39 every second . at each saccade
, a brief burst of neural activity in premotor command neurons changes the actual firing rate .
we assume that this change is such that immediately after each saccade , the actual firing rate equals the desired firing rate . between saccades
, we assume that no input is applied . at its desired level between saccades ,
which is consistent with experimental observations . ]
if the neural integrator is perfectly tuned , then the actual firing rate will remain constant between saccades and equal to the desired firing rate ( eyes are fixed ) .
if the neural integrator is not perfectly tuned , then the actual firing rate will deviate from the desired firing rate ( eyes drift ) until a new saccade occurs which brings the actual firing rate to its new desired value .
fig .
[ f2 ] shows the results of a simulation where the adaptation law satisfies the compatibility condition of the previous paragraph . in the beginning of the simulation
, we have mis - tuned the neural integrator .
clearly , after a short transient , the adaptation law achieves excellent tuning and the drift between two successive saccades becomes negligible .
we have thus shown that an adaptation law can tune a neural integrator with great accuracy to its bifurcation point . in order to achieve perfect tuning
, however , the adaptation law itself needs to satisfy a compatibility condition .
it seems that we have merely moved the problem of fine - tuning from the neural integrator to the adaptation law . the crucial observation and one of the main contributions of the present paper , however , is that this results in a significant decrease in sensitivity . _
the adaptation law is robust with respect to perturbations in its parameters . _ in order to illustrate this significant increase in robustness , let us first summarize the well - known @xcite sensitivity properties of neural integration .
experiments suggest that the actual time constant obtained in a tuned neural integrator circuit is typically greater than @xmath40 ; that is , @xmath41 .
this requires for the fine - tuning of @xmath2 a relative precision @xmath42 ranging from @xmath43 to @xmath44 , depending on whether the intrinsic time constant @xmath45 equals @xmath46 or @xmath47 ( typical values suggested in the literature ) . the required precision for @xmath2
should be contrasted with the required precision for the parameters of the adaptation law proposed in the present paper .
the simulations of fig .
[ f3 ] show that , in order to have @xmath41 as observed in experiments , the parameters of the adaptation law need to be tuned with a precision of @xmath48 , independent of the intrinsic time constant @xmath45 . comparing this with the originally required precision for the synaptic feedback strength @xmath2
, we conclude that _ the proposed adaptation mechanism could improve the robustness of neural integration with a factor ranging from @xmath49 to @xmath50_. we have studied a simple model for neural integration and proposed a class of feedback adaptation rules that could explain the experimentally observed robustness of neural integration with respect to perturbations . the analysis tools that we have introduced extend to the study of fine - tuning involved in other systems such as hair cell oscillations in the ear @xcite .
consider the nonlinear oscillator equation @xmath51 , which captures some of the essential features of hair cell oscillations @xcite .
inspired by our previous analysis , we consider a feedback adaptation law for the parameter @xmath2 of the form @xmath52 , with @xmath53 a positive variable characterizing the magnitude of oscillations and related to @xmath5 and @xmath54 via the expression @xmath55 .
fig .
[ f4 ] shows that , in the absence of the stimulus @xmath7 , this type of adaptation law is indeed able to bring and keep the bifurcation parameter close to its critical value , resulting in the spontaneous generation of oscillations . |
when quantum particles interact by a short - range potential with a scattering length much larger than the potential range , they may form universal bound states whose properties are independent of microscopic physics @xcite . besides universal @xmath2-boson bound states in one dimension @xcite and in two dimensions @xcite , the most remarkable example is the efimov effect in three dimensions , which predicts the emergence of an infinite tower of three - boson bound states with orbital angular momentum @xmath3 whose binding energies obey the universal exponential scaling @xcite .
we also solved the two coupled integral equations ( [ eq : residue ] ) numerically with @xmath1 at mass ratios of @xmath4 , @xmath5 , and @xmath6 and observed that the obtained binding energies asymptotically approach the predicted doubly exponential scaling for each mass ratio .
see table [ tab : binding ] for the obtained binding energies at @xmath7 for two identical bosons corresponding to the upper sign in eq .
( [ eq : residue ] ) .
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the form of the effective - range expansion up to the second order in @xmath8 is not affected as long as the underlying potential @xmath9 vanishes faster than @xmath10 at @xmath11 . for the van der waals potential @xmath12 , the denominator of eq .
( [ eq : t - matrix ] ) suffers a logarithmic correction of @xmath13\,k^2\ln|k|$ ] , which , however , disappears right at the @xmath0-wave resonance @xmath14 and thus does not affect the conclusion of this article .
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a * 78 * , 063616 ( 2008 ) . | we study two species of particles in two dimensions interacting by isotropic short - range potentials with the interspecies potential fine - tuned to a @xmath0-wave resonance .
their universal low - energy physics can be extracted by analyzing a properly constructed low - energy effective field theory with the renormalization group method .
consequently , a three - body system consisting of two particles of one species and one of the other is shown to exhibit the super efimov effect , the emergence of an infinite tower of three - body bound states with orbital angular momentum @xmath1 whose binding energies obey a doubly exponential scaling , when the two particles are heavier than the other by a mass ratio greater than 4.03404 for identical bosons and 2.41421 for identical fermions . with increasing the mass ratio
, the super efimov spectrum becomes denser which would make its experimental observation easier .
we also point out that the born - oppenheimer approximation is incapable of reproducing the super efimov effect , the universal low - energy asymptotic scaling of the spectrum . |
physicists have long speculated that the fundamental constants might not , in fact , be constant , but instead might vary with time .
dirac was the first to suggest this possibility @xcite , and time variation of the fundamental constants has been investigated numerous times since then . among the various possibilities , the fine structure constant and the gravitational constant have received the greatest attention , but work has also been done , for example , on constants related to the weak and strong interactions , the electron - proton mass ratio , and several others .
it is well - known that only time variation of dimensionless fundamental constants has any physical meaning .
here we consider the time variation of a dimensionless constant not previously discussed in the literature : @xmath0 .
it is impossible to overstate the significance of this constant .
indeed , nearly every paper in astrophysics makes use of it .
( for a randomly - selected collection of such papers , see refs .
@xcite ) .
.the value of @xmath0 measured at the indicated location at the indicated time . [ cols=">,>,>,>,>",options="header " , ] in the next section , we discuss the observational evidence for the time variation of @xmath0 . in sec .
iii , we present a theoretical model , based on string theory , which produces such a time variation , and we show that this model leads naturally to an accelerated expansion for the universe . the oklo reactor is discussed in sec .
iv , and directions for future research are presented in sec . v.
the value of @xmath0 has been measured in various locations over the past 4000 years . in table 1
, we compile a list of representative historical measurements @xcite .
we see evidence for both spatial and time variation of @xmath0 .
we will leave the former for a later investigation , and concentrate on the latter . in fig . 1
, we provide a graph illustrating the time variation more clearly .
= 3.8truein = 3.8truein the values of @xmath1 show a systematic trend , varying monotonically with time and converging to the present - day measured value .
the evidence for time variation of @xmath0 is overwhelming .
inspired by string theory @xcite , we propose the following model for the time variation of @xmath0 .
consider the possibility that our observable universe is actually a 4-dimensional brane embedded in a 5-dimensional bulk . in this case , slices " of @xmath0 can leak into the higher dimension , resulting in a value of @xmath0 that decreases with time .
this leakage into a higher dimension results in a characteristic geometric distortion , illustrated in fig .
such leakage " has been observed previously in both automobile and bicycle tires .
however , it is clear that more controlled experiments are necessary to verify this effect .
it might appear that the observational data quoted in the previous section suggest a value of @xmath0 that increases with time , rather than decreasing as our model indicates .
since our theoretical model is clearly correct , this must be attributed to 4000 years of systematic errors .
now consider the cosmological consequences of this time variation in @xmath0 .
the friedmann equation gives @xcite @xmath2 where @xmath3 is the scale factor and @xmath4 is the total density . at late times
@xmath4 is dominated by matter , so that @xmath5 .
hence , if @xmath0 increases faster than @xmath3 , the result will be an accelerated expansion .
of course , our model gives the opposite sign for the time - variation of @xmath0 , but this is a minor glitch which is probably easy to fix .
this model for the time variation of @xmath0 has several other consequences .
it provides a model for the dark matter @xcite , and it can be used to derive a solution to the cosmological constant coincidence problem @xcite .
further , it can be developed into a quantum theory of gravity @xcite .
no discussion of the time - variation of fundamental constants would be complete without a mention of the oklo natural fission reactor .
this investigation clearly opens up an entirely new direction in the study of the time variation of fundamental constants . the next obvious possibility is the investigation of the time variation of @xmath6 . following this
, there is a plethora of other constants that could be examined : the euler - mascheroni constant @xmath7 , the golden ratio @xmath8 , soldner s constant , and catelan s constant .
more speculatively , one might consider the possibility that the values of the integers could vary with time , a result suggested by several early fortran simulations .
this possibility would have obvious implications for finance and accounting .
a number of colleagues were kind enough to comment on the manuscript .
for some reason they did not want me to use their names , so i will identify them by their initials : s. dodelson , a.l .
melott , d.n .
spergel , and t. j. weiler . | we examine the time variation of a previously - uninvestigated fundamental dimensionless constant .
constraints are placed on this time variation using historical measurements .
a model is presented for the time variation , and it is shown to lead to an accelerated expansion for the universe .
directions for future research are discussed . |
although interstellar extinction has been discussed in many papers and quantitatively determined by dedicated missions ( iue , 2mass and others ) , there is a lack of proper handling in the field of eclipsing binaries .
the usually adopted approach is to calculate the amount of reddening from the observed object s coordinates and its inferred distance and to subtract it uniformly , regardless of phase , from photometric observations .
this paper shows why this approach may be inadequate , especially for objects where interstellar extinction and the color difference between both components are significant .
atmospheric extinction is a better - posed problem : similarly as interstellar extinction depends on @xmath4 , atmospheric extinction depends on air - mass , which is a measurable quantity , whereas @xmath4 has to be estimated .
to estimate the effect of reddening on eclipsing binaries , we built a synthetic binary star model , consisting of two main sequence g9
f5 v stars with @xmath5 , @xmath6 and @xmath7 , @xmath8 and 1 day orbital period .
the simulation logic is as follows : for the given phase , we calculate the effective spectrum of the binary by convolving doppler - shifted individual spectra of the visible surfaces of both components . to this intrinsic spectrum
we rigorously apply interstellar and atmospheric extinctions ( both as functions of wavelength ) .
we then convolve this reddened spectrum with instrumental response function ( composed of the filter transmittivity and detector response functions ) and integrate over the bandpass wavelength range to obtain the flux .
in contrast , we use the same intrinsic spectrum without rigorously applying the reddening . to simulate the subtraction of a reddening
_ constant _ from photometric observations , we simply divide the intrinsic spectrum by the flux that corresponds to this constant .
finally , we calculate the flux in the same manner as before and compare it to the flux obtained by applying rigorous reddening . for building synthetic light curves we use phoebe ( pra & zwitter , 2004 ; in preparation ) .
each light curve consists of 300 points uniformly distributed over the whole orbital phase range . to be able to evaluate the impact of reddening on photometric light curves exclusively ,
all second - order effects ( limb darkening , gravity brightening , reflection effect ) have been turned off .
color index on the g9 v
v ( 5500 k6500 k ) temperature interval , calculated by integrating the spectrum over both filter bandpasses.,title="fig:",width=226,height=113 ] color index on the g9 v
f5 v ( 5500 k6500 k ) temperature interval , calculated by integrating the spectrum over both filter bandpasses.,title="fig:",width=226,height=113 ] + we take kurucz s synthetic spectra ( @xmath9 ) from precalculated tables by munari et al . (
2004 ; in preparation ) .
the used @xmath10 response data ( filter @xmath11 detector ) are taken from adps @xcite , where we apply a cubic spline fit to obtain the instrumental response function . for interstellar extinction
, we use the empirical formula ( fig .
[ redlaw ] ) , where @xmath12 was assumed throughout this study .
interstellar dust catalog was used to obtain the maximum color excess @xmath13 values for different lines of sight .
for atmospheric extinction we use the equation triplet for rayleigh - ozone - aerosol extrinction sources given by and summarized by .
the observatory altitude @xmath14 km and the zenith air - mass are assumed throughout the study . to rigorously deredden the observations for the given @xmath15 and @xmath13
, it is necessary to determine the reddening for each wavelength of the spectrum . correcting differentially and integrating over the filter bandpass then yields the dereddened flux of the given filter .
however , without spectral observations , it is difficult to calculate properly the flux correction .
since formula depends on the wavelength , the usually adopted approach found in literature is to use the effective wavelength @xmath16 of the filter transmittivity curve to calculate the reddening correction .
we demonstrate the implications in the following section .
) of the johnson b transmittivity curve .
right : overplotted light curves with the subtraction constant calculated so that the magnitudes in quarter phase are aligned .
there is still a _
measurable _ difference in eclipse depth of both light curves .
@xmath17 is assumed.,title="fig:",width=226,height=113 ] ) of the johnson b transmittivity curve .
right : overplotted light curves with the subtraction constant calculated so that the magnitudes in quarter phase are aligned .
there is still a _
measurable _ difference in eclipse depth of both light curves .
@xmath17 is assumed.,title="fig:",width=226,height=113 ] + is assumed.,title="fig:",width=226,height=113 ] is assumed.,title="fig:",width=226,height=113 ] + by comparing the rigorously calculated fluxes against intrinsic fluxes with a simple constant subtracted , we come to the following conclusions : * 1 ) * taking the effective wavelength of the filter bandpass should be avoided .
since the flux is the integral over the filter bandpass , @xmath16 has a _ conceptually _ different meaning .
furthermore , @xmath16 of the given filter depends heavily on the effective temperature of the observed object and on the color excess @xmath13 ( fig .
[ weff ] ) . to determine the subtraction constant
, one has to make sure that _ the integral _ ( rather than any particular wavelength ) of the both curves is the same .
[ discrepancy ] shows the discrepancy between the properly calculated light curve and the one obtained by subtracting a @xmath16-calculated constant .
table [ analysis ] summarizes the differences between the proper treatment and other approaches . *
2 ) * even if the subtraction constant is properly calculated , the light curves still exhibit measurable differences in both minima ( figs .
[ discrepancy ] , [ minima ] ) .
this is due to the effective temperature change of the binary system during eclipses . for the analysed case , the difference in b magnitude is @xmath2mag , which is generally observable .
* 3 ) * if light curves in three or more photometric filters are available , it is possible to _ uniquely _ determine the color excess value @xmath13 by comparing different color indices in - and - out of eclipse .
the reddening may thus be properly introduced to the fitting scheme of the eclipsing binary analysis program .
this was done in phoebe .
.the summary of different approaches to calculate the wavelength to be used for the dereddening constant .
@xmath16 is determined by requiring that the area under the spectrum on both sides is equal .
@xmath18 is the value of extinction in b filter and @xmath19 is the deviation from the rigorously calculated value .
all values are calculated for @xmath17 at quarter phase .
note that @xmath18 is smaller than @xmath20 , since our simulated binary is cooler than 10000k .
[ cols="<,^,^,>",options="header " , ] atmospheric extinction is comprised of three different sources : the rayleigh scattering , the aerosol scattering and ozone absorption @xcite .
it depends on the wavelength of the observed light and on the air - mass of obsevations , which is of course the same for both binary components ; we are thus left only with wavelength dependence at some inferred air - mass . to assess the impact on the photometric data , we compare the flux change imposed on the intrinsic spectrum by reddening and by atmospheric extinction ( fig .
[ atmosphere ] ) .
we conclude that for weakly and moderately reddened eclipsing binaries the atmospheric extinction dominates the blue parts of the spectrum due to rayleigh scattering , but for larger color excesses ( @xmath21 ) the reddening is dominant throughout the spectrum .
by not properly taking reddening into account , we introduce systematics of @xmath2mag into the solution for our simulated binary .
one may ask : is this difference worth bothering with ? to demonstrate that it is , let us simulate a bit more exotic eclipsing binary with blue b8 main - sequence ( @xmath22k , @xmath23 ) and red k4 type iii giant ( @xmath24k , @xmath25 ) components with orbital period of 15 days . fig .
[ minima ] ( right ) shows that the discrepancy between the rigorously calculated light curve and the constant - subtracted one is as large as @xmath3mag , which means @xmath26% error in the determined distance to the observed object . on the other hand ,
the reddening effect may be reduced by using narrower filter - sets , e.g. strmgren _ ubvy _ set .
furthermore , typical observed color excesses rarely exceed few tenths of the magnitude , which additionally diminishes this effect
. however , future space scanning missions such as gaia @xcite will acquire measurements of eclipsing binaries without spatial bias .
thus interstellar extinction should be carefully and rigorously introduced into the reduction pipeline for eclipsing binaries . | interstellar and atmospheric extinctions redden the observational photometric data and they should be handled rigorously .
this paper simulates the effect of reddening for the modest case of two main sequence @xmath0k and @xmath1k components of a detached eclipsing binary system .
it is shown that simply subtracting a constant from measured magnitudes ( the approach often used in the field of eclipsing binaries ) to account for reddening should be avoided .
simplified treatment of the reddening introduces systematics that reaches @xmath2mag for the simulated case , but can be as high as @xmath3mag for e.g. b8 v k4 iii systems . with rigorous treatment ,
it is possible to _ uniquely _ determine the color excess value @xmath4 from multi - color photometric light curves of eclipsing binaries
. andrej.prsa@fmf.uni-lj.si |
the susy n=1 kdv model is related to the following l - operator : @xmath1 where @xmath2 , @xmath3 , @xmath4 are the chevalley generators of twisted affine lie superalgebra @xmath5 , @xmath6 is a superderivative , the variable @xmath7 lies on a cylinder of circumference @xmath8 , @xmath9 is a grassmann variable , @xmath10 is a bosonic superfield with the following poisson brackets : @xmath11 . making a gauge transformation of the l - operator
we obtain a new superfield @xmath12 = @xmath13 , where @xmath14 and @xmath15 generate the superconformal algebra under the poisson brackets : @xmath16 the susy n=1 kdv system has an infinite number of conservation laws and the first nontrivial one gives the susy n=1 kdv equation : @xmath17 the integrals of motion are generated by the logarithm of the supertrace of the corresponding monodromy matrix , which has the following form : @xmath18 @xmath19 \big).\ ] ] its quantum generalization can be represented in the quantum p - exponential form ( for the explanation of this notion see below and @xcite for details ) : @xmath20 vertex operators @xmath21 are defined in the following way @xmath22 .
the universal r - matrix with the lower borel subalgebra represented by @xmath23 is equal to @xmath24 . due to this fact @xmath25 satisfies the rtt - relation : @xmath26 where @xmath27 , @xmath28 mean that the corresponding object is considered in some representation of @xmath29 .
thus the supertraces of the monodromy matrix ( `` transfer matrices '' ) @xmath30 commute , providing the quantum integrability .
it is very useful to consider the evaluation representations of @xmath29 , @xmath31 , where now the symbol @xmath27 means integer and half - integer numbers . denoting @xmath32 as @xmath33
we find that @xmath34 commute : @xmath35=0.\nonumber $ ] the expansion of @xmath36 in @xmath37 ( the transfer matrix in the fundamental 3-dimensional representation ) is believed to give us as coefficients the local i m , the quantum counterparts of the mentioned i m of susy @xmath38 kdv .
using the super q - oscillator representations of the upper borel subalgebra of the quantum affine superalgebra @xmath39 we define the @xmath40 operators ( see @xcite for details ) .
the transfer - matrices in different evaluation representations can be expressed in such a way : @xmath41 where @xmath27 runs over integer and half - integer nonnegative numbers .
@xmath40 operators satisfy quantum super - wronskian relation : @xmath42 one should note , that we use only @xmath43-dimensional `` @xmath44-induced '' representations ( sometimes called atypical ) of @xmath45 .
it allows , however , to construct the fusion relations , see below . to construct the relations like baxter s ones we introduce additional `` quarter''-operators , constructed `` by hands '' from the @xmath46-operators : @xmath47 for odd integer @xmath48 .
the baxter s relations are : @xmath49 @xmath50 the fusion relations have the following form very similar to the @xmath51 case : @xmath52 but they are only `` fusion - like '' because the `` quarter''-operators do not seem to correspond to any representation of @xmath45 . the truncation of these relations for different values of @xmath53 , being the root of unity : @xmath54 , @xmath55 , @xmath56 has the following form : @xmath57 in the case when @xmath58 , where @xmath59 , @xmath60 there exists an additional number of truncations : @xmath61 @xmath62 these relations allow us to rewrite the fusion relation system in the thermodynamic bethe ansatz equations of @xmath63 type .
in this paper we studied algebraic relations arising from the integrable structure of cft provided by the susy @xmath64=1 kdv hierarchy .
the construction of the @xmath46-operator as a `` transfer''-matrix corresponding to the infinite - dimensional q - oscillator representation could be also applied to the lattice models .
the relations like baxter s and fusion ones will be also valid in the lattice case because they depend only on the decomposition properties of the representations . + in the following we also plan to study the quantization of @xmath65 susy kdv hierarchies , related with super - w conformal / topological integrable field theories .
i am very grateful to my supervisor prof .
p. kulish .
it is a pleasure to thank prof .
m. semenov - tian - shansky and f. smirnov for useful discussions and prof .
l. baulieu , b. pioline and lpthe , univ .
paris 6 for support and hospitality .
this work was supported by dynasty foundation and crdf grant rum1 - 2622-st-04 .
kulish , a.m. zeitlin , zapiski nauchn .
seminarov pomi , 291 ( 2002 ) 185 ( in russian ) , english translation in : journal of mathematical sciences ( springer / kluwer ) * 125 * ( 2005 ) 203 ; hep - th/0312158 .
kulish , a.m. zeitlin , phys . lett .
* b 581 * ( 2004 ) 125 ; hep - th/0312159 ; p.p .
kulish , a.m. zeitlin , theor .
* 142 * , 2005 , in press ; hep - th/0501018 .
kulish , a.m. zeitlin , phys . lett . * b 597 * ( 2004 ) 229 ; hep - th/0407154 .
p. p. kulish , a.m. zeitlin , nucl . phys .
* b * , 2005 , in press ; hep - th/0501019 . | the quantum susy n=1 hierarchy based on @xmath0 twisted affine superalgebra is considered . the construction of the corresponding baxter s q - operators and fusion relations is outlined .
the relation with the superconformal field theory is discussed .
one of the most famous integrable systems ( is ) is the korteweg - de vries hierarchy .
it is related with the superconformal field theory because its poisson brackets give the virasoro algebra and the involutive family of integrals of motion ( i m ) providing the integrability of the conformal field theory ( cft ) . since the late 1980s
the supersymmetric and fermionic extensions of the kdv system have been known ( see e.g. @xcite , @xcite , @xcite and references therein ) , which in turn are related with superconformal field theory ( scft ) . during the following years they were extensively studied on both the classical and the quantum level .
+ however , up to the present nobody has applied the most successful method in the theory of integrable systems , the so - called quantum inverse scattering method ( qism ) to these is . in this short paper
we demonstrate some algebraic tools giving possibility to study susy n=1 kdv via qism . |
in spite of the considerable efforts to explain the experimental raman spectra of cuprate superconductors , the @xmath0 superconducting response is not yet completely understood . it has been shown that the theoretical description of the @xmath0 raman response was very sensitive to small changes in the raman vertex harmonic representations , yielding peak positions varying between @xmath1 and 2@xmath1 @xcite . however , the data show peaks consistently slightly above @xmath1 for both ybco and bscco . in this paper we present calculations suggesting that the @xmath0 peak position is largely controlled by a collective spin fluctuation ( sf ) mode near 41 mev , consistent with inelastic neutron scattering ( ins ) observations @xcite .
we show that the @xmath0 response is strongly modified by the sf term and is not sensitive to small changes in the raman vertex .
the experimental peak position is well reproduced by our model whereas the @xmath2 and @xmath3 response remain essentially unaffected by the sf mode .
the cuo@xmath4 bilayer is modeled by a tight binding band structure with a nearest ( @xmath5 ) and a next nearest neighbor hopping ( @xmath6 ) parameter and an inter - plane hopping given by @xcite @xmath7 ^ 2 .\ ] ] @xmath8 can be 0 or @xmath9 , for bonding or anti - bonding bands of the bilayer , respectively .
the spin susceptibility ( @xmath10 ) is modeled by extending the weak coupling form of a @xmath11 superconductor to include antiferromagnetic spin fluctuations by an rpa form with an effective interaction @xmath12 ; i.e. @xmath13 where @xmath14 is the simple bubble in the d - wave state .
this form of the spin susceptibility is motivated by the fact that it contains a strong magnetic resonance peak at @xmath15 which was proposed @xcite to explain the ins resonance at energies near 41 mev in ybco @xcite and bscco @xcite .
the raman response function in the superconducting state is evaluated using nambu green s functions .
the spin fluctuations contribute to the raman response via a 2-magnon process as shown in fig .
[ total ] @xcite where a schematic representation of the feynman diagrams of the sf and the bubble contribution is plotted . for the electronic propagators
we have used the bare bcs green s functions and a d - wave superconducting gap @xmath16/2 $ ] .
the total raman response is calculated in the gauge invariant form which results from taking into account the long wavelength fluctuations of the order parameter @xcite .
the total raman susceptibility is thus given by @xmath17 where @xmath18 is determined according to fig .
[ total ] . the analytical continuation to the real axis
is performed using pad approximants .
we have used several different forms for the raman vertex @xmath19 which possess the correct transformation properties required by symmetry .
our calculations show that the sf term yields vanishingly small corrections to the response in the @xmath2 and @xmath3 channels , but contributes substantially to the @xmath0 channel .
the shape of the total response in the @xmath0 geometry is mainly dependent on the value of the effective interaction @xmath12 .
variations of @xmath12 change the relative magnitude of the two diagrams summed in fig .
[ total ] , changing the position of the peak in @xmath0 geometry .
importantly , we find that the @xmath0 response shows little dependence on the form used for the vertex : @xmath20 , or the vertex calculated in an effective mass approximation .
these results can be explained by symmetry reasons given that the sf propagator is strongly peaked for @xmath21 momentum transfers .
we compare the calculated raman response with the experimental spectra of an optimally doped bi-2212 sample @xcite in fig .
[ fig22 ] . adding the sf contribution leads to a shift of the peak position from near @xmath22 for @xmath23 to higher frequencies , allowing a better agreement with the experimental relative positions of the peaks in @xmath0 and @xmath2 geometries . for the fit we have adjusted @xmath5 to achieve a good agreement with the @xmath2 channel , obtaining @xmath24 mev , and then adjusted @xmath12 to match both the @xmath0 peak position as well as the peak in the sf propagator to be consistent with the ins peak at 41 mev . from this work
we conclude that including the sf contribution in the raman response solves the previously unexplained sensitivity of the @xmath0 response to small changes in the raman vertex .
whereas the sf ( two - magnon ) contribution controls the @xmath0 peak , the @xmath2 and @xmath3 scattering geometries are essentially unaffected and determined by the bare bubble alone . 9 t.p .
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rev . lett . * 72 * , 3291 ( 1994 ) . | while the low frequency electronic raman response in the superconducting state of the cuprates can be largely understood in terms of a d - wave energy gap , a long standing problem has been an explanation for the spectra observed in @xmath0 polarization orientations .
we present calculations which suggest that the peak position of the observed @xmath0 spectra is due to a collective spin fluctuation mode . |
combining published data with high - quality vlt / fors spectroscopy of sample of fornax s0s ( bedregal et al .
2006a ) we have carried out a combined study of the tully - fisher relation and the stellar populations of these galaxies . despite the relatively small sample and the considerable technical challenges involved in determining the true rotation velocity @xmath1 from absorption line spectra of galaxies with significant non - rotational support ( see mathieu et al .
2002 ) , some very interesting results arise .
s0s lie systematically below the spiral galaxy tully - fisher relation in both the optical and near - infrared ( figure 1 ) . if s0s are the descendants of spiral galaxies , this offset can be naturally interpreted as arising from the luminosity evolution of spiral galaxies that have faded since ceasing star formation .
moreover , the amount of fading implied by the offset of individual s0s from the spiral relation seems to correlate with the luminosity - weighted age of their stellar population , particularly at their centres ( figure 2 ) .
this correlation suggests a scenario in which the star formation clock stopped when gas was stripped out from a spiral galaxy and it began to fade into an s0 .
the stronger correlation at small radii indicates a final last - gasp burst of star formation in this region .
see bedregal , aragn - salamanca & merrifield ( 2006b ) for details .
-band tully - fisher relation ( tfr ) for s0 galaxies using different samples from the literature ( open symbols ) and our vlt fornax data ( filled circles ) .
the solid and dashed lines show two independent determinations of the tfr relation for local spirals . on average (
dotted line ) , s0s are @xmath2 times fainter than spirals at similar rotation velocities ( bedregal , aragn - salamanca & merrifield 2006b ) . , width=384,height=316 ]
-band spiral tfr versus the stellar population age at the galaxy centre ( left panel ) , at @xmath3 ( middle panel ) and at @xmath4 ( right panel ) .
the lines show models for fading spirals .
note that the correlation is strongest for the central stellar populations of the galaxies , suggesting that the last episode of star formation took place there ( bedregal , aragn - salamanca & merrifield 2006b ) .
, height=158 ]
entirely consistent and independent evidence comes from our recent studies of the properties of the globular cluster ( gc ) systems and stellar populations of sos ( arag - salamanca , bedregal & merrifield 2006 ; barr et al .
if interactions with the intra - cluster medium are responsible for the transformation of spirals into s0s , the number of globular clusters in these galaxies will not be affected .
that is probably not true if more violent mechanisms such as galaxy - galaxy interactions are the culprit ( see , e.g. , ashman & zepf 1998 ) .
if we assume that the number of globular clusters remains constant , the gc specific frequency ( @xmath5number of gcs per unit @xmath6-band luminosity ) would increase due to the fading of the galaxy . on average ,
the gc specific frequency is a factor @xmath7 larger for s0s than it is for spirals ( aragn - salamanca et al .
2006 ) , meaning that in the process s0s become , on average , @xmath7 times fainter than their parent spiral .
furthermore , in this scenario the amount of fading ( or increase in gc specific frequency ) should grow with the time elapsed since the star formation ceased , i.e. , with the luminosity - weighted age of the s0 stellar population .
figure 3 shows that this is indeed the case , adding considerable weight to the conclusions reached from our tully - fisher studies .
in bedregal et al . ( 2007 ) we show that the central absorption - line indices in s0 galaxies correlate well with the central velocity dispersions in accordance with what previous studies found for elliptical galaxies . however , when these line indices are converted into stellar population properties , we find that the observed correlations seem to be driven by systematic age and alpha - element abundance variations , and not changes in overall metallicity as is usually assumed for ellipticals .
these correlations become even tighter when the maximum circular velocity is used instead of the central velocity dispersion .
this improvement in correlations is interesting because the maximum rotation velocity is a better proxy for the s0 s dynamical mass than its central velocity dispersion .
finally , the @xmath8-element over - abundance seems to be correlated with dynamical mass , while the absorption - line - derived ages also correlate with these over - abundances .
these correlations imply that the most massive s0s have the shortest star - formation timescales and the oldest stellar populations , suggesting that mass plays a large role in dictating the life histories of s0s .
the stellar populations , dynamics and globular clusters of s0s provide evidence consistent with these galaxies being the descendants of fading spirals whose star formation ceased .
however , caution is needed since significant problems could still exist with this picture ( see , e.g. , christlein & zabludoff 2004 ; boselli & gavazzi 2006 ) .
moreover , the number of galaxies studied here is still small , and it would be highly desirable to extend this kind of studies to much larger samples covering a broad range of galaxy masses and environments . of the luminosity - weighted ages is gyr vs. the globular cluster specific frequency ( @xmath9 ) of s0s .
the line shows the evolution expected for a fading galaxy according to the stellar population models of bruzual & charlot ( 2003 ) .
the correlation between the fading of the galaxies ( or increase in @xmath9 ) and the spectroscopically - determined age of their stellar populations is clearly consistent with the predictions of a simple fading model .
note that the @xmath9 value for ngc3115b is very unreliable and almost certainly severely overestimated due to contamination from the gc systems of neighbouring galaxies .
see barr et al .
( 2007 ) for details .
, height=278 ] | the stellar populations in the bulges of s0s , together with the galaxies dynamics , masses and globular clusters , contain very interesting clues about their formation .
i present here recent evidence suggesting that s0s are the descendants of fading spirals whose star formation ceased .
lenticular , or s0 , galaxies make up some 25% of large galaxies in the local universe ( dressler 1980 ) , so understanding how they form must constitute a significant element of any explanation of galaxy evolution .
their location at the crossroads between ellipticals and spirals in hubble s tuning - fork diagram underlines their importance in attempts to develop a unified understanding of galaxy evolution , but also means that it is not even clear to which of these classes of galaxy they are more closely related .
one often - cited piece of evidence comes from the fact that the proportion of s0s is substantially smaller in distant ( @xmath0 ) clusters than in nearby ones , while spirals show the opposite trend ( dressler et al .
1997 ) , strongly suggesting a transformation from one to the other . however , even if this scenario is accepted , it does not answer the question as to whether s0s are more closely related to spirals or ellipticals , which is intimately connected to the mechanism of transformation . if the transformation simply involves a spiral galaxy losing its gas content through ram pressure stripping ( gunn & gott 1972 ) or `` strangulation '' ( larson et al .
1980 ) , so ceasing star formation and fading into an s0 , then clearly s0s and spirals are closely related .
however , it is also possible that mergers can cause such a transformation : while equal - mass mergers between spirals create elliptical galaxies , more minor mergers can heat the original disk of a spiral and trigger a brief burst of star formation , using up the residual gas and leaving an s0 . in such a merger scenario , the mechanism for creating an s0 is much more closely related to that for the formation of ellipticals .
clues to which mechanism is responsible are to be found in the `` archaeological record '' that can be extracted from spectral observations of nearby s0s
. in particular , the present - day stellar dynamics should reflect the system s origins , with the gentle gas stripping of a spiral resulting in stellar dynamics very similar to the progenitor spiral , while the merger process will heat the stars , resulting in kinematics more dominated by random motions , akin to an elliptical .
in addition , the absorption line strengths can be interpreted through stellar population synthesis to learn about the metallicity and star formation histories of these systems .
even more interestingly , these dynamical and stellar properties can be compared to see if a consistent picture can be constructed for the formation of each system .
i present here some recent evidence suggesting that such a consistent picture is indeed emerging . |
the hilbert space , @xmath8 associated with a quantum system composed of three qubits is spanned by basis vectors @xmath9 where @xmath10 or @xmath11 and @xmath12 . here
@xmath13 is the dimension of hilbert space associated with @xmath14 qubit . to simplify the notation we denote the vector @xmath15 by @xmath16and write a general three qubit pure state as @xmath17to measure the overall entanglement of a subsystem @xmath18 we shall use twice the negativity as defined by vidal and werner @xcite , and call it global negativity @xmath19 .
it is an entanglement measure based on peres - hororedecki @xcite npt ( negative partial transpose ) sufficient criterion for classifying bipartite entanglement .
the global partial transpose of @xmath20 ( eq.([1 ] ) ) with respect to sub - system @xmath21 is defined as@xmath22the partial transpose @xmath23 of an entangled state is not positive . global negativity , defined as@xmath24is an entanglement monotone and measures the entanglement of subsystem @xmath21 with its complement in a bipartite split of the composite system .
global negativity vanishes on ppt - states and is equal to the entropy of entanglement on maximally entangled states .
the @xmath0way partial transpose of @xmath25 with respect to subsystem @xmath21 is obtained by transposing the indices of subsystem @xmath21 in those matrix elements , @xmath26 , that satisfy the condition @xmath27 , where @xmath28 to @xmath2 .
a typical matrix element of the three qubit state operator @xmath20 involves a change of state of @xmath4 subsystems , where @xmath28 to @xmath2 .
for example , a matrix element involving a change of state of two subsystems looks like @xmath29 ( @xmath30 ) .
the set of two distinguishable subsystems that change state while one of the sub - systems does not , can be chosen in three distinct ways .
in general , the number of spins that are flipped to get a vector @xmath31 from the vector @xmath32 is @xmath33 .
the operator @xmath20 can be split up into parts labelled by @xmath4 ( @xmath34 ) and written as @xmath35with@xmath36here @xmath37 .
the @xmath5way and @xmath6way partial transpose with respect to qubit @xmath21 are defined as@xmath38and@xmath39
the @xmath0way negativity calculated from @xmath0way partial transpose of matrix @xmath40 with respect to subsystem @xmath21 , is defined as @xmath41where @xmath42 is the trace norm of @xmath43 . using the definition of trace norm and the fact that @xmath44 , we get@xmath45 , @xmath46 being the negative eigenvalues of matrix @xmath47 . the negativity
, @xmath48 ( @xmath49 ) , depends on @xmath0 way coherences and is a measure of all possible types of entanglement attributed to @xmath0 way coherences .
intuitively , for a system to have pure @xmath3partite entanglement , it is necessary that @xmath3way coherences are non - zero . on the other hand ,
@xmath3partite entanglement can be generated by @xmath50 way coherences , as well .
for a three qubit system , maximally entangled tripartite ghz state is an example of pure tripartite entanglement involving @xmath6way coherences .
the global negativity @xmath51 , for maximally entangled three qubit ghz state .
maximally entangled w - state is a manifestation of tripartite entanglement due to @xmath5way coherences . for pure states as well as those mixed states for which the density matrix is positive , entanglement of a subsystem is completely determined by global negativity @xmath19 and the hierarchy of negativities @xmath52 (
@xmath53 calculated from @xmath47 associated with the @xmath54 sub - system . for three qubit system , @xmath55 , @xmath56 , and @xmath19 ( @xmath49 )
quantify the coherences present in the composite system .
a natural question is , how much of global negativity comes from @xmath5way transpose and how much has its origin in @xmath6way transpose , for a given qubit ?
the operator @xmath57 in its eigen basis is written as @xmath58where @xmath59and @xmath60 ( @xmath61and @xmath62 ) are the positive ( negative ) eigenvalues and eigenvectors , respectively .
as such the negativity of @xmath23 is given by @xmath63it is easily verified that @xmath64substituting eq .
( [ 13 ] ) in eq .
( [ 12 ] ) and recalling that @xmath65 is a positive operator with trace one , we get@xmath66where @xmath67 is the contribution of @xmath0way partial transpose to @xmath19 .
the set of states that can be transformed into each other by local unitary operations lie on the same orbit and have the same entanglement as the canonical state expressed in terms of the minimum number of independent vectors @xcite .
construction of pure three qubit canonical state has been given by acin et al @xcite .
it is easily verified that for the states reducible to the canonical state by local unitary operations , although @xmath19 is invariant under local operations , @xmath68 varies under local unitary operations .
for a canonical state , @xmath69 @xmath70 , is found to lie at a minimum with respect to local unitary rotations applied to any of the three qubits .
we conjecture that for a canonical state , @xmath71 is a measure of genuine @xmath6way entanglement of three qubit system .
bipartite entanglement of qubit one with qubit two equals the negativity @xmath72 of @xmath73 where the reduced operator , @xmath74 in case no w - like tripartite entanglement is present , the bipartite entanglement of a qubit is given by @xmath75 .
for a canonical state , the measure @xmath76 contains information about the w - like as well as pairwise entanglement of qubit @xmath77
three qubit greenberger - horne - zeilinger state @xmath78is a maximally entangled state having genuine tripartite entanglement . for this state @xmath79 , and @xmath80 , for @xmath49 . on the other hand
there exists a class of tripartite states akin to maximally entangled w - state given by @xmath81for the pure state @xmath82 , @xmath83 , for @xmath49 .
bipartite entanglement of qubit one and two in the state @xmath82 is measured by the negativity of partially transposed reduced density operator @xmath84 ( @xmath85 , which is found to be @xmath86 @xmath87 the total pairwise entanglement of a qubit in w - state is twice the value of @xmath88 ( = @xmath89 and is less than @xmath76(@xmath90 ) .
the residue accounts for the w - type tripartite entanglement of the system and generates @xmath91 .
the three qubit state @xmath92has no genuine tripartite entanglement and no @xmath6way coherences as such @xmath93 .
analogous to the case of state @xmath82 , we have @xmath94 and @xmath95 . fig . 1 displays @xmath96 @xmath97 and @xmath98 as a function of parameter @xmath99 .
to decipher the interplay of genuine tripartite entanglement and entanglement generated by @xmath5way coherences , we examine the coherences of single parameter pure states @xmath100and @xmath101for a given value of @xmath99 , the state may have bipartite , genuine tripartite as well as w - type entanglement as seen by @xmath102 and @xmath103 displayed in figs . ( 2 ) and ( 3 ) .
bipartite entanglement ( @xmath94 ) of a qubit in a state @xmath104 is obtained by calculating the negativity @xmath105 of partially transposed @xmath106 defined as @xmath107 .
we may remark that the states @xmath108 and @xmath109 share the same @xmath110 with @xmath105 going to zero at @xmath111
( 2 ) and ( 3 ) also display total bipartite entanglement of a qubit , @xmath112 , total tripartite entanglement @xmath113 and w - state like entanglement @xmath114 , of a qubit in the states @xmath115 and @xmath116 , respectively .
tripartite entanglement @xmath117 , resulting from the interference of @xmath118 and @xmath119 shows that @xmath5way coherence is partially annihilated by the @xmath6way coherence and vice versa .
wooters three tangle calculated from concurrence @xcite has been used to study tripartite entanglement of three qubit system in ref .
three tangle for states @xmath120 , calculated from @xmath121 and tripartite entanglement @xmath117 are plotted as a function of @xmath99 in fig .
the two measures show similar trend and match at @xmath122 , but in general @xmath6tangle overestimates the tripartite entanglement of the three qubit composite system in comparison with @xmath123 . in summary ,
we have extended the use of ppt criteria to the case of tripartite entanglement by splitting the partially transposed three qubit state operator into matrices that contain information either about bipartite or about genuine tripartite entanglement .
the set of @xmath5way , @xmath6way and global negativities make clear distinction between the coherences and nature of entanglement present in three qubit composite states . the three qubit canonical states may be classified using @xmath124 and @xmath125 ( @xmath126 to @xmath127 , @xmath128 ) as labels .
these entanglement measures are easy to calculate using standard diagonalization routines .
we believe that the relations between @xmath0way negativities and entanglement of a composite quantum system , a ) with sub - system dimensions higher than two , and b ) with more than three subsystems , can be figured out .
we expect our results to help advance the investigation of multipartite entanglement .
financial support from national council for scientific and technological development ( cnpq ) , brazil and state university of londrina , ( faep - uel ) , brazil is acknowledged . | in this letter we propose to quantify three qubit entanglement using global negativity along with @xmath0way negativities , where @xmath1 and @xmath2 . the principle underlying the definition of @xmath0way negativity for pure and mixed states of @xmath3subsystems is ppt sufficient condition .
however , @xmath0way partial transpose with respect to a subsystem is defined so as to shift the focus to @xmath0way coherences instead of @xmath4 subsystems of the composite system .
[ multiblock footnote omitted ] quantum entanglement is not only a fascinating aspect of multipartite quantum systems , but also a physical resource needed for quantum communication , quantum computation and information processing in general .
bipartite entanglement is well understood , however , many aspects of multipartite entanglement are still to be investigated
. peres @xcite and the horedecki @xcite have shown a positive partial transpose ( ppt ) of a bipartite density operator to be a sufficient criterion for classifying bipartite entanglement .
negativity zycz98,vida00 based on peres horodecski criterion has been shown to be an entanglement monotone @xcite .
negativity is a useful concept being related to the eigenvalues of partially transposed state operator and can be calculated easily . in this letter
, we define @xmath5way and @xmath6way negativities and propose a classification of three qubit states based on measures related to global , @xmath5way and @xmath6way negativities .
general definition of @xmath0way negativities for pure and mixed states of @xmath3subsystems is given in ref @xcite .
the @xmath0way partial transpose with respect to a subsystem is defined so as to shift the focus to @xmath0way coherences instead of @xmath4 subsystems of the composite system . while pure @xmath0 partite entanglement of a composite system is generated by @xmath0way coherences , @xmath0partite entanglement can ,
in general , be present due to @xmath7 way coherences as well . |
we use the resonance lines of h - like o and of h - like and he - like ne to estimate the ne / o abundance ratio . in hot ( @xmath10-@xmath11 k ) coronal plasma these lines
are formed predominantly by radiative de - excitation of levels excited by collisions with thermal electrons .
the flux , @xmath12 , from such a transition @xmath13 in an ion of an element with abundance @xmath14 can be written as @xmath15 \;dt % \overline{n_e^2}(t)\ , \frac{dv(t)}{dt } \;dt \,\,\ , \mbox{erg cm$^{-2}$ s$^{-1}$ } \label{e : flux}\ ] ] where @xmath16 describes the line _ emissivity_the product of the relative population of the ion in question and the excitation rate of the transition as a function temperature , @xmath17 .
the kernel @xmath18the emission measure distribution describes the excitation power of the plasma as a function of temperature , which is proportional to the mean of the square of the electron density , @xmath19 , and the emitting volume @xmath20 , @xmath21 .
if we can choose o and ne lines whose @xmath16 functions have very similar temperature dependence , an abundance ratio by number , @xmath22 , can be derived simply from the ratio of their observed line fluxes , @xmath23 and @xmath24 , since all the temperature - dependent terms in equation [ e : flux ] cancel : @xmath25 an early study of ne / o ratios in solar active regions@xcite used the ratio of ne ix @xmath26 to o viii @xmath27 .
this ratio does , however , have some significant residual dependence on temperature.@xcite here we remove much of this temperature dependence by addition of ne x @xmath28 ; our combined ne @xmath16 function is @xmath29 . the resulting ratio @xmath30 is illustrated as a function of temperature in figure [ f : emissrat ] .
we have verified the small residual temperature sensitivity evident in the lower panel of figure [ f : emissrat ] to be negligible for our analysis by integrating the products of @xmath31 and @xmath32 with empirically - derived emission measure distributions , @xmath18 , for different stars,@xcite and for functions @xmath33 , with @xmath34 : the integrated emissivity ratio from these tests was @xmath35 .
we conclude that the line ratio method is robust and the higher ne / o abundance ratio found here will not be significantly changed through performing full emission measure distribution modelling .
we measured ne and o line fluxes ( listed in table 1 ) from _ chandra _ hetg x - ray spectra obtained directly from the chandra public data archive ( http://cda.harvard.edu ) .
final listed fluxes for ne x include small reductions ( @xmath36% for 17 out of 21 or our stars , and 25 - 37% for the remainder ) to account for a weak blend of fe xvii at 12.12 .
the fe xvii 12.12 contribution was estimated by scaling the observed strengths of unblended fe xvii lines at 15.26 , 16.77 , 17.05 and 17.09 ( the strong 15.01 resonance line was omitted to avoid potential problems with its depletion through resonance scattering ) by their theoretical line strengths relative to the 12.12 line as predicted by the chianti database .
minor blending in the wings of the ne ix 13.447 line was accounted for by fitting simultaneously with the neighbouring weaker lines , comprised of a fe xix - xxi blend at 13.424 and fe xix 13.465 , following a detailed study of these features in the capella binary system.@xcite since these blend corrections are generally very small , the uncertainties in these procedures have negligible ( @xmath37% ) influence on the derived ne / o abundance ratios .
jjd was supported by a nasa contract to the _ chandra x - ray center_. pt was supported by a chandra award issued by chandra x - ray center , which is operated by sao for and on behalf of nasa .
jjd thanks the nasa aisrp for providing financial assistance for the development of the pintofale package .
we thank drs .
g. share , r. murphy , w. ball and d.garcia-alvarez for useful discussions and comments .
.spectral line fluxes and derived ne / o abundance ratios for the stars analysed in this study .
line fluxes were measured from the medium energy grating ( meg ) component of _ chandra _ hetg spectra by line profile fitting using the package for interactive analysis of line emission ( pintofale ) software@xcite ( freely available from http : hea - www.harvard.edu / pintofale ) .
the effective collecting area of the instrument was accounted for using standard _
chandra _ calibration products and techniques ( see http://cxc.harvard.edu/ciao/ for details ) .
ne / o abundance ratios were obtained assuming the o / ne line emissivity ratio of @xmath38 , as described in methods . stated flux and
abundance ratio uncertainties correspond to @xmath39 limits .
[ cols="<,^,^,^,^,^,^ , < " , ] , vs. the coronal activity index @xmath1 .
error bars represent quadrature addition of @xmath39 uncertainties of line flux measurement .
also shown using hollow symbols are literature values@xcite for the stars procyon ( f5 iv ) and @xmath2 eri ( k2 v ) observed using the _ chandra _ low energy transmission grating spectrometer ( letgs ) to better represent the lower ranges of coronal activity .
the error - weighted mean ne / o abundance ratio is @xmath41 , or 2.7 times the currently assessed value@xcite which is illustrated by the dashed horizontal line .
the recommended value from comprehensive earlier assessments in common usage@xcite are also illustrated.,scaledwidth=100.0% ] , of the o viii @xmath28 line , and @xmath32 of the ne ix @xmath42 and ne x @xmath28 lines combined as @xmath29 .
the lower panel shows the logarithmic ratio @xmath30 .
emissivities are based on electron excitation rates and ion populations@xcite compiled in the chianti database,@xcite as implemented in pintofale.@xcite , scaledwidth=80.0% ] | the interior structure of the sun can be studied with great accuracy using observations of its oscillations , similar to seismology of the earth .
precise agreement between helioseismological measurements and predictions of theoretical solar models@xcite has been a triumph of modern astrophysics . however , a recent downward revision by 25 - 35% of the solar abundances of light elements such as c , n , o and ne@xcite has broken this accordance :
models adopting the new abundances incorrectly predict the depth of the convection zone , the depth profiles of sound speed and density , and the helium abundance.@xcite the discrepancies are far beyond the uncertainties in either the data or the model predictions.@xcite here we report on neon abundances relative to oxygen measured in a sample of nearby solar - like stars from their x - ray spectra .
they are all very similar and substantially larger than the recently revised solar value .
the neon abundance in the sun is quite poorly determined .
if the ne / o abundance in these stars is adopted for the sun the models are brought back into agreement with helioseismology measurements.@xcite harvard - smithsonian center for astrophysics , 60 garden street , cambridge ma 02138 , usa .
mit kavli institute for astrophysics and space research , massachusetts institute for technology , 70 vassar street , cambridge , ma 02139 , usa .
the role of the sun as a fundamental benchmark of stellar evolution theory , which itself underpins much of astrophysics , renders the `` solar model problem '' one of some importance .
the schism between helioseismology and models with a revised composition has arisen because abundant elements such as c , n , o and ne provide major contributions to the opacity of the solar interior , which in turn influences internal structure and the depth at which the interior becomes convective .
uncertainties in the calculated opacities themselves appear insufficient to bridge the gap,@xcite and the propriety of the recent abundance revisions has therefore been questioned .
the revised abundances are demanded by new analyses of the visible solar spectrum that take into account convection and associated velocity fields and temperature inhomogeneities through 3-d hydrodynamic modelling , and that relax the assumption of local thermodynamic equilibrium for computing atomic level populations.@xcite the solar abundances of c , n and o can be measured accurately based on their absorption lines .
however , ne lacks detectable photospheric lines in cool stars like the sun , and the ne abundance is therefore much less certain .
the solar ne content is assessed based on observations of neon ions in nebular and hot star spectra , and on measurements of solar energetic particles.@xcite measurements are generally made relative to a reference element such as o ; the downward revision of the solar o abundance therefore required a commensurate lowering of the ne abundance for consistency.@xcite expressed as the ratio of the number of ne atoms in the gas to those of o , this abundance is @xmath0 , which is very similar to values adopted in earlier studies.@xcite however , it has recently been pointed out that the solar model problem could be solved were the true solar ne abundance to be at least a factor of 2.5 times higher than recently assessed.@xcite we are motivated by the solar model problem to investigate ne abundances in nearby stars . while not detected in the optical spectra of cool stars , emission lines of highly ionised ne are prominent features of their x - ray spectra.@xcite the ne / o abundance ratio can be derived directly from the ratio of observed fluxes of the hydrogen - like and helium - like ions of o and ne ( see methods for details ) ; we adopt a slightly refined version of the method applied to the analysis of earlier solar x - ray spectra.@xcite in this way , we have obtained the ne / o abundance ratios for a sample of 21 stars lying within 100pc of the sun that have been observed by the _
chandra _ x - ray observatory using the high energy transmission grating spectrometer.@xcite a representative x - ray spectrum , that of the m1 v star au mic , is presented in figure [ f : megspec ] .
the stars studied , together with observed ne and o line fluxes and derived ne / o abundance ratios are listed in table 1 .
the ne / o abundance ratios are illustrated as a function of the ratio of logarithmic x - ray and bolometric luminosities , @xmath1a commonly used index of stellar coronal activity in figure [ f : abunrats ] . since our star sample contains more objects toward higher @xmath1 , we have added ne / o ratios for two stars of somewhat lower activity level ( procyon , an f5 subgiant , and @xmath2 eri , a k2 dwarf ) from the literature.@xcite there is no trend in the ne / o abundance ratio with @xmath1 , and the error - weighted mean ratio is @xmath3 .
our derived ne / o abundance ratio is 2.7 times higher than the currently recommended solar value@xcite but is consistent with the abundance inferred from helioseismology.@xcite solving the solar model problem by raising the ne abundance alone would require a minimum ratio @xmath4 , or 3.44 times larger than recommended.@xcite however , raising the c , n , o and fe abundances upward within their estimated uncertainty range of @xmath5% ( adjusting them all together is not unreasonable because the recent downward revisions are correlated ) would require an ne abundance higher by only a factor of 2.5@xcite quite within our estimated range . extensive review articles on the coronae of the sun and stars document evidence that coronal chemical compositions often appear different to those thought to characterise the underlying photospheres.@xcite
the differences appear to relate to element first ionisation potentials ( fips ) : the solar corona appears enhanced in elements with low fip ( @xmath6 ev ; e.g. , mg , si fe),@xcite whereas much more coronally active stars appear to have depletions in low fip elements and perhaps enrichments of high fip elements ( @xmath6 ev ; e.g. , o , ne , ar).@xcite at first sight , this chemical fractionation might render direct interpretation of coronal abundance ratios in terms of the composition of the underlying star problematic .
however , that we are seeing the same ne / o ratio in a wide variety of stars sampling a large range of different coronal activity levels indicates that there is no significant fractionation between o and ne in disk - integrated light from stellar coronae .
this leads us to conclude that the results represent the true ne / o abundance ratios of these stars .
this conclusion is bolstered by findings of a constant ne / o abundance ratio , in good agreement with our value , for a small sample of single and active binary stars observed by _ xmm - newton_.@xcite in view of the consistency of the ne / o abundance ratio in nearby stars , it seems likely that the solar ratio should be similar .
there are no recent full - disk integrated light ne / o measurements for the sun ; existing studies are instead based on observations of particular regions and structures of the solar outer atmosphere . we provide a tabular summary and bibliography of some of the different ne / o estimates since 1974 as supplementary information to this letter .
while the preponderance of estimates appear consistent with current assessments , the underlying observations do not sample photospheric material .
the ne / o ratio is in fact observed to differ substantially between the different observations . in particular , the highest measured ne / o ratios based on x - ray lines are 2 - 3 times the accepted solar value and are compatible with the abundance ratio we find for nearby stars .
these higher ne / o ratios were obtained for hotter active regions@xcite that are likely to dominate the solar full - disk x - ray emission .
these measurements are the most directly compatible with the ones presented here based on full - disk integrated light x - ray spectra of stars .
similarly `` high '' ne / o ratios have also been seen in @xmath7-ray observations of flares , @xmath8he - rich solar energetic particle events , and in the decay phase of long duration soft x - ray events.@xcite the @xmath7-ray measurements probe material that is irradiated by downward - flowing accelerated particles.@xcite the particle beams penetrate through to the chromosphere which is likely more representative of the underlying photospheric material than coronal regions .
recent sophisticated models of heliospheric pickup ion and anomalous cosmic ray populations from the local interstellar medium are also inconsistent with the current solar ne / o abundance ratio , but could be reconciled by raising this to @xmath9,@xcite which is similar to our values for nearby stars .
this ne / o ratio is somewhat higher than the mean from studies of ionized nebulae in the milky way and other galaxies , but falls within the scatter of results at solar metallicity.@xcite the implication of our study , then , is that the higher of the observed solar ne / o abundance ratios are the ones representative of the underlying solar composition .
this scenario is in accordance with our observations of ne / o in nearby stars and reconciles solar models with helioseismology . |
the catalog and atlas of cataclysmic variables ( edition 1 - @xcite and edition 2 - @xcite ) has been a valuable source of information for the cataclysmic variable ( cv ) community .
one of the goals of the catalog was to have the basic information on the objects ( i.e. coordinates , type , magnitude range , and finding charts ) in one central location , thus making it easy for observers to obtain data on the objects .
however , the impracticality of reprinting the finding charts in their entirety means that , with each new edition , they are spread among more publications , taking us further from our goal of a central location .
furthermore , as new objects are discovered , and known ones examined in greater detail , the printed editions can not keep pace with discovery , a `` living '' edition is therefore highly desirable , so that observers can access a complete and current list of cvs at any time . for the above reasons , as well as
the need to simplify the tracking of the objects ( there are over 1200 objects in the catalog ) , we have decided to generate a web - based version of the catalog .
this version will have all the information ( as well some additional information detailed below ) from the first two editions , plus information on over 150 new objects discovered since 1996 may .
those objects with revised finding charts will only have one chart presented , thus eliminating a possible confusion which necessarily exists when `` paper '' catalogs are generated .
the web site will also allow for easy searching of the catalog , and for generation of basic statistics ( e.g. how many dwarf novae , how many cvs have _ hubble space telescope _
data , etc . ) .
the catalog consists of ( as of 2000 december ) 1034 cvs , and another 194 objects that are non - cvs ( objects originally classified erroneously as cvs ) .
most of the objects are dwarf novae ( 40% ) , with another 30% being novae , and the rest mostly novalike variables . a large fraction ( 90% ) of the cvs have references to published finding charts , while 64% of the objects have published spectra ( 49% quiescent spectra and 15% outburst spectra ) .
we have taken this opportunity to make several enhancements to the catalog . in conjunction with hans ritter and ulrich kolb
, we have added orbital period data to the catalog ; about one - third of the objects have periods .
the period information is from @xcite , plus updated and additional values . in conjunction with hilmar duerbeck @xcite , we now include finding charts of novae ( when possible ) , and have measured coordinates for many in the _ hubble space telescope _ gsc v1.1 guide star reference frame ( as is the case for the non - novae ) . finally , in the first edition we introduced ( out of necessity ) a pseudo - gcvs name for certain objects ( e.g. phe1 ) , which was continued in the second edition . with the web - based catalog ,
these names are no longer needed , so we will cease generating new ones .
for those objects that already had such names ( some of which have appeared in subsequent papers in the literature ) and now have a formal gcvs designation , we will adopt the formal gcvs name , although we will keep the pseudo - gcvs name in the `` other name '' field for continuity .
the site can be reached via : http://icarus.stsci.edu/@xmath0downes/cvcat/ and is described in detail below . the home page ( figure [ fig1 ] ) for the catalog contains six links : * * search * - a link to the search page , from which the catalog may be accessed . * * description * - a description of the catalog , following the format of the previous editions . a description of all the fields is given . * * references * - a complete listing of the references mentioned in the catalog .
note that from each individual object page , you can go directly to the reference of interest . * * statistics * - a listing of a fixed set of basic statistics from the catalog , generated in real - time . *
* ascii report * - a listing of the entire catalog in the format of the previously published versions ( i.e. containing most but not all of the fields ) , sorted by right ascension
. this output can be down - loaded to an ascii file . *
* change log * - a listing , by object , of the changes made since the initial release of this edition the search page ( figure [ fig2 ] ) is the main page for access to the catalog .
it allows the user to search the catalog on any field or combination of fields .
the following text fields can be searched in a case - insensitive manner : gcvs name , other name , and the five reference fields ( coordinate , chart , type , spectrum , and period ) ; the object type and notes fields can be searched in a case - sensitive manner .
all textual searches support the use of wildcards .
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numerical searches ( supporting a `` @xmath1 '' and `` @xmath2 '' capability ) can be performed for the following fields : galactic latitude , minimum and maximum magnitude , outburst year ( for novae ) , and period . finally , a search for space - based observations using any of 10 observatories can be performed .
an on - line help file is available detailing the search capabilities for each field , as well as providing instructions for the use of wildcards .
after a search is initiated , the search results page ( figure [ fig3 ] ) presents the results of the search .
this page indicates the number of objects in the catalog that match the selection criteria , and presents an abbreviated view of the catalog entries for such entries , showing the basic information such as the coordinates , type , magnitude range , and period . to obtain the full information ( including the finding chart ) , one clicks on the object of interest .
the individual object page ( figure [ fig4 ] ) presents the complete information on the selected object . for the finding charts , the field size , source ( dss ( digitized sky survey ) , _ hubble space telescope _
data , ground - based image ) , filter / emulsion , and exposure time are given to allow the user to estimate the depth of the image .
for most objects , the dss image is used .
however , for particularly crowded fields ( such as in globular clusters ) , _ hubble space telescope _
data is used when available .
similarly , for particularly faint targets , ground - based ccd images are provided when possible . on this page
, one may click on any of the reference codes to go directly to the full reference on the references page .
we plan to update the site with new objects and information on a continual basis , although period and spaced - based updates will occur roughly every six months .
we encourage users to inform us of any updates that should be implemented ( e.g. revised identifications , new objects , etc . ) , and if appropriate to send us improved / original finding charts ( as either postscript or jpeg images ) .
the charts will be particularly useful for recent novae recovered in quiescence , and for the faintest objects where deep ccd imaging clearly reveals the correct identification .
we wish to thank anne gonnella , steve hulbert , calvin tullos , and mike wiggs for the excellent work in creating the site .
we also wish to thank matt mcmaster for assistance in generating the multitude of finding charts , and the director s discretionary research fund at stsci for financial support .
paula szkody , john thorstensen , and steve howell provided helpful comments on the initial version of the site .
rfw gratefully acknowledges the support of nsf grant ast-9618462 , and sabbatical support from stsci .
hwd acknowledges the hospitality and support of stsci . | the catalog and atlas of cataclysmic variables ( edition 1 - @xcite and edition 2 - @xcite ) has been a valuable source of information for the cataclysmic variable ( cv ) community . however , the goal of having a central location for all objects is slowly being lost as each new edition is generated .
there can also be a long time delay between new information becoming available on an object and its publication in the catalog . to eliminate these concerns , as well as to make the catalog more accessible , we have created a web site which will contain a `` living '' edition of the catalog .
we have also added orbital period information , as well as finding charts for novae , to the catalog . |
in this section we derive the flat band projected hamiltonian , eq .
( 2 ) , by adding interactions to eq . ( 1 ) .
we first use the tight - binding limit to show that flat bands arise from spin - orbit coupled fermions in optical lattices in the absence of interactions . using the tight - binding limit
we transform the interaction to an on - site hubbard interaction .
we then project the interaction into the flat band to derive eq .
we conclude by showing the limits in which the flat band approximation is accurate .
we start by noting that eq .
( 1 ) yields bloch bands that can be accurately fit with a tight - binding model ( for @xmath113 ) : @xmath114 where the nn hopping @xmath24 is tuned to fit the exact band width of the first bloch band of eq .
( 1 ) with @xmath115 and @xmath116 .
@xmath24 is therefore a single - particle hopping derived entirely from optical lattice parameters . here
@xmath117 creates a fermion spinor .
we use eq .
( [ tightbindh ] ) to derive the flat band projected hamiltonian .
( [ tightbindh ] ) can be solved exactly .
the two lowest eigenvalues are given by : @xmath118 fig .
1 plots the lowest bands .
we have tuned the spin - orbit coupling strength @xmath119 and the zeeman field strength @xmath9 to generate flat bands . for both lattice depths
we chose the same band flatness ratio @xcite , @xmath120 , where @xmath105 is the band width and @xmath121 is the separation between the two lowest bands . the unitary matrix that diagonalizes eq .
( [ tightbindh ] ) is : @xmath122 where @xmath123 and @xmath22 .
the eigenstates of eq .
( [ tightbindh ] ) can then be written in terms of the original fermi operators : @xmath124 where @xmath35 and @xmath125 denote operators for lower and upper bloch bands , respectively ( fig . 1 ) .
we can now use the unitary transformation to study the addition of interactions to the non - interacting model within the lowest flat band .
occupancy of just the lowest flat band implies that eq .
( [ tightbindh ] ) acts as an irrelevant constant , to a first approximation .
the relevant term in the model derives from interactions between atoms . in the tight - binding limit
, @xmath4-wave contact interactions thus yield an on - site projected hubbard interaction : @xmath126 where @xmath16 is the on - site repulsion , @xmath127 projects particles into the lowest flat band , @xmath128 creates a fermion at lattice site @xmath129 in the @xmath130 spin state , and the constant is the zero - point energy of the flat band . by using eq .
( [ utransform ] ) in combination with a fourier transform , eq .
( [ hinteraction ] ) becomes : @xmath131 where @xmath132 is defined in the main text .
this shows that eq .
( 2 ) , follows from the on - site interactions operating in a flat spin - orbit band .
we now argue that all fermions occupy the lowest band in realistic parameter regimes .
this is a valid assumption provided the system only partially fills the lowest band and @xmath121 is greater than inter - band interaction matrix elements .
the @xmath4-wave interaction can be tuned far from a feshbach resonance to ensure that @xmath121 is larger than characteristic interaction energies .
this maintains the flat band condition provided @xmath133 .
we note that , in a perfectly flat band @xmath134 , the problem remains strongly correlated even for small @xmath16 because the interaction becomes the only term in the hamiltonian .
the main text shows that @xmath106 is satisfied for @xmath100k . in this section
we show that application of a small staggered chemical potential spontaneously selects one of the degenerate wigner crystal states .
the lowest energies from our exact diagonalization study of @xmath48 [ eq .
( 2 ) ] are 2-fold degenerate at total momentum @xmath54 .
each state is uniform in the absence of symmetry breaking .
if the ground state is truly uniform , a small perturbation ( much smaller than the gap ) should leave the ground state density intact .
if , however , the ground state prefers to spontaneously select an inhomogeneous configuration , a small non - uniform perturbation should drive the system into one of the crystal configurations . in our case ( eqs . 2 or 3 )
a logical choice for degenerate crystal configurations has one particle at every other site in real space .
the crystals are defined as linear combinations of the two lowest energy states at @xmath54 .
to identify crystalline order we perturb the ground state with a small staggered chemical potential term @xmath55 to check if the ground state tends to spontaneously choose one of the two degenerate wigner crystal states .
( this is similar to methods employed in the numerical determination of staggered magnetization in the antiferromagnetic ising model , where a small staggered magnetic field is also applied to pick one of the two staggered magnetization directions . )
care needs to be taken in the limiting procedure . for a fixed small positive @xmath135
, one needs to extrapolate first the lattice size , i.e. , @xmath136 .
the ground state selected as @xmath137 then denotes one of the two broken - symmetry wigner crystal states .
the other wigner crystal state can be detected in the limit @xmath138 .
this shows that spontaneous breaking of the discrete sublattice symmetry inherent in wigner crystals can be detected with the application of a small staggered chemical potential . in this section
we prove the formulas for the emergent luttinger parameters @xmath87 and @xmath88 in the main text , eqs .
( 6 ) and ( 7 ) . to do this
we apply luttinger liquid theory to the effective model in the main text .
we first bosonize the non - interacting part of the model and then the interacting part .
the effective model [ eq . ( 4 ) ] is : @xmath139 where the non - interacting part can be rewritten in terms of two - component vectors in @xmath18-space : @xmath140 here the prime indicates summation over @xmath141 , @xmath142 is the identity matrix , @xmath143 is the @xmath144-component of the pauli matrix , and @xmath145 .
we diagonalize @xmath146 with a unitary transformation , defined on a restricted momentum range @xmath141 : @xmath147 here @xmath148 and @xmath149 label the two bands established by the sublattice dependent hopping in the effective model .
the non - interacting hamiltonian then becomes : @xmath150 where @xmath151 is defined in the main text [ eq .
( 5 ) ] . to begin the bosonization process
, we pass to the continuum limit and expand the field operators into left and right moving fermion fields around the two fermi points : @xmath152 where we have taken into account the opposite slopes in linearizing the effective single - particle energy dispersions of two bands ( see fig . 3 ) .
we can use the left / right ( l / r ) decomposition to bosonize eq .
( [ hochi ] ) using the usual bosonization methods .
we define four bosonic fields , @xmath153 with @xmath154 and @xmath155 , such that : @xmath156 where `` @xmath157 '' indicates normal ordering , @xmath158 , and @xmath159 corresponds to @xmath160 . with these transformations , eq .
( [ noninteractionh ] ) becomes : @xmath161 this form for @xmath146 explicitly reveals the four - component nature of the excitations near the fermi points : two bands ( @xmath154 ) and two directions of motion ( @xmath162 ) .
we now apply the bosonization procedure to the interacting term in eq .
( [ hexthubeq ] ) , @xmath86 .
we first fourier transform the interaction , apply the unitary transformation using eq .
( [ chi_transform ] ) , and we finally substitute the fourier transform for @xmath163 into @xmath86 . this gives rise to 256 terms .
most of these terms cancel to yield : @xmath164 where we have made use of the fact that even and odd sites correspond to bands 1 and 2 , respectively .
we are now able to bosonize the interaction term using the same transformations as those used above .
we first note that , since we are working at half filling ( @xmath83 ) , umklapp terms will vanish .
we further define two conjugate fields for the two bands : @xmath165 the total bosonized hamiltonian can then be written in terms of the conjugate fields : @xmath166\end{aligned}\ ] ] where @xmath167 and @xmath168 .
the two matrices @xmath169 and @xmath170 are given by : @xmath171 to find the emergent luttinger parameters we diagonalize the matrices defining the above hamiltonian with the following transformation @xcite : @xmath172 where the transformation matrices satisfy @xmath173^t=[t_{\theta}]^{-1}$ ] .
this guarantees that the new conjugate fields that hybridize the bands into normal modes , @xmath174 and @xmath175 , satisfy the canonical commutation relations .
it can be checked that the above condition is fulfilled by the following choice of transformation matrices : @xmath176 where the unitary matrix @xmath177 diagonalizes the rescaled matrix @xmath178 , i.e. , @xmath179 and the diagonal rescaling matrices @xmath180 and @xmath181 are given by : @xmath182 here @xmath183 and @xmath184 are defined in the main text .
the total hamiltonian defining the effective luttinger liquid theory can then be written in terms of the diagonalized hamiltonian for each of the normal modes
@xmath185 : @xmath186,\ ] ] where the emergent luttinger parameters @xmath187 and @xmath188 are given in the main text , eqs .
( 6 ) and ( 7 ) . | recent ultracold atomic gas experiments implementing synthetic spin - orbit coupling allow access to flatbands that emphasize interactions . we model
spin - orbit coupled fermions in a one - dimensional flat band optical lattice .
we introduce an effective luttinger - liquid theory to show that interactions generate collective excitations with emergent kinetics and fractionalized charge , analogous to properties found in the two - dimensional fractional quantum hall regime .
observation of these excitations would provide an important platform for exploring exotic quantum states derived solely from interactions .
_ introduction .
_ emergent quantum states derived from interactions can exhibit rich structure because they are , by definition , not adiabatically connected to the underlying single - particle states .
two - dimensional ( 2d ) electron gases placed in a strong magnetic field offer seminal examples . in the absence of a magnetic field , 2d
electrons typically demonstrate fermi - liquid behavior , but a strong magnetic field , the fractional quantum hall ( fqh ) limit @xcite , would seem to prevent the formation of a fermi liquid .
this regime is defined by an absence of single - particle kinetic energy that leaves inter - particle interactions to generate many - body quantum states in a flatband ( the lowest landau level ) .
however , it is now well known that interesting properties , such as fractional charge from screening and other kinetic effects @xcite , emerge from interactions in the fqh regime . the remarkable fact that application of an external field first suppresses single - particle properties to leave interactions to generate similar emergent properties leads to a natural question : can these emergent mechanisms manifest in other contexts ? flatbands in one dimension offer a logical analogue @xcite .
the luttinger - liquid paradigm @xcite captures the physics of many one - dimensional ( 1d ) models .
it predicts excitations with , e.g. , fractionalized charge arising from competition between interactions and kinetic energy .
external fields could , in analogy to 2d magnetic fields , be constructed to quench kinetics in 1d , but the absence of kinetics in 1d flatbands would appear to rule out luttinger - liquid behavior . in this letter , we show that kinetics , fractionalized charge excitations , and other luttinger - liquid - like properties emerge solely from interactions in experimentally feasible 1d flatband models .
our proposal relies on recent experimental progress @xcite in engineering synthetic spin - orbit coupling ( soc ) for ultracold atomic gases @xcite .
these experiments show that raman beams can be used to dress atoms with spin - dependent momentums .
rashba ( and/or dresselhaus ) socs governing these dressed states @xcite are tunable to extremes not possible in solids .
recent work shows that rashba coupling in a 1d optical lattice @xcite or gas @xcite can be tuned to yield flatbands , a new limit that could play a role analogous to the lowest landau level @xcite , but interaction effects in a 1d flat rashba soc band remain unexplored .
we study the impact of interactions between two - component fermions in a flat soc band in 1d optical lattices .
we find that the soc elongates single - particle basis states to generate highly non - trivial nearest neighbor ( nn ) interactions @xcite . the extended interactions lead to wigner crystals of spinors with dispersive collective modes .
these excitations are unexpected because they imply kinetics that emerge purely from interactions .
we predict that these excitations also exhibit fractionalized charge even in the flatband limit . to show this
, we must contend with the fact that the absence of single - particle kinetic energy prevents direct application of the luttinger - liquid theory .
we find , instead , that emergent kinetics allows us to introduce an effective luttinger - liquid theory .
we compute the emergent velocities and fractionalized charge of excitations as experimentally verifiable observables .
we also estimate the experimental parameters for observing these excitations .
detection of kinetics and fractionalized charge derived solely from interactions in 1d would have important consequences for the study of emergent luttinger - liquid behavior , in analogy to emergent fractional charge found in the 2d fqh regime . _ model .
_ we consider an equal population of @xmath0 two - component fermions in a 1d optical lattice .
we start with a non - interacting hamiltonian that adds rashba soc to the optical lattice potential @xcite : @xmath1 where @xmath2 is the momentum of particles of mass @xmath3 , @xmath4 is the optical lattice strength , @xmath5 is the optical lattice wave vector , @xmath6 is the recoil energy , @xmath7 is the soc strength , @xmath8 are pauli matrices , and @xmath9 is the zeeman field strength .
we work in units of the lattice spacing , @xmath10 .
figure [ vvsd ] plots the eigenvalues of @xmath11 , @xmath12 to show that eq .
( [ spinorbith ] ) yields flatbands .
we project into the lowest flatband .
projection , achieved by considering only @xmath13 particles , is warranted in the presence of an energy gap between the @xmath13 and @xmath14 bands at low densities @xcite .
we can derive a low - energy hubbard model of interactions operating in such a flat rashba band in the tight binding limit .
in the wannier basis the inter - atom interactions ( e.g. , @xmath4-wave contact interactions between alkali atoms ) become purely on site . after projection to the lowest flatband , the on - site hubbard interaction defines the hamiltonian of the entire system , and is therefore the focus of our study @xcite : @xmath15 where @xmath16 is the on - site hubbard repulsion that defines the only energy scale , @xmath17 creates a fermion at wave vector @xmath18 in the lowest band , and @xmath19 . here
the kronecker delta implies momentum conservation up to a reciprocal lattice vector and @xmath20 is the number of lattice sites .
equation ( [ hflatband ] ) is written in terms of lowest flatband - projected particles using a unitary transformation between the original fermions and flatband fermions , so that @xmath13 particles are defined as spinors of the original atoms @xcite .
we define the unitary transformation in terms of optical lattice parameters : @xmath21/\omega$ ] with @xmath22 , and @xmath23 . here
@xmath24 is the nn hopping @xcite .
of several lowest bloch bands due to soc for @xmath25 ( left ) and @xmath26 ( right ) .
the ratio of the lowest energy gap to the bandwidth is tuned to @xmath27 @xcite in both panels by setting @xmath28 , @xmath29 for @xmath25 , and @xmath30 for @xmath26 .
( inset ) diagonal interaction between @xmath13 particles as a function of inter - site distance , @xmath31 , for @xmath32 and @xmath33 yielding @xmath34.,width=364 ] projection to @xmath13 particles generates non - trivial delocalized single - particle basis states . to see this we fourier transform @xmath35 to real space
the on - site interaction between the original atoms becomes a longer range interaction between @xmath13 particles .
the leading diagonal interaction , @xmath36 , is between nn . here
@xmath37 . the inset of fig .
[ vvsd ] shows the interaction strength , @xmath38 , between @xmath13 particles .
different optical lattice depths lead to the same interaction , where @xmath38 falls off quickly with the dominant interaction given by @xmath39 , provided the band remains flat @xcite .
the inset shows two key results : ( 1 ) the interaction is longer range , and ( 2 ) the form of the interaction is robust over a wide range of @xmath4 . in the following we
can therefore focus on @xmath26 without loss of generality .
equation ( [ hflatband ] ) contains a large number of terms , but by considering a few of the largest terms ( with strengths @xmath40 , @xmath41 , and @xmath42 ) , we argue for intriguing low - energy states .
leading off - diagonal terms in eq .
( [ hflatband ] ) are given by conditional next nearest neighbor ( nnn ) hoppings of @xmath13 particles , i.e. , @xmath43 and @xmath44 , where @xmath45 and @xmath46 .
we note that conditional hopping originates entirely from interactions .
the @xmath40 term is the strongest and should generate crystal states of spinor @xmath13 particles , but conditional hoppings can give rise to emergent kinetics in excitations .
we verify this picture below by combining diagonalization with an effective model . _ numerical results .
_ to more rigorously study eq .
( [ hflatband ] ) , we use numerics to explore the low - energy hilbert space and confine our study to half filling , @xmath47=1/2 .
we note that the absence of kinetic energy excludes the direct use of luttinger - liquid theory .
( scattered symbols ) for the flatband - projected hamiltonian @xmath48 on various lattice sizes , @xmath20 .
the solid curve is the fit of @xmath49 with the effective extended hubbard model [ eq .
( [ hexthub ] ) ] that highlights a dispersive collective mode .
( right panel ) the energy dispersion of a similar classical model [ eq .
( [ dhexthub ] ] with @xmath50 , showing no dispersive collective modes .
the four schematics denote representative ground and excited state configurations of spinor @xmath13 particles ( encircled arrows ) in real space .
, width=336 ] ( scattered symbols ) for the flatband - projected hamiltonian @xmath48 on various lattice sizes , @xmath20 .
the solid curve is the fit of @xmath49 with the effective extended hubbard model [ eq .
( [ hexthub ] ) ] that highlights a dispersive collective mode .
( right panel ) the energy dispersion of a similar classical model [ eq .
( [ dhexthub ] ] with @xmath50 , showing no dispersive collective modes .
the four schematics denote representative ground and excited state configurations of spinor @xmath13 particles ( encircled arrows ) in real space . , width=115 ] ( scattered symbols ) for the flatband - projected hamiltonian @xmath48 on various lattice sizes , @xmath20 .
the solid curve is the fit of @xmath49 with the effective extended hubbard model [ eq . ( [ hexthub ] ) ] that highlights a dispersive collective mode .
( right panel ) the energy dispersion of a similar classical model [ eq .
( [ dhexthub ] ] with @xmath50 , showing no dispersive collective modes .
the four schematics denote representative ground and excited state configurations of spinor @xmath13 particles ( encircled arrows ) in real space .
, width=120 ] ( scattered symbols ) for the flatband - projected hamiltonian @xmath48 on various lattice sizes , @xmath20 .
the solid curve is the fit of @xmath49 with the effective extended hubbard model [ eq .
( [ hexthub ] ) ] that highlights a dispersive collective mode .
( right panel ) the energy dispersion of a similar classical model [ eq .
( [ dhexthub ] ] with @xmath50 , showing no dispersive collective modes .
the four schematics denote representative ground and excited state configurations of spinor @xmath13 particles ( encircled arrows ) in real space .
, width=120 ] ( scattered symbols ) for the flatband - projected hamiltonian @xmath48 on various lattice sizes , @xmath20 .
the solid curve is the fit of @xmath49 with the effective extended hubbard model [ eq . ( [ hexthub ] ) ] that highlights a dispersive collective mode .
( right panel ) the energy dispersion of a similar classical model [ eq .
( [ dhexthub ] ] with @xmath50 , showing no dispersive collective modes .
the four schematics denote representative ground and excited state configurations of spinor @xmath13 particles ( encircled arrows ) in real space .
, width=134 ] we numerically diagonalize eq .
( [ hflatband ] ) using the lanczos algorithm .
translational symmetry allows us to work within a fixed total momentum sector , @xmath51 .
the left panel of fig .
[ fitband ] shows the four lowest total energies , @xmath49 , as a function of the total momentum for several system sizes .
we find data collapse for @xmath52 .
our numerics therefore apply to the thermodynamic limit .
the ground state of @xmath48 is a spinor wigner crystal shown schematically in the left panel of fig .
[ fitband ] , set to @xmath53 .
two wigner crystals ( both with particles at every other site ) are defined in momentum space by a linear combination of wave functions at @xmath54 .
we verify the crystalline nature of the ground state by breaking the degeneracy with a small , staggered chemical potential , @xmath55 , added to eq .
( [ hflatband ] ) . in the @xmath56 limit
, the density shows that the system spontaneously picks one of the two degenerate wigner crystal ground states @xcite .
we have also calculated the charge structure factor @xmath57 .
we find that @xmath58 has well - defined peaks at @xmath59 , indicating wigner crystals .
we , for comparison , numerically solve a diagonal ( classical ) hamiltonian known @xcite to yield wigner crystals : @xmath60 the right panel of fig .
[ fitband ] shows the many - body energy spectrum .
the ground states of @xmath61 coincide with those of eq .
( [ hflatband ] ) , i.e. , at @xmath54 , further showing that the ground states of eq .
( [ hflatband ] ) are classical wigner crystals of spinor @xmath13 particles .
the first excited state of @xmath61 , however , is non - dispersive and lies at an energy @xmath40 .
this is the energy cost of moving one particle in the wigner crystal to a nn site ( see the schematic of this classical excitation in fig .
[ fitband ] , right panel ) .
a comparison of the left and right panels shows that while the ground states are essentially the same , the excited states of eq .
( [ hflatband ] ) are fundamentally different from those of eq .
( [ dhexthub ] ) .
the excited states of eq .
( [ hflatband ] ) , the left panel of fig .
[ fitband ] , exhibit a gap @xmath62 above the ground state .
the conditional hopping terms cause the otherwise degenerate excited band to form a dispersive collective mode .
the off - diagonal conditional hopping terms superpose the classical configurations of @xmath13 particles .
( see the schematic , left panel of fig .
[ fitband ] ) . to better understand the nature of the excited states , we construct an effective model . .fitting parameters ( @xmath63 and @xmath64 ) , the resulting energy differences ( @xmath65 ) , and the wave - function overlaps between @xmath66 and @xmath48 for an @xmath67 system for the ground state at each total momentum sector , @xmath51 , with @xmath68 . here
the small energy differences and high wave - function overlaps indicate the quality of the effective model in capturing the essential physics of the original model . [ cols="^,^,^,^,^",options="header " , ]
_ effective luttinger - liquid theory .
_ we construct an effective model of eq .
( [ hflatband ] ) by adding hopping terms to eq .
( [ dhexthub ] ) .
the effective hopping terms are emergent because they represent kinetics not present in the original model [ eq .
( [ hflatband ] ) ] .
we verify the accuracy of the effective model by comparing energetics and by taking wave - function overlaps .
the effective model is then studied using luttinger - liquid theory on the emergent degrees of freedom .
we capture the effects of conditional hopping with ordinary single - particle hopping terms in an effective extended hubbard model : @xmath69(\chi_i^{\dagger}\chi_{i+2}^{\vphantom{\dagger}}+h.c.)+h_{d } , \label{hexthub}\ ] ] where @xmath70 and @xmath71 are fitting parameters quantifying emergent nnn hopping .
figure [ eklinearlization ] illustrates @xmath70 and @xmath71 in real space .
note that @xmath63 and @xmath64 scale with @xmath16 because we added these parameters to capture the properties of excited states generated entirely by interactions in the original hopping - free model , eq .
( [ hflatband ] ) .
we vary @xmath70 and @xmath71 and numerically solve eq .
( [ hexthub ] ) to get the best fit of @xmath49 while maximizing overlap of the corresponding wave functions .
table [ bestfitparameters ] shows representative ( @xmath67 ) fits for the lowest eigenstates .
the energy differences between @xmath66 and @xmath48 are all within @xmath72 , and the wave - function overlaps for the lowest states are all above @xmath73 with almost @xmath74 overlaps at @xmath75 and @xmath76 .
the overlap between the ground states ( @xmath54 ) is above @xmath77 .
we plot the first excited state of eq .
( [ hexthub ] ) from the best fit parameters as the black curve in the left panel of fig .
[ fitband ] for comparison .
the overlap and energetic comparison show that eq .
( [ hexthub ] ) captures the essential properties of eq .
( [ hflatband ] ) at low energies .
we can therefore use eq .
( [ hexthub ] ) as an effective theory to make predictions for experiments .
we now show that eq .
( [ hexthub ] ) exhibits excitations with fractionalized charge quantified by luttinger - liquid theory .
we first diagonalize the hopping terms in eq .
( [ hexthub ] ) @xcite .
the emergent `` single - particle '' energy dispersion has two energy bands @xmath78 : @xmath79\cos(2k ) , \label{singleparticlenergy}\ ] ] with fermi velocities @xmath80\sin(2k_{f})|$ ] . for @xmath81
each dispersion crosses the fermi level at two fermi points @xmath82 ( fig . [ eklinearlization ] ) .
low - energy excitations near the fermi points therefore consist of two left movers and two right movers . ) .
the dashed and dotted lines cross at fermi points , @xmath83 .
linearization is shown as solid straight lines .
differing slopes indicate asymmetric bands , i.e. , @xmath84 .
inset : schematic of hopping terms used in eq .
( [ hexthub]).,width=355 ] ) .
the dashed and dotted lines cross at fermi points , @xmath83 .
linearization is shown as solid straight lines .
differing slopes indicate asymmetric bands , i.e. , @xmath84 .
inset : schematic of hopping terms used in eq .
( [ hexthub]).,width=192 ] we bosonize eq .
( [ hexthub ] ) to study interaction effects .
we linearize the dispersion at the fermi points @xcite , as depicted in fig .
[ eklinearlization ] .
the elementary excitations near @xmath85 are bosonic and , in the absence of the interacting term in eq .
( [ hexthub ] ) , have the charge of the original flat band particles .
we include @xmath86 and find the normal modes of the bosonized hamiltonian using a unitary transformation and rescaling of the bosonic fields @xcite .
the emergent normal mode luttinger parameters , i.e. , velocity , @xmath87 , and the charge fractionalization ratio , @xmath88 , are given by @xcite @xmath89 where @xmath90 denotes the two normal modes , @xmath91 , @xmath92 , and @xmath93/2 $ ] .
for @xmath94 , we have @xmath95 indicating that the charge has fractionalized for this normal mode . to see this we write the effective charge , @xmath96 , in terms of the original charge , @xmath97 , as @xmath98 where @xmath96 can be inferred from particle number conductance @xcite .
@xmath99 found here can be continuously tuned below unity .
this should be contrasted with fractionally charged excitations in the fqh regime , where the fractions are only rational @xcite .
the luttinger - liquid analysis therefore shows that low - energy collective modes of eq .
( [ hexthub ] ) can be thought of as fractionalized quasiparticles moving along a spinor wigner crystal .
this result , while known in standard luttinger - liquid theories @xcite , is surprising here since the single - particle eigenstates of the physical atoms are inert ( flatband ) particles that derive emergent kinetics from interactions . the close connection between eqs .
( [ hexthub ] ) and ( [ hflatband ] ) also indicates that these modes should be experimentally observable . _ experimental requirements and observables .
_ low temperatures and low atomic losses are , in general , difficult requirements for proposals to engineer strongly correlated quantum states with atomic gases .
most proposals require maximizing @xmath16 by tuning a feshbach resonance to enter strongly correlated regimes . however , feshbach resonances contribute to unwanted heating and losses @xcite , particularly in soc atomic gases @xcite . the flat band regime studied here
circumvents the need for strong @xmath16 ( and therefore a feshbach resonance ) because the system is automatically strongly correlated in the absence of kinetic energy .
we can estimate realistic parameters to show that the flatband regime is attainable .
@xmath100k is one of the best candidates for strong soc with low losses @xcite . for @xmath100k in a 1d optical lattice with @xmath26 , @xmath28 , @xmath30 , and a perpendicular confinement of lattice depth @xmath101 , we find @xmath102 , @xmath103 and @xmath104 , where @xmath105 is the bandwidth .
this shows that even bare @xmath4-wave scattering implies a strongly interacting flatband problem with @xmath106 .
note that the last inequality is a very stringent flatband requirement .
an accurate ( but weaker ) requirement assumes the many - body energy gap @xmath107 ( left panel , fig .
[ fitband ] ) to be larger than the single - particle hopping @xmath108 .
this implies that partial filling of the lowest band allows us to treat eq .
( [ spinorbith ] ) as an irrelevant constant for realistic system parameters .
parabolic confinement will compete with the many - body energy gap to diminish the size of the wigner crystal near the trap center
. the central crystal will give way to edge states when the parabolic trapping potential energy reaches the gap , i.e. , @xmath109 .
the crystal will then be as large as @xmath110 sites .
trapping potentials therefore place a lower bound on the size of the energy gap ( and therefore @xmath16 ) . for @xmath100k we find that even the bare @xmath4-wave scattering length allows significant crystal sizes , @xmath111 sites for realistic trapping strengths , @xmath112hz .
larger interaction strengths will increase the size of the crystal .
observations of the states proposed here are in principle possible with currently available methods .
the spinor wigner crystal state manifests as a peak in the static structure factor of the original fermions , observable with demonstrated probes : noise correlations @xcite or atomic matter wave scattering @xcite .
luttinger - liquid parameters have also been observed by interfering bose - einstein condensates @xcite . detecting fractionalized charge
is more challenging . in the current context
, it could be measured by , e.g. , detection of partial backscattering from an impurity @xcite , optical methods @xcite , or charge pumping @xcite .
_ summary .
_ we predict a set of intriguing collective states of matter in experiments with atomic fermi gases confined to 1d optical lattices and in the presence of synthetic soc .
we constructed and studied a model where the atomic interactions operate in a flatband .
we found that the single - particle basis states are delocalized spinors .
our analysis predicts that flatband spinor particles have surprising properties generated by on - site interactions among the original atoms : nn interactions and effective nnn hopping .
the many - body ground state was found to be a wigner crystal of spinors .
we find that an effective luttinger - liquid theory parametrizes emergent kinetics and fractionalized charge [ eq .
( [ luttingerparameters ] ) ] in the low - energy collective modes , in direct analogy to the mechanism of emergence found in the fqh regime .
we acknowledge helpful comments from i. spielman and support from the aro ( w911nf-12 - 1 - 0335,w911nf-12 - 1 - 0334 ) , afosr ( fa9550 - 11 - 1 - 0313 ) , and darpa - yfa .
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eclipsing binary systems provide a good opportunity for studying the presence of an unresolved third body by observing their minima times because of the light - time effect ( hereafter lite ) .
it was explained by irwin and its necessary criteria have been mentioned by frieboes - conde & hertzeg and also by mayer .
presence of the third body in the system is possible only if the times of minima behave in agreement with a theoretical lite curve , the resultant mass function has reasonable value and corresponding variations in radial velocities are measured . in the last decade also
a confirmation by astrometry seems to be plausible . in each case
we have calculated new light elements of the eclipsing pair and also the parameters of the predicted third body orbit .
the tables i. and ii .
present results for each system , where @xmath0 are masses of components , @xmath1 computed period of the unresolved body , @xmath2 semiamplitude of lite , @xmath3 eccentricity , @xmath4 length of periastron , @xmath5 mass function and @xmath6 minimum mass ( for @xmath7 = 90@xmath8 ) of predicted body , respectively .
the subscript 3 and 4 denotes the parameter of the third and fourth body , respectively .
we have derived new linear light elements : + oo aql : @xmath9 + v338 her : @xmath10 + t lmi : @xmath11 + rv lyr : @xmath12 + tw lac : @xmath13 + v396 mon : @xmath14 + in the case of oo aql and t lmi also the hypothesis of the fourth body was suggested .
the results are in table ii . in the system
oo aql the components are very similar and also similar to the sun ( sp . g5v , t @xmath15 5700 k ) , so the second order variations could also be caused by the the magnetic cycles , similar to the sun ( so - called applegate mechanism , see eg .
applegate ) . in the system
v338 her also the quadratic term was applied , so the mass transfer in the system could be present . from this hypothesis ( with mass transfer parameter @xmath16 )
we have derived the mass transfer rate to @xmath17 . in a few cases here
the potential third body would be detectable in the detailed light curve analysis .
especially the cases where the third body has not negligible mass , comparing with the eclipsing pair .
regrettably we have no information about the distance of individual systems and also about the absolute magnitudes , so the determination of the angular separation of the possible third body is very uncertain .
references : [ 1 ] al - naimiy et.al . ; [ 2 ] budding ; [ 3 ] cester ; [ 4 ] budding ; [ 5 ] halbedel ; [ 6 ] yang & liu .
we have derived new lite parameters for six eclipsing binaries by means of an @xmath18 diagram analysis . in two cases ,
oo aql and t lmi , another variation was found , so there is a possibility of a presence of the fourth body in the system , or magnetic activity in them .
but we have not enough data to make a final decision .
so the consequence is , that for the confirmation of presence of the lite in these systems , we need detailed photometric , spectroscopic or astrometric data of these binaries .
this research has made use of the simbad database , operated at cds , strasbourg , france , and of nasa s astrophysics data system bibliographic services .
this investigation was supported by the czech - greek project of collaboration rc-3 - 18 of ministry of education , youth and sport and by the grant agency of the czech republic , grant no . | the period changes of six eclipsing binaries have been studied with focus on the light - time effect . with the least squares method we also calculated parameters of such effect and properties of the unresolved body in these systems . with these results we discussed probability of presence of such bodies in the systems with respect to possible confirmation by another method . in two systems we also suggested a hypothesis of fourth body or magnetic activity for explanation of the `` second - order variability '' after subtraction of the light - time effect of the third body . |
the relativistic heavy ion collider ( rhic ) is under construction at brookhaven national laboratory .
in addition to its primary purpose , the search for quark - gluon plasma , a proposal to explore spin physics at rhic @xcite has been approved .
the major goals of the spin physics program at rhic are : * elucidation of the spin structure of the nucleon , and * precision tests of symmetries .
rhic offers a unique opportunity for those studies because of its capability of accelerating polarized proton beams up to @xmath0= 500 gev at the luminosity of 2@xmath110@xmath2@xmath3sec@xmath4 or more with large polarizations of @xmath570% .
obviously we will reach the high - energy frontier for polarized proton - proton collisions at rhic .
the phenix detector is one of the two large detectors at rhic @xcite .
its basic design concept is to detect photons , leptons , and hadrons with high momentum resolution and strong particle identification .
it consists of two spectrometers covering the central rapidity region ( central arms ) , which include an electromagnetic ( em ) calorimeter with fine segmentation ( @xmath6 ) , and two endcap muon spectrometers ( muon arms ) .
since hadron reactions with photonic or leptonic final states such as prompt photon production and lepton production from weak boson decays play major roles in spin physics program , phenix is very well suited to spin physics at rhic .
the studies are done by measuring the spin asymmetries in the cross sections for various reactions . by use of the spin rotators located upstream and downstream of phenix experimental hall , any combination of beam polarizations is possible .
thus we can measure the _ helicity _ dependent cross sections @xmath7 , @xmath8 , @xmath9 , and @xmath10 separately , where @xmath11 and @xmath12 represent positive and negative helicity states of the beam particles as well as _
dependent cross sections , @xmath13 , @xmath14 , @xmath15 , @xmath16 , where @xmath17 and @xmath18 represent transverse polarization of the beam particles . among these asymmetries
, we will discuss only two asymmetries in this presentation , a double longitudinal - spin asymmetry , @xmath19 , and a single longitudinal - spin asymmetry , @xmath20 ; @xmath21 the quantity @xmath19 is often used to extract helicity dependent structure functions ; @xmath20 extracts parity violation effects in the reaction . in the following section
, the sensitivity of our measurements is calculated assuming integrated luminosity of 320 pb@xmath4 and 800 pb@xmath4 for @xmath22 gev and 500 gev , respectively , which corresponds to 10 weeks of running with 70% machine efficiency .
the results of polarized muon scattering off a polarized proton target reported by the emc collaboration have stimulated both experimental and theoretical works to elucidate the spin structure of the proton .
the fraction of the proton spin carried by quarks , @xmath23=0.12@xmath240.09@xmath240.14 , was amazingly small comparing to the canonical expectation 0.60@xmath240.12 @xcite .
post - emc experiments , which have provided data with much better statistics including deuterium and @xmath25he targets , have confirmed the emc results .
a recent global analysis gives @xmath26 , and thus still shows a significant deficit .
phenix will measure the gluon polarization @xmath27 and anti - quark polarization @xmath28 with flavor @xmath29 identified not only to search for the origin of the deficit but also to check the validity of assumptions in the analysis to obtain @xmath23 , e.g. su(3)@xmath30 .
these measurements will be described in the following subsections .
one of the most reliable channels to measure the gluon polarization is high @xmath31 prompt photon production .
the production is dominated by the gluon compton process , followed in significance by the annihilation process . by neglecting the contribution from annihilation channel , ( which is justified in several predictions @xcite )
, the asymmetry @xmath19 can be written at the leading order ( lo ) as a function of photon @xmath31 , @xmath32 here @xmath33 , @xmath34 stands for the scattering angle of partons in their cms , and @xmath35 represents the double longitudinal spin asymmetry for the parton cross sections .
it should be noted that the phenix acceptance ( @xmath36 ) strongly selects the samples in symmetric quark - gluon scattering at @xmath37 and this selection allows great simplification of the expression in eq .
( [ e : asymdg ] ) @xcite .
since @xmath38 is calculated in qcd and @xmath39 has been measured in lepton scattering experiments , @xmath27 can be extracted from the measured @xmath19 . to overcome experimental difficulties due to
the huge background from hadron decays , phenix s finely segmented em calorimeter plays a crucial role in avoiding the fake prompt photon signal that results from the merging of two photons from a high-@xmath31 @xmath40 .
since the phenix calorimeter is as fine as @xmath6 , the prompt photon can be identified up to 30 gev/@xmath41 or more without serious background .
the yield for the assumed integrated luminosities has been calculated using pythia for the phenix acceptance and listed in table [ t : dg_sensitiv ] for both @xmath42200 gev and 500 gev . in addition
, the sensitivity of the measurement of @xmath27 has been evaluated using @xmath35 and the measured @xmath39 .
the listed errors are statistical only .
we have identified the origin of the systematic errors and have begun studies to minimize them . in addition , studies of @xmath27 measurements with other channels such as @xmath43 , open charm / beauty , and heavy quarkonium production are in progress .
.sensitivity summary for the measurements of gluon polarization via prompt @xmath44 production .
[ cols="^ , < , < , < , < , < , < " , ] [ t : dg_sensitiv ] the polarized - dis experiments are sensitive to neither differences between anti - quarks and quarks nor their flavors , since the photon couples to the square of their electric charge . therefore the measurement of anti - quark polarization and the flavor decomposition will improve the knowledge on the spin of the nucleon significantly .
the parity violating asymmetry @xmath45 for @xmath46 production is presented at lo as @xmath47 for @xmath48 production , the @xmath49 and @xmath50 should be exchanged in this expression .
the asymmetry is just the linear combination of the quark and anti - quark polarizations .
furthermore , the flavor in the reaction is almost fixed .
thus flavor decomposition is possible .
the asymmetry converges to @xmath51 at the limit of @xmath52 and to @xmath53 at the limit of @xmath54 .
@xmath55 production can be identified by the detection of muon with @xmath5620 gev/@xmath41 . with the assumed @xmath57=800 pb@xmath4
, we expect about 5000 events for each of @xmath58 and @xmath59 .
furthermore the @xmath60-range can be selected using the muon momentum as shown in figure [ f : polpdf](a ) . using the muon sample which is divided into several energy bins ( and thus @xmath60-bins ) ,
we have estimated the sensitivities of our measurements ( with statistical errors only ) .
the results are plotted on two model predictions for polarized structure functions @xcite in figure [ f : polpdf](b ) .
error bars indicate the expected statistical errors
. our measurement will be quite sensitive to both quark and anti - quark helicity distributions , although further studies are needed to minimize the background contributions .
the nature of parity non - conservation itself can be directly probed using the polarized beams at rhic , with @xmath45 the measure of violation .
taxil and virey studied various possibility to find new physics at rhic through the measurements of @xmath20 using polarized proton beams @xcite .
while their predictions are only for single - jet production , which is not detectable with phenix , we expect that such asymmetries , however , should persist with inclusive or leading particle production .
sensitivity studies for phenix are underway .
the phenix physics scope has been extended to include the spin physics program .
it will provide measurements of @xmath27 and @xmath28 that greatly reduce the uncertainties in the knowledge of the spin structure of the nucleon .
precision tests of symmetries in the standard model is foreseen .
the first collision of the polarized proton beams is expected to start in the year of 2000 .
we thank the technical staffs of the participating institutions for their vital contributions . this detector construction project is supported by the department of energy ( u.s.a . ) ,
sta and monbu - sho ( japan ) , ras , rmae , and rms ( russia ) , bmbf ( germany ) , frn and the knut & alice wallenberg foundation ( sweden ) , and mist and nserc ( canada ) . 9 d.p .
morrison for the phenix collaboration , these proceedings .
proposal on spin physics using the rhic ( r5 ) , aug .
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b364 * ( 1995 ) 181 , * b383 * ( 1996 ) 355 . | the phenix experiment at rhic has extended its scope to cover spin physics using polarized proton beams .
the major goals of the spin physics at rhic are elucidation of the spin structure of the nucleon and precision tests of the symmetries .
sensitivities of the spin physics measurements with the phenix detector system are reviewed . |
we report the results of the optical interferometry of the interacting system of galaxies ngc 7769 , 7770 , 7771 and 7771a and analyze their kinematics .
a detailed description of the morphological features of the galaxies as well as photometry and color analysis of ngc 7769 are presented in the complete version of the study : @xcite .
we also discuss the influence of interaction on the kinematics , dynamics and star formation in the system .
known models of galaxy interactions are based mostly on statistical observational data .
we try to illustrate how and to what extend these models can be applied to explain the features of the galaxies in this system . in order to study the velocity fields of the galaxies ,
the observations were carried out at the 2.6 m telescope of the byurakan astrophysical observatory ( bao , armenia ) on 8 november 1996 , with the byufosc ( byurakan faint object spectral camera ) in the interferometric mode , attached at the prime focus of the telescope . based on the h@xmath0 velocity fields ( the right - hand panels of figure [ vrot_velr ] ) , we calculated the rotation curves of the galaxies ( the left - hand panels of figure [ vrot_velr ] ) by using data points within sectors along the maximal gradient direction , see isovelocity contours in the right - hand panels of figure [ vrot_velr ] .
maximal rotational velocity of ngc 7769 is observed at the radius of around 15 arcsec from the galaxy nucleus .
the rotational velocities in figure [ vrot_velr ] are in good agreement with the hi measurements ( @xmath1 ) in @xcite .
our measurements of velocities , having a better spatial resolution compared with those of the previous studies ( @xcite ) , reveal weak perturbations of the rotation curve of ngc 7769 , which may be caused by interaction with ngc 7771 .
the same can not be said about the velocity field of ngc 7771 .
figure [ vrot_velr ] shows that there are perturbations and large dispersion in radial velocities at the distances larger than about 10 - 15 arcsec from the nuclei .
this distance is about half radius of the bar .
evidently , this scatter of radial velocities can be explained by the fact that part of the arms are included in the sector used to calculate radial velocities ( sector angle is @xmath2 ) .
however the asymmetric profile along the major axis suggests that northern and southern arms do not have the same radial velocity profiles .
the asymmetric tidal forces of ngc 7769 and ngc 7770 affecting on ngc 7771 , seem to be a natural cause of that . the rotation curve of ngc 7770 is significantly skewed .
this is probably because of the strong harassing interaction with the more massive ngc 7771 , see @xcite .
the rotation curve of ngc 7771a is typical for a late type sm galaxy .
@xmath3{vrot_7769.eps } & \includegraphics[width=0.52\hsize]{ngc7769.eps}\\ \includegraphics[width=0.47\hsize]{vrot_7770.eps } & \includegraphics[width=0.52\hsize]{ngc7770.eps}\\ \includegraphics[width=0.47\hsize]{vrot_7771.eps } & \includegraphics[width=0.52\hsize]{ngc7771.eps}\\ \includegraphics[width=0.47\hsize]{vrot_7771a.eps } & \includegraphics[width=0.52\hsize]{ngc7771a.eps } \end{array}$ ] by analyzing velocity fields , sizes , and shapes of spiral arms of ngc 7771 and ngc 7769 , in @xcite it has been suggested that ngc 7771 and ngc 7769 , which have a 2:1 mass ratio , appear to be having a prograde - retrograde interaction , with ngc 7769 being the retrograde one .
our better data support this conclusion .
this conclusion is in agreement with the latest models of galaxy collisions ( ) showing that during direct collisions tidally induced spiral arms are much longer and brighter than those during retrograde collisions .
we can conclude that galaxies ngc 7769 and ngc 7771 already have passed the first pericenter stage , however , probably the second encounter has not happened yet .
the first pericenter distance should have been large enough ( around few sizes of the galaxies ) , so that large disturbances in rotation curves have not appeared yet .
the quartet of galaxies ngc 7769 , 7770 , 7771 and 7771a is a system of interacting galaxies . here
, we present a fabry - perot imaging study of the system in h@xmath0 line .
we came to the following main conclusions : * close interaction between the component galaxies of the system has produced morphological features that are characteristic of the interactions .
we have detected features such as tidal arms , spiral arms induced by close interaction , bars and induced star formation . * from the results of our interferometric observations , we obtained the radial velocity profiles of galaxies .
the rotation curve of ngc 7769 is weakly distorted .
the rotation curve of ngc 7771 is strongly distorted by the tidal arms caused by direct flyby of ngc 7769 and flyby of a smaller neighbor ngc 7770 .
the rotation curve of ngc 7770 is significantly skewed because of the interaction with much massive ngc 7771 . *
the radial velocity profiles and morphological disturbances suggest that the ngc 7769 and ngc 7771 have passed the first pericenter stage , however , probably the second encounter has not happened yet .
study of such systems with methods combining photometric and visual analysis is an effective way to clarify features of star formation in different stages of interaction .
ongoing and future surveys using integral field spectroscopy will allow also to explore the spatial distribution of star formation in interacting systems .
alonso - herrero , a. , rosales - ortega , f. f. , snchez , s. f. , et al .
2012 , mnras , 425 , l46 chengalur , j. n. , salpeter , e. e. , & terzian , y. 1993 , apj , 419 , 30 di matteo , p. , combes , f. , melchior , a .-
semelin , b. 2007 , a&a , 468 , 61 nordgren , t. e. , chengalur , j. n. , salpeter , e. e. , & terzian , y. 1997 , aj , 114 , 77 yeghiazaryan , a. a. , nazaryan , t. a. , & hakobyan , a. a. 2015 , arxiv:1510.00193 | we performed the fabry - perot scanning interferometry of the quartet of galaxies ngc 7769 , 7770 , 7771 and 7771a in h@xmath0 line and studied their velocity fields .
we found that the rotation curve of ngc 7769 is weakly distorted .
the rotation curve of ngc 7771 is strongly distorted with the tidal arms caused by direct flyby of ngc 7769 and flyby of a smaller neighbor ngc 7770 .
the rotation curve of ngc 7770 is significantly skewed because of the interaction with much massive ngc 7771 .
the rotation curves and morphological disturbances suggest that the ngc 7769 and ngc 7771 have passed the first pericenter stage , however , probably the second encounter has not happened yet . |
nuclear lattice effective field theory ( nleft ) is a first - principles approach , in which chiral eft for nucleons is combined with numerical auxiliary - field quantum monte carlo ( afqmc ) lattice calculations .
nleft differs from other _ ab initio _
methods @xcite in that it is an unconstrained monte carlo calculation , which does not rely on truncated basis expansions or many - body perturbation theory , nor on prior information about the structure of the nuclear wave function . as in chiral eft ,
our calculations are organized in powers of a generic soft scale @xmath5 associated with factors of momenta and the pion mass @xcite .
we denote @xmath6 as leading order ( lo ) , @xmath7 as next - to - leading order ( nlo ) , and @xmath8 as next - to - next - to - leading order ( nnlo ) contributions .
the present calculations are performed up to nnlo .
we define @xmath9 as the lo lattice hamiltonian , and @xmath10 as the equivalent hamiltonian with the pion - nucleon coupling @xmath11 and contact interactions that respect wigner s su(4 ) symmetry . in our nleft calculations ,
@xmath9 is treated non - perturbatively ( see ref .
@xcite for a review ) .
the nlo contribution to the two - nucleon force ( 2nf ) , the electromagnetic and strong isospin - breaking contributions ( emib ) , and the three - nucleon force ( 3nf ) which first enters at nnlo , are all treated as perturbations .
it should be noted that our `` lo '' calculations use smeared short - range interactions that capture much of the corrections usually treated at nlo and higher orders @xcite . at nnlo ,
the 3nf overbinds nuclei with @xmath12 due to a clustering instability which involves four nucleons on the same lattice site .
a long - term objective of nleft is to remedy this problem by decreasing the lattice spacing and including the next - to - next - to - next - to - leading order ( n3lo ) corrections in chiral eft . in the mean time , the overbinding problem has been rectified by means of a 4n contact interaction , tuned to the empirical binding energy of either @xmath13he or @xmath14be @xcite . while this provides a good description of the alpha nuclei up to @xmath15 including the hoyle state @xcite , the overbinding is found to increase more rapidly for @xmath16 . in ref .
@xcite , a non - local 4n interaction which accounts for all possible configurations of four nucleons on adjacent lattice sites was introduced , and adjusted to the empirical binding energy of @xmath3 mg .
a detailed study of the spectrum and electromagnetic properties of @xmath1o ( with inclusion of the effective 4n interaction ) has been reported in ref .
the hoyle state is a resonance with spin - parity quantum numbers @xmath17 in the spectrum of @xmath0c , which plays an important role in resonantly enhancing the reaction rate for the so - called triple - alpha process , which is responsible for the production of carbon in massive stars that have reached the red giant stage in their evolution .
this reaction represents a significant bottleneck in the stellar nucleosynthesis , as @xmath14be is an unstable ( though relatively long - lived ) resonance .
for @xmath0c to form , a third alpha particle must combine with the @xmath14be resonance to form the hoyle state , which subsequently decays electromagnetically to the ground state of @xmath0c .
this reaction may then proceed further ( non - resonantly ) to form @xmath1o through addition of a fourth alpha particle .
however , the temperature of the stellar plasma at which the triple - alpha process takes place depends exponentially on the energy @xmath18 of the hoyle state above the triple - alpha threshold , which is experimentally known to be @xmath19 kev .
stellar model calculations @xcite have shown that only a narrow window of @xmath20 kev exists in @xmath18 where sizable amounts of carbon and oxygen can be produced simultaneously . .lattice results at leading order ( lo ) and available experimental values for the root - mean - square charge radii and quadrupole moments of the @xmath0c states .
[ hoyle1 ] [ cols="^,^,^",options="header " , ] in table [ oxygen2 ] , we note that the lo charge radius @xmath21 of the ground state of @xmath1o is smaller than the empirical value @xmath22 .
this leads to a systematic deviation , which arises from the overall size of the second moment of the charge distribution . to compensate for this overall scaling mismatch
, we have also calculated `` rescaled '' quantities multiplied by powers of the ratio @xmath23 , according to the length dimension of each observable . with such a scaling factor included , we find that the nleft predictions for the @xmath24 and @xmath25 transitions are in good agreement with available experimental values .
we have presented an overview of the central nleft results for the low - lying even - parity spectra of @xmath0c and @xmath1o .
this includes the hoyle state of @xmath0c which plays a central role in the stellar nucleosynthesis of life - essential elements .
we have also shown that the electromagnetic properties and transition rates of @xmath0c and @xmath1o are in agreement with available experimental data .
while the long - term objectives of nleft are to decrease the lattice spacing and include higher orders in the eft expansion , we also find that the missing physics up to @xmath4si can be approximated by an `` effective '' 4n interaction .
these results represent an important step towards more comprehensive nleft calculations of medium - mass nuclei in the near future .
we are grateful for the help in automated data collection by thomas luu .
partial financial support from the deutsche forschungsgemeinschaft ( sino - german crc 110 ) , the helmholtz association ( contract no .
vh - vi-417 ) , bmbf ( grant no .
05p12pdfte ) , and the u.s .
department of energy ( de - fg02 - 03er41260 ) is acknowledged .
this work was further supported by the eu hadronphysics3 project , and funds provided by the erc project no .
259218 nucleareft .
the computational resources were provided by the jlich supercomputing centre at the forschungszentrum jlich and by rwth aachen .
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t. a. lhde , e. epelbaum , h. krebs , d. lee , ulf - g .
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meiner , and g. rupak , _ to appear in physical review letters _ , arxiv:1312.7703 [ nucl - th ] | we review the calculation of the hoyle state of @xmath0c in nuclear lattice effective field theory ( nleft ) and its anthropic implications for the nucleosynthesis of @xmath0c and @xmath1o in red giant stars .
we also review the extension of nleft to the regime of medium - mass nuclei , with emphasis on the determination of the ground - state energies of the alpha nuclei @xmath1o , @xmath2ne , @xmath3 mg and @xmath4si by means of euclidean time projection . finally , we review recent nleft results for the spectrum , electromagnetic properties , and alpha - cluster structure of @xmath1o . |
the milagro observatory@xcite has made long term observations of the cygnus arm .
they report an excess of over 5.5@xmath1 over a 5.9@xmath2 square bin in ra and dec.@xcite .
this excess is inconsistent with a point source and may be due to a giant molecular cloud(gmc ) located in the same region as the excess .
this cloud has been reported by dame et .
al.@xcite to be at a distance of 1.7 pc with a estimated mass of @xmath3 .
the angular extent of the cloud is 44 square degrees .
diffuse emission of @xmath0 rays at tev energies have long been speculated to be the result of cosmic ray interactions with giant molecular clouds@xcite . in this scenario , galactic cosmic rays
interact with hydrogen and produce neutral pions .
these pions quickly decay and produce @xmath0 rays .
predictions by aharonian and atoyan @xcite have indicated that the flux from these gmc should follow the galactic cosmic ray flux ( excluding enhancements by local sources ) and would be proportional to the gmc mass over the square of the distance to the gmc .
the cygx cloud is a good target since it is close and very massive .
the whipple 10 meter atmospheric cherenkov telescope utilizes the well proven imaging technique to reject cosmic ray background events and to determine source geometry@xcite .
this method uses the shape of the shower image ( fitted to an ellipse ) to determine if the shower was initiated by a @xmath0 primary or a cosmic ray primary .
additionally , if the source is assumed to be at the center of the field of view ( fov ) , the angle between the major axis of the ellipse and the line formed by the centroid of the image and the center of the fov(@xmath4 angle ) , can be used to eliminate events not coming from the source location .
the energy threshold for the whipple 10 meter is 390 gev for a crab like spectrum@xcite extensions of this method have been made to make observations for objects that may not be in the center of the fov .
this is often the case when searching for new sources , diffuse emission , or sources that have been identified by other experiments with relatively low angular resolution . in this two dimensional analysis @xcite , the source location
is geometrically constrained to lie along the major axis of the shower image ( as it the case with the one dimensional analysis ) , but no requirement is made of the @xmath4 angle with respect to the center of the camera .
the distance from the image centroid to the source location along the major axis is estimated using @xmath5 where the _ width _ refers to the size of the minor axis , _ length _ refers to the size of the major axis , _ d _ is the distance along the major axis , and @xmath6 is a scaling parameter that must be determined . to break the ambiguity as to which direction along the major axis the source lies , the skewness in the image is used .
the @xmath6 parameter was determined by examining the crab supernova remnant @xcite .
the two dimensional analysis was applied to on - source crab data . to optimize the @xmath6 parameter ,
the value of @xmath6 was varied in steps of @xmath7 .
the optimal value was determined by the maximum signal at the source location the optimal value was determined to be @xmath8 .
once the @xmath6 parameter has been determined the data can binned and the point spread function ( psf ) for the method can be determined . here
we have used a 0.36@xmath9 square bin in ra and dec .
this bin size was found to optimize the significance of the on source crab observations .
the binning of the data is shifted six times in ra and dec . in steps of 0.06@xmath2 in order to compensate for edge effects in the binning . applying this analysis to the on source crab data we get a maximum significance of 11.6@xmath1 from 5.6 hours of on source data ( 4.9@xmath1/@xmath10 ) .
the psf of the excess in ra and dec . is fit to a gaussian distribution with a @xmath11 = 0.18@xmath2 for points source off axis ( that is to say , not in the center of the field ) the psf becomes broader as the source moves further away from the center of the fov . while the radial spread to the psf stays roughly the same , the azimuthal spread increases slightly from 0.18@xmath2 to 0.21@xmath2 at one degree offset .
the behavior of the psf as function off offset was determined by analyzing crab data taken at 0.3 , 0.5 , 0.8 and 1.0 degree offsets from the center of the field .
data used in this work was taken during the months of august 2004 through november 2004 .
the observation window for this object is small as the whipple 10 meter generally suspends observations in the summer months due to poor weather conditions in southern arizona . in this analysis
we have used 12 on / off pairs of 28 minutes each .
the total number of events in the on / off field after shape cuts is 14406/14594 ( on / off ) .
the coordinates of the observations are ra = 20:40:7.9 ( 310.03@xmath2 ) and dec = 42:39:51.12 ( 42.66@xmath2 ) in j2000 coordinates .
these coordinates were chosen to overlap with the morphology of the milagro excess @xcite as well as overlap with large values of neutral hydrogen column densities in the region @xcite .
the above analysis fails to find strong evidence for a point source of @xmath0-rays within the 2-d fov of the observations .
figure 2 shows the excess map and sigma map from the field .
the significance was calculated using the standard li and ma method @xcite .
the most significant bin in the map ( figure 2 ) is located at ra=310.8@xmath2 and dec=41.3@xmath2 .
the pretrial significance is 3.8@xmath1 in this bin . to account for trials factors associated with the binning and the oversampling we simulated 30,000 data sets for this field .
we find the chance probability of getting one bin with a sigma of 3.8 or higher is 12% as no compelling point source was found within this field of view , we must conclude that the milagro source@xcite must be rather diffuse in this region , or must be at a flux level below the sensitivity of this analysis .
diffuse sources are rather difficult with the whipple 10 meter , particularly if size of the regions of emission is extends beyond the fov
. such emission would be difficult to detect with this technique , which is optimized for point - source detection .
upper limits have been calculated in the manner prescribed by helene@xcite .
once the upper limit is known , the values are adjusted to correct for the changing sensitivity across the fov .
this changing sensitivity has been determined by the analysis of offset crab data .
figure 3 shows the upper limits for each bin in ra and dec .
the field has been limited to the inner 1@xmath2 of the fov . beyond this
the sensitivity decreases such that it is difficult to set meaningful upper limits .
we find no point source within in the inner half degree of the fov above 3.0% , 3.3% , and 4.0% of the crab at the 90% , 95% and 99% confidence level . in the outer half degree
we find no point source above 15.0%,16.7% and 20% of the crab at the 90% , 95% , 99% confidence level .
these upper limits are obtained by comparing the total number of excess counts at a given offset ( within the 2@xmath1 ellipse of the psf ) with the expected excess counts from the crab at the same offset .
this method assumes a crab like spectrum and point source distribution .
further work is being done to convolute the psf with the expected source distribution , obtained from neutral hydrogen column density maps of the region .
this will allow us to set stronger upper limits for the region and may reveal a sources that the point source analysis failed to detect .
we are also continuing to collect data on this target and are also observing other fields within the region of the milagro excess .
this work supported by the us national science foundation grant ( # phy 0099580 ) , the u.s .
department of energy , the smithsonian institute , the nserc in canada , the science foundation of ireland and by pparc in the united kingdom . | the whipple 10 meter atmospheric cherenkov telescope has made observations of the region known as the cygnus arm .
this region has been recently reported by the milagro experiment to contain a diffuse tev @xmath0-ray source centered at ra=308 and dec=42 .
we report upper limits ( using the whipple 10 m telescope ) obtained during the fall 2004 observing season centered on ra=310 and dec=42.65 . |
transmembrane voltage is often recorded during physiological study of biological neurons . however , voltage - gated ion channel activity and neurotransmitter levels are quite difficult to measure directly and are usually unobserved in such studies . in addition , there is a great diversity of neuron morphology , protein expression , and plasticity which may affect voltage dynamics and synaptic transmission @xcite . early development and senescence
may also be major determinants of voltage response profiles @xcite .
synaptic tuning in particular is thought to be an essential mediator of learning , stimulus response integration , and memory .
there is evidence that memory and learning may depend critically on several distinct types of dynamic behavior in the voltage of neurons .
the ml model reproduces the voltage of a single neuron and , depending on parameterization and initial conditions , can exhibit many of the experimentally observed behaviors of biological neurons @xcite . in this paper , we explore a simple neural network consisting of two biologically identical , reciprocally coupled ml neurons . @xcite have shown that this modest model can exhibit a wide range of oscillating or non - oscillating voltage depending on the values of just a few parameters , specifically in this study , @xmath1and @xmath0 . in the absence of noise , the model can predict synchronous or asynchronous firing , as well as either equal or unequal action potential amplitudes .
additionally , in the presence of even small noise in the applied current and weak synaptic coupling , the system can exhibit mixed - mode oscillations ( mmo ) characterized by periods of small amplitude oscillation interrupted by large amplitude excursions .
in further work with the two ml neuron model , @xcite explored two synaptically decoupled neurons driven by both common and independent intrinsic noise terms .
they found that shared common noise promotes synchronous firing of the two neurons , while separate intrinsic noise terms promoted asynchronous firing .
the relative scaling of the two noise sources was observed to be key in predicting the degree of synchrony .
in addition , while they did not specifically look at mmo , they hypothesized that such synchrony in a synaptically coupled network would increase the probability of mmo , by facilitating longer residence times within the unstable periodic orbits adjacent to the system s stable periodic orbits .
indeed , in this paper we will detail the relative positions of these parameter regions as they are of key importance to our conditioned likelihood approach . specifically , we will provide a quick look - up table for the region in parameter space where stable periodic orbits are possible .
@xcite develop a expectation - maximization ( em ) stochastic particle filter method to estimate the parameters in a single ml neuron based on observation of voltage only .
a key aspect of their approach is that they assume both the voltage and the channel gating variables are in an oscillatory regime , but stochastically perturbed . these perturbations are considered nuisance parameters which their method marginalizes away . specifically , they treat the unobserved channel gating variable from the model as a completely latent variable . starting from estimates of the initial conditions for the voltage and channel gating variables , they iteratively predict the gating variable and voltage and
then update the predicted voltage to the next time step using a modification of the well - known euler differential equation solver .
they discuss that an assumption of stationarity in their method limits applicability to only short time windows over which current input can be considered constant ( e.g. 600ms ) .
they also note that certain parameters , conductances and reversal potentials in particular , are sensitive to choice of tuning parameters required by the method .
these studies demonstrate the active progress as well as the challenges of model parameter estimation for biological neuronal models and , more generally , for relaxation oscillator models .
each of these studies derives asymptotic approximations or general forms for model likelihood , but use fundamentally different techniques and assumptions in doing so . in each
study the approach is specifically crafted to the model . in this paper
we attempt to develop a convenient bayesian estimation scheme with only a few tuning parameters and relatively few mild assumptions .
we focus our attention on deterministic synaptically coupled ml neurons .
application of our method to stochastically coupled ml neurons is on - going work in our group . in the case of ml , estimation of @xmath1and @xmath0is non - trivial due to the diversity of possible dynamic behavior and the abrupt transitions among these seen with just small changes in these parameters values .
however , we can better understand the critical values of these parameters by studying the system s bifurcation structure .
we are able to locate parameter regimes where dramatic changes in the system appear .
the neurons analyzed in this study are classified as type ii neurons , characterized by discontinuous drastic shifting between behavioral states . because there is a distinct switch in behavior , bifurcation analyses determine a closed region of parameter space over which the relevant dynamics may occur .
sampling over such a feasibility region amounts to conditioning the inference on an _ a priori _ assumed class of dynamics ( e.g. stable node , limit cycle , steady state etc . ) . while facilitating conditioning the likelihood on feature statistics of the voltage
, this may translate into increased confidence and reduced bias in the parameter estimates .
our goal is parameter inference based on the temporal voltage response of two synaptically coupled neurons which are deterministically coupled to voltage - gated ionic conductance dynamics @xcite .
a single ml model has a two - dimensional phase space and is known to reproduce many of the behaviors experimentally observed in biological neurons @xcite .
therefore , systems of coupled ml neurons may offer a reasonable starting point for developing statistical inference methods for models of neuronal networks .
the ml network we study is , @xmath2 where @xmath3 note that in the stochastic version of this model @xmath4 and @xmath5 and @xmath6 are standard independent wiener process variables . in this paper , however we will be concerned with the deterministic version of this model where @xmath7 . .
* * [ cols="^,^,^,^",options="header " , ] let @xmath9 have fourier series representation , + @xmath10 + then , it it to be shown that @xmath11 proof is by induction and is adapted from @xcite . in the base case ( @xmath12 ) , @xmath13 t & \overbrace{+2\pi^3 a_1 ^ 2 \phi_1 ^ 3 \sin(4\pi\phi_1 t)}^{g(t)~\in{~{\ensuremath{\mathcal{o}\!\left(1\right)}\xspace}}}\\ & - 2\pi^3 b_1 ^ 2 \phi_1 ^ 3 \sin(4\pi\phi_1t)\\ & + 16\pi^4a_1b_1\phi_1 ^ 4 \cos^2(2\pi\phi_1t)\\ & -16\pi a_1b_1\phi_1 ^ 4\end{aligned}\ ] ]
then it is supposed that for @xmath14 we have , @xmath15t+g_{n-1}(t)\end{aligned}\ ] ] @xmath16t+g_n(t)\end{aligned}\ ] ] next , collecting the @xmath17 terms from the summation yields , @xmath18 expanding the square in the previous result gives , @xmath19 the first term is recognized as the induction hypothesis and so has the form @xmath20 .
the integrals of the remaining terms are evaluated with the extensive use of trigonometric identities .
@xmath21t+g_{n-1}(t)\\ & + \sum_{k=1}^{n-1}8\pi^3a_ka_n\phi_k^2\phi_n^2\left[\frac{\sin(2\pi(\phi_k-\phi_n)t)}{\phi_k-\phi_n}+\frac{\sin(2\pi(\phi_k+\phi_n)t)}{\phi_k+\phi_n}\right]\\ & -\sum_{k=1}^{n-1}8\pi^3b_ka_n\phi_k^2\phi_n^2\left[\frac{\cos(2\pi(\phi_k-\phi_n)t)}{\phi_k-\phi_n}+\frac{\cos(2\pi(\phi_k+\phi_n)t)}{\phi_k+\phi_n}\right]\\ & + \sum_{k=1}^{n-1}8\pi^3b_ka_n\phi_k^2\phi_n^2\left[\frac{1}{\phi_k-\phi_n}+\frac{1}{\phi_k+\phi_n}\right]\\ & -\sum_{k=1}^{n-1}8\pi^3a_kb_n\phi_k^2\phi_n^2\left[\frac{\cos(2\pi(\phi_k-\phi_n)t)}{\phi_k-\phi_n}+\frac{\cos(2\pi(\phi_k+\phi_n)t)}{\phi_k+\phi_n}\right]\\ & + \sum_{k=1}^{n-1}8\pi^3a_kb_n\phi_k^2\phi_n^2\left[\frac{1}{\phi_k-\phi_n}+\frac{1}{\phi_k+\phi_n}\right]\\ & + \sum_{k=1}^{n-1}8\pi^3b_kb_n\phi_k^2\phi_n^2\left[\frac{\sin(2\pi(\phi_k-\phi_n)t)}{\phi_k-\phi_n}+\frac{\sin(2\pi(\phi_k+\phi_n)t)}{\phi_k+\phi_n}\right]\\ & \mathbf{+8\pi^4\phi_n^4a_n^2t}+2\pi^3a_n^2\phi_n^3\sin(4\pi\phi_nt)\\ & + 16\pi^4a_nb_n\phi_n^4\sin^2(2\pi\phi_nt)\\ & \mathbf{+8\pi^4\phi_n^4b_n^2t}-2\pi^3b_n^2\phi_n^3\sin(4\pi\phi_nt)\end{aligned}\]]the bold terms may be combined with the leading term of the induction hypothesis raising the upper bound of the summation from @xmath14 to @xmath22 .
the remaining terms , only containing @xmath23 as arguments of sines and cosines , can be merged with @xmath24 from the induction hypothesis . calling
this merger @xmath25 completes the induction .
cumulative power has been written in the desired form @xmath26t+g_n(t)\\ & = & c\cdot{t } + { \ensuremath{\mathcal{o}\!\left(1\right)}\xspace}\end{aligned}\ ] ] since clearly @xmath27 @xmath28 it follows that their sum @xmath29 proving the desired result .
this report summarizes work that was done as part of the summer undergraduate research institute of experimental mathematics ( suriem ) held at the lyman briggs college of michigan state university .
we are very grateful to the national security agency and the national science foundation for funding this research .
we would also like to thank our advisor , professor daniel p. dougherty , for his guidance throughout the summer , and our graduate assistant , joseph e. roth , for his assistance . | the morris - lecar ( ml ) model has applications to neuroscience and cognition .
a simple network consisting of a pair of synaptically coupled ml neurons can exhibit a wide variety of deterministic behaviors including asymmetric amplitude state ( aas ) , equal amplitude state ( eas ) , and steady state ( ss ) .
in addition , in the presence of noise this network can exhibit mixed - mode oscillations ( mmo ) , which represent the system being stochastically driven between these behaviors . in this paper , we develop a method to specifically estimate the parameters representing the coupling strength ( @xmath0 ) and the applied current ( @xmath1 ) of two reciprocally coupled and biologically similar neurons .
this method employs conditioning the likelihood on cumulative power and mean voltage .
conditioning has the potential to improve the identifiability of the estimation problem .
conditioning likelihoods are typically much simpler to model than the explicit joint distribution , which several studies have shown to be difficult or impossible to determine analytically .
we adopt a rejection sampling procedure over a closed defined region determined by bifurcation continuation analyses .
this rejection sampling procedure is easily embedded within the proposal distribution of a bayesian markov chain monte carlo ( mcmc ) scheme and we evaluate its performance .
this is the first report of a bayesian parameter estimation for two reciprocally coupled morris - lecar neurons , and we find a proposal utilizing rejection sampling reduces parameter estimate bias relative to naive sampling .
application to stochastically coupled ml neurons is a future goal . |
this work was supported by the national science foundation through nsf grant phy-9503384 , and by a nasa theory grant .
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there is a small but nontrivial discrepancy between our results and the those of shi _ et al .
_ @xcite . for @xmath62 ,
the atmospheric neutrino solution would imply @xmath63 according to shi _
@xcite , while we obtain @xmath64 .
the difference is due to our use of the @xmath25 correction to @xmath2 , and the fact that we numerically integrate the altered n@xmath14p reactions in our bbn calculation .
thus the constraints on the @xmath17 atmospheric neutrino solution are not as relaxed as might have been expected for d / h @xmath53 . | studies of limits on active - sterile neutrino mixing derived from big bang nucleosynthesis considerations are extended to consider the dependance of these constraints on the primordial deuterium abundance .
this study is motivated by recent measurements of d / h in quasar absorption systems , which at present yield discordant results .
limits on active - sterile mixing are somewhat relaxed for high d / h . for low d / h ( @xmath0 ) , no active - sterile neutrino mixing is allowed by currently popular upper limits on the primordial @xmath1he abundance @xmath2 .
for such low primordial d / h values , the observational inference of active - sterile neutrino mixing by upcoming solar neutrino experiments would imply that @xmath2 has been systematically underestimated , unless there is new physics not included in standard bbn .
upper limits on the abundance of @xmath1he produced in big bang nucleosynthesis ( bbn ) have been used to limit mixing between active ( @xmath3 , @xmath4 , or @xmath5 ) and sterile ( @xmath6 , no standard model interactions ) neutrinos @xcite . in this paper , we point out and discuss how these constraints are dependant on the adopted primordial deuterium abundance .
previous limits on sterile neutrino mixing have assumed a value for the lower bound on the baryon - to - photon ratio @xmath7 derived from interstellar medium and solar system measurements of deuterium ( d ) and @xmath8he , and models of chemical and galactic evolution .
recent measurements of d / h in quasar absorption systems ( qas ) have yielded discordant values of this ratio , some higher than previously derived ranges @xcite , and some lower @xcite .
several factors make an investigation of the primordial d / h dependance of bbn constraints on active - sterile neutrino mixing timely : the discordant qas measurements of d / h ; the fact that future solar neutrino experiments may be able to distinguish and identify @xmath9 mixing @xcite ; and the use of sterile neutrinos in schemes for neutrino masses and mixings that explain all available data @xcite . as is well known ( _ e.g. _ , ref
@xcite ) , the abundance of @xmath1he produced by bbn is essentially determined by the ratio of neutron to proton number densities ( n / p ) at `` weak freeze - out '' ( wfo ) .
wfo occurs when the reactions that interchange neutrons and protons proceed too slowly relative to the expansion rate of the universe to keep n / p at its equilibrium value of n / p @xmath10 . here
@xmath11 mev is the neutron - proton mass difference , and @xmath12 is the photon temperature . mixing between active and sterile neutrinos increases ( n / p)@xmath13 , and therefore the primordial @xmath1he mass fraction @xmath2 , in two ways .
first , active - sterile neutrino mixing effectively brings more degrees of freedom into thermal contact , increasing the energy density and hence the expansion rate of the universe .
second , active - sterile mixing especially @xmath9 mixing depletes the electron neutrino and antineutrino populations , reducing the rates of the n @xmath14 p interconversion reactions .
both of these effects cause n / p to freeze out at a lower temperature . using a neutrino ensemble evolution formalism @xcite that includes both neutrino oscillations ( with matter effects ) and neutrino collisions , previous authors
@xcite have produced exclusion plots in the @xmath15-@xmath16 plane for both @xmath9 and @xmath17 mixing @xcite . here
@xmath15 and @xmath16 are the difference of the squares of the neutrino vacuum mass eigenvalues and a measure of the vacuum mixing angle , respectively , associated with two - flavor neutrino mixing .
these studies showed that for @xmath18 , both the @xmath17 solution to the atmospheric neutrino problem @xcite and the @xmath9 large - angle msw solution to the solar neutrino problem @xcite are excluded for @xmath19 . in our study of the primordial d / h dependance of bbn constraints on active - sterile neutrino mixing ,
we have employed the same neutrino evolution formalism @xcite as previous authors .
we have neglected any net lepton number contributed by the neutrinos .
( the recently reported effect of active - sterile neutrino mixing _ generating _ net lepton number does not occur in the regions of parameter space we consider here @xcite . ) in this case , the neutrino and antineutrino sectors evolve identically .
a fermi - dirac momentum distribution for all neutrinos is assumed , but allowance is made for non - equilibrium number densities .
the differential equations in the formalism yield @xmath20 , and @xmath12 as functions of time . here
@xmath21 denotes the fraction of a full fermionic degree of freedom contributed by neutrino species @xmath22 , which we shall hereafter call the `` number density parameter '' of neutrino species @xmath22 . in the equations below we will take @xmath23 , since we are working under the assumption that the net lepton number contributed by the neutrinos is negligible . in our bbn computation
we have employed the kawano @xcite update of the wagoner @xcite code , with the latest world average neutron lifetime , @xmath24 s @xcite ; the reaction rates of ref .
@xcite ; and a correction of @xmath25 to @xmath2 due to finite nucleon mass and timestep - dependant effects @xcite .
we have altered the kawano code to use the ` temperature series ' of neutrino number density parameters , @xmath26 , to compute the energy density contributed by neutrinos and the n @xmath14 p interconversion rates .
the neutrino energy density is @xmath27 the n @xmath14 p rates are @xmath28 @xmath29 @xmath30 @xmath31 @xmath32 @xmath33 in these expressions @xmath34 , where @xmath35 and @xmath36 are the total electron ( or positron ) energy and rest mass , respectively ; @xmath37 ; @xmath38 , where @xmath39 is the appropriate neutrino temperature ; @xmath40 ; and @xmath41 is a constant obtained by solving the equation @xmath42 for @xmath41 , where @xmath43 is the experimentally measured neutron lifetime .
the lower limit on @xmath7 obtained from a standard bbn calculation with @xmath44 is not appropriate for bbn with active - sterile mixing .
this is because the lower limit on @xmath7 depends on the expansion rate , often codified as an effective number of neutrino generations @xmath45 @xcite . since active - sterile mixing increases @xmath45 ( at least for the range of parameter space we consider here @xcite ) ,
it affects the lower bound on @xmath7 .
therefore , we will plot our results as a function of the primordial d / h value the experimentally determined quantity rather than as a function of @xmath7 .
these considerations are most important for the @xmath17 atmospheric neutrino mixing solution , and much less important ( nearly negligible ) for the @xmath9 small - angle msw solution to the solar neutrino problem . in fig .
1 , a representative @xmath17 atmospheric neutrino mixing solution ( @xmath46 ev@xmath47 , @xmath48 @xcite ) is _ assumed _ , and the resulting bbn @xmath1he yield is plotted as a function of the bbn d / h yield .
the value of @xmath7 ( given as @xmath49 ) at various values of d / h is also indicated on the figure . for a given value of d / h
, the implied abundance of @xmath1he can be interpreted as the observational upper limit required to constrain the solution .
alternatively , a _ detection _ of these neutrino mixing parameters by , for example , future atmospheric neutrino experiments would yield an independent determination of the primordial @xmath1he abundance , so long as d / h were known from qas studies .
this could be very interesting , given the recent emphasis on the systematic uncertainties in the determination of the primordial @xmath1he abundance as derived from helium recombination lines in extragalactic hii regions @xcite .
the conclusions reached from fig . 1 are essentially the same over the range of @xmath15 ( @xmath50 ev@xmath47 ) for the proposed @xmath17 mixing explanation of the atmospheric neutrino problem .
fig .
2 is similar to fig . 1 , but with a representative @xmath9 small - angle msw solution to the solar neutrino problem assumed ( @xmath51 ev@xmath47 , @xmath52 @xcite ) .
these mixing parameters have only a very small effect on the bbn @xmath1he yield . for the most recent data from solar neutrino experiments and the standard solar model @xcite ,
there is no @xmath9 large - angle msw solution to the solar neutrino problem @xcite .
some of the qas data suggest d / h @xmath53 @xcite .
figs .
1 - 2 show the range d / h @xmath54 , as determined in ref .
@xcite .
this range of d / h implies a lower bound on @xmath7 that is significantly lower than that used in previous studies .
since a lower @xmath7 implies a lower @xmath1he yield , high d / h relaxes constraints on any effect that increases the expansion rate , including mixing with sterile species @xcite .
fig .
1 shows , however , that for d / h @xmath53 , the @xmath17 atmospheric neutrino solution is still somewhat constrained @xcite if current observational inferences @xcite of primordial @xmath1he are correct : @xmath55 , where the first error is statistical and the second systematic .
of course , if this @xmath17 atmospheric neutrino mixing solution were inferred from atmospheric neutrino experiments , and qas studies confirm d / h @xmath53 , the implied @xmath1he abundance of @xmath56 would be significantly higher than the central value of @xmath57 cited above .
the @xmath9 small - angle msw solar neutrino solution is allowed for d / h @xmath53 .
fig .
2 shows that an observational upper bound of @xmath58 would be required to restrict this small angle solution if such a high d / h is indeed the primordial value .
other very high quality qas data arguably better @xcite for the determination of d / h than that used in ref .
@xcite suggest d / h @xmath0 @xcite .
this value of d / h is incompatible with standard bbn with @xmath44 @xcite for current observational inferences of the primordial @xmath1he abundances @xcite , and any mixing with sterile neutrinos would only exacerbate the problem .
as mentioned previously , however , it has been argued that @xmath2 has been systematically underestimated , and a more appropriate upper limit on @xmath2 may actually be @xmath59 @xcite .
it is unlikely that the systematic error in @xmath2 is enough to allow the @xmath17 atmospheric neutrino solution for d / h @xmath0 .
however , observation of the @xmath9 small - angle msw solar neutrino solution , together with a solid determination of d / h @xmath0 , would require that @xmath2 has been systematically and significantly underestimated by about @xmath60 ( see fig .
2 ) , unless there is non - standard physics during the bbn epoch @xcite .
this is a somewhat trivial point , since the mixing parameters of the small - angle @xmath9 msw solution produce only very slightly more @xmath1he than the standard bbn picture with @xmath44 , for which the `` crisis '' at low d / h is well - known @xcite .
useful constraints on the @xmath9 small - angle msw solar neutrino solution would require very precise observational knowledge of @xmath7 and @xmath2 .
this may , however , still be interesting in view of the fact that future solar neutrino experiments may be able to distinguish the sterile neutrino oscillation - based solution from other solutions @xcite . also , many models that seek to satisfy all available constraints on neutrino properties employ the @xmath9 small - angle msw solar neutrino solution @xcite ( but see ref .
@xcite ) . |
in a recent paper kitano @xcite gives a formula for the reidemeister torsion of the 3-manifold obtained by a dehn surgery on the figure eight knot . in this paper
we generalize his result to all twist knots .
specifically , we will compute the reidemeister torsion of the complement of a twist knot in @xmath0 and that of the 3-manifold obtained by a dehn surgery on a twist knot .
let @xmath1 be the link in figure 1 , where @xmath2 denote the numbers of half twists in the boxes .
positive ( resp .
negative ) numbers correspond to right - handed ( resp .
left - handed ) twists .
note that @xmath1 is a knot if and only if @xmath3 is even .
the knot @xmath4 , where @xmath5 , is known as a twist knot . for more information on @xmath1 , see @xcite . in this paper
we fix @xmath6 .
let @xmath7 be the complement of @xmath8 in @xmath0 .
the fundamental group of @xmath7 has a presentation @xmath9 where @xmath10 are meridians and @xmath11 .
a representation @xmath12 is called nonabelian if the image of @xmath13 is a nonabelian subgroup of @xmath14 .
suppose @xmath15 is a nonabelian representation .
up to conjugation , we may assume that @xmath16 \quad \text{and } \quad \rho(b ) = \left [ \begin{array}{cc } s & 0 \\ -u & s^{-1 } \end{array } \right]\ ] ] where @xmath17 is a root of the riley polynomial @xmath18 , see @xcite .
let @xmath19 and @xmath20 .
let @xmath21 be the chebychev polynomials of the second kind defined by @xmath22 , @xmath23 and @xmath24 for all integers @xmath25 .
[ main1 ] suppose @xmath15 is a nonabelian representation .
if @xmath26 then the reidemeister torsion of @xmath7 is given by @xmath27 now let @xmath28 be the 3-manifold obtained by a @xmath29-surgery on the twist knot @xmath8 .
the fundamental group @xmath30 has a presentation @xmath31 where @xmath32 is the canonical longitude corresponding to the meridian @xmath33 .
[ main2 ] suppose @xmath15 is a nonabelian representation which extends to a representation @xmath34 .
if @xmath35 then the reidemeister torsion of @xmath28 is given by @xmath36 \(1 ) one can see that the expression @xmath37 is actually a polynomial in @xmath38 .
\(2 ) theorem [ main2 ] generalizes the formula for the reidemeister torsion of the 3-manifold obtained by a @xmath29-surgery on the figure eight knot by kitano @xcite .
\(1 ) if @xmath39 , then @xmath40 is the trefoil knot . in this case
the riley polynomial is @xmath41 , and hence @xmath42 \(2 ) if @xmath43 , then @xmath44 is the figure eight knot . in this case
the riley polynomial is @xmath45 , and hence @xmath46 the paper is organized as follows . in section [ section - chev ]
we review the chebyshev polynomials of the second kind and their properties . in section
[ nab ] we give a formula for the riley polynomial of a twist knot , and compute the trace of a canonical longitude . in section
[ section - r ] we review the reidemeister torsion of a knot complement and its computation using fox s free calculus .
we prove theorems [ main1 ] and [ main2 ] in section [ section - proof ] .
recall that @xmath21 are the chebychev polynomials defined by @xmath22 , @xmath23 and @xmath24 for all integers @xmath25 .
the following lemma is elementary .
[ chev ] one has @xmath47 let @xmath48 .
[ p_k ] one has @xmath49 we have @xmath50 the lemma follows .
[ p^2_k ] one has @xmath51 let @xmath52 we have @xmath53 since @xmath54 , we obtain @xmath55 for all integers @xmath25 .
hence @xmath56 .
[ formulas ] suppose @xmath57 \in sl_2(\bc)$ ] .
then @xmath58 , \label{power}\\ \sum_{i=0}^k v^i & = & \left [ \begin{array}{cc } p_{k}(t ) - d p_{k-1}(t ) & b p_{k-1}(t)\\ c p_{k-1}(t ) & p_{k}(t ) - a p_{k-1}(t ) \end{array } \right ] , \label{sum - power}\end{aligned}\ ] ] where @xmath59 .
moreover , one has @xmath60
since @xmath61 , by the cayley - hamilton theorem we have @xmath62 .
this implies that @xmath63 for all integers @xmath25 .
hence , by induction on @xmath25 , one can show that @xmath64 . since @xmath65 $ ] , follows . since @xmath66
, follows directly from . by lemma [ p^2_k ]
we have @xmath67 then follows from lemma [ p_k ] .
in this section we give a formula for the riley polynomial of a twist knot .
this formula was already obtained in @xcite .
we also compute the trace of a canonical longitude .
recall that @xmath68 and @xmath69 .
the fundamental group of @xmath7 has a presentation @xmath70 where @xmath10 are meridians and @xmath11 .
suppose @xmath15 is a nonabelian representation .
up to conjugation , we may assume that @xmath16 \quad \text{and } \quad \rho(b ) = \left [ \begin{array}{cc } s & 0 \\ -u & s^{-1 } \end{array } \right]\ ] ] where @xmath17 is a root of the riley polynomial @xmath18 .
we now compute @xmath18 . since @xmath71,\ ] ] by lemma [ formulas ] we have @xmath72,\ ] ] where @xmath73 .
hence , by a direct computation we have @xmath74\ ] ] where @xmath75 it is known that the canonical longitude corresponding to the meridian @xmath33 is @xmath76 , where @xmath77 is the word in the letters @xmath10 obtained by writing @xmath78 in the reversed order .
we now compute its trace .
this computation will be used in the proof of theorem [ main2 ] .
[ s^2 ] one has @xmath79 since @xmath17 is a root of the riley polynomial @xmath18 , we have @xmath80 .
lemma [ chev ] then implies that @xmath81 by replacing @xmath82 into the first factor of the above expression , we obtain the desired equality . [ longitude ] one
has @xmath83 by lemma [ formulas ] we have @xmath84.\ ] ] similarly , @xmath85.\ ] ] hence , by a direct calculation we have @xmath86 the lemma then follows from lemma [ s^2 ] .
in this section we briefly review the reidemeister torsion of a knot complement and its computation using fox s free calculus . for more details on the reidemeister torsion ,
see @xcite .
let @xmath87 be a chain complex of finite dimensional vector spaces over @xmath88 : @xmath89 such that for each @xmath90 the followings hold * the homology group @xmath91 is trivial , and * a preferred basis @xmath92 of @xmath93 is given .
let @xmath94 be the image of @xmath95 . for each @xmath96 choose a basis @xmath97 of @xmath98 .
the short exact sequence of @xmath88-vector spaces @xmath99 implies that a new basis of @xmath93 can be obtained by taking the union of the vectors of @xmath97 and some lifts @xmath100 of the vectors @xmath101 .
define @xmath102 $ ] to be the determinant of the matrix expressing @xmath103 in the basis @xmath92 .
note that this scalar does not depend on the choice of the lift @xmath100 of @xmath101 .
the _ torsion _ of @xmath87 is defined to be @xmath104^{(-1)^{i+1 } } \ \in \bc\setminus\{0\}.\ ] ] once a preferred basis of @xmath87 is given , @xmath105 is independent of the choice of @xmath106 .
let @xmath28 be a finite cw - complex and @xmath34 a representation .
denote by @xmath107 the universal covering of @xmath28 .
the fundamental group @xmath30 acts on @xmath107 as deck transformations .
then the chain complex @xmath108 has the structure of a chain complex of left @xmath109$]-modules .
let @xmath110 be the 2-dimensional vector space @xmath111 with the canonical basis @xmath112 . using the representation @xmath13
, @xmath110 has the structure of a right @xmath109$]-module which we denote by @xmath113 .
define the chain complex @xmath114 to be @xmath115 } v_{\rho}$ ] , and choose a preferred basis of @xmath114 as follows .
let @xmath116 be the set of @xmath96-cells of @xmath28 , and choose a lift @xmath117 of each cell .
then @xmath118 is chosen to be the preferred basis of @xmath119 .
a representation @xmath13 is called _ acyclic _ if all the homology groups @xmath120 are trivial .
the reidemeister torsion @xmath121 is defined as follows : @xmath122 let @xmath123 be a knot in @xmath0 and @xmath124 its complement .
we choose a wirtinger presentation for the fundamental group of @xmath124 : @xmath125 let @xmath126 be a representation .
this map induces a ring homomorphism @xmath127\to m_2(\bc)$ ] , where @xmath128 $ ] is the group ring of @xmath129 and @xmath130 is the matrix algebra of degree @xmath131 over @xmath132 .
consider the @xmath133 matrix @xmath134 whose @xmath135-component is the @xmath136 matrix @xmath137 where @xmath138 denotes the fox calculus .
for @xmath139 , denote by @xmath140 the @xmath141 matrix obtained from @xmath134 by removing the @xmath142th column .
we regard @xmath140 as a @xmath143 matrix with coefficients in @xmath88 .
then johnson showed the following .
@xcite [ johnson ] let @xmath126 be a representation such that @xmath144 .
then the reidemeister torsion of @xmath124 is given by @xmath145
suppose @xmath15 is a nonabelian representation which extends to a representation @xmath34 . recall that @xmath32 is the canonical longitude corresponding to the meridian @xmath33 .
if @xmath173 , then by @xcite ( see also @xcite ) the reidemeister torsion of @xmath28 is given by @xmath174 | we compute the reidemeister torsion of the complement of a twist knot in @xmath0 and that of the 3-manifold obtained by a dehn surgery on a twist knot . |
the b - v colour difference between young and evolved bars is 0.4 mag , which can be translated to an age difference of 10 gyr .
this means that bars can be robust structures , in agreement with recent @xmath0-body simulations and observations of barred galaxies at higher redshifts .
the young bars in our sample have an average length of 5.4@xmath11.6 kpc , while the evolved bars have an average length of 7.5@xmath11.2 kpc , consistent with recent theoretical expectations that bars grow longer while aging .
young bars are preferentially found in late - type spirals , indicating that bar recurrence may be more frequent in gas - rich , disk - dominated galaxies .
we also found that agn are preferentially hosted by galaxies with young bars , suggesting that the fueling of agn by bars happens in short timescales and that a clearer bar - agn connection would be found in a sample of galaxies with young bars .
we have also found that bar colours might be used as a proxy for bar ages .
enlarging the sample of bars with measured ages is paramount to calibrate this relation , confirm these results , compare in more detail observations and models , and better understand secular evolution .
see ( * ? ? ?
* ; * ? ? ?
* gadotti & de souza ( 2005 , 2006 ) ) for further details . | in an effort to obtain further observational evidences for secular evolution processes in galaxies , as well as observational constraints to current theoretical models of secular evolution , we have used bvri and ks images of a sample of 18 barred galaxies to measure the lengths and colours of bars , create colour maps and estimate global colour gradients .
in addition , applying a method we developed in a previous article , we could distinguish for 7 galaxies in our sample those whose bars have been recently formed from the ones with already evolved bars .
we estimated an average difference in the optical colours between young and evolved bars that may be translated to an age difference of the order of 10 gyr , meaning that bars may be long standing structures .
moreover , our results show that , on average , evolved bars are longer than young bars .
this seems to indicate that , during its evolution , a bar grows longer by capturing stars from the disk , in agreement with recent numerical and analytical results . |
subluminous stars are considered to be core helium - burning objects with a thin hydrogen shell . in a hertzsprung - russell diagram these stars are situated on the extreme horizontal branch ( ehb ) .
the evolution of these stars is far from understood since the formation of an sdb requires extreme mass loss at the tip of the rgb .
the mechanism leading to this mass loss is still under debate @xcite .
formation scenarios can be roughly divided into single star and binary channels . about 50% of the known sdbs
are found to be in close binaries with periods of few hours to several days consistent with the predictions of the common envelope ( ce ) ejection scenario @xcite .
two main sequence stars evolve in a binary system .
the more massive one becomes a red giant first and fills its roche lobe .
unstable mass transfer leads to the formation of a ce .
the core of the red giant and the immersed secondary experience a loss of orbital energy which is deposited in the ce causing a spiral - in towards the center of mass .
as soon as enough energy has been deposited in the envelope , it is ejected .
the inwards migration stops and the stars end up in a close binary .
this scenario is well suited to explain the known population of close sdb binaries .
but how are the apparently single sdbs formed ?
@xcite suggested that substellar companions like planets or brown dwarfs may be able to trigger ce ejection as well . in order to survive the common envelope and avoid evaporation inside the envelope
the secondary should have a minimum mass of more than @xmath0 .
a similar scenario has been proposed for the formation undermassive single white dwarfs by @xcite .
several substellar objects have recently been detected around hot subdwarfs in wide orbits , showing that planets can survive the red giant phase of their host star @xcite .
recently , @xcite discovered a candidate substellar companion in close orbit around the bright sdb hd 149382 , which perfectly fits into the pattern predicted by soker .
systems like that may be quite common and could have remained unnoticed up to now .
the goal of our project is to put constraints on the population of the lowest mass companions to sdb stars by measuring accurate radial velocities from multi - epoch high resolution spectra .
up to now our survey consists of 23 bright sdb stars with visual magnitudes ranging from @xmath1 to @xmath2 .
we used high resolution spectra obtained with the feros instrument mounted at the eso / mpg-2.2 m telescope .
the spectra have a resolution of @xmath3 .
spectra were reduced , calibrated and corrected for earth s orbital motion with the midas package using the feros reduction pipeline .
all programme stars are single - lined objects without visible spectral features of a companion . furthermore , those binaries with high radial velocity variability have been excluded because their unseen companions must be quite massive .
each star has been observed several times ( from 2 to 15 ) on timescales ranging from one day to some years . as an example we display a section of the feros spectrum of hd 205805 in fig [ fig : uebers ] .
subdwarf b stars are hot ( @xmath4 ) and their atmospheres radiative .
the dominant features in their spectra are the hydrogen balmer and several helium lines .
but the spectra also show up to @xmath5 weak metal absorption lines which are very sharp .
these features are best suited to measure radial velocities .
the atmospheres of sdb stars are considered to be stable .
the only known effect other than a close companion that may lead to rv variability are pulsations .
pulsating sdb stars do indeed exist with periods between @xmath6 and @xmath7 and rv amplitudes of few km s@xmath8 at most @xcite .
for our analysis we chose a set of sharp , unblended metal lines situated between @xmath9 and @xmath10(see fig .
[ fig : fitlines ] , left hand panel ) .
accurate rest wavelengths were taken from the nist database .
depending on the atmospheric parameters of the star and the quality of the data a subset was used .
the rv measurement consisted of two steps .
first , we fitted gaussian and lorentzian profiles to each line separately using the spas routine ( hirsch priv .
the resulting mean rvs had standard deviations ranging from @xmath11 to @xmath12 .
second , we used the fitsb2 routine @xcite to perform a simultaneous fit of all suitable lines .
uncertainties were calculated using a bootstrapping algorithm and found to range from @xmath13 to @xmath14 , while both methods result in consistent mean rvs .
the scatter of individual rv values measured with spas around the mean value was reviewed for possible systematic effects like correlations with certain elements .
no such effects could be found . to check the wavelength calibration for systematic errors we used telluric features as well as night - sky emission lines ( see fig .
[ fig : fitlines ] , right hand panel ) .
having their origin on earth these lines should have zero rv .
the feros instrument turned out to be very stable .
usually corrections of less than @xmath14 had to be applied . in order to derive the degree of precision we can achieve with our method , we generated synthetic model spectra with realistic atmospheric parameters including poisson noise with @xmath15 and determined the rv in the way described above .
the uncertainties turned out to range from @xmath13 to @xmath16 .
the accuracy is limited by the resolution of the spectrograph , the number of features and the intrinsic thermal broadening of the lines .
four stars were found to be significantly ( another two marginally ) rv variable .
the shift in rv ranges from @xmath17 to @xmath18 ( see tab . [ tab : a ] ) . as an example we display two rv measurements for ec 20106@xmath195248 in fig .
[ fig : bigshift ] .
follow - up photometry and high resolution spectroscopy are necessary to exclude pulsational variability and derive the orbital parameters of these binaries .
constraints can then be put on the companion masses . even if there are no low mass companions we would expect to find a certain fraction of systems with small rv shifts caused by more massive secondaries seen at low inclination .
a larger sample is needed to exclude this case .
.summary of rv results for our sample obtained using two different algorithms .
rvs for each line were measured independently and averaged with spas ( left hand column ) whereas fitsb2 uses a simultaneous fit ( right hand column ) . [ cols="<,>,>,>,>",options="header " , ] no rv variations were found in the rest of the sample .
nevertheless , the errors allow to derive upper limits for @xmath20 of undetected substellar companions in close orbits from the binary mass function ( eq . [ eq:1 ] ) .
@xcite suggested that these objects should be more massive than @xmath0 and should have orbital periods shorter than @xmath22 .
adopting this maximum period and assuming a canonical mass of @xmath23 for the sdb , the errors of the rv measurements can be used as upper limits for the rv semi - amplitudes ( @xmath24 ) . as can be seen in fig .
[ fig : const ] , tight constraints can be put on possible substellar companions , if the measurement error is smaller than @xmath25 .
given data of perfect quality , close - in planets with masses as low as @xmath26 can be firmly excluded if our assumption about the orbital period is correct . in conclusion @xmath27 of the apparently single stars in our sample show small rv variations .
the true fraction is expected to be higher , because the accuracy is limited by the s / n in most cases .
companions with masses higher than @xmath0 in close orbits ( @xmath28 ) can be excluded for @xmath29 of our sample .
one possible explanation for the fractions found is that the assumed ce scenario is not the only channel for sdb formation .
however , even with ce ejection being the only channel , the companion may be destroyed during the ce phase either by evaporation or a merger with the stellar core leaving behind a single sdb .
better statistics and reliable predictions from theoretical calculations are required to solve this problem .
s. geier , h. edelmann , u. heber , and l. morales - rueda , _ apj _ * 702 * , l96l99 ( 2009 ) .
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r. napiwotzki , l. yungelson , g. nelemans , t. r. marsh , b. leibundgut , r. renzini , d. homeier , d. koester , s. moehler , n. christlieb , d. reimers , h. drechsel , u. heber , c. karl , and e. pauli , `` double degenerates and progenitors of supernovae type ia , '' in _ spectroscopically and spatially resolving the components of the close binary stars _ , edited by r. w. hilditch , h. hensberge , & k. pavlovski , 2004 , vol .
318 of _ astronomical society of the pacific conference series _ , pp . 402410 .
s. qian , l. zhu , s. zola , w. liao , l. liu , l. li , m. winiarski , e. kuligowska , and j. m. kreiner , _ apj _ * 695 * , l163l165 ( 2009 ) .
r. silvotti , s. schuh , r. janulis , j. solheim , s. bernabei , r. stensen , t. d. oswalt , i. bruni , r. gualandi , a. bonanno , g. vauclair , m. reed , c. chen , e. leibowitz , m. paparo , a. baran , s. charpinet , n. dolez , s. kawaler , d. kurtz , p. moskalik , r. riddle , and s. zola , _ nature _ * 449 * , 189191 ( 2007 ) . | the origin of subdwarf b ( sdb ) stars is not fully understood yet since it requires high mass loss at the red giant stage .
sdbs in close binary systems are formed via common envelope ejection , but the origin of apparently single sdb stars remains unclear .
substellar companions may be able to trigger common envelope ejection and help forming sdbs that appear to be single .
+ using a sample of high resolution spectra we aim at detecting small radial velocity ( rv ) shifts caused by such low mass ( sub-)stellar companions .
the rvs are measured with high accuracy using sharp metal lines .
our goal is to test the theoretical predictions and put constraints on the population of the lowest mass companions to sdb stars .
address = dr .
karl remeis observatory & ecap , astronomical institute , friedrich - alexander university erlangen - nuremberg , sternwartstr .
7 , d-96049 bamberg , germany address = dr .
karl remeis observatory & ecap , astronomical institute , friedrich - alexander university erlangen - nuremberg , sternwartstr
. 7 , d-96049 bamberg , germany address = dr .
karl remeis observatory & ecap , astronomical institute , friedrich - alexander university erlangen - nuremberg , sternwartstr
. 7 , d-96049 bamberg , germany address = australian astronomical observatory , po box 296 , epping , nsw , 1710 , australia |
to prove our main theorem , we need to maximize the bell expression @xmath93,\,\boldsymbol{b}\in[\boldsymbol{o}],\,x\in[m],\,\boldsymbol{y}\in[\boldsymbol{m } ] } \beta_{a,\boldsymbol{b},x,\boldsymbol{y } } \ , \operatorname{tr}\left[m^{(a)}_x\,\sigma_{\boldsymbol{b}|\boldsymbol{y}}\right ] \label{eq : bell_ineqexcltriv}\ ] ] over all sets @xmath94}$ ] of generic dichotomic povm measurements @xmath95 by alice . however , it is known that , to test for bell nonlocality of quantum states , it suffices to examine only von neumann ( rank-1 projective ) measurements @xcite . the same happens for bell violations by assemblages , which we formalize with the following lemma .
[ lemma : excludingpovms ] let @xmath32 be an arbitrary well - behaved bell inequality with dichotomic outputs for alice and @xmath44 a generic qubit assemblage .
then , the maximal violation of @xmath32 by @xmath44 is attained under rank-1 projective measurements . _
proof_. our proof strategy consists in showing that , for an arbitrary assemblage @xmath44 and generic dichotomic povm measurements @xmath96 , the distribution @xmath97 , given by eq . , obtained from @xmath44 under @xmath96 , is equivalent to a distribution @xmath98 obtained from @xmath44 under a set @xmath99 of von neumann measurements followed by a local mixing of alice s outputs . since @xmath32 is well behaved , this implies that if @xmath97 violates @xmath32 , i.e. if @xmath53 , then @xmath100 , which implies that the maximal violation is always attained under von neumann measurements . since , for all @xmath101 $ ] , the povm measurement operators @xmath102 and @xmath103 are both non - negative , they can be diagonalized as @xmath104 where @xmath105 and @xmath106 are rank-1 orthonormal projectors i.e. @xmath107 , being @xmath108 the kronecker delta acting on @xmath19 , and where @xmath109 for @xmath110 . hence , @xmath111}$ ] defines a set of von neumann measurements . substituting eq . into eq .
( [ eq : pofsigma ] ) , we find that @xmath112 \ , \end{aligned}\ ] ] where we have introduced @xmath113 now , defining the distribution @xmath98 such that @xmath114 $ ] , we can write @xmath115 } q(a|a',x ) \
\tilde{p}_{\xi}(a',\boldsymbol{b}|x,\boldsymbol{y } ) \ .
\label{eq : povmaslocmix}\ ] ] this , as evident from eq .
( [ eq : deflocmix ] ) , is the expression of a local mixing of alice s outputs applied to the von neumann measurement distribution @xmath116 .
we can now continue with the proof of criterion [ crit : conds_for_bell_viol ] .
due to lemma [ lemma : excludingpovms ] , we need to maximize the bell expression only over the set of von neumann measurements , i.e. , with measurement operators @xmath47 of the form @xmath117 where each ( unit ) vector @xmath118 represents a direction on the bloch sphere .
recall that @xmath119 is the pauli - operator vector with respect to a fixed basis of @xmath19 of one s preference , so that the vectors @xmath120 are the only variables of the optimization .
using eqs . and
, and the fact that the vectors @xmath121}$ ] are all independent , we get @xmath122 } \left\{\sum_{a\in[2],\,\boldsymbol{b}\in[\boldsymbol{o}],\,\boldsymbol{y}\in[\boldsymbol{m}]}\frac12\beta_{a,\boldsymbol{b},x,\boldsymbol{y } } \ , p(\boldsymbol{b}|\boldsymbol{y } ) + \max_{\{\hat{\boldsymbol{s}}_x\}}\operatorname{tr}\left [ b_x\ , \hat{\boldsymbol{s}}_x \cdot \boldsymbol{\sigma}\right]\right\}\ .
\label{eq : recastexpr}\ ] ] where , for each @xmath101 $ ] , we have introduced the hermitean operator on @xmath19 @xmath123,\,\boldsymbol{y}\in[\boldsymbol{m } ] } \frac12(\beta_{0,\boldsymbol{b},x,\boldsymbol{y}}-\beta_{1,\boldsymbol{b},x,\boldsymbol{y}})\ , \sigma_{\boldsymbol{b}|\boldsymbol{y } } \ .
\label{eq : bobops}\ ] ] note that @xmath124 coincides with the expression inside the brackets of eq .
( [ eq : main_result_r ] ) . then , using that @xmath125 = \boldsymbol r(b_x)\cdot\hat{\boldsymbol{s}}_x$ ] , with the vector function @xmath126 defined in eq .
( [ bldef ] ) , the maximization is finally reduced to @xmath127= \max_{\{\hat{\boldsymbol{s}}_x\}}\ \boldsymbol{r}(b_x)\cdot { \hat{\boldsymbol{s}}}_x \ .
\label{eq : reducedprob}\ ] ] clearly , the maximum is @xmath127=\|{\boldsymbol s}^{\rm opt}_x\| \ , \label{bestexpr}\ ] ] with @xmath128 for all @xmath101 $ ] , attained by @xmath129 substituting eq . into eq .
, one obtains the optimal measurement settings of eq .
( [ eq : opt_meas ] ) . using , in turn , eqs . and
, one sees that eq .
is equivalent to the left - hand side of eq.([eq : main_result ] ) . in the 2-input , 2-output scenario , by virtue of fine s theorem @xcite ,
bell nonlocality is equivalent to the violation of the chsh inequality , given by @xmath130 or any of its 8 symmetries ( defined by swapping around the minus sign with the other terms , by applying an overall sign change , or by doing both ) .
so , it suffices to show that a violation of eq .
( [ eq:222 ] ) is equivalent to the violation of any of the 8 symmetries of the chsh inequality . in the notation of eq .
( [ eq : bell_ineq ] ) , and omitting the subindices from @xmath74 and @xmath75 , the chsh inequality is expressed as @xmath131 with @xmath132 .
its symmetries , in turn , are obtained by replacing @xmath28 or @xmath133 by their negations modulo 2 , by applying an overall sign change to @xmath134 , and by applying any composition of the three .
substituting eq . in eqs .
( [ eq : main_result ] ) and ( [ eq : main_result_r ] ) leads to eqs .
( [ eq:222 ] ) and ( [ eq:222_r ] ) , as the reader can straightforwardly verify .
this shows that the violation of eq .
( [ eq:222 ] ) , with the measurement direction @xmath135 given by eq .
( [ eq:222_r ] ) , is equivalent to the violation of the chsh inequality .
now , note that any of the other symmetries mentioned above either does not explicitly introduce any change in eqs .
( [ eq:222 ] ) and ( [ eq:222_r ] ) or simply corresponds to the relabelings @xmath136 , @xmath137 , for all @xmath138 $ ] , or compositions of the two .
none of the latter alters the statements of criterion [ crit : twotwo ] .
that is , the violation of eq .
( [ eq:222 ] ) , with @xmath135 given by eq .
( [ eq:222_r ] ) , is equivalent to the violation of any of the symmetries of the chsh inequality , which finishes the proof .
for any assemblage @xmath139}$ ] , the correlator @xmath140 , where @xmath141 is a @xmath85-valued unknown observable of bob s subsystem and @xmath142 is a pauli operator on @xmath56 , is given by @xmath143 } ( -1)^{b } \operatorname{tr}\left ( \sigma_{b|y } \ , a_{\alpha } \right)\\ & = \boldsymbol r \left(\sum_{b\in[2 ] } ( -1)^{b } \ , \sigma_{b|y } \right)\cdot \hat{\boldsymbol v}_{\alpha } \ , \label{eq : corrbasic2}\end{aligned}\ ] ] where @xmath144 is a unit vector in the bloch sphere in the direction of @xmath145 . using eqs .
, ( [ eq:222_r ] ) and the linearity of the vector function @xmath126 , one sees that @xmath146 and , analogously , @xmath147 it is now straightforward to see , from the definition of the euclidian norm , that eq .
( [ eq:222 ] ) is equivalent to eq .
( [ eq:222meas ] ) for @xmath148 the pauli operators in any orthonormal basis of @xmath19 .
furthermore , one can also see that the lhs of inequality ( [ eq : australians ] ) is equivalent to the lhs of inequality ( [ eq:222 ] ) evaluated at the projections of @xmath89 and @xmath88 onto the plane orthogonal to @xmath149 , instead of at @xmath89 and @xmath88 themselves . | we study multipartite bell nonlocality in a framework native of multipartite einstein - podolsky - rosen ( epr ) steering scenarios with a single trusted measurement device .
we derive a closed - form necessary and sufficient criterion for systems composed of a qubit and @xmath0 untrusted black - box measurement devices to violate under general dichotomic measurements on the qubit a generic bell inequality from a broad family of linear inequalities with arbitrarily many outputs for the @xmath0 untrusted devices and inputs for all @xmath1 parties .
the optimal quantum measurements for maximal violation are also obtained . for two users , and two inputs and two outputs per user , our criterion becomes necessary and sufficient for bell nonlocality .
furthermore , in that setting , its form generalizes recently obtained steering inequalities , which allows us to provide useful feedback from nonlocality to the detection of steering .
our findings constitute a practical tool for the study of the interplay between epr steering and bell nonlocality , with potential applications in multipartite information processing . _
introduction_. composite quantum systems can display exotic forms of non - classical correlations , a phenomenon known under the generic name of _ quantum nonlocality_. quantum nonlocal correlations can appear in three main variants .
the first one is _ entanglement _ , which refers to inseparability of quantum states ( described by density matrices ) @xcite .
the second one is _
bell nonlocality _
@xcite , the impossibility of explaining measurement statistics ( described by joint probability distributions ) with local hidden - variable ( lhv ) models @xcite .
the third variant is called _ einstein - podolsky - rosen ( epr ) steering _
@xcite , after the famous epr paper @xcite .
this is an effect by which ensembles of quantum states are remotely prepared by local measurements at distant labs @xcite .
the observable data in steering experiments thus consist of the measurement statistics at one lab and quantum states at another .
these data are compactly described by a joint mathematical object called _ assemblage _ , composed of a conditional probability distribution and a set of density matrices .
hence , epr steering constitutes an intermediate notion between entanglement and bell nonlocality @xcite .
apart from their fundamental importance , nonlocal correlations are of practical relevance : they are resources for physical tasks such as quantum key - distribution ( qkd ) @xcite and quantum random - number generation @xcite .
entanglement is a resource @xcite in the device - dependent ( dd ) scenario , defined by well - characterized , trusted quantum measurements .
bell nonlocality is useful for device - independent ( di ) protocols @xcite , i.e. , where the experimenters possess untrusted black - box measurement apparatuses . in turn
, epr steering has been identified @xcite as a resource @xcite for one - sided di situations , where one of the parties has an untrusted black - box device while the other possesses a trusted quantum platform .
fully ( both - sided ) di protocols relax the need for device characterization totally , but at the expense of being very demanding experimentally @xcite .
one - sided di implementations offer a middle - path alternative , relaxing device characterization only on one side but having , in return , less stringent experimental requirements @xcite for security @xcite than in fully di ones .
this is relevant to any asymmetric situation involving users with different levels of quantum control .
these developments motivated a great amount of work on the interplay between the different forms of quantum nonlocality .
the first problem tackled was that of entangled versus bell nonlocal states ( those capable of exhibiting bell nonlocality ) .
all pure entangled states were proven to be bell nonlocal @xcite , but mixed bell local entangled states were found @xcite .
later , necessary and sufficient conditions for arbitrary 2-qubit states to be bell nonlocal were derived @xcite .
a long list of works then followed these pioneering results ( see , for instance , sec .
iii.a of @xcite and refs .
therein ) .
the second problem was that of entangled versus steerable states ( those capable of exhibiting epr steering ) .
unsteerable entangled states , as well as bell local steerable ones , were found @xcite .
this led to an impressive amount of work : the sets of entangled , steerable and bell nonlocal states were proven inequivalent under general measurements @xcite ; necessary criteria for a two - qubit state to be steerable were found @xcite ; and constructive methods to test for unsteerability of a state were developed @xcite .
finally , in addition to the many known steering inequalities @xcite , a necessary and sufficient criterion for epr steering has been recently obtained for minimal - dimension assemblages
@xcite . here
we consider a third problem : steerable versus bell nonlocal multipartite assemblages .
we derive a closed - form necessary and sufficient criterion for an @xmath1-partite assemblage with a single trusted device , in possession of a user called alice , to violate , under general measurements [ positive operator - valued measure ( povm ) ] by her , a bell inequality .
the optimal measurements for the maximal violation are also given .
our theorems assume that alice s measurements are dichotomic and that her system is a qubit , but are otherwise general in the number of outcomes for the untrusted devices and of settings for all parties .
furthermore , we make only minimal assumptions on the bell inequalities treated , namely , that they are linear and that their violations do not increase under probabilistic local mixings of alice s outputs .
thus , many of the most popular bipartite @xcite and multipartite @xcite bell inequalities are within the range of applicability of our criterion . in addition , for @xmath2 users with 2 inputs and 2 outputs per user , our criterion unambiguously characterizes all bell nonlocal assemblages .
interestingly , in that setting , our criterion generalizes , in form , recently obtained steering inequalities @xcite . by virtue of this , we provide insight into the detection of epr steering within the framework of bell nonlocality and explain formal links between the two problems .
finally , we suggest potential connections of our findings with information - theoretic protocols with asymmetric levels of quantum control among the users involved . _
preliminaries_. we consider @xmath1 space - like separated parties , alice , in possession of a trusted measurement device , and @xmath0 users , @xmath3 , @xmath4 , @xmath5 , in possession of untrusted devices @xcite .
this is the @xmath6-sided di scenario .
each @xmath7-th untrusted device , for @xmath8 , is treated as a black box with unknown internal functioning , which , given an input @xmath9 $ ] , outputs @xmath10 $ ] , where @xmath11 and the notation @xmath12\coloneqq\{0,\ldots , n-1\}$ ] , for any @xmath13 , is introduced .
in addition , we will also use the short - hand notation @xmath14\coloneqq[n]^{n-1}$ ] .
alice s subsystem , in turn , is a qubit , on which she can perform any quantum measurement of her choice .
the joint system state is specified by an @xmath6-partite conditional probability distribution @xmath15 of the output string @xmath16 given the input string @xmath17 , associated to a ( normalized ) conditional single - partite quantum state @xmath18 on alice s subsystem s hilbert space @xmath19 .
these can be conveniently encapsulated in the _ assemblage _ @xmath20,\ , \boldsymbol{y}\in[\boldsymbol{m}]},\ ] ] of ( subnormalized ) conditional quantum states @xmath21 on @xmath19 , with @xmath22 .
in other words , @xmath23 provides a concise description of all the observable information in @xmath6-sided di experiments . on the other hand , in the fully di scenario of all @xmath1 users possessing black - box devices ,
the joint system behavior is described by an @xmath1-partite conditional distribution @xmath24,\ , \boldsymbol{b}\in[\boldsymbol{o } ] , \ , x\in[m],\ , \boldsymbol{y}\in[\boldsymbol{m}]},\ ] ] where @xmath25 is the probability of output values @xmath26 and @xmath27 conditioned on input values @xmath28 and @xmath29 . for ease of notation , we assume throughout that the numbers of inputs and outputs , @xmath30 and @xmath31 , respectively , are the same for all @xmath1 users , but all our results are also valid otherwise .
bell inequalities offer a practical tool to test for bell nonlocality in a given distribution @xcite .
every linear bell inequality is represented by a pair @xmath32 , with @xmath33,\ , \boldsymbol{b}\in[\boldsymbol{o } ] , \ , x\in[m],\ , \boldsymbol{y}\in[\boldsymbol{m}]}$ ] and @xmath34 , such that @xmath35,\ , \boldsymbol{b}\in[\boldsymbol{o}]\\ x\in[m],\ , \boldsymbol{y}\in[\boldsymbol{m } ] } } \beta_{a,\boldsymbol{b},x,\boldsymbol{y } } \ , p(a,\boldsymbol{b}|x,\boldsymbol{y } ) \leq \beta_{\rm l}\ ] ] for all bell local @xmath36 .
furthermore , in the multipartite scenario , bell inequalities can also be tailored so as to test for different forms of multipartite bell non locality @xcite .
our criterion below holds for all linear bell inequalities whose violations do not increase under local probabilistic mixings of alice s outputs , to which we refer , for short , as _
well - behaved bell inequalities_. more precisely , local mixings map @xmath36 into a distribution @xmath37 with elements @xmath38 } q(a|a',x ) \
p(a',\boldsymbol{b}|x,\boldsymbol{y } ) \ , \label{eq : deflocmix}\ ] ]
where @xmath39 , with @xmath40 } q(a|a',x)=1 $ ] , characterizes the mixing probability for each input @xmath28 and output @xmath41 .
hence , @xmath32 is well behaved if @xmath42 for all @xmath36 for which @xmath43 .
local mixings can map local distributions only into local distributions .
so , that a bell violation does not increase under such mixings is a basic reasonable property typically satisfied by known inequalities ( including all tight ones and , more generally , all those for which a constant local weight @xcite implies a constant violation ) .
examples not satisfying this property can be found among reducible inequalities that have superfluous terms @xcite .
finally , we say that @xmath44 _ violates a bell inequality _
@xmath32 if there exists a set @xmath45}$ ] of measurements @xmath46}$ ] , with non - negative measurement operators @xmath47 on @xmath19 fulfilling @xmath48 } m^{(a)}_x=\mathbb1 $ ] for all @xmath49 $ ] , @xmath50 being the identity operator on @xmath19 , such that the distribution @xmath51,\ , \boldsymbol{b}\in[\boldsymbol{o}],\ , x\in[m],\ , \boldsymbol{y}\in[\boldsymbol{m}]}$ ] , defined by @xmath52\ \forall\ , \substack{a\in[o],\ , \boldsymbol{b}\in[\boldsymbol{o}]\\ x\in[m],\ , \boldsymbol{y}\in[\boldsymbol{m } ] } , \end{aligned}\ ] ] violates @xmath32 , i.e. if @xmath53 .
our definition of bell violation by an assemblage considers only non - sequential measurements on a single copy of the assemblage . for quantum states
, it is known that measurements on multiple copies of the state @xcite , or sequential measurements ( filterings ) @xcite on a single copy , can produce bell violations by entangled states that would otherwise yield local correlations .
in fact , it has even been suggested @xcite that every entangled state might be bell nonlocal in this broader sense .
something similar is expected to happen with steerable assemblages .
however , the conditions for bell violations of assemblages in more general measurement scenarios are outside the scope of the present contribution .
_ conditions for bell violations_. before our first theorem , we need to introduce some notation .
since @xmath54 , any hermitian operator @xmath55 on @xmath56 can be decomposed in a bloch - sphere - like form @xmath57 $ ] , where @xmath58\in\mathbb{r}$ ] , @xmath59 is the pauli - operator vector , and @xmath60,\operatorname{tr}[o\,y],\operatorname{tr}[o\,z]\right)\in\mathbb{r}^3.\ ] ] if @xmath55 is a state , @xmath61 represents its bloch vector . finally , for any @xmath62 , we denote its euclidean norm by @xmath63 .
we can now present our main result , which we prove in the appendix [ app : main ] : [ crit : conds_for_bell_viol ] let @xmath44 be a generic assemblage given by eq . and let @xmath32 be a well - behaved bell inequality with dichotomic measurements for alice .
then , @xmath44 violates @xmath32 if , and only if , @xmath64}\left[\sum _ { \substack{a\in[2],\ , \boldsymbol{b}\in[\boldsymbol{o}]\\ \boldsymbol{y}\in[\boldsymbol{m } ] } } \frac12\beta_{a,\boldsymbol{b},x,\boldsymbol{y } } \ , p(\boldsymbol{b}|\boldsymbol{y } ) + \left\|{\boldsymbol s}^\mathrm{opt}_x\right\|\right ] > \beta_l , \label{eq : main_result}\end{aligned}\ ] ] where @xmath65 , \ , \boldsymbol{y}\in\boldsymbol{[m ] } } \frac12\left(\beta_{0,\boldsymbol{b},x,\boldsymbol{y}}-\beta_{1,\boldsymbol{b},x,\boldsymbol{y}}\right)\sigma_{\boldsymbol{b}|\boldsymbol{y}}\right ) \label{eq : main_result_r}\end{aligned}\ ] ] furthermore , the maximal violation is given by von - neumann measurements along the bloch - sphere directions @xmath66 , i.e. , with @xmath67 eqs .
, , and are the solutions to optimizations over general ( povm ) dichotomic measurements .
the criterion applies to many of the most - widely used bell inequalities . in the bipartite case ,
these include , for instance , the chsh @xcite and chained @xcite inequalities , as well as the @xmath68 inequality ( together with its variants for more outputs for bob or more inputs for both ) @xcite . in the multipartite scenario , in turn , criterion [ crit : conds_for_bell_viol ] covers important multipartite @xcite and genuinely multipartite @xcite bell inequalities , as well as @xmath6-sided di genuine @xmath1-partite entanglement witnesses @xcite .
for instance , svetlichny s inequality is obtained by taking @xmath69 and @xmath70 otherwise @xcite ; whereas the svetlichny - like chained inequality introduced in ref .
@xcite is given by @xmath71_2}$ ] , where @xmath72_2 $ ] stands for @xmath26 modulo @xmath73 .
violation of the former implies genuinely multipartite nonlocality ( gmn ) , while violation of the latter implies a strong form of gmn that , for large @xmath30 , is a resource for di quantum secret - sharing protocols against generic nonsignaling ( even post - quantum ) eavesdroppers @xcite .
quantum secret sharing ( qss ) is an intrinsically multipartite cryptographic protocol with remarkable security properties @xcite .
interestingly , unconditional security of qss has been recently proven in the @xmath6-sided di scenario we consider but in the continuous - variable regime @xcite . _
conditions for 2-input 2-output bipartite bell nonlocality_. as a crucial application of criterion [ crit : conds_for_bell_viol ] , we focus on the case of two parties , each one with dichotomic inputs and outputs . in this scenario ,
bell nonlocality is equivalent to a chsh violation @xcite , thus , applying criterion [ crit : conds_for_bell_viol ] to the chsh inequality , we automatically get a necessary and sufficient condition for nonlocality .
this is formalized by the following corollary , whose proof we leave for the appendix [ app:2input2output ] .
for ease of notation , from now on we omit the subindex 1 from the untrusted party s input @xmath74 and output @xmath75 .
[ crit : twotwo ] let @xmath44 be a generic assemblage with @xmath76 inputs and @xmath77 outputs per party and @xmath2 .
then , @xmath44 is bell local if , and only if , @xmath78 } \left\|{\boldsymbol t}^\mathrm{opt}_x\right\| \leq 2 \ , \end{aligned}\ ] ] where @xmath79}(-1)^{b+x\,y}\ , \sigma_{b|y}\right ) .
\label{eq:222_r}\end{aligned}\ ] ] furthermore , if inequality is violated , the maximal violation is given by von neumann measurements along the bloch - sphere directions @xmath80 .
_ connection to steering inequalities_. in ref .
@xcite a bipartite steering inequality for correlators was derived .
there , it was shown that , if @xmath81 and @xmath82 are any two out of the three pauli operators in an arbitrary basis of @xmath56 and @xmath44 is unsteerable , then @xmath83}\langle a_x\,(b_0 + b_1)\rangle^2 } + \sqrt{\sum_{x\in[2]}\langle a_x\,(b_0 - b_1)\rangle^2 } \leq 2 \ .\end{aligned}\ ] ] @xmath84 and @xmath3 are unknown , @xmath85-valued observables of the untrusted part ; and @xmath86}(-1)^a\operatorname{tr}[\sigma_{b|y}\,a_x]$ ] for all @xmath87 $ ] . later on , in ref .
@xcite , the authors found out that a violation of eq .
implies not only that @xmath44 is steerable but also that it violates the chsh inequality ( under some measurements for alice not necessarily corresponding to @xmath81 and @xmath82 ) . and @xmath88 onto some fixed chosen plane ( represented in red ) instead of at @xmath89 and @xmath88 themselves . for assemblages for which the plane ( represented in light blue ) of @xmath89 and @xmath88 happens to coincide with the chosen one , are equivalent . for any other assemblage
, inequality is more effective than inequality . ]
we can explain this implication in light of criterion [ crit : twotwo ] . to this end ,
we note ( see appendix [ app : steerineq ] ) that eq .
can be recast as @xmath90}\langle a_x\,(b_0 + b_1)\rangle^2 } + \sqrt{\sum_{x\in[3]}\langle a_x\,(b_0 - b_1)\rangle^2 } \leq 2 \ , \label{eq:222meas}\end{aligned}\ ] ] where @xmath91 is the third pauli operator complementary to @xmath81 and @xmath82 . clearly , the left - hand side ( lhs ) of eq .
is greater or equal than that of eq . .
as a consequence , a violation of eq . implies a violation and , therefore , of , consistent with the findings of ref .
@xcite .
the difference , of course , is that a violation of eq . does not in general
imply a violation of eq . .
so , while the former gives a necessary and sufficient condition for 2-input 2-output bipartite bell nonlocality , the latter only provides a sufficient one .
interestingly , one can show ( see appendix [ app : steerineq ] ) that , in the basis of @xmath56 in which the bloch - sphere directions associated to @xmath81 and @xmath82 are contained in the plane of @xmath89 and @xmath88 , the correlators involving @xmath91 in eq .
vanish and the lhss of eq . and
. thus coincide ( see fig .
[ compaust ] ) .
this implies that for every 2-input 2-output bipartite bell nonlocal assemblage there exists a pair of mutually unbiased bases for which the steering in the assemblage is witnessed via a violation of .
the implication was also recently formally proven in refs .
@xcite by other reasonings
. the pair of bases is readily obtained from eq . . _
concluding remarks_. we have unambiguously characterized all the @xmath1-partite assemblages with a single trusted ( qubit ) system that can violate under general dichotomic trusted measurements on the qubit bell inequalities from a very broad family . indeed , most widely used inequalities are within the firepower of our criterion .
in addition , the optimal povms for maximal violation were also provided and turn out to always be rank-1 projective von neumman measurements .
furthermore , for the important particular case of two users with 2 inputs and 2 outputs per user , our criterion unambiguously characterizes all bell nonlocal correlations in the assemblage . in that setting
, we showed that the form of the criterion generalizes that of recently obtained steering inequalities , and sheds light back onto the problem of steering detection within the framework of bell nonlocality .
our results hold for the usual scenario of non - sequential measurements on a single copy of the assemblage .
hence , our criterion is to qubit assemblages what the famous analytical criterion of ref .
@xcite for violation of the chsh inequality @xcite is to 2-qubit states , with the difference that we handle more generic bell inequalities and in the multipartite case .
we leave as an open question the conditions for bell violations and bell nonlocality in more general measurement scenarios @xcite .
it is important to remark that the optimizations we have solved can also be solved with semi - definite programming ( sdp ) @xcite .
however , while sdp can , in general , only give , for each problem instance , a numeric solution , we provided closed - form analytic expressions for the general case .
analytic solutions both carry more information and are more practical than numeric ones .
this is specially relevant in , e.g. , security proofs , which are naturally formulated symbolically and rarely admit numeric manipulations . in turn , here , it was precisely having closed - form solutions what made the conceptual connections with steering inequalities possible .
we have developed a practical toolbox to study of the interplay between epr steering and bell nonlocality , with potential implications in multipartite cryptographic protocols such as quantum secret sharing @xcite .
a further interesting prospect would be to explore possible connections with random - number generation schemes that are one - sided di in that they treat photon sources as black boxes @xcite . _
we thank s. p. walborn for his support .
we acknowledge financial support from the brazilian agencies cnpq ( national council for scientific and technological development ) , faperj ( research support foundation of the state of rio de janeiro ) , capes ( coordination for the improvement of higher education personnel ) , and inct - iq ( national institute for science and technology of quantum information ) .
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* appendix * |
there have been many generalizations of conway s `` game of life '' ( gol ) since its invention in 1970 @xcite .
almost all attributes of the gol can be altered : the number of states , the grid , the number of neighbors , the rules .
one feature of the original gol is the glider , a stable structure that moves diagonally on the underlying square grid .
there are also `` spaceships '' , similar structures that move horizontally or vertically .
attempts to construct gliders ( as we will call all such structures in the following ) , that move neither diagonally nor straight , have led to huge man - made constructions in the original gol .
an other possibility to achieve this has been investigated by evans @xcite , namely the enlargement of the neighborhood .
it has been called `` larger than life '' ( ltl ) . instead of 8 neighbors
the neighborhood is now best described by a radius @xmath0 , and a cell having @xmath1 neighbors .
the rules can be arbitrarily complex , but for the start it is sensible to consider only such rules that can be described by two intervals .
they are called `` birth '' and `` death '' intervals and are determined by two values each .
these values can be given explicitly as the number of neighbors or by a filling , a real number between 0 and 1 .
in the first case , the radius has to be given , too , in the last case , this can be omitted .
the natural extension of evans model is to let the radius of the neighborhood tend to infinity and call this the continuum limit .
the cell itself becomes an infinitesimal point in this case .
this has been done by pivato @xcite and investigated mathematically .
he has called this model `` reallife '' and has given a set of `` still lives '' , structures that do not evolve with time .
we take a slightly different approach and let the cell not be infinitesimal but of a finite size .
let the form of the cell be a circle ( disk ) in the following , although it could be any other closed set .
then , the `` dead or alive '' state of the cell is not determined by the function value at a point @xmath2 , but by the filling of the circle around that point .
similarly , the filling of the neighborhood is considered .
let the neighborhood be ring shaped , then with @xmath3 our state function at time @xmath4 we can determine the filling of the cell or `` inner filling '' @xmath5 by the integral @xmath6 and the neighborhood or `` outer filling '' @xmath7 by the integral @xmath8 where @xmath9 and @xmath10 are normalization factors such that the filling is between 0 and 1 . because the function values of @xmath11 lie also between 0 and 1 the factors simply consist of the respective areas of disk and ring .
the radius of the disk or `` inner radius '' is given by @xmath12 which is also the inner radius of the ring .
the outer radius of the ring is given by @xmath13 .
in the original gol the state of a cell for the next time - step is determined by two numbers : the live - state of the cell itself , which is 0 or 1 , and the number of live neighbors , which can be between 0 and 8 .
one could model all general rules possible by a @xmath14 matrix containing the new states for the respective combinations .
it could be called the transition matrix . now in our case this translates to the new state of the point @xmath2 being determined by the two numbers @xmath5 and @xmath7 .
the new state is given by a function @xmath15 .
let us call it the transition function .
it is defined on the interval @xmath16 \times [ 0,1]$ ] and has values in the range @xmath16 $ ] . to resemble the corresponding situation in gol ,
typically @xmath17 is chosen ( the diameter of the neighborhood is 3 cells wide ) .
as simple as the theoretical model is , it is not immediately obvious , how to implement it on a computer , as a computer can not handle infinitesimal values , continuous domains , etc .
but it can handle real numbers in the form of floating point math , and as it turns out , this is sufficient .
we also can model the continuous domain by a square grid , the ideal data structure for computation .
so we will be able to implement our function @xmath3 as a @xmath18 array . when implementing the circularly shaped integrals we run into a problem .
pixelated circles typically have jagged rims .
so either we let the radius of the circle be so huge , that the pixelation due to our underlying square grid is negligible .
then the computation time will be enormous .
or we use another solution used in many similar situations : anti - aliasing .
consider for example the integration of the inner region .
for the cell @xmath2 function values are taken at locations @xmath19 .
let us define @xmath20 . with an anti - aliasing zone around the rim of width @xmath21
we take the function value as it is , when @xmath22 . in the case
when @xmath23 we take 0 . in between
we multiply the function value by @xmath24 . similarly for the inner rim of the ring and the outer rim . in this way
the information on how far the nearest grid point is away from the true circle , is retained .
typically , @xmath25 is chosen .
we also have to construct the transition function @xmath15 explicitly .
luckily we can restrict ourselves like ltl , for the beginning , to four parameters : the boundaries of the birth and death intervals . to make things smooth and to stay in the spirit of the above described anti - aliasing we use smooth step functions instead of hard steps .
we call them sigmoid functions to emphasize this smoothness
. for example we could define @xmath26 @xmath27 @xmath28 then we can define the transition function as @xmath29 where birth and death intervals are given by @xmath30 $ ] and @xmath31 $ ] respectively .
the width of the step is given by @xmath32 .
as we have two different types of steps we have an @xmath33 and an @xmath34 .
note that neither the anti - aliasing nor the smoothness of the transition function are necessary for the computer simulation to work .
they just make things smoother and allow one to choose smaller radii for neighborhood and inner region and so achieve faster computation times for the time - steps .
so far we have made everything smooth and continuous but one thing : the time - steps are still discrete . at time
@xmath4 the function @xmath15 is calculated for every @xmath2 and this gives the new value @xmath35 at time @xmath36 .
if we think of the application of @xmath15 as a nonlinear operator @xmath37 we can write @xmath38 \ , f(\vec x , t)\ ] ] to give us the ability to obtain arbitrarily small time steps , we introduce an infinitesimal time @xmath39 and reinterpret the transition function as a rate of change of the function @xmath3 instead of the new function value
. then we can write @xmath40 \ , f(\vec x , t)\ ] ] where we have defined a new @xmath15 , that has values in the range @xmath41 $ ] instead of @xmath16 $ ] .
if the transition function in the discrete time - stepping scheme was @xmath42 then the smooth one is @xmath43 .
the formula above is also the most trivial integration scheme for the integro - differential equation @xmath44 \ , f(\vec x , t)\ ] ] this equation however leads to a different form of life .
the same generic gliders can not be found at the same birth / death values as in the version with discrete time - stepping , but it also leads to gliders , oscillating and stable structures .
we have described a model to generalize conway s `` game of life '' to a continuous range of values and a continuous domain .
the @xmath14 transition matrix of the gol has been generalized to the @xmath15 transition function . the 8 pixel neighborhood and 1 pixel cell of gol
have been generalized to a ring shaped neighborhood and a disk shaped cell .
the rule set has been generalized to four real numbers : the boundaries of the birth and death intervals .
the last remaining discrete attribute was the time - stepping .
we proposed a method for continuous time - stepping which reinterprets the transition function as the velocity of change .
the technique with two radii has been used in other contexts @xcite , but no gliders were described .
there has also been a computer implementation of a continuous version of gol , but without the inner radius technique , and no gliders were found at that time @xcite .
the goal of finding a glider that can move in arbitrary directions has been achieved . of the original gol it resembles both the glider and the spaceship at the same time .
it also resembles similar structures found in ltl .
so we think we have found the generic , generalized glider , and call it the `` smooth glider '' . | we present what we argue is the generic generalization of conway s `` game of life '' to a continuous domain .
we describe the theoretical model and the explicit implementation on a computer . |
radial abundance gradients in the milky way disk are among the main constraints of models of the chemical evolution of the galaxy .
the study of the gradients comprises the determination of their magnitudes along the disk , space variations and their time evolution ( see for example henry & worthey 1999 , maciel & costa 2003 ) .
probably the most interesting property of the gradients is their time evolution , which is a distinctive constraint of recent chemical evolution models .
maciel et al . ( 2003 ) suggested that the o / h gradient has been flattening during the last few gyr , on the basis of a large sample of planetary nebulae ( pn ) for which accurate abundances are available , and for which the ages of the progenitor stars have been individually estimated .
this work has been recently extended ( maciel et al . 2005 ) to include the s / h ratio in planetary nebulae , [ fe / h ] metallicities from open clusters and cepheid variables , as well as some young objects , such as ob associations and hii regions . in this work
, we review the main characteristics of the work by maciel et al .
( 2005 ) and analyze the uncertainties involved in the determination of the gradients . in particular , we investigate whether the derived uncertainties support either a systematic variation of the abundances with the galactocentric distance , as assumed by our work , or simply a dispersion of the abundances around some average value .
the main results for the time variation of the gradients as derived from planetary nebulae , open clusters , and cepheids are shown in tables 1 and 2 . adopting average linear gradients , which can be taken as representative of the whole galactic disk ,
the abundances can be written in the form where @xmath2(o / h ) + 12 or @xmath2(s / h ) + 12 for pn , hii regions and ob stars , and @xmath3 [ fe / h ] for open clusters and cepheids . for planetary nebulae , we have taken into account both o / h and s / h determinations and evaluated the gradient in the galactic disk according to the ages of the progenitor stars . for comparison purposes
, we can also derive the [ fe / h ] metallicities from the o / h abundances , on the basis of a [ fe / h ] @xmath4 o / h correlation derived for disk stars ( see maciel 2002 and maciel et al .
2005 for details ) .
the ages follow from the age - metallicity relation by edvardsson et al .
( 1993 ) , which also depends on the galactocentric distance . in this way
, we can divide the sample of pn into different age groups , each one having a characteristic gradient .
table 1 shows representative examples of 3 age groups for o / h and 2 age groups for s / h .
the table gives the gradient @xmath5 ( dex / kpc ) as defined by equation ( 1 ) .
all gradients in this paper have been calculated assuming @xmath6 kpc for the galactocentric distance of the lsr . for detailed references on the pn data the reader
is referred to maciel et al .
( 2003 , 2005 ) .
it should be mentioned that the pn age groups shown in table 1 are typical groups , arbitrarily defined .
in fact , we have extended this procedure by taking into account a variety of definitions of the age groups , with similar results .
column 2 of table 4 shows the estimated values of @xmath7 and @xmath8 [ within brackets ] assuming average values , that is , no linear variations .
the results for pn show that the probability is very low in all cases , so that the data points are probably not distributed according to a gaussian distribution around some average value .
however , it is interesting to note that , if we restrain the galactocentric distances to a smaller range , such as from @xmath9 kpc to 8 kpc , or @xmath10 kpc to 10 kpc , the probability @xmath8 increases , showing that , for a given galactocentric bin , the abundances show a better agreement with the gaussian distribution around some average value . for the open clusters , the table shows a generally better agreement with the gaussian distribution around a mean value , both for the friel and chen samples , in agreement with our conclusions in sect .
however , for cepheid variables we have the same results as for the pn , that is , the cepheid data are apparently not consistent with a gaussian distribution around a mean value .
we can also estimate @xmath8 in each case taking into account the derived linear correlations which are displayed in tables 1 and 2 .
here we have @xmath11 for the number of degrees of freedom , so that we can estimate @xmath7 and @xmath8 provided we have a reliable estimate of the uncertainty of the data . for planetary nebulae , recent discussions by pottasch et al .
( 2005 ) of objects with iso data suggest that the abundances of the beststudied elements are probably correct within 20% , which corresponds to 0.10 dex for oxygen .
this is probably a lower limit for other nebulae for which no infrared data is available , so that their abundances depend more heavily on ionization correction factors .
we may then adopt @xmath12 dex for o / h and @xmath13 dex for s / h as realistic estimates for planetary nebulae .
the latter can also be attributed to the open clusters , in view of the heterogeneity of the data and the use of photometric abundances . for cepheid variables , which have the best determinations , an average uncertainty
@xmath14 seems appropriate .
the results are shown in column 3 of table 4 , under the heading linear . again
the probabiliy is given within brackets .
we can see that in all cases the @xmath7 values are lower than the corresponding values for the averages , so that the probability @xmath8 is higher for the linear correlation than for the simple averages .
in fact , these probabilities are very close to unity in most cases , especially if we consider the more realistic , higher uncertainties .
it can also be seen that for cepheid variables the probability given in column 3 is essentially unity , reinforcing our conclusion about systematic abundance variations with the galactocentric distance . | we have studied the time variation of the radial abundance gradients using samples of planetary nebulae , open clusters , cepheids and other young objects . based on the analysis of o / h and s / h abundances for planetary nebulae and metallicities of the remaining objects
, we concluded that the gradients have been flattening out in the last 8 gyr with an average rate of the order of 0.005 0.010 dex kpc@xmath0 gyr@xmath0 .
we have estimated the errors involved in the determination of the gradients , and concluded that the existence of systematic abundance variations is more likely than a simple statistical dispersion around a mean value .
address = iag / usp , so paulo , brazil |
the author thanks prof .
j. katriel for helpful comments on the manuscript .
j. katriel , through correspondence , showed that eqns [ eq : evenzeros ] and [ eq : oddzeros ] agree with the first asymptotic term from dominici for @xmath51 for low @xmath40 ( not for maximal @xmath40 ) .
the latter result makes sense from the point of view that the maximal @xmath8 is always close to the edge of the hilbert space where the wavefunction goes to zero for any finite @xmath12 whereas that of the harmonic oscillator decays forever .
[ eq : evenzeros ] and [ eq : oddzeros ] do not agree with higher order terms ( w.r.t .
@xmath52 ) in dominici s asymptotic expansion . | the holstein - primakoff representation for spin systems is used to derive expressions with solutions that are conjectured to be the zeros of hermite polynomials @xmath0 as @xmath1 .
this establishes a correspondence between the zeros of the hermite polynomials and the boundaries of the position basis of finite - dimensional hilbert spaces .
the hermite polynomials are prevalent in many fields .
they can be defined as @xmath2 in the physics community , they are perhaps best recognized as the gaussian - weighted eigenfunctions ( in position representation ) of the quantum harmonic oscillator ( with @xmath3 , a convention that will be used for the rest of the paper ) : @xmath4 as such , they are orthogonal over the gaussian - weighted whole domain , @xmath5 .
this last property allows their use in gaussian quadrature , a useful and popular numerical integration technique where @xmath6 is approximated as @xmath7 where @xmath8 are the zeros of @xmath0 and @xmath9 is a well - behaved function . for this and many other reasons ,
an analytic formula for the asymptotic zeros of hermite and other orthogonal polynomials has been a subject of much interest@xcite , especially in the applied mathematics community and the field of approximation theory . in this paper ,
i examine the position state representation of the eigenstates of finite dimensional @xmath10-spin systems , as expressed in the holstein - primakoff transformation . as @xmath11 ,
the system becomes the infinite dimensional harmonic oscillator .
this association allows me to derive the simple main results presented in eqs [ eq : evenzeros ] and [ eq : oddzeros ] , with solutions that i conjecture become the asymptotic zeros of the hermite polynomials ( as @xmath1 ) .
furthermore , i numerically show that this convergence is rather quick and so the expressions can frequently be used , in many instances of finite - precision application , as the effective zeros of @xmath0 with finite @xmath12 , such as in applications of gaussian quadrature . in a more aesthetic sense
, these results also establish a beautiful correspondence between the boundaries of equal area partitions of circles with radii that are increasing in a certain manner and the hermite polynomial zeros .
spin systems are defined by the fundamental commutation relations between operators @xmath13 , @xmath14 and @xmath15 : @xmath16 = \hat s^+ , \hskip 5pt \left [ \hat s^z , \hat s^- \right ] = \hat s^- , \hskip 5pt \left [ \hat s^+ , \hat s^- \right ] = 2 \hat s^z.\ ] ] associating a spin with a boson @xmath17 , holstein and primakoff showed that to satisfy these commutation relations , the operators can be expressed as@xcite @xmath18 @xmath19 this is a very useful association and has found many applications in the condensed matter field s study of many - body spin systems .
each boson excitation represents the `` ladder up '' finitesimal excitation away from the spin s extremal @xmath10 state .
the hilbert space is finite - dimensional and possesses @xmath20 states @xmath21 .
in fact , considering eq .
[ eq : hkladder ] it is clear that the hilbert space outside this defined space is not even hermitian . transforming from the holstein - primakoff bosonic representation to position ( and its conjugate momentum ) space ( using the relations @xmath22 and @xmath23 ) reveals that the trivial hamiltonian is the harmonic oscillator : @xmath24 .
moreover , transformation of the @xmath14 and @xmath15 in eq .
[ eq : hkladder ] reveals that the hilbert space spans the domain @xmath25 .
just as in the @xmath26 representation , @xmath27 states all with the same area must exist within this domain .
fig .
[ fig : qbasis ] sketches out what they look like for the @xmath28-spin systems .
-basis representation of a ) @xmath29 , b ) @xmath30 and c ) @xmath31 systems is shown .
the radius of the hilbert space s domain is equal to @xmath32 and so grows along with the number of allowed basis elements . ] for a particular @xmath10-spin system , the lowest eigenstate must have the same sign at all @xmath33-basis elements since it must be nodeless . on the other hand
, the highest eigenstate must have @xmath34 nodes and so the @xmath33-basis elements must alternate in sign such that the eigenfunction passes through zero between them .
this latter behavior is sketched in fig .
[ fig : qbasis ] in red by the hermite polynomial @xmath0 denoting the value of the overlying @xmath33-basis element for the highest eigenstate . for @xmath11 ,
the hilbert space becomes infinite - dimensional and the hamiltonian becomes that of the harmonic oscillator defined over @xmath35 with the associated eigenfunctions proportional to @xmath36 .
it therefore follows that as @xmath11 , the boundaries of the @xmath33-basis elements become the zeros of the hermite polynomial @xmath0 where @xmath37 since the highest eigenstate must still have alternating sign with each @xmath33-basis element .
hermite polynomial zeros @xmath8 are real and symmetric around @xmath38 . to determine these boundary points , the @xmath39-dimensional hilbert space s circular shape in position space can be exploited . for even @xmath39 , the area of the all the @xmath33-basis elements up until the @xmath40th boundary ( measuring from the origin ) is @xmath41 . for odd @xmath39 ,
the area is @xmath42 .
this is illustrated in fig .
[ fig : areas ] . or b ) @xmath43 @xmath33-basis elements that approximately determine the @xmath40th zero of the hermite polynomial @xmath0 for @xmath12 even and odd respectively is shaded in blue .
the approximate @xmath40th zero is at the right boundary of these regions . ] using simple relations for the area of circle sectors and rectangles , it is possible to relate these @xmath33-basis element areas to @xmath8 ; the equation involving the approximate zeros of hermite polynomials @xmath44 with @xmath12 even is : @xmath45 while for odd @xmath12 it is : @xmath46 where @xmath47 and @xmath48 . solving these equations for @xmath8 yields the approximate @xmath40th zero for the @xmath12th hermite polynomial .
the results for the zeros of the first @xmath49 hermite polynomials are compared to the exact zeros in fig .
[ fig : zeros ] . in both cases ,
eqs .
[ eq : evenzeros ] and [ eq : oddzeros ] converge to the zeros of the hermite functions quite quickly and [ eq : oddzeros ] agree with the first asymptotic term from dominici@xcite for @xmath1 for low @xmath40 ( not for maximal @xmath40 ) .
the latter result makes sense from the point of view that the maximal @xmath8 is always close to the edge of the hilbert space where the wavefunction goes to zero for any finite @xmath12 whereas that of the harmonic oscillator decays forever .
eqs .
[ eq : evenzeros ] and [ eq : oddzeros ] do not agree with higher order terms ( w.r.t .
@xmath50 ) in dominici s asymptotic expansion . ] . th zeros of the hermite polynomials @xmath0 for @xmath12 a ) even and b ) odd compared to those obtained from solving eqs .
[ eq : evenzeros ] and [ eq : oddzeros].,title="fig : " ] th zeros of the hermite polynomials @xmath0 for @xmath12 a ) even and b ) odd compared to those obtained from solving eqs .
[ eq : evenzeros ] and [ eq : oddzeros].,title="fig : " ] the finding that the boundaries of equal area partitions of growing circles correspond to the asymptotic zeros of the hermite functions appears to be a novel one from a search of the literature .
it is all the more surprising that the origin of this one - to - one correspondance stems from the holstein - primakoff representations for finite - dimensional spin systems .
furthermore , on a practical level , the apparently rapid convergence of these solutions suggests that they may be useful for more efficient determination of hermite polynomial zeros for large - dimensional implementations of gaussian quadrature . |
in this section , we justify remark 3 above and outline the derivation of a result analogous to theorem [ kesten - ipc-1 ] for the random walk on h. kesten s _ incipient infinite cluster _ ( iic ) . for cylinder events @xmath74 , the iic measure is defined by @xmath716 it was shown in @xcite that the limit ( [ eq : iic - def ] ) exists and that the resulting set function extends to a measure .
note that the connected cluster of the origin , @xmath717 , is @xmath718-almost surely unbounded
. we will refer to this cluster as the iic .
we have the following result : let @xmath719 denote a simple random walk on the incipient infinite cluster started at @xmath7 .
let @xmath0 denote the first exit time of @xmath720 from @xmath35 .
there exists @xmath36 such that , for @xmath718-almost every @xmath38 and almost - every realization of @xmath721 , there is a ( random ) @xmath15 such that @xmath39 for @xmath1 greater than @xmath15
. we can proceed along the lines of the proof of estimate ( [ eq : w1bound ] ) , and consider a suitable modification of the random walk whose distribution coincides with that of @xmath162 from the first hitting time @xmath722 of @xmath723 to the first hitting time of @xmath517 after time @xmath722 , @xmath724 . to use the argument leading to ( [ eq : w1bound ] ) in our case
, we merely need to show that we can prove an estimate equivalent to the one obtained for @xmath725 in section [ section - comparison ] .
we will show that there are constants @xmath726 and @xmath34 such that @xmath727 by the argument given in the proof of lemma [ pisztoraslemma ] , there exists @xmath726 and @xmath34 such that @xmath728 let us denote the event on the left by @xmath729 .
clearly @xmath730 @xmath729 depends only on the status of edges inside @xmath731 .
write the conditional probability in the definition of @xmath718 as a ratio : @xmath732 for @xmath733 , we have , by independence and monotonicity @xmath734 now @xmath735 and by quasi - multiplicativity @xmath736 using this in ( [ eq : iic3prod ] ) , we have , by ( [ eq : pcbound ] ) : @xmath737 from which ( [ eq : piicbound ] ) follows at once
. * acknowledgements . *
we thank t. kumagai for suggesting the problem of proving a quenched analogue of kesten s subdiffusivity theorem and for comments on a previous verion .
we are very grateful to a. fribergh for comments that led to a substantial reorganization of the presentation .
j. h. and p. s. thank m. aizenman for advising and thank the organizers of the workshop `` current topics in mathematical physics '' at the erwin schrdinger institute , where some of this work was done .
pisztora , a. , _ scaling inequalities for shortest paths in regular and invasion percolation _ , carnegie - mellon cna preprint , available at http://www.math.cmu.edu/cna/publications/publications2000/001abs/00-cna-001.pdf | we derive quenched subdiffusive lower bounds for the exit time @xmath0 from a box of size @xmath1 for the simple random walk on the planar invasion percolation cluster . the first part of the paper is devoted to proving an almost sure analog of h. kesten s subdiffusivity theorem for the random walk on the incipient infinite cluster and the invasion percolation cluster using ideas of m. aizenman , a. burchard and a. pisztora .
the proof combines lower bounds on the instrinsic distance in these graphs and general inequalities for reversible markov chains . in the second part of the paper
, we present a sharpening of kesten s original argument , leading to an explicit almost sure lower bound for @xmath0 in terms of percolation arm exponents .
the methods give @xmath2 , where @xmath3 depends on the instrinsic distance and @xmath4 can be taken to be @xmath5 on the hexagonal lattice .
[ [ section ] ] h. kesten has proved @xcite that the simple random walk @xmath6 started at @xmath7 on the incipient infinite cluster ( iic ) @xcite in two - dimensional bernoulli bond percolation is subdiffusive in the sense that there exists @xmath8 such that the family @xmath9 is tight .
the purpose of the current work is to explain how a `` quenched '' version of this result can be obtained and extended to the random walk in an environment generated by a related two - dimensional model , invasion percolation .
( the model is defined in the next section ) .
we present some refinements of kesten s method , which provides a general framework for proving subdiffusivity of random walks in stochastic geometric models . in the case of two - dimensional invasion percolation ( as well as the incipient infinite cluster ) , the ideas in @xcite can be used to give explicit bounds on @xmath10 from in terms of known critical exponents ( see and below ) .
our main result is the following : [ main - theorem ] let @xmath11 be a simple random walk on the invasion percolation cluster ( ipc ) , and @xmath0 the first time @xmath12 exits the box @xmath13 ^ 2 $ ] : @xmath14 there exists @xmath3 such that , for almost every realization of the random walk , and almost every realization of the ipc , there is a ( random ) @xmath15 such that @xmath16 @xmath4 is a constant that can be estimated in terms of the behaviour of the one - arm and two - arm probabilities in critical percolation ( with measure @xmath17 ) : @xmath18 where @xmath19 are exponents such that @xmath20 and @xmath21 for some constants @xmath22 and @xmath23 . if one repeats the arguments of this paper in the setting of random walk on the iic or ipc of the hexagonal lattice , one can use the exact values of the one - arm and two - arm exponents to give a stronger bound on @xmath4 .
indeed , it is not necessary in that case to use the van den berg - kesten inequality @xcite in , therefore giving @xmath24 using the conjectured value @xmath25 @xcite , we get a lower bound @xmath26 . without using this value , but using @xmath27 @xcite , we get @xmath28 . this result is stronger than the corresponding theorem for the random walk on the iic stated in ( * ? ? ?
* theorem 1.27 ) , but it is derived by a modification of the strategy used there . in particular , kesten proves that @xmath29 for the `` averaged '' measure @xmath30 , which incorporates averaging with respect to the iic measure constructed in @xcite .
closer examination of his proof reveals that one can take @xmath31 , and that the estimates in @xcite are sufficient to establish a `` quenched '' result by a simple application of the borel - cantelli lemma . a substantial part of the present paper is concerned with presenting arguments to overcome the difficulties in adapting kesten s proof to the invasion percolation cluster .
the second result of the paper concerns a simple derivation of subdiffusivity of random walk on the ipc using results in @xcite and @xcite concerning the length of the shortest path from the origin to @xmath32 ( the _ chemical distance _ ) in near - critical percolation .
the work of these authors implies that for large @xmath1 this length is of order at least @xmath33 , where @xmath34 .
although theorem [ kesten - ipc-1 ] is contained in theorem [ main - theorem ] , it is of interest because its proof represents a significant reduction in complexity from the original argument of kesten .
[ kesten - ipc-1 ] let @xmath0 be the time for a random walker on the invasion percolation cluster to exit @xmath35 .
there exists @xmath36 such that , for @xmath37-almost every @xmath38 and almost - every realization of the random walk , @xmath39 for @xmath1 greater than some random @xmath15 . a similar , but simpler
, argument applies to the incipient infinite cluster and gives an alternative proof that the random walk on the iic is almost surely subdiffusive .
see the appendix for details .
@xmath8 in the statement of theorem [ kesten - ipc-1 ] depends on the value of @xmath40 obtained by the methods of aizenman - burchard and pisztora .
@xmath40 is both very small and difficult to calculate explicitly .
kesten s comparison argument ( explained in section [ section - comparison ] ) yields an improvement of the estimate for @xmath0 in the previous theorem by a factor of the form @xmath41 , which leads to theorem [ main - theorem ] .
we note that any explicit bound on @xmath40 would be directly reflected in that theorem . indeed , if one has upper and lower bounds ( with high enough probability )
@xmath42 then one can get the lower bound @xmath43 for any @xmath44 satisfying @xmath45 on the hexagonal lattice , this can be improved as above to @xmath46 one can actually show @xmath47 can be taken to be @xmath48 , which yields the improved bound ( assuming again the exact value of @xmath49 ) @xmath50 the improvement due to @xmath51 and @xmath47 in the previous remark comes from choosing @xmath52 larger in .
it is actually a common misconception that kesten s original `` lost in the bushes '' argument gives a lower bound for @xmath0 proportional to the ratio of volume of the iic to the volume of its backbone .
the reason this is false is that it is not clear how to increase @xmath52 to order @xmath1 .
the parameter @xmath52 gives the scale at which volume estimates can be applied .
there has been little success with rigorous results for random walks on low - dimensional critical models ( for instance , the iic and ipc ) .
one notable example is the work of d. shiraishi @xcite on random walk on non - intersecting two - sided random walk trace . for results in high dimensions , we mention the recent work of g. kozma and a. nachmias @xcite on the iic in dimensions @xmath53 and of m. barlow , a. jrai , t. kumagai and g. slade on the iic for oriented percolation @xcite . on a critical galton - watson tree ,
kesten @xcite found the asymptotics of @xmath0 and constructed a scaling limit for random walk on the iic ( see also @xcite and @xcite ) .
later , o. angel , j. goodman , f. den hollander and g. slade @xcite found similar results for random walk on the ipc on a regular tree . after setting some notation below ,
we give the definition of the invasion percolation model in section [ section - invasion ] , and recall some useful properties of the ipc derived in previous literature . we then prove theorem [ kesten - ipc-1 ] in section [ section - backbone ] , and explain how kesten s volume comparison argument is used to obtain theorem [ main - theorem ] in section [ section - comparison ] .
section [ section - volume ] contains the derivation of estimates used in the proof of theorem 1 . for convenience , we work on the square lattice @xmath54 , but our results extend to planar lattices for which the russo - seymour - welsh estimates hold true . in this section ,
we give notation used throughout the paper for future reference . for any vertex ( lattice point )
@xmath55 , @xmath56 is the box @xmath57\times[v_2-n , v_2+n])\cap \mathbb{z}^2\\ & = \{x\in \mathbb{z}^2 : |x - v|_\infty \le n\}.\\ the box @xmath58 , centred at the origin .
@xmath59 refers to the internal vertex boundary of @xmath56 : @xmath60 we also define @xmath61 where ipc is defined in the next section . for a graph @xmath62 , the set of edges is denoted by @xmath63 .
for each @xmath64 $ ] , the independent bond percolation measure @xmath65 is an infinite product of bernoulli measures with parameter @xmath66 indexed by the edges of @xmath54 . for a finite set @xmath67 , and a vector @xmath68
we have @xmath69 a _ configuration _
@xmath70 is an element of @xmath71 .
an edge @xmath72 is said to be _ open _ in the configuration @xmath70 if @xmath73 , and _ closed _ otherwise . if @xmath74 and @xmath75 are subsets of @xmath54 we denote by @xmath76 the probability of the event that @xmath74 and @xmath75 are connected by a path of open edges .
the notation @xmath77 is defined analogously .
we write @xmath78 to denote the probability that @xmath74 and @xmath75 are connected by @xmath66-open edges .
we will use the connection probabilities @xmath79 and @xmath80 defined as @xmath81 these probabilities refer to independent bond percolation with parameter @xmath66 .
when no parameter is specified , it is understood that @xmath82 ; that is , @xmath83 we denote by @xmath37 the invasion percolation measure on bond configurations in @xmath54 . throughout
, @xmath38 will denote a realization of the ipc ; that is , a subgraph of @xmath54 sampled from @xmath37 .
for each such @xmath38 , we denote by @xmath84 the probability measure associated with the simple random walk on the invasion cluster in the realization @xmath38 ( which by definition contains the origin @xmath85 ) . for @xmath86 ,
we denote by @xmath87 the logarithm of @xmath88 in base 2 . throughout the paper , @xmath89 will denote constants chosen independent of @xmath1 .
we use the notation @xmath90 if there exists a constant @xmath91 such that @xmath92 this notation is only used if the implicit constant @xmath91 is deterministic ; that is , it does not depend on the realization of the ipc or of the random walk .
the notation @xmath93 is used to emphasize that the implicit constant depends on the parameter @xmath94 .
the notation @xmath95 denotes the existence of two positive constants @xmath96 and @xmath97 such that @xmath98 if @xmath99 and @xmath100 are two positive sequences , we use the notation @xmath101 to mean @xmath102 [ [ section - invasion ] ] the planar invasion percolation cluster is a random subgraph of the lattice @xmath54 which can be constructed from the familiar coupling of the independent bond percolation measures @xmath65 , @xmath103 . to every edge @xmath72 of @xmath54 , viewed as a graph , associate a random variable @xmath104 , uniformly distributed in @xmath105 $ ] , @xmath106 and @xmath107 being independent for @xmath108 .
an edge is called @xmath66-_open _ if @xmath109 and is @xmath66-_closed _ otherwise .
the distribution of the set of @xmath66-open edges is that of a bernoulli bond - percolation process at density @xmath66 .
the distribution of @xmath110 is a product measure which will be denoted by @xmath30 .
the ipc consists of a union of subgraphs of @xmath54 constructed by an iterative process : we start at the origin @xmath111 . at every stage
, we form @xmath112 by adding to the current ( finite ) graph @xmath113 the edge @xmath72 with the least weight @xmath104 among @xmath114 as well as the endpoints of @xmath72 . the ipc is defined to be the union @xmath115 .
since the percolation probability @xmath116 at @xmath117 is zero @xcite , the ipc contains infinitely many edges @xmath72 with @xmath118 . on the other hand , for any @xmath119 , by the russo - seymour - welsh theorem , the ipc will intersect the ( unique ) @xmath66-open infinite cluster almost surely ( see @xcite for general @xmath120 ) . by construction ,
once an edge @xmath72 in the @xmath66-open infinite cluster has been added , all edges added to the ipc after @xmath72 have weight no bigger than @xmath66 .
we will later require bounds for the probability that the ipc intersects the @xmath66-open infinite cluster , for some fixed @xmath66 , by the time it reaches an annulus of size @xmath1 .
such estimates can be found in @xcite , @xcite .
an important notion in this context is the _ finite - size scaling length _
@xmath121 . to define it ,
consider for @xmath119 the probability @xmath122\times[0,m]).\ ] ] then @xmath121 is defined to be @xmath123 from @xcite , it is known that @xmath124 for @xmath125 , so we shall fix @xmath126 and henceforth simply refer to @xmath127 .
we note the following properties of @xmath128 : 1 .
@xmath129 is right - continuous , non - increasing in @xmath130 and @xmath131 as @xmath132 .
2 . taking @xmath133 small enough , there exists @xmath134 such that ( * ? ? ? * ( 2.8 ) ) : @xmath135 3 . again from ( * ? ? ?
* eq . 2.10 ) , there exists @xmath136 independent of @xmath66 such that @xmath137 let @xmath138 be the @xmath139-th iterate of @xmath140 , and @xmath141 define , for @xmath142 and @xmath143 @xmath144 @xmath145 is a constant to be determined later .
note that if @xmath146 , then @xmath147 when @xmath148 is sufficiently large . by ( 3 ) above , there exists @xmath149 such that @xmath150 item ( 2 ) in the list above implies ( * ? ? ? * ( 2.21 ) ) @xmath151 the measure @xmath30 refers to the coupling of the @xmath66-bernoulli measures described earlier .
if the event @xmath152 occurs , the ipc intersects the @xmath153-open infinite cluster by the time it reaches @xmath154 .
the bound ( [ eq : jarai - bound ] ) plays a role in estimates derived in section [ section - volume ] .
[ [ section - backbone ] ] we begin by giving a brief sketch of the main idea .
the first step is to consider a restriction of the random walk to a certain subset of the ipc , the backbone .
the exit time for this walk from a box of size @xmath1 is controlled using the varopolous - carne inequality .
this inequality implies that the exit time is at least of order @xmath155 , where @xmath120 is the chemical ( instrinsic ) distance to the boundary of the box of size @xmath1 through the ipc . in lemma [ pisztoraslemma ] , we outline an argument of a. pisztora that proves that @xmath120 grows superlinearly with @xmath1 .
all of these estimates are tight enough to apply borel - cantelli and close the proof of subdiffusivity .
the simple random walk started at @xmath7 on the ipc is the markov chain @xmath11 with the set of sites in the ipc as its state space , such that @xmath156 , and with transition probabilities given by @xmath157}{\operatorname{deg}(x,\mathrm{ipc})}.\ ] ] the random variable @xmath158 denotes the number of sites @xmath159 such that the edge @xmath160 belongs to the @xmath161 .
below , it will be convenient to work with a modification of @xmath162 that is reversible on @xmath163 .
thus , we let @xmath164 be the markov chain started at the origin and defined by the transition probabilities @xmath165}{\operatorname{deg}(x,\lambda(n))}.\ ] ] note that the distribution of @xmath166 coincides with that of @xmath12 for @xmath167 , where @xmath168 moreover , the distribution of @xmath169 is equal to the distribution of the exit time @xmath170 defined in terms of the `` full '' random walk @xmath162 on @xmath163 .
thus , it will suffice to obtain bounds on @xmath169 .
the `` backbone '' @xmath171 of @xmath163 is the set of sites in @xmath163 connected in the invasion cluster to @xmath7 and to @xmath32 by two disjoint paths . a simple argument ( see ( * ? ? ?
* lemma 3.13 ) ) shows that whenever @xmath172 leaves the backbone , it must return at the site where it left before it reaches @xmath173 .
thus the random walk @xmath172 on @xmath163 induces a random walk @xmath174 on @xmath171 which moves only when @xmath172 is in @xmath171 .
that is , if we define @xmath175 then @xmath174 is a random walk on the backbone @xmath171 , with transition probabilities given by @xmath176}{\operatorname{deg}(x,\lambda(n ) ) } , & y\neq x\\ \frac{\operatorname{deg}(x,\lambda(n ) ) -\operatorname{deg}(x , b(n))}{\operatorname{deg}(x,\lambda(n ) ) } , & x = y . \end{cases}\ ] ] here , @xmath177 is defined as the number of edges @xmath160 in @xmath163 such that @xmath178 . irrespective of the geometry of @xmath171 , @xmath174 must travel at least @xmath1 steps in @xmath171 to reach @xmath173 , because the distance between any two points in @xmath179 is no less than the corresponding chemical distance in @xmath54 .
this fact was used by kesten to conclude that the time spent by the walker on the backbone is of order at least @xmath180 with high probability .
the carne - varopoulos bound ( @xcite , @xcite ; see also ( * ? ? ?
* theorem 13.4 ) ) allows us to obtain a better estimate by considering the chemical distance on @xmath171 .
it implies that the reversible markov chain @xmath174 has at most diffusive speed in the intrinsic metric of the backbone .
if @xmath181 is the stationary measure for the walk @xmath174 ( @xmath181 depends on @xmath38 ) , then @xmath182 the right side of this expression refers to the chemical distance in the backbone @xmath171 .
the ratio appearing on the right can be bounded independently of the realization @xmath38 of the invasion percolation , since the stationary measure @xmath181 satisfies @xmath183 for any @xmath184 . since @xmath179
, we have the inequality of graph distances : @xmath185 summing this bound over @xmath186 , we find @xmath187 suppose we restrict our attention to realizations @xmath38 of the environment such that the chemical distance in @xmath171 satisfies @xmath188 for some @xmath189 and some deterministic constants @xmath34 , @xmath190 .
for such @xmath38 , @xmath191 and @xmath1 sufficiently large , we have : @xmath192 it follows from work of aizenman and burchard @xcite that the chemical distance inside a large box in independent bond percolation with parameter @xmath117 is bounded below by a power @xmath34 of the euclidean distance in @xmath54 with high probability .
pisztora @xcite showed how to extend this result to @xmath119 suitably close to @xmath193 , and to the invasion percolation cluster .
we reproduce the argument leading to his result , in a form that suits our needs , in the lemma below .
theorem [ kesten - ipc-1 ] follows from these results and the considerations above .
[ pisztoraslemma ] there exist @xmath194 and @xmath34 such that @xmath195 the models considered in @xcite are defined by families @xmath196 of probability measures on collections of curves in a compact region @xmath197 .
for each @xmath198 , @xmath199 is supported on unions of polygonal curves with step size @xmath198 .
the realizations in the support of @xmath199 are denoted by @xmath200 . a truncated version of capacity is used to obtain lower bounds on the minimal number @xmath201 of sets of diameter @xmath198 required to cover a given set @xmath202 : @xmath203 where @xmath204 the infimum is over borel probability measures supported on @xmath74 . under the assumption `` hypothesis h2 , '' the authors of
@xcite obtain uniform bounds for @xmath205 : if there exist some @xmath206 , and @xmath207 such that for every @xmath208 and collection of @xmath208 rectangles @xmath209 of lengths @xmath210 and cross - section @xmath70 , and satisfying @xmath211 for all @xmath139 , we have @xmath212 then the capacity @xmath205 of macroscopic curves is bounded below for some @xmath34 ( * ? ? ?
* theorem 1.3 ) : all curves @xmath213 in @xmath200 with @xmath214 satisfy @xmath215 @xmath216 is a random variable which is _ stochastically bounded below _ in the sense that @xmath217 uniformly in @xmath198 as @xmath218 .
we will apply the results in @xcite , with @xmath219 to bond percolation on the rescaled lattice @xmath220 ^ 2.\ ] ] for @xmath64 $ ] , let @xmath221 denote the independent bond percolation measure with parameter @xmath66 on the edges of @xmath222 .
@xmath221 induces a probability measure on configurations @xmath223 of curves in @xmath224 ^ 2 $ ] : the percolation configuration is a union of connected paths of @xmath66-open edges , each edge being identified with a line segment of length @xmath225 . in the case of independent percolation
, hypothesis h2 reduces to the existence of a cross - section @xmath70 and @xmath226 such that the probability that there exists an open - crossing of a rectangle of cross - section @xmath70 is less than @xmath80 . by the russo - seymour - welsh estimates , hypothesis h2 is satisfied for @xmath227 .
the lower bound ( [ eq : lowerbd ] ) gives an estimate for the chemical distance in @xmath223 between any two sets in @xmath228 ^ 2 $ ] .
any @xmath229-open path in @xmath222 connecting subsets @xmath74 and @xmath75 of @xmath222 at euclidean distance @xmath230 contains at least @xmath231 bonds .
denote by @xmath232 the ( random ) number of bonds in the shortest @xmath229-open path connecting @xmath74 and @xmath75 in @xmath222 . by ( [ eq : straight - runs ] ) ,
given any @xmath8 , we can choose @xmath233 such that for all @xmath1 , @xmath234 the scaling @xmath235 defines a measure - preserving bijection between @xmath236 and @xmath237 .
it follows that for each @xmath8 , there exists a constant @xmath233 such that for all subsets @xmath74 , @xmath75 of @xmath35 at euclidean distance @xmath238 from each other , @xmath239 note that if @xmath75 cuts @xmath74 from @xmath154 in @xmath54 , the restriction that the path be contained in @xmath35 is superfluous .
this point will be relevant below .
the observation in @xcite is that the aizenman - burchard bounds remain valid for @xmath119 as long as @xmath1 is smaller than the correlation length @xmath129 .
the estimate used to obtain ( [ eq : straight - runs ] ) depends only on @xmath70 and @xmath80 @xcite .
it follows from the definition of @xmath129 and the russo - seymour - welsh estimates that there exists @xmath226 such that for rectangles of cross - section ratio @xmath240 , say , with long side @xmath241 , @xmath242\times[0,n/3])\le \rho.\ ] ] thus ( [ eq : straight - runs ] ) remains true uniformly for @xmath243 . repeating the argument above
, we see that we can choose @xmath233 independent of @xmath244 to make the probability @xmath245 smaller than an arbitrary @xmath8 .
the distance refers to the chemical distance in the union of all percolation clusters in the box @xmath246 .
since @xmath131 as @xmath132 , for any fixed @xmath10 , @xmath129 is much greater than @xmath233 , and so the estimate on the distance is not vacuous .
more precisely , we find @xmath247 a block argument with blocks of size @xmath248 converts the initial estimate ( [ eq : c - epsilon ] ) into an exponential bound for the macroscopic chemical distance in near - critical percolation ( see the proof of ( * ? ? ? * theorem 1.3 , pp .
12 - 14 ) ) .
there exist constants @xmath249 , @xmath250 such that if @xmath66 is sufficiently close to @xmath229 : @xmath251 with this in hand , ( [ eq : invasion - chemical ] ) follows from the construction described in section [ section - invasion ] .
we outline the argument .
the occurrence of the event @xmath252 implies that all edges of the ipc in @xmath253 are @xmath254-open .
@xmath255 the final inequality follows from ( [ eq : renormalized - ab ] ) and ( [ eq : jarai - bound ] ) . recalling ( [ eq : l_p ] ) : @xmath256 and choosing @xmath257 suitably large in the definition of @xmath258 , we find : @xmath259 by slightly lowering @xmath40 to absorb the logarithm , the probability on the left of ( [ eq : finaldistance ] ) can be made to match the form of the left side of ( [ eq : invasion - chemical ] ) .
the final part of the proof of lemma [ pisztoraslemma ] shows that for any @xmath260 and any @xmath261 , one can find constants ( depending on @xmath262 and @xmath208 ) such that @xmath263 here @xmath34 is the constant appearing in ( [ eq : invasion - chemical ] ) .
such a statement will be used in section [ section - comparison ] below . for @xmath34 ,
let @xmath264 be the event @xmath265 by ( [ eq : invasion - chemical ] ) in the previous lemma , we have @xmath266 for some @xmath34 . applying the borel - cantelli lemma and choosing @xmath267 we can use ( [ eq - strong - cv ] ) with @xmath268 and @xmath269
; for some @xmath270 we have : @xmath271 a second application of the borel - cantelli lemma leads to theorem [ kesten - ipc-1 ] .
note that for the argument above it was not necessary to consider @xmath174 .
however ,
the decomposition of the ipc into a backbone and `` dangling ends '' will be central in the derivation of theorem [ main - theorem ] below .
the proof of theorem [ kesten - ipc-1 ] shows that @xmath174 alone already contributes at least @xmath272 steps to @xmath0 .
[ [ section - comparison ] ] our modification of kesten s argument compares the volume of sites in the invasion percolation cluster ( ipc ) to the volume of sites on the backbone to conclude that the walk must be subdiffusive .
we assume for simplicity of notation that @xmath273 , @xmath274 .
we introduce two stopping times : @xmath275 by definition , we clearly have : @xmath276 hence , it will suffice to obtain a lower bound on the right side of the previous expression .
@xmath277 is a simple random walk on ipc ; now define @xmath278 to be the simple random walk on ( the possibly disconnected ) @xmath279 with initial point @xmath280 .
letting @xmath281 be the hitting time of @xmath282 by the walk @xmath278 , we note that @xmath281 has the same distribution as @xmath283 .
a key tool in kesten s argument is the following result from @xcite , expressing the spatial `` smoothness '' of the local times for a reversible markov chain .
[ continuity ] let x , y be two sites in @xmath284 , and let @xmath285 be the local time at a site @xmath88 of the walk @xmath278 . then
, for some @xmath286 and any @xmath287 : @xmath288 in @xcite , lemma [ continuity ] is stated in terms of the intrinsic distance on the incipient infinite cluster .
replacing @xmath289 , @xmath290 and @xmath291 in the proof of lemma 3.18 in @xcite by @xmath292 , @xmath293 and @xmath294 , respectively , we obtain lemma [ continuity ] above .
we also modify our definition of the backbone .
@xmath295 is defined to be the set of sites in @xmath284 connected by two disjoint paths ( in @xmath284 ) to @xmath32 and @xmath296 .
@xmath297 is the induced walk on @xmath298 , defined analogously to @xmath174 in section [ section - backbone ] .
we let @xmath299 be the number of steps @xmath297 takes between @xmath300 and @xmath281 ; @xmath299 is the time spent by @xmath278 on @xmath295 .
kesten s comparison argument will be applied to @xmath278 .
the idea is to consider a `` thickening '' of the backbone of size @xmath52 . by lemma [ continuity ] , if a box @xmath301 of size @xmath52 contains a site @xmath302 with @xmath303 , the random walk visits all accessible sites of @xmath284 inside @xmath301 at least @xmath304 times , with high probability .
if it is traversed by a portion of the random walk , the box @xmath301 typically contains @xmath305 sites of @xmath284 , and at most @xmath306 sites of @xmath295 .
thus the time spent by @xmath278 in @xmath307 up to @xmath281 is larger than the time @xmath297 spends there by a factor of at least @xmath308 . by choosing @xmath52 appropriately ,
the set of sites @xmath159 on the backbone which do not satisfy the lower bound of order @xmath309 on @xmath310 will make a contribution bounded by a fraction of the total time spent on the backbone .
to realize the strategy just described , we tile @xmath311 by squares of size @xmath312 for a constant @xmath313 to be determined . here
@xmath49 is the exponent appearing in ( [ eq : twoarmprob ] ) .
we note for future reference that @xmath314 this bound on @xmath49 can be proved using the method of ( * ? ? ?
* cor . 3.15 ) .
for the details , the reader can see a standard sketch of a similar inequality ( for crossings of an annulus ) under equation later in the paper . for @xmath315 ,
define @xmath316\times [ q ( j_2 - 1 ) , q(j_2 + 2)].\end{aligned}\ ] ] given a realization @xmath38 of ipc and a realization of the walk , we follow the path of @xmath317 until @xmath281 by introducing two sequences @xmath318 and @xmath319 first setting @xmath320 and @xmath321 to be the index such that @xmath322 and then defining @xmath323 by @xmath324 @xmath278 may reach @xmath282 before leaving @xmath325 , in which case @xmath323 ends the sequence .
we let @xmath326 denote the component of @xmath327 containing @xmath328 .
@xmath278 may return several times to the same square , so @xmath329 may be equal to @xmath326 for @xmath330 .
enumerating the @xmath326 without repetition as @xmath331 , with @xmath332 the component of @xmath325 where @xmath333 is such that @xmath334 we define @xmath335 since any @xmath88 belongs to at most @xmath336 different @xmath337 squares , we have : @xmath338 we now state volume estimates analogous to those obtained in @xcite for the incipient infinite cluster ; they will be derived in the next section .
we will only be concerned with those indices in the set @xmath339 the first estimate is for the number of backbone sites in any @xmath337 square ; for any @xmath340 , we have : @xmath341 the second provides , with high probability , a lower bound for the number of sites of the ipc in a box @xmath342 , @xmath343 , given that there is a crossing of @xmath344 : @xmath345 here @xmath94 is arbitrary but the implicit constant depends on the choice of @xmath94 .
we now define the events @xmath346 , @xmath348 and @xmath349 , @xmath350 .
the ratio @xmath351 will be bounded below by @xmath308 on the event @xmath352 .
1 . @xmath353 2 .
@xmath354 3 .
@xmath355 4 .
@xmath356 5 . [ item - w1 ] @xmath357 6 .
@xmath358 7 .
@xmath359 by the remark following the proof of lemma [ pisztoraslemma ] , there exists @xmath360 such that @xmath361 with the same constant @xmath40 as in ( [ eq : invasion - chemical ] ) . for any @xmath362 and @xmath363
, we use the carne - varopoulos estimate ( [ eq - strong - cv ] ) , applied to the symmetric chain @xmath297 , to show as in the proof of theorem [ kesten - ipc-1 ] : @xmath364 giving the bound ( [ kesten - ipc-1 ] ) . recall the definition of @xmath52 in .
we have @xmath365 noting that there are , up to a constant , at most @xmath366 indices @xmath367 in @xmath368 , and choosing @xmath369 in ( [ eq - backbone - bound ] ) , and accordingly in the definition of @xmath370 , we find @xmath371 by the estimate ( [ eq - big - backbone - bound ] ) in section [ section - volume ] : @xmath372 by ( [ eq - crossing - bound ] ) ( for some @xmath94 large enough ) , we have @xmath373 finally , we have @xmath374 uniformly in @xmath38 . indeed ,
suppose @xmath88 and @xmath159 are two sites as in the description of @xmath375 , then , for @xmath1 sufficiently large , @xmath376 for any @xmath38 .
using @xmath377 for any @xmath378 in the ipc , we find that on @xmath379 , for some pair @xmath378 either @xmath380 and @xmath381 or @xmath382 and @xmath383 the first case is contained in the event appearing in ( [ eq : localtimes ] ) .
so is the second case , after reversing the roles of @xmath88 and @xmath159 in that event . using @xmath384 applying lemma [ continuity ] , and taking the union over all pairs , @xmath385
, we find that , whatever @xmath38 in the support of @xmath37 : @xmath386 a similar argument applies to @xmath387 . applying the borel - cantelli lemma to @xmath388
we find that for @xmath37-almost every @xmath38 , there exists @xmath270 such that @xmath389 holds when @xmath390 .
for any such @xmath38 , a further application of the borel - cantelli lemma shows that , @xmath84-almost surely , @xmath391 holds for @xmath1 large enough .
it remains to show that whenever all the events above hold , we have the subdiffusive bound of theorem [ main - theorem ] .
first , on @xmath392 , if we denote by @xmath393 the sum over indices @xmath394 such that @xmath395 contains a site @xmath396 with @xmath397 then , assuming @xmath398 also occurs , adjusting the constant @xmath313 in the definition of @xmath52 ( see ): @xmath399 it follows that @xmath400 on @xmath375 , letting @xmath401 be the lexicographically earliest point of @xmath295 in @xmath395 , we have for those indices @xmath402 occurring in @xmath393 : @xmath403 for each @xmath404 , @xmath405 contains an invaded crossing in @xmath395 .
thus , on @xmath406 , we can write : @xmath407 bounding every @xmath408 term below individually in ( [ eq : sumcompare ] ) and using the bk inequality ,
we find : @xmath409 on @xmath398 , we have @xmath410 . recalling the definition of @xmath52 from , we have the following bound for @xmath411 .
@xmath412 choosing @xmath3 such that @xmath413 , we obtain : @xmath414 the desired result .
[ [ section - volume ] ] in this section we prove the volume estimates ( [ eq - backbone - bound ] ) and ( [ eq - crossing - bound ] ) . we show that the following moment bounds hold for @xmath415 : @xmath416 for @xmath417 and constants @xmath418 , @xmath419 .
the estimate ( [ eq - small - moments ] ) implies the existence , for @xmath420 small enough , of the exponential moment : @xmath421 applying chebyshev s inequality with @xmath422 yields ( @xmath423 ) . from ( [ eq - big - moments ] ) , we obtain the finiteness , for sufficiently small @xmath424 , of the stretched exponential moment : @xmath425 using chebyshev s inequality with @xmath426 , we obtain , for each @xmath427 : @xmath428 to derive ( [ eq - small - moments ] ) and ( [ eq - big - moments ] ) , we follow the method introduced by jrai @xcite to estimate the moments of the volume @xmath429 of the ipc in a box . we will instead apply this argument to the volume of a backbone , and then combine it with an inductive argument of nguyen @xcite .
we begin with the first moment ( @xmath430 ) in ( [ eq - small - moments ] ) . if @xmath431 , the number of sites of @xmath342 with two disjoint connections in the ipc to @xmath432 provides an upper bound for the volume of @xmath433 .
let @xmath434 denote the set of sites in @xmath342 with two @xmath435-open connections to @xmath432 .
note that @xmath436 on @xmath437 ( defined in ) , every edge of the ipc in @xmath438 is @xmath435-open , as noted at the end of section [ section - invasion ] , and thus : @xmath439 the first term is bounded up to a constant factor by : @xmath440 the terms of the sum are estimated using the harris - fkg inequality : @xmath441 by decomposing @xmath342 according to the distance @xmath442 to @xmath432 , we find : @xmath443 .
\nonumber\end{aligned}\ ] ] by the same argument used for ( 7 ) in @xcite ( see remark ( 37 ) there ) , the sum up to @xmath444 in ( [ eq : smoothsum ] ) is bounded up to a constant by @xmath445 the proof in @xcite is carried out for @xmath446 , but the implicit constants that appear are due to applications of rsw theory and thus are uniformly bounded in @xmath119 . by comparability of the arm exponents below @xmath129 @xcite ( see also ( * ? ? ?
* theorem 26 ) ) , we have @xmath447 thus , finally , in ( [ eq : nsquared ] ) , we have ( since @xmath448 ) @xmath449 where in the first step we have used the inequality @xmath450 for @xmath451 .
a similar inequality for @xmath452 was used in @xcite , where the author indicates that it can be proved by the argument in ( * ? ? ?
* corollary 3.15 ) .
the proof of ( [ eq : twoarms ] ) follows the same general strategy , but does not use the van den berg - kesten inequality : let @xmath453 , @xmath454 and consider the annuli @xmath455 .
the inner squares of these @xmath456 annuli are adjacent .
the event that there exists a @xmath229-open left - right crossing of @xmath457 has probability bounded below uniformly in @xmath458 , and implies that one of the annuli is crossed by two disjoint @xmath229-open paths . by quasi - multiplicativity ,
this probability is comparable to @xmath459 , and ( [ eq : twoarms ] ) follows by a union bound . inserting ( [ eq : rholessthanlog ] ) into and then into ( [ eq : small - moment - sum ] ) , we find @xmath460 the final term corresponds to @xmath461 , which is @xmath462 by .
using , we may choose @xmath257 large enough to make the first term @xmath463 an important point made in @xcite is that choosing @xmath257 possibly larger , we may bound the contribution from the sum in the parentheses by a constant .
indeed , we have @xmath464 this establishes ( [ eq - small - moments ] ) for @xmath430 . to deal with the higher moments , we use the following general lemma : [ nguyenslemma ] let @xmath465 , @xmath466 , and @xmath467 be the set of sites of @xmath35 with two disjoint @xmath66-open connections to @xmath32 .
there exists a constant @xmath468 independent of @xmath1 and @xmath66 such that , for any @xmath469 , the following inductive bound holds : @xmath470 the result is essentially due to nguyen @xcite , who proved that for @xmath471 and @xmath472 , @xmath473 where @xmath474 is the set of sites in @xmath475 connected to @xmath476 by a @xmath66-open path .
@xmath477 is a constant uniform in @xmath478 and @xmath66 . when @xmath479 , the proof in @xcite is easily adapted to the variables @xmath480 .
we define the event @xmath481 the idea is to write @xmath482 where we have set @xmath483 letting @xmath484 we have @xmath485 uniformly in @xmath442 and @xmath1 ( even for @xmath486 ) . by the fkg inequality : @xmath487 @xmath488 is the event that @xmath489 is connected to @xmath32 by two disjoint open paths outside of @xmath490 . by independence ,
the last quantity on the right is bounded , up to a constant , by @xmath491 for any @xmath442 , we have : @xmath492 returning to ( [ eq : nguyen - triple - sum ] ) , we find @xmath493 if @xmath479 , we have @xmath494 and the estimate ( see the remark concerning ( [ eq : smoothsum ] ) above ) @xmath495 leads to the inductive estimate claimed above .
if @xmath496 , we split the sum as we did in the treatment of the first moment of @xmath497 : @xmath498 here we have used that for @xmath499 , @xmath500 .
this follows by a variant of the argument presented in ( * ? ? ?
* section 7.4 ) .
this establishes the lemma . using lemma [ nguyenslemma ] , induction and the fact that @xmath501 for large @xmath148 , we obtain @xmath502 thus , arguing as for ( [ eq : small - moment - sum ] ) : @xmath503 using the value of @xmath52 from and choosing @xmath504 , we use ( [ eq : logsconverge ] ) to get ( [ eq - small - moments ] ) in the case where @xmath367 is such that @xmath505 . for a general @xmath415 ,
the intersection @xmath506 is a union of at most two rectangles with side lengths @xmath507 and @xmath508 , @xmath509 . repeating the arguments above
, we see that the size of the intersection of each of these rectangles with the backbone @xmath295 enjoys the moment bounds ( [ eq - small - moments ] ) , with @xmath510 replaced by @xmath511 , with @xmath512 . using ( [ eq : twoarms ] ) , we obtain the upper bound @xmath513 . for the higher moments , moment bounds of the form ( [ eq - small - moments ] ) with a larger ( but still uniform in @xmath514 ) constant @xmath418 are valid .
( [ eq - backbone - bound ] ) now follows by a union bound .
the proof of ( [ eq - big - moments ] ) follows a very similar pattern to the above . instead of @xmath434 , we consider the sets @xmath515 of points of @xmath516 with two disjoint @xmath435-open connections to @xmath517 . repeating the steps for the case @xmath430 gives for the first moment . in the case of higher moments ,
we need to modify inequality , replacing it with @xmath518 for some @xmath519 .
decomposing as before over the events @xmath520 and choosing @xmath521 leads to . in its general outline ,
the proof is similar to that of ( 3.24 ) in @xcite , with some parameters chosen differently because we wish to bound only logarithmic deviations from the mean .
however , the estimates in @xcite are carried out for critical percolation ( @xmath522 ) , and the proof of the initial estimate ( [ eq : pre - peierls - bound ] ) below in the supercritical case introduces an additional technical difficulty . as explained in section [ section - invasion ]
, the entire ( finite ) @xmath229-open cluster of any site in the ipc also belongs to the ipc .
thus , for any crossing @xmath458 ( in the ipc ) of @xmath344 , the number of sites in @xmath344 connected to @xmath458 by @xmath229-open paths provides a lower bound for the quantity in ( [ eq - crossing - bound ] ) .
the starting point is the following : let @xmath458 be a deterministic path crossing @xmath523 .
let @xmath524 be the set of sites @xmath229-connected to @xmath458 inside this annulus .
we have the lower bound : @xmath525 for some constants @xmath526 independent of @xmath1 .
the proof is essentially that of ( 56 ) in @xcite .
kesten s idea is to compute the first and second moments of the number @xmath527 of sites in @xmath528 connected to open circuits in @xmath529 and @xmath530 ( and thus to @xmath458 ) and use the harris - fkg inequality and the second moment method .
fix some @xmath531 ( to be chosen later ) .
we first show that , for any @xmath532 ( entailing in particular @xmath533 by ) , and any coordinates @xmath534 such that @xmath535\times [ -t+v_2,4t+v_2]\subset s(3m)\setminus ( s(m))^\circ,\ ] ] we have , for some constants @xmath536 , @xmath537 , @xmath538\times [ v_2,3t+v_2 ] \\\text{such that } \sharp \{x \in t(\mathbf{v } ) :x \xrightarrow{p_c } r \text { in } t(\mathbf{v})\ } \le { \mathbf{c}}_{24 } t^2\pi(t)/(\log
t)^\delta \end{gathered}\right)\\ \lesssim \frac{1}{(\log t)^{{\mathbf{c}}_{25}}}.\end{gathered}\ ] ] for any crossing @xmath458 of @xmath539 , let @xmath540 the probability on the left of ( [ eq : pre - peierls - bound ] ) equals @xmath541 the precise meaning of `` @xmath458 intersects no @xmath229-open crossing of @xmath539 '' is that no site in @xmath539 is a common endpoint of an edge in @xmath458 and an edge in some horizontal @xmath229-open crossing of @xmath539 . in particular , @xmath458 is edge - disjoint from all @xmath229-open crossings . both terms on the right in ( [ eq : crossings - union - bound ] ) will be bounded , up to a constant factor , by @xmath542 .
we begin by estimating the first term in ( [ eq : crossings - union - bound ] ) . for any crossing lattice path @xmath543 of @xmath539 , let @xmath544 be the set of edges with an endpoint that can be connected to @xmath545 \times \{v_2\}$ ] by a path in @xmath539 that does not touch @xmath543 ( below @xmath543 ) .
note that @xmath544 may include edges not entirely contained in @xmath539 .
the lowest @xmath229-open crossing @xmath546 of @xmath547 is defined as the horizontal crossing of the rectangle by @xmath229-open edges such that the component @xmath548 , is minimal .
@xmath549 is defined inductively as the lowest crossing of @xmath550 ( defined analogously
see ( * ? ? ?
* prop .
2.3 ) for the existence of @xmath549 and precise definitions ) . for a given ( lattice path ) crossing @xmath543 of @xmath539 , write @xmath551 for the sigma algebra generated by the status of edges in @xmath552 .
we define @xmath553 to be the maximal @xmath208 such that @xmath549 exists .
the veracity of the following string of inequalities is then evident : @xmath554 on @xmath555 , we have the following uniform estimate for the conditional probability given @xmath556 : @xmath557 to see this , consider the left endpoint @xmath558 of the crossing @xmath559 , and annuli @xmath560 for @xmath561 , the existence of circuits @xmath562 around @xmath558 in @xmath563 and @xmath564 in @xmath565 , all of whose edges outside @xmath566 are @xmath229-open implies that any site in @xmath567 connected to @xmath568 is @xmath229-connected to the crossing @xmath559 .
thus , using the harris - fkg inequality , independence of the edge configurations in @xmath566 and @xmath569 and the second moment method as in the discussion preceding ( [ eq : pre - peierls - bound ] ) , there exist constants @xmath536 , @xmath570 , such that for each @xmath139 with @xmath571 : @xmath572 there are @xmath573 admissible indices @xmath139 , and so by independence of the configuration in the different annuli , we find @xmath574 which is the same as ( [ eq : oneoverlogdelta ] ) . returning to the double sum of : @xmath575 by the russo - seymour - welsh method , the @xmath17 probability of a dual vertical crossing of @xmath539 is bounded below by some @xmath576 .
thus , by disjointness of the @xmath549 s and the bk inequality , @xmath577 this allows us to bound the sum in ( [ eq : k - sum ] ) by @xmath578 .
we now estimate the second term on the right in ( [ eq : crossings - union - bound ] ) .
denote by @xmath579 the event that there exists a @xmath258-open crossing @xmath458 of @xmath539 such that @xmath458 intersects no @xmath229-open crossing of @xmath539 . for any @xmath580
, we have @xmath581 as previously , @xmath553 denotes the maximal number of disjoint @xmath229-open crossings of @xmath539 .
we will choose @xmath582 , so as to give the following bound for the second term above : @xmath583 where the constant @xmath584 is a constant such that @xmath585 .
for the first term , we have the union bound @xmath586 it will be shown below ( see lemma [ crossing - lemma ] ) that there is a constant @xmath587 such that , for any @xmath588 with @xmath589 , @xmath590 where @xmath591 is the `` alternating 4-arm probability , '' associated to the event that @xmath592 is connected to @xmath593 by two disjoint @xmath229-open paths and its dual edge is connected to @xmath593 by two disjoint @xmath229-closed dual paths .
thus @xmath594 for a constant @xmath595 .
the factor @xmath596 is @xmath597 ) .
indeed , it was shown in @xcite that , uniformly for @xmath119 sufficiently close to @xmath229 : @xmath598 applying this to @xmath599 , and using ( [ eq : l_p ] ) and @xmath600 ( * ? ? ? * proposition 12 ) , we find , for @xmath601 large enough : @xmath602
thus we have @xmath603 here we have used @xmath604 . using ( [ eq : nearcritical ] ) again
, we have : @xmath605 by quasimultiplicativity ( * ? ? ?
* proposition 12 ) : @xmath606 where @xmath607 is the probability that there are four arms of alternating occupation status connecting @xmath32 to @xmath608 in @xmath609 . using reimer s inequality @xcite and the ( exact ) scaling of the 5-arm exponent ( see ( * ? ? ?
* theorem 23 ) or @xcite ) , we have : @xmath610 here , @xmath611 is the one - arm probability , that @xmath32 is connected to @xmath608 by an open path .
since the one - arm probability satisfies the power - type upper bound @xmath612 for some @xmath613 ( apply the bk inequality to the bound on @xmath49 in ) , we find that @xmath596 is bounded , up to a constant , by @xmath614 since we assume @xmath615 , and @xmath616 ( see ( [ eq : twoarmbound ] ) ) , we find @xmath617 for some @xmath427 . returning to ( [ eq : bound - for - pgek ] ) ,
we have the bound : @xmath618 it remains to prove ( [ eq : fourarms ] ) .
this is done in lemma [ summation - lemma ] below .
before proceeding , let us introduce a definition : a _ @xmath229-closed arm with @xmath208 defects _ is a path of dual edges , all of which except for @xmath208 of them are @xmath229-closed .
the proof of lemma [ summation - lemma ] depends on the following : [ crossing - lemma ] let @xmath579 be the event that there exists a @xmath258-open crossing @xmath458 of @xmath539 such that @xmath458 intersects no @xmath229-open crossing of @xmath539 , and @xmath553 be the maximal number of horizontal @xmath229-open crossings of @xmath539 .
suppose @xmath619 ; then there exists an edge @xmath620 such that 1 .
@xmath72 has two disjoint @xmath258-open arms to @xmath621 $ ] ( the left side of @xmath539 ) and @xmath622 $ ] ( the right side of @xmath539 ) , respectively .
2 .
@xmath623 , the dual edge to @xmath72 , has two disjoint @xmath229-closed arms , each with at most @xmath208 defects to @xmath545\times \{v_2\}$ ] ( the bottom side of @xmath539 ) and to @xmath545\times \{v_2 + 3t\}$ ] ( the top side of @xmath539 ) , respectively .
3 . @xmath624 $ ] . on the event @xmath625 , menger s theorem ( * ?
? ?
* section 3.3 ) implies that there is a dual path @xmath40 joining the top of @xmath539 to the bottom , all of whose edges , with exactly @xmath208 exceptions , are closed and which moreover does not intersect itself .
this path must intersect the horizontal @xmath258-open crossing @xmath458 ( * ? ? ?
* prop .
2.2 ) along a @xmath258-open edge @xmath72 .
this edge must then be @xmath229-closed .
the dual edge @xmath623 , being part of the non - self - intersecting @xmath40 with @xmath208 defects , has two dual arms joining it to the top and bottom of @xmath539 .
( see figure [ fig : menger ] . )
moreover , the total number of defects on these two arms is @xmath208 .
this establishes the lemma . .
the dotted path has @xmath208 defects , shown as empty circles .
the solid black path represents a @xmath258-open crossing and the grey paths represent disjoint @xmath229-open crossings . ] the proof of ( [ eq : pre - peierls - bound ] ) is concluded by the following lemma , which establishes the estimate ( [ eq : fourarms ] ) : [ summation - lemma ] there is a constant @xmath587 such that , for each @xmath469 , the following bound holds : @xmath626 it suffices to estimate the probability that there is an edge in @xmath539 satisfying the two conditions in lemma [ crossing - lemma ] . to that end , we will show that the expected number of such edges in @xmath539 is bounded by the quantity on the right side of equation ( [ eq : xibound ] ) . for @xmath620 ,
let @xmath627 be the event that @xmath72 satisfies the conditions of lemma [ crossing - lemma ] .
the key step is the existence of a constant @xmath595 such that @xmath628 where @xmath629 is the event that @xmath72 has two disjoint @xmath229-open arms joining it to the left and right sides of @xmath539 respectively , and @xmath623 has two disjoint @xmath229-closed dual arms to the top and bottom sides of @xmath539 .
the effect of the arms with defects is to produce the logarithmic factor indicated in the equations above : @xmath630 where @xmath631 is defined analogously to @xmath629 above , except that the open connections are required to be @xmath258-open rather than @xmath66-open .
this follows from the argument in ( * ? ? ?
* prop .
17 ) , where it is shown that if @xmath632 denotes the probability that the origin is connected to @xmath32 by @xmath139 paths , with @xmath120 defects in total , whose occupation status is specified by the sequence @xmath633 , then @xmath634 @xmath635 is the event that there are @xmath139 arms ( without defects ) to @xmath32 ( with occupation status as in @xmath70 ) .
inspection of the proof in @xcite reveals that the constant implicit in ( [ eq : defects ] ) is of the form @xmath636 . separating the four arms as in @xcite verifies that ( [ eq : defectbound ] ) holds .
it now remains to show that @xmath637 ; that is , that for @xmath1 sufficiently large , we can change the @xmath258-open arms in the definition of @xmath631 to be @xmath229-open at the cost of a constant probability factor . for edges
@xmath72 at distance @xmath638 from the boundary , this follows immediately from ( * ? ? ?
* lemma 6.3 ) .
we briefly sketch how the proof given there can be adapted to the case where @xmath72 is close to the boundary .
we write @xmath639 as @xmath640 , where for @xmath641 , @xmath642 denotes the event that @xmath72 has two disjoint @xmath66-open arms to opposite vertical sides of @xmath539 and @xmath623 has two disjoint @xmath52-closed dual arms to the top and bottom of @xmath539 . using russo s formula as in ( * ? ? ? * ( 39 ) ) , we find @xmath643 @xmath644 is the event that @xmath623 has two disjoint @xmath229-closed dual connections to the top and bottom of @xmath539 , and @xmath645 is the event that there exist three disjoint @xmath66-open paths joining , respectively , one vertical side of @xmath539 to one endpoint of @xmath72 , the other endpoint of @xmath72 to an endpoint of @xmath646 , and the other endpoint of @xmath646 to the remaining vertical side of @xmath539 .
note that our notation differs somewhat from the one in @xcite .
for the purposes of illustration , we will henceforth suppose that @xmath647 for some @xmath648 ; that is , @xmath72 is close to the left side of @xmath539 .
the sum on the right of ( [ eq : damronetalsum ] ) can be rewritten as : @xmath649 @xmath650 denotes the left endpoint of the edge @xmath72 , if @xmath72 is a horizontal edge , and its bottom endpoint if @xmath72 is a vertical edge .
the first sum is bounded by @xmath651 @xmath652 denotes the probability that there are four arms of alternating occupation status joining @xmath32 to @xmath608 ; @xmath653 is the event that there are two @xmath229-closed arms , as well as a @xmath66-open arm connecting @xmath654 to @xmath593 . using gluing constructions similar to those in proofs of quasi - multiplicativity , and the fact that we may change the length of any connections involved by constant factors at the cost of constant factors in the probabilities
, we have : @xmath655 for @xmath656 , we can use ( * ? ? ? * theorem 27 ) to assert @xmath657 we can now follow @xcite exactly ( see equations ( 42 ) and ( 43 ) and the surrounding discussion ) to show that the sum in ( [ eq : fourfactors ] ) is bounded by : @xmath658 to deal with the second sum in ( [ eq : triplesum ] ) , we note that when @xmath659 the conjunction of the events @xmath660 and @xmath645 appearing in the probability on the right of the equation implies that @xmath72 has 2 @xmath66-open , and @xmath623 two @xmath229-closed arms to distance @xmath661 , that @xmath646 has four alternating arms with parameter @xmath66 to the boundary of the intersection of @xmath662 with @xmath539 , three of which reach to distance @xmath661 , and finally that @xmath663 has two @xmath229-closed arms to the top and bottom of @xmath539 and a @xmath66-open arm to the right side of @xmath539 , and all these connections occur inside @xmath539 . using these observations , an argument similar to the previous case and a summation analogous to that in the proof of (
* ? ? ?
* lemma 6.2 ) , shows that we can estimate ( in addition , using the remarks in ( * ? ? ?
* section 4.6 ) to change the @xmath66-open and closed arms in a half - plane to @xmath229-open and closed arms ) @xmath664 is at distance @xmath442 from the left boundary of @xmath539 and the distance between @xmath646 and @xmath72 is @xmath139 , a number between @xmath665 and @xmath442 .
the dark dotted curve represents a @xmath229-closed dual path ( given by menger s theorem ) and the dark solid curve represents a @xmath258-open path , connecting @xmath646 and @xmath72 to each other and to the left and right sides of @xmath539 .
the grey dotted curve represents a @xmath258-closed dual path connecting the edge dual to @xmath646 with the top and bottom of @xmath539 . ] turning to the final sum on the left in ( [ eq : triplesum ] ) , we can again closely follow @xcite to bound this term by @xmath666 the estimates outlined above for the left side of ( [ eq : damronetalsum ] ) imply @xmath667 integrating this from @xmath229 to @xmath258 and using ( [ eq : nearcritical ] ) , we find @xmath668 which is what we wanted to prove
. we have thus established ( [ eq : ecomp ] ) ; that is , we have shown @xmath669 \lesssim ( { \mathbf{c}}_{30}\log t)^{2k}\cdot(p_n(1)-p_c ) \sum_{e\in j(\mathbf{v})}\mathbb{p}(b_e),\ ] ] where @xmath670 is the number of edges satisfying the conditions in lemma [ crossing - lemma ] .
note that @xmath629 is equal to the event that the edge @xmath72 is _ pivotal _ for the existence of a left - right @xmath229-open crossing of @xmath539 . following ( * ? ? ?
* lemma 6.2 ) , we can show @xmath671 this concludes the proof of the lemma .
we use a block argument and a peierls argument to upgrade ( [ eq : pre - peierls - bound ] ) to ( [ eq - crossing - bound ] ) .
the annulus @xmath344 , centred at @xmath672 is tiled with smaller squares of side length @xmath673 the existence of a @xmath258-open crossing of @xmath344 implies that of a crossing @xmath674 of @xmath675 along edges of @xmath54 , with @xmath676 and @xmath677 .
the reason for considering this smaller annulus will become clear below .
we can now introduce sequences @xmath678 , and @xmath679 relative to the sequence @xmath680 and squares of size @xmath601 ; that is , @xmath681 @xmath682 the first observation is that we have a lower bound on @xmath424 due to the difference in scales : @xmath683 implying @xmath684 the second observation is that @xmath685 where @xmath686 denotes the @xmath208-th coordinate of the vector @xmath367 . from this , for each fixed @xmath424 , given @xmath687 , there are at most 16 choices for @xmath688 and so at most @xmath689 choices for the sequence @xmath690 .
the first factor is an estimate for the number of choices of squares @xmath691 with @xmath692 .
the third observation is that @xmath693 must contain , between @xmath694 and @xmath695 , a `` short '' crossing @xmath696 of a @xmath697 or @xmath698 rectangle @xmath262 ( that is , the crossing is between the long sides ) .
denote by @xmath699 the @xmath700 or @xmath701 rectangle around @xmath262 , as in ( [ eq : pre - peierls - bound ] ) .
then @xmath702 and so @xmath703 where @xmath704 is the number of points in @xmath699 connected to @xmath696 by a @xmath229-open path in @xmath699 .
it follows that @xmath705 the sum is over all possible finite sequences of squares @xmath706 , for all @xmath707 .
this quantity is controlled by choosing a subsequence of @xmath708 disjoint @xmath699 : each rectangle intersects a fixed number of other such rectangles .
the events appearing in the last probability are independent for disjoint @xmath699 s .
their probability can be bounded using ( [ eq : pre - peierls - bound ] ) ( with @xmath709 ) , since our choice of @xmath601 implies @xmath710 for large @xmath52 .
moreover , one can use the bound on the number of sequences of @xmath367 s ( there are at most 4 choices of @xmath262 for a given @xmath687 ) to control the entire sum : the last line in ( [ eq : peierls - sum ] ) is bounded up to a constant factor by : @xmath711 for @xmath52 large enough , this sum is bounded ( again up to a constant ) by : @xmath712 for any @xmath427 . on @xmath713 , any crossing @xmath458 in the portion of the ipc @xmath284
consists of @xmath714-open edges .
since any site @xmath229-connected to a site in the ipc also belongs to the ipc , we find that the probability in ( [ eq - crossing - bound ] ) is bounded by : @xmath715 choosing @xmath257 appropriately in the definition of @xmath258 ( depending on the parameter @xmath94 in ( [ eq - crossing - bound ] ) ) establishes the claim . |
a total coloring of a graph @xmath2 is an assignment of colors to the vertices and edges of @xmath2 such that every pair of adjacent / incident elements receive different colors .
a @xmath3-total coloring of a graph @xmath2 is a total coloring of @xmath2 from a set of @xmath3 colors .
the minimum positive integer @xmath3 for which @xmath2 has a @xmath3-total coloring , denoted by @xmath4 , is called the total chromatic number of @xmath2 .
it is easy to see that @xmath5 for any graph @xmath2 by looking at the color of a vertex with maximum degree and its incident edges .
the next step is to look for a brooks - typed or vizing - typed upper bound on the total chromatic number in terms of maximum degree .
it turns out that the total coloring version of maximum degree upper bound is a difficult problem and has eluded mathematicians for nearly 50 years .
the most well - known speculation is the total coloring conjecture , independently raised by behzad @xcite and vizing @xcite , which asserts that every graph of maximum degree @xmath6 admits a @xmath7-total coloring .
this conjecture remains open , however , many beautiful results concerning it have been obtained ( cf .
in particular , the total chromatic number of all outerplanar graphs has been determined completely by zhang et al .
@xcite and that of all series - parallel graphs @xcite has been determined completely by wu and hu @xcite .
a graph is pseudo - outerplanar if each of its blocks has an embedding in the plane so that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another .
for example , @xmath8 and @xmath9 are both pseudo - outerplanar graphs .
this notion was introduced by zhang , liu and wu in @xcite , where the edge - decomposition of pseudo - outerplanar graphs into forests with a specified property was studied . in this paper , we prove that the total chromatic number of every pseudo - outerplanar graph with maximum degree @xmath0 is exactly @xmath1 and thus the total coloring conjecture holds for all pseudo - outerplanar graphs .
to begin with , let us review an useful structural property of pseudo - outerplanar graphs which was proved in @xcite .
we shall prove the theorem by induction on @xmath31 .
so we assume that @xmath2 is 2-connected and thus @xmath32 .
set @xmath33 . in the next ,
we complete the proof by verifying the following four claims ; they imply a contradiction to lemma [ struc ] .
suppose , to the contrary , that @xmath36 .
consider the graph @xmath37 . by induction
, @xmath38 has an @xmath39-total coloring @xmath40 .
denote the other neighbor of @xmath41 by @xmath42 .
now erase the color on @xmath41 and color the edge @xmath11 with @xmath43 .
this is possible since @xmath44 .
denote the extended coloring at this stage still by @xmath40 .
then at last we color @xmath41 with @xmath45 , which is also possible since @xmath46 .
otherwise , @xmath48 admits an @xmath39-total coloring by induction and every edge of the 4-cycle has at least two available colors since it is incident with at most @xmath49 colored elements .
this implies that one can extend the coloring of @xmath38 to the four edges @xmath50 and @xmath51 since every 4-cycle is 2-edge - choosable . at last ,
the two vertices @xmath41 and @xmath52 can be easily colored since they are both of degree two .
suppose , to the contrary , that @xmath2 contains such a 4-cycle .
we consider the graph @xmath37 that has an @xmath39-total coloring @xmath40 by induction .
one can find that the only obstacle of extending @xmath40 to @xmath11 is the case when @xmath54 and @xmath55 . without loss of generality ,
let @xmath56 and @xmath57 . if @xmath58 , then we recolor @xmath41 by 4 ( note that @xmath59 ) and then color @xmath11 by 5 .
so we assume @xmath60 .
similarly we can prove @xmath61 .
therefore , we can recolor @xmath52 by 1 and then color @xmath11 by 6 , a contradiction .
suppose , to the contrary , that @xmath2 contains such a 7-path .
we consider the graph @xmath62 , which admits an @xmath39-total coloring @xmath40 by induction .
denote by @xmath63 the set of available colors to properly color an edge @xmath64 .
it is easy to see that @xmath65 and @xmath66 , moreover , @xmath67 and @xmath68 .
if @xmath69 , without loss of generality assume that @xmath70 and @xmath71 , then color @xmath72 with 1 and @xmath73 with @xmath47 .
if @xmath74 , then color @xmath75 with @xmath76 and @xmath77 with 1 .
if @xmath78 , then color @xmath77 with @xmath79 and @xmath75 with @xmath80 .
in each case we have @xmath81 .
thus we can color @xmath51 with @xmath82 and @xmath83 with @xmath84 such that @xmath85 . at this stage ,
the three vertices @xmath86 and @xmath42 can be easily colored since they are all of degree two .
so we assume that @xmath87 .
now we firstly color @xmath72 and @xmath73 by 1 .
if @xmath78 , then color @xmath77 with @xmath88 and @xmath75 with @xmath80 .
the current extended coloring satisfies that @xmath81 .
therefore , we can color the remaining elements similarly as before .
so we assume that @xmath74 .
this implies that @xmath89 and thus we can exchange the colors on @xmath72 and @xmath90 . by doing
so we obtain a new coloring satisfying @xmath91 and therefore we can extend this partial coloring to @xmath2 by a same argument as above .
the graph ( a ) in figure [ fig ] is a pseudo - outerplanar graph , since it has a pseudo - outerplanar drawing as ( b ) .
one can easy to check that it is a graph with maximum degree 3 and total chromatic number 5 .
thus the upper bound for @xmath6 in corollary [ main ] , although probably not the best possible , can not be less than 4 . | a graph is pseudo - outerplanar if each of its blocks has an embedding in the plane so that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another . in this paper , the total coloring conjecture
is completely confirmed for pseudo - outerplanar graphs . in particular , it is proved that the total chromatic number of every pseudo - outerplanar graph with maximum degree @xmath0 is @xmath1 .
+ keywords : pseudo - outerplanar graph , total coloring , maximum degree . |
the database was meticulously constructed by visually inspecting images of purported pne ( sourced from major catalogues , simbad and other literature sources ) in the supercosmos halpha survey ( shs , parker et al .
2005 ) , the int photometric halpha survey ( iphas , drew et al .
2005 ) and the supercosmos sky surveys ( sss , hambly et al .
2001 ) . during this process
both pne and obvious non - pne ( e.g. galaxies , photographic flaws , hii regions , duplicates , etc . ) were added to the database .
the non - pne were flagged as different object types to keep them separate and to keep track of objects already considered for inclusion in the database .
this laborious process allowed us to make the most accurate estimate to date of the total number of galactic pne , @xmath1 , instead of the usual approach of adding total numbers found in new catalogues .
we find @xmath2 , though this upper limit may be reduced as more non - pne are identified after we further check and refine our database before its first release .
our value is based on currently published catalogues and will also be revised upwards once future pn discoveries are published .
figure [ fig : fig1 ] depicts the current galactic distribution of pne .
the database will be accessible through a web - based interface that allows users to search for pne by object name ( iau pn g and common name ) and j2000 coordinates ( single or list format ) .
static releases will also be made intermittently to vizier for virtual observatory compatibility .
results from the web - interface may be browsed in a gallery format with thumbnail images of each pn , as well as a variety of other formats .
groups of pne may be selected based on their coordinates , object size , name , catalogue and so on .
links to online resources are also made available including e.g. the eso archive , vizier and staralt .
the first release of the database is planned for early 2012 and is intended to produce a working database with the cleanest and largest set of entries for published pne .
this will allow for queries to be made , selecting large samples of pne to be studied in a unified fashion with accurate coordinates and pn g designations .
the initial dataset will include at least image cut - outs from the shs and sss ( in jpeg and fits format ) , secgpn spectra in fits format ( acker et al .
1992 ) , radial velocities ( e.g. durand et al . 1998 ) and central star magnitudes ( e.g. tylenda et al .
additions to the database will mostly include large observational datasets in refereed publications that are of broad interest to the whole sample .
acker , a. , marcout , j. , ochsenbein , f. , stenholm , b. , & tylenda , r. 1992 , garching : european southern observatory , 1992 drew , j. e. , greimel , r. , irwin , m. j. , et al .
2005 , mnras , 362 , 753 durand , s. , acker , a. , & zijlstra , a. 1998 , a&as , 132 , 13 hambly , n. c. , macgillivray , h. t. , read , m. a. , et al .
2001 , mnras , 326 , 1279 jacoby , g. h. , kronberger , m. , patchick , d. , et al .
2010 , pasa , 27 , 156 miszalski , b. , et al .
2008 , mnras , 384 , 525 parker , q. a. , phillipps , s. , pierce , m. j. , et al .
2005 , mnras , 362 , 689 parker , q. a. , et al .
2006 , mnras , 373 , 79 tylenda , r. , acker , a. , raytchev , b. , stenholm , b. , & gleizes , f. 1991 , a&as , 89 , 77 viironen , k. , et al .
2009 , a&a , 504 , 291 | since the unifying strasbourg - eso catalogue of galactic planetary nebulae + ( secgpn ) a large number of new discoveries have been made thanks to improved surveys and discovery techniques .
the increasingly heterogeneous published population of galactic pne , that we have determined totals @xmath0 2850 pne , is becoming more difficult to study on the whole without a centralised repository .
we introduce a consolidated and interactive online database with object classifications that reflect the latest multi - wavelength data and the most recent results .
the extensible database , hosted by the centre de donnees astronomique de strasbourg ( cds ) , will contain a wealth of observed data for large , well - defined samples of pne including coordinates , multi - wavelength images , spectroscopy , line intensities , radial velocities and central star information .
it is anticipated that the database will be publicly released early 2012 . |
in recent years a number of authors have investigated the microlensing of extended stellar sources .
( hereafter snw95 ) have shown that the light curves of extended sources can exhibit a significant chromatic dependence , essentially because limb darkening renders the effective radius of the star a function of wavelength .
thus , in addition to improving constraints on the lens parameters , modelling the microlensing of extended sources provides a powerful tool for gravitational imaging stellar surfaces . in this contribution
we describe how microlensing light curves of extended sources may be used as a test of stellar atmosphere models .
we generate artificial light curves , assuming a particular model atmosphere , and use the backus - gilbert numerical inversion method to estimate the radial stellar intensity profile from the observed light curves .
the ( time dependent ) integrated flux , @xmath0 , from an extended stellar source of radius , @xmath1 , lensed by a point lens is given by ( c.f .
snw95 ) @xmath2 where @xmath3 is the projected distance from the lens to the element of the stellar surface and the amplification function , @xmath4 , takes its usual analytic form for a point source .
( note that @xmath3 is a function of @xmath5 , @xmath6 and @xmath7 ) .
if we assume that the projected stellar surface displays circular symmetry
i.e. @xmath8 we may write the above equation in the form @xmath9 where @xmath10 and the kernel function , @xmath11 , is obtained by integrating over @xmath6 .
a solution to this integral equation for @xmath12 can be obtained by applying the backus - gilbert inversion procedure .
this method takes account of the smoothing effect of the kernel function and reconstructs a regularised estimator of @xmath12 which optimises the trade - off between the bias and variance of the estimator . for details of the method
in order to test the feasibility of reconstructing stellar surface profiles , we assumed a simple , linear , limb darkening law with coefficient , @xmath13 .
we generated microlensed light curves of typically 100 data points , with an even sampling rate , and gaussian noise added to the photometry at a level of typically 2% of the baseline flux .
in most cases we took the impact parameter and the einstein radius equal to the stellar radius .
we considered two model atmospheres : @xmath14 ; @xmath15 ; @xmath16 , @xmath17 @xmath18 ; @xmath19 ; @xmath20 , @xmath21 where the johnson @xmath22 and @xmath23 band linear limb darkening coefficients are from .
we carried out inversions of the @xmath22 and @xmath23 band intensity profiles for these models , and investigated the effect of changing the impact parameter , stellar radius , light curve sampling rate and photometric accuracy .
figure [ fig : fig1 ] illustrates the reconstructed @xmath23 band profile for model ( 2 ) , for the case of the impact parameter and stellar radius equal to the einstein radius of the lens .
the errors on the recovered solution are determined from the covariance matrix of the backus - gilbert estimator .
we can see that the reconstructed profile is significantly biased for @xmath24 , due to the smoothing effect of the kernel function , but the true profile is well recovered over the interval @xmath25 .
we now briefly summarise the results of varying the stellar and lens parameters .
* the inversions generally recover the true profiles well over the interval @xmath25 , for a wide range of @xmath26 , @xmath27 and @xmath13 . *
increasing the einstein radius improves significantly the inversions for @xmath28 , but no improvement is seen for @xmath24 . * reducing the impact parameter ( i.e. a transit event ) significantly improves the accuracy of the reconstruction for @xmath28 and reduces the bias for @xmath24 . for impact parameters greater than the stellar radius , however , the reconstruction deteriorates rapidly for all @xmath29 . *
even with photometric errors of 10% a reasonable recovery of @xmath12 is still obtained over the interval @xmath25 ; on the other hand , reducing the errors to only 0.2% does _ not _ improve the recovery for @xmath24 , however .
this is because the bias is primarily due to the ill - posedness of the kernel over this range , and not due to the photometric errors . *
the reconstructions become unacceptably noisy when the number of light curve data points is reduced to @xmath30 , but there is little further improvement in accuracy above @xmath31 .
our results indicate that with realistic light curve sampling and photometric errors one can accurately reconstruct , at least in part , the multicolour radial intensity profiles of extended stellar sources from their microlensed light curves , provided that the impact parameter of the lens is comparable to the stellar radius .
the smoothing properties of the kernel function result in a biased solution for @xmath24 , unless the lensing event is a transit with small impact parameter .
nevertheless , the accurate recovery over the interval @xmath25 is a robust result over a wide range of stellar temperatures and limb darkening coefficients . despite the narrow width of this ` good fit ' annulus , it is still adequate to usefully discriminate between different model atmospheres
e.g. two models with the same temperature but with @xmath15 and @xmath32 .
thus , we conclude that broad band microlensed photometric light curves are a powerful tool for investigating extended stellar sources and testing model stellar atmospheres , and form a useful adjoint to spectroscopic and polarimetric microlensing signatures .
we are currently investigating the application of inversion techniques to more realistic model atmospheres and stellar intensity profiles . | we investigate the feasibility of reconstructing the radial intensity profile of extended stellar sources by inverting their microlensed light curves . using a simple , linear , limb darkening law as an illustration
, we show that the intensity profile can be accurately determined , at least over the outer part of the stellar disc , with realistic light curve sampling and photometric errors .
the principal requirement is that the impact parameter of the lens be less than or equal to the stellar radius .
thus , the analysis of microlensing events provides a powerful method for testing stellar atmosphere models .
gravitational lensing , stars : atmospheres |
plasma inhomogeneities across the magnetic field in the presence of finite - size charged grains causes a wide class of instabilities of an inhomogeneous dusty plasma called gradient instabilities .
such instabilities can be studied in the approximation on magnetic field where we have parallel straight field lines in order to simplify our treatment .
we look for instabilities in the very low frequency regime where a new spectrum instabilities and waves appear , induced by the dust collective dynamics : dust - acoustic - waves ( daws ) , dust - ion - acoustic - waves ( diaws ) , etc .
the frequency of daws are around 10 hz as determined in the laboratory and lower in astrophysical plasmas [ 1,2 ] . in the case that grains are in the micron range we expect a non - ideal behavior due to the fact that the particulate are highly charged and intermolecular forces could play certainly an important role . in order to discuss this problem
we compare the ideal properties with the simple hard - core model and in a next work we will use a better model by means of of the square - well model and the pad rational approximant to the equation of state [ 3 ] for hard - sphere gas , that in our knowledge is more realistic as the simple application of the van der waals equation of state [ 4 ] . in this paper
we show an analysis of the electrostatic waves and instabilities growth rates in a weakly non - ideal magnetized dusty plasma with density and temperature gradients , ignoring charge fluctuation .
as introduced before , the non - ideal behavior is characterized by the hardcore model defined by @xmath0 or in similar manner by the square - well model given by the ree and hoover expression [ 5 ] .
in this paper we introduce a new numerical treatment in combination with a more realistic formulation of the equation of state to simulate weak non ideal effects in order to analyze inhomogeneous vlasov - dusty plasma systems where a linearized dispersion relation is obtained . due to the lower frequency range ( @xmath1 )
, enough energy can be transferred from the particle to the wave and instabilities can be generated . in order to get an adequate linear dispersion relation with a magnetic field given by @xmath2 for maxwellian multi - species plasmas ( electron , ion and dust )
, we introduce our well known and very accurate multipolar approximation [ 6 ] for the @xmath3 dispersion function . in the presence of a magnetic field
we have the distribution function of the species @xmath4 , solution for the kinetic equation @xmath5 in the time dependent following form[7,8 ] @xmath6 { \bf \nabla } \phi ` ( r(t^{\prime } ) ) \cdot \frac{\partial f_{o\alpha } } { \partial { \bf v(}t^{\prime } { \bf ) } } dt^{\prime } \ ] ] where @xmath7 now , the dispersion relation in terms of the dielectric susceptibilities , in the low frequency approximation ( @xmath1 ) is @xmath8 where , @xmath9\ ] ] with : @xmath10 further , in order to simplify our expressions , we use : @xmath11 now , using the following identity for the dispersion function @xmath3 @xmath12,}$\nonumber}\ ] ] we obtain after several cumbersome algebraic manipulations the dielectric susceptibility in the form @xmath13 \right ) \right\ } \right ] \,\ ] ] in order to put our dispersion relation in a dimensionless form , we introduce following suitable definitions : @xmath14 now , using those results and assuming that @xmath15 we can write down eq.(3 ) as @xmath16 in the non ideal case ( dust ) we introduce a relation that in principle express the non ideal behavior of the system in terms of the pressure in the form @xmath17 given by the hard - core model . this model is taken for simplicity . a better model , as mentioned before , will be introduced in a future work .
now , following definitions are also useful @xmath18 those relations are very convenient by writing the full dispersion relation[4 ] .
in fact we have @xmath19 for the non - ideal case . for the ideal one ,
we use the well known relation @xmath20 , and in a similar way we get @xmath21 where @xmath22 .
two special cases can be worked out : + a ) density gradient equal to zero @xmath23 , that means , @xmath24 + +
b ) temperature gradient equal to zero @xmath25 , that means , @xmath26 + further we can introduce following relations in order to express dielectric susceptibilities in a suitable forms @xmath27 @xmath28 using those relations we arrive to the dispersion relation for the case b where we get : @xmath29\ ] ] @xmath30\ ] ] @xmath31\ ] ] where @xmath32\lambda_{p } $ ] and @xmath33 . in a similar way , it is possible to include the terms for case a , where we shall have @xmath34 introducing now the multipolar approximation to @xmath35 we can get a polynomial expression in the well known form[9 ] @xmath36 where coefficients @xmath37 and @xmath38 are functions of the system parameters .
such an expression is easy to solve and with high accuracy to find roots of the numerator .
an analysis of these solutions spectra permit us to give the imaginary parts @xmath39 in function of @xmath40 , which represent the growth rate instabilities .
the quasi - neutrality equation for dusty plasmas can be approached by a simplified one due to the high state of charge of the dust grains @xmath41 and the electron susceptibility can be neglected in the dispersion relation .
the range of the main parameters in the study of the low frequency oscillation of dust grains is established by the approximations that conduced to the simplified dispersion relation @xmath42 unstable dust oscillations ( @xmath43 ) are found for @xmath44 , @xmath45 . at the present time , we only give the results for the density gradient case ( _ i.e. _ @xmath46 ) . for slightly inhomogeneous plasmas with normalized density gradient length @xmath47 ,
the shape of the dust instability ( @xmath48 ) curve as function of the perpendicular to magnetic field wavelength ( @xmath49 ) is similar to that for ions , previously studied [ 8 ] . ) and for a relatively inhomogeneous one ( @xmath50).,width=359,height=359 ] the maximum value of the instability increases and narrows with the state of charge of the dust @xmath51 but decreases and get wider with the mass . for typical laboratory light dusty plasmas (
@xmath52 , @xmath53 ) the instability of dust acoustic or electrostatic waves is narrower and smaller than that for ions in figure 1 the peak of the left corresponds to the typical shape of instability of slightly inhomogeneous plasmas , while the right region of instability appears for density gradient lengths of the order of a hundred of debye lengths ( @xmath54 ) . for higher density gradients ( @xmath55 ) , this new instability region is wider and so high as the typical one . for even higher density gradients ( @xmath56 ) ,
figure 2 shows that the new right region gives a higher instability .
this figure also shows the effect of the non ideality of the plasma .
necessary condition for the exhibition of dust acoustic waves . for typical laboratory dust radius the @xmath57 parameter of the hard core potential equation of state , is of the order of @xmath58 . and for typical values of ion density of @xmath59 ( and corresponding @xmath60 , by quasi - neutrality relation ) , it appears a new intermediate instability region which can reach a maximum for denser plasmas ( @xmath61 ) or larger dust particles ( @xmath62 )
. this maximum is limited for the relation for dust collective behavior @xmath63
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icppp * 1 * , 94 - 97 ( 1996 ) | in this paper we introduce an algebraic form of the dispersion relation for a non ideal inhomogeneous dusty plasma in order to improve drastically the calculation of the drift instability growth rate .
this method makes use of the multipole approximation of the z dispersion function , previously published , and valid for the entire range . a careful analysis of the solutions spectra of this kind of polynomial equation permits us to calculate easily the growth rate of the drift instability for the ion - dust and dust acoustic mode .
the value of the parallel to magnetic field wavelength for which the instability reaches the maximal value is carefully localized and discussed .
the unstable dust - ion and dust acoustic mode are discriminated and analyzed in function of the density gradient , te / ti - ratio , and dust grain radius . |
observations of a sample of three late - type galaxies with low surface - brightness and the radio - weak edge - on galaxy ngc 5907 ( all with a low sfr ) revealed that they all have an unusually high thermal fraction and weak total and regular magnetic fields ( chyy et al .
2007 , dumke et al .
however , these objects still follow the total radio - fir correlation , extending it to the lowest values measured so far .
hence , these galaxies have a lower fraction of synchrotron emission than galaxies with higher sfr .
it is known that the thermal intensity is proportional to the sfr .
our findings fit to the equipartition model for the radio - fir correlation ( niklas & beck 1997 ) , according to which the nonthermal emission increases @xmath0 and the _ total _ magnetic field strength @xmath1 increases @xmath2 .
+ no similar simple relation exists for the _ regular _ magnetic field strength .
we integrated the polarization properties in 41 nearby spiral galaxies and found that ( independently of inclination effects ) the degree of polarization is lower ( @xmath3 ) for more luminous galaxies , in particular those for @xmath4 ( stil et al .
the radio - brightest galaxies are those with the highest sfr .
though a dynamo action needs star formation and supernova remnants as the driving force for velocities in vertical direction , we conclude from our observations that stronger star formation seems to reduce the magnetic field regularity . on kpc - scales , chyy ( 2008 ) analyzed the correlation between magnetic field regularity and sfr locally within one galaxy , ngc 4254 . while he found that the total and random field strength increase locally with sfr , the regular field strength is locally uncorrelated with sfr .
we determined the exponential scale heights of the total power emission at @xmath5 cm for four edge - on galaxies ( ngc 253 , ngc 891 , ngc 3628 , ngc 4565 ) for which we have combined interferometer and single - dish data ( vla and the 100-m effelsberg ) . in spite of their different intensities and extents of the radio emission , the vertical _ scale heights _ of the thin disk and the thick disk / halo are similar in this sample ( 300 pc and 1.8 kpc ) ( dumke & krause 1998 , heesen et al .
we stress that our sample includes the brightest halo observed so far , ngc 253 , with a very high sfr , as well as one of the weakest halos , ngc 4565 , with a small sfr . for ngc 253 heesen et al .
( this volume ) argued that the synchrotron lifetime ( which is @xmath6 ) mainly determines the vertical scale height of the synchrotron emission and estimated the cosmic ray bulk velocity to @xmath7 km / s .
as this is similar to the escape velocity , it shows the presence of a galactic wind in this galaxy . the fact that we observe similar averaged scaleheights at @xmath5 cm for
the four galaxies mentioned above imply that the galactic wind velocity is proportional to @xmath8 , and hence proportional to @xmath9 .
in a larger sample of 11 edge - on galaxies we found in all of them ( except the inner part of ngc 4631 , see krause 2009 ) mainly a disk - parallel magnetic field along the galactic midplane together with an x - shaped poloidal field in the halo .
our sample includes spiral galaxies of different hubble types and sfr , ranging from @xmath10 .
the disk - parallel magnetic field is the expected edge - on projection of the spiral magnetic field within the disk as observed in face - on galaxies .
it is generally thought to be generated by a mean - field @xmath11-dynamo for which the most easily excited field pattern is the axismmetric spiral ( ass ) field ( e.g. beck et al .
the poloidal part of the ass dynamo field alone , however , can not explain the observed x - shaped structures in edge - on galaxies as the field strength there seems to be comparable to that of the large - scale disk field .
model calculations of the mean - field @xmath12-dynamo for a disk surrounded by a spherical halo including a _ galactic wind _ ( brandenburg et al . 1993 ) simulated similar field configurations as the observed ones .
new mhd simulations are in progress ( see e.g. gressel et al . this volume , hanasz et al .
this volume ) which include a galactic wind implicitely .
a galactic wind can also solve the helicity problem of dynamo action ( e.g. sur et al .
2007 ) . hence
, a galactic wind may be essential for an effective dynamo action , and to explain the observed similar vertical scale heights and x - shaped magnetic field structure in edge - on galaxies . | from our radio observations of the magnetic field strength and large - scale pattern of spiral galaxies of different hubble types and star formation rates ( sfr ) we conclude that though a high sfr in the disk increases the total magnetic field strength in the disk and the halo the sfr does not change the global field configuration nor influence the global scale heights of the radio emission .
the similar scale heights indicate that the total magnetic field regulates the galactic wind velocities
. the galactic wind itself may be essential for an effective dynamo action . |
we collected and analyzed more than 4.5 million time - stamped emails from students at a globally top - ranked mba program , focusing specifically on the relationship between students evolving communication networks and their subsequent career outcomes .
this data is available in the form of email logs recorded and stored by the university , along with registrar data on each student before and during their matriculation in the program .
included in the dataset is a record of each email sent by an mba student between fall 2006 and spring 2008 .
the record includes the date and time at which the email was sent and received and the ( anonymous ) numeric ids of the sender and receiver of the message .
academic records ( gmat scores , grades , extra - curricular activities , prior work experience and job titles ) , and demographic data ( age , race and nationality ) , were merged with the network data to connect email transmissions with personal characteristics .
there are approximately 11.5 million e - mails and 4.5 million student - student e - mails in the data .
an important characteristics of the data is its randomized design .
students are randomly assigned to sections within the school , minimizing selection effects .
since observations began when students first met each other , we eliminated the left censoring that typically occurs when network data are captured after ties have already been formed .
in order to understand the causal relationship between network effects and job rank , we used coarsened exact matching ( cem ) ( see @xcite ) to construct a reduced , matched sample ( matching students on all characteristics possible , i.e. , age , gpa , industry experience ) . then on the matched data , we further examine the significance of network effects on students job rank . additionally , in order to demonstrate that our observations can not be explained by a random network process
, we compare real observations to the null model where the degree sequence of network is preserved and links are placed completely randomly .
three important findings emerge from the data .
first , a student s likelihood of securing a coveted , high paying job after graduation is strongly associated with the type of network they develop during their time in the program .
students with a higher network degree , greater network centrality , and more balanced communication among alters tend to have the highest post - graduation salaries , even when matching students along several covariates ( figure [ fig_1 ] ) .
second , and somewhat surprisingly , we find that the structural characteristics of a student s ego network emerge as early as one month into the program and remain remarkably stable thereafter ( figures [ fig_1 ] [ fig_2 ] ) .
the above findings suggest that , when exposed to a new social system , actors initial network configurations may provide `` early warning signals of success '' a finding that has important practical and policy - level implications .
lastly , we observe robust patterns of `` rich - club '' behavior at the level of the global network , where highly central students are more densely connected among themselves than students of a lower degree @xcite @xcite . while the students who comprise the rich - club networks were more likely to have a higher salary post - graduation , they were also less likely to have the highest rank on other objective indicators of ability , such as gmat scores and gpa ( figure [ fig_3 ] ) .
we interpret these findings as evidence of a trade - off between the development of human capital and social capital during an mba program : on the one hand , students can choose to invest in building technical skills and domain - specific knowledge to enhance their career prospects ; on the other hand , they can choose to invest in building their social capital by developing new ties and fostering a robust community of peers . though these choices are certainly not mutually exclusive , the economic benefits of the former appear to significantly outweigh the benefits of the latter .
despite the fact that business schools advertise their role in fostering a valuable , life - long network ( haas mba 2013 ) , we find considerable differences in the types of networks that students actually develop - differences that are strongly linked to their future job placement and , ultimately , their access to the inner circles of the managerial elite .
our findings have important implications for both mba students and the firms that hire them . from the standpoint of a student
, the data suggest that important resources exist within an emerging mba network that ultimately improve one s attractiveness to employers . while we can only speculate about the mechanism underling this pattern of results i.e. specific networks may foster the development of particular social skills or , instead
, they may provide access to employer networks outside of the program itself we nevertheless find that it quite literally `` pays '' to develop one s social network in the early stages of a program . from the standpoint of prospective recruiters , the subtle signals that allow one to reliably infer a student s network structure @xcite may provide valuable insight into the qualities that the applicant will bring with them to their new job .
michael useem .
classwide rationality in the politics of managers and directors of large corporations in the united states and great britain .
administrative science quarterly vol .
27 , no . 2 ( jun . , 1982 ) , pp . | the `` business elite '' constitutes a small but strikingly influential subset of the population , oftentimes affecting important societal outcomes such as the consolidation of political power @xcite , the adoption of corporate governance practices , and the stability of national economies more broadly .
research has shown that this exclusive community often resembles a densely structured network , where elites exchange privileged access to capital , market information , and political clout in an attempt to preserve their economic interests and maintain the status quo @xcite .
while there is general awareness that connections among the business elite arise because `` elites attend the same schools , belong to the same clubs , and in general are in the same place at the same time '' , surprisingly little is known about the network dynamics that emerge within these formative settings .
here we analyze a unique dataset of all mba students at a top 5 mba program .
students were randomly assigned to their first classes ; friendship among students prior to coming into the program was rare ; and the network data email transmissions among students were collected for the year 2006 when students almost entirely used the school s email server to communicate , thereby providing an excellent proxy for their networks . after matching students on all available characteristics ( e.g. , age , grade scores , industry experience , etc . ) i.e. creating `` twin pairs '' we find that the distinguishing characteristics between students who do well in job placement and those who do not is their network .
further , we find that the network differences between the successful and unsuccessful students develops within the first month of class and persists thereafter , suggesting a network imprinting that is persistent .
finally , we find that these effects are pronounced for students who are at the extreme ends of the distribution on other measures of success students with the best expected job placement do particularly poorly without the right network ( `` descenders '' ) , whereas students with worst expected job placement pull themselves to the top of the placement hierarchy ( `` ascenders '' ) with the right network . |
this work has been funded in canada by nserc ( grant sapin 386432 ) , cfi - lof and orf - sif ( project 24536 ) , and by the france - canada research fund ( project `` listening to scintillating fractures '' ) .
prof . c. dujardin of lyon , france , kindly provided access to his equipment to measure x - ray emission spectra .
we thank m. chapellier for stimulating discussion , and a. b. mcdonald for comments on our manuscript .
32ifxundefined [ 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 http://arxiv.org/abs/1101.5205 [ ( ) ] , link:\doibase 10.1016/s0927 - 6505(02)00111 - 1 [ * * , ( ) ] link:\doibase 10.1016/j.physleta.2006.03.059 [ * * , ( ) ] arxiv : hep - ex/0011064 [ * * , ( ) ] , link:\doibase 10.1140/epjc / s10052 - 012 - 1971 - 8 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1016/s0022 - 2313(00)00230 - 1 [ * * , ( ) ] link:\doibase 10.1088/0022 - 3727/17/1/016 [ * * , ( ) ] link:\doibase 10.1063/1.339650 [ * * , ( ) ] link:\doibase 10.1007/bf00555916 [ * * , ( ) ] link:\doibase 10.1038/nature07378 [ * * , ( ) ] http://arxiv.org/abs/astro-ph/0106200 [ ( ) ] , http://repository.dl.itc.u-tokyo.ac.jp/dspace/handle/2261/12045 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.79.3202 [ * * , ( ) ] link:\doibase 10.1007/s10704 - 006 - 0051 - 1 [ * * , ( ) ] @noop _ _ , ed .
( , ) @noop _ _ , ed .
( , ) @noop * * , ( ) link:\doibase 10.1016/j.nima.2005.01.303 [ * * , ( ) ] link:\doibase 10.1016/0266 - 3538(87)90086 - 8 [ * * , ( ) ] link:\doibase 10.1103/physrevb.52.9224 [ * * , ( ) ] link:\doibase 10.1016/s0022 - 2313(01)00412 - 4 [ * * , ( ) ] link:\doibase 10.1063/1.1662183 [ * * , ( ) ] link:\doibase 10.1006/adnd.1993.1013 [ * * , ( ) ] link:\doibase 10.1016/0022 - 2313(85)90114 - 0 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.68.1244 [ * * , ( ) ] @noop _ _ , ed .
( , ) link:\doibase 10.1007/bf00726989 [ * * , ( ) ] \doibase doi:10.1063/1.1735524 [ * * , ( ) ] http://arxiv.org/abs/0708.4315 [ ( ) ] link:\doibase 10.1016/j.astropartphys.2012.05.005 [ * * , ( ) ] link:\doibase 10.1209/0295 - 5075/99/28003 [ * * , ( ) ] | prompted by intriguing events observed in certain particle - physics searches for rare events , we study light and acoustic emission simultaneously in some inorganic scintillators subject to mechanical stress .
we observe mechanoluminescence in , and , in various mechanical configurations at room temperature and ambient pressure .
we analyze the temporal and amplitude correlations between the light emission and the acoustic emission during fracture .
a novel application of the precise energy calibration of provided by radioactive sources allows us to deduce that the fraction of elastic energy converted to light is at least @xmath0 .
rare - event searches in particle physics , like those looking for particle dark matter @xcite , employ draconian measures to reduce all forms of background that could hide their signal .
the main background is usually of radioactive nature , however other forms are possible .
for instance , the first , calorimetric phase of the cresst experiment measured phonons in sapphire crystals cooled to around @xmath1 to detect particle interactions @xcite .
initially , thermal contraction of the holders caused fractures in the crystals , limiting sensitivity of the experiment by three orders of magnitude in rate above the radioactive background until the problem was identified @xcite .
certain more recent experiments use a coupled measurement of phonons and scintillation to determine the nature of the interacting particles and reject the radioactive background @xcite .
however , mechanoluminescence , i.e. the emission of light by a solid subject to mechanical stress , occurs in many materials @xcite , including plastic scintillators @xcite .
in addition , correlated acoustic emission ( ae ) and visible - photon emission have been observed during mechanical deformation of colored alkali halides @xcite .
other types of emission , including electrons @xcite , positive ions @xcite , and x - rays @xcite , have also been studied in various materials under mechanical stress .
nevertheless , there is an opinion that fracture events in scintillation - phonon detectors would produce little or no light ( e.g. @xcite ) .
prompted by recent intriguing events observed by the cresst ii dark matter search using cryogenic scintillation - phonon detectors @xcite , we study mechanoluminescence as a form of photon - producing background in such devices . we also argue that photon emission can help characterize the rupture dynamics in a manner analogous to ae @xcite , and can provide complementary information about the fracture energy .
to investigate these two points , we have carried out what we believe are the first experiments to correlate the acoustic and light emission from common scintillators under mechanical stress , and to quantify the conversion of elastic energy into light .
our studies involve the measurement of acoustic phonons and luminescence as a piece of scintillating crystal is stressed to the point of rupture .
the inorganic scintillators studied here are widely used in particle detection @xcite : bismuth germanate ( , a.k.a .
bgo ) , zinc tungstate ( ) and cadmium tungstate ( ) . except where specified
, all samples were kindly supplied by crystal manufacturing lab ltd . , novosibirsk .
we have tested two loading geometries , at room temperature and ambient pressure .
initially , @xmath2 rectangular prisms were indented by a steel bead driven by a manual screw , causing the material to break in multiple fragments . in order to better control the rupture process
, we subsequently adopted the double cleavage drilled compression ( dcdc ) geometry @xcite ( fig .
[ fig_fig_1 ] ) .
these samples were @xmath3 rectangular prisms , polished to optical quality , with a @xmath4 diameter circular hole drilled perpendicularly in the middle of the @xmath5 face . under compression along the long axis ,
cracks formed reproducibly on either side of the hole in the plane parallel to the @xmath6 face . unlike bgo , tungstates have a cleavage plane @xcite ; orientation of the crystals was chosen so that this plane was parallel to the dcdc fracture direction .
the dcdc samples were pressed against a backstop by a pushrod that was driven by an actuator ( controlled manually in early experiments , and by computer in later ones ) .
this geometry provides some control of crack velocity via the applied compressive stress , up to a critical length at which point the sample cleaves abruptly @xcite . a force gauge in the pushrod measured the load imposed by displacements of the actuator .
the acoustic activity was recorded by two piezo - electric pinducers in contact with opposite sides of the crystal via ultrasound gel .
light emission was recorded by a standard bialkali photomultiplier ( pm ) covering the crystal .
the output of the pm was integrated with a time constant ranging from @xmath7 to @xmath8 depending on the sample to produce a signal that could be digitized more slowly than the raw pm output .
all four channels ( force , 2 ae , pm ) were digitized at a rate of @xmath9 samples per second in a continuous stream .
the setpoint of the computer - controlled actuator was also recorded .
the fracture procedure involves advancing the actuator by small steps ( down to @xmath10 ) .
each step is followed by a waiting time of at least @xmath11 .
the whole procedure lasts @xmath12@xmath13 during which we continuously monitor ae and pm activity .
figure [ fig_mainresult ] shows typical observations for the tungstate crystals .
spurts of activity on the ae and pm channels are correlated to drops in the force .
these abrupt force relaxations are a direct consequence of crack growth in the samples .
there is also a regime in which the force decreases slowly , and during which other processes , such as rearrangements and dislocations , produce ae and pm signals @xcite .
similar correlations have been observed for other materials , such as colored @xcite , composites @xcite , and silica glass @xcite .
indentation of bgo samples ( from fibercryst , france ) also produces correlated ae and light signals .
these correlations can be analyzed more quantitatively by looking at the statistical properties of individual acoustic and light events .
in this analysis , individual events have been determined on each channel by requiring that the signal surpass a threshold , and that it be separated from the previous event by at least a minimum time .
figure [ fig_wait_extra ] shows that the distributions of waiting times between events are similar for ae and light over two decades . for light events , it is possible to obtain a much smaller minimum separation time ( @xmath14 ) than for ae ( @xmath15 ) , since the former pulses are shorter than the latter . to quantify the amount of light emitted and to investigate the emission mechanism , we next compared the spectra of light emitted by bgo ( sample from fibercryst ) during fracture and during scintillation stimulated by x - rays . to avoid afterglow from the high dose of x - rays ,
the fracture spectrum was measured first , and then one of the fragments was measured while being irradiated by x - rays .
the measurements of the light spectra were carried out with a monochromator and a ccd camera . in the x - ray excitation configuration , a generator provided x - rays with a broad spectrum up to a few tens of . over the recorded range of wavelengths , both resulting spectra ( fig .
[ fig_spectra ] ) show a very similar shape ( as is the case for many materials @xcite ) , indicating a common underlying luminescence mechanism , at least for the last stages of light emission .
the likeness of our spectra to previous work on bgo indicates this last stage involves @xmath16 electronic transitions of the luminescence centers @xcite .
this measurement does not determine if the fractures excite the luminescence centers directly .
it can not be excluded that the crystal is indirectly excited by particles produced by fracto - emission @xcite interacting in the scintillator .
for instance , if x - rays or electrons are produced , the bgo sample would be quite efficient at detecting them itself ( attenuation length @xmath17 for @xmath18 x - rays @xcite ) , as opposed to needing an external x - ray detector @xcite .
alternatively , any uv light ( potentially from arcs in the atmosphere near the fracture ) might be re - absorbed in the scintillator since bgo absorbs wavelengths below @xmath19 @xcite .
we then calibrated the light channel for a dcdc bgo sample using radioactive @xmath20-ray sources as we would calibrate a regular scintillator used for particle detection .
calibration was carried out in the main dcdc setup and with the standard daq ( fig .
[ fig_fig_1 ] ) .
the analysis involves extracting individual events as described earlier , and then building the distribution of the event integrals .
the integral of an event , like its amplitude , is a proxy for the energy deposited by the @xmath20 , so we can identify the peaks in the distribution corresponding to the energies of the radioactive sources and of backscattering @xcite , as shown in fig .
[ fig_calibr ] . over the energy range available with these sources ,
the crystal and readout behave in a linear manner with respect to the energy deposited in the scintillator , and we extrapolate this calibration to higher energies .
calibrations before and after the fracture are compatible to within @xmath21 . from the standpoint of the scintillation mechanism , @xmath20-rays and x - rays can interact in inorganic scintillators via the photoelectric effect or through compton scattering , creating a primary electron - hole pair which eventually transfers a portion of the deposited energy to the luminescence centers ( for bgo ) . for a given light signal ,
calibrations provide the equivalent amount of energy deposited by a @xmath20-ray .
this can be converted to the actual energy of the emitted light by knowing the light yield of the scintillator ( @xmath22 for @xmath20-rays in bgo @xcite ) , and the energy of individual scintillation photons ( @xmath23 c.f .
fig [ fig_spectra ] ) .
the calibrated bgo crystal was then fractured in the main dcdc setup , using the protocol described earlier .
we focus on the last drop in force , occurring when the crack splits the sample in two .
figure [ fig_bgobreak ] shows that the overall behavior is similar to that observed for the tungstates .
the photon channel displays a wide distribution of event amplitudes .
they in fact reach up to the saturation level of the integrator , equivalent to @xmath20-ray energy deposits of @xmath24 a testimony to the amount of light emitted .
there is also a hint of increased photonic activity before the main fracture occurs .
the correlations are better quantified in fig .
[ fig_corr_extra ] illustrating the cumulative number of events on the acoustic channels and those on the photon channel .
minimum separation time required between photon events was @xmath8 , and @xmath25 between ae events .
on all three channels , the cumulative number of events rises steadily with a slight increase in slope until the fracture , as determined by the drop in the force .
this drop marks a sharp increase in the slope of cumulative events that eventually peters out .
when normalized to the total number of events on each channel , all curves are very similar .
lastly , event amplitudes for all channels are power - law distributed ( fig . [ fig_hist_spectra ] ) . for both ae channels ,
the exponents are @xmath26 ( errors are one standard deviation ) . assuming the ae energy is proportional to the amplitude squared
, the power - law distribution of the energy would have as exponent @xmath27 .
this is compatible with the exponent obtained for the light channel , @xmath28 , for which amplitude is proportional to energy .
a more precise measurement might check if the absolute value of the ae exponent is in fact larger than that of the light channel , an indication of dissipation in the acoustic channel @xcite .
we can also use our technique to estimate certain contributions to the energy budget of the rupture process in bgo .
elastic energy is converted into broken bonds , phonons ( that are either measured as acoustic emission or decay into heat ) and light .
for the emitted light , the calibrations with @xmath20-ray sources establish that the total amount of emitted light is equivalent to that caused by @xmath20-ray energy deposits of at least @xmath29 , in other words a total energy converted to light greater than @xmath30 . only a lower bound
is set because of integrator saturation mentioned previously .
eighty - five percent of this light is emitted in a @xmath31 window centered on the main fracture .
we next estimate the drop in elastic energy of the crystal during the main fracture . neglecting the hole in the dcdc geometry ,
the elastic energy in the crystal of section @xmath32 ( perpendicular to the force @xmath33 ) and of volume @xmath34 is @xmath35 , where @xmath36 is young s modulus . applying a standard acoustic measurement of longitudinal and shear waves @xcite to bgo
, we find @xmath37 , somewhat above the range of published values ( up to @xmath38 @xcite and references therein ) .
assuming the loaded volume and surface vary little , the drop in force of @xmath39 is equivalent to a drop in elastic energy of @xmath40 .
since the surface created during the fracture is roughly @xmath41 , this amounts to a surface creation energy of the order of @xmath42 , consistent with the scales typically found for crystals ( e.g. @xmath43 for , @xmath44 for @xcite ) .
furthermore , the conversion efficiency of elastic to luminous energy is therefore at least @xmath45
. a fuller understanding of the energy budget requires knowledge of the energy transformed into phonons , or heat .
the standard room - temperature technique to study brittle fracture only involves measurement of acoustic phonons using acoustic emission .
we are not aware of any work quantifying the fraction of energy going into acoustic phonons , or calibrating the energy scale of acoustic phonons .
the harder - to - implement technique of cryogenic calorimetry would allow measurement of all the phonons and would therefore provide more insight into the energy budget @xcite .
we have evidenced correlated mechanical - stress - induced emission of photons and of phonons in several common inorganic scintillators used for particle detection ( , and ) , at room temperature and in a normal atmosphere .
at least for bgo , both emissions share similar distributions of waiting times over several decades . also for bgo ,
the two energy distributions are similar , and the wavelength of fractoluminescence is the same as that of scintillation . in a mechanically - stressed scintillation - only detector , the power - law distribution of mechanoluminescence energies means that spurious low - energy events may mimick dark matter ones .
it seems reasonable to assume our results apply to other tungstates , and it would be interesting to study the light - phonon correlations of individual fracture events in cresst ii - like conditions of vacuum and low - temperature ( other background - based explanations of the intriguing cresst ii events have been proposed @xcite ) .
in addition , the re - absorption of products of fracto - emission could affect other types of solid - state particle detectors ( e.g. ionization or ionization - phonon ones @xcite ) . from the standpoint of fracture physics , in contrast to previous studies by acoustic emission or by spectrally - resolved luminescence ( e.g. @xcite ) , our use of the light channel offers a precisely calibrated energy scale .
this allows us to quantify the amount of elastic energy converted into light , and is a step towards a better fundamental understanding of fracture mechanisms . |
99 bohigas o 1991 _ random matrix theories and chaotic dynamics _ , in giannoni m j , voros a and zinn - justin j ( eds ) , _ proceedings of the 1989 les houches summer school on chaos and quantum physics _ , pages 88 - 199 .
( amsterdam : elsevier ) . | the spectral statistics of the circular billiard with a point - scatterer is investigated . in the semiclassical limit
, the spectrum is demonstrated to be composed of two uncorrelated level sequences .
the first corresponds to states for which the scatterer is located in the classically forbidden region and its energy levels are not affected by the scatterer in the semiclassical limit while the second sequence contains the levels which are affected by the point - scatterer .
the nearest neighbor spacing distribution which results from the superposition of these sequences is calculated analytically within some approximation and good agreement with the distribution that was computed numerically is found .
classical dynamics may be illuminating for the understanding of the corresponding quantum mechanical systems .
one of the most studied aspects of the relation between classical and quantum mechanics is the connection between the spectral statistics of the quantum system and the dynamical properties of its classical counterpart .
classically integrable systems typically exhibit poisson - like spectral statistics @xcite while classically chaotic systems exhibit spectral statistics of random matrix ensembles @xcite .
the spectral statistics of integrable and chaotic systems are universal , that is , they do not depend on specific details of the system but rather on the type of motion and its symmetries .
there are systems which are intermediate between integrable and chaotic ones and their spectral properties are not known to be universal .
such systems are of experimental relevance .
the spectral statistics of mixed systems , for which the phase space is composed of both integrable and chaotic regions , were studied by berry and robnik @xcite .
the spectrum can be viewed as a superposition of uncorrelated level sequences , corresponding to the various regions , which are either chaotic or integrable .
the nearest neighbor spacing distribution ( nnsd ) of such a superposition of sequences was calculated in @xcite .
the resulting statistics are , in some sense , intermediate between those of integrable and chaotic systems .
other types of systems with intermediate statistics include pseudointegrable systems and integrable systems with flux lines or point - scatterers @xcite .
the spectral statistics of billiards with flux lines @xcite and of some pseudointegrable billiards @xcite were recently studied .
a possible route towards an understanding of the spectral statistics of these systems is based on their classical periodic orbits .
it is possible to compute the correlation function of the energy levels from these orbits by using trace formulae @xcite . for the billiards with flux lines or for pseudointegrable billiards ,
the orbits include not only the periodic orbits but also diffracting orbits which are built from segments that start and end at some singularity of the system .
while the contributions from periodic orbits were easily calculated , those from diffracting orbits turn out to be much more involved .
the spectral statistics of these systems , that can be obtained numerically , appear to be intermediate between those of integrable and chaotic systems .
in particular , the nnsd show level repulsion at small spacings and an exponential falloff at large spacings . since the contributions of diffracting orbits to the spectral statistics of pseudointegrable systems and of billiards with flux lines is far from being understood , it is of interest to study simpler systems which exhibit intermediate statistics . a class of such systems is given by integrable systems with a point - scatterers .
a point - scatterer is the self adjoint extension of a `` @xmath0-function potential '' in two or three dimensions @xcite .
the spectral statistics of an integrable system with such a point - scatterer was first studied by eba @xcite .
it is a rectangular billiard with the perturbation at its center .
this `` eba billiard '' also exhibits intermediate statistics which differs from that of psedointegrable systems .
integrable systems with point - scatterers are much easier to study analytically compared to integrable systems with flux lines or to pseudointegrable systems .
the contributions of diffracting orbits to the correlation function of the energy levels were recently calculated for the rectangular billiard with a point - scatterer @xcite .
exact results for the nnsd were also obtained @xcite .
one of the intriguing features of the spectral statistics of the rectangular billiard with a point - scatterer ( and dirichlet boundary conditions ) is its dependence on the location of the scatterer . if the coordinates of the scatterer ( divided by the sides of the rectangle ) are rational numbers @xmath1 ( where @xmath2 ) then the spectral statistics depend in a non trivial way on @xmath3 @xcite .
in contrast , for typical locations , the spectral statistics seem to be location independent .
the cause for this dependence on location is that many wavefunctions vanish at rational values of the coordinates . at these locations
there are many degeneracies in the lengths of the diffracting orbits ( including repetitions ) .
since such dependence on location is atypical , it is of interest to study the dependence of the spectral statistics on the location of the perturbation in other systems .
for instance , it may be possible that for typical systems the location of the point scatterer affects the spectral statistics only smoothly ( and does not depend on the rationality of the coordinates of the scatterer ) .
the system that is studied in this work is the circle billiard perturbed by a point - scatterer and the dependence of the spectral statistics on its location is studied .
the ( two dimensional ) circle billiard with radius @xmath4 is described by the schrdinger equation @xmath5 with dirichlet boundary conditions @xmath6 .
( the units where @xmath7 are used in most of this work . )
the hamiltonian with the point - scatterer is the self - adjoint extension of a hamiltonian where one point , say @xmath8 , is removed from its domain .
it can be considered as the self - adjoint extension of a hamiltonian with a @xmath0-function potential at @xmath8 . given the eigenvalues ( and eigenfunctions ) of the unperturbed system @xmath9 ( and @xmath10 ) ,
namely the system in absence of the @xmath0-scatterer , the eigenvalues of the system with the point - scatterer are given by the roots of @xcite @xmath11 where @xmath12 and @xmath13 are two parameters .
for a more complete discussion regarding this equation , the roles of the parameters as well as the method of its numerical solution see , for example , @xcite .
this equation turns out to be very convenient for numerical solution since every root is located between two eigenvalues of the unperturbed system .
the ( unnormalized ) eigenfunctions of the circle billiard are @xmath14 where @xmath15 are bessel functions of the first kind and the angular momentum @xmath16 is any nonnegative integer .
the energy levels @xmath17 are determined by the boundary condition @xmath18 .
it is obvious that all the energy levels with @xmath19 are doubly degenerate . as a result
there is a linear combination of the two degenerate wavefunctions which vanishes at @xmath8 and an orthogonal linear combination which does not vanish at @xmath8 .
the perturbation breaks this degeneracy . the linear combination which vanishes is also an eigenfunction of the hamiltonian _ with _ the point - scatterer and
thus @xmath20 is an eigenvalue of the perturbed problem .
therefore , half of the spectrum is unchanged by the perturbation . to avoid this trivial part of the spectrum we choose to work with the non vanishing
linear combinations ( which are eigenfunctions of the unperturbed hamiltonian ) and only the half of the spectrum which is affected by the perturbation will be considered in this work . for convenience ,
the location of the perturbation is chosen at @xmath21 and therefore the eigenfunctions of the unperturbed hamiltonian which do not vanish there are @xmath22 the spectrum is determined by substituting these eigenfunctions and the corresponding energies @xmath17 in equation ( [ solvept ] ) .
we are interested in the dependence of the spectral statistics on the location of the scatterer .
this dependence can be easily understood in terms of the properties of the wavefunctions of the unperturbed system .
the quantum numbers @xmath23 correspond to a state with an angular momentum @xmath24 and an energy @xmath25 . in the semiclassical limit where @xmath26 , but @xmath27 and @xmath28 are kept fixed ,
the values of the wavefunctions are small in the classically forbidden region @xmath29 .
therefore , one can divide the states into two groups .
the first consists of states for which the point - scatterer is located in the classically forbidden region .
as will be demonstrated , the eigenvalues of these states change only slightly due to the perturbation ( and do not change at all in the semiclassical limit ) .
the second group includes the states for which the perturbation is located in the classically allowed region .
these states will be ( strongly ) affected by the perturbation .
this separation of the spectrum into a superposition of two sequences is the cause for the dependence of the spectral statistics on the location of the scatterer .
this separation into ( semiclassically ) affected and unaffected states is justified in the following . to demonstrate that some of the eigenvalues are almost unchanged by the perturbation one should solve equation ( [ solvept ] ) and show that for ( exponentially ) small eigenfunctions the corresponding eigenvalues are almost unaffected . for simplicity , instead of equation ( [ solvept ] ) it is sufficient to consider the finite sum @xmath30 where @xmath31 is assumed to be slowly varying function of @xmath32 .
we denote @xmath33 . to further simplify the argument
let us assume that @xmath34 and @xmath35 are of order unity while for @xmath36 the wavefunctions on the scatterer @xmath37 are all small .
the solutions of ( [ finitept ] ) are to a good approximation given by the solutions of @xmath38 for @xmath39 and by @xmath40 for @xmath36 .
this is true since substituting a solution of the form @xmath41 in ( [ finitept ] ) leads to @xmath42 expanding with respect to @xmath43 and then solving to the leading order in @xmath44 results in @xmath45 when @xmath46 , as is the case in the semiclassical limit , @xmath43 also approaches @xmath47 . for the other eigenenergies one can substitute @xmath48 ( with @xmath36 ) and find that in the leading order @xmath49 which also vanish when @xmath46 .
note that we have just found @xmath50 ( approximate ) solutions which are all the solutions between @xmath51 to @xmath52 .
it is not hard to generalize this calculation for more wavefunctions which are of order unity . in the semiclassical limit
, the resulting spectrum always consist of such affected and unaffected components . in this limit
the values of the wavefunctions at the scatterer are exponentially small if it is located in the classically forbidden region and the corresponding eigenvalues can be treated as unchanged by the perturbation for any semiclassical consideration . consider a circle billiard with the point - scatterer at @xmath53 .
the spectral statistics of its energy levels in an energy window of width @xmath54 around @xmath55 are studied in what follows .
assume that @xmath55 is very large compared to @xmath54 , i.e. that all the levels in the window have similar energies which are high enough to be considered semiclassical . a natural question to ask is how many of these levels are affected by the point - scatterer and how many are unaffected by it .
as was argued , the semiclassically unaffected levels are those for which the classical turning point , @xmath56 , satisfies @xmath57 .
equivalently , for a given energy @xmath55 , for a state to be unaffected , the angular momentum @xmath28 should satisfy @xmath58 ( for the estimate of @xmath59 , the values of the energies are approximated by @xmath55 ) .
the fraction of such states was calculated in @xcite where it was used to determine how many levels are poorly approximated in the wkb method .
this fraction is @xmath60.\ ] ] it is clear that when @xmath61 then @xmath62 , as expected , while for @xmath63 , @xmath64 .
as one approaches the semiclassical limit these states are less and less affected .
the predictions of equation ( [ xunaff ] ) can be checked numerically .
the number of unaffected levels was calculated for two energy windows as a function of the location of the perturbation .
the results are presented in figure [ unfig ] . , width=453,height=377 ] the radius of the billiard was chosen so that the mean level spacing is @xmath65 .
therefore both energy windows contain @xmath66 levels .
a level was counted as unaffected if its difference from an energy level of the unperturbed system was less than @xmath67 of the mean level spacing .
this criterion is somewhat arbitrary , since in the semiclassical limit the difference can be taken to be arbitrarily small .
figure [ unfig ] indicates that equation ( [ xunaff ] ) correctly describes the number of unaffected levels .
there is a slight deviation which is smaller for the levels from the higher energy window .
this deviation is caused by the fact that for any finite energy the wavefunctions do not vanish at the turning point @xmath56 but rather exhibit an airy - like structure ( in the radial direction ) near the turning point .
this means that for states with @xmath57 , for which @xmath56 is close to @xmath68 , @xmath69 might not be small at a finite ( but large ) energy .
the deviations are expected to vanish in the semiclassical limit as indicated by figure [ unfig ] .
the spectrum of the circle billiard with a point - scatterer can therefore be viewed as composed of two uncorrelated components .
one is unaffected by the point - scatterer and its relative fraction is @xmath70 while the other is affected and its relative fraction is @xmath71 .
the nnsd of a spectrum which is composed of several uncorrelated level sequences was computed by berry and robnik @xcite and is applied to the circle billiard with the point - scatterer in what follows .
the unaffected spectrum consists of many levels with different angular momentum quantum numbers and thus its statistics are poissonian @xcite . since the density of levels in this sequence is @xmath70 , and the radius was chosen so that the total level density is unity , its nnsd is @xmath72 the second level sequence contains the levels which are influenced by the point - scatterer and their density is @xmath71 .
the exact form of its nnsd is unknown and an exact computation of this nnsd is complicated and beyond of the scope of this letter . instead , following experience with other systems @xcite
, we will _ assume _ that the nnsd can be _ approximated _ by a semi - poisson distribution , that is , @xmath73 this distribution exhibits level repulsion at small spacings and exponentially small probability to find large spacings . in these works
it was found numerically to describe the distribution of spacings reasonably well but there is no analytical justification for its use .
note that even if the semi - poisson distribution is an approximation for the nnsd it may not approximate other spectral measures well . for instance , the form factor , which is the fourier transform of the energy - energy correlation function , satisfies @xmath74 for the semi - poisson distribution @xcite while for a billiard with point - scatterer one expects to find @xmath75 @xcite . in particular , for the rectangular billiard with a point - scatterer , the nnsd was computed analytically under some assumptions in @xcite and was found
_ not _ to be given by the semi - poisson or the poisson distributions .
the nth neighbor spacing distributions were also calculated there and found to be those of the poisson distribution at large spacings .
following @xcite the nnsd of the circle billiard with the point - scatterer , obtained by superposing the two sequences , is given by @xmath76 e^{-(2-x_{un})s}.\ ] ] when the perturbation is at the center , @xmath62 , and @xmath77 approaches the poisson distribution .
alternatively , when the perturbation is near the boundary , and @xmath64 , the nnsd approaches the semi - poisson distribution .
note that the distribution ( [ totalps ] ) does not exhibit complete level repulsion since its value at @xmath78 , @xmath79 , does not vanish and it is a manifestation of the existence of an infinite class of states that are unaffected by the perturbation . the nnsd of ( [ totalps ] ) is compared to numerical results in figure [ pofs ] . , ( c ) @xmath80 and ( d ) @xmath81 .
[ pofs],width=453,height=377 ] the nnsd was computed using the levels @xmath82-@xmath83 for three locations of the point - scatterer as well as for the circle billiard without the perturbation .
it is clear that the agreement is very good .
the main features of the distribution are captured by the simple argument leading to ( [ totalps ] ) .
there are slight deviations from the predictions of equation ( [ totalps ] ) , mainly at large @xmath84 .
these can be attributed to the fact that the nnsd of the affected spectrum differs from the semi - poisson distribution .
the results presented in figures [ unfig ] and [ pofs ] suggest that the spectrum of the circle with a point - scatterer consists of a superposition of two uncorrelated level sequences .
the relative densities of these sequences are determined by the way the classical tori of the integrable system are projected into coordinate space .
states for which the perturbation is in the classically allowed region are affected while states for which the perturbation is in the classically forbidden region are nearly unaffected .
we expect this behavior to be typical of systems where the scatterer affects only a fraction of the tori of the otherwise classically integrable systems .
this differs from the rectangular billiard where all tori are affected by the scatterer .
another important difference compared to the rectangle billiard results of the different nature of the wavefunctions . for the rectangular billiard
there are infinite classes of wavefunctions that have common zeros at rational points . consequently ,
if the scatterer is placed at such a location the wavefunctions are not affected , and the distribution depends strongly on the rationality of the location of the scatterer . for the circle billiard studied in the present work , on the other hand , there is one class of eigenfunctions that vanish on the scatterer since they are antisymmetric in @xmath85 .
these are not considered in the present work
. the symmetric eigenfunctions always satisfy @xmath86 . to obtain many functions , symmetric in @xmath85 , that vanish at the same location
is equivalent to finding many bessel functions which satisfy @xmath87 and @xmath88 for integer @xmath16 , @xmath89 and for @xmath90
. finding an infinite number of such solutions , for the same @xmath91 ( corresponding to the same location of the perturbation ) , is unlikely .
however , since any bessel function of large argument is asymptotically given by a cosine one can find states with close zeros , that is where @xmath92 and @xmath93 are zeros of @xmath15 , while @xmath94 is a zero of @xmath95 and @xmath96 is close to a zero of @xmath95 . in this case when the scatterer is at a zero of one of these states , the square of the wave function of the other state is much smaller there than its average value .
many such close zeros should exist to affect the spectral statistics .
this question is beyond the scope of the present letter and is left for future research .
our numerical results are not sensitive enough to resolve this issue .
the numerical results are used here just to verify that the mean dependence on the location of the scatterer is given by equation ( [ totalps ] ) .
we believe that the behavior of the circle billiard rather than that of the rectangular billiard is typical of integrable systems perturbed by a localized potential . in summary ,
the spectral statistics of the circle billiard , perturbed by a point - scatterer are intermediate between those of the poisson distribution , characteristic of integrable systems , and of the semi - poisson distribution .
the spectrum was shown to be composed of two uncorrelated components .
the first contains energy levels which are nearly unaffected by the perturbation , since the point - scatterer is in a classically forbidden region where the wavefunctions are exponentially small .
the relative fraction of such states was computed analytically and found to depend smoothly on the location of the point - scatterer .
the second contribution is from states which are affected by the perturbation .
the exact statistics of this level sequence are complicated but can be approximated by the semi - poisson statistics .
the nearest neighbor spacing distribution results of combination of the two and is a manifestation of the berry - robnik statistics .
other integrable systems should also exhibit qualitatively similar behavior when a localized perturbation is added to them .
this research was supported in part by the us - israel binational science foundation ( bsf ) and by the minerva center of nonlinear physics of complex systems . |
20 p. blaha , k. schwarz , g. k. h. madsen , d. kvasnicka and j. luitz , _ an augmented plane wave plus local orbitals program for calculating crystal properties _ , vienna univ . of technology , austria ( 2001 )
isbn 3 - 950131 - 1 - 2 | we have undertaken a study of diluted magnetic semiconductors @xmath0 and @xmath1 with @xmath2 , using the all electron linearized augmented plane wave method ( lapw ) for different configurations of mn as well as cr .
we study four possible configurations of the impurity in the wurtzite gan structure to predict energetically most favorable structure within the 32 atom supercell and conclude that the near - neighbor configuration has the lowest energy .
we have also analyzed the ferro - magnetic as well as anti - ferromagnetic configurations of the impurity atoms .
the density of states as well as bandstructure indicate half metallic state for all the systems . @xmath3 has also been estimated for the above systems .
introduction gallium nitride is one of the most promising materials among the diluted magnetic semiconductor ( dms ) material for application in spintronics . by doping transition metal ( tm ) atoms , mn or cr , local magnetic moment
are introduced in semiconductor which mediate ferromagnetically .
( ga , cr)n based dms was predicted to show high @xmath3 @xcite for high enough concentration of cr and further hashimoto _
et al . _
@xcite observed that ( ga , cr)n based dms grown by ecr molecular beam epitaxy showed @xmath3 above @xmath4k . cr@xmath5-implanted gan , studied by photoluminescence and superconducting quantum interference device ( squid ) reveal that the implanted cr@xmath5 incorporates substitutionally at ga site and the ferromagnetic order is retained upto @xmath6k @xcite .
takeuch _
et al . _
@xcite . have reported a systematic study of changes in the occupied and unoccupied n - partial density of states ( dos ) and confirm the wurtzite n @xmath7 dos and substitutional doping of cr into ga sites using sxes and xas .
recently , ferromagnetism above @xmath8k was reported in cr - gan thin films @xcite .
theoretically it was predicted that the ferromagnetic ( fm ) interaction in ( ga , mn)n may be retained upto room temperature @xcite .
the initial reports of high @xmath3 in ( ga , mn)n were followed by controversial results where the reported @xmath3 varied between @xmath9k - @xmath10k @xcite .
zajac and coworkers observed mn ions in ga@xmath11mn@xmath12n ( @xmath13 ) crystals coupled anti - ferromagnetically ( afm ) @xcite .
electronic structure and magnetic properties of zinc blende ga@xmath11mn@xmath12n for several values of @xmath14 with varied spatial distribution of dopant atoms to understand the magnetic interaction for explanation of fm - afm competition is discussed by uspenskii _
et al . _
@xcite where the calculations were done using the tight binding lmto method in the local spin density approximation .
sanyal and mirbt @xcite have studied mn doped gaas and gan dms using the _ ab - initio _ plane wave code ( vasp ) within density functional theory ( dft ) .
they have determined the interatomic exchange interactions by substituting mn in various positions in the unit cell and have attributed the origin of ferromagnetism in ( ga , mn)n to double - exchange mechanism involving the hopping of mn@xmath15 electrons .
raebiger _
et al . _
@xcite used the full potential linearized augmented plane wave ( fp - lapw ) method to investigate the interplay between clustering and exchange coupling in magnetic semiconductor ga@xmath11mn@xmath12as .
they have studied all possible arrangements of the two mn atoms on ga sublattice for @xmath16 and found that clustering of mn atoms at near neighbour ga sites is energetically preferred .
our analysis of the wurtzite gan doped with mn or cr is motivated by the latter study .
method and computational details we have employed the spin - polarized linearized augmented plane wave method ( fp - lapw ) as implemented in the wien2k package @xcite with the generalized gradient approximation ( gga ) for the exchange - correlation potential proposed by perdew , burke and ernzerof ( pbe96 ) @xcite .
this is state - of - the - art electronic structure method , which does not use any shape approximation for the potential , to solve the kohn - sham type of equations self - consistently .
gan normally occurs in the wurtzite structure with lattice constants @xmath17 and @xmath18 , giving @xmath19 ratio of 1.62 .
each ga is tetrahedrally bonded to n atoms at an average distance of @xmath20 and each n in turn is surrounded by four ga neighbors .
the calculations for dms were performed within a 32 atom supercell , constructed from @xmath21 standard unit cell of wurtzite structure wherein the dopant is substituted at various cation sites , since it has been shown that the formation energy for interstitial mn doping is higher than substitutional doping @xcite .
the supercell approach is used to restrict the dopant concentration to a small value , which is of interest for studying magnetic properties of the system , without altering the original underlying lattice structure .
our interest was in observing the changes in the electronic structure of the dms with respect to the possible different geometries of the dopants within the host semiconductor .
self - consistent electronic structure calculations were performed using the apw + local orbitals ( lo ) basis set for the valence and semi - core electrons with @xmath22 , @xmath23 and total energy convergence of @xmath24 the muffin - tin radii for ga , mn and cr were kept at @xmath25 and that for n at @xmath26 .
spin - polarized calculations were carried out to observe the effect of spin - splitting and to calculate the on - site magnetic moment at tm site .
we have studied wurtzite gan doped with one tm atom impurity , which is @xmath27 doping and two identical tm atoms in the @xmath28 atom unit cell amounting to @xmath29 doping . to simulate different surroundings for the transition metal ( tm ) atoms we have spanned certain geometries within the @xmath28 atom unit cell wherein the distance between the dopants is varied . in the case of the single impurity substitution ,
the nearest distance between two tm atoms is @xmath30 in plane and @xmath31 along the @xmath32-axes .
we have studied four different geometries of two tm atom substitutions at @xmath33 , near neighbor ( nn ) , @xmath34 , @xmath35 and @xmath36 separations .
when two near neighbor ( nn ) ga atoms are substituted by tm atoms the in - plane tm - tm atoms distance is @xmath33 and along the @xmath32-axes it is @xmath31 . for the second case
the out - of - plane distance is @xmath34 and the in - plane distance between the dopants is @xmath37 . in the third case ,
two ga atoms lying one above the other , along the @xmath38-axes , separated by a distance of @xmath35 are substituted by tm atoms and the in - plane tm atoms are at @xmath30 .
the last case is such that the in - plane separation ( @xmath30 ) and out of plane separation ( @xmath39 ) between the tm atoms is comparable . for estimating the magnetically favorable system , the spins of the dopants are aligned along the same direction , corresponding to the fm configuration , and aligned in the opposite directions corresponding to afm configuration .
the self consistency was achieved on a mesh of @xmath40 k - points .
structural relaxation for the tm site and the nn n sites was carried out to observe changes in the bond lengths between tm and the first shell of n atoms .
very small change ( @xmath41 ) was observed in the bond lengths and no significant changes were seen in the band structure in agreement with the earlier reported results @xcite .
thus the calculations reported here are for systems without allowing any relaxation .
results in the wurtzite gan semiconductor , each ga ( n ) is tetrahedrally bonded to 4 n ( ga ) atoms .
pure gan is a direct band gap semiconductor with top of the valence band consisting of n _ p_-states and the bottom of the conduction band having ga _
sp_-character .
the ga _
d _ levels are deep and do not take part in the bonding .
thus they are treated as core states .
the band gap of gan , which is underestimated by density functional theory within the approximation used for the exchange correlation energy functional , is @xmath42ev .
the experimentally determined band gap of undoped
gan is @xmath43ev .
das _
et al . _
@xcite have shown that , for mn atoms to couple ferromagnetically , they need to be kept apart by more than the critical distance of @xmath44 . similar calculations on clusters of ( gan)cr indicate that the critical cr - cr distance is @xmath45 @xcite . in all our calculations the distance between the dopants was greater than the corresponding critical distances .
mn doped systems localized magnetic moments are introduced within the gan system by substituting the cations with tm impurity atom(s ) .
mn atom with @xmath46 and @xmath47 electrons in the valence region replaces ga atom with valency @xmath48 . on substitution mn atoms contributing five @xmath49 levels per atom
are thus expected to contribute to the observed magnetic moment . 1.0 in 2.0 in 0.2 in 2.0 in + 0.02
cm + 2.1 in ( a ) 0.2 in 1.5 in since three of the valence electrons from mn go into compensating the three electron states of substituted ga , one hole per mn is introduced into the system .
figure [ mnet2 ] shows the mn - projected @xmath50 and @xmath51 majority spin electronic structure for @xmath52 .
the mn-@xmath49 states lie at the top of the valence band and cross the @xmath53 in some places .
these are split into @xmath51 and @xmath50 states , the @xmath50 level is two thirds filled and the @xmath51 is almost occupied .
the minority spin levels are empty and lie above the @xmath53 indicating @xmath54 spin - polarized states .
the mn induced states lie in the gap region of gan .
the top of the valence band in gan is composed of the n-@xmath55 levels and the unique properties , particularly the half metallic state of dms , thus arise from the tm _
d _ and host _ p _ interactions that couple the two subsystems .
mn - projected @xmath49-dos in @xmath56 ( a ) with mn - mn distance equal to @xmath33 and @xmath35 and ( b ) @xmath57 and @xmath37 .
the upper and lower panels represent the majority and minority spin dos respectively.,title="fig:",height=321 ] + ( a ) + mn - projected @xmath49-dos in @xmath56 ( a ) with mn - mn distance equal to @xmath33 and @xmath35 and ( b ) @xmath57 and @xmath37 .
the upper and lower panels represent the majority and minority spin dos respectively.,title="fig:",height=321 ] + ( b ) in order to understand the variation of exchange interaction among the tm impurity with the distance between the tm atoms , the concentration of mn atoms was increased to @xmath58 , equivalent to introducing @xmath59 mn atoms in the supercell .
self consistent calculations were carried out for two different magnetic configurations of mn electrons in which the electrons are parallel or antiparallel corresponding to fm or afm configuration .
for all the geometries of the dopants , systems , as described in section ii , with fm configuration of the mn atoms were found to have lower energy .
since the mn-@xmath49 levels are responsible for observed half metallic behavior , a comparison of mn-@xmath49 dos in various geometries is shown in figure [ mndos ] , ( a ) for separations @xmath33 and @xmath60 and ( b ) for @xmath57 and @xmath37 .
here the half metallic state is evident in all the cases .
the tm - tm distance of @xmath37 corresponds to single tm doping ( @xmath61 ) .
the mn @xmath62dos is broad for substitution at nn distance . in all the other cases
the band is split indicating that the majority spin @xmath49-bands of the two mn atoms at nn overlap . on increasing mn - mn distance , @xmath49-band splitting takes place implying a reduction in the interaction between the tm atoms .
it may be noted that the minority spin conduction band overlaps with the majority spin band for nn substitution .
a gap of @xmath63 is present between the minority spin conduction band and majority spin band for mn - mn distance greater than nn . the minority spin valence band
as well as conduction band is far apart from the @xmath53 thus retaining the highly spin polarized state also seen in the single mn doped @xmath64 system .
the down spin gap is @xmath65 for nn configuration and increases to @xmath66 at larger separations .
on increasing the distance between the mn atoms , splitting of d - level increases .
this is consistent with the observation that in single impurity doping , tm - tm atom distance is @xmath57 and splitting of the mn-@xmath49 band is larger as seen in figure [ mndos](b ) .
the magnetic moment at mn - site does not depend on the distance between the dopant atoms and has a value @xmath67 for all the geometries as indicated in table [ table1 ] . @llll system & total@xmath68 & dopant@xmath68 & n@xmath68 + @xmath52 & 4.00 & 3.33 & 0.001 + @xmath69 & 8.00 & 3.34 & 0.001 + mn - mn = @xmath33 & & & + @xmath69 & 8.00 & 3.34 & 0.005 + mn - mn = @xmath34 & & & + @xmath69 & 8.00 & 3.33 & 0.006 + mn - mn
= @xmath35 & & & + @xmath69 & 8.00 & 3.34 & 0.005 + mn - mn = @xmath57 & & & + the presence of localized moment influences the near neighbor n atoms within the gan system , such that dos of the nn n - atoms around the impurity atom becomes as shown in figure [ ndos](a ) . due to the @xmath70 interaction , induced states are seen on n atoms .
the magnitude of the induced states is maximum at nn n atoms and decreases as the distance from the tm atom increases .
this again indicates a localized nature of the tm states .
figure [ ndos](b ) shows the spin charge density ( scd ) in a plane containing three nn n atoms for single mn doping .
the plot shows that the scd on the n atoms lying above the mn atom is negative whereas it is positive for the n atom lying below the mn atom .
average magnetic moment on the nn n atoms is positive as shown in table [ table1 ] for all the different geometries .
( a ) variation in partial - dos at 3 n - sites in @xmath52 .
the solid lines : nn - n along @xmath71axis .
dotted lines :
nn - n in plane .
dashed - dotted lines : next nn - n ( b ) scd plot in a plane of three nn n atoms.,title="fig:",height=245 ] + ( a ) + 0.2 cm ( a ) variation in partial - dos at 3 n - sites in @xmath52 . the solid lines : nn - n along @xmath71axis .
dotted lines : nn - n in plane .
dashed - dotted lines : next nn - n ( b ) scd plot in a plane of three nn n atoms.,title="fig:",width=188,height=188 ] ( a ) variation in partial - dos at 3 n - sites in @xmath52 . the solid lines : nn - n along @xmath71axis .
dotted lines :
nn - n in plane .
dashed - dotted lines : next nn - n ( b ) scd plot in a plane of three nn n atoms.,title="fig:",width=75,height=188 ] + ( b ) cr doped systems electronic structure calculation for substitutional doping of cr in the gan system was also done and is analyzed in a similar fashion . for each cr
doped in the @xmath28 atom supercell , equivalent to @xmath72 doping , there are five spin up @xmath49-states which are introduced in the gan band gap .
since the cr atom has @xmath73 valence electrons , only three of the electron states out of the five @xmath74 levels are occupied , thus creating two hole states per cr substitution .
single cr doping into the @xmath28 atom supercell at cation site results in the cr-@xmath49 levels appearing in the band gap of the semiconductor host as seen in figure [ cret2 ] . 1.0 in 2.0 in 0.2 in 2.0 in 0.02 cm + 2.1 in ( a ) 0.2 in 1.5 in the cr d@xmath75 levels split ( figure [ cret2](a ) ) , out of which two energy levels lie below the fermi level ( @xmath76 ) and are occupied .
the third level which is @xmath77ev above is unoccupied .
one of the d@xmath78 level is occupied and the other lies just above the @xmath76 as seen in figure [ cret2](b ) .
however , the hybridization of @xmath50 and @xmath51 majority spin states is negligible and these levels are well separated as opposed to mn doped case . as in the case of mn doping ,
the cr minority spin @xmath49-states are above the @xmath76 and so the impurity states at the @xmath76 are @xmath54 spin polarized .
cr - projected @xmath49-dos in @xmath79 for cr - cr separation of ( a ) @xmath33 and @xmath35 and ( b ) @xmath57 and @xmath37.,title="fig:",height=321 ] + ( a ) + cr - projected @xmath49-dos in @xmath79 for cr - cr separation of ( a ) @xmath33 and @xmath35 and ( b ) @xmath57 and @xmath37.,title="fig:",height=321 ] + ( b ) in the @xmath79 system , figure [ cr2 ] shows that even though the cr atoms are substituted at nn sites there is a gap seen between the split cr-@xmath49 levels , unlike in the nn mn - doped system . the minority spin conduction band overlaps with the majority spin band for nn cr case , as in nn mn case . for all the geometries of the dopant atoms the system is half metallic .
scd plot in a plane of three nn n atoms around the tm atom in @xmath80.,title="fig:",width=188,height=188 ] scd plot in a plane of three nn n atoms around the tm atom in @xmath80.,title="fig:",width=75,height=188 ] the band gap for the minority spin , in case of two cr substitution at @xmath33 is @xmath81ev which is larger than the corresponding mn case which has a gap of @xmath82ev . when the tm - tm distance is @xmath35 , in the cr case the gap is @xmath83ev whereas for mn substitution it is @xmath84ev .
the lowest energy configuration for cr substitution occurs for cr - cr distance of @xmath33 and for all the geometries studied the fm configuration of tm atoms has lower energy compared to afm configuration .
the magnetic moments at the cr site in various geometries is as shown in table [ tab2 ] and it is seen that the magnetic moment does not show much variation depending on the distance , indicating that the direct interaction between the cr atoms is minimal . @llll system & total@xmath68 & dopant@xmath68 & n@xmath68 + @xmath85 & 3.00 & 2.47 & -0.025 + @xmath86 & 6.00 & 2.48 & -0.031 + cr - cr = @xmath33 & & & + @xmath86 & 6.00 & 2.47 & -0.022 + cr - cr = @xmath34 & & & + @xmath86 & 6.00 & 2.48 & -0.025 + cr - cr = @xmath35 & & & + @xmath86 & 6.00 & 2.48 & -0.026 + cr - cr = @xmath57 & & & + magnetic moments observed at the various dopant sites from our calculation are shown in the table [ tab2 ] .
the total magnetic moment per unit cell per mn atom is @xmath87 and the average magnetic moment on the nn n atoms in case of mn - doping is parallel to the mn - moment .
this can be understood as penetration of the spin - polarized mn states to the neighboring host which does not have any of its own states in the gap region .
the magnetic moment per unit cell per cr atom is @xmath88 .
the average magnetic moment on the nn n atom is anti - parallel to cr - moment .
the difference in the orientation of the average magnetic moment on nn n atoms of mn and cr is due to the difference in the @xmath55 dos of the nn n along the z axis ( figure not shown here ) compared to the nn n atoms lying in the xy plane above the tm atoms .
there is not much variation of magnetic moment with increase in cr - cr distance and shows a similar trend as mn doped systems .
the magnitude of average nn - n magnetic moment is greater in case of cr substitution , which contribute one less electron to the hybridized valcen band .
scd in figure [ crscd ] on all of the nn - n atoms of the single cr ( only 3 nn n atoms shown in figure [ crscd ] ) doped system is negative thus showing that the tm atom and the nn - n are anti - ferromagnetically coupled .
estimation of @xmath3 we have predicted the @xmath3 for the dms based on ga@xmath89mn@xmath90n@xmath89 and ga@xmath89cr@xmath90n@xmath89 considering the mean field approximation .
@xmath91 for mn and cr doping .
inset shows the mean field @xmath3 variation with distance between dopants.,height=321 ] figure [ evsd ] shows the @xmath91 for the mn / cr doped systems , where @xmath92 is the total energy for the antiferromagnetic ( afm ) configuration and @xmath93 is the total energy for the ferromagnetic ( fm ) configuration
. observed variation @xmath94 vs distance for mn substituted dms agrees with the one reported by sanyal @xcite .
@xmath94 is a measure of the exchange interaction in the system .
highest @xmath94 is seen for the case where the tm atoms are substituted as near neighbors , signifying larger overlap of the magnetic impurity orbitals . for mn doping at nn @xmath95ev ,
this compares well with the value calculated for dimer substitution by uspenskii _ et . al _
@xcite which compared qualitatively with the high @xmath96k measured @xcite . as for the identical cr case @xmath97 also compares well with the observed @xmath3 but is lower than the @xmath8k observed by liu and co - workers @xcite . from figure [ evsd ]
it also emerges that the exchange interaction decreases sharply as the distance between the tm atoms increases .
thus the exchange interaction is short range and could be interpreted as the double exchange mechanism .
summary and conclusions we have analyzed the electronic structure of gan doped with tm mn and cr with @xmath72 and @xmath58 doping for various possible geometries to replicate the situation where the tm atoms would appear either to cluster or be separated .
the self consistent fp - lapw calculations predict half metallic state for @xmath72 as well as @xmath58 doping . comparing the total energies of the fm and afm configurations for @xmath58 doping ,
the fm state is found to be lower in energy and is predicted to be the preferred state .
on - site magnetic moment at the tm site shows insignificant variation with distance between the dopants .
the near neighbor n atoms contribute to the states in energy gap of the semiconductor due to the influence of the tm atoms .
average magnetic moment at nn n site is parallel to the mn magnetic moment where as it is anti - parallel to the cr atoms .
we observe that both the systems with nn substitution of mn / cr atom would show high @xmath3 .
the energy gap between the minority spin band in mn is @xmath98 lower than in cr doped system and we think this could be an important factor in determining a suitable system .
but since the magnetic moment at mn site is higher than cr , it would be of interest to study mixed systems of mn and cr to incorporate the salient features of both tm atoms . |
the nuclear many - body problem of @xmath0 interacting nucleons can be solved exactly only in very specific cases or for very small particle numbers .
this is due to the large number of degrees of freedom involved in such a complex system .
let us for instance consider particles interacting through n hamiltonian written as @xmath1 then the exact ground state energy can be written as @xmath2 where @xmath3 , @xmath4 , ... denote the one- , two- , ... body density matrices that contain all the information on the one- , two- ... body degrees of freedom respectively .
a natural way to reduce the complexity of this problem is to assume that at a given level , the @xmath5body ( and higher - order ) density matrices becomes a functional of the lower - order ones .
this is what is done for instance in the hartree - fock ( hf ) approximation where all @xmath6-body density matrices ( with @xmath7 ) become a functional of @xmath8 .
unfortunately , the hf theory applied to the nuclear many - body problem in terms of the vacuum hamiltonian is a poor approximation and many - body theories beyond hf are necessary .
the introduction of energy density functional ( edf ) approaches in the 70 s was a major breakthrough ( see for instance @xcite for a recent review ) . in its simplest form ,
the edf formalism starts with an energy postulated as a functional of @xmath8 , the latter being built out of a slater determinant .
then the ground state energy is obtained by minimizing the energy with respect to @xmath8 , i.e. @xmath9 parameters are generally adjusted on specific experimental observations and therefore encompass directly many - body correlations .
current edf uses a generalization of eq .
( [ eq : simpleedf ] ) obtained by considering quasi - particle vacua as trial states . by making explicit use of symmetry breaking , such a functional called hereafter single - reference ( sr- ) edf is able to account for static correlation associated with pairing and deformation .
actual sr - edf takes the form : @xmath10 where @xmath11 denotes the anomalous density . to restore symmetries and/or incorporate dynamical correlations , guided by the generator
coordinate method ( gcm ) , a second level of edf implementation , namely multi - reference ( mr- ) edf is introduced .
recently , difficulties with the formulation and implementation of have been encountered in mr - edf .
a minimal solution has been proposed in ref . @xcite . besides these problems , the authors of ref . @xcite
have pointed out the absence of a rigorous theoretical framework for the mr edf approach . at the heart of the problem
is the possibility to break symmetries in functional theories and then restore them using configuration mixing .
this issue needs to be thoroughly addressed in the future . in this context , it is interesting to see if extensions of the functional used at the sr - edf level can grasp part of the effects that for standard functionals require the mr level .
it is worth realizing that , in the canonical basis for which @xmath12 , we have @xmath13 = \frac{1}{4 } \sum_{i , j } \bar v^{\kappa \kappa}_{i\bar i j \bar j } \sqrt{n_i ( 1-n_i ) } \sqrt{n_j ( 1-n_j ) } , \label{}\end{aligned}\ ] ] and therefore , the energy can be regarded as a functional of natural orbitals @xmath14 and occupation numbers @xmath15 . as a matter of fact , for electronic systems , gilbert has generalized the kohn - sham theory and shown that the exact energy of a system can be obtained by minimizing such a functional @xcite leading to the so - called density matrix functional theory ( dmft ) . the possibility to consider occupation numbers as building blocks of the nuclear energy functional
has recently been discussed in ref .
two levels of theory can be developed along the line of gilbert s idea ( i ) either , functionals in the strict gilbert framework can be designed . in that case , since the density identify with the exact density at the minimum , it should respect all symmetries of the bare hamiltonian .
( ii ) or we exploit the concept of symmetry breaking . in the latter case , similarly to the sr - edf , strictly speaking we can not anymore rely on the theorem , but we may gain better physical insight with relatively simple functionals .
the descriptive power of dmft is illustrated here in the two - level lipkin model @xcite . in this model , the hartree - fock ( hf ) theory fails to reproduce the ground state energy whereas configuration mixing like generator coordinate method ( gcm ) provides a suitable tool @xcite .
therefore , the two - level lipkin model is perfectly suited both to illustrate that dmft could be a valuable tool and to provide an example of a functional for system with a `` shape '' like phase - transition . in this model ,
one considers @xmath0 particles distributed in two n - fold degenerated shells separated by an energy @xmath16 .
the associated hamiltonian is given by @xmath17 where @xmath18 denotes the interaction strength while @xmath19 , @xmath20 are the quasi - spin operators defined as @xmath21 , @xmath22 and @xmath23 .
@xmath24 and @xmath25 are creation operators associated with the upper and lower levels respectively . due to the specific form of the lipkin hamiltonian , @xmath8 simply writes in the natural basis as @xmath26 . introducing the angle @xmath27 between the state @xmath28 and @xmath29 , leads to the following mean - field functional @xcite @xmath30 where @xmath31 .
this expression is easily obtained by generalizing the hartree - fock case ( recovered here if @xmath32 ) .
the main challenge of the method is to obtain an accurate expression for @xmath33 . to get the functional , clearly identified cases from which properties of the functional could be inferred have been used@xcite , namely the @xmath34 case and the large @xmath0 limit . in the two - particles case ,
the correlation energy can be analytically obtained and reads @xmath35 a simple extension of the @xmath34 case for larger number of particles is to assume that each pair contributes independently from the others leading to @xmath36 { \cal e}^{^{{n=2}}}_{\rm cor}$ ] .
however , such a simple assumption leads to a wrong scaling behavior in the large @xmath0 limit .
indeed , in this case , @xmath37 as @xmath0 tends to infinity while a @xmath38 scaling is expected @xcite .
to obtain the correct limit , a semi - empirical factor @xmath39 can be introduced such that @xmath40 with @xmath41 .
the value @xmath42 has been retained using a fitting procedure .
examples of results obtained by minimizing the functional given by eqs .
( [ eq : emflipkin ] ) and ( [ eq : coreta ] ) are shown in fig .
[ fig : chi ] for different particle numbers and interaction strengths . in all cases , a very good agreement ,
much better than the hf case is found . for @xmath43 to @xmath44 resp .
from top to bottom . in each case , the corresponding hf ( dashed line ) and dmft ( filled circle ) minimum energy are shown .
the dmft calculation is performed using the mean - field and correlation energy resp . given by eq .
( [ eq : emflipkin ] ) and eq .
( [ eq : coreta ] ) with @xmath45 ( adapted from @xcite ) . ]
the lipkin example suggests that dmft can be a valuable tool for describing ground state of a many - body system when symmetry breaking plays a significant role .
the functional designed here is exact only in the @xmath34 .
note that the functional proposed here breaks signature symmetry and therefore enters into the level ( ii ) of functional discussed in the introduction .
the lipkin model is however rather schematic and can not be used as a guidance for realistic situations .
the possibility to design a new accurate functional for nuclei remains a challenging problem .
the author thanks m. assi , b. avez , m. bender , t. duguet , c. simenel , o. sorlin and p. van isacker for enlightening discussions at different stages of this work and t. papenbrock for useful remarks on the scaling behavior in the lipkin model .
d. lacroix , t. duguet , and m. bender , phys . rev . *
c79 * , 044318 ( 2009 ) .
m. bender , t. duguet , and d. lacroix , phys . rev . *
c79 * , 044319 ( 2009 ) .
t. duguet , m. bender , k. bennaceur , d. lacroix , and t. lesinski , phys . rev . *
c79 * , 044320(2009 ) . | the possibility to use functionals of occupation numbers and natural orbitals for interacting fermions is discussed as an alternative to multi - reference energy density functional method . an illustration based on the two - level lipkin model
is discussed .
address = ganil , cea and in2p3 , bote postale 5027 , 14076 caen cedex , france |
let @xmath0 be a locally finite covering of the plane @xmath6 by closed unit discs .
the doubly covered region of @xmath6 by @xmath0 consists of the sets of points @xmath7 that are contained in at least two elements of @xmath0 .
a few years ago gabor fejes - tth posed the following question : if @xmath8 are two points at distance @xmath3 apart and contained in the doubly covered region of @xmath6 , what is the length of the shortest path @xmath9 joining @xmath1 and @xmath2 that is completely contained in the doubly covered region ? in a sense , this is the dual of a problem by laszlo fejes tth about the length of a path avoiding a packing of discs @xcite .
gabor fejes - tth conjectured that when the centres of the circles in @xmath0 form a unit square lattice , the length of @xmath9 is maximal .
for any two points @xmath1 and @xmath2 in this example , @xmath10
. a general upper bound of @xmath11 for @xmath12 is not difficult to obtain .
baggett and bezdek proved in @xcite that when the centres of the circles of @xmath0 form a lattice , then the unit square lattice is indeed the extreme case .
in this short note we give an upper bound for any locally finite covering .
@xmath13 this is still closer to @xmath14 than to @xmath15 , but we hope that our methods may help others to continue improving this bound .
let @xmath16 be the segment joining @xmath1 and @xmath2 and @xmath17 be a minimal sub - cover of @xmath16 .
suppose that @xmath16 is horizontal and that @xmath1 is to the left of @xmath2 .
the elements of @xmath18 can be ordered as @xmath19 such that if @xmath20 then the centre @xmath21 of @xmath22 is to the left of the centre @xmath23 of @xmath24 .
we may assume that @xmath1 is the leftmost point of the intersection of the line @xmath25 with @xmath26 and that @xmath2 is the rightmost point of the intersection of @xmath25 with @xmath27 .
if this were not the case then we may extend the length of @xmath16 by at most @xmath28 and move each one of @xmath1 and @xmath2 through paths of length at most @xmath29 so that they end up in this way .
this contributes a term @xmath30 to the length of the curve we find with respect to the original length of the segment @xmath16 . since the family @xmath0 is locally finite , every point in the boundary of a circle @xmath31
is doubly covered . for @xmath31 , define @xmath32 as the closure of a connected component of @xmath33 that does not contain the centre of @xmath34 . since @xmath18 is a minimal covering of @xmath16 then @xmath35 if and only if @xmath36 .
let @xmath37 and @xmath38 be the points of intersection of the boundary of @xmath22 with @xmath16 such that @xmath37 is to the left of @xmath38 and let @xmath39 be the midpoint of the segment @xmath40 for @xmath41 , @xmath42 and @xmath43 .
note that @xmath44 . .
] now we construct the path @xmath9 with the algorithm below , an example is shown in figure [ fig : gamma ] .
the path starts at @xmath45 . assuming the path has been constructed up to @xmath39 , let @xmath46 be the largest integer such that the sets @xmath47 are all on the same side of @xmath16 . without loss of generality
we assume they are all above @xmath16 .
the path then continues vertically upwards until it reaches the boundary of @xmath48 .
from here it continues towards the right while staying contained in the boundary of @xmath49 until it is vertically aligned with @xmath50 .
finally it goes vertically downwards until it reaches @xmath50 .
this is repeated until the path ends at @xmath51 .
it is easy to see that @xmath9 is well defined and completely contained in the doubly covered region , however it is not so easy to directly bound its length .
this path could be shortened by taking diagonal lines instead of vertical ones , but these two paths coincide in the extreme case . in order to bound the length of this path , for every @xmath52
we construct a new path @xmath53 .
the paths @xmath53 may not be contained in the doubly covered region but they satisfy @xmath54 .
assume for simplicity that @xmath55 is above @xmath16 , then the path @xmath53 starts at @xmath56 , goes upwards until it intersects the boundary of @xmath55 at a point @xmath57 , then goes to the right staying contained in the boundary of @xmath55 until it is vertically aligned with @xmath39 at the point @xmath58 , and finally goes downwards until it reaches @xmath39 .
this path is shown in figure [ fig : gammai ] .
it is not difficult to see that @xmath54 . and
@xmath59 . ]
now it is enough to prove the following .
[ lem : lemma ] @xmath60 we may assume that @xmath61 is the diameter of @xmath22 , otherwise let @xmath32 be a circle with diameter @xmath61 and @xmath59 be the curve defined on @xmath32 in the same way that @xmath53 is defined on @xmath22 .
this new curve clearly has larger length than @xmath53 ( see figure [ fig : gammai ] ) .
let @xmath62 and @xmath63 , then @xmath64 recall also that @xmath65 .
if we fix the value of @xmath66 , then by using lagrange multipliers we obtain that @xmath67 is maximum when @xmath68 , @xmath69 or @xmath70 .
since the cases @xmath68 and @xmath69 are symmetrical , we have essentially two cases . now we only need to determine the maximum of @xmath71 as a function of @xmath72 .
this occurs when @xmath73 and corresponds to the case @xmath70 .
this gives @xmath74 which proves the lemma .
the method we use only considers discs that intersect the segment @xmath25 , we construct a path contained in the boundary of these discs and in their doubly covered region . considering only these circles it is impossible to obtain the bound fejes - tth conjectured .
below we construct an example considering only these circles such that @xmath75 .
let @xmath26 and @xmath76 be intersecting circles and assume @xmath77 and @xmath78 .
if we only allow the path @xmath9 to be in the intersection of the circles and their boundary , then there are only @xmath28 possible choices for @xmath9 .
this is a simplified version of our problem , as the boundary of a circle in @xmath0 is always doubly covered .
but because we only consider two circles , this new problem can be solved precisely to obtain @xmath79 .
here the number @xmath80 is an approximation obtained as the maximum of a function involving trigonometric functions .
the example that gives this value can be extended to families with more circles and have arbitrarily large @xmath3 .
this is done by using several translated copies of this example ( see fig .
[ fig : example ] ) . .
] thus , it is essential to consider elements of @xmath81 further away from the segment @xmath25 .
however an improvement on our algorithm would surely lower our bound .
to obtain our bound we consider only @xmath82 discs at a time , this is precisely what is done in lemma [ lem : lemma ] .
a this bound could potentially be lowered if we were to consider @xmath83 or more discs instead .
the algorithm would have to be modified or changed to take advantage of this .
we tried this approach together with alexey garber , however we were unable to obtain any substantial improvements .
a problem we encountered was that the functions to be minimised became extremely complicated . | given a covering of the plane by closed unit discs @xmath0 and two points @xmath1 and @xmath2 in the region doubly covered by @xmath0 , what is the length of the shortest path connecting them that stays within the doubly covered region ?
this is a problem of g. fejes - tth and he conjectured that if the distance between @xmath1 and @xmath2 is @xmath3 , then the length of this path is at most @xmath4 . in this paper
we give a bound of @xmath5 . |
in order to study the effects of the tensor and spin orbit interactions in nuclei we use a simple interaction v = v@xmath5 + x v@xmath6 + y v@xmath7 where c @xmath8 central , s.o . @xmath8 spin - orbit and t @xmath8 tensor . for x=1 , y=1 we select v so as to be close to a realistic g matrix like bonn a. we can turn the spin orbit interaction off ( on ) by setting x=0 ( x=1 ) .
likewise we can turn the tensor interaction off ( on ) by setting y=0(1 ) .
this allows us to study behaviors as a function of x and/or of y. this interaction @xcite is a modification and correction of a previous interaction @xcite .
this change does not affect calculations purely in the p shell but there are some changes when core excitations are included .
to avoid confusion we call the current interaction v(2005 ) and the previous one v(1991),i.e after the year of publication .
the details and reasons for the changes are given in ref .
@xcite .
our main thesis will be that the tensor interaction given by a bare g matrix is too strong in the isospin t=0 channel .
by simply making it weaker we can correlate a lot of data and be rid of a lot of anomalies .
we conclude by presenting simple arguments by gerry brown to justify using a weaker tensor interaction in the valence space .
care must be taken in that alternate explanations could give the same result as a weaker tensor interaction e.g. a stronger spin - orbit interaction .
the topics we discuss are : 1 . the quadrupole moment of the j=1@xmath9 state of @xmath0li .
2 . the t=1 t=0 energy splitting of 0@xmath1 states in @xmath20 .
the near vanishing of the gamow - teller matrix element for a=14 @xmath3o(j=0 t=1 ) @xmath10
@xmath3n(j=1 t=0 ) .
4 . the effect of the tensor interaction on single particle energies in open shell nuclei - @xmath4c and @xmath3n . some of these results have been discussed in previously @xcite- @xcite .
the nucleus @xmath0li is often described in cluster models as a deuteron plus an alpha particle . however , the quadrupole moment of the deuteron is positive @xmath12 e @xmath13 whereas that of @xmath0li is negative , @xmath14 e @xmath13 . that the deuteron has a quadrupole moment leads to it having a j=1@xmath11 ground state and hence the isospin must be t=0 . without a tensor interaction the quadrupole moment of the deuteron
would be zero .
.static quadrupole moments e @xmath13 for various model spaces and tensor interaction strengths ( y ) using the bare electric charges @xmath15 and @xmath16 .
all calculations are done with the full spin - orbit strength x=1 . [
cols="^,^,^,^ " , ] the second order contribution to the central interaction is negative definite and for the tensor it is positive definite .
hence whether there is destructive or constructive interference depends on the sign of the first order term as given above .
we start from a moszkowski - scott cutoff radius of 1 fermi corresponding to x of 0.7 .
they argued that the part of the attraction up to the cut - off radius of 1 fermi cancels out the short - range repulsion whose range is about 0.4 fermi .
@xcite the first order central even is negative so adding the second order tensor contributions makes it more negative .
this is reasonable from the existence of the t=0 even channel bound state for two nucleons , namely the deuteron .
the bare central interaction in this channel is not deep enough to support a bound state so the tensor interaction has to contribute .
we can see this in figure [ fig : vce ] the first order tensor even interaction is negative ( when the -3 factor is included ) so the combination of first and second order terms must be less negative or weaker .
the sign of the first order tensor interaction is determined phenomenologically by the positive sign of the quadrupole moment of the deuteron .
also the sign is consistent with the one pion exchange potential .
this supports all the conclusions of the previous sections where we see repeatedly that the bare g matrix tensor interaction is too strong in the t=0 channel and needs weakening . for the odd channels we have first in figure [ fig : vco ] the central odd potential . here
the attractive contribution of the second order term pulls an initially repulsive first order term down sufficiently so that it is slightly attractive . for the tensor odd interaction in [ fig : vto ] ,
the inclusion of the second order term again makes the total tensor portion less attractive .
we emphasize that the main point of this work is to show that there are clear experimental signatures that require that in the spin triplet channel renormalization relative to a bare g matrix are required .
this is especially true for even l states where not only is the effective central part of the interaction made deeper but also the effective tensor interaction is weakened ( screening effects ) .
relative to the use of only a first order tensor interaction , the combined first and second order tensor interaction helps to explain the the smaller energy splitting of t=1 and t=0 0@xmath23 states in @xmath2o and the vanishing of the gamow - teller matrix element in the @xmath3c beta decay .
also the anti - spin orbit effects in open shell nuclei like @xmath4c are reduced although they are still substantial .
we still have a problem with @xmath0li .
although we have shown that one needs the tensor interaction to get a negative quadrupole moment we get it to be increasingly negative with increasing model space .
in closing we note that the shell model works very well in the p shell as noted by the many works of cohen and kurath @xcite .
one purpose here is to understand why it works so well .
we see that although the explicit configurations involving higher shells are not present in most calculations , their implicit presence is of crucial importance for the success of the model .
we have adopted a low - tech approach to illustrate this point .
for more trustworthy quantitative results , the high powered ab initio shell model methods or other equivalent methods need to be employed .
nevertheless we feel that the more qualitative methods used here are of considerable value in providing insight into the physics behind these more complex approaches . | we show several examples were the tensor interaction of the lowest order g matrix in a nucleus is too strong .
the examples include the quadrupole moment of @xmath0li , the isosplitting of the lowest 0@xmath1 states in @xmath2o , the near vanishing gamow - teller matrix element in the weak decay of the j=0 t=1 state of @xmath3o to the j=1 t=0 ground state of @xmath3n , and the magnitude of the deformation of @xmath4c .
it would appear that we could get better results by decreasing the tensor interaction strength by about a factor of two .
we then examine the simple estimates of gerry brown concerning second order tensor effects .
we note that for the triplet even channel the combination of first and second order tensor does indeed yield an effective weaker tensor interaction and helps to get better agreement with experiment . |
we appreciate the assistance of the technical staff and operators at the 88-inch cyclotron .
this work was supported by the director , office of science , office of nuclear physics , u.s .
department of energy under contract no .
de - ac02 - 05ch11231f .
part of this work was performed under the auspices of the u.s .
department of energy lawrence livermore national laboratory under contract de - ac52 - 07na27344 .
sm fed by ground and isomer state decays of @xmath0eu .
the minimum chi - squared fits with eq .
[ eq : oscdec ] ( red ) and exponential decay ( blue ) are shown with the data .
the 768 kev data shows decay of both the 2.4 s ground state and the 1.22 m ( @xmath63 ) state of @xmath0eu.,width=340 ] when the data are fit with just exponential decay , without an oscillating term , assuming the known lifetimes of the ground and isomeric states in @xmath0eu .
the fourier transform is taken of residuals ( data minus fit ) to search for a frequency peak not accounted for by an exponential decay term .
no peak is resolved at either 0.14 hz ( 7 seconds ) or 0.2 hz ( 5 seconds ) in either data set.,width=340 ] nd k@xmath3 x - rays .
the number of counts in a window surrounding the k@xmath3 peak is plotted as a function of time after production of @xmath0pm .
the best fit exponential decay curve is also shown . ,
width=340 ] for the first 40 seconds of decay , as examined in ref .
the best fit to eq .
[ eq : oscdec ] is shown ( red ) , finding @xmath48 and @xmath64 s , as is a curve ( green ) calculated with an oscillation amplitude @xmath65 and @xmath66 s , suggested by ref . | we have searched for time modulation of the electron capture decay probability of @xmath0pm in an attempt to confirm a recent claim from a group at the gesellschaft fr schwerionenforschung ( gsi ) .
we produced @xmath0pm via the @xmath1sn(@xmath2na , 5n)@xmath0pm reaction at the berkeley 88-inch cyclotron with a bombardment time short compared to the reported modulation period .
isotope selection by the berkeley gas - filled separator is followed by implantation and a long period of monitoring the @xmath0nd k@xmath3 x - rays from the daughter .
the decay time spectrum of the x - rays is well - described by a simple exponential and the measured half - life of 40.68(53 ) seconds is consistent with the accepted value .
we observed no oscillatory modulation at the proposed frequency at a level 31 times smaller than that reported by litvinov _
et al . _
( phys .
lett .
b 664 ( 2008 ) 162 ) . a literature search for previous experiments that might have been sensitive to the reported modulation uncovered another example in @xmath0eu electron - capture decay .
a reanalysis of the published data shows no oscillatory behavior .
a recent paper reported observation of an oscillating decay rate of the isotopes @xmath4pr and @xmath0pm when electron capture decays were measured using highly charged ions with a schottky mass spectroscopy technique @xcite .
the authors concluded that the decay activity oscillated in time with a period of about 7 seconds and a relative amplitude of 20% for both isotopes , and attributed the oscillating behavior to interference between neutrino mass eigenstates in the two - body kinematics for electron capture decay of the hydrogen - like ions .
if confirmed , this effect is surprising and might offer a new avenue for studying neutrino mixing .
references @xcite suggest quantitative explanations for the observations of ref .
@xcite in terms of two - species neutrino mixing .
other authors argue strongly in refs .
@xcite that associating the claimed decay rate modulation with neutrino oscillation is inconsistent with well - established principles of quantum mechanics .
we do not wish to enter the debate about the consistency of the experimental claims with the theoretical treatment of neutrino oscillations and quantum mechanics in the present paper .
instead we focus on the experimental issues and describe our attempt to confirm the findings of the gsi group .
the experiment at gsi exploited several unique features of the heavy ion synchrotron and storage ring facility @xcite .
the measurement was made on hydrogen - like ions of @xmath0pm and @xmath4pr produced in flight and then mass separated . after a short period of stochastic cooling , one or two ions at a time are captured , and the cyclotron frequency is measured as the ions coast in the ring .
the relevant parameter is the time to electron capture following ion injection . with this elegant technique , the daughter ions ( which have the same charge but slightly different masses )
are also trapped and the time of electron capture is determined as a discontinuous change of the cyclotron frequency .
the authors find that their decay data are well - described by @xmath5 .
% \end{equation}\end{aligned}\ ] ] according to the model in refs .
@xcite , the oscillation period is @xmath6 , where @xmath7 is the mass of the daughter nucleus , @xmath8 is the lorentz factor of the moving stored ions ( @xmath9 for the stored ions in the gsi experiments ) , and @xmath10 is the squared mass difference between the two participating neutrino mass eigenstates .
we have considered the possibility that the oscillating decay rate effect might have been missed in previous experiments studying electron capture or even ordinary beta decay using more traditional radiation detectors and implanted sources of neutral atoms .
we searched the literature for decays with relatively high electron capture probability and with half - lives within a range suggested by the model in @xcite ( @xmath11 ) .
the modulation period depends linearly upon the mass of the decaying nucleus , and in nuclei around a=150 , the period ( at rest in the lab frame ) is about 5 seconds .
there are numerous examples of measured decays in this mass region that have characteristics similar to @xmath0pm and @xmath4pr . in most experiments to measure decay properties , a typical procedure is to bombard the target for a time comparable to the lifetime of the isotope of interest .
this timing optimizes the data collection rate , but reduces sensitivity to a time - modulated decay rate because the modulation for nuclei produced at different times would destructively interfere .
an oscillating effect would be most clearly seen when decay time and modulation period are very similar . on the other hand
, the effect would be difficult to see for lifetimes much shorter than the modulation period .
for @xmath4pr and @xmath0pm , typical long bombardment and production times would preclude the observation of a 5 to 10 second modulated decay time spectrum .
we found one earlier experiment which did have the necessary short time bombardment followed by a long counting period : a study of the isomeric decay of @xmath0eu to states in @xmath0sm @xcite . in that work , the @xmath0eu
was prepared using short irradiations from 0.1 to 2.0 s , which would preserve a 5 to 10 second oscillation , but with a diminution of the modulation amplitude from the suggested 20% to between 19.9% and 15% ( depending on the irradiation time ) .
the decays of the ground @xmath12 ( @xmath13 s ) and @xmath14 isomer ( @xmath15 m ) states were observed by subsequent gamma decay of excited states of @xmath0sm populated uniquely by ground or isomer decays .
figure 1 from ref .
@xcite shows the decay of the gamma activity produced by decays of @xmath0eu@xmath16 , with 1 second time bins , extending to 50 seconds after bombardment .
the electron capture probability for the 1.22 m @xmath14 isomer decay is 17% . the data from figure 1 of ref .
@xcite show no obvious oscillations
. a fit of that data ( shown in this paper in fig .
[ fig:142eudatfit ] ) using eq .
[ eq : oscdec ] finds a minimum of @xmath17 with an oscillation period of 3.540(53 ) seconds , an amplitude of @xmath18 , and a phase @xmath19 rad for the 1023 kev data . for the 768 kev data ,
we find a minimum @xmath20 with an oscillation period of 4.854(73 ) seconds , @xmath21 , and @xmath22 rad .
when performing fits to the data using eq .
[ eq : oscdec ] we restrict the phase to @xmath23 and ensure @xmath24
. fits to simple exponential decay find @xmath25 for the 1023 kev data and @xmath26 for the 768 kev data .
the fourier transform power spectra in fig .
[ fig:142eufft ] are calculated for the residuals to simple exponential fits .
the fourier spectra show no well - resolved peaks corresponding to 5 second oscillation ( or 7 second oscillation , under the assumption that the oscillation period is not proportional to the lorentz factor @xmath8 ) .
our @xmath0eu analysis would limit an oscillatory term to be a factor of about 3 smaller than that reported in ref .
@xcite , if we assume that our uncertainty in @xmath27 ( @xmath28 ) represents our sensitivity to an oscillating term , and if we correct for the electron capture branching ratio and account for the small reduction in an oscillation amplitude from the ( up to ) 2 s bombardment time .
our best - fit period and phase parameters disagree strongly with the results of @xcite .
the electron capture branching ratio also depends on the ionization state of the parent atom , as shown in refs .
@xcite , a distinction between the experiments in ref .
@xcite and ref .
@xcite which used stopped , presumably neutral atoms .
there is no reason _ a priori _ to expect that an electron capture decay rate oscillation would have the same amplitude in eu compared to pm and pr .
although this result would seem to disfavor the time modulation of electron capture decay , the uncertainty in the possible oscillation amplitude is large and the statistical power of this data is limited . despite the negative evidence from @xmath0eu and because of the importance of the gsi claim , we performed a test in @xmath0pm , one of the two isotopes reported to show a positive effect .
we studied the electron capture and beta decay rate from @xmath0pm , using a source of stopped ions in a foil . a thin target (
400 @xmath29g/@xmath30 @xmath1sn backed with carbon ) was bombarded with 95 mev @xmath2na@xmath31 ( average beam intensity 100 pna ) from the 88-inch cyclotron at lawrence berkeley national laboratory .
this beam energy was selected to produce predominantly @xmath0pm , and calculations with pace-2 @xcite indicate that other @xmath32 isotopes have production rates about a factor of 9 lower than @xmath0pm .
the reaction products moved through the berkeley gas - filled separator ( bgs ) @xcite , which separated the @xmath0pm from the beam and other products by their different magnetic rigidities .
the @xmath0pm stopped in a 25 @xmath29 m thick aluminum foil at the focal plane of the bgs .
an intrinsic germanium
clover " detector @xcite is located just outside a 2 mm thick aluminum vacuum window and counted the x - ray and gamma - ray emissions from the stopped @xmath0pm . after a short bombardment time of 0.5 s ,
the primary beam was shut off , and events from the germanium detector were recorded for 300 s before repeating the beam / count cycle .
approximately 190 bombardment cycles were performed over a 32 hour run . with the cyclotron beam off ,
the germanium detector recorded roughly 300 events per second above the 20 kev threshold .
figure [ fig : pmgloenergy ] is the energy spectrum from the clover germanium detector , summed over all beam bombardment cycles and counting time bins .
events during the beam bombardment are shown separately from the spectrum measured after the beam has been shut off .
events showing energy deposition in multiple leaves " of the clover detector have been excluded ( which reduces the compton background under the x - ray portion of the spectrum ) .
k@xmath33 and k@xmath34 from the daughter @xmath0nd are not resolved at 37.36 and 36.85 kev , but k@xmath3 is distinguished from k@xmath35 at about 42.5 kev in the beam off " spectrum .
the peaks at 433 , 381 , 241 , 208 , and 43 kev are cascaded from a 67 @xmath29s ( @xmath36 ) state populated in the production reaction @xcite , and confirm that @xmath0pm was indeed produced and well - distinguished from other isobars .
these peaks disappear in the beam off " spectrum , confirming that the cyclotron beam was turned off , and confirming that the production time of the @xmath0pm was short compared to a 5 to 10 second oscillation period .
the events were sorted into time - binned energy spectra for every 0.5 s. to measure the electron capture decays of @xmath0pm , we measured the rate of detected k - shell x - rays ( k@xmath33 , k@xmath34 ) .
electron capture decay strongly favors k - shell x - ray emission compared to positron decay .
the electron capture branching ratio for neutral @xmath0pm is 22% , of which about 85% is k - shell capture @xcite .
the x - ray fluorescence yield for a k - shell vacancy ( k@xmath3 plus k@xmath35 ) for nd is 91% @xcite .
positron emission by @xmath0pm can in principle produce k - shell vacancy and hence k - shell x - rays via shakeoff and direct collision .
however , these processes strongly favor outer shell vacancy production , particularly in high z atoms .
the k - shell vacancy probability after @xmath37 decay should be very small about @xmath38 for @xmath0pm @xcite . a very conservative upper limit on the amount of k - shell x - rays arising from @xmath37 emission would be 1.5% , assuming that all the auger k electron yield from @xmath0pm decay @xcite was attributable to @xmath37 .
a detected k - shell x - ray therefore has a greater than 92% probability to have been created by electron capture decay , and a less than 8% probability to have come from positron emission .
detection of a k - shell x - ray therefore strongly selects electron capture decays .
figure [ fig : pmkadecay ] shows the time decay of the nd k@xmath3 x - rays for the entire 300 s counting time along with a best - fit exponential decay curve .
for each 0.5 s time bin after the beam bombardment , we histogram the number of counts in a 5 kev window surrounding the k@xmath3 peak .
no oscillation is apparent in fig .
[ fig : pmkadecay ] .
we performed @xmath39 minimizing fits of the decay data to the function in eq .
[ eq : oscdec ] to search for a resolved oscillatory time dependence .
fixing @xmath40 and allowing @xmath41 , @xmath42 , and @xmath43 to vary for a simple exponential decay gave @xmath44 with 594 degrees of freedom . allowing @xmath41 , @xmath42 , @xmath43 , @xmath27 , @xmath45 , and @xmath46 to vary in the fits , we find a minimum @xmath47 , with @xmath48 , @xmath49 , and an oscillation period of 3.178(36 ) s. to evaluate the statistical significance of the fits using eq .
[ eq : oscdec ] compared to a simple exponential , we performed identical fits of monte - carlo generated data which had similar statistical power .
the monte - carlo generated data contained only exponential decay with the same time - bin structure as the experimental data , with no oscillating terms .
repeated trials of the fit procedure to eq .
[ eq : oscdec ] on several hundred randomized exponential decay data sets found an average oscillation amplitude @xmath50 by 2.5 standard deviations .
moreover , the @xmath39 parameter of the fits using oscillation terms improved , according to a fisher f - test , at a confidence level of 5% . comparing the fits of the real data using a fisher f - test
, we find a confidence level of 29.5% for the `` null hypothesis '' that the oscillating terms are real .
that is , the hypothesis that there _ is _ an oscillating term in the real @xmath0pm decay data would only be justified at a 29.5% confidence level , compared to the 5% confidence level generated by the fits to simulated , non - oscillatory data , and compared to the 0.6% confidence claimed in @xcite ) .
this supports the conclusion that the fits to our data from @xmath0pm do not include statistically significant oscillations .
the fisher f - test confidence levels are misleading in this instance .
fourier transforms of the residuals to a simple exponential fit to the data are shown in fig .
[ fig : fftresid ] , and show no statistically significant peaks at 1/(5 s ) ( or 1/(7 s ) , assuming that the oscillation period does not depend on @xmath8 ) . in all fit cases ,
we find @xmath51 s , which agrees with the accepted lifetime of @xmath0pm ( 40.5(5 ) s @xcite ) .
figure [ fig : pmcompare ] shows the first 40 seconds of our decay data , roughly the time period considered in @xcite for @xmath0pm , along with a calculated decay curve with @xmath52 , @xmath53 s , and @xmath54 rad found in @xcite .
these parameters are inconsistent with the data .
we conclude that any oscillation of 5 ( or 7 ) seconds , if present , must have an amplitude smaller than that found in ref .
@xcite ( @xmath55 ) by about a factor of 31 , using the uncertainty in the oscillation amplitude fit parameter as our sensitivity .
our beam bombardment time of 0.5 s would reduce our sensitivity to an oscillation amplitude , but a simple calculation ( integrating oscillating exponential decay and production ) suggests that this would reduce our sensitivity by 2% ( i.e. if @xmath52 , this experiment would have measured @xmath56 ) .
our best - fit oscillation amplitude is not statistically significant , and in any case has an amplitude a factor of 16 smaller than that proposed in @xcite , with a much different period and phase . as a cross - check on our data , we measured the decay of 511 kev positron annihilation activity . reference @xcite suggests that the decay rate for electron capture events will oscillate while the @xmath37 decay rate might not , since the three - body phase space density for positron emission has a large neutrino momentum distribution , which would average out any coherence between final neutrino mass states
. reference @xcite concludes that the positron decay of hydrogen - like ions will oscillate on a time scale too fast to have been observed in our data .
the 511 kev decay data was fit to a simple exponential decay , finding a half - life of @xmath57 s with @xmath58/596 . fitting this data to eq .
[ eq : oscdec ] we find a minimum @xmath59 for @xmath60 , @xmath61 and period @xmath62 s. the confidence level for this fit compared to simple exponential decay is 2.8% , suggesting that the oscillatory fit parameters improve the fit , more so than in the electron capture data set analyzed here , and slightly more so than in our monte - carlo simulated data which containted no oscillations .
this is contrary to the expectation under the neutrino mixing hypothesis that there should be no oscillation in the positron decay activity .
this result supports the conclusion that our oscillation fit parameters in the electron capture decay data are not statistically significant .
it might be argued that our experiment using neutral atoms would be insensitive to the proposed neutrino oscillation effect , since the participation of the remaining atomic electrons could provide a decoherence of the neutrino momentum states in the larger phase space of the final atomic states after the decay .
however , the usual description of electron capture decay suggests that the neutrino and recoil nucleus momenta are determined largely by the structure of the weak interaction hamiltonian . in the case where multiple atomic electrons are present ,
there can in principle be interference between contributions from different atomic electronic states . in calculating the total decay rate , or the ratio of k - shell to l - shell electronic capture , or the ratio of electron capture to positron decay ,
the neutrino ( and recoil nucleus ) momentum is determined by an overlap integral of the atomic electrons wave functions , summed over electron states @xcite .
but these calculations , particularly in high - z nuclei such as nd , are dominated by the k - shell contribution , and multiple electron terms represent corrections to the total decay rate of only a few percent @xcite . moreover , our experiment detected k - shell x - rays , meaning that the captured electron was indeed a k - shell electron with a similar wavefunction to the the hydrogenic ions investigated in @xcite .
the subsequent atomic de - excitation processes do not greatly influence the generation of the neutrino or recoil momenta .
if multiple electron effects destroy the coherence of the mixed neutrinos momenta in the final state , this would be apparent in data from the gsi group comparing the decay time spectrum of hydrogen - like and helium - like stored ions , similar to data reported in @xcite .
a further desirable confirmation of the data from the gsi group would be to examine the @xmath37 decays of the hydrogen - like ions , which should show no oscillations on the timescales available for examination , according to ref .
@xcite . to summarize
, no convincing oscillation was observed in the decay time spectrum of electron capture decays of @xmath0pm ( or in @xmath0eu using published data ) when dressed with their full complement of electrons and at rest in a solid metal matrix .
any 5 second oscillation not resolved in this experiment must have an amplitude a factor of 31 times smaller than that reported in ref .
@xcite .
the proposed oscillating decay rate could in principle be attributable to the truly two - body nature of the final state in the hydrogen - like decays observed in @xcite , although this would require an unconventional explanation with respect to electron capture decay in neutral atoms . under this hypothesis
, data on the decay of helium - like stored ions would show much different oscillation behavior . |
the recent discovery of the transiting extrasolar planetary system wasp-18 ( * ? ? ?
* hereafter h09 ) lights the way towards understanding the tidal interactions between giant planets and their parent stars .
wasp-18b is one of the shortest - period ( @xmath5d ) and most massive ( @xmath6@xmath1 ) extrasolar planets known .
these properties make it an unparallelled indicator of the tidal dissipation parameters @xcite for the star and the planet @xcite . a value similar to that observed for solar system bodies ( @xmath7@xmath8 ; @xcite ) would cause the orbital period of wasp-18 to decrease at a sufficient rate for the effect to be observable within ten years ( h09 ) .
in this work we present high - precision follow - up photometry of wasp-18 , obtained using telescope - defocussing techniques @xcite which give a scatter of only 0.47 to 0.83 mmag per observation .
these are analysed to yield improved physical properties of the wasp-18 system , with careful attention paid to statistical and systematic errors .
the quality of the light curve is a critical factor in measurement of the physical properties of transiting planets @xcite . in the case of
wasp-18 the systematic errors arising from the use of theoretical stellar models are also important , and are a limiting factor in the understanding of this system .
[ cols="<,^,^,^,^,^,<,^,^,^",options="header " , ] the physical properties of a transiting planetary system can not in general be calculated purely from observed quantities . the most common way to overcome
this difficulty is to impose predictions from theoretical stellar evolutionary models onto the parent star .
we have used tabulated predictions from three sources : _ claret _
@xcite and _ cambridge _ @xcite .
this allows the assessment of the systematic errors caused by using stellar theory .
we began with the parameters measured from the light curve and the observed velocity amplitude of the parent star , @xmath10ms@xmath11 ( h09 ) .
these were augmented by an estimate of the velocity amplitude of the _ planet _ , @xmath12 , to calculate preliminary physical properties of the system .
we then interpolated within one of the grids of theoretical predictions to find the expected radius and @xmath13 of the star for the preliminary mass and the measured metal abundance ( @xmath14}}= 0.00 \pm 0.09 $ ] ; h09 ) .
@xmath12 was then iteratively refined to minimise the difference between the model - predicted radius and @xmath13 , and the calculated radius and measured @xmath13 ( @xmath15k ; h09 ) .
this was done for a range of ages for the star , and the best overall fit retained as the optimal solution .
finally , the above process was repeated whilst varying every input parameter by its uncertainty to build up a complete error budget for each output parameter @xcite .
a detailed description of this process can be found in @xcite .
table[tab : absdimall ] shows the results of these analyses .
the physical properties calculated using the _ claret _ and _ y@xmath9 _ sets of stellar models are in excellent agreement , but those from the _ cambridge _ models are slightly discrepant .
this causes systematic errors of 4% in the stellar mass and 2% in the planetary mass , both of similar size to the corresponding statistical errors .
the quality of our results is therefore limited by our theoretical understanding of the parent star .
our final results are in good agreement with those of h09 ( table[tab : absdimall ] ) , but incorporate a more comprehensive set of uncertainties .
these results give the equilibrium temperature of the planet to be one of the highest for the known planets : @xmath16 where @xmath17 is the bond albedo and @xmath18 is the heat redistribution factor . this equilibrium temperature , and the closeness to its parent star , make wasp-18b a good target for the detection of thermal emission and reflected light .
we have presented high - quality observations of five consecutive transits by the newly - discovered planet wasp-18b , which has one of the shortest orbital periods of all known transiting extrasolar planetary systems ( teps ) .
our defocussed - photometry approach yielded scatters of between 0.47 and 0.83 mmag per point in the final light curves .
these data were analysed using the jktebop code , which was modified to include the spectroscopically derived orbital eccentricity in a statistically correct way .
the light curve parameters were then combined with the predictions of theoretical stellar evolutionary models to determine the physical properties of the planet and its host star .
a significant source of uncertainty in our results stems from the use of theoretical models to constrain the physical properties of the star .
further uncertainty comes from observed @xmath13 and @xmath19 $ ] , for which improved values are warranted .
however , the systematic error from the use of stellar theory is an important uncertainty in the masses of the star and planet .
this is due to our fundamentally incomplete understanding of the structure and evolution of low - mass stars . as with many other transiting systems ( e.g. wasp-4 ;
@xcite ) , our understanding of the planet is limited by our lack of understanding of the parent star .
we confirm and refine the physical properties of wasp18 found by h09 .
wasp-18b is a very massive planet in an extremely short - period and _ eccentric _ orbit , which is a clear indicator that the tidal effects in planetary systems are weaker than expected ( see h09 ) .
long - term follow - up studies of wasp-18 will add progressively stricter constraints on the orbital decay of the planet and thus the strength of these tidal effects .
we now split the full sample of known ( i.e. published ) teps into two classes according to planetary mass .
the mass distribution of transiting planets shows a dearth of objects with masses in the interval 2.03.1@xmath1 .
there are nine planets more massive than this and 46 less massive .
seven of the nine high - mass teps have eccentric orbits ( hat - p-2 , @xcite ; hd17156 , @xcite ; hd80606 , @xcite ; wasp-10 , @xcite ; wasp-14 , @xcite ; wasp-18 ; xo-3 , @xcite ) , and the existing radial velocity observations of the remaining two can not rule out an eccentricity of @xmath20 or lower ( corot - exo-2 , @xcite ; ogle - tr - l9 , @xcite ) . by comparison ,
only four of the 46 low - mass teps have a _ significant _ @xcite orbital eccentricity measurement .
these numbers imply that the more massive teps are a different population to the less massive ones ; fisher s exact test @xcite returns a probability lower than @xmath21 of the null hypothesis ( although this does not account for our freedom to choose the dividing line between the two classes ) .
this indicates that the two types of teps have a different internal structure , formation mechanism , or evolution , a suggestion which is supported by observations of misalignment between the spin and orbital axes of @xmath22@xmath1 teps @xcite .
there is , however , a bias at work here .
the more massive teps cause a larger radial velocity signal in their parent star ( @xmath23 ) , so a given set of radial velocity measurements can detect smaller eccentricities ( see also * ? ? ?
the eccentricity of the wasp-18 system is in fact below the detection limit of existing observations of most teps .
we therefore advocate the acquisition of additional velocity data for the known low - mass teps , in order to equalise the eccentricity detection limits between the two classes of teps .
these observations would allow acceptance or rejection of the hypothesis that more massive teps represent a fundamentally different planet population to their lower - mass brethren .
the observations presented in this work will be made available at the cds ( http://cdsweb.u-strasbg.fr/ ) and at http://www.astro.keele.ac.uk/@xmath24jkt/. the operation of the danish 1.54 m telescope was financed by the danish natural science research council ( fnu ) .
we thank dr .
j. eldridge for calculating the _ cambridge _ set of stellar models used in this work .
jsouthworth and dra acknowledge financial support from stfc in the form of postdoctoral research assistant positions .
astronomical research at the armagh observatory is funded by the northern ireland department of culture , arts and leisure ( dcal ) .
dr ( boursier fria ) , ff and jsurdej acknowledge support from the communaut franaise de belgique actions de recherche concertes
acadmie wallonie - europe .
the following internet - based resources were used in research for this paper : the eso digitized sky survey ; the nasa astrophysics data system ; the simbad database operated at cds , strasbourg , france ; and the ar@xmath25iv scientific paper preprint service operated by cornell university . | we present high - precision photometry of five consecutive transits of wasp-18 , an extrasolar planetary system with one of the shortest orbital periods known . through the use of telescope defocussing we achieve a photometric precision of 0.470.83 mmag per observation over complete transit events .
the data are analysed using the jktebop code and three different sets of stellar evolutionary models .
we find the mass and radius of the planet to be @xmath0@xmath1 and @xmath2@xmath3 ( statistical and systematic errors ) respectively .
the systematic errors in the orbital separation and the stellar and planetary masses , arising from the use of theoretical predictions , are of a similar size to the statistical errors and set a limit on our understanding of the wasp-18 system .
we point out that seven of the nine known massive transiting planets ( @xmath4@xmath1 ) have eccentric orbits , whereas significant orbital eccentricity has been detected for only four of the 46 less massive planets .
this may indicate that there are two different populations of transiting planets , but could also be explained by observational biases .
further radial velocity observations of low - mass planets will make it possible to choose between these two scenarios . |
some strongly coupled lattice fermion - gauge models with a charged scalar field , which break chiral symmetry dynamically , might be considered to be a possible alternative to the higgs mechanism for mass generation , as discussed in @xcite .
let us concentrate on a prototype with @xmath0 gauge group , a scalar of fixed modulus and one staggered fermion ( corresponding to 4 flavors ) , where both the scalar and fermion have charge one .
the action has been described in @xcite with three bare parameters @xmath1 .
the dynamical mass generation is meaningful only in the chiral limit @xmath2 .
we consider here the phase transition line * net * between two phases @xcite : \(1 ) dynamical mass generation ( nambu ) phase , below the * net * line , where chiral symmetry is spontaneously broken ( @xmath3 ) due to the strong gauge fluctuations so that the fermion mass @xmath4 is dynamically generated ; \(2 ) higgs phase , above the * nets * line , where the higgs mechanism is operative , but @xmath5 .
the scalar field induces a second order chiral phase transition * ne * line which opens the possibility for approaching the continuum . whether such a model can replace the higgs mechanism depends crucially on the existence and renormalizability of the continuum limit . to search for such a continuum theory and grasp its nature
, we need to make precise determination of the second order phase transition point with divergent correlation lengths .
for such a purpose , we have done extensive simulations using hybrid monte carlo ( hmc ) algorithm and developed some new methods for locating the * ne * line .
the hmc simulations have been done on @xmath6 and @xmath7 , where on @xmath6 , we have better statistics ( 1024 - 6500 trajectories ) for different @xmath1 .
the detailed results for the spectrum are reported in @xcite .
we have measured the following local observables : plaquette energy @xmath8 , link energy @xmath9 and chiral condensate @xmath10 , where for @xmath10 we use the stochastic estimator method .
however , it is very difficult to use the local quantities at finite @xmath11 to detect a critical behavior on the * ne * line , since they show smooth behavior as a function of @xmath12 or @xmath13 .
( one could expect the critical behavior only in the infinite volume and chiral limit . ) for @xmath14 near the point * e * , the peaks of susceptibility for different quantities develop and coincide , while the boson mass @xmath15 gets smaller .
concerning the location of the * et * line , on the @xmath6 and @xmath7 for @xmath16 or @xmath17 and @xmath18 , we find explicit two state signals from the thermo - cycle , time history and histogram analysis of the local quantities . on the * ne * line
, the @xmath19 meson shows more obviously the phase transition than other quantities . in the nambu phase
, the @xmath19 meson should obey the pcac relation . in the symmetric phase ,
the @xmath19 meson is no longer a goldstone boson , and one should observe a deviation from pcac . at @xmath20 , these properties are nicely seen in fig .
[ fig1 ] , from which one sees that for @xmath21 where the system is in the broken phase , we have goldstone bosons .
however , on @xmath6 , even at @xmath22 ( possibly in the chiral symmetric phase ) , a linear extrapolation leads to @xmath23 . for larger @xmath12
, the extrapolated result gets smaller ( e.g. at @xmath24 , @xmath25 ) and is expected to vanish in the @xmath26 limit . of course
, one should not expect the linear extrapolation to be valid at the critical point .
@xmath27 is not a convenient order parameter for the chiral transition of a finite system due to the sensitivity of chiral extrapolation .
we employ a different method for determining the chiral transition , namely we calculate the chiral susceptibility in the chiral limit , defined by @xmath28 if there is a second order chiral phase transition , @xmath29 should be divergent ( in other words , @xmath30 should be zero ) at the critical point and in the thermodynamical limit . in the chiral limit ,
the chiral susceptibility in the nambu phase is difficult to obtain , but it is calculable in the chiral symmetric phase @xcite .
it can be shown that in the symmetric phase @xmath30 , defined in eq .
( [ def ] ) , is the same as @xmath31 where @xmath32 are the positive eigenvalues of the massless fermionic matrix . approaching the * ne * line from the symmetric phase by fixing @xmath13 , @xmath33 should behave as @xmath34 corresponding to the divergent correlation length at the second order phase transition point in the thermodynamical limit @xmath26 . in the nambu phase
, it can also be shown that eq .
( [ order ] ) is equivalent to @xmath35 in the @xmath26 limit .
then in such a limit , @xmath33 should be zero since @xmath36 in the nambu phase .
therefore , @xmath33 defined in eq .
( [ order ] ) is a suitable order parameter for the chiral phase transition : it is zero in the broken phase , and it is nonzero in the symmetric phase .
let us again focus on the results at @xmath20 . to perform the calculation
, we generalize mfa @xcite , in which the chiral limit @xmath2 is accessible , to the fermion - gauge - scalar models . from fig .
[ fig2 ] , we observe that on @xmath37 the chiral transition appears at @xmath38 , being consistent with the observation of fig .
[ fig1 ] .
= 7.5 cm = 7.5 cm
the location of the * ne * line on the available lattices obtained by the above methods is summarized in fig .
[ fig3 ] , where the point * n * is plotted by interpolation .
we have determined the phase transition line * ne * with high precision and demonstrated that this second order chiral transition line joins the higgs phase transition line at the end point * e * being around @xmath39 , separating the higgs and nambu phases .
no finite size scaling analysis has been done , and larger lattices are required for such a purpose . from the spectroscopy @xcite , we know that @xmath40 scales to zero when crossing the chiral transition line * ne*. nevertheless , the susceptibility for @xmath9 and correlation length for the composite scalar @xmath41 remain finite on the whole * ne * line except approaching the end point * e*. therefore , the end point , hopefully being a second order point with divergent correlation lengths , is the most suitable candidate for the continuum limit .
further work to be done is to study the finite size effects , analyze the dependence of the end point * e * on the bare fermion mass , investigate the scaling properties and understand the nature of the end point , which is underway @xcite .
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piedrafita , g. di carlo , a. galante , a.f .
grillo , l.a .
fernandez and a. vladikas , phys .
d48 ( 1993 ) 402 .
w. franzki , c. frick , j. jersk and x.q .
luo , in preparation . | we report the recent results from the computer simulations of a fermion - gauge - scalar model with dynamical chiral - symmetry breaking and chiral transition induced by the scalar field .
this model might be considered to be a possible alternative to the higgs mechanism of mass generation .
a new scheme is developed for detecting the chiral transition .
our results show with higher precision than the earlier works that the chiral transition line joins the higgs phase transition line , separating the higgs and nambu ( chiral - symmetry breaking ) phases .
the end point of the higgs transition with divergent correlation lengths is therefore suitable for an investigation of the continuum limit . |
we use a pre - computed database of 46800 synthetic flux and polarization spectra for field strengths of 1400 mg and a wide range of effective temperatures to generate synthetic zeeman spectra , which we subject to an automatic optimization scheme with the aim to recover the parameters describing the original magnetic field configuration .
we adopt input field geometries which involve the sum of non - aligned dipole and quadrupole components .
we also provide for off - centre shifts of the configuration . by adding gaussian noise we create input spectra with and 20 , respectively .
= 0.80 off - centred dipole - quadrupole combinations are reliably recovered even for by our code , which can handle up to 12 free parameters but becomes inefficient if higher multipoles are included .
the poor convergence is caused by the complexity of the @xmath0-landscape , which develops an increasing number of secondary minima for more complex field geometries ( for a detailed investigation , see euchner et al . , 2002 ) .
we find that a given set of phase - resolved zeeman spectra can be reproduced within the noise by quite different formal representations of the field , which , however , all seem to describe rather similar actual field geometries
. we are confident , therefore , that observed phase - resolved zeeman spectra can provide quite definite information on the field structure .
= 0.85 the magnetic field structure over the surface of accreting white dwarfs in cataclysmic binaries becomes accessible to observation only in states of low or switched - off accretion .
following the first detection of zeeman lines in the polar eferidani ( @xmath1 = 81 min ) by wheatley & ramsay ( 1998 ) , we obtained phase - resolved flux and circular polarization spectra of this object with fors1 at the eso vlt on 22 nov 2000 .
fig.1 shows the spectra collected in four almost equal phase bins after removal of a slight rotational temperature variation .
there is no obvious variation of the zeeman structures with phase .
fits of inclined multipole field models with the angle @xmath2 between magnetic and rotational axis left free yielded .
the lack of phase dependence of the zeeman features further suggests that the tesseral / sectoral multipole components are weak .
we fitted the mean flux and polarization spectra , therefore , with an expansion including only the zonal components up to a maximum degree .
fig.2 shows the dipole+quadrupole fit , which is substantially better than the pure dipole ( not shown ) .
fig.3 depicts the result for the expansion up to .
the dipole is replaced by large coefficients for the higher - order multipole components which substantially improve the fit . while the higher - order multipole fit accounts for many of the observed details all the wiggles in the blue are genuine zeeman features
there are still noticeable discrepancies : ( i ) the observed lack of circular polarization around 5700 requires that a larger fraction of the field occurs at negative @xmath3 ( angle between line of sight and field direction ) , and ( ii ) the observed narrow spectral feature at 5120 can be fitted by additional field contributions with @xmath4 between 60 and 100 mg pointing away from the observer , possibly generated by a steeper rise of @xmath4 towards the lower ( unseen ) pole than provided by the model . | we have developed a new method to derive the magnetic field distribution on the surfaces of rotating magnetic white dwarfs from phase - resolved flux and circular polarization spectra .
an optimization code based on an evolutionary strategy is used to fit synthetic zeeman spectra for a variety of model geometries described in the framework of a truncated multipole expansion .
we demonstrate that the code allows the reconstruction of relatively complex fields using noise - added synthetic input spectra . as a first application
, we analyze flux and circular polarization spectra of the polar eferi in a low state of accretion taken with fors1 at the eso vlt . |
this work was supported by the national basic research program of china under grants nos .
2012cb821305 , 2010cb923200 and 2013cb922403 , the national natural science foundation of china under grants nos . 11374375 , 11204043 , 11274399 and 61078027 , and the ph.d .
programs foundation of ministry of education of china under grant nos .
b.a.m . appreciates hospitality of the sun yat - sen university ( guangzhou , china ) . | we demonstrate that in - bulk vortex localized modes , and their surface half - vortex ( horseshoe " ) counterparts self - trap in two - dimensional ( 2d ) nonlinear optical systems with @xmath0-symmetric photonic lattices ( pls ) .
the respective stability regions are identified in the underlying parameter space .
the in - bulk states are related to truncated nonlinear bloch waves in gaps of the pl - induced spectrum .
the basic vortex and horseshoe modes are built , severally , of four and three beams with appropriate phase shifts between them .
their stable complex counterparts , built of up to 12 beams , are reported too .
nonlinear spatially periodic systems support diverse types of self - trapped in - gap states .
in particular , spatial gap solitons @xcite originate from the interplay between the periodicity and nonlinearity .
further , surface gap solitons @xcite appear at the interface between a uniform medium and a photonic lattice ( pl ) built into a nonlinear material .
extended self - trapped waves with steep edges also exist in these settings , being related to truncated nonlinear bloch waves @xcite .
modes of the latter type provide a link between extended nonlinear bloch waves @xcite and tightly localized gap solitons @xcite .
recently , a great deal of interest has been drawn to the realization of parity - time ( @xmath0 ) symmetry in optics .
originally , this concept was developed in quantum mechanics , where it was demonstrated that , beyond the conventional hermitian hamiltonians , their @xmath0-symmetric non - hermitian counterparts may also give rise to purely real ( hence physically relevant ) spectra @xcite .
following the similarity between quantum mechanics and paraxial optics @xcite , @xmath0-symmetric optical systems with complex refractive indices @xcite have been extensively studied theoretically @xcite and experimentally @xcite . in this context ,
@xmath0-symmetric pls play an important role .
taking into account the non - orthogonality of the respective eigenmodes , their coupled - mode description had to be reformulated via the variational principle @xcite .
the light propagation in @xmath0-symmetric pls embedded into linear media were analyzed preliminarily @xcite .
further , it has been found that 1d and 2d spatial gap solitons exist in pls built into a nonlinear material @xcite .
although bloch waves @xcite and gap solitons @xcite were studied before in the context of some @xmath0-symmetric pls , the comprehensive study of self - trapped states in 2d @xmath0-symmetric systems combining lattices and nonlinearity was not reported yet .
in particular , such self - trapped nonlinear states may serve as a necessary link between spatially localized gap solitons and extended nonlinear bloch waves under the @xmath0 symmetry .
self - trapped vortices and surface modes are of great interest in the context of the @xmath0-symmetric settings .
indeed , the study of nonlinear surface modes pinned on the interface of a @xmath0-symmetric system opens a way to explore the interplay between surface effects , the nonlinearity , and the @xmath0-symmetry . on the other hand ,
the analysis of localized vortices supported by @xmath0-symmetric pls should shed light on the cooperation and competition of the @xmath0-symmetry with the azimuthal instability and spatial periodicity . in this work ,
we show the existence of in - bulk and surface self - trapped states in 2d nonlinear systems with @xmath0-symmetric pls .
in particular , we report in - bulk solitary vortices and novel half - vortex surface modes .
stable half - vortex surface modes appear as horseshoes `` pinned on the interface between a uniform linear medium and a nonlinear medium with built - in @xmath0-symmetric pl .
the in - bulk vortices and surface horseshoes '' have a common linear stability region at intermediate values of propagation constants .
we consider the light propagations in two nonlinear systems with @xmath0-symmetric pls : a uniform setting of the nonlinear @xmath0-symmetric pl , and a composite setting of the nonlinear @xmath0-symmetric pl at the left side ( @xmath1 ) and a uniform linear medium at the right side ( @xmath2 ) . assuming that the light propagates along the @xmath3-axis , the amplitude of the electromagnetic field is written as @xmath4 , with carrier wavenumber @xmath5 and frequency @xmath6 . with the effective refractive index including contributions from the complex pl and the kerr effect , @xmath7
, the amplitude obeys the nonlinear schrdinger equation with the complex potential , @xmath8 e \notag \\ & & + 2k_{0}^{2}\left [ n_{0}^{\mathrm{pl}}\left ( n^{\mathrm{r}}+in^{\mathrm{i}}\right ) + n_{0}^{\mathrm{pl}}n^{\mathrm{nl}}|e|^{2}\right ] e=0 .
\label{eq : one}\end{aligned}\]]here , @xmath9 is a constant , @xmath10 is the background refractive index , @xmath11 and @xmath12 are real and imaginary ( gain / loss ) parts of the spatial modulation of the local index , and @xmath13 is the kerr coefficient .
similarly , the light propagation in the linear uniform medium obeys the paraxial equation @xmath14 e=0 , \label{eq : two}\]]with the respective real refractive index , @xmath15 .
we normalize the equations by defining @xmath16 , @xmath17 , @xmath18 , and @xmath19 z}$ ] with an arbitrary scaling factor @xmath20 .
the accordingly rescaled form of eqs .
( [ eq : one ] ) and ( [ eq : two ] ) is@xmath21with @xmath22 , and @xmath23 $ ] .
the complex @xmath0-symmetric potential , @xmath24 , is chosen as@xmath25 , \\
w(\xi , \eta ) & = & \theta \{\sin [ \sqrt{2}(\eta -\xi ) ] + \sin [ \sqrt{2}(\eta+\xi ) ] \},\end{aligned}\]]with amplitudes @xmath26 and @xmath27 of the modulation of the real and imaginary parts of the refractive index .
this pl is a @xmath28 counterclockwise rotation of the one considered in refs .
@xcite .
the configurations of pls are shown by the white - blue circles in the @xmath29-plane .
its band - gap structure can be derived by using the plane - wave expansion method based on the floquet - bloch theorem , see fig .
[ fig : one ] ( a ) .
-symmetric photonic lattice : @xmath30 is the propagation constant , and @xmath31 , @xmath32 are bloch wavenumbers in @xmath33 and @xmath34 directions .
( b , c ) intensity profiles of self - trapped modes for propagation constant @xmath35 in the uniform and truncated systems , respectively .
the white dot - dash line depicts the interface in the truncated system .
( d ) power @xmath36 and ( e ) the real part of the instability growth rate , @xmath37 , of the self - trapped modes versus the propagation constant , @xmath38 .
green triangles and red circles in ( d ) correspond to the modes shown in ( b ) and ( c ) , respectively .
the green dashed and red solid lines in ( e ) represent , severally , in - bulk and surface self - trapped modes .
( f ) the density profile at @xmath39 , evolved from the initial self - trapped mode ( c ) with @xmath40 noise .
parameters are @xmath41 , @xmath42 and @xmath43 . ] to combine eqs .
( [ eq : seven ] ) and ( [ eq : eight ] ) into a single equation , we define a step function , @xmath44 at @xmath45 and @xmath46 at @xmath47 : @xmath48 gq+u(\xi ) |q|^{2}q=0 .
\label{eq : jia_1}\end{aligned}\]]the stationary solution with real propagation constant @xmath38 is looked for as @xmath49 , where complex function @xmath50 obeys equation @xmath51 gu \nonumber\\ & + & u(\xi)|u|^{2}u - bu=0.\label{eq : five}\end{aligned}\ ] ] to find the stationary self - trapping solutions , we used numerical simulations with the modified squared - operator method @xcite . while the existence and stability of the simplest single - beam solitons in the present setting is quite evident , as the first step of the analysis we produce double - beam self - trapped states . for @xmath35 and @xmath43 , the in - bulk and surface double modes
are displayed in fig .
[ fig : one](b , c ) . due to the presence of the interface
, the intensity of the surface self - trapped states is larger than the in - bulk ones , at the same propagation constant .
the dependence of the total power , @xmath52 , on the propagation constant @xmath38 demonstrates that the power of the surface modes is also larger than that of the in - bulk ones , see fig .
[ fig : one](d ) .
different from the single - beam solitons @xcite , both surface and in - bulk self - trapped states in the semi - infinite gap do not exist near the first bloch band in fig .
[ fig : one](d ) .
stability of the self - trapped modes was investigated by means of the linearization for small perturbations . to a given stationary state ,
@xmath53 , the perturbation is added as @xmath54e^{ib\zeta } $ ] with infinitesimal @xmath55 @xcite , where @xmath56 and @xmath57 are two perturbation eigenfunctions , @xmath58 is the corresponding growth rate , and the star ( @xmath59 ) stands for the complex conjugate . from eq .
( [ eq : jia_1 ] ) , the following linearized equations are derived : @xmath60f \nonumber\\ & & + u(\xi ) u^{2}g , \\ + i\delta g&=&\left [ \nabla _ { \perp } ^{2}+u(\xi ) r^{\ast } ( \xi,\eta)u+\psi+2u(\xi ) |u|^{2}\right]g \nonumber\\ & & + u(\xi)\left(u^{\ast}\right)^{2}f,\end{aligned}\ ] ] with @xmath61g - b$ ] . as usual , the self - trapped mode is linearly unstable if there is an eigenvalue with @xmath62 . as seen in fig .
[ fig : one](e ) , the two - beam in - bulk and surface self - trapped modes have a common stable region , with @xmath63 , at intermediate values of propagation constants @xmath38 .
the predicted stability of the modes has been verified in direct simulations of eq .
( [ eq : jia_1 ] ) with @xmath40 random noise added as an initial perturbation , see an example for @xmath35 in fig .
[ fig : one](f ) . .
( c , d ) the intensity profile and phase distribution for the three - beam surface self - trapped state at @xmath64 .
( e ) power @xmath36 of the self - trapped states versus @xmath38 .
( f ) the real part of the instability growth rate , @xmath65 , versus @xmath38 .
green triangles and the red circles correspond to the modes shown in ( a ) and ( c ) , respectively .
parameters are @xmath41 , @xmath42 , and @xmath43 . ]
adding more beams with phase shifts between them , one can construct composite vortices .
for an example , a composite vortex with the total phase circulation of @xmath66 may appear as a four - beam complex with the off - site vortex core in the center and the phase shifts @xmath67 between adjacent beams @xcite .
we have found that the composite vortices can exist in the system of the uniform setting . the intensity profile and phase distribution of a typical stable four - beam vortex in the uniform setting system
are shown in fig .
[ fig : two](a ) and ( b ) for @xmath64 . near the interface in of the composite setting system , there are no complete vortex modes , while there appear essentially new surface modes , in the form of _ half - vortices _ ( horseshoes `` ) , built of three beams , see figs .
[ fig : two](c ) and ( d ) .
the dependence of the power @xmath36 on the propagation constant @xmath38 shows that , although the half - vortex mode ( c ) is built of three beams , its power @xmath36 is larger ( near the first bloch band ) than that of the in - bulk vortex mode ( a ) , which is composed of four beams , see fig .
[ fig : two](e ) .
the linear stability analysis shows that there exists a common stability region at intermediate values of propagation constants @xmath38 for the in - bulk vortices and surface horseshoes '' , see fig .
[ fig : two](f ) . , @xmath68 ( b ) .
profiles of the 7-beam surface states : ( c ) at @xmath69 , ( d ) at @xmath68 .
( e ) power @xmath36 of the states versus @xmath38 .
( f ) the real part of the instability growth rate , @xmath37 , versus @xmath38 .
green triangles correspond to ( a ) and ( b ) , and red circles to ( c ) and ( d ) .
parameters are @xmath41 , @xmath42 and @xmath43 . ] on top of the simple few - beam self - trapped states , like the conservative 2d nonlinear systems @xcite , our @xmath0-symmetric systems can also support complex multi - beam ones built of up to 12 beams , see fig .
[ fig : three ] . due to the interaction between individual beams ,
their intensity is larger at the center of the structure , the intensity difference gradually vanishing with the increase of propagation constant @xmath38 . near the interface in the truncated system
, there are no beams located in the linear medium , while the near - interface beams become stronger , see fig .
[ fig : three](c , d ) .
power @xmath36 increases with propagation constant @xmath38 for both the in - bulk and surface self - trapped states , see fig .
[ fig : three](e ) . results of the linear - stability analysis for these states , displayed in fig .
[ fig : three](f ) , reveal a common stability region for the in - bulk and surface modes , at intermediate values of propagation constants @xmath38 . noise .
( a ) the density profile at @xmath70 evolves from the the in - bulk vortex self - trapped nonlinear waves in fig .
[ fig : two ] ( a ) .
( b ) the corresponding phase distribution for ( a ) .
( c ) the density profile at @xmath70 evolves from the in - bulk multi - beam self - trapped mode in fig .
[ fig : three ] ( b ) .
( d ) the density profile at @xmath70 evolves from the surface multi - beam self - trapped mode in fig .
[ fig : three ] ( d ) . ] by simulating the beam propagation with @xmath40 random noise , we have verified the stability of the vortex modes , as shown in fig .
[ fig : two ] , and of multi - beam ones , see fig .
[ fig : three ] . in particular , fig .
[ fig : four](b ) demonstrates that the phase distribution of the input vortex mode keeps the phase - winding structure .
thus , the direct simulations corroborate the predictions of the linear - stability analysis . in conclusion
, we have found several novel species of in - bulk and surface self - trapped states in 2d kerr - nonlinear optical systems with @xmath0-symmetric pls ( photonic lattices ) .
these include stable in - bulk localized vortices and surface half - vortices ( horseshoes " ) .
the self - trapped modes are related to truncated nonlinear bloch waves , the surface modes being linked with the truncated in - bulk ones .
along with the basic vortex and half - vortex states , which are built , respectively , of four and three constituent beams .
the stable multi - beam self - trapped states , composed of up to 12 constituents , have been found too .
the formation of these surface modes results from the interplay of the surface effects , nonlinearity , and the @xmath0-symmetry . due to the surface - enhanced reflection ,
the discrete diffraction is stronger in the direction perpendicular to the interface than in the direction parallel to it @xcite , therefore the surface modes feature stronger nonlinearity , which is necessary to balance the diffraction . |
the ac stark shift of the atomic ground state level and the phase shift of the transmitted field are both governed by the response of the two - level atomic system to the laser field , and in particular , the mean optical coherence induced in the ions .
we denote the lower and upper levels @xmath85 and @xmath86 , and we assume that we only excite the ions very weakly , i.e. , the ground state population @xmath87 to first order in the rabi frequency @xmath88 , where @xmath89 is the dipole matrix element and @xmath90 denotes the driving field amplitude .
this leads to the equation of motion for the optical coherence @xmath91 where the decoherence rate @xmath40 may include both radiative ( @xmath92 ) and non - radiative contributions , and where we have defined @xmath93 as the detuning , measured in radians per second .
the steady state solution reads , @xmath94 , and the resulting macroscopic polarization of the doped cantilever ( with an ion density @xmath55 ) is then given by @xmath95 , leading to the optical phase shift and energy @xmath96 analyzed in the article .
the purpose of this appendix is to analyze the case where @xmath7 performs oscillatory motion , @xmath97 , and the strain induced coupling leads to a temporally modulated detuning , @xmath98 . due to the long lifetime of the ion coherence and excited state
, we can not merely assume that the coherence @xmath99 attains the stationary solution @xmath100 , evaluated at the time dependent value of the detuning .
we can assume , however , that the ions reach a periodic steady state , and for weak modulation , we make the ansatz , @xmath101 applying this ansatz in , and isolating terms @xmath102 leads to the relations , @xmath103 , @xmath104 where @xmath105 . inserting these results in the equation for the non - oscilating terms in the periodic solutions to , we obtain , @xmath106 , which holds to first order in the small modulation amplitude of the detuning @xmath107 . if @xmath109 , and if @xmath110 , the expression simplifies , @xmath111 \simeq \frac{-\omega } { 2\delta_r(t)},\ ] ] and the coherence , indeed , follows the stationary solution adiabatically in this coupling regime . when @xmath112 is comparable to or larger than @xmath113 , we see a more complicated dependence of the real ( dispersive ) and imaginary ( absorptive ) part of the coherence , and also a weaker dependence on the position in the high frequency limit , where the coherence is unable to follow the time dependent steady state .
i. yeo , p .-
de assis , a. gloppe , e. dupont - ferrier , p. verlot , n. s. malik , e. dupuy , j. claudon , j .- m .
g ' erard , a. auffves , g. nogues , s. seidelin , j. poizat , o. arcizet , and m. richard , nat . nanotechnol .
9 , 106 ( 2014 ) . | by spectrally hole burning an inhomogeneously broadened ensemble of ions while applying a controlled perturbation , one can obtain spectral holes that are functionalized for maximum sensitivity to different perturbations .
we propose to use such hole burnt structures for the dispersive optical interaction with rare - earth ion dopants whose frequencies are sensitive to crystal strain due to the bending motion of a crystal cantilever .
a quantitative analysis shows that good optical sensitivity to the bending motion is obtained if a magnetic field gradient is applied across the crystal during hole burning , and that the resulting opto - mechanical coupling strength is sufficient for observing quantum features such as zero point vibrations .
the ability to control and probe physical systems at the quantum mechanical level has been demonstrated in a variety of examples , ranging from single photons and atoms , over currents and voltages in electronic circuits to the motion of mechanical devices .
many of these systems are promising candidates for sensitive and high precision measurements and for transmission , processing and storage of quantum information , but one single system is typically not adequate for all of these functions .
this has spurred the interest in so - called hybrid system @xcite which combine physical components which are separately optimized for different tasks . outside the technical challenge of handling physically very different systems in a single laboratory experiment
, the mismatch of their physical properties ( resonance excitation frequencies , spatial overlap , and coherence time scales ) presents a main obstacle against the efficient transfer of quantum states between them .
for instance , single atoms have microscopic dipole moments and interact only weakly with physical observables of mesoscopic quantum systems which occupy orders of magnitude larger spatial volumes . one successful remedy to this weak coupling is to use ensembles of many atomic particles with correspondingly increased coherent coupling strength .
this is , indeed , the rationale behind the use of atomic ensembles for optical interfaces and memories and of spin ensembles in conjunction with superconducting circuits @xcite . in this article , we address the application of rare - earth ion doped crystals for hybrid quantum technologies .
such crystals have found rich applications in quantum communication protocols where their strong inhomogeneous broadening is favorable for speed and bandwidth .
one particular hybrid technology which holds promise for an efficient coupling between radically different degrees of freedom relies on opto - mechanical interactions @xcite .
major achievements such as the ability to prepare a mechanical oscillator in the quantum ground state @xcite have spurred ambitious goals to prepare non - classical states of motion and use such systems in precision measurements and quantum information applications .
while mechanical oscillators can be coupled via light beams to other systems , such as atomic ensembles @xcite we propose a simpler set - up in which the bending motion of an inorganic crystal is coupled to the optical transitions in an ensemble of rare - earth ion dopants in the same crystal .
the crystal strain generated by the harmonic motion of a bulk mechanical oscillation provides an intrinsically stable coupling to the dopant optical properties as , contrary to other coupling mechanisms to externally imposed fields , it circumvents instabilities arising from drifts in position of field source and oscillator .
weak strain coupling to single emitters has been previously observed @xcite , but using a rare - earth ions ensemble , we benefit from their strong collective coupling as well as their narrow homogeneous linewidths and long coherence times , allowing unambiguous observations of quantum features .
we suggest to use spectral hole burning to prepare a transmission window with no optical absorption and obtain the opto - mechanical coupling through the off - resonant , dispersive interaction with ions with transition frequencies outside the hole . as a concrete application
, we will focus on @xmath0 ions in a @xmath1 host matrix as they exhibit record - narrow optical transitions among solid - state emitters . despite its long coherence time @xcite ,
the 580 nm optical transition @xmath2 @xmath3 in @xmath0 ions is sensitive to crystal strain @xcite , making it an attractive candidate system for strain - coupled optomechanics .
we investigate a crystal in the shape of a micro - meter scale cantilever whose bending produces a local strain - induced perturbation to the ions resonant frequencies .
schematics of a cantilever anchored in the foreground of the drawing , and with the far end ( a ) in its equilibrium position , and ( b ) bent upwards .
the colors on the cantilever indicate different values of the strain : green corresponds to a non - strained material , blue and red to compressive and tensile strain , respectively .
the white dashed lines indicate planes of equal strain , neglecting variations along @xmath4 .
the bottom graphs illustrate how a single emission line is shifted due to the strain ( arbitrary units).,width=302 ] during bending , there will be a strain gradient across the cantilever , ranging from tensile to compressive strain ( see fig .
[ peda_fig ] ) , and ions with identical frequencies in the unbent crystal ( a ) experience different frequency shifts in a bent cantilever ( b ) . in the latter case , ions located in three different layers ( subject to identical strain within each layer ) experience different strain and the emission lines shift accordingly : if the oscillator is bent upwards , emitters on the top face experience a compression , whereas emitters on the bottom face are subject to a tensile strain .
as the cantilever vibrates , the collective line shape periodically broadens and narrows , and the motion should be readily detected by an optical probe .
however , if static local perturbations of the emitters lead to additional inhomogeneous broadening with a very wide absorption line shape , the strain induced frequency shifts can not be probed dispersively . to circumvent this
, we propose to prepare the ion ensemble by burning of narrow spectral holes with the cantilever at rest , and subsequently use their optical response to the cantilever motion .
spectral holes are formed by resonant optical excitation and decay of the ions into other long - lived ( dark ) states , typically other hyperfine state within the electronic ground state manifold .
due to the narrow homogenous linewidths , one can scan the excitation laser over a finite frequency interval and burn a spectral hole which must be narrower than the ground state hyperfine splitting to avoid repopulating the hole by subsequent excitation of the dark ions .
a weak optical field with a frequency at the center of the hole burnt interval will subsequently interact dispersively with the ions with frequencies outside the spectral hole , and we will in the following propose a method to make this dispersive coupling sensitive to the cantilever motion . as indicated by the example in fig .
[ peda_fig ] , bending of the crystal shifts the transition frequency by an amount proportional to the strain experienced by the individual ions .
this implies that , to lowest order , the weak probe field is not sensitive to the bending , unless we can ensure an asymmetric response of the ion frequency distribution .
such an asymmetry can be prepared by a particular hole burning protocol as discussed in the following sections . for our analysis
we assume a spatially homogeneous ion distribution , and we also consider the frequency distribution due to the inhomogeneous broadening to be constant within a wide detuning range .
we assume that the bending of the beam shifts the resonance frequency of all ions by a quantity proportional to the strain at their position in the crystal .
the bending of the cantilever resonator with thickness @xmath5 and length @xmath6 is represented by a single collective coordinate @xmath7 , measured as the vertical displacement of the cantilever tip shown in fig .
[ peda_fig ] .
the ions are all located near the anchoring of the cantilever with spatial coordinates @xmath8 , where @xmath9 is in the same direction as @xmath7 and runs from @xmath10 to @xmath11 , and @xmath4 is along the beam axis , and takes values from @xmath12 to @xmath6 . from the euler - bernouilli theory of beams , we know that the local strain of the matrix around ion @xmath13 is proportional to both @xmath7 , @xmath9 and @xmath14 .
we optically address only atoms near the base of the cantilever , and we therefore neglect the @xmath4 dependence , so the strain induced frequency shift reads @xmath15 , with @xmath16 a constant depending on the beam geometry and physical properties of the doped material .
to enhance the position sensitivity of the cantilever , we prepare our system by applying conventional spectral hole burning while the unbent crystalline cantilever is subject to a magnetic gradient .
we assume a convenient choice of light polarization and magnetic field directions with reference to the crystalline axes so that all ions are shifted by the linear zeeman effect such that the overall transition frequency of the @xmath17 ion at position @xmath8 can be written as @xmath18 where @xmath19 is the chosen central frequency for our hole burnt structure , @xmath20 is the intrinsic inhomogeneous frequency shift of the @xmath17 ions ( without the effect of cantilever motion or magnetic field ) , @xmath21 is the externally applied magnetic gradient and @xmath22 the linear zeeman effect sensitivity .
we finally also assume that @xmath20 and @xmath9 are uncorrelated .
our functionalized hole burning procedure consists of two steps : * 1 ) * apply a static magnetic gradient @xmath23 and scan the hole burning laser across the spectral range @xmath24 $ ] within the inhomogeneous linewidth . the value of @xmath25 can be chosen arbitrarily , but must fulfill the condition @xmath26 .
this step is represented in fig .
2a . *
2 ) * apply the reverse static magnetic gradient @xmath27 and scan the hole burning laser across the interval @xmath28 $ ] .
this step is represented in fig .
2b . as shown in fig .
2 , after the magnetic gradient has been switched off , we obtain an inhomogeneous absorption profile of the ions with a transmission hole around the central frequency @xmath19 , and with a width that depends on the location @xmath9 in the cantilever but is on average close to @xmath29 .
we are then ready to probe and interact with the system by transmission of a weak laser beam with frequency @xmath19 , at the center of the hole . in fig .
2c , we show how the frequency shift @xmath15 due to bending of the cantilever transforms the shape of the transmission window .
in particular , we note that our hole burning procedure ensures a non - vanishing phase shift of the probe beam due to changes in @xmath7 , because contributions from ions with positive @xmath9 are not canceled by the ions with negative @xmath9 as they are further detuned . at a finite temperature , corrections to the hole shape
arise because the thermal brownian motion of the cantilever amplitude during hole burning exposes the whole inhomogeneous profile to strain distortions like the one shown in fig .
2c .
these distortions lead to excitation and decay into dark states of those ions that are brought into resonance with the hole burning laser during their random brownian motion .
we estimate the consequences of this mechanism by assuming that ions with frequencies within @xmath30 of the spectral hole in fig .
2a are also transferred to dark states , where @xmath31 is of same order of magnitude as the root - mean - square thermal excursions of @xmath7 .
fig .
2d shows how the spectral hole is modified by the thermal brownian motion under hole burning and effectively excludes ions at vertical position @xmath9 that are detuned with respect to @xmath19 by an amount @xmath32 $ ] , where @xmath33 and @xmath34 for positive ( negative ) values of @xmath9 . to describe the interaction between the probe laser field and the hole burnt ensemble
, we use a semi - classical approach based on the maxwell - bloch equations , which , on the one hand , accounts for the state of the individual atoms by solution of the quantum optical master equation and , on the other hand , describe the damping and dispersion of the maxwell electric field amplitude due to the interaction with the mean atomic dipole density . due to the spectral hole ,
the laser is far off - resonant from all atomic transitions , and we are in the dispersive and perturbative regime of linear excitation of the atoms .
the cantilever is a mechanical resonator of eigenfrequency @xmath35 , for which we assume a small modulation amplitude . under this assumption , and imposing the extra condition that @xmath36 for adiabatic following ( see appendix ) , the phase shift of the probe field with wavelength @xmath37 and cross section @xmath38 interacting with a two - level system with detuning @xmath39 and radiative decay rate @xmath40 is given by @xmath41 per atom .
the phase change @xmath42 of the classical ( coherent state ) field amplitude is equivalent to a phase change of the same value occurring on each single photon due to the presence of the ion .
the combined state of the ion and the stream of photons thus accumulates a quantum phase @xmath43 that changes at a rate @xmath44 , where @xmath45 ( @xmath46 ) is the rate of photons passing through @xmath38 with intensity @xmath47 .
the ion is left in its ground state and the photons leave with unchanged frequency , and the time evolving phase due to the light - matter interaction is equivalent to the light induced ac stark shift of the ion due to the laserfield , @xmath48 where @xmath49 . the expression for @xmath50 has , indeed , the familiar form @xmath51 , valid for large detunings .
we note that we applied a simple two - level description and to evaluate the accurate value of the optical phase shift and atomic energy shift , one must specify the actual transition , the light polarization , and also the index of refraction of the host material .
since the ion transition frequencies are modified by the crystal strain , their ac stark shifts become functions of @xmath7 as well as a of the optical intensity , i.e. , we obtain the desired optomechanical coupling . the total energy shift of all ions - due to the dispersive light matter interaction , is the sum of eq.([phaseshift ] ) over all ions .
schematics of the result of the hole burning procedure on the ground state ion distribution as a function of the ion position and `` intrinsic '' inhomogeneous frequency detuning @xmath39 .
the lines denote the border between the zones with large @xmath52 values where the ions are left in the ground state , and the central zones where they transferred to the dark state during burning . in the first step of the hole burning procedure ( a )
, the hole is burned in the @xmath24 $ ] interval , indicated with dashed lines , while the magnetic gradient @xmath23 is on .
the solid lines show the hole after the magnetic gradient is turned off , realizing the `` right '' edge of the final hole .
the second step ( b ) realizes the `` left '' edge by hole burning the interval @xmath28 $ ] while the magnetic gradient @xmath27 is on .
the solid lines show the final hole after the 2 steps are completed and the magnetic gradient is turned off .
panel ( c ) shows the asymmetric effect of bending after burning both edges ( in black , the unbent cantilever , and in red , the cantilever bent by a positive amount @xmath7 ) .
panel ( d ) shows the effect on the hole due to thermal brownian motion during the burning procedure : dashed lines correspond to burning at zero temperature and solid lines to burning with finite brownian motion of the cantilever .
, width=302 ] to evaluate the dependence of the total interaction energy on the collective cantilever coordinate @xmath7 , we integrate the contributions over the thickness @xmath5 of the cantilever and the corresponding distribution of @xmath53 from all ions outside the spectral hole : @xmath54.\end{gathered}\ ] ] here , @xmath55 is the density of emitters per unit of frequency and unit vertical distance , and the interaction limits which correspond to the edges of the spectral holes are expressed in terms of the functions defined previously in the text . by applying a first order taylor expansion of @xmath56 in @xmath57 ,
the integrals can be readily evaluated and we obtain @xmath58.\end{gathered}\ ] ] note that for temperatures below 100 mk during hole burning , the brownian motion is negligible , @xmath59 , and the expression for @xmath60 simplifies , and a third order taylor expansion in @xmath61 yields @xmath62 the interaction strength given by this expression is explicitly proportional to @xmath7 and @xmath47 as desired for the opto - mechanical coupling .
note that it is also proportional to the bias magnetic gradient @xmath23 , applied during hole burning .
the parameter @xmath23 thus serves as an ( adjustable ) gain coefficient for the optomechanical coupling . to obtain a maximum coupling strength , it is desirable to match the bias magnetic gradient to the width of the spectral hole @xmath63
. the coupling of the lightfield with the emitters causes an ac stark shift that depends on the bending of the cantilever and hence results in an displacement @xmath64 of the resonator equilibrium position given by @xmath65 where @xmath66 is the angular frequency of oscillation of the resonator .
conversely , the coupling between the light field and the emitters produces a phase shift on the laser .
the same proportionality applies between the optical phasehift and interaction energy for the whole system as for a single ion , and we can thus directly write the phase shift of the optical beam as @xmath67 where the dependence on the intensity @xmath47 cancels due to the explicit proportionality between @xmath60 and @xmath47 in the dispersive coupling regime .
this phase - shift can be detected interferometrically , which provides a readout of the position of the resonator .
in particular , for the light intensity that leads to the displacement expressed in eq . [ staticdisplacement ] , the corresponding phase shift of the carrier optical frequency is @xmath68 .
the vibratory movements , on the other hand , can be detected on the transmitted laser field by observing spectral sidebands at the mechanical oscillator frequency @xmath69 from the carrier .
thermal excitation of the cantilever motion yields a total integrated power of sideband phase fluctuations equal to @xmath70 , which converges to the value @xmath71 at zero temperature . to detect such sidebands
requires their amplitude to exceed the shot noise limit for the given optical power and integration time ( measurement bandwidth ) .
the resolution of the phase fluctuation is thus ultimately limited by photon absorption and transfer of ions to other long lived states , which gradually `` overburns '' the spectral hole and destroys it . the transition line in eu : yso , however , has a very small natural linewidth @xmath40 and therefore , even for high optical power , overburning will not occur before a typical time @xmath72 , which sets the limit for the integration time .
increasing optical power within the integration time @xmath73 can lead to arbitrarily high resolution , although of course practical consideration ( heat dissipation , etc ... ) limits the maximum power to a few mw .
further improvement of the resolution can be obtained by repeating the experiment ( erasing , re - imprinting and probing the spectral hole ) and averaging the successive results . as an example we consider a single - clamped cantilever with the dimensions @xmath74 interacting with a laser beam traversing the cantilever near its fixed end for maximum strain as illustrated in fig .
[ holes ] .
we consider a cantilever which consists of @xmath1 ( young modulus of 135 gp ) with an effective mass @xmath75 kg , and of which the first excited mechanical mode vibrates at @xmath35= 890 khz .
the cantilever contains a 0.1 % doping of @xmath0 ions , with a @xmath2 @xmath3 transition centered at 580 nm and with @xmath76 hz ( and @xmath77 typically @xmath78 khz ) at t=3 k. we choose a power of 1 mw and a hole width of @xmath29=6 mhz to fulfill the condition for large detuned adiabatic following , see appendix . to calculate the proportionality constant @xmath16 we use the values from @xcite -211.4 hz / pa for crystal site 1 , which is the most sensitive of the two non - equivalent sites .
the sensitivity to magnetic field is @xmath79 khz / g @xcite . in order to maximize the coupling while fulfilling the condition @xmath80 we choose @xmath81 t / m , which seems possible to achieve in a cryogenic environment over short distances ( mm or less ) using small superconducting coils . in this configuration , the static displacement of the tip of the resonator due to the light field amounts to @xmath64=0.4 pm and
the corresponding phase shift of the laser ( the carrier ) equals 0.2 mrad .
this shift is easily observable as , for the 1 mw laser power , the shot noise limited phase resolution is 0.45 @xmath82rad within the allowed detection time , before hole - overburning becomes non - negligible ( approximately 16 ms for the 122 hz linewidth ) .
each of the spectral sidebands due to brownian motion at 3 k contains an integrated phase of 0.11 mrad , while the phase resolution for an integration time equal to the inverse of the thermal linewidth ( 25 @xmath82s ) is 14 @xmath82rad , hence the sidebands are readily observed .
moreover , by increasing the integration time , it is possible to observe the detailed shape of the brownian motion induced sidebands .
the sidebands due to zero point fluctuations contain a total integrated phase of 0.4 @xmath82rad , a value close to the phase resolution corresponding to the maximum integration time before hole - overburning .
the shot noise limited resolution is therefore sufficient to observe the effect of the quantization of the mechanical resonator .
note that the resolution can be further increased by repeating the full hole imprinting and measurement sequence several times , or use optical repumpers to preserve the spectral hole .
the effect of the zero point fluctuations can be observed by measuring the total integrated phase fluctuations in the sidebands and/or their amplitude and linewidth at a given temperature . at the shot - noise limited highest resolution achievable ,
the few percent deviation from the purely brownian - motion induced sidebands due to zero - point fluctuations can be observed even at 3 k temperature .
several measurements at various temperatures may also be used to estimate the various necessary parameters ( exact temperature of the resonator , q factor , ... ) within a few percent accuracy .
furthermore , we emphasize that even though , for technical simplicity , we evaluated the different parameters at 3k , a dilution fridge with temperature around 0.1k or lower can also be used , reducing the thermal fluctuations in order to make quantum features even more easily discernible .
a last potentially perturbing effect arises due to fluctuations of the laser power .
as the static displacement corresponding to 1mw of laser power is approximately 0.4 pm , this laser power must be stable to within @xmath83 10@xmath84 to ensure a perturbation much smaller than the zero point fluctuations ( @xmath83 1 fm ) , a power stability requirement well within reach of standard stabilization techniques . in summary
, we have proposed a hybrid scheme which provides an efficient optomechanical strain based coupling between a mechanical resonator and an ensemble of narrow linewidth ions . the coupling mechanism via a functionalized spectrally burnt hole does not suffer from inhomogeneous broadening of the emitters , nor from the inhomogeneity of the strain across the resonator during displacement .
it is even possible to further increase the coupling by exploiting not just one but an ensemble of functionalized spectral holes within the inhomogeneous linewidth , which can be obtained by using a comb of optical frequencies .
the narrow linewidth and dispersive nature of the coupling allows use of a relatively large optical power without destruction of the functionalized hole and observation of the cantilever motion at the quantum level .
all authors thank stefan krll and philippe goldner for discussions , and ss thanks daniel estve for help and discussions .
km acknowledges support from the villum foundation , ylc from the ville de paris emergence program and ss from la r ' egion rhne - alpes ( cmira explora - pro ) . |
there are several reasons to believe that a population of intergalactic globular clusters ( igcs ) should exist outside of galaxies : \(1 ) the jeans mass at recombination was @xmath0 solar masses , and hence globular cluster sized objects could have formed wherever the local density of matter was high enough .
\(2 ) many galaxies may have met their demise over a hubble time as a result of collisions and tidal disruption .
globular clusters are likely to survive the disruption of their parent galaxy , resulting in the gradual accumulation of a population of igcs . intergalactic stars , planetary nebulae , supernovae and hii regions have already been found ; it would be surprising if there were no igcs .
\(3 ) the existence of igcs might explain high specific frequencies , bimodal globular cluster metallicity distributions and other current puzzles in the study of globular cluster systems .
jordn et al .
( 2003 ) reported a tentative detection of igcs in the center of the rich galaxy cluster a1185 ( @xmath1 ) based on @xmath2-band images obtained with wfpc2 on the hubble space telescope .
we ( ct , jordn , marzke , west ) recently obtained very deep , multicolored ( @xmath3 and @xmath2 ) images of the same a1185 field using hst with the new acs .
the goals of these new observations are to 1 ) detect the peak of the assumed universal gaussian - like globular cluster luminosity function ( which should occur at @xmath4 at a1185 s distance ) and thereby confirm that these candidate igcs are bona fide globular clusters and 2 ) use color information to infer their metallicities .
preliminary analysis indicates that we are reaching sufficiently faint magnitudes to reliably detect the luminosity function turnover .
the number and colors ( metallicities ) of igcs will provide constraints on the number and types of galaxies that have been destroyed or stripped over a hubble time . using the keck telescope , we ( ferguson , gregg , tanvir , von hippel , west ) recently measured the redshift of a candidate igc in the nearby virgo galaxy cluster that was found serendipitously on an hst image obtained for another project .
preliminary data reductions show that this object , which is slightly resolved in the hst image and appears to be a distant globular cluster , has a recessional velocity of @xmath5 km / s , and hence is most likely in the virgo cluster .
its apparent magnitude , @xmath6 , is consistent with it being a bright globular cluster .
using telescopes on mauna kea we have since obtained optical and nir colors of this object , as well as a medium - resolution spectrum that should yield its velocity dispersion .
these data are presently being analyzed . | we confirm and extend our previous detection of a population of intergalactic globular clusters in abell 1185 , and report the first discovery of an intergalactic globular cluster in the nearby virgo cluster of galaxies .
the numbers , colors and luminosities of these objects can place constraints on their origin , which in turn may yield new insights to the evolution of galaxies in dense environments .
# 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in |
* model parameters . * the single fermion hamiltonian eq .
( [ fh ] ) is obtained by adiabatically integrating out all the electronic excitation states of the atoms in the rotating wave frame .
the parameters in eq .
( [ fh ] ) are related to the experimental tunable parameters as @xmath116 , @xmath117 and @xmath118 . here
@xmath119 is the pumping field strength , @xmath120 is the pumping laser frequency detuning with respect to electronic transitions of atoms , and @xmath121 is the coupling strength between the cavity mode and the fermions . * mean field equation for fermi superfluids . *
when the lattice induced by the pumping field is not strong , we can approximate @xmath122 as a constant .
the mean field gap equation becomes @xmath123 together with the number equation @xmath124 , or more explicitly , @xmath125 we can determine @xmath41 and @xmath126 self - consistently for a given pumping strength @xmath89 and given density @xmath7 .
* instability condition for superradiant phase transition .
* the mean field value of the cavity field @xmath127 satisfies @xcite @xmath128 where @xmath56 is the fermion density order parameter . the introduced decay rate @xmath62 is to model the weak leakage of electromagnetic field from the high-_q _ cavity .
in a steady state , @xmath129 ; we have @xmath130 which locks the cavity field to the fermion density order parameter .
both @xmath131 and @xmath132 are zero in the normal phase and become nonzero in the superradiant phase . to the second order of @xmath131
, the effective free energy can be obtained as @xmath133 where @xmath134 with a specified @xmath131 . by substituting ( [ mean_alpha ] ) into eq .
( [ fa ] ) , we have @xmath135\eta_0 ^ 2\theta^2,\label{freeenergy}\end{aligned}\ ] ] which determines the superradiant transition when the quadratic coefficient of @xmath132 changes its sign .
* explicit expression for density - wave order susceptibility . * the explicit expressions for the density - wave order susceptibility within the bcs theory are @xmath136 in the bsc limit , the factor @xmath137 with @xmath138 the fermi - dirac distribution ; @xmath139 becomes the same as it is for free fermions @xcite . in the bec limit , @xmath140 and @xmath141 , @xmath142 which is the same as it is for condensed noninteracting bosons @xcite .
* determination of phase boundary .
* the boundary between the non - superradiant and superradiant phases is determined by eq .
( [ cri ] ) .
since @xmath143 , @xmath144 , and @xmath87 is a dimensionless function of dimensionless parameters @xmath89 and @xmath98 , we could recast eq .
( [ cri ] ) in the form @xmath145 by introducing @xmath146 .
we take typical experimental values @xmath147 and @xmath114 .
thus at each given pumping strength @xmath89 we can obtain the critical strengths of the cavity detuning @xmath63 . | in this letter we consider the superradiant phase transition of a two - component fermi gas in a cavity across a feshbach resonance .
it is known that quantum statistics plays a crucial role for the superradiant phase transition in atomic gases ; in contrast to bosons , in a fermi gas this transition exhibits strong density dependence .
we show that across a feshbach resonance , while the two - component fermi gas passes through the bec - bcs crossover , the superradiant phase transition undergoes a corresponding crossover from a fermionic behavior on the weakly interacting bcs side , to a bosonic behavior on the molecular bec side .
this intricate statistics crossover makes the superradiance maximally enhanced either in the unitary regime for low densities , in the bcs regime for moderate densities close to fermi surface nesting , or in the bec regime for high densities .
recent experiment has combined atomic bose - einstein condensates and cavity quantum electrodynamics together where atom - light interactions are strongly enhanced @xcite . a superradiant phase transition driven by external pumping field has been observed , across which atoms form a density - wave order @xcite , and roton mode softening has been found in the vicinity of this superradiant phase transition @xcite .
theoretical studies have extended to investigate noninteracting fermi gases inside a cavity @xcite .
it is shown that the fermi statistics plays a crucial role in the superrandiant phase transition at moderate and high densities @xcite . at moderate densities
, fermi surface displays a nesting structure and strongly enhances superradiance , when the momentum of the cavity light field matches the nesting momentum .
while at high densities , pauli blocking effect forbids a large part of atom - light scattering processes , and consequently , strongly suppresses superradiance .
the strong density dependence marks the major difference between superradiances in ideal fermi gases and bose gases . during the past decade ,
another important development in cold atom physics is the study of strongly interacting two - component fermi gases and the bec - bcs crossover utilizing feshbach resonance @xcite .
the inter - atomic @xmath0-wave scattering length @xmath1 can be continuously changed by a feshbach resonance , and the dimensionless parameter @xmath2 ( @xmath3 is the fermi momentum in the noninteracting limit ) controls the bec - bcs crossover . in the bcs limit of the crossover @xmath4 , fermions form loosely bound cooper pairs and the low - energy response
is dominated by fermionic quasi - particles ; the system recovers a noninteracting fermi gas . in the bec limit
@xmath5 , cooper pairs transform into tightly bound bosonic molecules , and the system responses to external fields mainly as bosons . in between ,
when @xmath1 is so large that @xmath6 , the system is in a strongly interacting regime and its response shall exhibit both bosonic and fermionic characters .
so far , fermi gases with inter - atomic interactions in a cavity have been barely studied . in this work
we consider a two - component fermi gas in a cavity across a feshbach resonance .
given that the gas can be continuously tuned between the fermion limit and the boson limit , and that atoms with different statistics have been shown to behave differently in the superradiant phase transition @xcite , the motivation of our study is to address how the statistics crossover manifests itself in the superradiant phase transition across a feshbach resonance , and the physical consequence of this crossover .
in experiments , the superradiant phase transition is usually driven by increasing the strength of pumping field . in this work
we will reveal nontrivial dependence of the critical pumping strength on the density of fermions @xmath7 and the inter - atomic interaction strength characterized by @xmath2 .
our results represent a manifestation of the interplay between strong interactions from feshbach resonance and strong atom - light coupling in a cavity , and will provide insight for future experiments . _ model . _
our system is schematically shown in fig .
( [ setup ] ) . applied on the fermi gas is a pumping field that consists of two laser beams counter - propagating along the @xmath8 direction , with frequency @xmath9 and polarization in the @xmath10 direction .
the single - mode cavity field of interest varies in the @xmath11 direction , with frequency @xmath12 close to @xmath9 .
the system is described by the hamiltonian @xmath13 , where @xmath14 is the field operator for the cavity mode and @xmath15 is the cavity field detuning .
direction shown by the red arrows .
the cavity field is represented by the wiggled lines in the @xmath16 direction .
fermions of different spins are shown in different colors .
, width=226 ] the hamiltonian experienced by the fermions has two parts @xmath17 .
the free fermion part is @xcite @xmath18 where @xmath19 are the fermion field operators with ( peudo ) spin index @xmath20 .
the pumping field and the cavity field generate respectively the optical potentials @xmath21 and @xmath22 , and the coupling between the pumping field and the cavity field comes from an interference term @xmath23 with @xmath24 , @xmath25 is the wavevector magnitude of both the pumping field and the cavity mode @xcite .
the recoil energy @xmath26 is defined for latter use .
the inter - atomic interaction nearby a feshbach resonance is described by the hamiltonian @xmath27 the bare attractive inter - fermion interaction coupling @xmath28 is renormalized to the @xmath0-wave scattering length @xmath29 via @xmath30 with @xmath31 the momentum cutoff .
this attractive interaction between fermions lead to fermion pairing and a fermi superfluid ground state .
_ ground state in non - superradiant phase .
_ before entering the suprradiance phase , @xmath32 , fermions only experience a one - dimensional lattice @xmath33 along the direction of the pumping field , and the single - particle eigenstates are the bloch states @xmath34 satisfying @xmath35 . by expanding @xmath36 with @xmath37 and @xmath38 the gas volume , we introduce fermion pairing order parameter @xmath39@xmath40 . here
we assume the lattice @xmath33 is weak and we have ignored pairing at non - zero crystal momentum . with this assumption , the order parameter @xmath41 is determined by the gap equation and the number equation @xcite . in this fermi superfluid state ,
the single - particle green s functions are given by @xmath42 @xmath43 with @xmath44 and the fermionic matsubara frequencies @xmath45 for @xmath46 , and @xmath47 the inverse of temperature .
their diagrams are shown in fig .
( [ diagram])(a1 ) .
the components @xmath48 and @xmath49 describe the propagation of particles and holes , respectively , while @xmath50 and @xmath51 are the anomalous green s functions proportional to the pairing gap @xmath41 which we take to be real . , title="fig:",width=207 ] + _ condition for superradiant phase transition . _
the superradiant phase transition is determined by the instability of non - superradiance toward developing non - zero @xmath52 . as shown in ref .
@xcite , the superradiant phase transition occurs simultaneously with the formation of density - wave order of atoms with momentum @xmath53 , where @xmath54 is the momentum transfer between the cavity field and the pumping field .
that is to say , @xmath55 is proportional to the density - wave order parameter @xmath56 with @xmath57 . by integrating out the fermion fields ,
one can obtain the free - energy of the system in the form @xmath58 @xcite , where @xmath59 changing sign from positive to negative gives the critical pumping field strength @xmath60 for the superradiance transition @xcite @xmath61 here @xmath62 is the cavity mode decay rate , and @xmath63 is the shifted cavity mode detuning @xmath64 , which is assumed to be red - detuned ( @xmath65 ) . the most essential quality determining
this transition is the density - wave order susceptibility of the fermi superfluid defined as @xmath66.\end{aligned}\ ] ] here @xmath67 includes the integration of the spatial coordinates and the imaginary times , @xmath68 is the time ordered operator .
the expectation value of the fermion operators @xmath69 is taken in the non - superradiant fermi superfluid phase .
a larger @xmath70 means that the fermi gas has stronger tendency toward forming a density - wave order at a momentum @xmath53 , and it is easier for the fermi gas to enter the superradiant phase ; in another word , the critical pumping strength shall be smaller .
_ density - wave order susceptibility .
_ in order to capture both fermionic and bosonic responses of a fermi superfluid , the density - wave order susceptibility @xmath70 should be calculated by the random phase approximation .
this approximation maintains conservation laws @xcite and guarantees that one can recover the free fermion and the free boson results in the limits @xmath71 , respectively . within this approximation
, we have @xmath72 and @xmath73 where @xmath74 is the mode factor for cooper pair fluctuations .
the fermionic response @xmath75 is due to that the cavity field couples to the fermonic excitations of the fermi superfluid by breaking up cooper pairs
. the feynman diagrams corresponding to @xmath75 are shown in fig .
( [ diagram])(b ) .
the diagrams describe the process that a fermion with momentum @xmath76 is scattered to momentum @xmath77 , where the momentum transfer comes from the photon momentum change from the pumping field to the cavity field , as denoted by the vertex in fig .
( [ diagram])(a2 ) . since all fermions are paired in the fermi superfluid phase , this process must be accompanied by pair breaking . in the bcs limit where the pairing gap @xmath41 is small and pairs
are easy to break , @xmath75 is dominant in @xmath70 and could recover the transition for free fermions in the limit of vanishing pairing gap @xcite .
while in the bec limit this process is strongly suppressed because of large pairing gap . the bosonic response @xmath78 originates from the process that the cavity field excites nonzero momentum cooper pairs and corresponds to the diagram shown in fig .
( [ diagram])(c ) . in this process
, one of the two fermions in the cooper pair , say , the one with momentum @xmath76 , is scattered to momentum @xmath79 by a photon .
thus , the cooper pair acquires a finite momentum @xmath80 and propagates with this fixed momentum @xmath80 ( up to a reciprocal lattice vector along @xmath8 ) .
after another scattering with a photon , the cooper pair returns to zero - momentum . because of weak lattice @xmath81 we only take into account the contributions from the scattered cooper pairs of momentum @xmath82 .
the cooper pair propagator @xmath83 is given in eq .
( [ chib3 ] ) and its diagram in fig .
( [ diagram])(c ) which is a summation of ladder diagrams .
there are two ways for a cooper pair to propagate , through multiple scattering and through vacuum fluctuations , respectively . both
are included in eq .
( [ chib3 ] ) and in the bottom of fig .
( [ diagram])(c ) . in the bec limit
@xmath78 is dominant in @xmath70 and @xmath84 recovering the free boson result . while in the bcs limit
, @xmath85 is exponentially suppressed @xcite . ,
@xmath86 and @xmath87 vs @xmath88 are plotted in ( a ) , ( b ) , ( c ) respectively with the pumping strength @xmath89 fixed at @xmath90 and @xmath91 taking @xmath92 , @xmath93 and @xmath94 . the bottom row is a pictorial representation of the bec - bcs crossover .
[ fffbf],width=264 ] we plot in fig .
( [ fffbf ] ) the dimensionless susceptibility @xmath95 , as well as its fermionic and bosonic constituent @xmath96 and @xmath97 , as functions of the bec - bcs crossover controlling parameter @xmath2 , for different filling fractions @xmath98 .
first , fig .
( [ fffbf])(a ) shows that @xmath99 exhibits strong density dependence on the bcs side . around a moderate density of @xmath100 ,
the fermi surface nesting is optimal , and @xmath99 becomes much larger than the low - density limit value @xmath101 @xcite .
this is the regime where fermi surface nesting strongly enhances superradiance , as discussed in noninteracting fermi systems @xcite . on the other hand , for high densities ,
say , @xmath102 in fig .
( [ fffbf])(a ) , @xmath99 is much smaller than @xmath103 on the bcs side .
this is the regime where the pauli exclusion principle strongly suppresses superradiance . as approaching the bec side ,
@xmath99 is strongly suppressed for all densities .
second , as shown in fig .
( [ fffbf])(b ) , @xmath104 approaches the value of noninteracting bosons ( also @xmath105 ) in the bec limit , independent of densities . while on the bcs side , for all densities @xmath104 is strongly suppressed .
figure ( [ fffbf])(c ) shows the central result of this work .
the total @xmath87 exhibits different features for different densities as @xmath88 varies .
the most intriguing case is at relatively low - densities , say , @xmath106 , where @xmath87 displays a maximum in the unitary regime ( @xmath107 ) .
this maximum is because in this regime , the bosonic contribution already takes off while the fermionic contribution has not damped out . while for moderate densities of fermi surface nesting regime , @xmath87 monotonically increases as @xmath2 increases from the bec limit to the bcs limit , due to the fermi - surface nesting enhancement of @xmath70 on the bcs side .
in contrast , for high densities , @xmath87 monotonically decreases , due to the pauli blocking suppression of @xmath70 on the bcs side .
the total @xmath87 has strong density dependence on the bcs side where it is dominated by the fermionic behavior , and becomes less and less sensitive to density in the bec limit where it is dominated by the bosonic behavior .
this change of @xmath87 with @xmath2 between the two limits is the manifestation of statistics crossover in superradiance .
_ phase diagram . _
the boundary separating the normal and the superradiant phases can be obtained by solving eq .
( [ cri ] ) @xcite . in fig .
( [ pd ] ) , we plot the phase diagram in term of @xmath108 and @xmath109 for different densities and interaction strengths . in the bcs region ,
fig .
( [ pd ] ) ( a ) shows that the moderate density @xmath110 is the easiest to be superradiant . in the unitary region as shown in fig .
( [ pd ] ) ( b ) the low density @xmath111 is the easiest to be superradiant primarily due to the maximum of @xmath87 mentioned above in this part of the parameter space . on the bec side ,
fig .
( [ pd ] ) ( c ) shows that the density dependence diminishes since it shall be washed out completely in the bec limit .
( a ) , @xmath112(b ) and @xmath113 ( c ) , and with different densities @xmath111 , @xmath93 and @xmath94 .
for all cases , we take the typical experimental parameters @xmath114 and @xmath115.,width=302 ] _ conclusion . _
we have presented basic features of the superradiant phase transition of two - component fermi gases across a feshbach resonance .
the main results are : i ) on the bcs side of resonance the superradiant phase transition shows strong density dependence , similar as noninteracting fermi gas ; while on the bec side it gradually becomes density independent , similar as noninteracting bosons .
ii ) superradiance is mostly enhanced in the unitary regime for low density , in the bcs regime for moderate density , and in the bec regime for high density . in this work ,
we have only focused on the superradiant phase transition itself . inside the superradiant phase
, the additional lattice due to the cavity field will further modify the single - particle dispersion , which will feedback to the fermi superfluid .
furthermore , the quantum fluctuation of the cavity field will also generate additional effect on the fermi superfluid .
the properties of fermi superfluids in the superradiant phase would be a subject for future studies .
_ acknowledgements .
_ this work is supported by tsinghua university initiative scientific research program , nsfc under grant no . 11004118 , no . 11174176 , no . 11104157 , no .
11474179 and no . 11204152 , and nkbrsfc under grant no .
2011cb921500 .
j. mckeever , a. boca , a.d .
boozer , j.r .
buck and h.j .
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see supplementary material for ( i ) relations between model parameters and experimentally tunable parameters ; ( ii ) details of gap and number equation ; ( iii ) derivation of free - energy in term of order parameter ; ( iv ) asymptotic expression of @xmath75 in the bcs limit ; ( v ) asymptotic expression of @xmath78 in the bec limit ; ( vi ) details of solving the equation for phase boundary .
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we consider a two - dimensional configuration of @xmath0 particles with @xmath105 contacts and @xmath57 polygons . for convenience of notation ,
only single digit particle indices are used in this example , so that the notation @xmath106 means the cartesian @xmath10 component of the unit vector from the center of particle @xmath107 to that of particle @xmath108 . + and @xmath104 matrices are shown .
arrows represent the normal vectors used to construct the @xmath19 and @xmath104 matrices ( before normalization ) .
different arrow colors are for visualization purposes only . ]
the convention for ordering of the contacts is demonstrated in eq .
[ eq : c ] ( and see also fig .
[ fig : m_configuration ] ) : @xmath109 the @xmath19 matrix is used to describe the force balance condition ( eq . 1 in the main text ) and has dimension @xmath110 in the most general case when contact forces have both normal and tangential components .
each row is associated with a given particle @xmath21 and each column describes one contact and has non - zero entries corresponding only to the pair of particles @xmath21 and @xmath22 forming that contact .
its first @xmath0 rows store the @xmath10 components and the next @xmath0 rows store the @xmath11 components of unit normal vectors @xmath111 and unit tangential vectors @xmath112 ( counter - clockwise orthogonal to @xmath111 ) .
the first @xmath105 columns of @xmath19 correspond to the normal directions and the next @xmath105 columns correspond to the tangential directions ( which can also of course be expressed using the normal directions via a simple rotation transformation ) .
an example of some of the terms of the @xmath19 matrix for the configuration of fig .
[ fig : m_configuration ] is given in eq .
[ eq : m ] : the @xmath104 matrix is used to describe the torque balance condition ( see eq . 9 in the main text ) and is of dimensions @xmath114 . again
, the row indices correspond to particles and the column indices refer to contacts .
the non - zero entries in each column correspond to the radii of particles @xmath21 and @xmath22 forming that contact .
an example of some of the terms of the @xmath104 matrix for the configuration of fig .
[ fig : m_configuration ] is given in eq .
[ eq : t ] : @xmath115 when the external torque is zero , as in our loading protocol using compression , the radii are eliminated from the torque balance equation and the @xmath104 matrix can be further simplified to the form of eq .
[ eq : t_alt ] : @xmath116 the @xmath55 matrix ( cf .
eq . 7 in the main text ) is used to describe the presence of closed polygons formed by particles in contact and and is of dimensions @xmath117 . here
row indices correspond to polygons and column indices refer to the contacts .
non - zero entries in each row describe the unit normal directions joining two particles in contact which are members of a given polygon .
the first @xmath57 rows store the @xmath10 components and the next @xmath57 rows store the @xmath11 components of unit vectors @xmath111 .
an example for some of the terms of the @xmath55 matrix is given in eq .
[ eq : q ] ( and see fig .
[ fig : q_configuration ] ) : @xmath118 | the determination of the normal and transverse ( frictional ) inter - particle forces within a granular medium is a long standing , daunting , and yet unresolved problem .
we present a new formalism which employs the knowledge of the external forces and the orientations of contacts between particles ( of any given sizes ) , to compute all the inter - particle forces . having solved this problem
we exemplify the efficacy of the formalism showing that the force chains in such systems are determined by an expansion in the eigenfunctions of a newly defined operator . in a highly influential paper from 2005
majmudar and behringer @xcite wrote :
inter - particle forces in granular media form an inhomogeneous distribution of filamentary force chains .
understanding such forces and their spatial correlations , specifically in response to forces at the system boundaries , represents a fundamental goal of granular mechanics .
the problem is of relevance to civil engineering , geophysics and physics , being important for the understanding of jamming , shear - induced yielding and mechanical response . "
a visual example of such force chains in a system of plastic disks is provided in fig .
[ mahesh ] . in this letter
we present a solution of this goal .
to be precise , the problem that we solve is the following : consider a granular medium with known sizes of the granules , for example the 2-dimensional systems analyzed in ref .
@xcite and shown in fig . [ mahesh ] , of @xmath0 disks of known diameters @xmath1 . given the external forces , denoted below as @xmath2 and the external torques @xmath3 exerted on the granules , and given the angular orientations of the vectors connecting the center of masses of contacting granules ( but not the distance between them ! ) , determine all the inter - particle normal and tangential forces @xmath4 and @xmath5 . the method presented below applies to granular systems in mechanical equilibrium ; the issue of instabilities and abrupt changes in the force chains will be discussed elsewhere . for the sake of clarity and simplicity we will present here the two - dimensional case ; the savvy reader will recognize that the formalism and the solution presented will go smoothly also for the three - dimensional case ( as long as the system is in mechanical equilibrium ) .
the full formalism will be presented in a longer publication in due course .
the obvious conditions for mechanical equilibrium are that the forces and the torques on each particle have to sum up to zero @xcite .
the condition of force balance is usefully presented in matrix form using the following notation .
denote the ( signed ) amplitudes of the inter - particle forces @xmath6 as a vector @xmath7 , where the amplitudes @xmath8 appear first and then the amplitudes @xmath9 .
the vector of @xmath10 and @xmath11 components @xmath12 and @xmath13 is denoted as @xmath14 where all the @xmath10 components are presented in @xmath14 first and then all the @xmath11 components .
the vector @xmath7 has @xmath15 entries where @xmath16 is the number of contacts between particles .
the vector @xmath14 has @xmath17 entries where @xmath0 is the number of particles , with zero entries for all the particles on which there is no external force .
we can then write the force balance condition as @xmath18 where @xmath19 is a @xmath20 matrix .
the entries in the matrix @xmath19 contain the directional information , see supplemental material at [ url will be inserted by publisher ] for an example of an @xmath19 matrix .
denote the unit vector in the direction of the vector distance between the centers of mass of particles @xmath21 and @xmath22 by @xmath23 , and the tangential vector by @xmath24 orthogonal to @xmath23 .
then the entries of @xmath19 display the projections @xmath25 and @xmath26 or @xmath27 and @xmath28 as appropriate .
we thus guarantee that eq.([m ] ) is equivalent to the mechanical equilibrium condition @xmath29 as is well known , the friction - less granular system in the thermodynamic limit is jammed exactly at the isostatic condition @xmath30 @xcite . in the friction - less case @xmath19 is a @xmath31 matrix and as long as @xmath32 one can solve the problem by multiplying eq .
( [ m ] ) by the transpose @xmath33 , getting @xmath34 in this case the matrix @xmath35 has generically exactly three goldstone modes ( two for translation and one for rotation ) @xcite , and since the external force vector is orthogonal to the goldstone modes ( otherwise the external forces will translate or rotate the system ) , eq .
( [ mmt ] ) can be inverted with impunity by multiplying by @xmath36^{-1}$ ] .
in fact even when @xmath37 but the system is small enough to be jammed , this method can be used since there are enough constraints to solve for the forces .
this last comment is important for our developments below .
the problem becomes under - determined above isostaticity in the frictionless case , when force chains begin to build up that span from one boundary to the other .
with friction we anyway have twice as many unknowns and we need to add the constrains of torque balance .
the condition of torque balance is @xmath38 on every particle , where @xmath39 is the external torque exerted on the @xmath21th disk @xcite . for disks
, @xmath40 is in the normal direction , and therefore the torque balance becomes a condition that the sum of tangential forces has to balance the external tangential force .
this condition can be added to eq .
( [ m ] ) using a new matrix @xmath41 in the form @xmath42 the order of the extended matrix @xmath41 is @xmath43 , see supplemental material at [ url will be inserted by publisher ] for an example of @xmath44 . above jamming when the number of contacts increases @xmath45 .
the matrix @xmath46 is not square , and the matrix @xmath47 which is of size @xmath48 , has at least @xmath49 zero modes @xcite .
accordingly it can not be inverted and one can conclude that * the conditions of mechanical equilibrium are not sufficient to determine all the forces . *
obviously what is missing are additional constraints to remove the host of zero modes .
these additional constraints are _ geometrical _ constraints @xcite which can be read from those disks which describe connected polygons . in other words , since we know the orientation @xmath23 of each contact in our system , we can determine which granules are stressed in a triangular arrangement , and which in a square or pentagonal etc . , see fig .
[ geometry ] .
each such arrangement is a constraint on the radius vectors adjoining the centers of mass .
for example if particles @xmath50 and @xmath51 are in a triangular arrangement then @xmath52 , with the analogous constraint on squares , pentagons etc .
these constraints can be written in a matrix form by denoting the _ amplitudes _ of inter - particle vector distances as @xmath53 where we again arrange the @xmath10 components first and the @xmath11 components second : @xmath54 where the matrix @xmath55 again has entries @xmath25 or @xmath26 as appropriate to represent the vectorial geometric constraints , see supplemental material at [ url will be inserted by publisher ] for an example of @xmath56 . denoting the total number of polygons by @xmath57 the dimension of the matrix @xmath55
is @xmath58 .
of course @xmath53 has @xmath16 entries while @xmath7 had @xmath15 entries .
note that in generic situations there can be also disks which are not stressed at all .
these are referred to as rattlers " .
for example in the configuration shown in fig .
[ geometry ] there exist 14 rattlers . at this point
we specialize the treatment to hookean normal forces with a given force constant @xmath59 @xcite .
non hookean forces result in a nonlinear theory that can still be solved but much less elegantly . for the present case @xmath60 \ . \label{sig}\ ] ] denoting the amplitudes of the vectors @xmath61 as the vector @xmath62 ( again with first the @xmath10 and then the @xmath11 components ) , we can rewrite eq .
( [ q ] ) in the form @xmath63
having this result at hand we can formulate the final problem to be solved .
arrange now a new matrix , say g , operating on a vector @xmath7 , with a rhs being a vector , say @xmath64 , made of a stacking of @xmath14 , @xmath65 and @xmath66 , as before with @xmath10 and then @xmath11 components : @xmath67 using these objects our problem is now @xmath68 the dimension of the matrix @xmath69 is @xmath70 and the matrix @xmath71 has the dimension @xmath48 .
we can use now the euler characteristic @xcite to show that the situation has been returned here to the analog of the invertible matrix @xmath72 when @xmath73 : the euler characteristic in two dimensions requires that @xmath74 where @xmath75 is the number of rattlers " i.e. disks on which there is no force .
accordingly we find that @xmath76 consequently , the matrix @xmath77 has no zero eigenmodes .
thus the final solution for the forces can be obtained as @xmath78 where @xmath79 is the set of eigenfunctions of @xmath77 associated with eigenvalues @xmath80 .
we compared the inter - particle forces obtained from direct numerical simulations ( see below for details ) to those computed from eq .
( [ final ] ) .
both normal and tangential forces are of course identical to machine accuracy .
we reiterate that we did not need to know the distances between particles .
this is important in applying the formalism to experiments since it is very difficult to measure with precision the degree of compression of hard particles like , say , metal balls or sand particles . note also the remarkable fact that we never had to provide the frictional ( tangential ) force law in the formalism to obtain the correct forces ! at this point we can discuss the force chains . by definition
these are the large forces in the system that provide the tenuous network that keeps the system rigid . observing eq .
( [ final ] ) we should focus on the eigenfunction @xmath79 of @xmath77 that have the smallest eigenvalues and the largest overlaps with @xmath81 .
these can be found and arranged in order of the magnitude of @xmath82 independently of the calculation of @xmath7 . in fig .
[ order ] we show the contribution to the total energy @xmath83 , learning that about 20% of the leading eigenfunction are responsible for 90% of the energy .
we can therefore hope that the force chains will be determined by the same relatively small number of eigenfunctions .
this is not guaranteed ; due to contributions to the forces that oscillate in sign the convergence can be much slower than in the case of the energy where the sum is of positive contributions . in fig .
[ chains ] we show in upper left panel the force chains in the configuration of fig . [ geometry ] . in the other panels we show the prediction of the force chains using 100 , 200 and 300 of the ( energy )
leading modes .
we learn that with 100 out of the 864 modes the main force chains begin to be visible .
with 200 out of the 864 modes the full structure of the force chains is already apparent , although with 300 it is represented even better .
since the number of geometric constraints is very large , one can ask whether all the geometric constraints are necessary , as eq .
( [ euler ] ) shows that @xmath84 .
the answer is no , we could leave out constraints as long as we have enough conditions to determine the solution .
there is the obvious question then why do we have a unique solution when the number of equations is larger than the number of unknowns .
the answer to this question lies in the properties of the vector @xmath64 and the matrix @xmath85 which does have many zero modes .
a condition for the existence of a solution is that @xmath64 is orthogonal to all the zero modes of @xmath85 , as can be easily checked .
we have ascertained in our simulations that this condition is always met . in the near future
we will present an extension of this formalism to three dimensions and the use of the formalism to study the instabilities of the force networks to changes in the external forces . as a final comment we should note that in fact only _ one _ external force is necessary to determine _ all _ the inter - disk forces .
this single external force is necessary to remove the re - scaling freedom that this problem has by definition .
* simulations * : for the numerical experiments in 2-dimensions we use disks of two diameters , a ` small ' one with diameter @xmath86 and a ` large ' one with diameter @xmath87 . such @xmath0 disks are put between virtual walls at @xmath88 and @xmath89 .
these walls exert external forces on the disks .
the external forces are taken as hookean for simplicity . for disks near the wall at @xmath88
we write @xmath90 here @xmath91 denotes the @xmath10 component of the position vector @xmath40 of the center of mass of the @xmath21th disk , and we have a similar equation for the @xmath11 components with @xmath92 replaced by @xmath93 .
when two disks , say disk @xmath21 and disk @xmath22 are pressed against each other we define their amount of compression as @xmath94 : @xmath95 where @xmath96 is the actual distance between the centers of mass of the disks @xmath21 and @xmath22 . in our simulations
the normal force between the disks acts along the radius vector connecting the centers of mass .
we employ a hookean force @xmath97 . to define the tangential force between the disks we consider ( an imaginary ) tangential spring at every contact which is put at rest whenever a contact between the two disks is formed . during the simulation we implement memory such that for each contact we store the signed distance @xmath98 to the initial rest state . for small deviations we require a linear relationship between the displacement and the acting tangential force .
this relationship breaks when the magnitude of the tangential force reaches @xmath99 where due to coulomb s law the tangential loading can no longer be stored and is thus dissipated .
when this limit is reached the bond breaks and after a slipping event the bond is restored with a the tangential spring being loaded to its full capacity ( equal to the coulomb limit ) .
the numerical results reported above were obtained by starting with @xmath100 particles on a rectangular grid ( ratio 1:2 ) with small random deviations in space and no contacts .
we implement a large box that contains all the particles .
the box acts on the system by exerting a restoring harmonic normal force as described in eq .
( [ external ] ) .
the experiment is an iterative process in which we first shrink the containing box infinitesimally ( conserving the ratio ) .
the second step is to annul all the forces and torques , to bring the system back to a state of mechanical equilibrium .
we therefore annul the forces using a conjugate gradient minimizer acting to minimize the resulting forces and torque on all particles .
we iterate these two steps until the system is compressed to the desired state .
this work had been supported in part by an
ideas " grant stanpas of the erc .
we thank deepak dhar for some very useful discussions .
we are grateful to edan lerner for reading an early version of the manuscript with very useful remarks . 99 t.s .
majmudar and r.p .
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the existence of rattlers " that are not in close contact with other particles may increase the number of zero modes . to see this note that the matrices @xmath101 and @xmath102 have the same rank , but @xmath102 can not have more than @xmath103 non - zero eigenmodes .
therefore @xmath101 must have at least @xmath49 zero modes .
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s.v matveev , euler characteristic " , in hazewinkel , michiel ,
encyclopedia of mathematics , springer , isbn 978 - 1 - 55608 - 010 - 4 , ( 2001 ) . |
hgmn stars are chemically peculiar stars for which periodic variability has not been found as of yet .
searches for variability have been made mostly photometrically though some studies of spectral variability have also been attempted .
historically , several hgmn stars have been claimed to be variable but variability as yet to be confirmed in any of them @xcite .
a large number of hgmn stars were observed as part of the hipparcos mission but no periodic variability was detected .
the maximum permitted amplitude can in many cases be expected to be at most a few mmag .
recently , some spectral variability was claimed in @xmath0 andromed which were interpreted as possible surface chemical inhomogeneities @xcite .
the authors argued that such variability would be the exception rather than the rule in hgmn stars .
the pursuit of elusive evidence of variability , both spectroscopically and photometrically , is motivated by several unresolved questions : * pulsations is expected theoretically from current models , in other words confirmation of stability or the discovery of low amplitude pulsations can provide constraints on physical processes not accounted for in the models ( see turcotte & richard in these proceedings ) ; * rotational variability would provide evidence of surface inhomogeneities related to diffusion , mass loss and/or magnetism in the atmosphere of b stars ; * confirm or infirm that all hgmn stars are part of binary or multiple systems which could help answer the question as to how b stars can be slowly rotating in the absence of binarity or magnetism . in this short paper
we present preliminary results of the search of line profile variability in a substantial series of echelle spectra of four bright hgmn stars of the southern hemisphere .
these observations represent an unprecedented effort to study spectroscopic variability in hgmn stars and are expected to help put stronger constraints on pulsations in these stars .
the four program stars were the brightest southern hgmn stars visible during the periods of observation ( see next section ) .
three of the four are within the theoretical instability region for spb stars ( hd 11753 being right on the cool edge ) , the fourth ( hd 53244 ) being slightly too evolved ( figure [ fig : hrd ] ) .
-t@xmath1 diagram showing the program stars and the theoretical limit of the spb instability region @xcite along with a sample of other hgmn stars @xcite . ]
the spectra were taken over two campaigns of several days , from september 28@xmath2 to october 11th@xmath2 and from december 2@xmath3 to december 15@xmath2 2000 , with the coralie spectrograph at the 1.2 m telescope at la silla .
the observations are summarized in table [ tab : obs ] .
.summary of observations of the program stars [ cols="^,^,^,^,^,^ " , ] due to space constraints we henceforth discuss only the star for which the better results were obtained at this point in the analysis , hd221507 .
the spectra selected for this star after bad data was removed are shown in figure [ fig : spec ] .
we focused on the siii doublet at @xmath44128.053 and @xmath44130.884 for which the first moment was calculated , a procedure developed to study spb stars @xcite .
the variability was studied using the pdm method .
the models of hgmn stars suggest that they should pulsate in a similar way to spb stars , if at all .
four phase plots are shown in figure [ fig : phase ] .
the periods shown , 0.31 , 0.44 , 0.78 , 1.38 @xmath5 were the ones which would reproduce the best approximation to a sine wave .
the periods are in the range expected for spbs .
the scatter is evidently quite large in all cases and the variability , although somewhat suggestive , is far from clear .
this work was performed in part under the auspices of the u.s .
department of energy , national nuclear security administration by the university of california , lawrence livermore national laboratory under contract no.w-7405-eng-48 . | spectra of four non - magnetic chemically peculiar late b type stars ( hgmn ) stars are analysed to detect periodic spectral line variations ( lpvs ) .
a procedure developed to study lpvs in slowly pulsating b stars has been adopted as pulsational properties of hgmn stars should be expected to be similar . in the preliminary results discussed here no conclusive evidence for periodic lpvs was uncovered .
a more sensitive re - analysis of the data is under way . |
in this work we adopt a chemical evolution model ( see chiappini , matteucci , & romano 2000 ) that assumes two main accretion episodes for the formation of the galaxy : the first one forming the halo and bulge in a short timescale followed by a second one that forms the thin - disk , with a timescale which is an increasing function of the galactocentric distance ( being of the order of 7 gyrs at the solar neighborhood ) .
the present model takes into account in more detail than previously the halo density distribution and explores the effects of a threshold density in the star formation process , both during the halo and disk phases .
the model also includes the most recent nucleosynthesis prescriptions concerning supernovae of all types , novae and single stars dying as white dwarfs . in the comparison between model predictions and available data ,
we have focused our attention on abundance gradients as well as gas , star and star formation rate distributions along the disk , since this kind of model has already proven to be quite successful in reproducing the solar neighborhood characteristics .
we suggest that the mechanism for the formation of the halo leaves heavy imprints on the chemical properties of the outer regions of the disk , whereas the evolution of the halo and the inner disk are almost completely disentangled .
this is due to the fact that the halo and disk densities are comparable at large galactocentric distances and therefore the gas lost from the halo can substantially contribute to build up the outer disk . we also show that the existence of a threshold density for the star formation rate , both in the halo and disk phase , is necessary to reproduce the majority of observational data in the solar vicinity and in the whole disk . in particular , a threshold in the star formation implies the occurrence of a gap in the star formation at the halo - disk transition phase , in agreement with recent data .
@xmath0 the outer gradients are sensible to the halo evolution , in particular to the amount of halo gas which ends up into the disk .
this result is not surprising since the halo density is comparable to that of the outer disk , whereas is negligible when compared to that of the inner disk .
therefore , the inner parts of the disk ( @xmath1 @xmath2 @xmath3 ) evolve independently from the halo evolution . @xmath0
we predict that the abundance gradients along the galactic disk must have increased with time .
this is a direct consequence of the assumed `` inside - out '' scenario for the formation of the galactic disk .
moreover , the gradients of different elements are predicted to be slightly different , owing to their different nucleosynthesis histories . in particular , fe and n , which are produced on longer timescales than the @xmath4-elements , show steeper gradients .
unfortunately , the available observations can not yet confirm or disprove this , because the predicted differences are below the limit of detectability .
@xmath0 our model guarantees a satisfactory fit not only to the elemental abundance gradients but it is also in good agreement with the observed radial profiles of the sfr , gas density and the number of stars in the disk .
@xmath0 our best model suggests that the average @xmath5fe]@xmath6 ratios in stars slightly decrease from 4 to 10 kpcs .
this is due to the predominance of disk over halo stars in this distance range and to the fact that the `` inside - out '' scenario for the disk predicts a decrease of such ratios .
on the other hand we predict a substantial increase ( @xmath7 dex ) of these ratios in the range 1018 kpcs , due to the predominance , in this region , of the halo over the disk stars .
finally , we conclude that a relatively short halo formation timescale ( @xmath8 0.8 gyr ) , in agreement with recent estimates for the age differences among galactic globular clusters , coupled with an `` inside - out '' formation of the galactic disk , where the innermost regions are assumed to have formed much faster than the outermost ones , represents , at the moment , the most likely explanation for the formation of the milky way .
this scenario allows us to predict abundance gradients and other radial properties of the galactic disk in very good agreement with observations .
more observations at large galactocentric distances are needed to test our predictions . | we present theoretical results on the galactic abundance gradients of several chemical species for the milky way disk , obtained using an improved version of the two - infall model of chiappini , matteucci , & gratton ( 1997 ) that incorporates a more realistic model of the galactic halo and disk .
this improved model provides a satisfactory fit to the elemental abundance gradients as inferred from the observations and also to other radial features of our galaxy ( i.e. , gas , star formation rate and star density profiles ) .
we discuss the implications these results may have for theories of the formation of the milky way and make some predictions that could in principle be tested by future observations .
# 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in |
many models of galaxy formation and evolution in recent years take as a reference for the observational side the results coming from qso absorption studies and in particular those objects that show a large hi column density , namely , the damped lyman@xmath0 systems ( dlas ) with @xmath9 atoms 2 ( see for example pei et al .
dlas have been widely studied both because they are believed to be the progenitors of present day galaxies and because the large hi column density allows one to probe the chemical enrichment problem .
dlas constitute so far the best laboratory where to directly measure the heavy element enrichment in a large interval of evolutionary phases , and to understand the processes of star formation and metal pollution of the universe .
however , this kind of investigation requires a careful consideration of the effects of dust depletion suffered by dlas ( pei et al .
1991 ; pettini et al .
we present the analysis of a sample of dlas in the redshift range @xmath1 to investigate their chemical state .
we find that , after allowance for dust depletion corrections which are obtained with a very general approach , the dla population clearly shows a metallicity redshift evolution .
.mean element abundances relative to hydrogen [ cols="^,^,^,^,^,^,^,^ " , ] we have collected data from the literature for a dla sample , which includes objects . this sample represents the largest and most complete sample of dlas for which measurements of hi and heavy element column densities are available .
the ions considered for abundance measurements are feii , znii , crii , siii , mnii , sii , niii .
these ions are the dominant contributors to the abundances of the corresponding elements in hi clouds with high column densities , because they all have ionization potentials below 13.6 ev . in table
1 we give the mean metal abundances relative to hydrogen and iron . they are presented with the customary definition [ x / h]@xmath12 , where x@xmath13 and y@xmath13 are the ion column densities of element x and y. for comparison , the mean abundances for warm halo ( wh ) clouds ( savage and sembach 1996 ) and the small magellanic cloud ( smc , welty et al .
1997 ) are also shown .
we note that globally dlas show [ x / h ] and [ x / fe ] abundance ratios more similar to those of smc and wh clouds , respectively .
this suggests that metal abundances in dlas are the result of chemical enrichment processes similar from the ones operating in the smc and that the most common depletion pattern operating in dlas is similar to the one observed in wh clouds . indeed , to derive a complete picture of the dla chemical state ,
one must correct for dust depletion effects . since every element considered is affected by dust depletion differently , one must consider all measured species simultaneously . in the milky way ,
a number of depletion patterns have been identified , showing highest depletions in dense disk clouds and lowest depletions in low density , warm halo clouds ( savage & sembach 1996 ) .
we make a simplification assuming that the depletion patterns in dlas may be reproduced by one of the four depletion patterns identified for the mw : warm halo , warm halo + disk ( whd ) , warm disk ( wd ) and cool disk ( cd ) clouds ( savage & sembach 1996 ) , thus modifying the dust to metals ratio to obtain the best match with the observations . by means of a @xmath2 minimization procedure
we determine the best fit dla metallicities and the dust to metals ratios . fig .
1 shows the metallicity as a function of redshift .
filled symbols represent dlas with three or more measured elemental abundances for which it has been possible to obtain a proper best fit solution ( 37 dlas ) . for the cases with only two elements
observed , each fit has a zero formal error and , therefore , a reduced @xmath2 can not be calculated ; thus , the best fit is considered less significant ( 16 dlas , empty symbols and solid error bars ) . finally , for the cases where only one element is measured
, we estimate the metallicity assuming a wh depletion pattern ( 16 dlas , empty symbols and dotted error bars ) . the combination of the largest sample available ( dlas )
, a large redshift baseline ( @xmath14 ) and a more accurate dust correction applied have led to the unambiguous detection of the redshift evolution of metallicity in dla galaxies , with mean values around 1/30 of solar at @xmath4 to 3/5 of solar at @xmath6 .
we found a significant linear correlation of metallicity with redshift ( 99.99% significance level ) with a slope of @xmath15 , which is clearly not consistent with a null gradient , indicating genuine evolution of dla metallicity with redshift . in fig . 1 we also show six boxes centered on the weighted averages over @xmath16 intervals and whose vertical widths mark the corresponding @xmath17 weighted dispersion . in addition
, we note that the vertical dispersion of points greatly exceeds the total errors , indicating that , although all dlas align along an average trend , there are real differences among individual objects in either initial conditions , or time of formation , or metal enrichment efficiency , or all of the above .
pei & fall ( 1995 ) consider that a mean heavy element abundance in the interstellar medium of dla galaxies is given by the ratio of the total metal content to the total gas content ( i.e. @xmath18 ) , which means that large hi dlas dominate when calculating the global metallicity .
this kind of analysis has been performed on a sample of dlas using the znii absorption line and a null result has been found for the evolution ( pettini et al .
1999 ) , and it is not disproved if our sample is used
. however , the lack of evident evolution in this case appears to be due to the fact that those dlas with large hi column density are concentrated in the central redshift region ( 84% of dlas with @xmath19 are in the bin @xmath20 ) . in other words , we can not exclude the redshift evolution of metallicity in high hi dlas till when the considered sample will be fairly distributed in the considered redshift range .
if there were a metallicity evolution also for high hi dlas , dust obscuration could be too large at @xmath21 to make dlas detectable ( a dla with @xmath22 and metallicity @xmath23 would extinct about 1.5 magnitude in the ly@xmath0 ) .
moreover , this is where uv observations , required to detect ly@xmath0 absorption , are very sensitive to any appreciable dust opacity .
in 37 dlas for which we have obtained a best fit solution , we have found that the majority ( @xmath24% ) show a wh like depletion pattern , then @xmath25% and @xmath26% have a whd like and wd like depletion pattern , respectively .
the predominance of wh like conditions may be interpreted as evidence for a lower density in the absorbing clouds relative to clouds in the mw disk and/or to more efficient dust destruction processes operating in dla galaxies , possibly due to more active star formation that may produce a more turbulent , dust hostile environment . in our sample
there are 31 dlas with measured si and for which it has to perform @xmath2 minimization . for 8 over 31 systems
, there is evidence of deviation of si with respect the expectations , having a mean value of @xmath27 .
no correlation with redshift and metallicity is revealed .
this result may be indicative of @xmath0element enrichment in a fraction of dlas , or of a lower efficiency of silicate formation than is found in the milky way .
we like to stress that it is the combination of a large sample of dlas ( objects ) , a wide redshift baseline ( @xmath14 ) and a more accurate dust correction applied to all heavy elements measured , that has allowed us to demonstrate that the metallicity of dlas evolves with redshift .
the combined use of as many elements as possible allows one to much more efficiently track the metallicity evolution down to very metal poor gas clouds . | we have collected data for damped lyman@xmath0 ( dla ) systems , to investigate the chemical evolution of galaxies in the redshift interval @xmath1 . in doing that
, we have adopted the most general approach used so far to correct for dust depletion .
the best solution , obtained through @xmath2 minimization , gives as output parameters the global dla metallicity and the dust to metals ratio .
clear evolution of the metallicity vs. redshift is found ( 99.99% significance level ) , with average values going from @xmath3 solar at @xmath4 to @xmath5 solar at @xmath6 .
we also find that the majority of dlas ( @xmath7% ) shows dust depletion patterns which most closely resemble that of the warm halo clouds in the milky way , and have dust to metals ratios very close to warm halo clouds .
2@xmath8 # 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in |
we present in this section the update of the unitarity triangle ( ut ) analysis within the standard model ( sm ) , performed by the utfit collaboration following the method described in refs .
the constraints used in the analysis can be distinguished in side and angle constraints , where the latter do not rely on theoretical calculations of hadronic matrix elements .
the side constraints come from the measurement of direct cp - violation in the kaon sector ( @xmath9 ) , of @xmath3 and @xmath5 mixing ( @xmath10 , @xmath11 ) and of semileptonic b decays ( @xmath12 ) .
the angle constraints are cp - violating measurements for the @xmath3-system , performed with high statistics at b - factories : @xmath13 , @xmath14 , @xmath15 , @xmath16 , and @xmath17 . as shown in fig .
[ fig : sm_allconstr ] , the ckm matrix turns out to be consistently overconstraint .
the ckm parameters @xmath18 and @xmath19 result to be accurately determined : @xmath20 , @xmath21 @xcite . , @xmath22 @xcite . ]
the ut analysis has thus established that the ckm matrix is the dominant source of flavour mixing and cp - violation and that new physics ( np ) effects can at most represent a small correction to this picture .
-@xmath19 plane , including both angle and side measurements .
the closed contours at @xmath23% and @xmath24% probability are shown .
the full lines correspond to @xmath24% probability regions for the different constraints.__,title="fig : " ] + due to the redundant experimental constraints , interesting consistency checks can be performed by comparing various ut analyses where different constraints are used .
in particular , the ut analyses based on only angle ( utangle ) or only side ( utside ) constraints , shown in fig . [ fig : sm_anglevsside ] , provide well compatible results @xcite : @xmath25 , @xmath26 and @xmath27 , @xmath28 , respectively .
the @xmath29 difference between the two @xmath18 results is mainly a manifestation of the tension of the @xmath30 inclusive measurement , based on heavy quark effective theory parameters extracted from experimental fits with some model dependence , with the rest of the fit and with the @xmath30 exclusive measurement , relying on semileptonic form factors determined from lattice qcd or qcd sum rules .
in fact , the utangle analysis turns out provide an indirect determination of @xmath30 ( @xmath31 ) that is in perfect agreement with the @xmath30 exclusive measurement ( @xmath32 ) , while the utside analysis uses in input the inclusive - exclusive average for @xmath30 that is @xmath33 higher than the utangle indirect determination @xcite . -@xmath19 plane , including only angle ( left ) or side ( right ) measurements .
the closed contours at @xmath23% and @xmath24% probability are shown .
the full lines correspond to @xmath24% probability regions for the different constraints . _ _ ] -@xmath19 plane , including only angle ( left ) or side ( right ) measurements .
the closed contours at @xmath23% and @xmath24% probability are shown .
the full lines correspond to @xmath24% probability regions for the different constraints . _
_ ] the ( overconstraint ) ut analysis also allows to extract some hadronic quantities that can be compared to the results of lattice qcd calculations @xcite .
this comparison is shown in table [ tab : lattice ] for the hadronic parameters describing mixing in the @xmath1- , @xmath3- and @xmath5-meson sectors .
the remarkable agreement between the lattice calculations and the indirect ut analysis determinations provides additional evidence of the sm success in describing flavour physics and of the reliability of lattice qcd calculations .
it is interesting to note that an improvement of the accuracy of the lattice determinations of @xmath34 and @xmath35 would be important to increase the precision of the ut analysis .
.__values of the hadronic parameters that describe @xmath1-@xmath2 and @xmath36-@xmath37 mixing : @xmath34 , @xmath38 and @xmath39 , as obtained from the ut analysis including angle and @xmath40 constraints , and from lattice qcd calculations @xcite . _ _ [ cols="^,^,^,^",options="header " , ] v. lubicz and c. tarantino , nuovo cim . * 123b * ( 2008 ) 674 [ 0807.4605 [ hep - lat ] ] . m. bona _ et al . _ [ utfit collaboration ] , jhep * 0603 * ( 2006 ) 080 [ hep - ph/0509219 ] .
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f. j. botella , g. c. branco and m. nebot , 0805.3995 [ hep - ph ] . | we present the update of the unitarity triangle ( ut ) analysis within the standard model ( sm ) and beyond . within the sm , combining the direct measurements on sides and angles , the ut turns out to be overconstraint in a consistent way , showing that the ckm matrix is the dominant source of flavour mixing and cp - violation and that new physics ( np ) effects can appear at most as small corrections to the ckm picture . generalizing the ut analysis to investigate np effects , constraints on @xmath0 transitions
are also included and both ckm and np parameters are fitted simultaneously . while no evidence of np effects is found in @xmath1-@xmath2 and @xmath3-@xmath4 mixing , in the @xmath5-@xmath6 mixing an hint of np
is found at the @xmath7 level .
the ut analysis beyond the sm also allows us to derive bounds on the coefficients of the most general @xmath8 effective hamiltonian , that can be translated into bounds on the np scale .
ckm matrix 12.15.hh |
in a recent study @xcite we successfully calculated the mass of the @xmath1 meson using staggered fermions . according to conventional lore
, the @xmath1 receives a large portion of its mass from instanton effects . at the fermionic level
, this would mean receiving contributions from zero modes of the dirac operator .
these zero modes come with a definite chirality in the continuum and are associated with the topological charge ( @xmath2 ) of a gauge configuration via the index theorem : @xmath3 where @xmath4 and @xmath5 are respectively the number of right and left - handed zero modes . at finite lattice spacing ,
@xmath6 and @xmath7 do not exactly anti - commute , so there are no exact chiral zero modes and nor is the topological charge of gauge configurations well defined .
nevertheless , one might expect to find a few low lying eigenmodes carrying chiral charge .
smit and vink @xcite verified this sort of behavior for @xmath8 gauge theory in 2 dimensions . in this work , we set out to study the extent to which the lattice `` zero modes '' reproduce the continuum picture for qcd .
accordingly , we have constructed the lowest few eigenmodes of staggered @xmath7 on both quenched and dynamical @xmath0 gauge configurations ( 83 samples each ) of size @xmath9 corresponding to a lattice spacing of 0.1fm .
we have used the subspace iterations method @xcite to compute the eigenmodes .
the eigenvalues that we obtain have relative errors in the range @xmath10 for the smallest through @xmath11 for the highest .
the lattice analog of the @xmath8 axial anomaly equation gives @xmath12 where @xmath13 is the fermion propagator .
the mode by mode contribution to @xmath14 on all the dynamical configurations is shown in figure [ dg5 ] .
it appears to diminish as @xmath15 increases which can be verified quantitatively by calculating the width , shown in the bottom part of figure [ dg5 ] .
this behavior is consistent with the continuum where @xmath16 for all the non zero eigenvalues .
figure [ qg5 ] shows a similar set of plots for the quenched ensemble .
although both @xmath17 and its width show qualitatively similar behavior as before , they seem to drop more slowly with increasing @xmath15 .
the solid line present in all the figures shown so far represents the minimum value of the 32nd eigenvalue in the ensemble , ie .
the line below which we are sure to have found 100% of the modes .
[ dg5 ] [ qg5 ] = 2.5 in the sum in eqn([trg5sf1 ] ) for @xmath18 and 32 modes at @xmath19 on a typical dynamical configuration is shown in figure [ q05 ] .
the plateau in @xmath2 as the number of modes increases implies that the contribution of the higher modes to the sum in eqn [ trg5sf1 ] is becoming negligibly small .
although , the plot shown here is for one configuration , a similar behavior is seen on all the configurations , quenched and dynamical . in figure
[ q05 ] , the isolated point represented by a square is the value of @xmath2 obtained from pseudofermions .
the `` true '' value obtained by averaging over 82 copies of noise lies within the band marked by the dashed lines .
since the plateau in @xmath2 lies within this band it is a very good indication that these lowest modes are the topological zero modes which make the dominant contribution to the sum in eqn [ trg5sf1 ] .
we have compared @xmath2 obtained from pseudofermions and modes on all the dynamical configurations and find that both agree within errors .
on the quenched ensemble , the difference between the two estimators appears to be statistically significant .
this simply reinforces the inference reached by comparing figures [ dg5 ] and [ qg5 ] that more than 32 lowest eigenvalues are required to identify an upper limit for the number of shifted zero modes on the quenched configurations .
calculation of hadron correlators on the lattice normally involves inverting @xmath20 to obtain the quark propagator .
when resolved in terms of the eigenvalues and eigenvectors of @xmath7 , on a fixed background gauge field , it takes the form @xmath21 comparing the full propagator obtained using standard inversion procedures like cg , with that constructed out of the sum above with the available eigenmodes gives an insight into the role of the eigenmodes in determining hadronic quantities .
the @xmath1 meson , being a flavor singlet , has both connected and disconnected contributions to its correlator .
the discussion in section 2 suggests an extension of the analysis to the @xmath1 disconnected correlator which is given by @xmath22 .
we calculate this quantity using pseudofermions and with the available 32 modes .
the ratio @xmath23 of disconnected to the connected correlator for @xmath19 is shown in fig [ r01 ] .
for the connected correlator , we use the full result available from our previous study @xcite . on the dynamical configurations , @xmath23 is fit to the form @xcite @xmath24 $ ] . for staggered fermions , @xmath25 ,
the number of valence fermions equals 4 while @xmath26 .
it is evident from the figure that @xmath23 data with 32 modes agree within error with the full result .
= 3.1 in we find similarly good agreement with only 24 modes though 16 appears to be not quite enough .
the pattern is the same at the heavier quark masses , 0.02 and 0.03 .
we note that this behavior is not automatic as 32 modes are insufficient to yield a good approximation to the true quark propagator .
we find , for example , that the pion is off by a large factor .
based on our knowledge of the staggered eigenvalue spectrum and on recent work done by negele _
et al . _
@xcite with wilson fermions , one might expect that 100 - 200 modes would be needed to reproduce the pion propagator .
thus the @xmath1 is special in exhibiting an extraordinary sensitivity to the lowest few modes .
the quark propagator , constructed out of the available 32 modes , can be used as a starting seed for calculating @xmath27 using cg . [ residue ]
the benefit of using such approximate solutions as a starting iterate in the cg procedure is illustrated in figure [ residue ] .
the number of cg iterations required for the residual to converge to a given tolerance drops by more than @xmath28 at @xmath19 . at higher quark masses , this effect is not so pronounced owing to the fact that the lowest 32 eigenvalues are smaller than the quark mass itself .
these conclusions hold only for delta function sources or gauge fixed wall sources built out of delta functions .
quark propagators from noisy sources are dominated by short distance modes and hence there is virtually no improvement in the cg convergence rate when the starting seed is built out of long wavelength modes .
we find that it is possible to identify the shifted zero modes on the lattice since they make the dominant contribution to @xmath29 and @xmath30 . on our dynamical configurations , it seems that the number of topological zero modes must be less than 32 while on the quenched configurations the data indicates that there are more than this number . for most hadronic quantities ,
the approximate propagator constructed out of these topological zero modes is not accurate enough to be useful .
however , a remarkable result of our study is that we find that the @xmath1 flavor singlet interaction on the dynamical configurations is dominated entirely by the quasi - zero modes of @xmath7 .
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b286 ( 1987 ) 485 . t. l. ivanenko and j. w. negele , this proceedings . | we have constructed the lowest few eigenvectors of the staggered dirac operator on @xmath0 gauge configurations , both quenched and dynamical .
we use these modes to study the topological charge and to construct approximate hadronic correlators . |
and its sister source , , were the first objects dubbed `` micro - quasars '' .
their spectra are typical of galactic black hole candidates ( bhcs ) , and they are associated with time variable cores of double - lobed radio sources , reminiscent of extra - galactic radio sources .
this morphology , seen on a parsec scale within the milky way , earned them their nickname . and are the brightest persistent sources in the galactic bulge above @xmath550 kev @xcite .
their timing characteristics are typical of the black hole low / hard state @xcite , and they consistently emit near their brightest observed levels , although they vary over times of days to years .
their emission properties are readily likened to the canonical bhc , cyg x-1 .
in fact , together with cyg x-1 , they are the only known persistent , low - state bhcs , and all three sources have maximum luminosities around @xmath6ergs s@xmath7 .
radio jets have now been observed in cyg x-1 , furthering the similarity @xcite . and are , however , quite different from the galactic _ superluminal _ radio sources more typically thought of as micro - quasars : grs 1915 + 105 and gro j1655 - 40 .
the emission from these objects is much brighter and more spectacularly variable .
their radio jets , too , are much brighter and are highly variable , being unresolved or absent except during exceptional ejection events which last only weeks . in contrast , the radio lobes of and are quite stable @xcite .
[ f_lc ] .[t_obs]observations [ cols="<,^,^,^,^ " , ]
during more than 5 years monitoring with the _ rxte _ prior to 2001 march , the hard spectrum was always dominated by a hard power law with photon index @xmath8 @xcite with occasional appearance of a weak thermal component @xcite . as shown in figure [ f_lc ] , made an abrupt state change in 2001 march .
the hard flux dropped by an order of magnitude in a few days , leaving the thermal component seen in figure [ f_spec ] .
based on relative luminosity , however , the current soft state is not a _ high_/soft state . rather it is significantly less luminous than the low / hard state in this source .
this can be contrasted to cyg x-1 and the soft transients , where the _
high_/soft state is more luminous .
rather , this seems to be a low - luminosity state which is fading into quiescence ( figure [ f_lc ] ) . finally , we note that the measured column density is consistent with previous measurements @xcite since strong jet ejections are generally associated with the `` very high state '' and transitions from the `` off '' to high / soft states in transients @xcite , it is perhaps not surprising that no jet emission appeared in our low / hard state observations ( sep - oct 2000 ) and the recent transition observation ( mar 2001 ) .
perhaps our best opportunity will come when ( if ? ) makes a transition once again to its normal , low / hard state .
we have an approved _ chandra_cycle 3 proposal to monitor the morphology of and hope to observe a jet ejection . | we observed the `` micro - quasar '' four times with _ chandra_. two hrc - i observations were made in 2000 september - october spanning an intermediate - to - hard spectral transition ( identified with _ rxte _ ) .
another hrc - i and an acis / hetgs observation were made in 2001 march following a hard - to - soft transition to a very low flux state .
the accurate position ( j2000 ) of the source is ra @xmath0 18 01 12.40 , dec @xmath0 @xmath125 44 36.0 ( 90% confidence radius @xmath0 0.6 ) , consistent with the purported variable radio counterpart .
all images are consistent with being a point source , indicating that any bright jet is less than 1light - month in projected length , assuming a distance of 8.5kpc .
the march spectrum is well - fit with a multi - color disk - blackbody with an inner temperature of @xmath2kev , interstellar absorption of @xmath3 , and ( un - absorbed ) 1@xmath110kev luminosity of @xmath4 .
no narrow emission lines are apparent in the spectrum and upper limits to line equivalent widths are given .
# 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in |
in qcd or the massive schwinger model in @xmath3 vacuum @xcite , it is well known that the scalar operator condense while the pseudoscalar does not ; @xmath4 where the second equation follows from parity symmetry . however ,
if we have a non - zero @xmath5 term , which violates parity symmetry , both of them have non - zero expectation values ; @xmath6 which indicates that @xmath7 meson should have a long - range correlation as @xmath8 we would like to present our numerical results of the 2-flavor massive schwinger model with a @xmath1 term .
we investigate @xmath9 condensates and the @xmath0 meson correlators in each topological sector .
it is also found that they are non - trivially related each other to reproduce the @xmath5 dependence .
we find that their behavior is well understood by the intuitive picture based on the clustering decomposition and the statistical mechanics .
in particular , our study shows that the accurate contributions from higher topological sectors are essential in order to assure parity symmetry , which never allows the long - range correlation of the @xmath0 meson correlators .
moreover , it is also shown that the origin of the fluctuations of disconnected diagram is from pseudoscalar condensates in each topological sector .
our strategy to calculate the @xmath5 vacuum effects is to separate the integral of the gauge fields into topological sectors ; @xmath10 where @xmath11 denotes a coupling constant and @xmath12 and @xmath13 denote the expectation value and the partition function in a fixed topological sector respectively .
the expectation values with a fixed topological charge ; @xmath12 are evaluated by generating link variables with the following gauge action @xcite ; @xmath14 where @xmath15 denotes a plaquette and @xmath16 is a fixed constant .
this action impose the lscher s bound @xcite on the gauge fields , which realize an exact topological charge on the lattice , that is never changed in each step of the the hybrid monte carlo updation .
@xmath17 normalized by that of zero topological sector can be evaluated by decomposing it into three parts ; @xmath18 where @xmath19 denotes the classical minimum of the gauge action with topological charge @xmath20 , @xmath21 denotes the moduli integral , and @xmath22 , all of which are numerically calculable @xcite .
we choose the domain - wall fermion action with pauli - villars regulators for sea quarks .
the link variables are updated by the hybrid monte carlo algorithm .
the parameters are chosen as @xmath23 , @xmath24 .
we take @xmath25 and @xmath26 lattice where @xmath27 denotes the size of the extra dimension of domain - wall fermions .
50 molecular dynamics steps with a step size @xmath28 are performed in one trajectory .
configurations are updated every 10 trajectories . for each topological sector
, around 500 trajectories are taken for the thermalization staring from the initial configuration which is the classical instanton solution with topological charge @xmath20 .
we generate 300 configurations in @xmath29 sectors for the measurements and from 1000 to 10000 for the reweighting factors at various @xmath30 , where @xmath31 at @xmath32 and @xmath33 at @xmath34 .
the topological charge dependence of pseudoscalar condensates is derived by the anomaly equation ; @xmath35 where @xmath36 denotes the volume of the torus . as seen in fig .
[ fig : condaxq ] , our data show a good agreement with this equation .
then the @xmath0 meson correlators should have long - range correlations . from the clustering decomposition , this can be expressed as @xmath37 where @xmath38 means the expectation value with the topological charge @xmath39 in the region @xmath40 which denote the half of the large box , where the pseudoscalar operators reside , respectively and @xmath41 denotes the probability of the distribution where @xmath39 instantons appear in the box @xmath42 and @xmath43 appear in the box @xmath44 . in @xmath45 case , one obtains @xmath46 where we assume @xmath47 and use the anti - symmetry;@xmath48 as seen in fig .
[ fig : condaxq ] . on the other hand , at large @xmath20 ,
assuming the distribution @xmath41 to be gaussian around @xmath49 , the correlation can be evaluated as follows , @xmath50 where @xmath51 is a numerical constant . as seen in fig [ fig : longq ] , it is surprising that these very simple arguments describe the data quite well .
@xmath5 dependence of the @xmath0 correlators are evaluated by substituting the data into eq.([eq : exp ] ) .
[ fig : etaproptheta ] shows the result .
it is obvious that there are long - range correlations at @xmath1 while @xmath3 case is consistent with zero , which suggests our reweighting method works well at small @xmath5 .
we study @xmath20 and @xmath5 dependence of the pseudoscalar condensates and the @xmath0 meson correlators . we find
that pseudoscalar does condense in each topological sector ; @xmath52 and there exists a long - range correlation of @xmath0 meson ; @xmath53 which are well understood by the clustering properties .
it is also found that each contribution from different topological sectors plays very important role to produce non - trivial @xmath5 dependence of these observables . in particular , the cancellation the long - range correlation of @xmath0 meson requires accurate measurements of higher topological sectors .
it is also obvious that the fluctuation of the disconnected diagrams originates from these pseudoscalar condensates .
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b * 350 * , 92 ( 1995 ) . | we study the chiral condensates and the @xmath0 meson correlators of the massive schwinger model in @xmath1 vacuum .
our data suggest that the pseudoscalar operator does condense in a fixed topological sector and gives long range correlations of the @xmath0 meson .
we find that this is well understood from the clustering decomposition and statistical picture .
our result also indicates that even in @xmath2 case , the long range correlation of @xmath0 meson receives non - zero contributions from all the topological sectors and that their cancellation is non - trivial and requires accurate measurement of the reweighting factors as well as the expectation values .
it is then clear that the fluctuation of the `` disconnected '' diagram originates from the pseudoscalar condensates . |
in this supplementary section we present a list of virtual processes that contribute to the first term of eq.(9 ) in the main text .
table i corresponds to the single - level anderson model , whereas tables ii - iv dwell on the universal hamiltonian .
in addition , we present the dimensionless integral that gives rise to @xmath21 ( defined by eq .
15 in the main text ) : @xmath166 where @xmath167 and @xmath168 .
( [ eq : f ] ) can be derived by adding the @xmath169 transition amplitudes of tables ii - iv along with the second and third term of eq.(9 ) in the main text , with @xmath136 .
the derivation is simplified by exploiting time - reversal as well as particle - hole symmetry , although similar integrals may be derived in absence of particle - hole symmetry .
note that the sum and integral in eq .
( [ eq : f ] ) are uv - finite , even though numerous individual amplitudes in tables ii - iv are uv - divergent ; the delicate cancellation between different uv divergences adds considerable confidence on the veracity of our results .
moreover , we find that the main contribution to @xmath21 originates from states with @xmath170 and @xmath171 .
these observations together justify our assumption of energy - independent tunneling amplitudes and uniform energy - level spacings . in other words ,
our assumptions hold provided that the tunneling - amplitudes and the energy - level spacings vary slowly on energy scales of order @xmath14 , which is typically much smaller than the fermi energy . | we develop a general method to evaluate the kondo temperature in a multilevel quantum dot that is weakly coupled to conducting leads .
our theory reveals that the kondo temperature is strongly enhanced when the intradot energy - level spacing is comparable or smaller than the charging energy .
we propose an experiment to test our result , which consists of measuring the size - dependence of the kondo temperature . _
introduction. _ the kondo effect , a many - body phenomenon that emerges from the interaction between localized and itinerant fermionic degrees of freedom , is characterized by a low - temperature infrared ( ir ) divergence in perturbative calculations of physical observables such as resistivity and magnetic susceptibility@xcite .
this ir divergence is controlled by an ultraviolet ( uv ) cutoff @xmath0 that appears in the expression for the kondo temperature via @xmath1 , where @xmath2 is the kondo coupling and @xmath3 is the fermi level density of states per spin for itinerant carriers . such expression for @xmath4
is generally valid for @xmath5 and can be derived perturbatively starting from the venerable kondo hamiltonian @xcite , @xmath6
. an accurate microscopic theory of @xmath2 and @xmath0 provides crucial guidance for experimental explorations of strongly correlated electron systems . a precise way to quantify
@xmath4 is to work with a `` first - principles '' microscopic model that reduces to @xmath7 at energy scales below @xmath0 .
quite generally this first - principles hamiltonian can be written as @xmath8 , where @xmath9 captures the hybridization between the localized and itinerant degrees of freedom .
a perturbation theory calculation of physical observables in @xmath9 then yields hallmark kondo - like divergences , with @xmath0 and @xmath2 unequivocally determined in terms of the microscopic parameters of @xmath10 .
perhaps the first author to successfully implement the aforementioned scheme was haldane@xcite , who evaluated the magnetic susceptibility for the single - level anderson hamiltonian to fourth order in the hybridization amplitude @xmath11 . in the local - moment regime and for an infinite bandwidth in the continuum he obtained @xmath12 and @xmath13 , where @xmath14 is the coulomb charging energy . over time , haldane s formula @xmath15 has remained as the norm for the interpretation of experimental studies of the kondo effect in quantum dots@xcite , even though its applicability in these devices is _ a priori _ unclear .
a primary concern regarding haldane s formula is that it makes no reference to the multiple energy levels present in real dots .
this concern was first addressed by inoshita et al .
@xcite , who suggested that the dense energy spectrum of quantum dots should enhance @xmath4 by several orders of magnitude .
nevertheless , no such giant enhancement has been observed@xcite .
more recently , aleiner _
et al .
_ @xcite argued that , in real quantum dots with an average single - particle spacing @xmath16 , the main modification from haldane s formula should consist of replacing @xmath17 by @xmath18 .
the conclusions of refs .
[ , ] rely on effective kondo hamiltonians , and are thus less rigorous than the `` first - principles '' approach described above . in this paper
we follow the spirit of ref . [ ] and construct a precise theory for @xmath4 in real quantum dots that are weakly coupled to conducting leads .
we adopt the _ universal hamiltonian _ @xcite as an appropriate `` first - principles '' model for real quantum dots , and reach results that differ qualitatively from those of refs .
[ , , ] . in the infinite bandwidth limit we conclude that @xmath19 for @xmath20 , where @xmath21 is a function of @xmath22 ( fig .
1 ) . this result predicts an unconventional dependence of @xmath4 on the size of the quantum dot .
is a lengthscale defined in the text .
_ inset _ : the function @xmath21 of eq .
( [ eq : l_dot ] ) for a quantum dot with infinite equally - spaced energy levels . ] _ method. _ our calculation centers on spin - flip matrix elements of an effective hamiltonian , @xmath23 where @xmath24 and @xmath25 are degenerate eigenstates of @xmath26 , and @xmath27 is the effective hamiltonian derived from degenerate perturbation theory in @xmath9 .
@xmath24 and @xmath25 are tensor products of a target ( i.e. the localized degrees of freedom ) and a projectile ( i.e. an itinerant particle that scatters off the target ) .
both the spin of the projectile and the spin of the target are flipped in the course of spin - flip processes .
the calculation of eq .
( [ eq : a ] ) is considerably simpler than that of the magnetic susceptibility in ref .
[ ] because it requires neither partition functions nor external magnetic fields .
in spite of its relative simplicity , eq .
( [ eq : a ] ) is closely connected to the scattering t - matrix and thus to a physical observable , namely the scattering rate . in view of the above connection
, our approach exploits the long - known fact@xcite that in the kondo model the spin - flip matrix elements of the t - matrix produce the `` running '' kondo coupling @xmath28 where @xmath29 is the temperature .
the computation of eq .
( [ eq : a ] ) from a `` first - principles '' model and its subsequent identification with eq .
( [ eq : kondo ] ) produces the desired explicit expression for @xmath2 and @xmath0 in terms of microscopic parameters .
_ effective hamiltonian. _ given a hamiltonian @xmath8 , where @xmath26 has a degenerate energy spectrum , there exists a perturbative green s function technique@xcite to construct its exact eigen energies . according to this approach ,
the key eigenvalue equation to be solved is @xmath30 where @xmath31 is the projection operator onto the degenerate subspace spanned by the eigenvectors of the unperturbed energy @xmath32 , @xmath33 are the eigenvalues of @xmath10 and @xmath34 are the projections of the corresponding eigenvectors onto the projected subspace .
hence @xmath35 , which is hermitian on the projected subspace , may be identified with an effective hamiltonian . also , @xmath36 and @xmath37 for non - negative integers @xmath38 . in eq .
( [ eq : un ] ) , @xmath39 if @xmath40 and @xmath41 if @xmath42 .
in addition , @xmath43 .
@xmath44 is extended over all sets of non - negative integers @xmath45 satisfying the conditions @xmath46 @xmath47 and @xmath48 . from eq .
( [ eq : eigen ] ) it follows@xcite that @xmath49 _ single - level anderson model. _ in order to verify that eq .
( [ eq : a ] ) produces the correct @xmath0 and @xmath2 , we employ the simplest first - principles model for which rigorous results have been long established@xcite . using the standard notation ,
the anderson hamiltonian is @xmath8 , where @xmath50 we evaluate @xmath51 to fourth order in the hybridization amplitude @xmath11 .
we choose @xmath52 as the initial and final scattering states .
@xmath53 is the fermi sea in the continuum and @xmath54 denotes the empty state of the localized level .
the momentum of the projectile is assumed to be close to the fermi surface ( i.e. @xmath55 ) . @xmath24 and
@xmath25 can be connected only via spin - flip processes ; this choice is convenient in that it filters out spin - independent scattering .
the effective hamiltonian may be evaluated using eq .
( [ eq : messiah ] ) and noting that @xmath31 projects onto a two - dimensional subspace spanned by @xmath24 and @xmath25 ; the outcome reads @xmath56 with @xmath57 and @xmath58 where we have exploited @xmath59 and defined @xmath60 .
our @xmath27 connects states with equal energy and thus contains less information than the effective hamiltonian derived from a fourth - order schrieffer - wolff@xcite transformation , with which it agrees when @xmath61 . at any rate
, this limitation has no practical consequences because all observable properties are determined by @xmath62 and @xmath63 located at the fermi surface . from eq .
( [ eq : h_eff ] ) the lowest order contribution to @xmath51 reads @xmath64 where @xmath65 denotes virtual intermediate states that satisfy @xmath66 . also , @xmath67 and @xmath68 ( @xmath69 as we focus on elastic scattering ) .
each time @xmath9 acts on a state it changes the number of particles by one both in the continuum and in the localized level , yet it conserves the total number of particles and the total spin . accordingly @xmath70 and @xmath71 .
next , we compute the 4th order contribution to the scattering amplitude using @xmath72 in eq .
( [ eq : h_eff ] ) : @xmath73 where @xmath74 . the second and third terms in eq .
( [ eq : t4 ] ) were derived by inserting @xmath75 between two subsequent @xmath31 operators in eq .
( [ eq : h_eff ] ) .
in particular , the second term in eq .
( [ eq : t4 ] ) is uv divergent and plays a crucial role in ensuring that @xmath51 remains uv finite even when the bandwidth of the continuum states is taken to infinity .
\{@xmath76,@xmath77,@xmath78 } label intermediate states , which are collected in table i of the supplementary material .
summing over all contributions and assuming an infinite bandwidth in the continuum we obtain @xmath79 where @xmath3 is the fermi surface density of states in the continuum and @xmath80 is the infrared energy cutoff .
for the present zero - temperature calculation @xmath81 .
@xmath82 can be identified ( modulo a factor @xmath83 ) with eq .
( [ eq : kondo ] ) , which yields @xmath84 and @xmath85 .
these expressions agree with those of ref .
[ ] .
_ connection with scattering theory. _ here we show that eq .
( [ eq : a ] ) is closely linked to a physical observable . according to standard scattering theory@xcite
, the spin - dependent scattering amplitude in the anderson model is given by @xmath86 where @xmath87 and @xmath88 label the spin of the projectile , @xmath89 and @xmath90 are eigenstates of the full hamiltonian @xmath10 in absence of projectiles and @xmath91 is the exact ground state energy , i.e. @xmath92 and @xmath93 . in the local moment regime and for a large system containing an odd number of electrons
@xmath89 and @xmath90 are spin 1/2 ground states . at @xmath94
the spin resides on the localized level but for @xmath95 the magnetization is spatially delocalized@xcite .
below we use @xmath96 and @xmath97 to denote the spin direction ( @xmath98 or @xmath99 ) of @xmath89 and @xmath90 , respectively .
we evaluate spin - flip matrix elements perturbatively for the real part of eq .
( [ eq : t_lan ] ) with @xmath55 .
it is immediate to see that the leading order contribution agrees with @xmath100 .
the fourth order term involves expanding @xmath89 , @xmath90 , @xmath10 and @xmath91 to second order in @xmath9 ; the result agrees with eq .
( [ eq : t4_and ] ) .
in particular , the @xmath101 ( re)normalization of @xmath89 and @xmath90 @xcite coincides with the last term in eq .
( [ eq : t4 ] ) .
this easily - overlooked term is essential for the correct evaluation of the t - matrix . in sum , @xmath102 for @xmath55 .
the su(2 ) symmetry of the anderson hamiltonian dictates @xmath103 , where @xmath104 is a vector of pauli matrices and we ignore spin - independent scattering .
the imaginary part of the t - matrix , which quantifies the electronic scattering rate off the localized level , can then be extracted by virtue of the optical theorem : @xmath105 ^ 2 \propto j^2 + 2 j^3 \log(\lambda/\omega)+ ... $ ] _ multilevel quantum dots. _ we are now ready to evaluate @xmath0 and @xmath2 for real quantum dots via eq .
( [ eq : a ] ) . the universal hamiltonian of a quantum dot that is weakly connected to a conducting lead can be written as @xmath8 , where @xmath106 @xmath107 labels the discrete single - particle energy levels in the dot , @xmath108 is the number operator for dot electrons , @xmath109 is the gate charge , @xmath14 is the charging energy , and we have neglected intradot exchange interactions . for simplicity we take @xmath110 and @xmath111 for @xmath112 .
these simplifications are partly justified because our theory is uv - finite ( see below and the suppl .
material ) .
the unperturbed initial and final scattering states are @xmath113 , where @xmath53 is the fermi sea in the lead , @xmath114 creates a projectile in the lead just above the fermi surface , @xmath115 is an eigenstate of the dot containing @xmath116 electrons and @xmath117 creates an electron in the dot at level `` 0 '' located immediately above the highest ( @xmath118-th ) doubly - occupied level ( @xmath119 ) .
the unperturbed energy is @xmath120 , where @xmath121 is the kinetic energy of the filled fermi seas ( herein @xmath122 ) and @xmath123 is the kinetic energy for the singly - occupied level `` 0 '' ( tunable by a gate voltage ) .
@xmath124 is the coulomb energy cost for adding @xmath125 electrons to the dot ; we have chosen @xmath126 without loss of generality by shifting all @xmath127 by a constant .
we begin by recognizing that @xmath128 and that eq .
( [ eq : h_eff ] ) remains valid .
therefore @xmath100 and @xmath129 are given by eqs .
( [ eq : t2_0 ] ) and ( [ eq : t4 ] ) , respectively . for the former
we find @xmath130 where we used @xmath131 and defined @xmath132 .
next , we focus on @xmath129 .
its computation requires considering numerous sets of intermediate states ; these are listed in tables ii , iii and iv of the supplementary material . for simplicity
we start by separating out the contribution from the @xmath133 ( singly occupied ) level in the dot . assuming an infinite bandwidth in the lead we arrive at @xmath134 where @xmath135 .
eqs .
( [ eq : a2d ] ) and ( [ eq : m0_3 ] ) are independent of @xmath16 and essentially identical to those of the single - level anderson model . finally , we sum the contributions from @xmath136 levels .
these depend on @xmath16 and encode the influence of the multilevel energy spectrum in the kondo physics . tables ii and iii show that individual virtual processes involving @xmath136 levels are plagued with ir and uv divergences . remarkably , different divergences end up cancelling one another , partly assisted by the last two terms in eq .
( [ eq : t4 ] ) . on one hand ,
the cancellation of @xmath136 infrared divergences corroborates that kondo correlations arise only from processes involving the singly occupied level in the dot . on the other hand
, the cancellation of @xmath136 ultraviolet divergences confirms that high - energy excited states in the dot and lead do not alter the physics of the kondo effect . in spite of being divergence free ,
the influence of @xmath136 levels is important and makes the kondo coupling @xmath16-dependent . in the infinite bandwidth limit and in proximity to the particle - hole symmetric point ( @xmath137 )
we obtain @xmath138,\ ] ] where @xmath21 is a dimensionless function of @xmath22 evaluated numerically ( fig . 1 and suppl .
material ) . when @xmath139 , @xmath140 and multilevel effects are negligible
; in the opposite limit @xmath141 and multilevel effects are important .
the sum of eqs .
( [ eq : a2d ] ) , ( [ eq : m0_3 ] ) and ( [ eq : a4 ] ) can be arranged as @xmath142 .
for @xmath143 we obtain @xmath144 where @xmath145 is the kondo coupling corresponding to a single - level dot and @xmath146 is the width of the energy levels in the dot .
eq .
( [ eq : l_dot ] ) is valid for @xmath147 , i.e. @xmath148 , and constitutes the main result of this paper .
we selected @xmath0 on physical grounds so that it sets the energy scale below which ( i ) the universal hamiltonian maps onto the kondo hamiltonian , ( ii ) the renormalization group flow for @xmath2 is that of the simple kondo model . _
experimental implications. _ from eq .
( [ eq : l_dot ] ) , the kondo temperature for a multilevel quantum dot is @xmath149 , for any @xmath22 insofar as @xmath148 ( this condition implies that the broadening of the many - body energy eigenvalues of the isolated dot is much smaller than the energy spacing between them ) .
fig .
1 displays @xmath4 as a function of the linear dot dimension @xmath150 . introducing a lengthscale @xmath151 such that @xmath152 and @xmath153
, it follows that @xmath154 and @xmath155 .
@xmath156 is kept fixed ( independent of @xmath150 ) and we take @xmath157 and @xmath158 ; these are reasonable extrapolations based on available experimental data .
clearly haldane s single - level formula is accurate for smallest dots with @xmath139 ; in contrast , the multilevel enhancement of the kondo temperature becomes important for larger dots with @xmath159 . for @xmath160
, @xmath21 is so large that @xmath20 is possible only for a very small value of @xmath161 , which in turn results in an unmeasurably low @xmath4 .
therefore eq .
( [ eq : l_dot ] ) is experimentally relevant for dots with @xmath162 , wherein the multilevel enhancement is more modest yet still noticeable ( @xmath163 ) . in conclusion ,
we have developed a method to evaluate the kondo temperature of real quantum dots with unprecedented precission .
our theory predicts an unconventional and potentially measurable size - dependence of @xmath4 in dots with @xmath164 .
our formalism is valid and our results readily generalizable for models that incorporate energy - dependence in the dot - lead tunneling amplitude as well as non - uniform distribution of energy levels in the dot . _
acknowledgements. _ we are indebted to a. andreev , o. entin - wohlman , j. folk and l. glazman for helpful conversations .
this research has been supported by nserc and cifar .
50 for a review see e.g. p. coleman , _ many - body physics _ , http://www.physics.rutgers.edu/@xmath165coleman/mbody.html .
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w. marshall ( academic press , new york , 1967 ) .
see e.g. a. messiah , _ quantum mechanics ( vol .
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langreth , phys .
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( addison - wesley , reading , ma , 1994 ) . |
the experimentally clean signatures of @xmath0 and @xmath1bosons make the measurement of these processes in association with jets well suited to test perturbative qcd at the lhc .
the processes allow for comparisons of multi jet production with predictions either from the parton shower approach or from exact multi parton matrix elements ( @xmath5 ) matched with parton showers . in addition , full next to leading order ( @xmath6 ) calculations are also available for comparison with many of the results .
the @xmath7 processes also differ from pure qcd multi jet processes with respect to the scale of the hard interaction , due to the large mass of the electroweak gauge bosons .
measurements of @xmath0/@xmath1+jets are also important to control backgrounds to other measurements at the lhc . in the standard model context ,
one example is the top quark cross section measurements , where @xmath0+jet is often the dominant background .
also several beyond the standard model searches , such as the zero lepton susy search , suffer from irreducible background from either @xmath0+jets or @xmath1+jets , or both . here
we report on the atlas @xmath1+jets and @xmath0+jets cross section measurements @xcite based on data recorded during 2010 .
the analyses include the electron and muon decay channels and are based on an integrated luminosity of 33 pb@xmath8 .
the atlas detector systems were all fully operational during this data taking period and the detector acceptance considered was approximately determined by the following constraints .
electrons were used within the inner detector acceptance ( @xmath9 ) , whereas reconstructed muons were considered inside the acceptance of the trigger chambers ( @xmath10 ) .
the electron ( muon ) @xmath11 ( @xmath4 ) was required to be larger than 20 gev , in order to be well inside the highly efficient plateau of the triggers .
jets were reconstructed inside the main calorimeters ( @xmath12 ) and missing transverse energy was based on the full calorimeter acceptance ( @xmath13 ) .
the cross section measurements were made within the kinematic region defined by the event selection , which was well covered by the atlas detector acceptance , * @xmath14 * @xmath15 * @xmath16 * @xmath17 * @xmath18 note that here the jet @xmath4 requirement , as well as the rapidity variable , differs between the @xmath0 ( 20 gev ) and @xmath1 analysis ( 30 gev ) .
the @xmath19 criteria refers to the leptons and all selected jets and the @xmath1 selection also require the two leptons to be of opposite charge .
the results were then corrected for detector effects , using a bin by bin unfolding method , and compared with theory ( @xmath20 ) predictions inside the same kinematic region .
for the theory results , jets were reconstructed using the same algorithm based on all final state particles with a lifetime larger than 10 ps , except the leptons from the @xmath7 decays .
lepton momenta also included any photons radiated within @xmath21 .
a good agreement was generally found between the selected data candidates and predictions from mc , for both @xmath1 and @xmath0 events in both the electron and muon channels . regarding the background for these measurements , the background coming from qcd
was estimated based on a data driven method whereas the electroweak and top backgrounds were estimated from mc .
the background contamination of the selected @xmath1+jets samples was of the order of 1% for the muon channel , as well as 5% for the electron channel . in the @xmath0+jets samples ,
the background was in the order of 10% .
the main source of uncertainty in these measurements comes from the jet energy scale , which contributes with approximately 10% , followed by pile up corrections ( @xmath22% ) and luminosity ( @xmath23% ) .
more details about the analysis are found in @xcite .
the obtained number of events were then used to measure the differential cross section times branching ratio with respect to a number of different quantities .
all the results correspond to inclusive measurements , corrected for detector effects , within the kinematic region defined by the event selection above .
as function of the number of jets ( left ) .
cross section for @xmath24 as function of @xmath4 of the leading jet ( right).[fig : zw2],title="fig : " ] as function of the number of jets ( left ) .
cross section for @xmath24 as function of @xmath4 of the leading jet ( right).[fig : zw2],title="fig : " ] the differential cross section with respect of the number of selected jets was measured both using @xmath0 and @xmath1 events . the absolute cross section and the ratio of cross sections from events with @xmath25 jets over @xmath26 jets
were measured .
some of the uncertainties are reduced in the ratios .
figure [ fig : zw2 ] ( left ) shows the @xmath27 cross section as a function of number of selected jets .
the measured values show a good agreement with the nlo predictions , here represented by results obtained by mcfm .
the results are also in good agreement with expectations from the multi parton me programs ( alpgen and sherpa ) , which have been normalized to the inclusive nnlo cross sections obtained by the fewz program .
the results do on the other hand show poor agreement with the lo plus parton shower results ( pythia ) for events with more than one jet .
this is due to the combination of a not properly covered phase space ( @xmath1s are not produced by the parton shower ) together with using the parton shower approach for the hard jets .
the results are shown together with the corresponding ratios between the results obtained from data over predictions from the mc programs mcfm , alpgen and sherpa .
as function of @xmath4 of the second leading jet ( left ) .
cross section for @xmath28 as function of @xmath3 ( right).[fig : zw3],title="fig : " ] as function of @xmath4 of the second leading jet ( left ) .
cross section for @xmath28 as function of @xmath3 ( right).[fig : zw3],title="fig : " ] figure [ fig : zw2 ] ( right ) shows the differential cross section with respect to the leading jet @xmath4 for @xmath24 .
the measurement is performed separately for events with 1 to 4 jets .
the results from the @xmath0+jets measurements are also compared against nlo predictions from blackhat
sherpa , where @xmath0 + 3jets predictions at nlo is compared with lhc data for the first time .
the results are shown together with the corresponding ratios between results from mc over data , for events with 1 and 2 selected jets . again
a good agreement was found between the measurement and the mc predictions .
the differential cross sections were also measured with respect to the @xmath4 of the other selected jets in the event , using the 2 leading jets in @xmath1 events and up to 4 leading jets in the @xmath0 analysis . the cross section with respect to the @xmath4 of the second leading jet
is shown in figure [ fig : zw3 ] ( left ) for @xmath29 . a good agreement is shown and similar agreements were also found with respect to the other jets , both in the @xmath0 and @xmath1 results . in the @xmath0 analysis the cross section was also measured with respect to @xmath3 .
this quantity corresponds to the scalar sum of the @xmath4 from the jets as well as leptons , i.e. muon , electron and neutrinos , in the event , and this is the quantity which was used to characterize the scale of the hard process in the mc simulations .
the results from @xmath28 are shown in figure [ fig : zw3 ] ( right ) , which are produced separately for event with 1 to 4 jets .
the ratios between the results from mc over data are shown and a generally good agreement was found in both the electron as well as the muon channel . | we report on the measurements of inclusive @xmath0+jets and @xmath1+jets cross sections in proton proton collisions at @xmath2 tev with the atlas detector .
cross sections , in both the electron and muon decay modes of the bosons , are presented as a function of jet multiplicity , the transverse momentum of the jets and the quantity @xmath3 which is the scalar sum of the @xmath4 in the event .
measurements are also presented of the ratios of cross sections .
the measured cross sections are compared to different particle
level predictions , based on perturbative qcd , where the measured @xmath0 + 3jet cross section is for the first time compared with next to leading order calculations .
address = cavendish laboratory , university of cambridge , cambridge , uk . |
to study dynamics of super yang - mills theory , a lattice formulation is an attractive tool .
there are some approaches , however limited , not exact , realization has done @xcite .
why do we stress the exact symmetry ?
it is because the point of our approach , the scenario is that the exact fermionic symmetry on lattice with keeping @xmath0 symmetry in the continuum limit , then it s just susy if it is not brs symmetry .
so , the first step of our trial is to construct exact fermionic , not brs - like symmetry on lattice .
we present new lattice models with an exact fermionic symmetry .
we start with considering in the system with minimal degrees of freedom , realized as one - cell model , derive the fermionic transformation , which is expected to contain the continuum supersymmetry .
then it is extended to the entire lattice in a non - trivial way , it is multi - cell model . we perform it by introducing the particular lattice structure with a special pattern called ` ichimatsu ' pattern in japanese . also using that pattern , we give another model , which we call pipe model . finally , we reach cell and pipe mixed model .
first , we consider a fundamental lattice .
let us refer it as a cell .
the action consists of gauge and fermion part : @xmath1 @xmath2 the gauge action @xmath3 is the plaquette action , here @xmath4 is the gauge coupling constant . for the fermion part
, we put real staggered fermion on each site .
the fermion action @xmath5 consists of terms which corresponds to fermions located at neighbouring sites connected by link variables .
the coefficient @xmath6 is a sign factor for the staggered fermion .
we introduce a fermionic transformation for which mixes the fermion and the link variables .
we assume the form to realize the ordinary susy transformation in a continuum limit .
let us call our transformation as pre - susy transformation .
the susy transformation of a fermion is to give the field strength , @xmath7 , hence , our ansatz of the fermion transformation is given as @xmath8 where @xmath9 is a grassmann - odd parameter .
we chose the transformation of the link variables in order the operator from fermion action under the fermion transformation , to balance the one which produced from the change of the gauge action .
@xmath10 where @xmath11 is a grassmann - odd parameter .
now we are going to check the action invariance under pre - susy transformation .
the variation of the action under the pre - susy contains terms cubic and linear in the fermion variables . for the action to be invariant
, these two sets of terms should vanish separately : @xmath12
corresponding to these conditions , we obtain two types of relations between introduced parameter : @xmath13 @xmath14 we can check out exact symmetry in this one - cell model . solving these relations , we find there are @xmath15 independent parameters for each site .
then , we consider our model with pre - susy in an entire lattice space . since naive extension only produces @xmath16 symmetry , to be discussed later , here we take a sophisticated way .
we put a restriction on the plaquette variables . in the simplest 2-dimensional space ,
the allowed plaquette variables form ` ichimatsu ' pattern as fig .
[ 2d - ichimatsu ] .
contrary to the ordinary lattice theory , not all the possible plaquette variables are included in this extension .
all the fermion and link variables are included , while only the plaquette variables are restricted .
even in arbitrary space dimension , we can do this multi - cell extension in the way that there presents the same pattern on any two - dimensional surfaces . by taking same steps in the one - cell case
, one can find that the action invariance under pre - susy transformation in multi - cell included model with the ichimatsu pattern .
however , it is easy to suppose that in the cell model , the perturbative picture is not present , since the interaction between cells is introduced through the fermion variables , which belong to two neighbouring cells .
( details are reported in our previous paper @xcite . ) by the way , here we construct another model , by exchanging the allowed and discarded plaquette variables , since there are two ways to put the ichimatsu pattern on a two - dimensional surface .
now let us call this alternative model , pipe model , from its structure in 3-dim . case .
the pipe model is present for any dimension , ( anyhow this distinction is meaningless in 2-dim . )
its plaquette variables are arranged in the complimentary to the cell model .
it may say that we decomposed ordinary lattice space using the ichimatsu pattern .
+ this model differs from the cell model only in the gauge sector .
however , repeating the same procedure as the cell model , we find the presence of the pre - susy in the pipe model .
we do not encounter the difficulty in the cell model even when we remove the fermion variables .
however , we may show that the pipe model without fermion variables do not have an appropriate continuum limit ( can be shown equivalent to trivial model statistical mechanically ) .
so far , we consider these two models separately .
now we are going to consider its mixed model to overcome the difficulties . in defining the mixed model ,
we assign different coupling constants to two sets of plaquette variables in the gauge action , @xmath17 is for those of the cell model and @xmath18 is for the pipe model .
@xmath19 + s_{g}^{pipe}[\beta_p]\ ] ] while in the pre - susy transformation , the fermion transformation that produces the plaquette variables is influenced to be producing two sets of plaquette variables : @xmath20 where @xmath21 ( @xmath22 ) means plaquette variables of the cell ( pipe ) model .
although it is not so obvious whether we may keep the fermionic symmetry or not , we can find exact fermionic symmetry even in the mixed model .
we note two important features here .
first , the mixed model may be treated properly even in a perturbative way , since there included all the plaquette variable same as in the ordinary lattice .
second , when we introduced our pre - susy transformation , we assumed the form of the fermion transformation to be related to the continuum susy .
considering the naive continuum limit of eq .
( [ equ : dxi ] ) , we find it becomes @xmath23 where @xmath24 are grassmann - odd parameters related to the parameter @xmath25 .
it is proportional to the field strength , but is also proportional to the difference of the two coupling constants @xmath17 and @xmath18 .
this shows two couplings , @xmath17 and @xmath18 must differ to have proper continuum limit , is the reason for the transformation will be higher order and vanish in the limit in the ordinary lattice model .
if the fermionic symmetry is really related to the expected susy , the continuum limit is to be studied with the cell and pipe mixed model .
first result of pure gauge system with the ichimatsu structure is reported in @xcite .
we presented new lattice models with an exact fermionic symmetry , as a step towards the super yang - mills theory on lattice . in the last cell and pipe mixed model
, we find the crucial condition towards the continuum susy and the ichimatsu pattern plays an important role in our models .
we may confirm that our models satisfy reasonable requirements for a proper lattice theory , such as translational and rotational invariance and the reflection positivity , also by the property of the ichimatsu pattern .
we also work on the realization of the staggered majorana fermion , however the complete realization of spinor structure still remains as an unsolved question .
now , the recovery of the spinor structure and the doubling problem are crucially important and most difficult .
some discussions will be given in our forthcoming paper @xcite .
9 g. curci and g. veneziano , nucl .
b292 ( 1987 ) 555 .
d. b. kaplan and martin schmaltz , chin .
38 ( 2000 ) 543
. k. itoh , m. kato , h. sawanaka , h. so and n. ukita , prog .
( 2002 ) 363 .
k. itoh , m. kato , m. murata , h. sawanaka and h. so , _ vacuum structure of the ichimatsu - decomposed lattice models _ , these proceedings .
k. itoh , m. kato , h. sawanaka , h. so and n. ukita , in preparation . | we present the lattice models with exact fermionic symmetries relating fermions and link variables .
the plaquettes are distributed in an ichimatsu pattern ( chequered ) .
we explain this peculiar structure allows us to have a translation in the algebra of the fermionic symmetries . |
we consider ideal mhd in twodimensional space , @xmath2 the dependent variables @xmath3 denote the fluid s density , pressure , and velocity .
in addition to , the magnetic field @xmath4 satisfies @xmath5 the fluid is assumed to be polytropic , @xmath6 , and have a constant temperature @xmath7 , so that @xmath8 with constant sound speed @xmath9 . by scaling , we assume without loss of generality that @xmath10 we abbreviate as @xmath11 with @xmath12 using , we also write it as a symmetric hyperbolic system , @xmath13 with @xmath14 , @xmath15 , and @xmath16 applying the chain rule , we rewrite as @xmath17 where @xmath18 with @xmath19 note that , as we have used on the way from to , the matrices @xmath20 and @xmath21 in are _ not _ the jacobians of the fluxes @xmath22 and @xmath23 .
ideal mhd shock waves , in their prototypical form , have the structure @xmath24 where @xmath25 is the direction of propagation and @xmath26 the speed of the shock wave .
function being a weak solution of is equivalent to the rankine - hugoniot conditions @xmath27 due to rotational and galilean invariance it is without loss of generality that we henceforth assume that @xmath28 i. e. we exclusively consider shock waves of the form @xmath29 and the rankine - hugoniot conditions read @xmath30 note now first that for waves , as for any solutions of whose spatial dependence is only via @xmath31 , the divergence - free condition reduces to @xmath32 we assume and simply write @xmath33 instead of @xmath34 . in this paper , we are interested in lax shocks . following @xcite , two states @xmath35 that satisfy the rankine - hugoniot conditions constitute a @xmath36 and a @xmath37 two states do satisfy the rankine - hugoniot conditions
if and only if the two quadruples @xmath38 and @xmath39 have coinciding images under the mapping @xmath40 that @xmath22 induces by omitting its forth , trivial component , in other words if both quadruples satisfy the four equations @xmath41 for the same values of the four parameters @xmath42 . as simple arguments or @xmath43 give no lax shocks ] show , we lose no generality in assuming that @xmath44 using in and inserting the result and in then yields @xmath45 as for every solution @xmath46 of , relations , , provide unique associated values for @xmath47 and @xmath48 , understanding will give a complete picture .
one distinguishes two cases .
_ @xmath49 : parallel shocks .
_ in this case , has two solutions @xmath50 the corresponding states constitute a @xmath51 and a @xmath52 the fact that the value of @xmath53 has no influence on the @xmath54 components of parallel shocks is easily understood by noticing that they have @xmath55 and thus are purely gas dynamical .
_ @xmath56 : non - parallel shocks .
_ in this case , @xmath57 tends to @xmath58 not only for @xmath59 und @xmath60 , but also for @xmath61 .
thus for every @xmath62 has two solutions @xmath63 that consitute a slow shock .
similarly , for every @xmath64 there are two solutions @xmath65 that define a fast shock .
according to majda s theory @xcite on the persistence of shock fronts , the local - in - time stability of the planar discontinous wave is determined by the behaviour of the lopatinski determinant @xmath66 where @xmath67 . while _ uniform stability
_ corresponds to the non - vanishing of @xmath68 on all of @xmath69 , shocks with @xmath70 are _ neutrally stable _ or _ strongly unstable _ , respectively .
the ingredients of the lopatinski determinant are @xmath71 where @xmath72 denote @xmath73 .
the theory of hyperbolic initial - boundary value problems @xcite implies that @xmath74 are well - defined bundles of constant dimension . to be precise , it is on @xmath75 that the _ lopatinski matrices _ @xmath76 have constantly trivial neutral spaces and thus `` consistent splitting '' , i. e. , stable and unstable spaces of constant dimensions , so that in particular @xmath77 are constant ; for points @xmath78 with purely imaginary values of @xmath79 , the @xmath80 are defined as limits from the interior of @xmath69 @xcite . from the one - dimensional ` lax counting ' of characteristic speeds @xcite , we know that @xmath81 while @xmath82 the lopatinski determinant @xmath68 being degree - one homogeneous in @xmath83 , we from now on fix the transverse wave number to @xmath84 to avoid abundant notation , we also fix from now , again without loss of generality , @xmath85 and use the two parameters @xmath86 instead of the three paramters @xmath87 . for parallel shocks ,
our choice implies @xmath88 in this paper we concentrate on slow shocks .
the following is a key observation . for slow parallel mhd shocks in , with and @xmath89 , @xmath90 interesting manipulations
show that one can take @xmath91 together with @xmath92 , this yields @xmath93.\ ] ] , cf .
theorem 1 .
the black boundary is the lax condition : @xmath94 . ]
the situation of parallel shocks is degenerate as it possesses a reflectional symmetry in the transverse ( @xmath95-)direction . for the lopatinski determinant
this symmetry means that @xmath96 vanishes exactly if @xmath97 does .
perturbing the parameter @xmath9 away from @xmath98 breaks this symmetry , and the zero of @xmath68 that we found , for @xmath99 at @xmath100 , splits . for all values of @xmath101 that permit a ( then unique ) slow mhd shock wave , we write @xmath102 for the corresponding lopatinski determinant . starting from theorem 1 , we found the following .
there are an @xmath103 and two functions , @xmath104 both defined on @xmath105 , such that @xmath106 for some values of @xmath9 between @xmath107 and @xmath108 .
the red curve corresponds to @xmath99 and thus to the red curve in fig . 1 .
] a detailed description of the numerics is postponed to a later publication .
do unstable modes emerge in families of shock waves that correspond to parameter values which cross the critical manifold ?
the following is what we conclude from numerical computations . .
both from a physics perspective and as the evans function for non - ideal shock waves is intimately related to the lopatinski determinant for their non - ideal counterparts @xcite , one expects the _ gallopping instability _ described in this paper to occur also in the presence of viscosity and and electrical resistivity .
s. k. godunov : lois de conservation et intgrales dnergie des quations hyperboliques .
_ nonlinear hyperbolic problems , proceedings st .
etienne 1986_. 135 - 148 .
lecture notes in mathematics 1270 , springer : berlin etc . , 1987 . | this note studies classical magnetohydrodynamic shock waves in an inviscid fluidic plasma that is assumed to be a perfect conductor of heat as well as of electricity . for this mathematically prototypical material
, it identifies a critical manifold in parameter space , across which slow classical mhd shock waves undergo emergence of a complex conjugate pair of unstable transverse modes . in the reflectionally symmetric case of parallel shocks
, this emergence happens at the spectral value @xmath0 , and the critical manifold possesses a simple explicit algebraic representation .
results of refined numerical treatment show that for only _ almost _ parallel shocks the unstable mode pair emerges from _ two _ spectral values @xmath1 .
= 1 |
in this work we calculate the free energy of qcd on the lattice , up to three loops in perturbation theory .
we employ wilson gluons and the @xmath2 improved sheikholeslami - wohlert ( clover ) @xcite action for fermions .
the purpose of this action is to reduce finite lattice spacing effects , leading to a faster approach to the continuum .
dynamical simulations employing the clover action are currently in progress by the cp - pacs / jlqcd @xcite and ukqcdsf @xcite collaborations and therefore perturbative studies of properties of the qcd action with clover quarks are worthy of being undertaken .
the free energy , in the simpler case of wilson fermions , was studied in @xcite .
the free energy in qcd on the lattice can be related to the average plaquette .
the results find several applications , for example : a ) in improved scaling schemes , using an appropriately defined effective coupling which depends on the average plaquette ( see , e. g. , @xcite ) , b ) in long standing efforts , starting with @xcite , to determine the value of the gluon condensate , c ) in studies of the interquark potential @xcite , and d ) as a test of perturbation theory , at its limits of applicability . indeed , regarding point ( d ) above , the plaquette expectation value is a prototype for additive renormalization of a composite , dimensionful operator .
the vacuum diagrams contributing to such a calculation are power divergent in the lattice spacing and may well dominate over any nonperturbative signal in a numerical simulation .
starting from the wilson formulation of qcd on the lattice , with the addition of the clover ( sw ) fermion term , the action reads in standard notation : @xmath3 , \nonumber \\ s_f & = & \sum_{f}\sum_{x } ( 4r+m_b)\bar{\psi}_{f}(x)\psi_f(x ) \nonumber \\ & & -{1\over 2}\sum_{f}\sum_{x,\,\mu } \left [ \bar{\psi}_{f}(x)\left ( r - \gamma_\mu\right ) u_{\mu}(x)\psi_f(x+\hat{\mu})+ \bar{\psi}_f(x+\hat{\mu})\left ( r + \gamma_\mu\right ) u_{\mu}(x)^\dagger \psi_{f}(x)\right]\nonumber \\ & & + { i\over 4}\,c_{\rm sw}\,\sum_{f}\sum_{x,\,\mu,\,\nu } \bar{\psi}_{f}(x ) \sigma_{\mu\nu } { \hat f}_{\mu\nu}(x ) \psi_f(x ) \label{latact}\end{aligned}\ ] ] @xmath4 here @xmath5 is the usual product of @xmath1 link variables @xmath6 along the perimeter of a plaquette in the @xmath7-@xmath8 directions , originating at @xmath9 ; @xmath10 denotes the bare coupling constant ; @xmath11 is the wilson parameter , which will be assigned its standard value @xmath12 ; @xmath13 is a flavor index ; @xmath14 $ ] .
powers of the lattice spacing @xmath15 have been omitted and may be directly reinserted by dimensional counting .
the clover coefficient @xmath16 is a free parameter for the purposes of the present calculation and our results will be presented as a polynomial in @xmath16 , with coefficients which we compute .
preferred values for @xmath16 have been suggested by both perturbative ( 1-loop ) @xcite and non - perturbative @xcite studies .
we use the standard covariant gauge - fixing term @xcite ; in terms of the vector field @xmath17 @xmath18 $ ] , it reads : @xmath19 having to compute a gauge invariant quantity , we chose to work in the feynman gauge , @xmath20 .
covariant gauge fixing produces the following action for the ghost fields @xmath21 and @xmath22 @xmath23 + \frac{i\,g_0}{2 } \,\left[q_{\mu}(x ) , \delta^+_{\mu}\omega(x ) \right ] \nonumber\\ & & - \frac{g_0 ^ 2}{12 } \,\left[q_{\mu}(x ) , \left [ q_{\mu}(x ) , \delta^+_{\mu}\omega(x)\right]\right]\nonumber\\ & & - \frac{g_0 ^ 4}{720 } \,\left[q_{\mu}(x ) , \left[q_{\mu}(x ) , \left[q_{\mu}(x ) , \left [ q_{\mu}(x ) , \delta^+_{\mu}\omega(x)\right]\right]\right]\right ] + \cdots \bigr)\biggr\ } , \nonumber\\ & \delta^+_{\mu}\omega(x ) \equiv \omega(x +
{ \hat \mu } ) - \omega(x)&\end{aligned}\ ] ] finally the change of integration variables from links to vector fields yields a jacobian that can be rewritten as the usual measure term @xmath24 in the action : @xmath25 in @xmath26 and @xmath27 we have written out only terms relevant to our computation .
the full action is : @xmath28 the average value of the action density , @xmath29 , is directly related to the average plaquette . for the gluon part
we have : @xmath30 as for @xmath31 , it is trivial in any action which is bilinear in the fermion fields , and leads to : @xmath32 ( @xmath33 : number of fermion flavors ) .
we will calculate @xmath34 in perturbation theory : @xmath35 the @xmath36-loop coefficient can be written as @xmath37 where @xmath38 is the contribution of diagrams without fermion loops and @xmath39 comes from diagrams containing fermions .
the coefficients @xmath38 have been known for some time up to 3 loops @xcite ( also in 3 dimensions @xcite , where they are applied to `` magnetostatic '' qcd @xcite and to dimensionally reduced qcd @xcite ) .
independent estimates of higher loop coefficients have also been obtained using stochastic perturbation theory @xcite .
the fermionic coefficients @xmath39 are known to 2 loops for overlap fermions @xcite and up to 3 loops for wilson fermions @xcite ; in the present work we extend this computation to the clover action .
the calculation of @xmath40 proceeds most conveniently by computing first the free energy @xmath41 , where @xmath42 is the full partition function : @xmath43 \exp(-s ) \label{z}\ ] ] then , @xmath34 is extracted through @xmath44 in particular , the perturbative expansion of @xmath45 : @xmath46 leads immediately to the relations : @xmath47
a total of 62 feynman diagrams contribute to the present calculation , up to three loops .
the first 36 diagrams are totally gluonic , and the others have both gluon and fermion contributions ; these are shown in appendix a. the involved algebra of lattice perturbation theory was carried out using our computer package in mathematica . the value for each diagram is computed numerically for a sequence of finite lattices , with typical size @xmath48 .
certain diagrams must be grouped into infrared - finite sets , before extrapolating their values to infinite lattice size ( diagrams 11 + 12 + 13 , 22 through 36 , 43 + 53 , 44 + 52 + 58 , 46 + 56 , 51 + 57 , 55 + 60 , 61 + 62 ) .
extrapolation leads to a ( small ) systematic error , which is estimated quite accurately ; a consise description of the procedure is provided in ref .
@xcite .
diagrams in the shape of a triangular pyramid ( 18 , 19 , 20 , 49 , 50 ) are the most cpu demanding , since integration over the 3 loop momenta can not be factorized ; these diagrams were necessarily evaluated for smaller @xmath49 , but fortunately @xmath50 was already sufficient for a very stable extrapolation in these cases . diagram 40 vanishes identically by color antisymmetry .
we have calculated @xmath56 , @xmath57 for typical values of the lagrangian ( unrenormalized ) fermion mass parameter @xmath0 , which is connected to the familiar hopping expansion parameter @xmath58 .
our results are listed in appendix b. a complete _ per diagram _ breakdown of the results would be far too lengthy to present ; instead , for potential comparisons , we provide in appendix b a breakdown only for a particular value of @xmath0 .
we list below some examples of values for @xmath34 , setting @xmath61 .
for @xmath63 we have : @xmath64 for two degenerate flavors ( @xmath62 ) and @xmath65 ( corresponding to @xmath66 ) : @xmath67 c_{\rm sw } = 2.0 : \ & e_g = ( 1/3)\ , g^2 & + 0.013663456(3 ) g^4 & + 0.0110200(13 ) g^6 \end{array } \label{m-0.518106}\ ] ] for @xmath62 and @xmath68 : @xmath69 c_{\rm sw } = 1.3 : \ & e_g = ( 1/3)\ , g^2 & + 0.025219798(9 ) \ g^4 & + 0.0129659(5 ) \
g^6 , \\[0.5ex ] c_{\rm sw } = 2.0 : \ & e_g = ( 1/3)\ , g^2 & + 0.01800170(1 ) \ g^4 & + 0.012948(1 ) \ g^6 \end{array } \label{m0.038}\ ] ] it is seen that the 3-loop coefficients are quite pronounced for typical values of @xmath10 used in numerical simulations . for convenience , the behaviour of @xmath34 versus @xmath70 is also presented in fig .
3 , for the same parameter values as in eqs .
( [ m-0.518106],[m0.038 ] ) .
the detailed results , tabulated in appendix b for arbitrary values of @xmath71 , @xmath33 , @xmath16 , show a very smooth behaviour as a function of @xmath0 ; consequently , one is able to reconstruct @xmath34 also for arbitrary values of @xmath0 by naive interpolation , to excellent precision .
figure 4 depicts all diagrams contributing to the free energy at 1 loop ( diagram 1 ) , 2 loops ( 2 - 6 , 37 - 38 ) , and 3 loops ( 7 - 36 , 39 - 62 ) .
solid ( curly , dashed ) lines represent fermions ( gluons , ghosts ) , and the filled square is the contribution from the measure part of the action . the filled circle , corresponding to the non - fermionic part of the 1-loop gluon self - energy , is given in figure 5
. tables i - iv provide a _ per diagram _ breakdown of our results , at a given value of @xmath0 ( @xmath68 ) , in order to allow for potential comparisons and cross checks .
the total results for the coefficients @xmath56 , @xmath72 , @xmath73 , @xmath74 are listed in tables v - viii , respectively , for a wide selection of @xmath0 values which are used in the literature .
given the smooth dependence of all these coefficients on @xmath0 , interpolations to other intermediate values of @xmath0 can be performed with great accuracy . | we calculate the perturbative value of the free energy in lattice qcd , up to three loops .
our calculation is performed using wilson gluons and the sheikholeslami - wolhert ( clover ) improved action for fermions . the free energy is directly related to the average plaquette . to carry out the calculation ,
we compute all relevant feynman diagrams up to 3 loops , using a set of automated procedures in mathematica ; numerical evaluation of the resulting loop integrals is performed on finite lattice , with subsequent extrapolation to infinite size .
the results are presented as a function of the fermion mass @xmath0 , for any @xmath1 gauge group , and for an arbitrary number of fermion flavors . in order to enable independent comparisons
, we also provide the results on a _ per diagram _ basis , for a specific mass value .
lattice perturbation theory , free energy , average plaquette , clover action , 11.15.ha , 12.38.gc , 12.38.bx |
the @xmath0-function introduced by fox @xcite , will be represented and defined in the following manner : @xmath1 & = h_{p , q}^{m , n } \left\lbrack x \bigg| \begin{array}{@{}l@ { } } ( a_{1 } , \alpha_{1 } ) , \ldots , ( a_{p } , \alpha_{p})\\[.2pc ] ( b_{1 } , \beta_{1 } ) , \ldots , ( b_{q } , \beta_{q } ) \end{array}\right\rbrack\nonumber\\[.2pc ] & = \frac{1}{2 \pi i } \int_{l }
\frac{\prod_{j = 1}^{m } \gamma ( b_{j } - \beta_{j } \xi ) \prod_{j = 1}^{n } \gamma ( 1 - a_{j } + \alpha_{j } \xi)}{\prod_{j = m + 1}^{q } \gamma ( 1 - b_{j } + \beta_{j } \xi ) \prod_{j = n + 1}^{p } \gamma ( a_{j } - \alpha_{j } \xi ) } x^{\xi } \ { \rm d}\xi.\end{aligned}\ ] ] for the nature of contour @xmath2 in ( 1.1 ) , the convergence , existence conditions and other details of the @xmath0-function , one can refer to @xcite .
the general class of polynomials introduced by srivastava @xcite is defined in the following manner : @xmath3 = \sum\limits_{k = 0}^{[v / u ] } \frac{(-v)_{uk } a ( v , k)}{k ! } x^{k } , \quad v = 0 , 1 , 2 , \ldots,\ ] ] where @xmath4 is an arbitrary positive integer and coefficients @xmath5 are arbitrary constants , real or complex .
@xmath6^{-\nu } h_{p , q}^{m , n } [ y \ { x + a + ( x^{2 } + 2ax)^{1/2 } \}^{-\mu}]\nonumber\\ & \qquad\qquad \times s_{v}^{u } [ z \ { x + a + ( x^{2 } + 2ax)^{1/2 } \}^{-\alpha } ] \hbox{d}x\nonumber\end{aligned}\ ] ] @xmath7 } ( -v)_{uk } a ( v , k ) \frac{(z / a^{\alpha})^{k}}{k ! } h_{p + 2 , q + 2}^{m , n + 2}\nonumber\\[.2pc ] & \quad\ , \times \left\lbrack ya^{-\mu } \bigg| \begin{array}{@{}l@ { } } ( -\nu - \alpha k , \mu ) , ( 1 + \lambda - \nu - \alpha k , \mu ) , ( a_{1 } , \alpha_{1 } ) , \ldots , ( a_{p } , \alpha_{p})\\[.2pc ] ( b_{1 } , \beta_{1}),\ldots,(b_{q } , \beta_{q } ) , ( 1 - \nu - \alpha k , \mu ) , ( -\nu - \alpha k - \lambda , \mu ) \end{array } \!\right\rbrack,\end{aligned}\ ] ] where 1 .
@xmath8 , 2 .
@xmath9 . to obtain the result ( 2.1 ) , we first express fox @xmath0-function involved in its left - hand side in terms of contour integral using eq .
( 1.1 ) and the general class of polynomials @xmath10 $ ] in series form given by eq . ( 1.2 ) . interchanging the orders of integration and summation ( which is permissible under the conditions stated with ( 2.1 ) ) and evaluating the @xmath11-integral with the help of the result given below @xcite : @xmath12^{-\nu } { \rm d}x\\ & \quad\ = 2 \nu a^{-\nu } \left(\frac{1}{2}a\right)^{z } [ \gamma ( 1 + \nu + z)]^{-1 } \gamma ( 2z ) \gamma ( \nu - z),\quad 0 < \hbox{re } ( z ) < \nu,\end{aligned}\ ] ] we easily arrive at the desired result ( 2.1 )
if in the integral ( 2.1 ) we reduce @xmath10 $ ] to unity and fox @xmath0-function to gauss hypergeometric function @xcite , we arrive at the following result after a little simplification : @xmath13^{-\nu}\nonumber\\ & \qquad\ \times { _ { 2}f_{1 } } ( a , b ; c ; y ( x + a + ( x^{2 } + 2ax)^{1/2})^{-1}){\rm d}x\nonumber\\ & \quad\ = 2^{1 - \lambda } \nu \gamma ( 2 \lambda ) a^{\lambda - \nu } \frac{\gamma ( \nu - \lambda)}{\gamma ( \nu + \lambda + 1)}\nonumber\\ & \qquad\ \times { _ { 4}f_{3 } } ( a , b , \nu - \lambda , \nu + 1 ; c , \nu , \nu + \lambda + 1 ; y / a),\end{aligned}\ ] ] where @xmath14 the importance of the result given by ( 3.1 ) lies in the fact that it not only gives the value of the integral but also ` augments ' the coefficients in the series in the integrand to give a @xmath15 series as the integrated series
. a number of other integrals involving functions that are special cases of fox @xmath0-function @xcite and/or the general class of polynomials @xcite can also be obtained from ( 2.1 ) but we do not record them here .
the authors are thankful to the worthy referee for his very valuable suggestions .
the first author is thankful to the university grants commission , new delhi for providing necessary financial assistance to carry out the present work .
the authors are thankful to k c gupta , jaipur for his useful suggestions . | in the present paper we derive a unified new integral whose integrand contains products of fox @xmath0-function and a general class of polynomials having general arguments . a large number of integrals involving various simpler functions follow as special cases of this integral .
= msam10 at 10pt = mtgub at 10.4pt = tibi at 10.4pt [ theore]*theorem * [ theore]proposition [ theore]lemma [ theore]definition [ theore]corollary [ theore]remark [ theore]example |
1.2 cm in recent years , a number of experimental and theoretical studies have been made to understand the decay of light di - nuclear systems ( a @xmath5 60 ) formed through low - energy ( e@xmath6 @xmath5 10 mev / nucleon ) , heavy - ion reactions . in most of the reactions
studied , the properties of the observed , fully energy damped yields have been successfully explained in terms of either a fusion - fission ( ff ) mechanism or a heavy - ion resonance behavior @xcite .
the strong resonance - like structures observed in elastic and inelastic excitation functions of @xmath7mg+@xmath7 mg @xcite and @xmath0si+@xmath0si @xcite have indicated the presence of shell stabilized , highly deformed configurations in the @xmath8cr and @xmath9ni compound systems , respectively . in a recent experiment using eurogam ,
the present collaboration studied the possibility of preferential population of highly deformed bands in the symmetric fission channel of the @xmath9ni compound nucleus as produced through the @xmath0si+@xmath0si @xcite reaction at e@xmath10 mev .
the present work aims to investigate the possible occurence of highly deformed configurations of the @xmath9ni and @xmath11ca di - nuclei produced in the @xmath0si+@xmath0si and @xmath0si+@xmath2c reactions through the study of light charged particle ( lcp ) emission . in - plane coincidences of the lcp s with both evaporation residues ( er ) and ff fragments have been measured .
the lcp s emitted from ff fragments may provide informations on the deformation properties of these fragments .
moreover , the in - plane angular correlations data will be used to extract the temperatures of the emitters . in this paper
we will concentrate on the er results .
1.2 cm the experiments were performed at the ires strasbourg vivitron tandem facility using 112.6 mev @xmath0si beams on @xmath0si ( 180 @xmath12g/@xmath13 ) and @xmath2c ( 160 @xmath12g/@xmath13 ) targets .
both the heavy ions and their associated lcp s were detected using the * icare * charged particle multidetector array @xcite .
the heavy fragments ( er , quasi - elastic , deep - inelastic and ff fragments ) were detected in eight telescopes , each consisting of an ionization chamber ( ic ) followed by a 500 @xmath14 m si detector .
the in - plane detection of coincident lcp s was done using four triple telescopes ( si 40 @xmath14 m , si 300 @xmath14 m , 2 cm csi(tl ) ) placed at forward angles , 16 two - element telescopes ( si 40 @xmath14 m , 2 cm csi(tl ) ) placed at forward and backward angles and two telescopes consisting of ic s followed by 500 @xmath14 m si detectors placed at the most backward angles .
the ic s were filled with isobutane and the pressures were kept at 30 torr and at 60 torr for detecting heavy fragments and light fragments , respectively .
typical inclusive and exclusive ( coincidence with all er s detected at 15@xmath15 ) energy spectra of @xmath4 particles at 40@xmath3 for the @xmath0si+@xmath0si reaction are shown by solid histograms in fig . 1(a ) and 1(b ) , respectively .
exclusive @xmath0si+@xmath2c @xmath4 spectra measured at 40@xmath3 in coincidence with s and p er s at 15@xmath16 are also displayed in fig .
1.2 cm the data analysis was performed using cacarizo , the monte carlo version of the statistical - model code cascade @xcite .
the angular momenta distributions , needed as the principal input to constrain the calculations were taken from compiled @xmath0si+@xmath0si @xcite and @xmath0si+@xmath2c @xcite complete fusion data .
the other ingredients for the realistic statistical - model calculations such as the nuclear level densities and the barrier transmission coefficients , are usually deduced from the study of the evaporated light particle spectra . in recent years , it has been observed in many cases that the standard statistical model can not predict the shape of the evaporated @xmath4-particle energy spectra satisfactorily @xcite , with the measured average energies of the @xmath4 particles generally much lower than the corresponding theoretical predictions .
several attempts have been made to explain this anomaly either by changing the emission barrier or by using a spin - dependent level density .
the change in the emission barriers and consequently the transmission probabilities affects the lower energy part of the calculated evaporation spectra . on the other hand ,
the high - energy part of the spectra depends critically on the available phase space obtained from the level densities at high spin as well as the corresponding transmission coefficients . in hot rotating nuclei formed in heavy - ion reactions
, the level density at higher angular momentum should be spin dependent . the level density , @xmath17 , for a given angular momentum @xmath18 and energy @xmath19
is given by the well known fermi gas expression : @xmath20^{1/2 } ) , \label{lev}\ ] ] where @xmath21 is the level density parameter , @xmath22 is the pairing correction and e@xmath23 = @xmath24j(j+1 ) is the rotational energy , @xmath25 is the effective moment of inertia , @xmath26 is the rigid body moment of inertia and @xmath27 , @xmath28 are the deformation parameters @xcite . by changing the deformation parameters one can simulate the deformation effects on the level densities .
the cacarizo calculations have been performed using two sets of input parameters : one with a standard set and another with non - zero values for the deformation parameters .
the solid lines in fig .
1 show the predictions of cacarizo using the standard parameter set with the usual liquid drop model deformation .
it is clear that the average energies of the measured @xmath4 spectra are lower than those predicted by the standard statistical - model calculations .
the dashed lines show the predictions of cacarizo using @xmath27 = 3.2 x 10@xmath29 and @xmath28 = 2.2 x 10@xmath30 .
the shapes of the inclusive as well as the exclusive @xmath4 energy spectra are well reproduced after including the deformation effects . in the case of @xmath0si+@xmath2c
an interesting result is observed . in order to explain the inclusive energy spectra of @xmath4-particles it has been necessary to use the similar deformation parameters as for @xmath0si+@xmath0si system .
however , it was not possible to explain the exclusive energy spectra of @xmath4 particles obtained in coincidence with all of the er s using this set .
therefore , @xmath4 energy spectra have been generated in coincidence with individual s and p er s as shown in fig .
the shape of the @xmath4 spectrum ( solid histograms ) obtained in coincidence with s is completely different from the spectrum obtained in coincidence with p. the dashed lines in fig .
2 are the predictions of cacarizo using non - zero values of @xmath27 and @xmath28 .
the shape of the @xmath4 spectrum measured in coincidence with p is reasonably well reproduced by the theoretical curve .
however , the model could not predict the shape of the @xmath4 spectrum obtained in coincidence with s. this is due to the fact that in this case , there may be additional contributions to the @xmath4-particle spectra from the decay of unbound @xmath31be , produced through a binary decay such as asymmetric ff and/or an orbiting mechanism with @xmath11ca @xmath32 @xmath33s+@xmath31be .
this confirms the double - humped structure found in the inclusive @xmath33s velocity spectra measured by harmon et al . @xcite .
the question of the real nature ( ff or orbiting ) of this decay process remains open .
1.7 cm inclusive as well as exclusive energy spectra of @xmath4-particles have been measured for the reaction @xmath0si+@xmath0si and @xmath0si+@xmath2c , respectively .
the observed energy spectra of @xmath4 particles are not well reproduced by the standard statistical - model calculations with the usual liquid drop model deformation . however
, a satisfactory description of the measured energy spectra has been achieved by invoking the changes in the level density and barrier due to the onset of large deformation effects at high spins .
the @xmath4 spectra obtained in coincidence with s for the reaction @xmath0si + @xmath2c have an additional component which may come from the decay of @xmath31be , which is unbound and produced through the binary decay of @xmath11ca @xmath32 @xmath33s+@xmath31be .
work is in progress to analyse the proton energy spectra as well as the angular correlations of both the proton and @xmath4 in - plane angular correlations .
1.7 cm 99 k. farrar et al . , phys . rev .
* c 54 * , 1249 ( 1996 ) and references therein . r.w .
zurmhle et al . , phys . lett . *
129b * , 384 ( 1983 ) .
betts et al . , phys .
lett . * 47 * , 23 ( 1981 ) .
r. nouicer , ph.d .
thesis , strasbourg university , report * ires 97 - 35*. t. bellot , ph .
d. thesis , strasbourg university , report * ires 97 - 34*. g. viesti et al .
, phys . rev . * c 38 * , 2640 ( 1988 ) and references therein .
vineyard et al . , phys . rev .
* c 41 * , 1005 ( 1990 ) .
harmon et al . , phys . rev . * c 34 * , 552 ( 1986 ) .
vineyard et al . , phys . rev . * c 47 * , 2374 ( 1993 ) .
g. la rana et al . , phys . rev . * c 37 * , 1920 ( 1988 ) ; _ ibidem _ * c 40 * , 2425 ( 1989 ) .
govil et al . , phys . rev .
* c 57 * , 1269 ( 1998 ) ; _ ibidem _ phys
b307 * , 283 ( 1993 ) . | _ inclusive as well as exclusive energy spectra of the light charged particles emitted in the @xmath0si(e@xmath1 mev ) + @xmath0si,@xmath2c reactions have been measured at the strasbourg vivitron facility in a wide angular range 15@xmath3 - 150@xmath3 , using the icare multidetector array .
the observed @xmath4-particle energy spectra are generally well reproduced by the statistical model using a spin - dependent level density parameterisation .
the results suggest significant deformation effects at high spin . _
23 true cm 16 true cm 0 true cm 0 true cm -1 true cm 5.0 cm * abstract * 2.0 cm |
the ringing observed in fig .
2 is a manifestation of a non - adiabatic transition through the eit resonance .
landau - zener theory deals with this kind of transitions and gives analytic prediction to the population transfer between the levels . in the case of an eit in buffer gas
the best way to describe the system is using the dressed state picture . taking the hamiltonian of the bare three levels under the rotating wave approximation where @xmath83 are the rabi frequencies of the coupling and the probe fields respectively ,
@xmath25 is a constant two photon detuning and @xmath24 is one photon detuning where in the case relevant to us is @xmath84 . in order to see the resemblance to the landau - zener case
it is instructive to change to a new basis where @xmath85 the new hamiltonian will become @xmath86 the @xmath87 matrix of levels @xmath88 and @xmath89 is a landau - zener hamiltonian . under eit conditions @xmath90 hence it is possible to diagonalize this @xmath87 matrix with two new dressed levels with eigenvalues @xmath91 . in the simple case where @xmath47 these states are
just @xmath92 and @xmath93 with @xmath94 @xcite .
this landau - zener dynamics is interrogated by the probe field , meaning that the transition element @xmath95 we are measuring in the experiment , carries the dynamics described above as depicted in fig .
[ fig : dressed](a ) . in our experiment
a phase modulation sweep in time causes a periodic crossing between the two dressed levels .
when a magnetic field is applied the system is split into three sub - systems with three levels in each on of them as discussed in the main page and in @xcite .
each one of these sub - systems behaves exactly as a single eit system with the exception of a magnetic zeeman shift @xmath32 . as a consequence the energy levels of the sub - systems @xmath97 and @xmath98
are reversed with respect to the magnetic field ( with @xmath96 while the energy levels of the sub - systems @xmath60 is degenerate up to the interaction avoided level crossing as depicted in fig .
[ fig : dressed](b ) . one interesting characterization of the landau - zener transition is the transition time .
this time can be measured by the decay time of the oscillations after the transition @xcite .
the two parameters that determines the transition properties is the coupling rabi frequency and the chirp rate defined as @xmath99 . in the case of sinusoidal phase modulation , where @xmath100 , the chirp rate at @xmath47 is @xmath101 .
it is useful to define the transition using a dimensionless parameter @xmath102 .
figure [ fig : decay ] shows the decay time , @xmath103 , as a function of @xmath104 for our experimental results ( red squares ) as well as for our simulation results ( black circles ) . the decay time is found from an exponential fit to the ringing peaks as depicted in the inset in fig .
[ fig : decay ] .
it is possible to see that in the diabatic limit ( low @xmath104 ) the decay time is nearly constant and converging towards @xmath105 , while at the adiabatic limit ( high @xmath104 ) the decay is linear with @xmath104 .
similar theoretical results for the landau - zener theory have been reported before @xcite .
decay time of the ringing at as a function of @xmath106 .
black circles - simulation , red squares - experiment .
the decay time is calculated using an exponential fit to the peaks of the ringing as shown in the inset .
green dashed line - the eit decay according to @xmath105 .
blue dash dotted line - linear fit for the adiabatic case .
simulation parameters are similar to the one in fig .
3 with variable modulation index and modulation frequency . , width=325 ] figure [ fig : broad magnetic ] shows a broad scan of magnetic field vs. time .
this scan is done in the case of two photon resonance in the absence of magnetic field .
we can distinct clearly the functional behavior of the three eit lines for @xmath68 .
the @xmath60 line is not dependent upon magnetic field thus its phase is constant with a pulse every half a cycle .
both @xmath61 lines are sinusoidally modulated with a cycle equal to @xmath107 and a phase of @xmath108 between them .
each of these two lines behave exactly like the detuning sweep of one eit line ( with no magnetic field ) under phase modulation ( see for example fig .
this feature is understandable , as applying magnetic field can be translated to detuning via the larmor frequency zeeman shift .
adding a constant detuning or constant magnetic field creates a symmetric shift of the two sinusoids until reaching a field larger than @xmath109 . in this case
the two sinusoids get separated and the constant pulse of @xmath60 disappears .
since the two sinusoids do not intersect the interference pattern disappears .
the major consequence is that measuring a constant magnetic field accurately using this method is possible only for magnetic fields with larmor frequency smaller than @xmath109 .
as mentioned in the main text , the spectrum of eit under axial magnetic field creates a splitting to three sub - levels .
this is certainly verified by the three pulses seen in fig .
4(a ) . as a complementary measurement
we also measure the steady state spectrum of the eit under variable magnetic field as can be seen in fig .
[ fig : magnetic - splitting ] .
4ifxundefined [ 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 http://steck.us/teaching[__ ] , link:\doibase 10.1103/physreva.67.033810 [ * * , ( ) ] link:\doibase 10.1103/physreva.59.988 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.62.2543 [ * * , ( ) ] | a strong phase modulated coupling field leads to an amplitude modulation of a probe field in an electromagnetically induced transparency process .
time vs. detuning plots for different modulation frequencies reveals a transition between an adiabatic regime where a series of smooth pulses are created with a global phase dependent upon the detuning , and a non - adiabatic regime where a strong transient oscillating response is added to these pulses . in the extreme non - adiabatic regime , where the modulation frequency is higher than the transient decay time , a coherent interference pattern is revealed .
adding a magnetic field lifts the hyperfine level degeneracy , resulting in a separation of the original pulse to three pulses .
every pulse has now a different phase dependent upon the magnetic field causing an interference effect between the different magnetic level transients .
we explore the dynamics of the magnetic and non magnetic cases and show its resemblance to the landau - zener theory .
we also show that combining the global phase of the pulses with the transient interference allows for a wide magnetic sensing range without loosing the sensitivity of a single eit line .
electromagnetically induced transparency ( eit ) is a coherent process , where a strong coupling field creates a narrow transmission band in the probe spectrum in an otherwise fully absorptive medium @xcite .
the narrow linewidth of eit makes it suitable for applications in many fields such as extreme slow light , quantum storage devices , non linear optics and high sensitivity magnetic sensors . on the other hand
, this narrow linewidth directly limits the bandwidth of data that can be processed . a signal which has a broader bandwidth than the eit linewidth
will be absorbed spectrally , or will not be delayed in the time domain @xcite . in terms of magnetic sensing this means that although a very high sensitivity is possible using eit , it is a problem to probe with this sensitivity a broadband field .
a possible solution to this problem may arise from two directions .
one idea is to use a multimode eit system where each eit line has still a narrow bandwidth but spreading the signal across many systems allows for a broader signal .
such systems were devised spatially @xcite and spectrally @xcite for larger data capacity as well as for broadband magnetic sensing @xcite .
a different approach is to use dynamic eit where the transient response may have a much broader bandwidth than steady state eit .
the transient response of an eit media to a sudden switching @xcite as well as for ac magnetic field @xcite was explored theoretically and experimentally for various regimes . for a constant detuning
the decay of the transients is dictated by the eit linewidth , while the frequency of the transient oscillations equals the two photon detuning @xcite .
the detuning can be larger than the linewidth leading to an under - damped oscillator response . in the case of a linear sweep through the resonance the the frequency of the transient
is chirped @xcite and behaves similarly to a landau - zener transition @xcite .
transients were used also as a magnetic sensing technique to measure the earth s magnetic field with @xmath0 sensitivity @xcite . in this letter
we take both concepts of multimode eit and transient eit and combine them together . using a full mapping of the transient response as a function of the detuning we are able to show the transition from an adiabatic regime to a non - adiabatic regime .
we also show the complex interference pattern that arises when a magnetic field is applied . in this case
an interference between transients from three zeeman sub - levels is visible .
the problem of three level crossing in the context of landau - zener transitions was addressed both theoretically and experimentally @xcite .
experimentally it was shown using 2 two level system ( tls ) defects that have a similar energy gap inside a josephson junction , that interference occurs at certain phases when the junction frequency is swept @xcite . here ,
on the contrary , we have the ability to change the levels energy gap experimentally implementing the three levels landau - zener transition for variable level detuning .
moreover , we show that this interference can be useful as a high sensitivity magnetometer combined with broadband phase modulation sweep in order to achieve a wideband high sensitivity magnetometer .
we now describe the effect a phase modulated strong coupling field has upon the temporal shape of a probe pulse going through an eit media .
the coupling field can be written as follows : @xmath1 where @xmath2 is the amplitude of the coupling field , @xmath3 is the optical frequency and @xmath4 is the time dependent phase modulation . in order to describe the change in the probe field due to this modulated coupling field the full spatio - temporal maxwell - bloch equations for an eit system needs to be solved @xcite . in the case of long enough interaction media and perturbative probe intensity
the system can reach a steady state solution .
the eit susceptibility in this case has only temporal and no spatial dependence . for a susceptibility with no temporal dependence the probe transmission amplitude ,
@xmath5 is given by the convolution of the entering signal , @xmath6 , and the susceptibility , @xmath7 , hence @xmath8.$ ] in the case of a modulated field the susceptibility is time dependent , and this convolution is not valid .
a possible way of solving this problem is by taking the spectral decomposition of the susceptibility @xmath9 .
now the transmission is just @xmath10 $ ] @xcite .
we use a sinusoidal modulation , hence @xmath11 where @xmath12 is the modulation depth and @xmath13 is the modulation frequency .
the spectrum of such a modulated field can be described as a sum of bessel functions @xmath14 the spectrum of this modulation has narrow peaks separated by a frequency @xmath15 with a full modulation span of @xmath16 .
the transfer function in this case is just an infinite comb of single eit lines @xcite weighted by bessel functions : @xmath17 here @xmath18 is the two - level absorption coefficient , with @xmath19 the density of the atoms and @xmath20 is the transition dipole moment , @xmath21 is the homogeneous decay rate , @xmath22 is the decoherence rate of the two ground states , @xmath23 is the rabi frequency of the coupling field which is phase modulated , @xmath24 is the one photon detuning of the probe field and @xmath25 is the two photon detuning . in the case of the d1 line of warm @xmath26 vapor with buffer gas
the fwhm eit linewidth is @xcite @xmath27 where @xmath28 is the doppler broadening .
this linewidth is usually a few khz which is much narrower than the pressure broadened homogeneous linewidth ( @xmath29 ) and the doppler broadening ( @xmath30 ) , thus the probe two level susceptibility is effectively constant for the full modulation bandwidth as long as @xmath31 .
applying a magnetic field removes the zeeman degeneracy and the energy levels of the hyperfine levels will create a ladder according to the zeeman splitting of the two lower levels with a larmor frequency @xmath32 .
@xmath33 here is the magnetic field , @xmath34 is the bohr magneton and @xmath35 is the lande coefficient of the hyperfine level .
the eit susceptibility will be determined by the zeeman splitting with several eit peaks having a certain phase between them .
we can write the transfer function as follows : @xmath36 the experimental setup
is shown in fig .
[ fig : setup ] . for an eit @xmath37 scheme we use the hyperfine transitions of the d1 line of @xmath26 . a dfb laser locked to the @xmath38 transition
is split into probe and coupling beams using a polarizing beam splitter .
the phase modulation over the coupling field as well as the pulse creation of the probe is done using acousto - optic modulators .
in order to bring the probe to resonance with the @xmath39 transition an electro - optic modulator is used . the beams ( orthogonal polarization ) are combined together using a glan - taylor polarizer and pass through a 7.5 cm cell containing an isotopically pure @xmath26 with 10 torr ne as buffer gas heated to @xmath40 .
the cell is shielded from an outside magnetic field using a 3-layers of @xmath20-metal .
an axial magnetic field is created using a uniform solenoid .
after the cell another polarizer is used in order to filter the coupling field while the probe is detected using an amplified photodiode . the experimental setup .
dfb - distributed feedback laser , pbs - polarizing beam splitter , ap - aperture , aom - acousto - optic modulator , eom - electro - optic modulator , gt - glan - taylor polarizer , @xmath20ms - @xmath20- metal shield , bd - beam dump , pd - photodiode , sl - solenoid . ]
figure [ fig : ringing ] demonstrates the transmission temporal response of a square probe pulse with intensity of @xmath41 due to a phase modulated coupling field with intensity of @xmath42 in an eit media .
two major features are observed , one is a train of pulses that is created with a period and phase that is dependent upon the coupling field modulation frequency and the detuning @xcite .
the second feature is a transient ringing that is associated with the response of the media to a sudden change in the susceptibility .
this ringing has a chirped frequency as expected @xcite .
it decays with a characteristic time that depends upon @xmath43 and the chirp rate through the transition @xcite .
the instantaneous frequency of the coupling field due to the modulation is @xmath44 while the response has the spectral width of the eit linewidth , thus the relation between the modulation frequency and the eit width , sets the adiabaticity of the response .
transient oscillations of the probe amplitude due to coupling modulation with @xmath45 and @xmath46 .
red - @xmath47 , green - @xmath48 . , width=264 ] figure [ fig : spectrum ] shows experimentally and theoretically the transition between the adiabatic regime where the modulation frequency is lower than the eit linewidth ( @xmath49 ) and the non - adiabatic regime where @xmath50 .
for both regimes the phase of the pulses is determined by the instantaneous frequency hence we see a sine like plot as a function of the detuning with a period @xmath51 and an amplitude @xmath52 . in the adiabatic regime
the transients decay fast enough so they are hardly noticeable , but as the modulation frequency become comparable to the eit linewidth [ fig .
[ fig : spectrum](b ) ] the transient ringing is clearly observed . in the non - adiabatic regime
the modulation frequency is faster than the decay of the transient ringing creating an interference between consecutive pulses as can be observed in fig .
[ fig : spectrum](c ) .
simulation of these 2d patterns using eq .
[ eq : susc_no_mag ] are depicted in fig .
[ fig : spectrum](c - e ) showing a striking similarity to the results .
one aspect this linear response theoretical simulation fails to take into account is the smearing of the interference pattern when the probe pulse is turned on as can be visualized particularly in fig .
[ fig : spectrum](c ) .
the cause of this effect is the gradual build up of the dark state polariton and consequently the creation of the eit line that has a characteristic time of @xmath53 @xcite . integrating
the time domain reveals the steady state spectrum of the probe light .
figure [ fig : spectrum](f - h ) shows the integrated spectra of the experimental data ( green line ) as well as the simulation ( red line ) in the adiabatic and non - adiabatic regimes .
these spectra fit to the phase modulation spectrum according to the fourier of eq .
[ eq : bessel ] , meaning a delta functions separated by the modulation frequency , broadended due to the finite eit linewidth . another way of thinking about the interference shown in fig .
[ fig : spectrum](c ) is as a landau - zener - stuckelberg interference pattern where a transition is crossed repetitively faster than the transient decay time @xcite .
adiabatic to non - adiabatic transition .
2d mapping of the spectro - temporal response of the probe pulse is demonstrated in the case of ( a ) adiabatic regime , ( b ) intermediate regime and ( c ) non - adiabatic regime .
the parameters of the modulation are ( a ) @xmath54 and @xmath55 , ( b ) @xmath45 and @xmath46 , ( c ) @xmath56 and @xmath57 .
all the experiments are done with @xmath58 .
plots ( d - f ) show a simulation of the three regimes that takes into account eq .
[ eq : susc_no_mag ] with the parameters written above . plots ( g - i ) show the spectrum of the transmission taken as the time integral for each frequency . green - experimental spectrum , red - simulation spectrum . ,
width=325 ] figure [ fig : magnetic - results ] shows a 2d mapping of the temporal response of the probe for different magnetic fields ( the two photon detuning is on resonance with the magnetic insensitive transition ) . in the adiabatic regime ( fig .
[ fig : magnetic - results](a ) ) it is possible to see a splitting of the sole pulse in b=0 into three pulses .
these pulses correspond to three eit lines that are present in the spectrum . for the d1 line of rubidium ,
using an arbitrary magnetic field , up to 7 eit lines may appear @xcite .
due to the vectorial nature of the magnetic interaction , the relative strength of these lines depends on the angle between the beam direction and the magnetic field as well as the polarization of the pump and probe beams @xcite .
the specific configuration we use in our setup is @xmath59 with linear polarization . in this case
only three lines appear in the spectrum @xcite as can be seen experimentally @xcite .
the central pulse matches the @xmath60 and thus its phase is constant , while the other two pulses correlate with the @xmath61 , hence having a sinusoidal phase shift .
figures [ fig : magnetic - results](b - c ) show the non - adiabatic regime where every pulse has an oscillating tail with a certain phase causing an interference pattern . a simulation based on eq .
[ eq : mag ] is shown in fig .
[ fig : magnetic - results](d - f ) having the same basic features of the experimental results . temporal response of a probe pulse due to magnetic field for ( a ) adiabatic regime ,
( b ) intermediate regime and ( c ) non - adiabatic regime .
the parameters of the modulation are ( a ) @xmath62 and @xmath63 , ( b ) @xmath45 and @xmath64 , ( c ) @xmath65 and @xmath66 . plots ( d - f ) show a simulation of the three regimes that takes into account eq .
[ eq : mag ] with the parameters written above.,width=325 ] the observed interference may be understood in the following way . in the case of hyperfine eit in a buffer gas and under the condition of @xmath67 it is possible to treat the two ground states as a degenerate set of effective two level systems @xcite . adding a magnetic field removes the degeneracy and
splits each two level system according to the zeeman frequency @xcite . in our case due to the selection rules stated above the splitting
is to three groups with @xmath68 . as a consequence of this picture it is possible to think of the magnetic sweep in time as a chirp of the three tls s ( as depicted by the black dashed lines in fig .
[ fig : magnetic - results](a ) ) .
thus , the interference we measure is a direct consequence of a three landau - zener transitions degeneracy @xcite .
the dynamic pattern created by the phase modulation can be used for broadband magnetic sensing .
each magnetic field has a certain characteristic pulse timing associated with it .
the phase of the first pulse is a prominent feature for broad magnetic sensing as the total amplitude of the modulation is @xmath69 .
this corresponds to @xmath70 and @xmath71 for [ fig : magnetic - results](b ) and [ fig : magnetic - results](c ) respectively .
moreover , the interference pattern offers a way of measuring accurately the magnetic field in the area of the interference .
the sensitivity to magnetic field is dependent upon the signal ( @xmath72 ) to noise ( @xmath73 ) ratio and given by @xcite @xmath74 where @xmath75 is the measurement time and @xmath76 is the gradient of the integrated measured amplitude and the magnetic field . in our case , since the transient ringing is a complex multi frequency feature the best way to characterize the transients for different magnetic field is by using the correlation between them . using this method we measured the noise and the gradient of the correlation function and
estimate our sensitivity to be @xmath77 for the 5 khz modulation and @xmath78 in the case of 15 khz .
the ultimate sensitivity for a given system is @xcite @xmath79 where @xmath80 is the volume of the magnetometer . in our case
this sensitivity is @xmath81 well below the measured sensitivity .
the main reason for that is the electrical noise in the detector and amplifier which is used in order to observe the data in the oscilloscope .
better electronics may allow at least one order of magnitude improvement . in conclusion
, we show experimentally the transient response of an eit media to a phase modulated pump .
this response reveals explicitly the coherent nature of eit . in the non - adiabatic regime
where eit peaks are spectrally resolved it is possible to see interference between the different modes . albeit the interference between these modes does not contribute to eit spectral narrowing and as a consequence to a slower light propagation @xcite , it does create a transient behavior useful for broadband data transfer .
applying a magnetic field splits the eit line into three , allowing us to see an interference pattern caused by the landau - zener crossing of the three eit lines . along with the wideband sweep due to the high modulation index
it is shown to be a useful tool for sensitive wideband magnetometry .
we acknowledge helpful discussions with n. davidson , m. kiffner and t. dey and support of the bikura ( isf ) grant no .
1567/12 .
35ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ]
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[ * * , ( ) ] link:\doibase 10.1103/physreva.65.053802 [ * * , ( ) ] link:\doibase 10.1103/physreva.85.013820 [ * * , ( ) ] http://www.ncbi.nlm.nih.gov/pubmed/19862058 [ * * , ( ) ] link:\doibase 10.1103/physreva.69.023806 [ * * , ( ) ] link:\doibase 10.1103/physreva.85.063809 [ * * , ( ) ] link:\doibase 10.1103/physreva.55.2165 [ * * , ( ) ] http://arxiv.org/abs/1307.3878 [ ] link:\doibase 10.1088/0953 - 4075/45/21/215401 [ * * , ( ) ] link:\doibase 10.1088/0305 - 4470/19/7/017 [ * * , ( ) ] link:\doibase 10.1103/physreva.55.4418 [ * * , ( ) ] link:\doibase 10.1038/nphys2149 [ * * , ( ) ] link:\doibase 10.1103/physreva.77.023406 [ * * , ( ) ] link:\doibase 10.1038/ncomms1050 [ * * , ( ) ] http://stacks.iop.org/1751-8121/47/i=1/a=015301 [ * * , ( ) ] link:\doibase 10.1103/physreva.79.023829 [ * * , ( ) ] link:\doibase 10.1364/ol.31.002625 [ * * , ( ) ] @noop link:\doibase 10.1016/j.physrep.2010.03.002 [ * * , ( ) ] link:\doibase 10.1103/physreva.83.015801 [ * * , ( ) ] link:\doibase 10.1103/physreva.82.033807 [ * * , ( ) ] link:\doibase 10.1103/physreva.81.013833 [ * * , ( ) ] @noop link:\doibase 10.1103/physreva.55.648 [ * * , ( ) ] http://www.ncbi.nlm.nih.gov/pubmed/9913629 [ * * , ( ) ] link:\doibase 10.1103/physreva.67.033810 [ * * , ( ) ] link:\doibase 10.1038/nphys566 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.70.552 [ * * , ( ) ] |
the main motivation of our modeling work is to contribute to building realistic prominence models . for this
, we need an accurate knowledge of thermodynamic quantities ( temperature , densities , ) , level populations ( useful , e.g. , to infer the magnetic field properties from spectro - polarimetric observations ) , velocity fields , energy budget .
however these quantities still have large uncertainties associated with them .
observations of several different lines from different atoms / ions allow us in theory to measure these parameters . among these lines ,
the h and he lines are important as they are strong and largely contribute to the radiative losses .
however the prominence plasma being out of lte and optically thick in h and he resonance lines , the interpretation of line spectra or intensities in radially moving prominences is a non - trivial task .
therefore , non - lte radiative transfer calculations including velocity fields are needed to build realistic prominence models . here
we present such calculations and preliminary results .
the prominence is represented by a 1d plane - parallel slab standing vertically above the solar surface .
each prominence model is defined by a set of free parameters : the temperature @xmath0 , the gas pressure @xmath1 , the slab thickness @xmath2 ( or the total column mass ) , the height of the slab above the limb @xmath3 , the microturbulent velocity , and the radial speed . for this preliminary study
we consider isothermal and isobaric prominences , although the code allows for inclusion of a transition region between the cold prominence and the hot corona .
we first solve the pressure equilibrium , the ionization equilibrium , and the coupled statistical equilibrium ( se ) and radiative transfer ( rt ) equations for a 20 levels h atom .
then the se and rt equations are solved for other elements : ( 29 levels ) and ( 4 levels ) , and ( 5 levels ) .
more details on the modeling of the hydrogen , calcium , and helium spectra in quiescent prominences can be found in @xcite respectively , and references therein . for the modeling of active and eruptive prominences
, we use a velocity - dependent incident radiation as boundary conditions for the rt equations .
it has already been shown by @xcite in the case of the hydrogen lines that the doppler effect induces a frequency shift of the incident profile relative to the rest case , and also a distortion of the incident profile due to the variation of the doppler shift with the direction of the incident radiation .
it is also the case for the helium ( @xcite ) and calcium incident radiation .
we reproduce the results of @xcite who computed the hydrogen radiation emitted by a radially moving prominence , using partial redistribution in frequency ( prd ) for the lyman lines .
we obtain the same variation of the relative intensities ( intensities normalised to the line intensities when the prominence is at rest ) and the same line profiles for ly@xmath4 , ly@xmath5 , and h@xmath4 .
the main result is that there exists an important coupling between ly@xmath5 and h@xmath4 which causes these lines to be first doppler brightened , and then doppler dimmed , with increasing velocity , while there is only a doppler dimming effect on ly@xmath4 . figure [ nl - fig : intensities ] presents relative intensities as a function of velocity for the 584 , 304 , and 10830 lines ( left panel ) and k and 8542 lines ( right panel ) at two different temperatures ( 8000k and 15000k ) .
the 10830 line does not show any sensitivity to the doppler effect , which is mainly due to the very weak incident absorption line .
the 584 line is quite sensitive to the doppler effect .
its doppler dimming is more important at low temperature .
the resonance lines are the most sensitive to the radial velocity of the plasma ( the relative intensity of the 256 line , not shown , exhibits a similar variation as 304 ) , and the doppler dimming is strong at the temperatures considered in this study .
such a result was expected since the main mechanism of formation at these temperatures for these lines is the scattering of the incident radiation ( @xcite ) .
let us stress that in this preliminary study we have not included a transition region between the cold prominence and the hot corona ( pctr ) .
the presence of a hotter plasma in the pctr may somehow decrease the sensitivity of the resonance lines to the doppler effect as collisions will become more important in the formation processes of these lines .
this will be investigated in a future work .
the right panel of fig .
[ nl - fig : intensities ] indicates that there is no strong doppler effect on the resonance lines , while we observe some doppler brightening of the 8542 line ( and indeed of the other two infrared lines at 8498 and 8662 , not shown ) at low temperature .
if spectroscopic observations of erupting prominences are available , then a comparison between computed and observed line profiles can be made .
we show in figs .
[ nl - fig : profils he ] and [ nl - fig : profils ca ] the line profiles for the same helium and calcium lines considered in fig . [
nl - fig : intensities ] at two different temperatures ( solid line : 8000 k , dashed line : 15000 k ) , at four different velocities ( from top to bottom : 0 , 80 , 200 , and 400kms@xmath6 ) .
the doppler dimming effect is well observed in the helium resonance lines at 584 and 304 as the radial velocity is increased ( fig .
[ nl - fig : profils he ] ) .
we can observe asymmetries in the line profiles of these lines when the prominence plasma is moving radially , with some intensity enhancement which is especially visible in the red wing of the 584 line at low temperature .
this is explained as follows .
the radiation emitted by the disk center in our code is represented by symmetrical line profiles .
when the prominence is at rest , the incident radiation illuminating the prominence slab is symmetrical ; however when the prominence plasma is moving radially the incident line profile becomes distorted and shifted towards the red . as we used the prd standard approximation in our calculations of the resonance lines of helium ,
the resulting line profiles are asymmetrical .
this would not have been the case if we had considered complete redistribution in frequency ( crd ) instead of prd .
the line asymmetry is more visible for the 584 line as its wings are fairly bright , while the wing intensities of the 304line are low . despite the fact that the line asymmetry increases with speed for both lines , it is more visible at low speeds ( when the intensity in the wing is high enough ) . finally , it is more pronounced at low temperatures because the scattering of the incident radiation is relatively more important as compared to collisional processes than it is at higher temperatures .
figure [ nl - fig : profils ca ] shows that the intensities of the lines are lower at 15000k than at 8000k , an effect of to ionization ( @xcite ) .
k ( solid line ) and @xmath7k ( dashed line ) , with @xmath8 dyn@xmath9 , and @xmath10 km , at different velocities : 0 , 80 , 200 , and 400kms@xmath6 from top to bottom .
abscissa is @xmath11 in and vertical axis is specific intensity in ergs@xmath6@xmath9sr@xmath6@xmath6 . from left to right : 584 , 304 , and 10830.,scaledwidth=69.0% ] for k and 8542 lines.,scaledwidth=69.0% ]
we show here how the non - lte radiative transfer calculations can help us to infer the thermodynamic properties of a prominence observed by the sumer spectrometer on soho .
this prominence was actually a rather quiet prominence and we have not included any velocity fields in these calculations .
it was observed during the 13th medoc campaign held at ias in june 2004 .
we select a few pixels in the sumer slit which cut across the prominence and average the line profiles there .
we consider the line profiles of two h resonance lines ( ly@xmath5 and ly@xmath12 ) and the resonance line at 584 . for the comparison between computed and observed line profiles
we now include the presence of a pctr .
the temperature variation between the cold prominence core and the corona suggested by @xcite has been adopted for this study . by a trial and error process we selected the temperature profile shown in fig .
[ nl - fig : temp ] .
the other prominence parameters are @xmath13 dyn@xmath9 , @xmath14 km ( total column mass 2.4@xmath15g@xmath9 ) , @xmath16 km , and the microturbulent velocity @xmath17 kms@xmath6 .
we obtain a very good agreement between the computed profiles ( convoluted with the sumer instrumental profile ) and the observed profiles , as shown in fig .
[ nl - fig : match ] .
it is worth noting that fitting hydrogen _ and _ helium resonance lines simultaneously places strong constraints on the parameter space , and it was not possible to find another set of parameters for the prominence that would be significantly different than what is given above and that would lead to a satisfactory fit of the observed profiles .
( left ) , ly@xmath12 ( middle ) , and 584 ( right).,scaledwidth=69.0% ]
the non - lte radiative transfer modeling that we are developing is a key tool for interpreting observations and constructing realistic prominence models . the combination of lines from hydrogen , helium , and calcium , places strong constraints on the models .
imaging and spectroscopy must be used for comparisons with calculations to determine thermodynamic parameters and velocities . the radial velocity determined from the comparison between observed and computed line profiles , in combination with line - of - sight velocities , should allow us to infer the full velocity vector of the prominence plasma . in a future work we will compare our model results with simultaneous observations of , e.g. , h@xmath4 and 304 . | active prominences exhibit plasma motions , resulting in difficulties with the interpretation of spectroscopic observations .
these solar features being strongly influenced by the radiation coming from the solar disk , doppler dimming or brightening effects may arise , depending on which lines are observed and on the velocity of the plasma . interlocking between the different atomic energy levels and non local thermodynamic equilibrium lead to non - trivial spectral line profiles , and this calls for complex numerical modeling of the radiative transfer in order to understand the observations .
we present such a tool , which solves the radiative transfer and statistical equilibrium for h , , , and in moving prominences where radial plasma motions are taking place .
it is found that for isothermal , isobaric prominence models , the resonance lines are very sensitive to the doppler effect and thus show a strong doppler dimming .
the lines doppler effect for the prominence models considered here .
we illustrate how the code makes it possible to retrieve the plasma thermodynamic parameters by comparing computed and observed line profiles of hydrogen and helium resonance lines in a quiescent prominence .
this new non - lte radiative transfer code including velocities allows us to better understand the formation of several lines of importance in prominences , and in conjunction with observations , infer the prominence plasma thermodynamic properties and full velocity vector . |
this work is supported by the swedish research council ( vr ) under grant nos . 621 - 2012 - 3805 , and 621 - 2013 - 4323 and the gran gustafsson foundation .
we also thank d.s .
delion for discussions and for his efforts studying above - mentioned nuclei within the coherent state model and p. maris for his effort calculating those nuclei with the code mfdn .
the computations were performed on resources provided by the swedish national infrastructure for computing ( snic ) at pdc , kth , stockholm .
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these nuclei are of particular interest since their yrast spectra show a vibrational - like equally - spaced pattern but the few known e2 transitions show anomalous rotational - like behavior , which can not be reproduced by collective models .
our calculations reproduce well the equally - spaced spectra of those isotopes as well as the constant behavior of the @xmath0 values in @xmath1te .
the calculated @xmath0 values for neutron - deficient and heavier te isotopes show contrasting different behaviors along the yrast line .
the @xmath0 of light isotopes can exhibit a nearly constant bevavior upto high spins .
we show that this is related to the enhanced neutron - proton correlation when approaching @xmath2 .
the advent of large - scale radioactive beam facilities and new detection technologies have enabled the study of the spectroscopy and transition properties of @xmath3 nuclei just above the presumed doubly magic nucleus @xmath4sn @xcite .
several unexpected phenomena have recently been observed : large @xmath5 values of neutron deficient semi - magic @xmath6 isotopes have triggered extensive experimental @xcite and theoretical @xcite activities , in particular regarding the fundamental roles played by core excitations and the nuclear pairing correlation ( or seniority coupling ) .
the study of transition rates in isotopic chains just above @xmath7 may provide further information on the role of core excitations @xcite .
the limited number of valence protons and neutrons are not expected to induce any significant quadrupole correlation in this region @xcite .
the low - lying collective excitations of those nuclei were discussed in terms of quadrupole vibrations @xcite in relation to the fact that the even - even te isotopes between @xmath8 and 70 show regular equally - spaced yrast spectra ( c.f . , fig . 1 in ref .
@xcite ) .
if that is the case , the te isotopes will provide an ideal ground to explore the nature of the elusive nuclear vibration and the residual interactions that leading to that collectivity .
however , the available e2 transition strengths along the yrast line in @xmath9te show an anomalous rotational - like behavior , which can not be reproduced by collective models or the interacting boson model @xcite .
another intriguing phenomenon is the nearly constant behavior of the energies of the @xmath10 and @xmath11 states in te and xe isotopes and their ratios when approaching @xmath2 , in contrast to the decreasing behavior when approaching @xmath12 @xcite .
this was analyzed in ref .
@xcite based on the quasiparticle random phase approximation approach where an competition between the quadurople - quadrupole correlation and neutron - proton pairing correlation was suggested .
an enhanced interplay between neutrons and protons is expected in the @xmath4sn region since the protons and neutrons partially occupy the same quantum orbitals near the fermi level @xcite . in relation to that , there has also been a long effort searching for superallowed alpha decays from those @xmath3 isotopes @xcite .
the region is also expected to be the endpoint of the astrophysical rapid proton capture ( rp ) process @xcite .
the octupole correlation may also play a role here ( the coupling between the @xmath13 and @xmath14 orbitals ) @xcite .
still , compared to tin , the experimental information is less abundant in the isotopic chain of tellurium where little was known experimentally below the neutron midshell until recently . much more work is needed and further measurements are underway in order to map out the ordering and nature of single - particle states and two - body effective interactions in the region @xcite . in this work
we present systematic large - scale calculations on the spectroscopy and transition properties of even - even te isotopes .
the large - scale shell model , which takes into account all degrees of freedom within a given model space , is an ideal approach to study competition between collective and single - particle degrees of freedom .
it is however a challenge , especially in the midshell , due to the huge dimension of the problem ( c.f . ,
fig . 1 in ref .
@xcite ) . on the theoretical side ,
we have done systematic calculations on the @xmath15 values of even - even te isotopes @xcite .
the results are , however , rather sensitive to the truncation imposed .
now we are able to do full shell model calculations for all low - lying states of all te isotopes with further optimization of the shell - model algorithm .
a full shell model calculation for the spectroscopy of @xmath16te was done in ref .
@xcite .
a schematic calculation for @xmath17te in the @xmath18 subspace was presented in ref .
@xcite .
systematic calculations on the sn and sb isotopes were given in refs . and
ref .
@xcite , respectively . in ref .
@xcite , the possible onset of vibrational collectivity in te isotopes was discussed within an effective field theory framework .
we consider the neutron and proton orbitals between the shell closures @xmath19 ( and @xmath20 ) @xmath21 and 82 , comprising @xmath22 , @xmath14 , @xmath23 , @xmath24 and @xmath13 and assume @xmath4sn as the inert core .
the robustness of the @xmath25 shell closures , which has fundamental influence on our understanding of the structure of nuclei in this region , is supported by recent measurements @xcite .
the nearly degenerate neutron single - particle states @xmath26 and @xmath27 orbitals in @xmath28sn were observed by studying the @xmath29-decay @xmath30te @xmath31 @xmath28sn @xcite .
based on the assumption that the ground state of @xmath30te has spin - parity @xmath32 , the @xmath27 orbital was suggested to be the ground state of @xmath28sn instead of @xmath26 .
the excitation energy of the @xmath14 is taken as @xmath33 mev .
the energies of other states have not been measured yet .
they are adjusted to fit the experimental binding energies of tin isotopes .
the starting point of our calculation is the realistic cd - bonn nucleon - nucleon potential @xcite .
the interaction was renormalized using the perturbative g - matrix approach to take into account the core - polarization effects @xcite .
the @xmath34 part of the monopole interaction was optimized by fitting to the low - lying states in sn isotopes @xcite .
further optimization of the @xmath35 part of the interaction is also underway which , however , is still a very challenging task .
our calculations show that the present effective hamiltonian are already pretty successful in describing the structure and transition properties of sb , te , i , xe and cs isotopes as well as heavy nuclei near @xmath12 in this region .
the te isotopic chain is the heaviest and longest chain that can be described by the nuclear shell model .
the dimension for the mid - shell @xmath36te reaches @xmath37 for which the diagonalization is still a very challenging numeric task . in our previous calculations for the @xmath15 of mid - shell te isotopes @xcite , we restricted a maximum of four neutrons that can be excited from below the fermi surface to the neutron @xmath38 subshell and excluded proton excitation to @xmath38 due to limited computation power , which , as we understand now , is a rather severe truncation .
the model space was further extended to allow at most 8 particles to the @xmath38 subshell in ref .
@xcite , where the @xmath15 values show a much smoother parabolic behavior as a function of @xmath19 .
full shell - model calculations are done for all nuclei in the present work .
all shell - model calculations are carried out within the @xmath39-scheme where states with @xmath40 are considered .
diagonalizations are done with a parallel shell model program that we developed @xcite and with a slightly modified version of the code kshell @xcite .
all calculations are done on the supercomputers beskow and tegnr at pdc center for high performance computing at the kth royal institute of technology in stockholm , sweden . to test the validity of the effective interaction ,
we have done systematic calculations on the yrast spectra of isotopes @xmath41te .
the results for @xmath42te are plotted in fig .
[ sm ] in comparison with available experimental data @xcite .
an overall good agreement between theory and experiment is obtained .
noticeable difference is seen in the excitation energies of the @xmath43 states in @xmath44te and the @xmath45 states in @xmath17te .
all isotopes plotted in the figure show rather regular and vibrational - like spectra up to @xmath43 except @xmath17te . for that nucleus , the calculated spectrum still shows a equally - spaced pattern .
however , a smaller gap between @xmath46 and @xmath47 states is expected from recent measurement but the spin - parity assignments for those states are still tentative .
the equally - spaced pattern breaks down in isotopes heavier than @xmath48te where a gradual depression of the excitation energies of the @xmath49 states is seen .
comparison between theory and experiment for the energies of the @xmath10 and @xmath11 yrast states ( upper ) and the @xmath15 values ( lower ) for the te isotopic chain .
the open circles and open triangles in the lower panel correspond to the square of the neutron and proton matrix elements , @xmath50 and @xmath51 , respectively.,title="fig : " ] comparison between theory and experiment for the energies of the @xmath10 and @xmath11 yrast states ( upper ) and the @xmath15 values ( lower ) for the te isotopic chain .
the open circles and open triangles in the lower panel correspond to the square of the neutron and proton matrix elements , @xmath50 and @xmath51 , respectively.,title="fig : " ] a closer comparison between experiment and calculation on the excitation energies of the yrast @xmath10 and @xmath11 states are plotted in fig .
[ sm2 ] as a function of neutron number for all even - even te isotopes . in the lower panel of the figure , the calculated @xmath15 in tellurium isotopes
are compared to the most recent experimental data @xcite .
the b(e2 ) value is calculated as @xmath0 = @xmath52 where @xmath53 and @xmath54 are the proton and neutron matrix elements and we take effective charges @xmath55 = 1.5e and @xmath56 = 0.8e as were employed in @xcite .
the isospin dependence of the effective charges is not considered here , which is not expected to have large influence on the trend .
the model prediction agrees rather well with available data .
the largest deviations are seen in isotopes @xmath57te .
a recent measurement gave a value smaller than the adapted one in the former case .
in the figure we also plotted the square of the neutron and proton matrix elements , @xmath50 and @xmath51 , which represent the separate contributions to the @xmath0 values from the neutron and proton excitations . as can be seen from the figure , the parabolic behavior of the @xmath0 values , which looks similar to that of sn isotopes , is mostly due to the contribution from the neutron excitation .
the contribution from the proton excitation shows a rather smooth and slightly decreasing behavior as the neutron number increases .
as mentioned earlier , the shell - model calculations for mid - shell te isotopes , in particular @xmath36te , are quite sensitive to the filling of both the proton and neutron @xmath38 subshells .
both the proton and neutron transition matrix elements are enhanced when one goes from a small model space calculation with restricted number of particles in @xmath38 to the full shell - model calculation .
comparison between theory and experiment @xcite for @xmath58 values in @xmath1te along the yrast line .
the open circles and open triangles in the lower panel correspond to the square of the neutron and proton matrix elements , @xmath50 and @xmath51 , respectively .
the dashed lines correspond to the predictions of collective models @xcite . ]
the regularly - spaced level spectra in mid - shell te isotopes have been expected to be be associated with a collective vibrational motion . for a spectrum corresponds to a vibrator , there should be collective e2 transitions between states differing by one phonon .
the transition strengths should be linearly proportional to the spin of the initial states , i.e. , one has @xmath59=2 in the harmonic vibrator model .
unfortunately , there are very few data available for @xmath0 values in states beyond @xmath60 . as shown in ref .
@xcite , the measured @xmath0 values along the yrast line in @xmath1te show a rather anomalous constant behavior up to @xmath61 , which looks more like a rotor and is in contradiction with that for a vibrator .
our shell model calculations for those e2 transitions are shown in fig .
[ sm3 ] , which indeed exhibits a rather constant behavior up to higher spins .
moreover , both the proton and neutron matrix elements remain roughly the same along the yrast line . as can be seen fig .
[ sm3 ] , the ratio @xmath59 for @xmath1te is measured to be even slightly smaller than one .
this is not seen in the theory .
the ratio is calculated to be @xmath62 which actually agree well with the prediction for a rotor .
as discussed in refs .
@xcite , it happens rarely in open - shell nuclei that one has @xmath63 .
the reason why the ratio for @xmath1te is observed to be so small is still not clear .
calculated @xmath58 values along the yrast line for even - even te isotopes .
, title="fig : " ] calculated @xmath58 values along the yrast line for even - even te isotopes . ,
title="fig : " ] calculated @xmath58 values along the yrast line for even - even te isotopes .
, title="fig : " ] calculated @xmath58 values along the yrast line for even - even te isotopes .
, title="fig : " ] in ref .
@xcite , the ratios @xmath59 for isotopes @xmath64te are measured to be 1.640 , 1.500 and 1.162 , respectively .
the ratios calculated from the shell model @xmath0 values are 1.322 , 1.299 , and 1.301 , respectively , for above three nuclei , which agree reasonably with experimental data . in fig .
[ sm4 ] we plotted the calculated @xmath0 values for the yrast states of all even - even te isotopes between @xmath65 and 80 .
as can be seen from the upper panel of the figure , the @xmath0 values for the yrast states of @xmath66te , which are at the beginning of the shell , remain roughly constants up to spin @xmath67 and decrease significantly around @xmath68 , which indicates that the collectivity has collapsed there . on the other hand , as shown in the lower panel of the figure , the results for the isotopes @xmath69te at the end of the shell show a very different behavior : the @xmath0 values decrease dramatically after @xmath70 , which reach practically zero value for states up to @xmath71 in all nuclei except @xmath72te .
the results for the groups @xmath73 and @xmath74 are shown in the middle panels of fig .
[ sm4 ] . in the former group ,
the @xmath0 values show a rather constant behavior up to @xmath61 .
the results for @xmath71 diverge in relation to the fact that several low - lying @xmath75 states are predicted for those nuclei by the shell - model calculations and , in cases like @xmath76te shown in the panel , it is the second @xmath75 state that is connected to the yrast @xmath77 state with strong e2 transition . as a result ,
the @xmath78 value vanishes . in the latter group
, the @xmath0 values also show a large decrease after @xmath70 but to a extent that is much less than those in the fourth group , @xmath69te . to understand the behavior of the calculated @xmath0 values seen in fig .
[ sm4 ] , we notice that the ratio @xmath79 roughly equally to two for all known te isotopes below @xmath80 . but it decreases rapidly to around 1.2 in the semi - magic @xmath81te . moreover , the ratio @xmath82 starts to decrease already at @xmath83 , resulting in seniority - like spectra .
the seniority quantum number refers to the number of particles that are not paired to @xmath84 .
it is known that , for systems involving the same kind of particles , the low - lying states can be well described within the seniority scheme @xcite .
this is related to the fact that the @xmath34 two - body matrix elements is dominated by monopole pairing interactions with @xmath84 .
the seniority coupling may be broken by the neutron - proton correlation if both protons and neutrons are present .
this indeed happens in the most neutron deficient te isotopes close to @xmath85 , where the valence neutrons and protons are expected to occupy identical @xmath27 and @xmath26 orbitals and the neutron - proton correlation is expected to be strong . as a results , both the spectra and @xmath86 transition properties show rather regular collective behaviors . on the other hand , for nuclei @xmath69te at the other end of the shell ,
the normal seniority coupling may prevail since the neutron - proton correlation involves particles in different shells and is much weaker . as a result ,
the energy gap between @xmath49 and @xmath11 as well as the @xmath86 transition between the two states reduce significantly ( e.g. , @xmath86 transitions between states with the same seniority is disfavored ) .
the groups @xmath73 and @xmath74 fall between above two cases . if the dimension is not too large , it is possible to project the wave function as a coupling of the proton group and neutron group with good angular momenta in the form @xmath87 where @xmath88 and @xmath89 denote the angular momenta of the proton group and neutron neutron group ( see , e.g. , @xcite ) .
the ground state for a even - even nucleus will be represented by a single configuration with @xmath90 if there is no neutron - proton correlation .
the neutron - proton interaction induces contributions from configurations with higher angular momenta for the protons and neutrons as well as higher - lying configurations with the same angular momenta .
it is seen that , as expected , the @xmath16te ground state shows a high mixture of many component , among which one has 47% with @xmath90 , 30% with @xmath91 , 12.5% with @xmath92 , and 6.7% with @xmath93 . for @xmath17te ground state the results are 41.7% with @xmath90 , 42 % with @xmath91 , 11.9% with @xmath92 . for @xmath17te ground state
the contribution from @xmath90 decreases further to 36.3% while the contribution from @xmath91 increases to 46.6% .
the wave functions for other low - lying states show a similar complex structure . on the other hand , the wave functions for the low - lying states of the isotopes
@xmath69te show a much simpler picture and are dominated by either neutron or proton excitations in many cases .
the contributions from @xmath90 components are 65.2% , 74.4% and 85.5% for isotopes
@xmath94te . the @xmath60 state in @xmath95te
is dominated by @xmath96 whereas @xmath97 state is dominated by the proton excitation @xmath98 instead .
the low - lying states for @xmath99te show a similar result
. it may be interesting to mention that a similar picture with rotational - like @xmath0 transitions and vibrational - like spectrum is also predicted for @xmath85 nuclei like @xmath100pd @xcite in relation to the quest for the possible existence of np pairing in @xmath3 nuclei for which there is still no conclusive evidence after long and extensive studies ( see , recent discussions in refs .
@xcite ) .
moreover , the @xmath29 formation amplitude may increase as a result of of the strong neutron - proton correlation .
there has already been a long effort answering the question whether the formation probabilities of neutron - deficient @xmath3 te and xe isotopes are larger compared to those of other nuclei @xcite .
we have evaluated within the shell - model approach the @xmath29 formation amplitude @xcite .
if the neutron - proton correlation is switched on , in particular if a large number of levels is included , there can be indeed significant enhancement of @xmath29 formation amplitude . to summarize ,
we have done systematic calculations on the spectroscopy and transition properties of te isotopes within the large - scale configuration interaction shell model approach .
a monopole - optimized realistic interaction is used .
the calculations reproduce well the excitation energies of the low - lying states as well as the regular and vibrational - like behavior of the yrast specta of @xmath101te ( fig .
1 ) .
the energies of the first @xmath10 and @xmath11 states as well as their ratios show rather a rather constant behavior when approaching @xmath2 in relation to the enhanced neutron - proton correlation ( fig .
2 ) . on the other hand ,
a squeezed gap between the @xmath97 and @xmath47 states is expected when approaching @xmath12 , resulting in seniority - like spectra .
those structure changes are also reflected in the calculated and available experimental e2 transition strengths .
the calculated @xmath102 show a parabolic behavior as a function of @xmath19 , which is dominated by the contribution from the neutron excitation .
moreover , the calculations reproduced reasonably well the nearly constant behavior of the @xmath0 values of @xmath1te and @xmath64te along the yrast line ( figs . 3 & 4 ) . the anomalous constant behavior is related to the competition between the seniority coupling and the neutron - proton correlations . for neutron - deficient te isotopes , the constant behavior of @xmath0 values can be extended to high spin values around @xmath103 14 .
wheras for heavier isotopes , when the neutron - proton correlation gets weaker , the @xmath0 values can reduce significantly after @xmath70 and vanishes for the heaviest isotopes . |
we acknowledge the financial support of the royal society ( s.b.d . ) and the uk epsrc , and invaluable discussions with igor mazin , michelle johannes and zahid hasan .
this experiment was performed with the approval of the japan synchrotron radiation research institute ( jasri , proposal nos .
2005a0092-nd3a - np and 2005b0182-nd3a - np ) .
this work was partially supported by a grant - in - aid for scientific research ( no.18340111 ) from the ministry of education , culture , sports , science and technology , japan .
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* 91 * , 097003 ( 2003 ) . | the surprise discovery of superconductivity below 5k in sodium cobalt oxides when hydrated with water has caught the attention of experimentalists and theorists alike .
most explanations for its occurence have focused heavily on the properties of some small elliptically shaped pockets predicted to be the electronically dominant fermi surface sheet , but direct attempts to look for them have instead cast serious doubts over their existence .
here we present evidence that these pockets do indeed exist , based on bulk measurements of the electron momentum distribution in unhydrated and hydrated sodium cobalt oxides using the technique of x - ray compton scattering .
hydrated sodium cobalt oxides ( na@xmath0coo@xmath11.3h@xmath2o ) , for a certain range of na concentrations , exhibit superconductivity at a temperature of 5k @xcite .
although many analogies have been drawn with the high-@xmath3 cuprates ( for instance , as being possibly the only other example of a mott insulator becoming superconducting under doping @xcite ) , the sodium cobaltate system exhibits its own unique set of anomalous behaviour such as unusually high thermopower @xcite and @xmath4-linear resistivity @xcite ) , distinctly indicative of strongly correlated electron behaviour . in a conventional superconductor ,
electrons at the fermi surface form cooper pairs under an attractive interaction mediated by lattice vibrations .
the manner in which electrons form these pairs can be strongly influenced by the shape of the fermi surface .
questions regarding the origin of the pairing interaction and the nature of the superconductivity in the cobaltates has stimulated significant theoretical speculation , most of which has focused heavily on the properties of some small elliptically shaped pockets predicted to be the electronically dominant fermi surface sheet @xcite , but the outcome of direct attempts to look for them has instead cast serious doubt over their existence @xcite . here
we present evidence that these pockets do indeed exist , based on bulk measurements of the electron momentum distribution in unhydrated and hydrated sodium cobalt oxides using the technique of x - ray compton scattering .
the structure of na@xmath0coo@xmath2 comprises hexagonal planes of electronically active edge - sharing coo@xmath5 octahedra @xcite .
these planes are separated by insulating layers of na , and , in the hydrated samples , water , that serve as spacers ( resulting in electronic two - dimensionality ) and charge reservoirs .
the intercalation of water , at a concentration of @xmath6 , has a dramatic effect on properties of the compound .
it is accompanied by a near doubling of the @xmath7-axis lattice parameter and , for @xmath8 , the onset of superconductivity .
calculations of electronic structure based on the local - density - approximation ( lda ) predict , for concentrations @xmath90.6 in na@xmath0coo@xmath2 , a fermi surface composed of two hole sheets originating from co @xmath10-orbitals @xcite .
the larger sheet , of @xmath11 character , is centred at the @xmath12-point ( zone centre ) whereas the other sheet , of @xmath13 character , comprises six smaller elliptical cylinders located between @xmath12 and @xmath14 ( for @xmath15 , this band becomes completely filled and does not contribute to the fermi surface ) .
these @xmath13 pockets may be of considerable importance , since many of the models for superconductivity in na@xmath0coo@xmath16h@xmath2o are predicated upon their existence .
in particular , an analysis of permitted order - parameter symmetries has indicated that an unconventional @xmath17-wave ( @xmath18 ) would be the most probable if the pockets exist @xcite .
with 70% of the density - of - states at the fermi level , these pockets are central to several models of spin - fluctuation - mediated superconductivity , developing either via imperfect nesting between the fermi surface pockets @xcite , or as a consequence of these disconnected pieces of fermi surface @xcite . a series of angle - resolved photoemission spectroscopy ( arpes ) results
, the consensus of which report observation of the @xmath11 sheet at @xmath12 but crucially not the @xmath13 pockets , has fuelled the controversy surrounding the elliptical pockets .
indeed , they observe the @xmath13 band as lying completely below the fermi level @xcite .
however , it is well - known that surface effects and matrix elements can have a strong influence on photoemission results , although there are recent reports of measurements being made with an electron escape depth of 200 @xcite . here
we directly tackle this controversy by presenting a compton scattering study both of the fermi surfaces of a representative set of compositions of the unhydrated parent compound na@xmath0coo@xmath2 ( @xmath19=0.38 , 0.51 and 0.74 ) and a hydrated ( actually deuterated ) sample at a superconducting composition , na@xmath20coo@xmath21d@xmath2o .
for these measurements , single crystals of @xmath22 were grown in warwick using the floating - zone technique .
samples of lower sodium concentration were then obtained by a chemical deintercalation method , immersing the crystals in solutions of br@xmath23 and acetonitrile .
the lattice parameters were obtained by x - ray diffraction ; the relationship between na doping and the crystal structure is well characterised by powder neutron diffraction measurements and icp - aes techniques @xcite and so the sodium concentrations for the three crystals used in this study were determined as 0.74(1 ) , 0.51(1 ) and 0.38(1 ) .
the macroscopic properties ( magnetic susceptibility , heat capacity and transport measurements @xcite ) are identical to those previously reported for similar compositions @xcite .
the superconducting sample was produced by the chemical intercalation of deuterium oxide ( d@xmath2o was used rather than h@xmath2o to allow future neutron experiments on the same sample ) by submersion in liquid d@xmath2o for three months at 5@xmath24c @xcite .
subsequent to the compton experiment , a measurement of its magnetisation showed that it was superconducting at a temperature of 3.5k .
a compton profile represents a double integral ( one - dimensional projection ) of the full three - dimensional electron momentum density . for each composition , five compton profiles equally spaced between @xmath12-@xmath25 and @xmath12-@xmath14 were measured on the high - resolution compton spectrometer of beamline bl08w at the spring-8 synchrotron .
the unhydrated measurements were made at room temperature , while those on the hydrated sample were at 11k .
the hydrated sample was transferred from a d@xmath2o bath directly onto a cryostat precooled to @xmath26k under a helium atmosphere in order to preserve the hydration . on removal at the end of the measurement , the sample
was observed to be still in its hydrated state .
the spectrometer consists of a cauchois - type crystal analyser and a position - sensitive detector , with a resolution fwhm at the compton peak of 0.115 a.u ( 1 a.u . of momentum @xmath27 1.99 @xmath2810@xmath29 kg m s@xmath30 ) @xcite .
for each compton profile , @xmath31 counts in the peak data channel were accumulated , and each compton profile was corrected for possible multiple - scattering contributions .
a two - dimensional momentum density , representing a projection down the @xmath7-axis of the full three - dimensional density , was reconstructed from each set of five profiles using tomographic techniques @xcite and then folded back into the first bz using the lock - crisp - west procedure @xcite to obtain the occupation density from which the occupied parts of the bz could be inferred .
the occupation density is shown for each composition in fig .
[ expfs ] , where black represents the lowest occupancy and white the highest .
( color online ) the experimental fermi surface of na@xmath0coo@xmath2 for ( a ) @xmath32 , ( b ) 0.51 , and ( c ) 0.38 , and for na@xmath20coo@xmath21d@xmath2o ( d ) obtained from the reconstruction of five compton profiles for each composition .
the boundary of the first brillouin zone is indicated . ] considering first the unhydrated parent compound , the contours associated with the hexagonal @xmath11 hole sheet can be clearly identified for @xmath32 , but the hexagonal shape becomes progressively less clear and is significantly distorted in the @xmath33 data .
we shall argue that this distortion is strong evidence for the presence of the @xmath13 pockets .
fermi surface obtained by plotting the contour at the maximum of the first derivative of the occupation density for na@xmath34coo@xmath2 . ] to assess the size of the hexagonal fermi surface of the na@xmath34coo@xmath2 compound , a method using the extrema in the first derivative of the occupation density was employed ( see for example @xcite ) .
at this composition elliptical pockets are not expected , and so the determination using this method should unambiguously reveal the hexagonal fermi surface .
fig .
[ fderiv ] is the result , and shows a fermi surface in excellent agreement with lda calculations .
we can explain the distortion of the hexagonal shape for smaller na concentrations as being due to the presence of @xmath13 elliptical hole pockets close to the central @xmath11 fermi surface .
a simple geometric simulation of such a fermi surface is shown in fig .
[ simul ] together with the resulting occupation density ( convoluted with the experimental resolution ) , illustrating how the presence of small pockets distort the hexagonal appearance of the @xmath11 fermi surface . the experimental occupation density for the hydrated na@xmath20coo@xmath21d@xmath2o ( fig .
[ expfs ] ) is also strongly suggestive of the presence of small hole pockets , which can be discerned close to the central hexagon ; it is also worth remarking that the @xmath11 sheet retains a strong hexagonal shape , whereas electronic structure calculations in the hydrated structure ( although without the actual presence of water ) predict something more circular @xcite .
our results suggest that the effects of hydration on the fermi surface are in fact rather modest , and perhaps not as drastic as suggested by xiao _
et al .
_ @xcite .
an estimate of the areas of the hexagonal @xmath11 sheet and ( where appropriate ) the six elliptical @xmath13 pockets based on a comparison of simulations to the experimental data ( with the total area constrained by the appropriate na concentration ) is presented in table [ fsparam ] . that the pockets consistently appear rather close to the @xmath11 sheet is also noteworthy .
( color online ) a simulation of a fermi surface comprising a central hexagonal sheet ( representing the @xmath11 hole sheet ) and six @xmath13 elliptical hole pockets ( left ) together with the resulting occupation density convoluted with the experimental resolution ( right ) .
, title="fig : " ] ( color online ) a simulation of a fermi surface comprising a central hexagonal sheet ( representing the @xmath11 hole sheet ) and six @xmath13 elliptical hole pockets ( left ) together with the resulting occupation density convoluted with the experimental resolution ( right ) . , title="fig : " ] .
estimate of areas of the @xmath11 and e@xmath35 fermi surfaces ( in the case of the latter , this is the total area of all six pockets ) based on applying the simulation of the fermi surface as illustrated in fig . [ simul ] for the compositions studied , as a proportion of the hexagonal first brillouin zone . [ cols="^,^,^",options="header " , ] recent measurement of shubnikov - de haas oscillations in na@xmath36coo@xmath2 indicate the presence of some unidentified fermi surface pockets occupying approximately 0.6% and 1.4% of the bz @xcite , which is consistent with our estimate of the size of each pocket we observe occupying about 0.8% ( table [ fsparam ] ) .
in addition , an examination of phonon softening in this system by rueff _
et al . _
@xcite is interpreted by those authors as strong evidence for the existence of nested pockets .
the question of why the arpes experiments have consistently not observed these e@xmath35 pockets remains .
issues such as surface sensitivity , including possible surface relaxations of coo@xmath5 octahedral contractions that destroy the pockets @xcite , as well as matrix - element effects or na disorder @xcite must be possibilities .
however , at least in the superconducting compound , it is very difficult to reconcile the observed behaviour of the specific heat , or even understand the presence of superconductivity without the presence of the @xmath13 pockets @xcite .
moreover , in an attempt to take into account coulomb correlations not included in the lda , calculations @xcite based on the lda@xmath37 approach , have suggested that for a sufficiently large coulomb energy ( @xmath38 ev ) the @xmath13 band is pulled below the fermi level , eliminating these smaller fermi surface pockets for all na concentrations .
however , other studies have put an upper limit of about 2.3ev on @xmath39 @xcite , and when dynamical coulomb correlations are incorporated the effect is to stabilize the @xmath13 pockets @xcite .
the theoretical debate rages on , with predictions in support of @xcite and contrary to @xcite the existence of these pockets . in conclusion
, we have presented the fermi surface of several members ( @xmath33 , 0.51 and 0.74 ) of the unhydrated sodium cobalt oxide na@xmath0coo@xmath2 and of a hydrated composition na@xmath20coo@xmath21d@xmath2o .
reasonable _ qualitative _ agreement is observed between our experimentally determined fermi surfaces and the lda predictions , and there is clear evidence for the smaller e@xmath35 elliptical hole pockets which develop at lower na concentrations than na@xmath34coo@xmath2 . for na@xmath40coo@xmath2 , their presence
is clearly indicated in experimental maps of the occupancy within the brillouin zone .
most importantly , however , the occupancy map for na@xmath20coo@xmath21d@xmath2o also shows the presence of small e@xmath35 elliptical hole pockets . while alternative models that describe the superconductivity arising as a consequence of frustration on the triangular lattice @xcite , nesting across the large , hexagonal sheet @xcite , or spin fluctuations enhanced by the dopant dynamics @xcite are not ( and can not be ) ruled out , the observation of the pockets lends strong support to theories based on their special nesting properties |
.volume fraction for @xmath98 , rlp . [ cols="^,^,^,^,^,^,^,^,^,^,^",options="header " , ]
according to the theory , the average voronoi volume for a packing with a distribution of radius @xmath109 , is given by the following self - consistent equation : where the different quantities are calculated as follow : @xmath111 @xmath112 to simplify we denoted @xmath113 , @xmath114 and @xmath80 the step - function .
@xmath115 @xmath116 @xmath117 @xmath118 @xmath119 @xmath120 | we develop a model to describe the properties of random assemblies of polydisperse hard spheres .
we show that the key features to describe the system are _ ( i ) _ the dependence between the free volume of a sphere and the various coordination numbers between the species , and _
( ii ) _ the dependence of the coordination numbers with the concentration of species ; quantities that are calculated analytically .
the model predicts the density of random close packing and random loose packing of polydisperse systems for a given distribution of ball size and describes packings for any interparticle friction coefficient .
the formalism allows to determine the optimal packing over different distributions and may help to treat packing problems of non - spherical particles which are notoriously difficult to solve .
understanding the basic properties of spheres packings is a major challenge since this problem may provide valuable knowledge regarding low temperature phases in condensed matter physics @xcite .
the canonical example is perhaps the monodisperse sphere packing problem .
it has been mathematically proven that the optimum way to arrange monodisperse spheres is the face - centered cubic lattice ; a problem that has been solved recently by hales , @xmath0 400 years after the famous kepler conjecture on the issue . on the other hand , it is commonly observed that packings arrange in a random fashion at a lower density state called random close packing or rcp @xcite .
furthermore , packings are mechanically stable up to an even lower limit called random loose packing , rlp . in parallel with the large literature dealing with monodisperse sphere packings , a large body of experimental
, theoretical and numerical work has been devoted to the analysis of polydisperse systems ; the interest arising due to the simple fact that polydispersivity is omnipresent in most realistic systems and industrial applications @xcite . while previous approaches have focused on frictionless packings , an integrated analytical approach that brings together different observations for all packings from rlp to rcp and for any friction or coordination number is still lacking .
based on our previous statistical mechanics approach @xcite , here we build such a framework .
we show that the key aspect to solve this problem is the dependence of the various coordination numbers between the different species and the concentration of the species .
this is calculated here and shown to agree well with computer simulations .
this result is then incorporated into a statistical theory of volume fluctuations as in @xcite which calculates the free volume of a particle in terms of the coordination number .
the main result is the prediction of the rlp and rcp limiting densities for a given distribution of ball sizes as well as the prediction of densities for any packing in between those limits .
the formalism allows for a determination of the best packing fraction in terms of different distribution of ball sizes with specified constraints , as we show with a simple example .
we discuss possible generalization of the method to solve more difficult problems like the phase behavior of systems of non - spherical particles like rods or spherocylinders in any dimensions ; problems of long - standing history in condensed matter @xcite .
recent theoretical advances @xcite allow for the prediction of the density of rcp and rlp for equal - size ball packings using a relation between the average volume and the geometrical coordination number . following this approach ,
we here describe long - range spatial correlations through a mean - field background term .
this approximation makes the problem amenable to analytic calculations , and is shown to describe well our simulation results .
an explicit inclusion of such correlations is possible in our framework , but severely complicates any solution attempts .
thus , we believe that the present approach is accurate enough for many important properties , such as the volume fraction calculation . the above theoretical framework will guide the present formalism for polydisperse systems .
we first treat the case of binary mixtures of hard spheres of radius @xmath1 and @xmath2 in 3d and then generalize the problem to any distribution and dimension .
* calculation of @xmath3. * the key quantity to calculate is @xmath3 , denoting the mean number of contacts of a ball of radius @xmath4 with a ball of radius @xmath5 , versus the concentration of one of the species . we need a formula for @xmath3 as a function of @xmath6 , the later being the size ratio , @xmath7 is the fraction of small balls in the packing @xmath8 with @xmath9 the number of @xmath10 balls and @xmath11 is the global geometrical coordination number averaged over all the particles : @xmath12 where @xmath13 is the average coordination of each species .
the coordinations are determined by three equations : @xmath14 we assume that these coordinations are inversely proportional to the average solid angle extended by contacting balls @xmath1 and @xmath15 .
the average solid angles are denoted @xmath16 and are calculated in terms of the solid angle that a ball @xmath5 occupy on a ball @xmath4 according to @xmath17 with ( see fig .
[ voro]a ) @xmath18 thus , @xmath19 represents the mean occupied surface on a @xmath10 ball weighted by the concentrations @xmath7 and @xmath20 .
this represents an approximation since the real weights are @xmath21 and @xmath22 , respectively . then , @xmath23 leading to the following normalizations : @xmath24 thus , the system of eqs .
( [ z1221 ] ) is reduced to a system of three equations for four unknowns @xmath3 . to close the system
we assume proportional laws and deduce @xmath3 from @xmath13 by considering that @xmath3 is proportional to the number of contacts of the @xmath10 balls times the number of contacts of the @xmath25 balls : @xmath26 using the first equation in ( [ z1221 ] ) we find the constants @xmath27 and @xmath28 , leading to the solution : @xmath29 figure [ fig : zandp]a , compares this solution to numerical simulations of hertz packings jammed at rcp @xcite for @xmath30 and @xmath31 .
we find that the formulae are very accurate for size ratios below 1.5 and present small deviations up to size ratio 2 .
the results are also in agreement with @xcite . *
the voronoi cell. * the common way to divide a system into volumes associated with each particle is the voronoi tessellation . the voronoi cell for monodisperse particles @xcite is composed by all the points nearest to the center of the ball than to any other ball .
this definition has been extended in @xcite to the case of polydisperse systems and non - spherical particles : instead of considering the classical voronoi polyhedron defined by the center of the particle , one should consider all the points which are closer to the surface of a given particle .
such a construction is called the _
voronoi s region _ and tiles a system of nonspherical convex particles and polydisperse systems as can be seen in fig .
[ voro]b . following this approach
we calculate the average volume of a polydisperse voronoi cell , denoted @xmath32 .
the volume fraction is given by @xmath33 , where @xmath34 is the mean volume of a ball .
we first find the analytical formula of the voronoi s region .
the boundary of the voronoi cell in the direction @xmath35 of a @xmath10 ball next to a @xmath25 ball at position @xmath36 is ( fig .
[ voro]b ) : @xmath37 where @xmath35 and @xmath38 are unitary .
thus , the boundary of a voronoi cell of a ball @xmath10 in the direction @xmath35 is the minimum of @xmath39 over all the particles @xmath25 for any @xmath40 .
this leads to @xmath41 the volume of the cell of the ball @xmath10 is then given by @xmath42 .
we define the orientational voronoi volume , @xmath43 , along the direction @xmath35 by @xmath44 .
this leads to @xmath45 this definition leads to the results of @xcite when @xmath46 .
since the system is isotropic , the mean voronoi volume can be calculated as : @xmath47 * calculation of the mean voronoi volume. * having calculated the voronoi cell exactly in eq .
( [ voronoi ] ) , we now proceed to develop a probability theory of volume fluctuations in the spirit of the quasiparticle approximation used in @xcite to obtain the mean voronoi volume . for a given ball @xmath10
, the calculation of @xmath48 reduces to finding the ball @xmath49 that minimizes @xmath39 .
we call @xmath49 the voronoi ball for the ball @xmath10 .
we consider @xmath50 , @xmath51 and @xmath52 .
we have @xmath53 .
therefore , we just need to compute the inverse cumulative distribution function , denoted @xmath54 , to find all the balls @xmath25 with @xmath55 , and thus not contributing to the voronoi volume of the ball @xmath10 .
the average voronoi volume is then given by the expression @xmath56 we calculate the mean voronoi volume for the balls of radius @xmath1 and @xmath15 separately and then average them .
we denote @xmath57 and @xmath58 the inverse cumulative distributions respectively , and @xmath59 and therefore , @xmath60 * calculation of @xmath54. * there are three salient steps in the calculation of @xmath54 : _ ( i ) _ the separation of @xmath54 following eq .
( [ p_c ] ) . _
( ii ) _ the separation of each term @xmath61 , @xmath62 , into two contributions : a term taking into account the contact spheres , @xmath63 , and a bulk term , @xmath64 .
the contact term clearly depends on @xmath3 .
the bulk term averages over all spatial correlations of non - contact particles and , without significant loss of accuracy as shown below , we assume that it only depends on the average value of @xmath32 . in principle , it is possible to use a more realistic form for this term , but this would render the problem practically unsolvable . _ ( iii ) _ the separation of @xmath63 and @xmath65 into two terms @xmath66 and @xmath67 , @xmath68 , for each species .
an assumption of the theory ( to be tested a posteriori with computer simulations ) is that all of these terms are independent .
thus @xmath69 also , we work in the limit of large number of particles leading to boltzmann - like exponential distributions for each @xmath70 and @xmath71 @xcite .
four important quantities are then defined : _ ( i ) _ @xmath72 and _ ( ii ) _ @xmath73 : the excluded volume and surface on the ball , respectively , where no center of a ball @xmath25 can be located for a given ball @xmath10 and for a given @xmath74 . _ ( iii ) _ @xmath75 : the mean number of balls @xmath25 by the left free volume . _ ( iv ) _ @xmath76 : the mean number of balls @xmath25 by the left free surface on a ball @xmath10 .
these considerations lead to : @xmath77 next , we calculate these four quantities . to simplify we denote @xmath78 , @xmath79 and @xmath80 the step - function .
we obtain : @xmath81 where @xmath82 and @xmath83 .
the fourth quantity , @xmath76 , is the most difficult to calculate . in terms of the occupied areas
eqs .
( [ occ ] ) we have @xmath84
. however , for the contact terms , the analogy with the boltzmann - like exponential distribution of volumes is far from being exact .
this is because this form is based on the large number limit which in the case of contacting balls is no more than around 6 .
therefore , the exponential distribution is used as an ansatz with @xmath76 a variational parameter as in @xcite .
we denote @xmath85 the mean solid angle of the gaps left between the @xmath25 contacting neighbors of a @xmath10 ball ( fig .
[ voro]a ) .
we obtain : @xmath86 then , we perform numerical simulations to find @xmath87 .
we randomly generate balls of radius @xmath4 and @xmath5 with the proportion @xmath21 and @xmath22 respectively around a ball of radius @xmath4 and then evaluate the mean free surface @xmath88 and its inverse to obtain @xmath76 .
we find @xmath89 which is a generalization of the results of @xcite to polydisperse systems .
using eqs .
( [ pij ] ) , ( [ sstar ] ) and ( [ rhos ] ) into ( [ pc ] ) , @xmath54 can be calculated by solving the equations numerically .
figure [ fig : zandp]b depicts the comparison of the theoretical results of the probability of voronoi volumes @xmath54 with computer generated hertzian packings with @xmath31 for @xmath90 at rcp .
the calculated distribution is quite accurate for most of the range except for small deviations at high values of @xmath74 , which however , do not affect much the value of the average @xmath91 in eq .
( [ ww ] ) .
this shows that our mean - field approximation already captures the main contribution of the probability distribution function @xmath54 .
* calculation of @xmath32. * the above considerations lead to a final form to calculate @xmath92 using eq .
( [ pc ] ) into ( [ ww ] ) : @xmath93 notice that @xmath94 depends on @xmath32 , eq .
( [ sstar ] ) , and @xmath95 depends on the @xmath3 , eq .
( [ rhos ] ) , which in turn depends on the concentration @xmath7 and @xmath11 through eq .
( [ zs ] )
. therefore eq .
( [ final ] ) is a self - consistent equation to obtain @xmath96 , for a given @xmath97 . a numerical integration of eq .
( [ final ] ) is performed versus @xmath7 for a given value of @xmath11 . by considering the isostatic limits of @xmath31 and @xmath98 we predict the limits of rcp and rlp at zero friction and infinite friction between the spheres , respectively @xcite .
the results for the volume fraction at rcp versus @xmath7 are depicted in fig .
[ fig : jamcompare]a which also compares the results to numerically generated packings of hertz spheres @xcite .
we see a very good agreement indicating that the theory captures well the behavior of polydisperse packings and that the approximations used are reasonable . for size ratios larger than 2 deviations
are found indicating the limit of validity of the approach .
for any other value of interparticle friction between 0 and @xmath99 , the density is obtained by setting @xmath11 between the limiting isostatic values of 6 and 4 , respectively .
the resulting volume fraction is shown in fig .
[ fig : jamcompare]b .
the result for rlp for a given @xmath7 is a new prediction as this problem has not been investigated before .
our results promote new experiments to test the rlp limit of polydisperse systems shown in fig .
[ fig : jamcompare]b . the formalism can be extended to consider any distribution of sphere radius .
the main modification is that all the sums over the radius are replaced by integrations over the desired distribution of radius @xmath100 ( the binary case corresponds to two delta - functions at @xmath1 and @xmath15 ) . and
all the quantities are calculated for balls of internal radius @xmath101 and external @xmath102 and @xmath7 and @xmath20 are replaced by @xmath103 and @xmath104 , respectively , including the coordinations ( see supplementary information b ) .
this result allows to explore the space of radius distributions in search of the optimal packings for a given polydispersivity .
this analysis could be of industrial interest if one wishes to optimize the packing fraction by introducing different species in the mixture .
\(a ) versus @xmath7 for different values of @xmath97 as indicated .
error bars are std over 10 realizations of the packings with 10,000 balls .
( b ) three dimensional surface plot of @xmath105 as a function of @xmath11 and @xmath7 for @xmath106 .
the numerical results at rcp and rlp are provided in supplementary information a.,title="fig:",scaledwidth=25.0% ] ( b ) versus @xmath7 for different values of @xmath97 as indicated .
error bars are std over 10 realizations of the packings with 10,000 balls .
( b ) three dimensional surface plot of @xmath105 as a function of @xmath11 and @xmath7 for @xmath106 .
the numerical results at rcp and rlp are provided in supplementary information a.,title="fig:",scaledwidth=25.0% ] we calculate the volume fraction for several distributions @xmath100 constraint by ball radius in the range [ 1,2 ] , in search of the optimal packing .
we calculate the volume fraction for various @xmath100 ranging from uniform to two - peaked gaussian distributions of varied widths .
we find that the more small balls we add the better the packing until a certain point where the volume fraction starts to decrease .
this maximum can be rationalized assuming that the small balls always fill the gaps between the large ones as long as there are enough large balls .
further extensions of the theory to any dimension can be performed by replacing @xmath107 by d in eq .
( [ voronoi ] ) and developing a theory of volume fluctuation in d - dimensions .
we notice that many of the approximations employed in 3d may become exact for large d , thus we expect improved results in the mean field limit of infinite dimensions .
the method allows to treat more difficult problems .
for instance , the prediction of the volume fraction of a jammed system of non - spherical particles is a long - standing problem .
theoretical predictions of onsager @xcite are valid for large aspect ratios , like elongated rods .
experiments however , find interesting new physics for small aspect ratios . in this respect
, the present polydisperse theory could be mapped to the problem of ellipsoids , spherocylinders or rods .
a voronoi cell needs to be calculated as a function of the angles defining the orientation of the non - spherical particles in analogy of the calculation between two particles of different radii .
the integration over @xmath108 in eq .
( [ final ] ) is then replaced by integration over weighted orientational angles . the above analysis can also be extended to dimensions beyond three @xcite .
although many of the appproximations should work better in higher dimensions , some of the hypotheses ( for example , the contact term ansatz ) need to be reassessed .
thus , higher dimensions studies can not be addressed as trivial extensions and need to be handled with care . in summary
, a theoretical framework is presented that predicts the rlp and rcp limits of a system of polydisperse spheres and brings together distinct results into a common theoretical framework .
the formalism has the potential to solve other problems in condensed matter physics such as the mixing and phase behavior of systems of hard particles of different shapes and size . 99 a. coniglio , a. fierro , h. j. herrmann , m. nicodemi , eds , _ unifying concepts in granular media and glasses _
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the angular clustering of faint @xmath11-selected field galaxies has been studied extensively ( e.g. , efstathiou et al . 1991 ; roche et al . 1993 , 1996 ;
brainerd , smail & mould 1995 ; hudon & lilly 1996 ; lidman & peterson 1996 ; villumsen , freudling & da costa 1996 ; woods & fahlman 1997 ) , and a prime motivation of these studies has been to investigate the nature of the faint field population .
in particular , it is possible to infer the effective correlation length of the sample and the rate at which clustering evolves from a combination of the amplitude of the angular autocorrelation function , @xmath1 , and the redshift distribution of the faint galaxies , @xmath12 .
these observations can then be used to link properties of the faint field population with samples of local galaxies .
while the exact interpretation remains controversial , it is generally accepted that overall @xmath1 is fitted well by a power law of the form @xmath13 ( although see infante & pritchet ( 1995 ) for evidence of a flattening in the power - law coefficient at faint limits ) .
here we investigate the clustering of faint galaxies and focus on the behavior of @xmath1 at small angular separations .
we obtain a clear measurement of @xmath1 on scales of @xmath14 whereas previous investigations have been largely limited to scales of @xmath15 .
additionally , we use the clustering properties of the galaxies to estimate the number of pairs of galaxies that are physically close to each other in space ( separations of @xmath7 kpc ) .
the data consist of deep @xmath11-band imaging of 11 independent fields that were obtained in good conditions with the low resolution imaging spectrograph on the 10-m keck - i telescope .
each of the @xmath16 fields is centered on a high redshift quasar with high galactic latitude ; however , the presence of the quasar in the field is irrelevant to the present investigation ( i.e. , the presence of a small group of galaxies at the redshift of the quasar will not influence the results below ) .
the galaxy catalogs are complete to @xmath17 and the apparent magnitudes of the galaxies have been corrected for extinction . in order to reduce the stellar contamination in the object catalogs , only objects with @xmath18 are considered in the analysis below .
there is , of course , some residual stellar contamination of the galaxy catalogs at faint limits and we estimate that to be : @xmath1916% ( @xmath20 ) , @xmath1913% ( @xmath21 ) , @xmath1911% ( @xmath22 ) .
the integral constraints vary little from field to field due to the use of the same detector in all cases as well as the lack of very large , bright galaxies in the fields .
to compute the angular clustering of the faint galaxies we use the landy & szalay ( 1993 ) estimator : @xmath23 where @xmath24 , @xmath25 , and @xmath26 are the number of unique data - data , data - random , and random - random pairs within a given angular separation bin .
regions of the frame where faint galaxy detection was either lower than average or impossible ( e.g. , due to the presence of bright stars and galaxies ) were masked out when computing @xmath25 and @xmath26 .
raw correlation functions ( uncorrected for stellar contamination or the integral constraint ) were determined for each of the fields , from which a mean correlation function was computed .
the results for the mean raw correlation function are shown in figure 1 , where the error bars show the standard deviation in the mean . from top to bottom , the panels show the results for objects with @xmath20 , @xmath21 , and @xmath22 , respectively . also shown
are the formal best - fitting power laws of the form @xmath27 ( solid lines ) and the best - fitting power laws of the form @xmath13 ( dashed lines ) .
the power laws in the figure have been suppressed by the appropriate integral constraints and no correction for residual stellar contamination has been applied . the number of pairs of galaxies that we observe to be separated by @xmath28 is larger than the number predicted by the fiducial @xmath13 power law ( i.e. , the power law that is typically obtained from measurements that have been performed on scales of @xmath29 ) .
this is consistent with the results of carlberg et al .
( 1994 ) and infante et al .
( 1996 ) who both found @xmath1 to have a higher amplitude on small angular scales ( @xmath30 ) than a simple inward extrapolation of @xmath1 as measured at large angular scales .
as yet , however , it is unclear whether the steepening of @xmath1 is due to the existence of a population of `` companion '' galaxies ( which are not seen at the present epoch ) or luminosity enhancement ( e.g. , due to interactions ) of intrinsically faint galaxies that are in pairs . in the absence of significant luminosity enhancement
, we can estimate the number of pairs of galaxies that are physically close to each other simply by using the following probability : @xmath31 ( e.g. , burkey et al .
1994 ) , where @xmath32 is the number density of galaxies brighter than the faintest member in a pair of galaxies that is a candidate for close physical separation , @xmath33 is the observed angular separation between the galaxies , and @xmath34 is the smallest separation observed between all detected galaxies ( @xmath35 in our data ) . using eqn .
( 2 ) we compute the number of pairs of galaxies for which @xmath36 and @xmath37 in our data .
additionally , we use monte carlo simulations ( in which the magnitudes of the galaxies are shuffled at random ) to calculate the number of pairs of galaxies that would have @xmath36 and @xmath37 simply by chance .
the latter step allows the removal of random superpositions from the estimate of the `` true '' number of close pairs in the sample . below @xmath28
there are fewer pairs of galaxies with @xmath36 and @xmath37 in the actual data than are expected in a random distribution ( i.e. , based on the monte carlo simulations ) , indicating that we are undercounting the very closest pairs due to blending of the images . using our measured @xmath1 , however
, we can correct the faint pair counts on scales @xmath38 and estimate the fraction of galaxies in our sample that are in truly close physical pairs . based on a simple extrapolation of the cfrs redshift distribution
, we expect that the mean redshift of our galaxies is @xmath39 and , hence , if @xmath40 , physical pairs of galaxies that are separated by @xmath41 will be within @xmath42 kpc of each other .
the best - fitting power law form of @xmath1 ( corrected for stellar contamination and the integral constraint ) then yields an estimate of the pair fraction at this physical separation of @xmath6 for @xmath43 , which agrees with the results obtained by carlberg et al .
( 1994 ) for the fraction of galaxy pairs with separations @xmath44 kpc at @xmath8 .
this suggests , therefore , that little evolution in the merger rate of galaxies occurred between @xmath9 and @xmath10 .
a significant amount of work remains to be done on this project , including a clustering analysis of 7 additional independent fields and a more rigorous study of the number of pairs of faint galaxies located at close physical separation .
financial support under nsf contract ast-9616968 ( tgb ) and a boston university presidential graduate fellowship ( cjl ) are gratefully acknowledged .
the observations were obtained at the w. m. keck observatory , which is operated jointly by the california institute of technology and the university of california .
data analysis was performed exclusively on the origin2000 at boston university s scientific computing & visualisation facility .
brainerd , t.g . ,
smail , i. & mould , j.r . , 1995 , mnras , 275 , 781 burkey , j. m. , keel , w. c. , windhorst , r. a. , & franklin , b. e. , 1994 , apj , 429 , l13 carlberg , r. g. , pritchet , c. j. , & infante , l. 1994 , apj , 435 , 540 efstathiou , g. , bernstein , g. , katz , n. , tyson , j.a .
, & guhathakurta , p. , 1991 ,
apj , 380 , l47 hudon , j.d . &
lilly , s.j . , 1996 ,
apj , 469 , 519 infante , l. & pritchet , c.j . , 1995 ,
apj , 439 , 565 infante , l. , de mello , d. & menanteau , f. 1996 , apj , 469 , l85 landy , s.d . & szalay , a.s . , 1993 ,
apj , 412 , 64 lidman , c.e . & peterson , b.a . , 1996 , mnras , 279 , 1357 roche , n. , shanks , t. , metcalfe , n. , & fong , r. , 1993 , mnras , 263 , 360 roche , n. , shanks , t. , metcalfe , n. , & fong , r. , 1996 , mnras , 280 , 397 woods , d. & fahlman , g.g .
, 1997 , apj , 490 , 11 villumsen , j.v . ,
freudling , w. , & da costa , l.n . , 1996 , apj , 481 , 578 | we present a preliminary measurement of the angular clustering of faint ( @xmath0 ) field galaxies in which we concentrate on the behavior of @xmath1 on small angular scales ( @xmath2 ) .
the galaxies are strongly clustered and @xmath1 is well - characterized by a power law of the form @xmath3 .
the best - fitting value of the power law index , @xmath4 , is , however , steeper than the fiducial value of @xmath5 , indicating that there are more pairs of galaxies separated by @xmath2 in our sample than would be otherwise expected . using the best - fitting form of @xmath1 , we estimate that @xmath6 of the galaxies are in physically close pairs ( separations @xmath7 kpc ) .
this is a factor of order 2 larger than local galaxy samples but comparable to galaxy samples with @xmath8 .
the mean redshift of our galaxies is of order 0.95 , and , therefore , our result suggests that there was little or no evolution in the merger rate of galaxies between @xmath9 and @xmath10 . |
it has long been known that galaxies in the dense cluster environments systematically differ from those in the field in their morphology , stellar populations , gas fractions and gas distributions .
more interestingly , observations of galaxy clusters at intermediate redshift show that these properties change with redshift .
this could indicate that the dense cluster environment speeds up the evolution of galaxies .
various mechanisms affecting the evolution of galaxies have been suggested such as ram - pressure stripping , merging , tidal interaction , harassment or starvation . in spite of the abundance of statistical studies on clusters at intermediate redshifts ( e.g. * ? ? ?
* ; * ? ? ?
* ) there is a lack of indepth studies of individual galaxies which will further constrain the environmental effects in clusters . our goal is to do a detailed study of galaxies that are currently being affected by the cluster environment .
virgo is ideal for this purpose .
its nearness allows us to study details , and as a dynamically young cluster it shows a variety of processes at work to affect the galaxies .
different mechanisms can be traced by hi observations . since the first virgo survey in early 1980s
the sensitivity of the vla ( very large array ) has been significantly improved by almost a factor of 10 with a comparable resolution . as several virgo galaxies observed with a higher sensitisity show ( e.g. * ? ? ?
* ; * ? ? ?
* ; * ? ? ?
* ) , the newer hi data will provide much more detailed ( and possibly never seen ) structures .
we selected 41 spirals ( s0/a@xmath0sm ) in the virgo cluster ( figure 1 ) .
these galaxies cover a factor of 50 in mass of the cluster and span a wide range in star formation properties @xcite .
they are located throughout the cluster , from the dense region close to the center to the low density outer parts .
compared to previous surveys ( e.g. * ? ? ?
* ; * ? ? ?
* ) with brightness cutoff of @xmath1 in @xmath2 , our sample also contains fainter systems of @xmath3 mag which are more likely to be affected by stripping or gravitational interactions due to their low masses .
the observations of the 13 galaxies were made between february and may 2004 in vla c - short array .
we integrated @xmath48 hours on each source with a total bandwidth of 3.25 mhz and a channel separation of 24.4 khz ( 5.2 km s@xmath5 at 21 cm ) .
online hanning smoothing has been applied and the resulting velocity resolution is 10.4 km s@xmath5 .
we typically reached @xmath60.3 mjy per beam ( typically @xmath7 ) per channel ( 10.4 km s@xmath5 ) .
this corresponds to a 3@xmath8 surface density sensitivity of 4@xmath910@xmath10 @xmath11 . in some cases ( e.g. ngc 4294 , ngc 4299 , ngc 4383 and ngc 4694 )
the cubes were spatially convolved in order to bring out faint structures .
in figure 2 we present the total hi maps of 13 galaxies .
ngc 4351 , ngc 4396 , and ngc 4189 decline more sharply in hi surface brightness on the side toward the cluster center .
these galaxies are at intermediate distances from m87 ( 1.7@xmath04.3 deg ) and are likely to be experiencing on - going icm - ism pressure .
extended tails are seen in a number of galaxies .
ngc 4294/9 , ngc 4351 , ngc 4396 , ngc 4424 , ngc4651 , and ngc4698 are found at a range of distances ( @xmath12 deg from m87 ) .
no obvious companions are found around these galaxies and none of their tails except possibly ngc 4424 and ngc 4651 seems to be related to tidal interactions .
especially we note that ngc 4294 and ngc 4299 which are only 0.1 deg apart from each other and at similar redshifts , show tails in the same direction , unlike what happens in tidal interactions .
even though we do not have a coherent three dimensional picture of virgo , it is worth mentioning that all the tails except for one case ( ngc 4651 ) are pointing away from the cluster center .
the southern extension in hi of ngc 4424 could be related to the giant elliptical ngc 4472 which is located 1.56 deg away to south , either through a tidal interaction or an icm - ism interaction @xcite .
ngc 4651 is a peculiar case in a sense that its optical tail and the gas tail are extended in opposite directions , to east for the stellar extension and to west for the gas tail , suggesting a minor merger .
gas accretion or tidal interaction is also found in some galaxies such as ngc 4383 ( a small gas blob in se ) and ngc 4694 ( tidally interacting with a low surface brightness system vcc 2062 ) .
there are several galaxies with truncated hi disks such as ngc 4064 , ngc 4424 , ngc 4569 , and ngc 4580 .
ngc 4424 and ngc 4569 are located in high density regions in a sub - cluster or in the center of the cluster ( ngc 4569 is discussed further in the following section ) and icm - ism pressure likely has caused the truncation in the gas .
however ngc 4064 and ngc 4580 are exceptional in a sense that both of them are located in low density environments with projected distances of 8.8 and 7.2 degrees from m87 ( 3.1 and 2.5 virial radii ; * ? ? ?
* ) , respectively .
the most likely explanation is that they have gone through the center and are on their way out .
even if galaxies have not gone through the the highest density regions , icm - ism interactions could still happen at further distances from the cluster center when the galaxy interacts with locally enhanced icm due to sub cluster - cluster merging as suggested by @xcite for ngc 4522 .
deep optical and h@xmath13 images have already been taken .
more recently we have been granted galex ( galaxy evolution explorer ) time for the entire sample of 41 galaxies .
the uv data will allow us to trace the timescales on processes at work .
combined by multiwavelength data and also compared with simulations eventually we will get more clear understandings of galaxy evolutions in the cluster environments .
ngc 4569 is one of the closest galaxies to the cluster center in our sample and it is severely deficient in hi ( @xmath14% of the normal hi for its size and type ) .
both the h@xmath13 and hi are truncated , and only extend to 30% of its stellar disk . an anomalous arm is seen to the west both in hi and h@xmath13 ( figure 3 ) and the hi arm seems to be interrupted at the location of the h@xmath13 starburst outflow nebulosity .
the hi arm extends for 3 arcmin(@xmath413 kpc ) while the h@xmath13 nebulosity near the minor axis on the nw side extends up to 6 kpc from the nucleus .
some h@xmath13 filaments are also observed on the se side , but only within @xmath15 of the nucleus , and there is no large - scale h@xmath13 nebulosity in the se like that in the nw .
these diffuse structures seen along the minor axis must be a product of a nuclear starburst and have been disrupted by icm pressure ( nebulosity as the superbubble ) .
the icm pressure must be stronger on the se to see that the nebulosity is weaker on this side and the more extended hi structure on the other side seems to be consistent with this picture .
this arm thus must be extra - planar , because it appears to be interacting with the minor axis starburst outflow , and because it lacks significant dust extinction , suggesting that it lies behind the disk .
the arm resembles the features that are seen in some phases of icm - ism interaction simulations @xcite it has been suggested that the stripped gas can form one extra - planar arm by the combination of wind pressure and galaxy rotation ( see also * ? ? ?
ngc 4396 also show signatures of extra - planar gas .
the kinematics of this gas are especially interesting ( figure 4 ) .
note how the velocities in the component north of the disk are all redshifted w.r.t .
nearby disk on both sides of minor axis suggesting non - circular motions , possibly indication of the presence of extra - planar gas .
simulations , estimating icm pressure in this region or studying stellar population will be helpful for sorting out the interaction parameters and history .
some other cases of extra - planar gas in virgo cluster galaxies have been studied in highly inclined systems .
there are several pieces of evidence that ngc 4522 is experiencing ongoing icm - ism stripping despite its location far ( but similar to the distance of ngc 4396 ) from the cluster center @xcite .
see also the case of ngc 4402 by crowl et al .
( 2004 ) .
biller , b. a. , jones , c. , forman , w. r. , kraft , r. , & ensslin , t. 2004 , astro - ph/0406132 ( accepted to apj ) .
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apjs , 122 , 51 kenney , j. d. p. , van gorkom , j. , & vollmer , b. 2004 , aj , 127 , 3361 kenney , j. d. p. , hameed , s. , chung , a. , van gorkom , j. , & vollmer , b. 2004 , in preparation koopmann , r. , & kenney , j. d. p. 2004 , apj , in press ( october 1 , 2004 ) , astro - ph/0406243 phookun , b. , vogel , s. n. , & mundy , l. g. 1993 , apj , 418 , 113 phookun , b. , & mundy , l. g. 1995 , apj , 453 , 154 poggianti , b. m. et al .
1999 , apj , 518 , 576 schulz , s. , & struck , c. 2001 , mnras , 328 , 185 tully , r. b. , & shaya , e. j. 1984 , apj , 281 , 31 vollmer , b. , cayatte , v. , boselli , a. , balkowski , c. , & duschl , w. j. 1999 , a&a , 349 , 411 vollmer , b. , cayatte , v. , balkowski , c. , & duschl , w. j. 2001 , aj , 561 , 708 vollmer , b. , balkowski , c. , cayatte , v. , van driel , w. , & huchtmeier , w. 2004 , a&a , 419 , 35 vollmer , b. , beck , r. , kenney , j. d. p. , & van gorkom , j. 2004 , aj 127 , 3375 warmels , r. 1988 , a&as , 72 , 19 | we present preliminary results of vla hi imaging of selected virgo cluster galaxies .
the goal is to study environmental effects on galaxy evolution .
our sample of 41 galaxies is spread throughout the cluster and spans a wide range in star formation properties . here
we present the total hi maps of 13 galaxies .
we find a number of galaxies with extended hi tails , almost all pointing away from the cluster center .
truncated hi disks are found close to the center but also in the outer region .
some galaxies near the cluster center show compression of the gas on one side .
multiwavelength data of ngc 4569 and kinematics on ngc 4396 indicate that some of the hi is extra - planar .
these preliminary results on the hi morphology already suggest that a variety of environmental effects such as icm - ism interactions , harassment , tidal interactions or mergers may be at work to affect the evolution of galaxies . |
deep - inelastic scattering ( dis ) provides a wealth of information about nucleon structure . recently ,
very high momentum transfers in dis have been achieved at the hera collider , where 820 gev protons have been collided with 27.5 gev positrons for a center - of - mass energy @xmath4=300 gev . in the highest momentum transfer region , @xmath5 dis cross sections depend on proton parton densities and properties of the electroweak interaction .
the @xmath6 dis process is illustrated in fig .
[ fig : dis ] . the variables used to describe the process are @xmath7 , the struck parton momentum fraction , @xmath8 , the fractional energy transfer in the proton rest frame ( inelasticity ) and @xmath0 , the four - momentum transfer squared , where @xmath9 .
neutral - current dis events are characterized by the exchange of a photon or @xmath10-boson , and have a positron and a jet ( or jets ) in the final state .
the outgoing positron and the hadronic matter are balanced in transverse momentum . charged - current dis events are characterized by the exchange of a @xmath11-boson , and contain an undetected neutrino and a jet ( or jets ) in the final state .
the presence of the neutrino is detected as missing transverse momentum .
the dis data presented here were collected and analyzed by the zeus collaboration and correspond to an integrated luminosity of 47.7 pb@xmath1 taken from 1994 to 1997 .
zeus @xcite is a multi - purpose magnetic detector ; the primary components used in these analyses are the calorimeters ( rcal , bcal , fcal ) , the central tracking detector ( ctd ) , and the luminosity monitor .
the coordinate system is defined such that the @xmath12-axis follows the proton direction , and the origin is the nominal @xmath13 interaction point . the zeus detector is displayed in fig . [
fig : detector ] .
the zeus compensating uranium - scintillator calorimeter covers the polar angle region @xmath14 with full azimuthal coverage over this region .
its energy resolution for electromagnetic showers is @xmath15 , and for hadronic showers is @xmath16 , as measured under test - beam conditions .
the zeus ctd operates in a solenoidal 1.43 t magnetic field , and primarily provides vertex reconstruction , track momentum , and charge information for these analyses .
the luminosity is determined from the rate of bethe - heitler bremsstrahlung ( @xmath17 ) photons detected in an electromagnetic calorimeter at @xmath18 m. 13 truecm
to lowest order ( qed born level ) the neutral - current dis ( @xmath19 ) cross section is @xmath20\ ] ] where @xmath21 . in lowest - order qcd ,
the structure functions @xmath22 and @xmath23 are the sums over quark flavor of the product of quark couplings and momentum distributions .
the quark couplings depend on the quark charges , and the electroweak parameters @xmath24 , etc .
the qed born - level charged - current dis ( @xmath25 ) cross section is @xmath26\ ] ] where at lowest order the structure functions @xmath27 and @xmath28 contain sums and differences of quark and antiquark momentum distributions .
the neutral- and charged - current longitudinal structure functions , @xmath29 and @xmath30 , respectively , provide a small ( @xmath31 ) contribution in the kinematic range discussed here , and have been included .
electroweak radiative corrections to these born - level equations , including initial- and final - state radiation , vertex and propagator corrections , and two - boson exchange , are significant and have been included to at least lowest order @xcite .
the differential nc dis cross section @xmath32 is shown in fig . [
fig : nc_dsdq2 ] ; @xmath33 and @xmath34 are shown in fig . [
fig : nc_dsdx_dsdy ] .
the differential cc dis cross sections @xmath35 ,
@xmath36 , and @xmath37 , are shown in figs .
[ fig : cc_dsdq2 ] , [ fig : cc_dsdx ] . and [ fig : cc_dsdy ] ,
respectively . for both neutral and charged current , the data points are compared to the standard model predictions using the cteq4d @xcite parton distribution functions ( pdf s ) , shown by the solid curves , with estimated pdf uncertainty shown by the shaded bands .
5.1 truecm 5.1 truecm 9.7 truecm 9.7 truecm 6.7 truecm 6.7 truecm 6.4 truecm 6.4 truecm 6.4 truecm 6.4 truecm the pdf uncertainty is calculated from a nlo fit @xcite to world dis data , and includes statistical and systematic errors on these data , as well as variations in the assumed electroweak and qcd parameters . for the neutral - current cross section
, these uncertainties range from 2.5% at @xmath38 to 8% at @xmath39 . for the charged - current cross section ,
the extracted uncertainties range from 9% at @xmath38 to 17% at @xmath40 .
the larger cc uncertainty is due to the larger uncertainty in the @xmath41-quark pdf relative to the @xmath42-quark pdf .
both the nlo fit , which includes higher - twist effects , and a recent reanalysis of nmc and slac data @xcite yield a larger @xmath43 ratio at high-@xmath7 than the cteq4 pdf s , where the @xmath43 ratio is constrained to be zero at @xmath44 . within present experimental precision ,
any of these hypotheses can be accommodated ; their differences are not included in the pdf uncertainty band .
increasing the @xmath43 ratio at high-@xmath7 reduces the cc data excess at high-@xmath7 , but does not appreciably affect the nc cross section since the nc process is not sensitive to the @xmath41-quark . the charged - current dis reduced cross section @xmath45
is shown in fig .
[ fig : ccreduced ] along with the standard model ( cteq4d ) prediction . at high-@xmath7 , the valence @xmath41 and @xmath46 quarks ( dashed curves ) dominate @xmath47 , whereas at lower-@xmath7 the @xmath48 and @xmath49 sea quarks ( dotted curves ) dominate
. 12 truecm at momentum - transfer - squared close to the @xmath10 and @xmath11 masses squared , i.e. , @xmath50 gev@xmath3 , the cross sections become sensitive to contributions from these exchanges , and to the propagator masses in particular .
the sensitivity of the nc cross section to the @xmath10 mass is demonstrated in fig .
[ fig : zmass ] .
the measured cross sections are compared with the standard model predictions by varying @xmath51 while keeping the couplings fixed .
three mass values are considered , @xmath51 = 40 , 91 and @xmath52 gev . clearly , @xmath53 gev is favored , in agreement with the world average value @xmath51 = ( 91.187 @xmath54 0.007 ) gev @xcite . in the cc channel , the sensitivity of the cross section to the @xmath11 mass is better than in the nc case , as shown in fig .
[ fig : wmass ] . the @xmath55 may be extracted by a @xmath56 fit to the measured cross sections , leaving all other electroweak parameters fixed , with the result @xmath57 this may be compared with the world average value of @xmath55 = ( 80.41 @xmath54 0.10 ) gev @xcite . the agreement between these @xmath10 and @xmath11 masses and the masses extracted through timelike production in @xmath58 and @xmath59 data confirms the standard model prediction in the spacelike regime . 5.0 truecm 5.0 truecm 10 truecm
the zeus neutral - current and charged - current dis cross sections in the momentum transfer range @xmath60 gev@xmath3 , i.e. , for @xmath0 over more than two orders of magnitude , have been shown to be in good agreement with the standard model predictions . however
, a slight excess persists at highest @xmath0 . with the current data set , zeus has gained sensitivity to the electroweak propagator masses @xmath55 and @xmath51 .
in 1998 , hera switched from positron - proton collisions to electron - proton collisions .
in addition , the proton energy was increased from 820 gev to 920 gev , thus increasing the center - of - mass energy by 6% from 300 gev to 318 gev . as of january 1 , 1999
, zeus had already accumulated @xmath61 of @xmath62 data , nearly an order of magnitude more than accumulated during the only other @xmath62 running period 1992 - 1993 .
an additional @xmath63 is expected in the 1999 calendar year .
the hera upgrade begins in may 2000 , after which we foresee @xmath64/year .
these future data will improve our understanding of the electroweak interaction , and may even hold some surprises .
we anticipate an exciting future for high-@xmath0 physics at hera .
the kind assistance of my zeus colleagues in the preparation of this talk , and the great efforts of the conference organizers are greatly appreciated .
this work is partly funded by the u.s .
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j. * c3 * ( 1998 ) 1 . | neutral - current and charged - current deep - inelastic scattering at very high four - momentum transfer squared ( @xmath0 ) have been studied in positron - proton collisions at center - of - mass energy 300 gev using the zeus detector at hera .
an integrated luminosity of 47.7 pb@xmath1 was collected in the years 1994 - 1997 .
differential cross sections are presented for @xmath2 gev@xmath3 and compared to standard model predictions . |
direct photon production is widely recognized as a process that is potentially important in determinations of the gluon distribution function . the next - to - leading - order ( nlo )
cross section for direct photon production has been given in refs .
the role of higher - order soft - gluon corrections has also been addressed more recently .
threshold resummation studies for direct photon production have appeared in refs .
@xcite while a joint threshold and transverse momentum resummation formalism has been given in ref .
@xcite . in a previous paper
@xcite we presented analytical and numerical results for the next - to - next - to - leading - order ( nnlo ) next - to - next - to - leading - logarithm ( nnll ) soft - gluon corrections for direct photon production .
here we increase the accuracy of our previous calculation by including additional subleading soft corrections .
our approach follows ref .
@xcite which in turn is based on and extends previous work on threshold resummation @xcite .
at lowest order , the parton - parton scattering subprocesses are @xmath1 and @xmath2 .
we define the mandelstam invariants @xmath3 , @xmath4 , and @xmath5 , which satisfy @xmath6 at threshold .
note that the photon transverse momentum is @xmath7 . here
we calculate the cross section @xmath8 in single - particle - inclusive kinematics in the @xmath9 scheme .
the soft corrections to the cross section appear in the form of plus distributions _
l(s_4)_+ with @xmath10 at @xmath11th order in @xmath12 beyond the leading order , while the virtual corrections appear in @xmath13 terms .
we begin with the nlo soft and virtual corrections in the @xmath14 scheme . a somewhat different notation from that used in ref .
@xcite has been adopted here , so the previously calculated terms are repeated here , as well .
the corrections to the parton - level cross section , @xmath15 , can be written for either subprocess as e _
= ^b_f_i f_j \{c_3^f_i f_j d_1(s_4 ) + c_2^f_i f_j d_0(s_4 ) + c_1^f_i f_j ( s_4 ) } , where @xmath16 is the renormalization scale , and the born terms are given by ^b_q|q = e_q^2 ( + ) , ^b_qg= - e_q^2 ( + ) , where @xmath17 is the charge of a quark of type @xmath18 , and @xmath19 with @xmath20 the number of colors .
also @xmath21 , @xmath22 , c_2^q|q=- -2c_f ( ) , c_2^qg =- c_f-(c_f+c_a ) ( ) , where @xmath23 is the factorization scale , @xmath24 , and @xmath25 , with @xmath26 the number of quark flavors .
we also define for use below @xmath27 and @xmath28 .
finally we write @xmath29 .
for @xmath30 we have c_1^=c_f ( ) + ( ) , and @xmath31\ln(p_t^2/s ) -(\beta_0/4)\ln(\mu_r^2/s ) + { c'}_1^{q \bar q}$ ] where @xmath32 is defined in eq .
( 3.11 ) of ref .
for @xmath33 we have c_1^= ( ) + ( ) and @xmath34 \ln(p_t^2/s)-(\beta_0/4)\ln(\mu_r^2/p_t^2 ) + { c'}_1^{qg}$ ] where @xmath35 is defined in eq .
( 3.8 ) of ref .
@xcite .
note that the nlo @xmath36 coefficients have also been presented in ref .
the notation for @xmath37 and @xmath38 is the same as before , while the notation for splitting @xmath39 into @xmath40 and @xmath41 terms for each subprocess is new and useful in presenting the nnlo expressions below .
next , we turn to the nnlo soft and virtual corrections in the @xmath14 scheme . these corrections can be written for either channel as e _ = ^b_f_i f_j ^(2)_f_i f_j .
[ nnlom ] for the @xmath42 process we have ^(2)_q|q&= & ( c_3^q|q)^2 d_3(s_4 ) + d_2(s_4 ) + & & + \{c_3^q|q c_1^q|q + ( c_2^q|q)^2 -_2 ( c_3^q|q)^2 - t_2 ^q|q + c_3^q|q ( ) + 2 c_f k . + & & .
+ c_a - } d_1(s_4 ) + & & + \{c_2^q|q c_1^q|q -_2 c_2^q|q c_3^q|q + _ 3 ( c_3^q|q)^2 - t_1^q|q + c_2^q|q ( ) + g^(2)_q |q . + & & + c_f + & & .
+ c_a - ( ) } d_0(s_4 ) + & & + r^q|qg ( s_4 ) .
[ nnloqqbar ] here @xmath43 , @xmath44 , and @xmath45 .
the function @xmath46 denotes a set of two - loop contributions @xcite and is given by g^(2)_q |q = c_f c_a ( _ 3 + _
2- ) + n_f c_f ( -_2 + ) .
we determine in the virtual corrections @xmath47 only the terms that involve the renormalization and factorization scales , denoted as @xmath48 and given explicitly by r^ q|qg&= & ^2 ( ) \{^2 - 2 _ 2 c_f^2 + _ 0 c_f+ c_f ( ) } + & & + ( ) ( ) c_f+^2 ( ) + & & + ( ) \{c_f^2 ^2 ( ) -c_f . + & & - _ 0 c_f -c_f ( ) + c_f^2(-11 _ 3 + _ 2 - ) + & & .
+ c_f c_a ( _ 3-_2 - ) + n_f c_f ( + ) } + & & + ( ) \{-c_f ( ) + t_1^q|q + ( ) + } , where @xmath49 and _ q / q^(2)=c_f^2(-_2 + _ 3 ) + c_f c_a(-_3+_2 + ) + n_f c_f ( -- )
. for the @xmath50 process we have ^(2)_q g&= & ( c_3^qg)^2 d_3(s_4 ) + d_2(s_4 ) + & & + \{c_3^qg c_1^qg + ( c_2^qg)^2 -_2 ( c_3^qg)^2 - t_2 ^qg + c_3^qg ( ) + ( c_f+c_a ) k .
+ c_f - _ 0 c_f } d_1(s_4 ) + & & + \{c_2^qg c_1^qg -_2 c_2^qg c_3^qg + _ 3 ( c_3^qg)^2 - t_1^qg + c_2^qg ( ) + g^(2)_qg .
+ & & + ( c_f+c_a ) + c_f k ( ) + c_a k ( ) + & & . + c_f - c_f _ 0 ( ) } d_0(s_4 ) + & & + r^qgq ( s_4 ) .
[ nnloqg ] the function @xmath51 denotes a set of two - loop contributions @xcite and is given by g^(2)_q g&=&c_f^2(-+_2 -_3)+ c_f c_a ( _ 3 -_2- ) + & & + c_a^2 ( _ 3 + _
2- ) + n_f c_f ( _ 2 + ) + n_f c_a ( -_2 - ) .
finally , the terms in @xmath52 that involve the renormalization and factorization scales , denoted as @xmath53 , are given explicitly by r^ qgq&=&^2 ( ) \{^2- ( c_f+c_a)^2 .
+ & & . + } + & & + ( ) ( ) + ^2 ( ) + & & + ( ) \{^2 ( ) .
+ & & - + & & --c_f^2 ( _ 3 + ) -c_a^2(_3 + ) + & & .
-c_f c_a ( _ 3+_2 + ) + n_f c_f ( + ) + n_f } + & & + ( ) \{- ( ) + t_1^qg + } , where _ g / g^(2)=c_a^2(+_3 ) -n_f(+ ) . for both processes the coefficients of the @xmath54 , @xmath55 , and @xmath56 terms were given previously in ref .
the additional subleading @xmath57 and @xmath13 terms presented here are new .
data from the ua-6 collaboration @xcite at @xmath58 gev.,height=4 ] data for the rapidity distribution from the ua-6 collaboration @xcite at @xmath58 gev.,height=4 ] data from the ua-6 collaboration @xcite at @xmath58 gev.,height=4 ] gev / c.,height=4 ]
in order to show the effect of including the new nnlo terms , the same procedure employed in ref . @xcite has been used .
first , a complete nlo calculation of the appropriate cross section is performed using a program @xcite which employs the phase - space slicing technique described in ref .
the original nlo calculation has been extended to include a complete nlo treatment of the bremsstrahlung contribution .
the set 2 fragmentation functions of @xcite have been used along with the cteq6 m parton distribution functions @xcite . in all cases the factorization and renormalization scales have been set equal to a common scale @xmath59 which has been chosen to be proportional to the photon s transverse momentum .
once the nlo results have been obtained , the approximate nnlo contributions can be added to them .
several examples are discussed below , comparing the nlo and nlo + approximate nnlo results . in fig .
[ fig1 ] the nlo ( dashed curves ) and nlo plus approximate nnlo results ( solid curves ) are compared to data @xcite from the ua-6 collaboration for proton proton interactions . in each case
the upper ( lower ) curve corresponds to the scale choice of @xmath60 @xmath61 .
the pattern demonstrated previously in fig .
@xcite is still found to be true , even after the addition of the newly calculated terms presented in this work .
the scale dependence of the nlo result is greatly decreased by the addition of the nnlo terms .
furthermore , one can see that for the scale choice of @xmath62 the nnlo scale dependent terms give a negligible contribution .
even with the nnlo contributions , the results lie somewhat below the data at the lower values of @xmath63 . in fig .
[ fig2 ] the rapidity dependence is shown for the ua-6 proton proton data .
again , the scale dependent nnlo terms give a negligible contribution for the choice @xmath62 and the overall scale dependence is greatly reduced when the nnlo terms are added .
as noted previously , the curves lie below the data over the majority of the rapidity range shown .
of course , this distribution is dominated by the contributions from the low end of the @xmath63 range , so this is no surprise , given the results shown in fig .
[ fig1 ] . in fig .
[ fig3 ] the photon @xmath63 distribution is shown for the case of @xmath64 interactions and compared to data from the ua-6 collaboration @xcite . whereas the @xmath65 reaction is dominated by the @xmath50 subprocess , the @xmath64 reaction receives additional significant contributions from the @xmath66 subprocess .
nevertheless , a pattern similar to that in the previous two figures is apparent here as well .
note , however , that for the case of @xmath67 , the nnlo contribution is negative , further reducing the scale dependence shown in fig .
[ fig3 ] . as for the @xmath65 case ,
the band formed by the theoretical curves lies somewhat below the data at the lower values of @xmath63 . finally , in fig .
[ fig4 ] a similar comparison is made to data from the e-706 collaboration @xcite .
the same behavior seen in the previous figures is evident here , although the theoretical band now lies significantly below the data at the lower end of the range covered by the data .
we have extended the results of ref .
@xcite to include additional nnlo subleading logarithms resulting from soft gluon corrections for direct photon production .
the additional terms are numerically small and the results remain qualitatively the same as in the previous analysis . in particular , the reduced scale dependence relative to nlo calculations remains .
the research of n.k . has been supported by a marie curie fellowship of the european community programme `` improving human research potential '' under contract number hpmf - ct-2001 - 01221 .
the research of j.o . is supported in part by the u.s .
department of energy . | previous work on soft - gluon resummation for direct photon production is extended to include additional subleading logarithmic terms through @xmath0 and some representative comparisons are made to experimental results from the e-706 and ua-6 collaborations .
the additional terms are small in magnitude , indicating good convergence properties to the level of accuracy calculated .
the scale dependence remains much smaller than that of the next - to - leading - order calculation . |
in this paper , we consider the @xmath0c+@xmath0c reaction as a case study to point out the problems for the inelastic scattering states which have so far remained unsolved , and to address particularly the magnitude problem for the inelastic scattering data .
theoretical calculations using the coupled - channels ( cc ) method fail to correctly predict the magnitude of the single-2@xmath1 and mutual-2@xmath1 states data together with the elastic scattering data . in order to get the magnitude right
, many futile theoretical attempts have been made for these states .
previous theoretical works show that the shapes of the central real potentials are actually correct , since they explain the elastic scattering data and predict the resonances at the correct energies with reasonable widths .
it appears that the failure of the standard methods is mostly related to the inelastic scattering data : the magnitude of the theoretical cross - sections is much smaller than the measured experimental data . in this paper
, we make further applications of a new coupling potential @xcite in describing the scattering observables of the @xmath0c+@xmath0c system .
for the spherical nuclei , the nuclear shape and also the shape of the potential between projectile and target nuclei are characterized by a constant radius @xmath2 , which defines the distance of the center of the nucleus from the surface . however , for a deformed nucleus , the radius parameter is no longer constant but depends on the angular location of the point ( @xmath3 ) .
the nucleus @xmath0c we study in this paper is strongly deformed and its collective excitation is taken into account by using the standard deformation procedure based on the taylor expansion .
if the interaction potential between two nuclei is taken to be @xmath4 , the taylor expansion about @xmath5 yields , @xmath6 here , the first term is the usual diagonal optical potential that describes only the elastic scattering and the second and third terms are used to describe the inelastic scattering and to obtain the coupling potentials for the single-2@xmath1 and mutual-2@xmath1 states .
@xmath7 in equation [ pot ] is given as : @xmath8 with @xmath9 as the projectile @xmath10 or the target @xmath11 .
@xmath12 is the deformation parameter and it is -0.6 for the @xmath0c nucleus . in the phenomenological analysis
, the real nuclear potential has the square of the woods - saxon shape , and the imaginary potential has the woods - saxon volume shape .
the parameters of the real and imaginary parts are taken from ref .
@xcite . for the microscopic analysis ,
the nucleon - nucleon double - folding potential @xcite is @xmath13 where @xmath14 and @xmath15 are the nuclear matter distributions for projectile and target nuclei respectively , and they are given by @xmath16\ ] ] where @xmath17=0.1644 @xmath18 , @xmath19=0.4988 @xmath20 and @xmath21=0.3741 @xmath20 for projectile and @xmath22=0.207 @xmath18 , c=2.1545 fm , and a=0.425 fm for target nuclei .
the m3y nucleon - nucleon effective interaction is taken in the form @xmath23 where @xmath24 .
the real and imaginary potentials are shown in figure [ comp ] and the parameters are given in table [ param ] labelled as df . in the new coupled - channels model
, we have replaced the usual first derivative coupling potential by a second - derivative coupling potential in woods - saxon form which is multiplied by the diffuseness parameter ( @xmath10 ) to normalize the units in the calculations .
the parameters are given in table [ param ] .
we have used both the phenomenological and microscopic potentials to analyze the experimental data of the @xmath0c+@xmath0c reaction at e@xmath25=74.2 mev , 93.8 mev and 126.7 mev .
the experimental data is taken from ref .
the results of our analyzes are displayed in figure [ ground ] for the ground , [ single ] for the single-2@xmath1 , and [ mutual ] for the mutual-2@xmath1 states in comparison with experimental data .
both double - folding and phenomenological potentials provide excellent agreement with the experimental data for the ground state at all energies and a good fit to the single-2@xmath1 state data .
however , the mutual-2@xmath1 state prediction is much smaller than the measured one : the standard cc model using double - folding or phenomenological potentials underestimates its magnitude by a factor of 3 to 10 as it has been previously observed @xcite .
varying the parameters and changing the shape of the real and imaginary potentials do not provide a complete solution to the problems of this reaction @xcite . in order to solve this problem
we have used a new coupling potential .
this potential has a second - derivative of woods - saxon shape and it is compared in figure [ comp ] with the standard coupling potential .
this new coupling potential has a repulsive part at short distances and an attractive part at large distances which is related to the orientation of two @xmath0c nuclei at short and large distances @xcite .
we have been able to obtain excellent agreement with all the available experimental data for the ground , single-2@xmath1 and mutual-2@xmath1 states by using this new coupling potential .
the parameters are shown in table [ param ] .
this new approach solves the magnitude problem of the mutual-2@xmath1 state data , which has been an outstanding problem with this reaction .
the results for the ground , single-2@xmath1 and mutual-2@xmath1 states are compared with the standard ones in figures [ ground ] , [ single ] and [ mutual ] .
in the present work , we have demonstrated that a consistent solution could be obtained for the problems of the @xmath0c+@xmath0c reaction over a wide energy range .
however , we achieve this by using a coupling potential which has a non - standard form . within the standard formalism ,
our findings using folding or phenomenological potentials are in agreement with the previous works @xcite .
although , within standard approach , these potentials give excellent agreement with the experimental data for the ground state , they are unable to provide a consistent solution to the problems of the inelastic scattering data .
this work therefore clearly shows that the standard deformation procedure based on taylor expansion is inadequate in describing such highly deformed nuclei .
it is obvious that the standard formalism should be questioned further .
this work is supported by the turkish science and research council ( tbitak ) : grant no : tbag-2398 and erciyes university - institute of science : grant no : fbt-04 - 15 and fbt-04 - 16 .
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.the parameters of the real and imaginary potentials . for the new coupling potentials , the radius and diffuseness 0.7 fm and
the depth is 215.0 mev , 230 mev and 245 mev for @xmath26=74.2 mev , 93.8 mev and 126.7 mev . [
cols="<,>,^,^,^,>,>,>,>",options="header " , ] | we present the failure of the standard coupled - channels method in explaining the inelastic scattering together with other observables such as elastic scattering , excitation function and fusion data .
we use both microscopic double - folding and phenomenological deep potentials with shallow imaginary components .
we argue that the solution of the problems for the inelastic scattering data is not related to the central nuclear potential , but to the coupling potential between excited states .
we present that these problems can be addressed in a systematic way by using a different shape for the coupling potential instead of the usual one based on taylor expansion . |
determining the chemical composition of galaxies is of fundamental importance for tracing back the history of evolution of galaxies .
in particular , the lmr has been widely studied in the local universe .
metallicities are tightly related with the luminosities of galaxies in such a way , that brighter systems have higher abundances ( lamareille et al .
furthermore , recent studies have also suggested that this relation extends to intermediate redshifts but displaced towards lower metallicities and higher luminosities ( kobulnicky et al .
2003 ) .
when studying galaxy evolution , stellar mass is a better parameter than luminosity .
however , because of the difficulties in obtaining stellar masses , most studies have used luminosity as a surrogate .
recently , though , tremonti et al . (
2004 ) have estimated the relation between metallicity and stellar mass in the local universe .
the authors found a strong correlation extended over 2 dex in stellar mass and a factor of 10 in metallicity . in this work ,
we study the evolution of the mmr and the lmr by employing numerical chemo - dynamical simulations which allow to describe the non - lineal growth of structure simultaneously with the enrichment of the interstellar medium in a cosmological framework .
we have run numerical simulations by using the chemical gadget-2 of scannapieco et al .
( 2005 ) . a @xmath0cdm cosmological model ( @xmath4=0.3 , @xmath0=0.7 , @xmath5=0.04 and @xmath6=100 @xmath7 km s@xmath8 mpc@xmath8 with @xmath7=0.7 ) was assumed , according to which galaxies formed by the hierarchical aggregation of substructures .
we have analysed two realizations of the power spectrum in a 10 mpc @xmath9 side box , initially resolved with @xmath10 ( s160 ) and @xmath11 ( s80 ) particles , corresponding to mass resolutions of @xmath12 m@xmath13 and @xmath14 m@xmath13 for the gas phase and @xmath15 m@xmath13 and @xmath16 m@xmath13 for dark matter respectively .
a salpeter initial mass function has been assumed with upper and lower limits of 40 m@xmath17 and 0.1 m@xmath17 , respectively .
the chemical model includes metal - dependent radiative cooling , star formation and chemical enrichment by supernovae ii and ia ( scannapieco et al . 2005 ) .
galactic objects were identified by applying an identification algorithm that combines the friends - of - friends technique and the contrast density criterium of white , efstathiou & frenk ( 1993 ) .
dynamical and chemical properties were estimated at the optical radius calculated accordingly to the standard definition as the radius which contains the 83% of the baryonic mass of the system ( tissera et al .
colours and magnitudes of galactic systems were calculated by resorting to population synthesis models ( see de rossi et al .
2006 in preparation ) .
our simulations predict a linear correlation between oxygen abundance and luminosity which is in good agreement with the observational results .
we have also found an evolution in the lmr in such a way that the slope increases and the zero point decreases with redshift consistently with the findings of kobulnicky & kewley ( 2004 ) , among others .
in particular , we decided to work with the i - band because it is less affected by extinction and can be more directly related with the underlying mass distributions .
our results indicate that at a given chemical abundance , galactic systems are @xmath18 3 dex brighter at @xmath19 compared to @xmath20 , with the larger evolution at fainter magnitudes .
futhermore , we have encountered a mean evolution in the chemical abundances of galactic systems of @xmath18 1.6 dex for brighter magnitudes and @xmath18 2.5 dex for faint ones , from @xmath19 to @xmath20 .
we have also analysed the mmr for simulated galactic systems , obtaining similar trends to those found by tremonti et al .
( 2004 ) in the sloan digital sky survey ( sdss ) but with a displacement of -0.25 dex in the zero point .
this last difference may be explained taking into account that the sdss explored only the central regions of galaxies which could lead to an overestimation of their metal content .
galactic abundances derived from simulations tend to increase with stellar mass which is also consistent with the observed behaviour .
however , we obtained an excess of metals in the lower mass end which could be due to the absence of supernovae energy feedback in our model .
we have determined a characteristic stellar mass at @xmath21 m@xmath13 where a change in the curvature of the mmr occurs .
this characteristic mass , which corresponds to an oxygen abundance of @xmath18 8.7 dex , has been obtained by estimating where the residuals of the linear fits depart systematically from zero .
it is important to note that this mass is similar to the characteristic mass derived from the sdss by tremonti et al .
( 2004 ) and gallazzi et al.(2005 ) .
in addition , we have found that the mmr exhibits the same general patterns from @xmath19 to @xmath20 , but with a displacement towards higher abundances as redshift decreases .
the characteristic stellar mass @xmath22 remains almost unchanged with time and only its corresponding chemical abundance evolves by 0.05 dex in the same redshift range .
the major departure from the local mmr occurs for smaller systems which increase their chemical content by @xmath18 0.10 dex . on the other hand ,
massive systems show less evolution with variations of @xmath18 0.05 from @xmath19 to @xmath20 . we have also studied the metallicity - optical velocity relation ( mvr ) finding a well defined correlation from @xmath19 to @xmath20 .
however , a higher level of evolution has been found in the mvr when compared to the mmr .
fast rotators show an enrichment of @xmath18 0.18 dex from @xmath19 to @xmath20 while at lower metallicities the variations are of @xmath18 0.28 dex .
this significant evolution of the mvr is a consequence of the increase of the mean density of the universe at high redshift , so that at a fix stellar mass , systems need to concentrate more in order to fullfill the contrast density criterium and , hence , galaxies reach higher velocities .
the larger evolution showed by the simulated mvr , when compared with the mmr , shows that the second is more fundamental in the sense that it is not strongly dependent on the cosmic epoch ( see tissera , de rossi & scannapieco 2005 for details ) . by analysing the merger trees of systems at @xmath20 , we have encountered that the features of the mmr can be traced back in time taking into account the mergers and interaction history of galaxies within the hierarchical aggregation picture .
more massive systems transform most of their gas into stars at high redshifts and experience important mergers . at lower redshifts
, these galactic objects are saturated of stars , so that , while their stellar mass largely increases in a merger event , their metallicity remains basically unchanged .
on the other hand , less massive systems form their stars in a more passive way or by rich - gas mergers leading to a more tight correlation between metallicity and stellar mass .
we have estimated the mmr , the lmr and the mvr correlations finding similars trends to those detected in observations .
these relations evolve with redshift with the major changes driven by less massive systems in consistency with the downsizing scenario .
gallazzi , a. , charlot , s. , brinchmann , j. , white , s. d. m. , tremonti , c. , 2005 , mnras , accepted ( astroph/0506539 ) kobulnicky , h. a. , willmer , c. n. a. , phillips , a. c. , koo , d. c. , faber , s. m. et al .
2003 , apj 599 , 1006 kobulnicki , h. a. , kewley , l. j. 2004 , apj 617 , 240 lamareille , f. , mouhcine , m. , contini , t. , lewis , i. , maddox , s. 2004 , mnras , 350 , 396 scannapieco , c. , tissera , p. b. , white , s. d. m. , springel , v. , 2005 , mnras , accepted ( astroph/0505440 ) tissera , p. b. , de rossi , m. e. , scannapieco , c. , 2005 , mnras , 364 , 38l tremonti , c. a. , heckman , t. m. , kauffmann , g. , brinchmann , j. , charlot , s. et al .
2004 , apj 613,898 white , s. d. m. , efstathiou , g. & frenk , c. s. , 1993 , mnras , 262 , 1023 | in this work , we study the luminosity - metallicity relation ( lmr ) and the stellar mass - metallicity relation ( mmr ) of galactic systems in a hierarhical clustering scenario .
we performed numerical hydrodynamical simulations with the chemical gadget-2 of scannapieco et al.(2005 ) in a @xmath0cdm universe .
we found that our simulated galactic systems reproduce the observed local lmr and its evolution in zero point and slope .
the simulated mmr is also in agreement with recent observational results . from the analysis of the evolution of the mmr
, we found a characteristic mass at @xmath1 which separates two galactic populations with different astrophysical properties .
more massive systems tend to have their stars formed at @xmath2 and show less evolution than smaller systems .
hence , this characteristic mass is determined by the formation of the structure in a hierarchical scenario .
our results also suggest the need for efficient supernova feedback .
1.0 cm 0.3 cm en este trabajo , estudiamos la relacin luminosidad - metalicidad ( lmr ) y la relacin masa estelar - metalicidad ( mmr ) de los sistemas galcticos en un modelo de agregacin jerrquica .
realizamos simulaciones numricas hidrodinmicas con el cdigo qumico gadget-2 de scannapieco et al .
( 2005 ) en un universo @xmath0cdm .
encontramos que nuestros sistemas galcticos simulados reproducen la lmr local y su evolucin en el punto cero y la pendiente .
la mmr simulada est tambin en acuerdo con resultados observacionales recientes . a partir del anlisis de la evolucin de la mmr , hallamos una masa caracterstica en @xmath1 , la cual separa dos poblaciones galcticas con diferentes propiedades astrofsicas .
los sistemas ms masivos tienden a formar sus estrellas a @xmath3 y muestran menor evolucin que los sistemas pequeos .
entonces , esta masa caracterstica es determinada por la formacin de la estructura en un universo jerrquico .
nuestros resultados sugieren tambin la necesidad de un importante _ feedback _ por supernovas . |
heavy ion collisions have received significant attention in recent years . electromagnetic probes ( photons , dileptons etc )
have been proposed to be one of the most promising tools to characterize the initial state of the collisions @xcite . because of the very nature of their interactions with the constituents of the system they tend to leave the system almost unscattered .
photons are produced at various stages of the evolution process .
the initial hard scatterings ( compton and annihilation ) of partons lead to photon production which we call hard photons .
if quark gluon plasma ( qgp ) is produced initially , there are qgp - photons from thermal compton plus annihilation processes .
photons are also produced from different hadronic reactions from hadronic matter either formed initially ( no qgp scenario ) or realized as a result of a phase transition from qgp .
these apart , there exits another class of photon emission process via the jet conversion mechanism ( jet - plasma interaction ) @xcite which occurs when a high energy jet interacts with the medium constituents via annihilation and compton processes . in current heavy ion collision experiments ,
the temperature @xmath1 is not only the important scale , momentum scale , @xmath2 , ( of the partons ) is also important .
therefore running of the coupling in the high momentum regime ( @xmath3 ) has to be taken into account to calculate the cross sections and the energy - loss processes . in this work
we calculate photons from jet - plasma interaction taking into account running of qcd coupling and both collisional and radiative energy losses .
the plan of the article is as follows .
we discuss the formalism in the next section .
results will be discussed in the section 3 .
finally we will conclude .
the lowest order processes for photon emission from qgp are the compton scattering ( @xmath4 ) and annihilation ( @xmath5 ) process .
the differential photon production rate for this process is given by @xcite : @xmath6 where , @xmath7 represents the spin averaged matrix element squared for one of those processes which contributes in the photon rate and @xmath8 is the degeneracy factor of the corresponding process .
@xmath9 , @xmath10 and @xmath11 are the initial state and final state partons . @xmath10 and @xmath11 are the bose - einstein or fermi - dirac distribution functions .
in the photon production rate ( from jet - plasma interaction ) one of the collision partners is assumed to be in equilibrium and the other ( the jet ) is executing random motion in the heat bath provided by quarks ( anti - quarks ) and gluons . furthermore , the interaction of the jet is dominated by small angle scattering . in such scenario the evolution of the jet phase space distribution
is governed by fokker - planck ( fp ) equation where the collision integral is approximated by appropriately defined drag ( @xmath12 ) and diffusion coefficients @xcite .
the drag and diffusion coefficients are infrared singular .
the infra - red cut - off is fixed by plasma effects , where only the medium part is considered , completely neglecting the vacuum contribution leading to ambiguity in the energy loss calculation . if the latter part is taken into account the strong coupling should be running .
thus for any consistent calculation one has to take into consideration this fact . in that case @xmath13 ( @xmath14 in this case ) , and the above integrals must be evaluated numerically where the infra - red cut - off is fixed by debye mass to be solved self - consistently : @xmath15 here the strong coupling which we take as running , i. e. @xmath16 .
we chose the following parametrization of @xmath0 which respects the perturbative ultra - violet ( uv ) behavior and the 3d infra - red ( ir ) point @xcite : @xmath17 with @xmath18 in this case .
the parameters @xmath19 , @xmath20 and @xmath21 are given by @xmath22 , @xmath23 and @xmath24 gev .
for the limiting behavior ( @xmath25 ) of the coupling we choose , @xmath26 here @xmath27 and @xmath28 denote the values of the ir fixed point of @xmath29 yang - mills theory in @xmath30 and @xmath31 dimensions , respectively .
the remaining four parameters ( @xmath32 and @xmath33 ) fit the numerical results for pure yang - mills theory obtained from the rg equations in ref .
@xcite . in our calculation
we have considered both collisional and radiative energy losses in the following manner .
@xmath34\end{aligned}\ ] ] for running @xmath0 , the expressions for the collisional and radiative energy losses can be found in @xcite .
having known the drag and diffusion , we solve the fp equation using green s function techniques ( for details see ref .
@xcite ) . in order to obtain the space - time integrated rate we first note that the phase space distribution function for the incoming jet in the mid rapidity region is given by
( see ref .
@xcite for details ) @xmath35 with this jet parton phase space distribution function one can easily obtain jet photon yield from eqn .
( 1 ) : @xmath36
in order to obtain the photon @xmath37 distribution we numerically integrate eq .
( [ last ] ) .
the results for jet - photons for rhic energies are plotted in fig .
[ fig_rhic446 ] ( left ) where we have taken @xmath38 mev and @xmath39 fm / c .
we find that the yield is decreased with the inclusion of both the energy loss mechanisms as compared to the case when only collisional energy loss is considered .
it is to be noted that when one considers collisional energy loss alone the yield with constant @xmath0 is more compared to the situation when running @xmath0 is taken into account ( see fig .
[ fig_rhic446 ] left ) . in order to compare our results with high @xmath37 photon data measured by the phenix collaboration @xcite
, we have to evaluate the contributions to the photons from other sources , that might contribute in this @xmath37 range . in fig . [ fig_rhic446 ] ( right )
the results for jet - photons corresponding to the rhic energies are shown , where we have taken @xmath38 mev and @xmath39 fm / c .
the individual contributions from hard and bremsstrahlung processes @xcite are also shown for comparison .
we have calculated the transverse momentum distribution of photons from jet plasma interaction with running coupling , i. e. with @xmath40 where we have included both collisional and radiative energy losses .
using running coupling we find that the depletion in the photon @xmath37 spectra is by a factor of @xmath41 more as compared to the case with constant coupling for rhic energies phenix photon data have been contrasted with the present calculation and the data seem to have been reproduced well in the low @xmath37 domain .
the energy of the jet quark to produce photons in this range ( @xmath42 ) is such that collisional energy loss plays important role here .
it is shown that inclusion of radiative energy loss also describes the data reasonable well . | we calculate photons from jet - plasma interaction considering collisional and radiative energy loss of jet parton .
the phase space distribution of the participating jet is dynamically evolved by solving fokker - planck equation .
we treat the strong coupling constant ( @xmath0 ) as function of momentum and temperature while calculating the drag and diffusion coefficients .
it is observed that the quenching factor is substantially modified as compared to the case when @xmath0 is taken as constant .
it is shown that the phenix data is reasonably well reproduced when contributions from all the relevant sources are taken into account .
energy loss , running coupling , qgp |
knowing the spectrum of quantum mechanical states of an electron in magnetic field there are two ways of calculating the thermal gyroradius at given temperature @xmath24 .
either the gyroradius is calculated by averaging it over the distribution which is its expectation value @xmath13 , would be the exact way to do it , or one calculates the energy expectation value @xmath32 which provides the expectation value of the squared gyroradius @xmath33 whose root can also be taken to represent the thermal gyroradius .
this latter value will be calculated first .
the energy levels of an electron in a homogeneous magnetic field ( the case of interest here as a magnetic inhomogeneity provides only higher order corrections ) has been calculated long ago @xcite .
since the parallel dynamics of an electron is not of interest here , it can be taken classically .
then the energy of an electron in the magnetic field becomes @xmath34 with @xmath35 , quantum number @xmath36 , and @xmath37 the two spin directions .
the average distribution of electrons over these @xmath36 perpendicular energy states is given by the fermi distribution @xmath38 written here for the anisotropic case , with @xmath39 the fugacity which depends on the chemical potentials @xmath40 in parallel and perpendicular direction , and @xmath41 are the inverse temperatures ( in energy units ) .
at sufficiently large temperatures the unity in the denominator is neglected , a very well justified approximation which still takes into account the non - continuous energy distribution over discrete landau levels thus maintaining the quantum character of electrons .
the fugacities enter into the normalization constant now .
this is the case far away from eq .
( [ denslim ] ) which for plasmas interests us here . under these conditions
the expectation value of the ( average ) perpendicular energy of the electrons ( i.e.the perpendicular electron pressure ) is calculated from the integral @xmath42 the spin contribution in the perpendicular energy either compensates for the half landau energy level or completes it to the first order level .
thus the sum splits into two terms which both are geometric progressions which can immediately be done .
the final result , using the normalization of the integral to the average density of particles and dividing through @xmath43 thus yields for the average energy @xmath44 at the assumed large temperatures the exponentials must be expanded to first order yielding the very well known and expected classical result that the average energy is the temperature , @xmath45 .
hence , taking its root and inserting into the gyroradius we find what is expected in this case : @xmath46 this is the root - mean - square gyroradius , a well known quantity . at lower temperatures @xmath47 , still by far exceeding fermi energy .
the former expression for @xmath32 has to be used in this expression . however , the correct gyroradius is not its root mean square but the expectation value @xmath48 .
this is substantially more difficult to calculate .
there are two ways of finding the expectation value .
either one makes use of the landau solution of the electron wave function and refers to the wigner distribution of quantum mechanical states .
this , for our purposes would be overdoing the problem substantially .
we do not need the quantum mechanical probability distribution for the simple estimate we envisage here .
afterwards we would anyway have to combine the wigner function with a suitable momentum distribution .
the second and simpler way is to refer to the above fermi distribution and directly using the energy distribution as above . under the same conditions as in the calculation of the rms value
, this procedure circumvents the use of the wave function being interested only in the energy levels .
it , however , requires the solution of the integral - sum @xmath49 the sum , the integral contains , can not anymore be done in closed form as there is no known way to tackle the summation over the root quantum index @xmath36 in a non - geometric progression .
( we may note in passing that for a non - moving plasma calculating the average velocity moment would lead to a null result .
however , in calculating the expectation value of the gyroradius , the perpendicular velocity , being a radius vector in perpendicular velocity space , is always positive .
this is reflected by the summation over positive quantum numbers @xmath36 only .
it and the gyroradius are positive definite quantities which both do not average out . )
one thus needs to deviate from summing and to approximate the sum by an integral , which slightly overestimates the final result . transforming the energy into a continuous variable , which implies that the summation index becomes continuous , then simply leaves us with a gaussian integral of which we know that the mean value and the rms values are related by the classical formula @xmath50 the classical result for the mean , where by the @xmath51 sign we indicated that the integral yields an upper limit for the expectation value of the gyroradius .
the above estimates permit determining the thermal fluctuation ( thermal spread ) of the gyroradius .
this fluctuation is the difference between the mean square and squared expectation values . with the above results we find that the thermal fluctuation of the gyroradius amounts to @xmath52 this number falls into the @xmath0 range of `` freedom '' in choosing the thermal speed that is commonly used in the definition of the rms gyroradius as either @xmath53 or @xmath54 .
the expected result of the quantum mechanical calculation of the electron gyroradius of a hot plasma is that the root - mean - square value of the gyroradius is reproduced by quantum mechanics in the high temperature limit which clearly completely justifies its use not adding anything new except for the confirmation of well - known facts which can be generated in much simpler way by using the boltzmann calculation . for lower temperatures we obtained a slightly more precise expression for the average energy which should be used in the definition of the thermal gyroradius .
the expectation value of the gyroradius is different from the root mean square value , which is also known .
its value is lower given by the gaussian mean .
usually this value is not used in any classical calculation .
the difference between both values is within the commonly used @xmath0 freedom for choosing the rms thermal speed .
the above calculation is based on keeping the discrete distribution of all plasma electrons over landau levels .
since these are equally spaced up to infinity , the energy remains discontinuous even at large temperatures .
the correction is , however , unimportant .
it is sometimes claimed that the discrete distribution of electron energies is reflected in the dielectric function of a plasma which leads to collective cyclotron harmonics , i.e. bernstein modes .
however , landau levels have nothing in common with cyclotron harmonics .
they are single particle energy levels occupied at most by two electrons of oppositely directed spins , while cyclotron harmonics are eigenmodes of the magnetized plasma , channels which allow for propagation of electro(magnetic ) signals .
the discrete energy distribution of the electrons naturally disappears in the cacluation of the expectation values and is not reflected in any cyclotron harmonic waves .
the average gyroradius appearing in the arguments of the modified bessel functions in the dielectric expression of the harmonics in the combination @xmath55 is a
_ mere scaling factor _ of wavelengths related to the perpendicular thermal speed .
it can not be taken as definition of a gyroradius , which is a particle property .
in fact , calculation of the dielectric function from quantum mechanics accounting for the landau energy distribution is very difficult . in classical theory one integrates the vlasov equation with respect to time along classical particle orbits @xmath56 which , in quantum mechanics , is forbidden as _ no orbits exist_. orbits are replaced by probability distributions in space varying with time .
the calculation therefore needs to be performed via a feynman path integral approach which has not yet been done .
it is , however , reasonably expected that it will , in the classical limit reproduce the classical dielectric function and the harmonic structure of bernstein and cyclotron modes .
sonnerup , b. u. . : magnetic field reconnection , in : solar system plasma physics , vol .
45 - 108 , edited by : lanzerotti , l. t. , kennel , c. f. , and parker , e. n. , north - holland publ .
, amsterdam - new york , 1979 . | the average ( thermal ) gyroradius of charged particles is re - examined from a quantum - mechanical point of view .
the straight quantum - mechanical calculation clearly reproduces its conventionally used random - mean - square ( rms ) expectation value .
the quasi - classical approach reproduces its mean expectation value as well thus confirming the purely classical calculation .
it shows that the fluctuations in the gyroradius amount to 21.5% of its rms value .
this fluctuation is , however , within the usual @xmath0-``freedom '' range of choice in the definition of the rms gyroradius respectively the mean thermal speed on which the rms value is based , its correct ( nonrelativistic ) value @xmath1 and its simplified version @xmath2 .
the gyroradius @xmath3 of a particle of charge @xmath4 and mass @xmath5 in a magnetic field of strength @xmath6 is one of the fundamental parameters used in plasma physics .
its exact classical definition for a single particle of perpendicular velocity @xmath7 is @xmath8 written here for an electron with cyclotron frequency @xmath9 ( see any textbook on plasma physics , e.g. , * ? ? ?
* ) .
its ion equivalent is obvious .
single electrons have undefined temperature , usually assumed to be zero , hence the exactness of the given expression . in quantum mechanics electrons obey landau levels possessing a well defined magnetic length @xmath10 which corresponds to the gyroradius of an electron in the lowest landau energy level , @xmath11 .
in high temperature plasmas one defines the gyroradius through the thermal velocity , respectively the perpendicular temperature @xmath12 .
a more precise definition makes use of the velocity distribution function of the electrons to calculate the average gyroradius @xmath13 as the moment of the classical distribution with respect to the perpendicular particle velocity . in the random - mean - square ( rms ) limit this yields the above expression for @xmath14 with replacement @xmath15 , which is nothing but the well - known rms thermal gyroradius " .
occasionally freedom is taken for convenience in either keeping or dropping the @xmath0 value in the definition of the thermal speed which introduces a 29% indeterminacy of the thermal speed and rms gyroradius . for temperatures @xmath16 , i.e. @xmath17 , with @xmath18 fermi temperature ,
electrons become degenerate and one needs to refer to the fermi distribution . in this case the thermal speed is replaced by the fermi velocity @xmath19 , a _ collective _ velocity of degenerate electrons .
one may note that this does _ not _ reproduce the above quantum mechanical magnetic length of electrons , @xmath20 ! the latter being a single electron property in magnetic fields caused by the quantum character of the magnetic flux with flux quantum @xmath21 , not a collective effect .
it defines the radius of a magnetic field line through @xmath22 . at non - zero temperatures @xmath23 slightly exceeding @xmath18
a small correction is to be introduced on the fermi velocity entering the collective rms gyroradius . in many particle theory ,
starting from the single particle picture one should , for greater precision , make use however of the distribution of particles over the entire spectrum of energy states available and accessible to the population at a given temperature @xmath24 .
since only two particles of opposite spins can occupy any energy state there is a well - known competition between the available states and the number of particles at @xmath24 . if the temperature @xmath25 , with @xmath26 the fermi temperature , then the particles have effectively zero temperature , and the gyroradius is defined through their single velocity in the respective state .
this case is encountered in high density compounds at @xmath27}}^\frac{3}{2 } \quad \mathrm{m}^{-3}\ ] ] with temperature measured in ev .
( one may note that this number is remarkably close to the conditions encountered in the solar chromosphere ! ) .
when this condition holds , quantum effects become important ; it implies that the interparticle distance @xmath28 equals the thermal de broglie wavelength @xmath29 , here written with @xmath30 . at higher temperatures and lower densities the average gyroradius should be calculated by adding up all electrons in the available states . by assuming a classical distribution function this is circumvented and reduced to a simple integration yielding the above thermal rms value @xmath31 . here
we present the exact quantum mechanical calculation which naturally confirms the thermal rms gyroradius result but , in addition , yields the expectation value @xmath13 of the gyroradius and provides its width of uncertainty , i.e its thermal spread . |