bluuebunny/crossref_metadata_2025 huggingface dataset:
Kinetics of OH Radicals as Determined by Their Absorption Spectrum VI. Attempt to Trace OH Radicals in the Thermal Hydrogen-Oxygen Reaction
Authors: Oldenberg O. , Morris J. E. , Morrow C. T. , Schneider E. G. , Sommers H. S.
Published: January 2005

The steady thermal reaction of hydrogen and oxygen at a pressure of 60 cm and a temperature in the region of 550° C was searched by absorption spectrum technique for free OH radicals. None were detected, although the apparatus would have been sensitive to an OH radical pressure of 4×10−4 mm. This negative result indicates that the OH radicals have a very short lifetime in the reacting mixture.
The Mercury Photosensitized Reactions of 1-Butene and 2-Butene
Authors: Gunning H. E. , Steacie E. W. R.
Published: January 2005

An investigation has been made of the reactions of 1-butene and 2-butene with mercury (3P1) atoms at 30°C in a static system. The main products of the 1-butene reaction are 2-butene and liquid polymer. A large number of other products are also formed especially at low pressures. The rate of isomerization increases rapidly with pressure in the region of complete quenching, while the rate of polymerization falls off rapidly. The over-all rate of consumption of 1-butene increases very slightly with increasing pressure. The over-all quantum yield of 1-butene consumption is 0.32 at 20-mm pressure. The principal steps in the mechanism proposed for the reaction are: 1−C4H8+Hg(3P1)→1−C4H8*+Hg(1So),1−C4H8*+1−C4H8→2−C4H8+1−C4H8,1−C4H8*→C4H7+H,followed by polymerization reactions of the type 1−C4H8+C4H7→C8H15,etc.It is therefore possible for the first time, to obtain a direct measurement of the rate of collisional deactivation merely by determining the rate of formation of 2-butene. 2-butene is much less reactive than 1-butene. The over-all quantum yield of 2-butene consumption has a maximum value of 0.11 at an initial pressure of 1–2 mm. Above this pressure the rate of 2-butene consumption falls off rapidly. The main product of the reaction is a liquid polymer. Smaller amounts of hydrogen and methane are also formed. An activated molecule mechanism is proposed which accounts for the essential aspects of the reaction.
The Differential Entropy of Dilution in Aqueous Solutions of Amino Acids
Authors: Robinson A. L.
Published: January 2005

The relative partial molal entropy of water, S̄1 — S̄10, in aqueous solutions of seven amino acids is calculated from existing data. For all solutions, S̄1 — S̄10 = km2 approximately; k is positive for glycine solutions and negative for solutions of the two alanines and four aminobutyric acids. The S̄1 — S̄10 values are determined largely by the size and structure of the hydrocarbon part of the amino acid and only to a minor degree by its dipole moment. Positive S̄1 — S̄10 values are interpreted as indicating a structure breaking effect of glycine on water; negative S̄1 — S̄10 values indicate a structure strengthening effect which increases with size of the hydrocarbon residue. The data are consistent with the recent discussion of Frank. Heat capacity and viscosity values for some of these solutions qualitatively show expected behavior.
Ultrasonic Propagation in Liquids: I. Application of Pulse Technique to Velocity and Absorption Measurements at 15 Megacycles
Authors: Pellam J. R. , Galt J. K.
Published: January 2005

Equipment developed by the Massachusetts Institute of Technology Radiation Laboratory has been applied to the measurement of sound velocity and absorption in liquids at 15 mc/sec. Pulses of one microsecond duration are generated by a transducer, which also picks up the resultant echoes from a plane reflector. Velocity measurements are made by determining the distance the transducer must be moved to delay the received echoes by a specified increment. Absorption measurements are made by determining the attenuation necessary to keep the receiver signal constant as the transducer is moved. The attenuation factor can be measured to an accuracy of about 5 percent and sound velocity to about 0.05 percent. Measurements in homologous series of organic liquids are reported.
A Theoretical Evaluation of Normal Frequencies of Vibration of the Isomeric Octanes
Authors: Ahonen Charles O.
Published: January 2005

A theoretical evaluation of some of the normal frequencies of vibration of the isomeric octanes has been made. This report gives the skeletal frequencies for those of the eighteen isomeric octanes which lead to representations having 7 or fewer frequencies. The forces are assumed to be valence forces. The force constants used are 4.12×105 dynes per cm for the C–C bond stretching, and 0.37×105 dynes per cm for the bending between two C–C bonds, the values being based on computations by the author for n-propane, using the experimental data of E. F. Barker and Violet L. Wu. The angle between two C–C bonds was taken as 107° whenever possible. The results of this investigation are summarized by means of charts, showing the location of bands calculated. For the molecules reported all normal skeletal frequencies for a valence force model were evaluated, and attempts at correlation between experimental and theoretical bands have been made.
The Use of Punched Cards in Molecular Structure Determinations II. Electron Diffraction Calculations
Authors: Shaffer P. A. , Schomaker Verner , Pauling Linus
Published: January 2005

Special files of punched cards for use with the electrical accounting machines of the International Business Machines Corporation in evaluating the Fourier sums and integrals used in the determination of the structure of gas molecules from electron diffraction photographs are described.
The Mercury Photo-Sensitized Reaction Between Hydrogen and Oxygen
Authors: Volman David H.
Published: January 2005

The mercury sensitized reaction between hydrogen and oxygen in a flowing system was studied. The quantum yield of the reaction was found to be less than unity, and no evidence of a chain reaction at 40° was found. Hydrogen peroxide was the principal product, and yields of peroxide above 90 percent were found for many of the experiments. It was shown that in the gas phase reaction between ozone and hydrogen peroxide, water was rapidly formed. The yield of product was found to increase with increasing oxygen flow rate, with increasing total flow rate for a fixed hydrogen to oxygen ratio, and to increase and then decrease with increasing hydrogen to oxygen ratio. The reaction scheme proposed is Hg+hν→Hg*,Hg*+H2→H+H+Hg,Hg*+O2→O2+Hg,H+O2→HO2,H+M→HM,HM+HM→H2+M,HO2+HO2→H2O2+O2.This leads to the rate equation, d(H2O2)dt=k3/k4′′(O2)k3/k4′′(O2)+1Iak1/k2(H2)k1/k2(H2)+(O2),using the flow rates of hydrogen and oxygen. k1/k2 is the ratio of quenching efficiencies of hydrogen and oxygen for activated mercury or 1.62. k3/k4″ is taken equal to 1.2×10−3.
Thermodynamic Properties of Gaseous Boron Trifluoride, Boron Trichloride, and Boron Tribromide
Authors: Spencer Hugh M.
Published: January 2005

In this paper are presented tables of the thermodynamic functions heat content, free energy, entropy, and heat capacity of boron trifluoride, boron trichloride, and boron tribromide in the ideal gas state at 1 atmosphere from 298.16° to 1000°K. These functions were calculated from Raman and infra-red spectra in the case of the two first mentioned and from the Raman spectrum of liquid boron tribromide. A satisfactory calorimetric check of the entropy of boron trifluoride is presented.
Elastic Loss and Relaxation Times in Cross-Linked Polymers
Authors: Kirkwood John G.
Published: January 2005

The outlines of a theory of elastic loss in cross-linked polymers are developed on the basis of the ideas underlying the theory of dielectric loss of Fuoss and the writer and the three-dimensional network model of rubber structure proposed by James and Guth. The relaxation time distribution function for elastic loss is given in the approximation of free internal rotation of chain segments of the net. The influence of intermolecular hindering torques and network restraints on the relaxation time spectrum is discussed, but explicit calculations are postponed for a later report.
A New Approach to the Theory of Relaxing Polymeric Media
Authors: Green M. S. , Tobolsky A. V.
Published: January 2005

A molecular theory of relaxing media is presented which gives an expression for the stress in terms of the strain history. At any given time the strain history produces a distribution in internal strains which for mechanical properties can be characterized by a limited number of internal strain parameters. The second law of thermodynamics is used to define dissipation of energy at constant temperature and explicit expressions for dissipation of energy for any strain history are obtained. Inasmuch as relaxation during straining causes an essential reorganization of structure which is in fact the cause of dissipation, the kinetic theory of elasticity is extended to non-isotropic polymeric networks. A tensor expression for the stress-strain-time relations is thereby developed.
The Statistical Mechanical Theory of Transport Processes I. General Theory
Authors: Kirkwood John G.
Published: January 2005

Outlines are sketched for a general statistical mechanical theory of transport processes; e.g., diffusion, heat transfer, fluid flow, and response to time-dependent external force fields. In the case of gases the theory leads to the Maxwell-Boltzmann integro-differential equation of transport. In the case of liquids and solutions, it leads to a generalized theory of Brownian motion, in which the friction constant is explicitly related to the intermolecular forces acting in the system. Specific applications are postponed for treatment in later articles.
An Improved Hot Wire Cell for Accurate Measurements of Thermal Conductivities of Gases over a Wide Temperature Range Results with Air between 87° and 375°K
Authors: Taylor William J. , Johnston Herrick L.
Published: January 2005

An improved hot wire cell of the potential lead type is described, and theoretical treatments given for radiation, end conduction, potential lead conduction, and the ``temperature jump'' effect. The latter is found to be satisfactorily eliminated by linear extrapolation of reciprocal plots of ``apparent conductivities'' as a function of pressure. Convection effects are absent at pressures below about 20 cm of mercury. The emissivity from bright platinum was measured. The results confirm the measurements by Milverton. The thermal conductivity of air was measured at seventeen temperatures between 87° and 375°K. Our results, which are believed accurate to within ±0.5 percent, are generally lower than other results which have been reported in recent years, for air. Tabulation of our results is given, together with a critical analysis of possible sources of error. The advantage of our method over either the ``thick wire'' or ``compensating lead'' type of cells is pointed out.
A Test of Langmuir's Interpretation of the Jones-Ray Effect
Authors: Wood Lloyd A. , Robinson Lawrence B.
Published: January 2005

Jones and Ray observed a minimum in the surface tension-concentration curve of electrolyte solutions at about 0.001N. Langmuir interpreted this as an instrumental effect rather than a real minimum, and advanced a quantitative theory by which the surface tension data could be corrected. To compute the Langmuir correction factor the zeta-potentials of vitreous silica in contact with the solutions measured by Jones and Ray in their capillarimeter are required. Since the zeta-potentials with barium chloride solutions have now been measured, the Langmuir theory has been tested to see whether it eliminated the minimum observed with this solute, and this paper presents the results of this test. It was found that Langmuir's interpretation was strikingly substantiated.
On Phase Changes of Bose-Einstein Fluid Models
Authors: Goldstein Louis
Published: January 2005

It is shown that all Bose-Einstein (B.E.) fluid models investigated thus far undergo first-order transformations in phase space. The ideal fluid condenses smoothly, e.g., without any possibility of supersaturation, insofar as its thermodynamic characteristic functions and their first derivatives, with respect to the independent variables, are continuous along the transition or saturation curve. The non-ideal B.E. fluids considered, in which attractive interatomic forces averaged over the volume of the fluid are assumed to operate, undergo sudden condensation strictly similar to that exhibited by ordinary fluids in coordinate space. This process is accompanied by the occurrence of discontinuities of the first derivatives of the characteristic functions. By introducing a suitable repulsive interatomic force, smeared over the volume of the fluid, the smooth condensation of the ideal fluid is changed to a phase transformation of the third order. Here the second derivatives of the characteristic functions exhibit discontinuities along the transformation curve. The same interatomic force lifts the order of the sudden first-order transformation by changing it to a second-order one. This is accompanied by the appearance of the lambda-point type discontinuity of the constant pressure heat capacity along the transition or lambda line. The interplay of forces of parallel and opposing tendencies in modifying the nature of a phase transformation without changing its order and in lifting or lowering their order is thus brought out in these studies over B.E. fluid models. The bearing of these results on the problems connected with the thermal properties of liquid helium is touched upon briefly.
Pressure Dependence of Accommodation Coefficients
Authors: Amdur I.
Published: January 2005

The pressure dependence of accommodation coefficients of gases on metals has been explained by assuming that the accommodation coefficient varies linearly with the fraction of the surface covered with adsorbed gas. It is further assumed that the accommodation coefficient has a negligibly small value α0 on a gas-free surface and an asymptotic value α∞ on a saturated surface. The assumptions lead to an accommodation coefficient isotherm which reproduces the pressure dependence of 119 accommodation coefficient values for ten gases on platinum with an average absolute deviation of 1.5 percent.
Surfaces of Solids XVII A First- and a Second-Order Phase Change in the Adsorbed Film of n-Heptane on Graphite
Authors: Jura George , Harkins William D. , Loeser Edward H.
Published: January 2005

The films of n-heptane formed by adsorption on the surface of graphite between 25° and 40°C exhibit a first- and a second-order phase transition. The first-order transition occurs between the gaseous and liquid expanded phases. The second-order transition occurs between the liquid expanded and liquid intermediate phases, and between the gaseous and liquid intermediate phases above the critical temperature. Since the behavior of the film is the same as that observed with insoluble films on aqueous subphases, the nomenclature for the latter is used. The critical constants for the gaseous film were determined as: temperature, 31°C; area, 400A2 per molecule; and film pressure, 1.05 dyne cm−1. These values are different from those found for n-heptane on ferric oxide, which shows that the solid as well as the gas plays an important role in the determination of the adsorption isotherm. The heat of transition for the first-order change is estimated as 13,000±5000 cal. mole−1 at 25°C. It is found at corresponding temperatures, within this large experimental error, that the heat evolved in the first-order transition is the same on graphite and ferric oxide.
Thallous Sulfide Photo-Conductive Cells I. Experimental Investigation
Authors: von Hippel A. , Chesley F. G. , Denmark H. S. , Ulin P. B. , Rittner E. S.
Published: January 2005

An experimental study of thallous sulfide photo-conductive cells is presented. It includes an investigation of the sensitizing oxidation reaction; cell characteristics as function of voltage, temperature, light intensity, wave-length, modulating frequency, and composition; measurements of thermoeffect and optical absorption; x-ray and electron diffraction studies, and an examination of the photo-response to light flashes of very short duration. A theoretical discussion will be given as Part II in a subsequent paper.
Thallous Sulfide Photo-Conductive Cells II. Theoretical Discussion
Authors: von Hippel A. , Rittner E. S.
Published: January 2005

A theory is formulated which is in substantial agreement with all facts reported in a previous paper on thallous sulfide photo-conductive cells. The underlying physical picture is that light is absorbed in the Tl2S proper and in the presence of oxygen, positive ``holes'' and negative oxygen ions are formed. The ``holes'' and ions recombine at a rate proportional to the product of their concentrations. During their lifetime, the negative ions act as space charge compensators for the defect conduction and allow correspondingly higher currents to flow. This space charge compensating action, which is similar to the role played by positive ions in gas discharges, seems to be one essential mechanism of producing secondary photo-effects.
A Relation between Internuclear Distances and Potential Barriers of Methyl Groups
Authors: French F. A. , Rasmussen R. S.
Published: January 2005

An empirical relation is shown between contributions of various groups to the potential barriers of methyl groups and the minimum distance of approach of atoms of these groups to the methyl group. The use of this relation to derive barrier values for other molecules is demonstrated.
Time-Dependence of Boundary Tensions of Solutions I. The Role of Diffusion in Time-Effects
Authors: Ward A. F. H. , Tordai L.
Published: January 2005

The variation of surface tension of solutions with time has in some cases, where the change is over within a few seconds or less, been explained on the basis of diffusion. This paper attempts a rigorous mathematical analysis of the role that diffusion might play in such time-effects. The limitations of diffusion theories which have been proposed previously are discussed. A general theory of diffusion to the surface is derived, which allows for back-diffusion and which makes no special assumptions of a physical nature. It is possible to use Fick's equation to calculate the total amount of solute which diffuses from a semi-infinite bulk solution into the surface if the concentration immediately under the surface is known at various times throughout the process. It is shown how the latter information may be deduced from the variation of surface tension with time together with final equilibrium values of surface tension. The methods of this theory are applied to analyze recent data on time-effects of short duration. It is concluded that even in cases where the variation of the surface tension is over in less than a second the rate-determining process is not diffusion. Even for these very rapid changes one is therefore led to assume the existence of an activation barrier which determines the rate of adsorption.
Free Radical Determination in Biacetyl Photolysis
Authors: Ascah Ralph G. , Burton Milton , Ricci John E. , Davis Thomas W.
Published: January 2005

Biacetyl vapors have been decomposed by radiation of wave-length 2537A in a flow system. The total amount of decomposition was determined by collection and analysis of the gaseous products. The amount of methyl radical formed was determined by allowing the products to pass over and react with lead mirrors. The amount of lead transported establishes the maximum quantity of methyl formed assuming a ratio of CH3 to Pb of 4:1. The data show that approximately one methyl group is released for every two or three molecules of carbon monoxide in the products and consequently that about half of the total reaction proceeds without the primary formation of free radicals. A Spence and Wild type of reaction scheme will explain both the analytical data and observations on fluorescence.
The Reaction of Methyl Radicals with Butadiene and Isoprene
Authors: Gunning H. E. , Steacie E. W. R.
Published: January 2005

An investigation of the reaction of methyl radicals with butadiene and isoprene has been made by photolyzing admixed mercury dimethyl. In each case the amount of methane formed by the reactionCH3+RH→CH4+Rwas small relative to the accompanying polymerization. For butadiene the activation energy of methane formation is 6.5 kcal. This value is, however, rather unreliable on account of the small amount of methane formation.
The Mercury Photo-Sensitized Reactions of Diethyl Ether
Authors: Darwent B. deB. , Steacie E. W. R. , van Winckel A. F.
Published: January 2005

The mercury photo-sensitized reactions of diethyl ether have been investigated in a circulatory system at 35°C and between 9- and 193-mm pressure. The principal products are hydrogen and C8H18O2; smaller amounts of CO, CH4, C2H6, and aldehydes are formed. The quantum yield for ether disappearance is about 2.0 initially but decreases with the duration of the experiment. In low pressure experiments the hydrogen produced initially is consumed in the later stages of the experiment and ultimately vanishes. The decrease in rate and the consumption of hydrogen in the late stages of the reaction are not caused by the accumulation either of polymer on the wall or to hydrogen and/or CO in the system. The nature of the reaction is altered by interrupting the experiment thus indicating the occurrence of a dark reaction. The initial stages may be explained by either a C–H split followed by ``atomic cracking'' or by reactions involving an active ether molecule.
Note on the Conditions of Equilibrium for Systems of Many Constituents
Authors: Brinkley Stuart R.
Published: January 2005

An analytical criterion is given for the choice of independent components. An inclusive discussion of equilibrium in systems of many constituents is thereby achieved. By reason of symmetry, the resulting conditions are particularly suited to form the basis of a calculation of the equilibrium composition of the system.
Dynamics of Nonlinear Stochastic Systems
Authors: Kraichnan Robert H.
Published: January 2005

A method for treating nonlinear stochastic systems is described which it is hoped will be useful in both the quantum-mechanical many-body problem and the theory of turbulence. In this method the true problem is replaced by models that lead to closed equations for correlation functions and averaged Green's functions. The model solutions are exact descriptions of possible dynamical systems, and, as a result, they display certain consistency properties. For example, spectral components of Green's functions which must be positive-definite in the true problem automatically are so for the models. The models involve a new stochastic element: Random couplings are introduced among an infinite collection of similar systems, the true problem corresponding to the limit where these couplings vanish. The method is first applied to a linear oscillator with random frequency parameter. The mean impulse-response function of the oscillator is obtained explicitly for two successive models. The results suggest the existence of a sequence of model solutions which converges rapidly to the exact solution of the true problem. Applications then are made to the Schrödinger equation of a particle in a random potential and to Burgers' analog for turbulence dynamics. For both problems, closed model equations are obtained which determine the average Green's function, the amplitude of the mean field, and the covariance of the fluctuating field. The model solutions can be expressed as sums of infinite classes of terms from the formal perturbation expansions of the solutions to the true problems. It is suggested that correspondence to stochastic models may be a useful criterion to help judge the validity of partial summations of perturbation series.
On a Moving Boundary Problem
Authors: Katz Sylvan
Published: January 2005

An exact analytic solution is obtained for a uniformly expanding, neutral, infinitely conducting plasma sphere in an external uniform and constant magnetic field.
Synthetic Approach and Canonical Variables in a Nonlocal Field Theory
Authors: Tani Smio
Published: January 2005

The analysis of the axiomatic structure of quantum field theory establishes what we call the synthetic approach. The translation in time is completely defined by its transformation function; it defines also canonical variables. The Hamiltonian is discussed at a later stage in order to establish the correspondence between this approach and the conventional one. An application is made to analyze Kristensen-Møller's theory of a nonlocal field. Its field variables are shown to be noncanonical; two field operators referring to two different points at the same time are not independent from each other, when the distance between the two points is of the order of the extension of the form factor; however, this fact does not disturb the calculation of the S matrix. The situation is clearly understood when the Kristensen-Møller field operators are compared with the corresponding canonical variables.
On the General Theory of the Approach to Equilibrium. II. Interacting Particles
Authors: Henin F. , Résibois P. , Andrews F.
Published: January 2005

The general method described in a recent paper by Prigogine and Henin [J. Math. Phys. 1, 349 (1960), hereafter referred to as I] is applied to a system of interacting particles. A full use is made of the diagram technique due to Prigogine and Balescu. The distribution function is Fourier analyzed and each Fourier coefficient is decomposed into two parts: one (ρ′) whose evolution results from scattering processes and which obeys a diagonal differential equation; the second one (ρ″), whose evolution is due to direct mechanical interactions which build the correlation described by the Fourier coefficient. The ρ″ can be expressed in terms of functions ρ′ corresponding to lower correlations. We study first the velocity distribution function. Only scattering processes contribute to the evolution of this function. The equations obtained ensure evolution of this function to the correct equilibrium value at any order in the concentration C and the coupling constant λ. We then study the asymptotic behavior of the Fourier coefficients which describe correlations among the particles. The part ρ′ of these coefficients corresponding to scattering processes vanishes for large time. In other words, scattering processes play a fundamental role in the establishment of the correct velocity distribution function, but once this is achieved they become ineffective. The correct equilibrium correlations, which in equilibrium theory are described in terms of cluster diagrams, are built by direct mechanical interactions among the particles involved in the correlation. We give a detailed proof up to order C2. The extension to higher orders does not introduce any new feature.
On the General Theory of the Approach to Equilibrium. III. Inhomogeneous Systems
Authors: Andrews Frank C.
Published: January 2005

We find the evolution equation for a singlet distribution function in a fluid containing macroscopic inhomogeneities. The equation, a generalization of the Boltzmann equation, derived from the Liouville equation, may be written formally to any order in either concentration or strength of interaction. We find f1 to be a functional only of itself and other f1's. We then show that initially present correlations are destroyed during the same relaxation time as in homogeneous systems. We can then write formally to any order the equation for a reduced s-particle distribution function fs, which proves to be a functional of a product of f1's, all the time dependence of the fs being lodged in the f1's for times greater than the relaxation time.
Support of a Field in p Space
Authors: Dell'Antonio G. F.
Published: January 2005

The notion of generalized free field is introduced, as an obvious extension of a discrete superposition of independent free fields with different masses. The following assumptions are also made: there is an underlying Hilbert space H (positive-definite metric), the theory is Lorentz invariant, the vacuum belongs to H and is there unique, the spectrum of the energy-momentum operator is—apart from the origin—completely contained within the region p2≥ε2, p0>0. It is then shown that a necessary condition for a cyclic field to have support in p2 only on a finite interval of the positive real axis, is that A(x) be a generalized free field. In the Appendix a similar result is derived under slightly weaker conditions.
Nature of the Axioms of Relativistic Quantum Field Theory. I
Authors: Sudarshan E. C. G. , Bardakci K.
Published: January 2005

The formulation of field theories by means of Wightman functions is studied. It is shown that, given two field theories that satisfy all the axioms, one can construct a family of Wightman fields with the same properties by a process of superposition of Wightman functions. The condition of unitarity is formulated without reference to asymptotic conditions, and it is proved that the Wightman fields constructed by the above superposition process (starting with ``unitary'' fields) fail to preserve unitarity, and a fortiori, the standard asymptotic condition.
Principles of Limiting Absorption and Limiting Amplitude in Scattering Theory. I. Schrödinger's Equation
Authors: Odeh Farouk M.
Published: January 2005

The outgoing solution of the time-independent Schrödinger equation, with a suitably restricted real potential, is shown to be the uniform limit of the square-integrable solutions of the same equation with complex energy as the imaginary part of the energy tends to zero. Under further restrictions on the potential, it is also shown that the solution to the initial-value problem for the time-dependent Schrödinger equation tends to the outgoing solution as time increases indefinitely.
Symmetric Expansion of One- and Two-Center Coulomb Potentials
Authors: Fontana Peter R.
Published: January 2005

A one-center expansion of the electrostatic interaction energy of a discrete charge distribution is developed by making use of the algebra of irreducible tensors. The result is completely symmetric in the coordinates of the particles, and the relative magnitude of the vectors need not be specified. It is shown that a suitable interaction representation provides useful formulas for electrostatic and quantum mechanical applications. In addition, some transformation equations make it possible to refer any arbitrary number of vectors to a second origin, thus yielding general two-center expansions for overlapping charge distributions.
Probability Distribution for Classical Fluids
Authors: Broyles A. A.
Published: January 2005

The probability distribution in phase space is obtained for particles in a small part of the total volume of a classical monatomic fluid in thermal equilibrium. It is shown that the distribution reduces to that obtained from a grand canonical ensemble as this part of the volume increases in size. The Debye-Huckel pair distribution function is obtained in the proper limit for the Coulomb case. The distribution is in the form of a first approximation with an infinite series of correction terms.
Note on the Algebraic Aspect of the Integration of a System of Ordinary Linear Differential Equations
Authors: Wichmann Eyvind H.
Published: January 2005

In this note the Lie algebra generated by the coefficient matrix of a system of ordinary, linear, first-order differential equations is considered. A systematic discussion, based on some well-known results in the theory of Lie albegras, is given for the reduction of the problem of integration of such a system. For the purposes of this note the integration of a system of equations for which the coefficient matrix does not depend on the independent variable is regarded as ``elementary.'' It will be shown that the problem of integrating any system of linear ordinary differential equations can be reduced to the problem of integrating a set of such systems, each one of which has the property that the corresponding Lie algebra is simple, and in such a way that the sum of the dimensionalities of the Lie algebras of the reduced systems in the set does not exceed the dimensionality of the Lie algebra of the original system.

The application of the reduction principle to the equations of motion in classical mechanics and in quantum mechanics is considered. It is shown that the principle in question applies to a class of Hamiltonian equations of motion not customarily regarded as describing linear systems.
Simple Realizations of the Infinitesimal Generators of the Proper Orthochronous Inhomogeneous Lorentz Group for Mass Zero
Authors: Lomont J. S. , Moses H. E.
Published: January 2005

A realization of the infinitesimal generators for the mass-zero case of the proper orthochronous inhomogeneous Lorentz group is given explicitly for both continuous and discrete spin cases in terms of a uniform notation. The realization for the discrete spin case is unitarily equivalent to that given by Shirokov.

For the sake of completeness the infinitesimal generators for the case of nonzero mass, derived by Foldy, are also given. Hence the present paper contains realizations for all irreducible unitary representations of the inhomogeneous Lorentz group of physical interest.

Since the irreducible representations of the two-dimensional Euclidean group play an important role in the massless case, simple realizations of the irreducible representations of the generators of this group are also given.
Properties of the Irreducible Angular Momentum Tensors
Authors: Rose M. E.
Published: January 2005

The angular momentum operators define a set of irreducible tensors which are unique except for a normalization constant. The normalization is conveniently defined in terms of statistical tensors which describe oriented states. The properties of the tensors discussed here include: (1) the trace of products of components of such tensors, (2) symmetry properties of the traces, and (3) the expansion of products of components of these tensors into a sum of irreducible tensors. The corresponding expansion of commutators and anticommutators of these components is also discussed briefly.
The Quantum-Mechanical Scattering Problem
Authors: Jordan Thomas F.
Published: January 2005

The purpose of this paper is threefold: to provide a mathematically rigorous formulation of the quantum-mechanical scattering problem from the time-independent point of view as has been done by Jauch from the time-dependent point of view, to establish a union between the two formulations, and to investigate the necessity of the asymptotic condition which occurs as a postulate in the time-dependent formulation. The formulation of the problem depends only on the ``total'' and ``free'' Hamiltonian operators. Under the conditions necessary for the time-dependent formulation, the wave operators defined by the asymptotic limits provide a unique solution of this problem. The possibility that solutions can exist when the asymptotic conditions are not valid is investigated by defining wave operators by an integral representation. The conditions sufficient for these to provide a unique solution are shown to be possibly weaker than the asymptotic conditions; there may be a class of Hamiltonian operators for which such solutions exist but for which the asymptotic limits do not. An explicit characterization of such a set of Hamiltonian operators is not achieved, but this question of the necessity of the asymptotic condition has been reduced to a specific mathematical problem. It is hoped that this paper will find a reader who is able to carry the mathematical investigation further.
The Quantum-Mechanical Scattering Problem. II. Multi-Channel Scattering
Authors: Jordan Thomas F.
Published: January 2005

The study reported in an earlier paper of the single channel scattering problem is extended to include the multi-channel case.
Perturbation Theory and Lie Algebras
Authors: Murray F. J.
Published: January 2005

An approach to perturbation theory is considered based on the formula exp (—iB) A exp (iB) = A exp (iB]), where [AB] = AB − BA is the commutator. The operators A constitute a linear space A and the operators B considered are such that B] take A into itself. The present discussion considers the case where A is finite dimensional with coordinates c1, ···, cn; i.e., A is isomorphic with the set of n-dimensional vectors c. Under these circumstances sufficient conditions for the basic formula are given in terms of the ``analytic vectors'' of Nelson. The set, B, of B's available can be considered closed under the processes of taking linear combinations and forming the commutator. Thus B is a Lie algebra. Exponentiation leads to a Lie group of operators U, and A and A′ are said to be B equivalent if A′ = U−1AU. For A finite dimensional each B is associated with an n × n matrix, b, which specifies the operation B] relative to the vectors c. Effectively then, B is finite dimensional. The matrices b form a Lie algebra with a corresponding Lie group of matrices u such that A and A′ are B equivalent if and only if the corresponding two vectors c and c′ are u images; i.e., c′ = u c. Computationally, therefore, the set of A′ equivalent to a given A is obtained by considering the orbit of a given vector c under the Lie group; i.e., the set of u c. A neighborhood A′ of a given A consists of those operators A′ in the form A + δA where δA is arbitrary except for a restriction on the size of its coordinates, ci. Given A, a neighborhood A′ can be found for which one can obtain by a well-known construction on the orbits a set of functionally complete and functionally independent invariants for B equivalence. Computationally global and rational invariants are desirable and these can be obtained in the form of similarity invariants, provided that a can be mapped onto a set of n × n matrices a in such a way that if c corresponds to a, then b c corresponds to a′ = [ab]. If the sets A and B are identical such a mapping is immediately available and if, in addition, the corresponding Lie algebra is semisimple, in general, given A, the A's in some neighborhood are each equivalent to an A″ which is a function of A. This corresponds to a case in which a very simple perturbation of A levels occurs. Two examples are discussed.
A Procedure for Estimating Eigenvalues
Authors: Bazley Norman W. , Fox David W.
Published: January 2005

A new procedure is given for calculation of lower bounds to the eigenvalues of self-adjoint operators. Computation of the lower bounds is reduced to the solution of linear algebraic problems.
A Note on Perturbation Theory in Nearly Periodic Systems
Authors: Pytte Agnar
Published: January 2005

It is shown that for a particle in periodic or nearly periodic motion, an integrated form of the equations of motion may be the better starting point for an approximate calculation of the orbit. The motion of a charged particle in a static, inhomogeneous magnetic field is used to illustrate how this approach avoids the difficulty of the spurious secular terms to all orders of the approximation.
Stochastic Models for Many-Body Systems. I. Infinite Systems in Thermal Equilibrium
Authors: Kraichnan Robert H.
Published: January 2005

Some model Hamiltonians are proposed for quantum-mechanical many-body systems with pair forces. In the case of an infinite system in thermal equilibrium, they lead to temperature-domain propagator expansions which are expressible by closed, formally exact equations. The expansions are identical with infinite subclasses of terms from the propagator expansion for the true many-body problem. The two principal models introduced correspond, respectively, to ring and ladder summations from the true propagator expansion, but augmented by infinite classes of self-energy corrections. The latter are expected to yield damping of single-particle excitations. The eigenvalues of the ring and ladder model Hamiltonians are real, and they are bounded from below if the pair potential obeys certain conditions. The models are formulated for fermions, bosons, and distinguishable particles. In addition to the ring and ladder models, two simpler types are discussed, one of which yields the Hartree-Fock approximation to the true problem. A novel feature of all the model Hamiltonians (except the Hartree-Fock) is that they contain an infinite number of parameters whose phases are fixed by random choices. Explicit closed expressions are obtained for the Helmholtz free energy of all the models in the classical limit.
Stochastic Models for Many-Body Systems. II. Finite Systems and Statistical Nonequilibrium
Authors: Kraichnan Robert H.
Published: January 2005

In a preceding paper [J. Math. Phys. 3, 475 (1962)], some model Hamiltonians were proposed for quantum-mechanical many-body systems with pair forces. For infinite systems in thermal equilibrium, they led to temperature-domain propagator expansions which were formally summable and expressible by closed equations. These expansions were identical with infinite subclasses of terms from the propagator expansion for the true many-body problem. The two principal models corresponded to ring- and ladder-diagram summations from the true propagator expansion, augmented by infinite classes of self-energy corrections. The model Hamiltonians were called stochastic because they contained parameters whose phases were fixed by random choices. In the present paper, more general models are formulated which yield formally summable propagator expansions for finite systems. The analysis is extended to correlation and Green's functions defined for nonequilibrium ensembles. The nonequilibrium treatment is developed in the Heisenberg representation in such a way that unlinked diagrams do not arise. A basic convergence question associated with the formal closed equations for the model propagators and correlation functions is examined by means of finite-difference integration of the Heisenberg equations of motion. This procedure appears to converge independently of whether the perturbation expansions for the propagators and correlation functions converge. It yields substantial support for the validity of the formal closed model equations.
Excitation Spectrum of a Fermion System
Authors: Yanagawa S.
Published: January 2005

By the method of ``modes'' introduced by Sawada, the excitation spectra of a fermion system with singular interactions have been obtained in three cases; (1) with one additional particle above and and one hole below the Fermi surface, (2) with one additional particle above the Fermi surface, and (3) with two particles above the Fermi surface. The argument holds not only for the system of low density (nuclear matter with a hard-core repulsion), but also for the case of the interacting electron gas in the high-density region. The results have been compared with the Brueckner-Goldstone perturbation-expansion formulas, by using diagrams in the second- and third-order of the expansion in interaction strength. We show that in the approximation of our treatment, the occupation probability function at temperature T has the same form as the Fermi distribution function.
Studies in Nonequilibrium Rate Processes. V. The Relaxation of Moments Derived from a Master Equation
Authors: Shuler Kurt E. , Weiss George H. , Andersen Knud
Published: January 2005

A study has been made of the relaxation of the moments of probability distributions whose time evolution are governed by a master equation. The necessary and sufficient condition for the first moment, M1(t), to undergo a simple exponential relaxation is found to be ∑ n=0∞nAnm=βm+γ,where Anm is the transition probability per unit time for transitions from state m to n, and where β and γ are constants. The necessary and sufficient condition under which the first k moments, M1(t), M2(t), ···, Mk(t), satisfy a closed system of linear equations is found to be ∑ n=0∞nrAnm= ∑ i=0kβrimi.Near equilibrium, i.e., as t → ∞, all the moments Mr(t) obey, to a good approximation, a simple exponential relaxation law irrespective of the form of the Anm.

For systems described by the Fokker-Planck equation ∂P(x, t)∂t=−∂∂x [b1(x)P(x, t)]+12∂2∂x2 [b2(x)P(x, t)],the necessary and sufficient condition that the first moment M1(t) undergo a simple exponential relaxation is found to be b1(x) = βx + γ and the necessary and sufficient condition for the 2nd moment, M2(t) to have a simple exponential relaxation is 2xb1(x)+b2=β22x2+γ2. It is shown that these conditions are equivalent to the conditions on the Anm stated above.
On the Mutual Coherence Function in an Inhomogeneous Medium
Authors: Parrent George B. , Shore Robert A. , Skinner Thomas J.
Published: January 2005

The paper is concerned primarily with determining the mutual coherence function of the field produced by a plane quasi-monochromatic source in a region of variable refractive index. A scalar theory is used throughout. Section I presents a brief review of the conceptual background of coherence theory. Section II contains an outline of the mathematical formalism of coherence theory and shows that in a region of variable refractive index the mutual coherence function is propagated according to a pair of inhomogeneous scalar wave equations. In Sec. III, the pair of wave equations are solved using appropriate Green's functions to derive an expression for the mutual coherence function of a field produced by a plane quasi-monochromatic source. In Sec. IV the case of a statistically homogeneous medium is treated and an expression for the ensemble average of the mutual coherence function is obtained in terms of integrals of the two-point correlation function characterizing the medium.
On Scattering of Waves by Random Distributions. II. Two-Space Scatterer Formalism
Authors: Twersky Victor
Published: January 2005

The bulk parameters (propagation number K, etc.) of the coherent multiple scattered field in a slab region of randomly distributed scatterers are expressed functionally in terms of ``two-space'' isolated scatterer amplitudes. These amplitudes correspond to a single object excited by a wave traveling in ``K space'' (the medium associated with the coherent field) but radiating into ``k space'' (free space). The formalism is applied to small spheres of arbitrary ε′ and μ′, and the ``Lorentz-Lorenz'' form for each bulk parameter ε and μ is obtained. It is shown that ε and μ are independent of the direction and of the polarization of the incident field; thus, the coherent field for the distribution defines a unique Maxwellian medium. As a second approximation for small spheres, the ``loss terms'' corresponding to incoherent scattering are included; these appear in the bulk ε and μ in the roles of electric and magnetic ``conductivities.' As another illustration, we consider normal incidence on a slab of tenuous scatterers (parameters close to those of free space) large compared to wavelength; for this case the bulk parameters ε and μ are equal to the index of refraction. We consider the incoherent as well as the coherent effects.
Traces of Products of Angular Momentum Matrices. II. Spherical Basis
Authors: Ambler E. , Eisenstein J. C. , Schooley J. F.
Published: January 2005

Closed formulas are given for Tr JαpJβqJγr ··· where α, β, γ ··· are equal to + 1, 0, or −1, and p, q, r are non-negative integers for which p + q + r + ··· ≤ 9. The procedures used in evaluating the traces are described. In order to facilitate numerical evaluation from existing tables, simple relationships are given between traces in the spherical basis and traces in the Cartesian basis. The use of these tables for evaluating expectation values of certain operators is discussed in relation to other methods based on the coupling and recoupling theory of angular momentum.