bluuebunny/crossref_metadata_2025 huggingface dataset:
Spectrum of Er3+ in LaCl3
Authors: Eisenstein J. C.
Published: January 2005

The energy levels of the Er3+ ion in LaCl3 are calculated by finding the eigenvalues of the matrices for the combined Coulomb, spin—orbit, and crystal-field interactions within the f11 configuration. For suitably chosen values of the three Slater integrals and the spin—orbit coupling constant the root-mean-square deviation between the calculated and the experimental positions of 20 terms is 140 cm−1. Crystal-field parameters can be chosen which reduce the root-mean-square deviation between the calculated and the measured splittings of 72 levels to 3.84 cm−1. Zeeman splitting factors in the direction parallel to the crystal axis are also calculated. The results for Er3+ in LaCl3 are compared with those for Nd3+ in LaCl3.
Electric Moments and Molecular Orbitals for Methane
Authors: King W. T.
Published: January 2005

The applicability of simple molecular orbital theory as a basis for determining approximate empirical wavefunctions useful for correlating molecular properties is examined for the methane molecule. It is found that LCAO molecular orbitals provide a fairly good approximation to the molecular wavefunction of methane in its equilibrium configuration, but represent the changes in the wavefunction produced by molecular deformation too crudely to provide an interpretation of its dipole-moment derivatives determined by infrared intensity measurements on methane.
Analytical Expressions for Potentials of Neutral Thomas—Fermi—Dirac Atoms and for the Corresponding Atomic Scattering Factors for X Rays and Electrons
Authors: Bonham R. A. , Strand T. G.
Published: January 2005

Approximate analytical expressions for the electrostatic potentials of neutral Thomas—Fermi—Dirac atoms have been obtained by fitting a sum of three exponential terms to the tabulated values by least-squares methods. The six different parameters used for each atom are expressed as functions of the atomic number. The corresponding expressions for the radical electron density, the mean-square radius, the diamagnetic susceptibility, the atomic scattering factors for x rays and the atomic scattering factors for electrons according to the first Born approximation are given. The accuracy of the approximate expressions are discussed in relation to results obtained from the tabulated values of the electrostatic potential and from the corresponding electron density.
Reaction Rates of Hydrogen, Ammonia, and Methane with Mixtures of Atomic and Molecular Oxygen
Authors: Wong Edgar L. , Potter A. E.
Published: January 2005

The stirred-reactor technique was used to measure the reaction rates of H2, NH3, and CH4 with mixtures of atomic and molecular oxygen (O+O2) at temperatures ranging from 350° to 600°K. A mass spectrometer capable of detecting atomic oxygen and hydrogen was used to analyze the reacting mixture in the stirred reactor. It was discovered that atomic oxygen could be quantitatively monitored by the mass spectrometer at a mass-to-charge ratio of 8. The reaction of H2 with the O+O2 mixture was found to be a chain reaction, with four to six atoms of oxygen consumed for each hydrogen molecule. The rate constant for the disappearance of atomic oxygen was 3×1013 exp(−8300/RT) cc mole−1 sec−1. The products of the reaction were H2O and H. The reactions of NH3 and CH4 with the O+O2 mixture were complex chain reactions, with a minimum of eight atoms of oxygen disappearing for each molecule of NH3 or CH4. Rate constants for atomic-oxygen disappearance were 3×1012 exp (−4800/RT) cc mole−1 sec−1  and  2×1013 exp (−6900/RT) cc mole−1 sec−1,respectively. The products of the former reaction were NO, H2O, and small amounts of H2 and H. The products of the latter reaction were H2O, CO2, and small amounts of CO, H2, and H.
On the Recalculation of the Potential Curves for the Ground States of I2 and H2
Authors: Weissman Stanley , Vanderslice Joseph T. , Battino Rubin
Published: January 2005

A brief discussion of a program for evaluating the integrals which appear in the Rydberg—Klein—Rees (RKR) method of determining potential curves from spectroscopic data is given. Application is made to the ground states of H2 and I2 and the results compared to earlier calculations.
Structure of Zinc Oxide Nuclei
Authors: Sears G. W. , Powell R. , Donn B.
Published: January 2005

In the classical analysis of nucleation of a crystalline phase from a fluid phase it is assumed that the nucleus is a perfect single crystal. In the present paper it is shown that ZnO fourlings almost certainly nucleate as fourfold twins. Alternative nucleation mechanisms are described which would account for a polycrystalline nucleus.
Infrared Spectrum and Dipole—Dipole Coupling in Crystalline Cyanogen
Authors: Verderame Frank D. , Nebgen John W. , Nixon Eugene R.
Published: January 2005

The infrared spectrum of a polycrystalline film of cyanogen at 80°K is reported for the region from 4000 to 500 cm−1. In an attempt to account for the splitting observed for the ν3 fundamental, transition-dipole—transition-dipole interactions have been calculated by computing lattice sums for a variety of crystal shapes and sizes. It is concluded that dipole—dipole coupling can account for only a fraction of the observed splitting.
Dissociation of H2 and Other Diatomic Molecules by Neutron Bombardment
Authors: Clendenin W. W.
Published: January 2005

The perturbation treatment of neutron scattering introduced by Fermi is extended to the case of dissociation of diatomic molecules. In contrast to the usual determination of the kinematics of one fragment only, in the present calculation the detailed energy distributions of both fragments of the dissociated molecule are obtained. The theory has been applied in the cases of the ground states of H2, N214, Cl235 and HCl35.

The necessary conditions on perturbation theory to provide a correct treatment of recoil effects are examined and discussed.
Exciton Magnetic Resonance in Wurster's Blue Perchlorate
Authors: Thomas D. D. , Keller H. , McConnell H. M.
Published: January 2005

The paramagnetic resonance of a single crystal of Wurster's blue perchlorate (a solid ionic free radical) in the 20°—77°K temperature range provides evidence that the elementary electronic excitations are triplet excitons with excitation energy J=246 cm−1. Exciton motion is indicated by the absence of nuclear hyper-fine structure at low exciton concentrations where exciton—exciton spin exchange interaction is negligible. Exciton motion is thought to be primarily along linear free-radical chains; the rate of chain jumping is estimated to be greater than 50 Mc/sec but probably less than 103−104 Mc/sec.

At certain crystal orientations in a strong magnetic field the rate of exciton chain jumping is large compared to the difference in the fine-structure splitting of excitons on the two linear chains in the low-temperature monoclinic crystal; in this case the (three-dimensional) exciton spin Hamiltonian, H=E(Sx2−Sy2), with | E | = 212 Mc/sec, accounts for the observed splittings to within 2%. At other crystal orientations the fine-structure lines are lifetime broadened (due to chain jumping) so much that they are not observable.

The well-known sudden decrease in magnetic susceptibility of Wurster's blue perchlorate at 186°K is apparently due to an orthorhombic→monoclinic phase transition in which the Wurster's blue cations pair up to form dimers with ground singlet states and excited triplet states. The observed value of E can, however, be estimated theoretically with good accuracy in terms of the known intermolecular distances for the room-temperature orthorhombic lattice and a molecular orbital approximation for the spin distribution in the positive ions. This suggests that the dimerization involves only small molecular displacements in the crystal lattice.
Crystallography of Impurity Adsorption on Copper Surfaces
Authors: Robertson W. M. , Shewmon P. G.
Published: January 2005

The object of the investigation was to determine the sites at which impurity adsorption occurs on crystalline surfaces. This was done by measuring the variation of surface free energy (γ) with orientation (θ). A relation between the torque [τ=(∂γ/∂θ)μ], chemical potential (μ), surface excess (Γ), and orientation is derived: (∂τ/∂μ)θ = — (∂Γ/∂θ)μ. Methods for determining surface torques by measuring dihedral angles at grooves where twin boundaries and grain boundaries intersect surfaces are discussed.

Measurements were made of relative torques τ/γ on copper surfaces for the range of oxygen pressures from 10−22 to 10−13 atm at 1000°C using water—hydrogen mixtures. Increasing po2 was found to increase the magnitude of the torque on the (111) and (100) orientations. This indicates preferential adsorption at a range of orientations on the smooth, low-index (100) and (111) surfaces rather than on stepped surfaces of orientations different from the low-index orientations.

A calculation based on a simple model of the surface indicates that about a half monolayer of oxygen atoms is adsorbed on the (111) surface. The characteristic pressure for oxygen adsorption on (111) was calculated to be 3×10−17 atm oxygen at 1000°C.

Measurements of torques after annealing in an atmosphere containing hydrogen sulfide showed that sulfur adsorption decreases the torque near (100) while not affecting that near (111). This indicates preferential sulfur adsorption at steps near the (100) orientation.

Integration of the plots of torque as a function of orientation showed that the surface energies of (100) and (111) relative to higher index surfaces are lowered as the oxygen pressure is increased.
Molecular, Crystal, and Valence Structures of Iso-B18H22
Authors: Simpson Paul G. , Folting Kirsten , Dobrott Robert D. , Lipscomb William N.
Published: January 2005

A three-dimensional single-crystal x-ray diffraction study has established the formula and molecular structure, including all H-atom positions, of i-B18H22. The boron arrangement has the geometry of two B10 units from B10H14 which share a common B9–B10 edge so that the molecule has a twofold axis, and so that the two halves open up in opposite directions. Thus the geometry is similar to that in n-B18H22, in which the two halves are related instead by center of symmetry. There are four molecules of i-B18H22 in a unit cell having parameters a=9.199, b=13.180, c=12.515 Å, and β=109°35′. The space group is P21/c. The value of R=Σ‖ F0 |—| Fc‖/Σ| F0 |=0.108 for the 3195 observed diffraction maxima. In a study of the 6992 two-center and three-center resonance structures the charge distribution has been approximated in order to predict the nucleophilic and electrophilic sites of chemical activity. The shared B9 atom is the most positively charged B atom.
Continuum Infrared Radiation from High-Temperature Air and Nitrogen
Authors: Taylor Raymond L.
Published: January 2005

The absolute intensity of infrared radiation from air and nitrogen, shock-heated to an equilibrium temperature of 6000°—9000°K, has been measured at nearly atmospheric density over the wavelength range of 2–8 μ. This continuum radiation is interpreted as due to free—free scattering of electrons from neutral atoms and molecules. The temperature dependence of this radiation has been determined at 2.35 and 3.88 μ for air and at 3.07 μ for nitrogen. An analysis of the temperature-dependent data, using the classical Kramers' relation, has resulted in establishing the effective Coulombic charges, Z, for N2, N, and O, which indicate that in air the bulk of the free—free radiation is due to scattering from nitrogen atoms and molecules; oxygen gives a negligible contribution. A small change in slope of the wavelength data for air occurs around 3.6–4.0 μ. Speculation concerning the origin of this radiation is presented.
Criterion for Ordering the Bond Strengths of Neutral and Positively Charged Hydrides
Authors: Stanton R. E.
Published: January 2005

A simple criterion is developed for ordering the bond strengths of isoelectronic hydrides, particularly positive ions with the formula MZ+1Hn+ and the corresponding neutral molecules MZHn. For monohydrides the criterion states that the bond strength of the ion will be greater than that of the neutral molecule whenever the ionization potential of the atom MZ+1 is less than 13.6 eV. When this ionization potential is significantly greater than 13.6 eV the neutral molecule has the stronger bond. For polyhydrides the total dissociation energy of the ion is almost always greater than that of the neutral molecule. Extensive experimental evidence in support of this criterion is presented.
Gauge Properties of Propagators in Quantum Electrodynamics
Authors: Zumino Bruno
Published: January 2005

The effect of a change of gauge on the propagators is studied systematically for quantum electrodynamics. Various gauges are considered, among them the Coulomb, the Landau, the Feynman, and the Yennie gauges. The equivalence of the various formulations of the theory is demonstrated. For the relativistic gauges, the transformation of the wave function renormalization constant is described.
Mandelstam Representation for Potential Scattering
Authors: Klein Abraham
Published: January 2005

A proof of the Mandelstam representation for the nonrelativistic scattering amplitude is given when the potential is of the Yukawa form or (by obvious extension) a suitable linear combination of such forms. The analytic properties of the scattering amplitude as a function of momentum transfer are established by using only a finite sequence of equivalent definitions of the scattering amplitude. By studying the Born series for individual partial waves, it is shown in addition that there cannot be an essential singularity at infinity. Together, these results imply both dispersion relations for individual partial waves and the Mandelstam representation.
Poincaré Cycles, Ergodicity, and Irreversibility in Assemblies of Coupled Harmonic Oscillators
Authors: Mazur Peter , Montroll Elliott
Published: January 2005

The transport coefficients (diffusion constant, electrical conductivity, etc.) associated with irreversible processes in an assembly of particles can be expressed as integrals over certain time relaxed correlation functions between small numbers of variables of the assembly. The scattering of slow neutrons is also a measure of time relaxed correlation functions.

Irreversibility is a consequence of the vanishing of the correlation coefficients as the relaxation time becomes infinite. On the other hand these coefficients have Poincaré cycles so that any value which they take on is repeated an infinite number of times. It is shown that, in the case of fluctuations of 0(N−½) from zero (N being the number of degrees of freedom), the period of Poincaré cycles is of the order of the mean period of normal mode vibrations while for fluctuations of a magnitude independent of N the period is of the order of CN where C is a constant which is greater than 1.

The time relaxed correlation coefficients of a pair of particles separated by r lattice spacings decays as t−m/2, m being the number of dimensions of the assembly. The statistics of the decay of the momentum of a particle from a preassigned initial value to its equipartition value are discussed.
Dimensional Changes in Crystals Caused by Dislocations
Authors: Toupin R. A. , Rivlin R. S.
Published: January 2005

According to the classical linear elasticity theory, if one or more dislocations are introduced into a body of elastic material, the average value of each of the infinitesimal strain components is zero; in particular, the change in volume is zero. This result seems not to be in accord with experimental data on cold worked metals. In this paper we use nonlinear elasticity theory to show how changes in the average dimensions of elastic bodies, either isotropic or anisotropic, resulting from the introduction of dislocations, can be calculated. In particular, we derive an explicit relation between the resultant change in volume, the stored energy, and the pressure derivatives of the elastic moduli.
Solution of the Collisionless Boltzmann Equation using a Diagram Technique
Authors: Von Roos Oldwig
Published: January 2005

The diagram technique recently developed by the author (see footnote 1) for the solution of Liouville's equation is extended and suitably modified to cover the case of the collisionless Boltzmann equation for a plasma. The usefulness of the method is demonstrated by two problems. These are: first, the influence of a plane polarized electric wave on the electron distribution function of a low-temperature plasma, and second, the propagation of a (small) initial disturbance for the case of a plasma which is governed by the Vlasov equation (see footnote 8).
Variational Method for Studying the Motion of Classical Vibrating Systems
Authors: Luttinger J. M. , Thomas R. B.
Published: January 2005

A quantity Z (essentially the action integral) is shown to be stationary for small periodic deviations from the actual motion. By choosing approximations to the motion containing the frequency and some other parameters, it is possible to make this variational principle the basis of an approximate determination of the motion.

Quite simple trial functions are found to give surprisingly accurate values for the frequency and the Fourier components of the motion. Comparison between the exact solution and ours for some systems is given.
Convergence of the Born Expansion
Authors: Aaron Ronald , Klein Abraham
Published: January 2005

The convergence of the iterated Born series for the Green's function in nonrelativistic potential scattering is studied in n dimensions, thus generalizing a recent study of Zemach and Klein. For spherically symmetrical potentials the series is proved to converge at sufficiently high energies for a rather general class of potentials.
Analyticity of the Fourth Order Scattering Amplitude with Two Complex Invariants
Authors: Tarski Jan
Published: January 2005

The partial Feynman amplitude corresponding to a particular fourth order diagram is examined as a function of energy and momentum transfer with both of these variables complex. The region of regularity of this function is found, and the types of singularities at the remaining points are determined. An approach which requires only elementary calculations is indicated. The condition for the validity of Mandelstam's representation in the fourth order is obtained. Spectral representations for exchange scattering processes at fixed momentum transfer are discussed as another application of the principal results.
On the Many-Particle Structure of Green's Functions in Quantum Field Theory
Authors: Symanzik K.
Published: January 2005

The structure of the expectation values of retarded multiple commutators (r functions) is analyzed in terms of the number of particles in the decomposition of absorptive parts. As to the one-particle structure, it is found that an r function is a sum of a finite number of terms, each of them except one (that one being called one-particle irreducible) being in momentum space a product of one-particle irreducible factors, linked by one-particle propagation functions. As to the two-particle structure, it is found that a one-particle irreducible function is the solution of an inhomogeneous Bethe-Salpeter equation, whose kernel and inhomogeneous term both are two-particle irreducible functions. This structure, which could be generalized to higher particle numbers, closely resembles perturbation theory but is here derived from locality and the asymptotic condition alone, by converting the nonlinear system of integral equations for r functions stepwise into one in which neither one- or two-particle reducible functions, nor one- or two-particle intermediate states appear. The implication of such structure analysis for an interpretation of perturbation theory, improvements of present methods to derive analytic properties of scattering amplitudes, and a formalism with unstable particles are discussed, and the strength of singularities of various functions investigated.
Invariant Imbedding and Mathematical Physics. I. Particle Processes
Authors: Bellman Richard , Kalaba Robert , Wing G. Milton
Published: January 2005

With the use of invariance principles in a systematic fashion, we shall derive not only new analytic formulations of the classical particle processes, those of transport theory, radiative transfer, random walk, multiple scattering, and diffusion theory, but, in addition, new computational algorithms which seem well fitted to the capabilities of digital computers. Whereas the usual methods reduce problems to the solution of systems of linear equations, we shall try to reduce problems to the iteration of nonlinear transformations.

Although we have analogous formulations of wave processes, we shall reserve for a second paper in this series a detailed and extensive treatment of this part of mathematical physics.
Normal Form of Antiunitary Operators
Authors: Wigner Eugene P.
Published: January 2005

Antiunitary operators are characterized in a manner similar to the characterization of unitary operators by their characteristic vectors and characteristic values. It is shown that a complete orthonormal set of vectors can be defined, some of which are invariant under the antiunitary operator. The rest of the vectors, which are always even in number, form pairs in such a way that the antiunitary operator transforms each member of a pair into a multiple of the other member of the same pair [Eq. (11)]. The extent to which the vectors of the orthonormal set are determined by the antiunitary operator is ascertained and the number of free parameters in the various cases of degeneracy found.
Real Representations of Coordinate Rotations
Authors: Fano U.
Published: January 2005

Since irreducible tensorial sets that represent observables are of integral degree, their transformations under coordinate rotations have real representations. Real representations, with rows and columns classified by eigenvalues of the commuting operators Jz2 and exp (iπJy), are given explicitly, so that complex functions of rotation angles need not be used. The addition of angular momenta is worked out for sets in the real representation such as the sets of real orbital wave functions. Applications to the theory of angular distributions are discussed.
``Front'' Description in Relativistic Quantum Mechanics
Authors: Acharya R. , Sudarshan E. C. G.
Published: January 2005

The problem of introducing a Cartesian position operator canonically conjugate to the momentum operator into relativistic one-particle theories is investigated independent of any particular relativistic wave equation. The known result that such a description is possible for particles with nonvanishing mass is rederived. The general problem of introduction of canonical variables into relativistic theories is formulated and solved. The configurational indices so obtained correspond to directed plane wavefronts rather than point particles.
Structure of the Many-Channel S Matrix
Authors: Newton Roger G.
Published: January 2005

We are considering the nonrelativistic elastic and inelastic scattering of two particles with internal degrees of freedom, or reactions giving rise to two particles. It is shown under very general conditions that all elements of the S matrix can be simply obtained from a single analytic function of all channel momenta, the Fredholm determinant of the scattering and reaction integral equations. Its properties are investigated and the restrictions are established which are necessary and sufficient in order to assure that the unitarity condition is fulfilled. The square well and a superposition of Yukawa potentials are considered as examples.
Classification of Four-Dimensional Riemannian Spaces
Authors: Schell J. F.
Published: January 2005

A classification of four-dimensional Riemann spaces with signature +2 is given. The classification depends upon the differential as well as the algebraic properties of the Riemann tensor. The tool employed is the infinitesimal-holonomy group of the space. An introduction to the concept of the holonomy group is given, and the technique of classification is outlined. A comparison with the classification of empty spaces given by A. Z. Petrov and with the recent work of E. Newman is also given.
On the Capacity of the Icosahedron
Authors: Conlan James , Diaz J. B. , Parr W. E.
Published: January 2005

An application of Dirichlet's principle, using a simple trial function suggested by the symmetries of the problem, is shown to furnish fairly close, readily computable, upper bounds for the capacity of any regular solid.
Free Propagator Expansion in the Evaluation of the Lamb Shift. I
Authors: Layzer Arthur J.
Published: January 2005

This is the first of two papers concerned with ``order properties,'' with respect to the parameter αZ, of an expansion method for the evaluation of the bound electron self-energy ΔE and the application of these properties to the calculation of the new Lamb shift orders of α(αZ)6 ln2(αZ) and α(αZ)6 ln(αZ). The expansion method is the free-propagator expansion (FPE); that is, the formal algebraic expansion of the bound electron propagator or Green's function in ``powers'' of the external (Coulomb) potential. The principal result of the general mathematical analysis is a theorem which asserts that the FPE is an order expansion for (only) those terms of ΔE that are nonanalytic in the parameter w ≡ (αZ)2 and is thus particularly suitable for the calculation of this class of terms. A practical result of the theorem is that the new logarithmic orders arise from only the first four terms of the FPE. The nonanalytic part of a fixed term In of the FPE can be attacked directly through a consideration of Im+In, where Im+In denotes the imaginary part of In, regarded as a function of the complex variable w, on the upper side of a branch cut along the negative w axis. As an auxiliary result, boundedness properties in momentum space are derived for certain iterated operators related to the FPE of the bound nonrelativistic electron Green's function.
Ellipsoidal Distributions of Charge or Mass
Authors: Carlson B. C.
Published: January 2005

The Coulomb (or gravitational) energy is calculated for a distribution of charge (or mass) in which the surfaces of constant density are a family of similar concentric ellipsoids. The density can vary in any manner from one surface to another, and the ellipsoids need not have an axis of symmetry. Two examples are discussed: the charge distribution of a deformed atomic nucleus having a diffuse surface, and the mass distribution of a stellar galaxy. The energy is shown to be a product of two factors. One is the energy of the spherical distribution from which the ellipsoidal distribution can be obtained by a volume-preserving deformation. The other is an anisotropy factor that has a simple geometrical significance and depends only on the two eccentricities of the ellipsoids. Its values range from unity to zero and are tabulated numerically.
Group-Theoretical Treatment of Time- and Energy-Dependent Multiple Scattering with Application to the Slowing-Down of Neutrons
Authors: Guth E. , Inönü E.
Published: January 2005

Wigner developed a group-theoretical method for those problems of multiple scattering in which the elementary scattering law is invariant under a group of transformations. The integral transforms, used in more standard treatments for the reduction of convolutions, come in here more naturally through the representations of the groups of transformations. Wigner's work is extended here to include the time-dependent slowing-down of neutrons. In this case a group of linear transformations comes in, which does not yield orthogonality. Nevertheless, it is possible to determine all positive time moments of the distribution function and from them the distribution function itself. The conditions for the existence of the group are that the total scattering cross section is proportional to vγ (γ: any real number) and that either the ratio of absorption to scattering cross section is constant (including zero) or that the absorption cross section varies as 1/v. Moreover, it is assumed that the angular dependence can be anisotropic, but does not depend on energy in the center of mass system. For the special case of no absorption and spherically symmetric elastic scattering in center of mass system, our solution reduces to Waller's recent exact expression. As a further generalization, we discuss the group which, with the same assumptions about the cross sections, exists for the case of time-energy-space-direction dependence. Here also, the group-theoretical method yields naturally all positive moments of the distribution functions.
Strict Localization in Quantum Field Theory
Authors: Knight James M.
Published: January 2005

A definition of strict localization of states in quantum field theory is presented. This definition is based on considering products of field operators as the primary measurable quantities of the theory. An example of a localized state is given, showing that such a state arises when a free field interacts with an external current that is limited to a bounded region of space-time. It is shown by means of a graphical technique that a state having a finite number of particles cannot satisfy the definition of localization. A simple representation of localized states is investigated, and arguments are given to support its generality and uniqueness.
Characteristic Coulomb Green's Function and Its Eigenfunction Expansion
Authors: Mapleton Robert A.
Published: January 2005

The eigenfunction form of the Green's function is derived from the characteristic form of the Green's function for the nonrelativistic Coulomb operator. This derivation shows how the wave boundary condition of the characteristic form of the Green's function is related to the branch cut in the continuous part of the spectrum. An application of the eigenfunction expansion form of this Green's function is discussed.
Wave Equations for Scalar and Vector Particles in Gravitational Fields
Authors: Hjalmars Stig
Published: January 2005

An earlier representation of the Kemmer wave equation in Riemann space is modified so as to remove the matrices, which in general have to be added to the differential operators of the equation. It is pointed out that the possibility of this removal is equivalent to the fact, known from Maxwell's equations, that if only scalars, vectors, and antisymmetrical tensors are involved, the field equations can be written with ordinary derivatives without explicit use of the affine connection. The wave equation is written in component form, and the photon zero mass case is obtained by means of the most general matrix mass term, without any questionable limiting process.
Eigenvalues and Eigenvectors of a Symmetric Matrix of 6j Symbols
Authors: Rose Morris E. , Yang C. N.
Published: January 2005

A real orthogonal symmetrical matrix M is defined. It represents the transformation between two coupling schemes for the addition of the angular momenta b, a, b to form a′. The eigenvectors and eigenvalues of M are found. A physical application is briefly discussed.
Traces of Products of Angular Momentum Matrices
Authors: Ambler E. , Eisenstein J. C. , Schooley J. F.
Published: January 2005

Closed formulas are given for evaluating Tr JapJbqJcr ⋯ where a, b, c, ⋯ are equal to x, y, or z and p, q, r, ⋯ are non-negative integers for which p+q+r+ ⋯ ≤ 10. All possible combinations of the angular momentum components for p+q+r+ ⋯ ≤ 9 are included. Numerical values of the traces are given for J = ½, 1, ⋯ 10. The procedures used in evaluating the traces are described.
Evolution of a Crystal in an Oscillating Field
Authors: Blum Lesser
Published: January 2005

Using Prigogine and Henin's method for the asymptotic solution of Liouville's equation, a master equation for the anharmonic homogeneous crystal is obtained; this equation is valid up to the order γ2λ2t, where γ, and λ are the coupling parameters.
Causality and the R Matrix
Authors: Ebel M. E.
Published: January 2005

A proposed counterexample to the theorem that causality and unitarity alone will guarantee that the R matrix has no singularities off the real energy axis is considered and shown to come from an acausal interaction. The nature of the acausality in this model and in some trivial generalizations is also discussed.
Small Angle X-Ray Scattering from Randomly Oriented Cylinders of Arbitrary Cross Section
Authors: Miller Akeley , Schmidt Paul W.
Published: January 2005

At the larger angles of the small-angle region, asymptotic expansions are the appropriate means of calculating the intensity of the small angle x-ray scattering from a sample of identical particles with uniform electron density and random position and orientation. Since other workers have previously shown that this intensity can be expressed as a Fourier integral transform involving a function characteristic of the size and shape of the particles, standard techniques for asymptotic expansion of Fourier integrals can be employed. In order to make a more general study than would be possible by consideration of particles with a single shape, right cylinders of arbitrary cross section have been chosen for this investigation. These generalized cylinders permit a fairly wide range of choice of particle shape and are well adapted to the study of both elongated and flattened particles. An integral has been developed which gives the characteristic function for the generalized cylinder in terms of the characteristic function of the two-dimensional cross section of the cylinder. As the asymptotic expressions for the scattered intensity depend quite critically on the discontinuities in derivatives of the characteristic function, a study has been made of these derivatives. An expression for the scattered intensity from rectangular parallepipeds has been calculated. A treatment of the limiting case of highly elongated cylinders is given.
An Approach to the Eulerian-Lagrangian Problem
Authors: Lumley John L.
Published: January 2005

The generalization is developed, for n functions of n variables, of the Kac-Rice theorem on the occurrence of zeros of a random function, and is applied to a particular kind of vector integral equation in a single independent variable (time) by discretizing the time axis. In this way the functional probability for the solution of the equation is obtained. This technique is applied to the stochastic integral equation representing the motion of a tagged point in a fluid in turbulent motion.
Non-Rectilinear Current Flow in a Straight Conductor
Authors: Pipkin A. C. , Rivlin R. S.
Published: January 2005

The flow of current in a long, straight, isotropic, conducting rod of uniform, noncircular cross section, due to a potential difference between its ends, is considered. The electric field in the material is assumed to depend linearly on the current density and to be quadratic in the magnetic field, while the magnetic permeability of the material is assumed constant. It is shown in the case of a rod of elliptical cross section that the current has a circulatory component superposed on the rectilinear flow.
Long-Time Behavior of a Random Lattice
Authors: Hori Jun-ichi
Published: January 2005

The time-dependent behavior of a linear isotopic random lattice is studied. The asymptotic average value of the solution of the equation of motion for the amplitude zk of a normal mode of the virtual regular lattice, composed of atoms with mass equal to the harmonic average of those of the isotopes, is calculated in the limit N → ∞, t → ∞, γn → 0, and γnt=a finite constant (n=1, 2,⋯). Here N is the number of normal modes, and γn the nth cumulant of the distribution of inverse mass. The initial conditions are zj(0)=δkj and (dzj/dt)t=0=0(j=−N/2+1, −N/2+2,⋯, N/2−1, N/2). The result comes out to be 〈zk〉=exp(−iαωk+ ∑ n=1∞γ2nQknqn−1/n!)t,where αωk≡2f12Hωk (f: force constant, H2: average inverse mass) is the eigenfrequency of the kth normal mode, Qk≡8iωkf2/α3, and q=t/N, assumed to be a constant. If q → 0, there remains only the first term, and the result coincides with that for the case of a Gaussian distribution.
Molecular Collisions. IV. Nearly Spherical Rigid Body Approximation
Authors: Gioumousis George , Curtiss C. F.
Published: January 2005

Rotational transitions in the collisions of diatomic molecules are treated by means of a rigid body approximation. The deviation from sphericity is assumed to be sufficiently small that it may be treated as a perturbation. The differential cross section is calculated for the H2-HD collision in which the hydrogen remains in its ground state while the hydrogen deuteride goes from its ground state to the first excited rotational state.
A Criterion for Singularities in Perturbation Theory
Authors: Halpern Francis R.
Published: January 2005

A criterion is proposed to distinguish between the singular and nonsingular portions of the Landau surface on the physical sheet. The set of diagrams under consideration are those containing a single loop. A proof of the Mandelstam representation for the four-point function is given based on this criterion.
Atomic Many-Body Problem. II. The Matrix Components of the Hamiltonian with Respect to Correlated Wave Functions
Authors: Szász Levente
Published: January 2005

The wave function for an atom with N electrons in arbitrary configuration will be written in the form Ψ=Ψ0+ ∑ i=1N ∑ j=i+1Nf(ij),where ψ0 is a Slater determinant and f(ij) is the antisymmetrized product of (N − 2) one-electron spin-orbitals and one 2-electron function Φ(ij/12). The correlation between the two spin-orbitals φi and φi can be taken into account by introducing r12 (the interelectronic distance) explicitly into the 2-electron function Φ. The purpose of the paper is to analyze the structure of the matrix components of the Hamiltonian with respect to the wave function given above. Starting from exact, general formulas for the matrix components it will be shown that, all integrals which occur in the diagonal, as well as in the nondiagonal matrix components can be reduced to six basic integrals which are 2- , 3- , and 4-electron integrals, containing interelectronic distances. It will be indicated that, five of the six basic integrals can be calculated in closed form whereas one of them, (an exchange integral) can be given only in the form of an infinite series.
The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics
Authors: Dyson Freeman J.
Published: January 2005

Using mathematical tools developed by Hermann Weyl, the Wigner classification of group-representations and co-representations is clarified and extended. The three types of representation, and the three types of co-representation, are shown to be directly related to the three types of division algebra with real coefficients, namely, the real numbers, complex numbers, and quaternions. The author's theory of matrix ensembles, in which again three possible types were found, is shown to be in exact correspondence with the Wigner classification of co-representations. In particular, it is proved that the most general kind of matrix ensemble, defined with a symmetry group which may be completely arbitrary, reduces to a direct product of independent irreducible ensembles each of which belongs to one of the three known types.
Numerical Estimation of the Partition Function in Quantum Statistics
Authors: Fosdick Lloyd D.
Published: January 2005

A method for estimating the partition function of a quantum mechanical system is described. The method is based on a technique for evaluating the Wiener integrals in terms of which the partition function may be expressed. This technique involves first, an approximation of the Wiener integral by an n-dimensional integral and, second, a Monte Carlo estimation of the value of the n-dimensional integral. Application of the method to a harmonic oscillator and a pair of interacting particles in a box in two dimensions is described.
Uniqueness and Existence of the Solution to the Static London-Maxwell Equations in Two Dimensions
Authors: Odeh Farouk
Published: January 2005

A theorem is given for the existence and uniqueness of the time-independent solution of the exterior-interior problem associated with determining the distribution of superconducting current, according to the London model, in simply- or doubly-connected two-dimensional regions. The proof of the corresponding theorem in three dimensions is outlined. A discussion is also given of the relationship between two ``different'' solutions which already exist for rectangular regions.
Relativistic Invariance and the Square-Root Klein-Gordon Equation
Authors: Sucher J.
Published: January 2005

Although the usual operator invariance requirements and corresponding commutation conditions encountered in the study of the invariance of relativistic wave equations (and other equations of physics) are sufficient conditions for invariance, they are by no means necessary. More general conditions are given and illustrated with the square-root Klein-Gordon equation. A new proof is thereby given of the Lorentz invariance of this equation. The methods developed are extended to cover the presence of external fields, and it is proved that the usual gauge invariant modification of the relativistic Hamiltonian of a spinless particle which takes into account the presence of an external electromagnetic field leads, in the quantum mechanical case, to an equation which does not admit the proper Lorentz group. This theorem and its generalization are discussed in connection with Dirac's statement that the square-root equation cannot be extended to include interaction without losing Lorentz invariance.
Expectation Value Formalism in Quantum Field Theory. I
Authors: Bakshi Pradip M. , Mahanthappa Kalyana T.
Published: January 2005

A formalism has been developed to evaluate the expectation values of projection operators directly. The formalism is applied to the production of particles when the corresponding quantized fields are coupled to external sources and external electromagnetic fields.
A Summation Procedure for Certain Feynman Integrals
Authors: Babbitt Donald G.
Published: January 2005

A mathematically rigorous summation procedure is developed for Feynman integrals involving a repulsive potential. The central idea of the proof is to connect certain Wiener integrals with the Feynman integral in question by making use of holomorphic vector-valued functions and holomorphic semigroups.
Low-Energy Expansion of Scattering Phase Shifts for Long-Range Potentials
Authors: Levy Bertram R. , Keller Joseph B.
Published: January 2005

Phase shifts can be used to describe the quantum mechanical scattering of a particle by a spherically symmetric potential. They are odd functions of the wave number k, which is proportional to the square root of the energy of the incident particle. For short-range potentials, they are analytic at k = 0 and can be expanded in odd powers of k. However for long-range potentials they are not analytic at k = 0 so they cannot be expanded in powers of k. Since electron-atom, atom-atom, proton-neutron, and multipole-multipole potentials are of long range, it is important to consider such potentials. A method is presented for determining the appropriate expansions around k = 0. The method is applied to potentials which are O(r−v) as r → ∞, and the phase shifts for any angular momentum are obtained up to and including the first nonanalytic term. For L = 0 and 3 < v < 4 the next term is also obtained. The nonanalytic terms involve ln k and fractional powers of k. The results for v = 4 and the k dependence for v = 2L + 3 agree with those obtained in a different way by O'Malley, Spruch, and Rosenberg.

The method involves a function η(r) whose asymptotic value η(∞) is the desired phase shift. A nonlinear first-order ordinary differential equation is derived for η(r). This equation is solved by expansion and iteration methods. To expand the solution around k = 0, two simple theorems are proved concerning the asymptotic forms of certain integrals containing a parameter.
Reduced Width Amplitude Distributions and Random Sign Rules in R-Matrix Theory
Authors: Krieger T. J. , Porter C. E.
Published: January 2005

A multivariate reduced width amplitude distribution is derived from quite general assumptions of level independence and of functional form invariance of the distribution. The multivariate distribution reduces to the well-known Gaussian singlet distribution (one degree of freedom) and moreover allows for the possibility of channel-channel reduced width amplitude correlations. Such correlations, which cannot be represented by a singlet distribution, may be especially relevant to partial fission reduced width amplitudes. In particular, it is noted that in statistical theory complete correlation or anticorrelation between two random variables implies a linear relation between them. The multivariate distribution is then used as a basis for a more precise statement of the ``random sign'' rules of R-matrix theory than hitherto given. It is pointed out that these rules imply a linear averaging over the multivariate reduced width amplitude distribution (in addition to the usual linear energy average) and are not merely consequences of the equiprobability of positive and negative signs for the reduced width amplitudes. The influence of the energy eigenvalue distribution and of variable reduced width amplitude statistics is discussed. It is emphasized that the multivariate reduced width amplitude distribution is relevant to any reaction theory in which level widths are defined even though it is phrased here in the language of R-matrix theory.
Convergence of Fugacity Expansions for Fluids and Lattice Gases
Authors: Penrose Oliver
Published: January 2005

Upper and lower bounds are obtained for R(V), the radius of convergence of the Mayer expansion VΣl bl(V)zl expressing the logarithm of the classical grand partition function for a finite volume V as a power series in the fugacity z. The particles in V interact only through two-body forces whose potential φ(r) satisfies s−1 Σi<j≤s φ(xi − xj) ≥ const ≡ −Φ for all s, x1 ⋯ xs. The bounds are [e1+2Φ/kT ∫ |e−φ(r)/kT−1| d3r]−1≤R(V)≤|eΦ/kTl/(l−1) bl(V)|1/(l−1)for any l ≥ 2. For lattice gases the integral becomes a sum. The upper bounds, obtained from the theory of entire functions, include a subsequence converging to R(V) as l → ∞. The lower bound is obtained by using the Kirkwood-Salsburg integral equation to calculate upper bounds on the bl(V)'s and the coefficients in the fugacity expansions of the s-particle distribution functions. For hard-core potentials some of these bounds can be strengthened. For nonnegative potentials, 1/2|b2(V)| is an extra upper bound on R(V). The radius of convergence of the infinite-volume series Σ blzl is shown to be at least limV→∞ R(V), with equality for nonnegative potentials.
Existence and Uniqueness Theorems for the Neutron Transport Equation
Authors: Case K. M. , Zweifel P. F.
Published: January 2005

In an attempt to understand the conditions under which the neutron transport equation has solutions, and the properties of those solutions, a number of existence and uniqueness theorems are proved. One finds that the properties of the solution are closely related to the boundedness of the source as well as to certain velocity-space integrals of the scattering kernel. Both time-dependent and time-independent equations are considered as are also the time-dependent and time-independent adjoint equations. Although only a very few of all possible existence and uniqueness theorems for these equations are considered here, the work may serve as a guide to the treatment of similar problems.