Table of contents

Relativistic transformation formulae for thermodynamic quantities are obtained here without having recourse to integrations in the phase space. The controversial role of the Jacobians of transformation is thereby completely eliminated. Throughout the discussion we talk in terms of the distribution of particles over discrete quantum states, nevertheless, paying due regard to the relativity of temporal measurements (which enter into the time-averaging of every statistical quantity), we again obtain the old Planck-Einstein formulae.

The correspondence between the physical features of this approach and the one followed in an earlier paper on this subject has been elucidated.

Algebraic expressions are found for the density of electron eigenstates, degree of localization and probability distribution of the phase in a liquid chain of δ-function potential wells of strength -2κ₀(h²/2m), for positive electron energies h²κ²/2m. A `diffusion approximation' is used, in which the probability distribution of the spacings enters only through the mean u and the standard deviation σu. The approximation is valid provided mainly that the wells are weak (κ<sub>0</sub>/κ << 1) and the disorder small (κuσ << 1). Excellent agreement is obtained with the numerical results of Borland and Bird. The energy gap, when still present, is `eroded' by a fraction of order (σ²κ³u²/κ₀)^2/3. This `gap' is really a `quasi-gap', in which the density of states falls off exponentially, its dominant factor being exp{-θ|κ-κ_c|^3/2/κ^3/2κ₀^1/2σ²u^1/2 where 4/3 × 2^1/2 <or= θ <or= 2 and κ_c is the nearer edge of the gap for σ = 0.

A restriction on the choice of translation-invariant coordinates for a system of N identical fermions, interacting by pair forces, allows a lower bound to be established on the ground-state binding energy. This bound is the sum of the first N-1 eigenvalues of a two-particle problem. The quality of the method is found to depend on the choice of coordinates consistent with the initial restraint, the optimum set giving a bound which, in the case of Hooke's law pair forces, is [radical]3(N-1)/2(N+1) times the exact energy.

The Rainich conditions for electrovac space-times are derived using Rumer's form of Maxwell's equations. The analysis is based on the 2j + 1 representations of the Lorentz group directly, rather than on their tensor or two-spinor equivalents.

Lipkin's Zilch for a massless spin 2 particle is extended to generally covariant form and is shown to be covariantly conserved.

The work of Mittleman and Myerscough is extended to excited states of a finite dipole, thus giving values of the dipole moment required to bind an electron in a state (n_λ, n_μ, m) of a finite dipole.

An equation is derived for the first of the half-order terms, u_1/2, in the Z^-1/2 expansion of the orbital wave functions for the (1s)(1ś) ¹S ground state of helium and helium-like ions This equation is solved by expanding u_1/2 in a series of generalized Laguerre polynomials, and two cases are considered, in which 12 and 30 terms respectively are taken in the series. The effect of u_1/2 on the second- and third-order energy coefficients E₂ and E₃ is next studied. It is shown that the inclusion of u_1/2 accounts for 92% of the Z-independent term in the series expansion of the radial correlation energy, and that considerable improvement is made on the restricted Hartree-Fock value of E₃.

Relativistic Hartree-Fock self-consistent fields have been obtained for a number of atoms by solving numerically the equations given by Grant, but with the magnetic interaction terms and off-diagonal Lagrange parameters omitted. The results are discussed briefly, with those for Hg being considered in more detail. The off-diagonal parameters are included in some simple cases and their effect shown to be relatively small. The binding energies for several inner atomic electrons are calculated and compared with experiment.

Two theorems are proved which are of considerable use in understanding the situation which arises while applying the variation perturbation method for calculating the state vectors and the energies of stationary quantum systems. These theorems also enable us to ensure that the second-order energy calculated by the use of the method is a rigorous upper bound to the true value The earlier results of Sinanoulu and Miller follow as immediate corollaries of these theorems.

The perturbed stationary-state approximation is used to calculate the cross section for charge transfer between H⁺ ions and atomic hydrogen at ion energies between 1 eV and 10 keV, with the proper inclusion of exchange effects due to the identity of the protons. The resulting values diverge at low energies from previous theoretical calculations, and lie above the experimental measurements.

The threshold polarizations of the resonance lines of ⁶Li, ⁷Li and ²³Na, excited by electron impact, are calculated using the theory of Percival and Seaton. The results obtained are in good agreement with measured values of Hafner, Kleinpoppen and Krüger.

Electron impact ionization of N₂ below 70 eV has been studied under conditions favourable to the trapping of ions in the electron beam. The ionization of N₂ is observed to produce N₂⁺, N₂⁺(*), N₂²⁺, N⁺ + N(⁴S^o), N⁺ + N(²P^o), (N⁺ + N?), N²⁺ + N and (N²⁺ + N?), where product pairs in parentheses have excess energy distributed in an undetermined manner Electron collisions with N₂⁺ give N⁺, N₂²⁺ and N²⁺ + N, and collisions with N₂²⁺ give N²⁺ + N.

The problem of determining appearance potentials under these conditions is discussed, and a new method is presented and applied to the above processes. Comparisons with known or expected appearance potentials show that the error in determining thresholds is 0 37 ± 0 23 eV.

The differential incoherent scattering cross sections of 320 keV photons by K-shell electrons of lead, tantalum and samarium are experimentally determined employing a fast-slow coincidence scintillation spectrometer by the degraded photon K X-ray coincidence technique at angles of scattering 45°, 60°, 90° and 110°. The results are compared with the computed values and the trends of the incoherent scattering function are discussed.

Measurements of neutrons scattered inelastically from vanadium have been made and absolute cross sections determined over a range of energy and momentum transfers of 0-50 meV and 1-12 Å^-1 respectively. A frequency distribution of the normal modes of vibration in vanadium is evaluated from these data Peaks are observed at energy transfers of 8, 20 5, 25 5 and 28 5 meV (equal to 1 9, 4 9, 6 1 and 6 9 × 10^-12 sec^-1). Neutron spectra have been calculated using this distribution and good agreement obtained with experimental data. A comparison is made with the frequency distributions of niobium and tantalum.

Paramagnetic neutron scattering at zero temperature in a one-band itinerant electron model of a metal is discussed in terms of the random-phase approximation formula of Izuyama, Kim and Kubo for the q, ω dependent generalized susceptibility of an electron gas. The formula is evaluated numerically in the effective mass approximation, and qualitative comparison made with simple sum rules for a localized spin paramagnet. The formula is also applied to predict neutron scattering cross sections at fixed scattering angle. The itinerant character produces a reduction in the scattering intensity additional to that produced by the atomic form factor. The use of neutron scattering in the determination of the unrenormalized band mass of the d-band holes in palladium is discussed

Comparison is made with experiments of Cable and Wollan for scattering on palladium. It is suggested that the paramagnetic scattering should be observable at sufficiently small angles.

In this paper we consider the two-magnon interaction of neutrons in magnetically coupled systems, namely ferromagnetic and ferrimagnetic crystals. The interaction mechanism is relevant to magnetic systems having magnetic atoms, the states of which allow orbital transitions due to interaction with neutrons. The formulation of the Heisenberg-type exchange interaction in terms of atomic states, which are perturbed owing to the scattering with neutrons, gives rise to the interaction terms. In effect they represent two-magnon Raman processes along with the scattering of neutrons. The calculations are carried out for ferromagnetic and ferrimagnetic systems The scattering cross sections are evaluated in terms of the matrix elements of the neutron-magnon interaction between the combined neutron-magnon states. The final expressions for the scattering cross sections, evaluated in the long-magnon-wavelength and low-temperature approximation, contain temperature T in different powers.

The details of a similar calculation applicable to a simple antiferromagnet will be reported in a separate paper.

Spin-lattice relaxation time (T₁) measurements were carried out at 35 Gc/s on the ruby samples used previously by Standley and Vaughan at 9 3 Gc/s. Values of T₁ of the order of 20 msec were found at 4 2 °k. The results are compared with theoretical predictions of Donoho. Vapour-phase grown rubies show a greater angular variation of T₁ than Verneuil or fluxed-melt grown samples but magnitudes are only about one-quarter of those predicted and the observed angular variation is less than that calculated. This is in contrast with the 9 3 Gc/s results and the concentration independence observed at that frequency is now less obvious. It is shown that harmonic cross relaxation can play an important role at the higher frequencies in determining the absolute magnitudes of the measured relaxation parameter and its angular variation.

A calculation of the interaction of an electric field with the trivalent praseodymium ion at an yttrium site in yttrium ethyl sulphate is attempted and is found to be consistent with the results of a paraelectric resonance experiment. Some implications of the effect for pairs of interacting ions are discussed.

A detailed study of the nuclear magnetic resonance spectra and the proton spin-lattice relaxation times in solid 2,2- and 2,3-dimethylbutane have been made from 20 °K to their melting points. The interpretation of the data is consistent with thermal, x-ray and Raman studies. At low temperatures the spectra are narrower than expected for an effectively rigid lattice, and it is concluded that the tunnelling by the CH₃ rotors narrows the observed spectra. The proton spin-lattice relaxation mechanism is not sensitive to this motion.

The nuclear quadrupole relaxation time in liquid metals is calculated, assuming free-ion cores interacting by an oscillatory screened potential, and whose positions in the liquid are described by the time-dependent pair distribution function of Oppenheim and Bloom. The relaxation time depends on two- and three-particle correlation functions, and the three-particle correlations are treated by means of the superposition approximation. Uncertainties in the antishielding factor and the potential oscillations are eliminated by using the experimental value of the quadrupole coupling constant in the solid.

Detailed calculations are performed for Ga and In, and good agreement with the observed relaxation time is obtained if the three-particle terms are neglected. The calculated three-particle terms are not negligible, and it is suggested that the superposition approximation introduces serious errors.

Measurements of the critical surface currents above H_c2 in a rectangular sample of a Ta-Nb alloy have been made as a function of the magnitude of the applied magnetic field and its orientation with respect to the main surface plane. The results are similar to the corresponding data of Swartz and Hart; but the field dependence is different from, and the magnitudes are smaller than predicted by, the calculations of Abrikosov and Park. By a simple analysis of the results, the angular dependence of the surface nucleation field has been derived and is also compared with the theory. The reasons underlying the discrepancies between theory and experiment are discussed Surface roughness is a factor having appreciable influence on the observed critical currents.

High-frequency photoconductive phase responses in single-crystal n-type indium arsenide with net carrier concentrations in the range 1 × 10¹⁶ to 5 × 10¹⁶ electrons/cm³ have been studied. Carrier lifetimes have been derived from these responses and compared with those consistent with a direct interband Auger process of recombination.

Galvanomagnetic measurements as a function of temperature between 77 °K and 273 °K in single crystals of antimony are presented. Results are interpreted in terms of the two-carrier multi-valley band model to obtain carrier mobilities, densities and details of the nature of the Fermi surface. The temperature dependences of the mobilities of holes and electrons are found to be T^{-1 48} and T^{-1 42} respectively. Contrary to the usual assumption, the electrons are sited in pockets with a small angle of tilt (5°) away from the trigonal axis, while the extrema containing holes have a large tilt angle (24°). Carrier densities are almost independent of temperature, ranging from 3 9 × 10¹⁹ cm^-3 at 77 °K to 4 2 × 10¹⁹ cm^-3 at 273 °K for both electrons and holes. If, as recent work indicates, there are three pockets of electrons and six of holes, then at 273 °K each electron ellipsoid contains 1 4 × 10¹⁹ electrons/cm³ and each hole pocket 0 7 × 10¹⁹ holes/cm³.

A contribution to the electronic specific heats of alloys due to the simultaneous presence of the electron-phonon and electron-impurity interactions is theoretically investigated by means of the perturbation method. The electron-phonon interaction modifies the energy spectrum of an electron and the electron-impurity interaction gives rise to the finite lifetime on this quasi-particle. Because of the level broadening due to the uncertainty relation, all densities of states in the neighbourhood of the Fermi level, which are formed from the real part of the self-energy of the system, contribute to the specific heat. It follows that the enhancement of specific heats due to the electron-phonon interaction tends to decrease as the concentration of impurity atoms is increased, and becomes only 10% of that for a pure metal when the broadenings of levels lying near the Fermi level reach about four times k_BΘ_D, where k_B and Θ_D are Boltzmann's constant and the Debye temperature, respectively. On the basis of this theory, the recent data of the silver-gold alloy, which show an interesting behaviour with a minimum value for about 15 at.% Au in Ag, are analysed. It is found that the enhancement of specific heats for pure silver due to the electron-phonon interaction is about 7% of the ordinary contribution from the density of states formed by the band structure.

For the spin paramagnetism in alloys it is shown that there is no contribution from the electron-phonon interaction and the electron-impurity interaction has an effect resulting from the level broadening.

The Green function is derived for a small concentration of fermions in a weakly interacting Bose gas using mass operator perturbation theory. For small momentum it is found that the spectral weight function of the system consists of a quasi-particle peak associated with an effective mass m* as well as some weight at higher energies The momentum dependence of both parts of the spectral weight function is determined. The implications of the weight at higher energies in the case of dilute ³He-⁴He mixtures are discussed.

A simple shell model theory is developed in terms of the double-time Green function method to calculate the one-phonon infra-red absorption in a monatomic crystal with a small concentration of uncharged point defects. The theory allows for a defect change in atomic mass, change in core and shell charge, and change in internal core-shell force constant. The frequency-dependent absorption is found in terms of a one-phonon density of states; the integrated optical absorption is expressed in a closed form similar to that found by Doniach and Huggins. Fitting the theoretical parameters to data for krypton and xenon defects in solid argon results in a theoretical absorption of the order of magnitude found experimentally by Jones and Woodfine. Anharmonic effects in the experiments are thought to account for the principal discrepancies between theory and experiment.

The shear viscosity of insulating crystals is estimated using a recently derived relaxation time expression. The relaxation time per mode, identified with the inverse of the one-phonon half-width, is calculated for a simplified model of a face-centred cubic lattice with nearest-neighbour interactions. `The shear viscosity is found to be of the order of 10^-5 poise for rare-gas crystals.

As a first stage in the investigation of suitable potentials for the noble metals, the energy bands and Fermi surface of copper have been calculated by the Green function method. The potentials of Herman and Skillman used by many workers are found to be unsuitable; the d bands are too narrow and too low and the Fermi surface does not contact the boundary of the Brillouin zone. Attempts to modify the potential between the muffin-tin spheres failed to improve the results. The amount of exchange energy included in a Gáspár-type potential is found to affect the band structure considerably. Satisfactory agreement with experimental and other theoretical results is nevertheless obtained if the exchange is reduced to about 30-40% of the amount proposed by Gáspár.

It is shown that by means of a projection operator a modified Green operator may be defined for periodic potentials. The operator obeys a modified Dyson equation and is related by a simple expression to the Green operator of the periodic lattice. An application of the modified Green operator to the problem of determining the energy levels of an impurity in an otherwise periodic lattice is given.

The Goldstone theorem requires that a many-body system with broken symmetry has an excitation branch, whose frequency tends to zero in the limit of infinite wavelength. We treat a system where the broken symmetry comes from the terms which give rise to the Jahn-Teller effect. Both the excitation branches we discuss in detail have finite frequencies at infinite wavelength when there is no Jahn-Teller term; the introduction of this term forces one branch to have zero frequency at infinite wavelength, in agreement with the Goldstone theorem The main point of this paper is this striking illustration of Goldstone's conjecture. Some of the simpler features of the excitation branches are discussed, they do not appear to have been treated in detail in the literature. Systems of ions in twofold degenerate E ground states may exhibit such excitations, which will have a characteristic velocity considerably less than that of sound.

In this paper the properties of athermal monomer-dimer liquid mixtures are considered on the basis of the quasi-crystalline model, the lattice considered being the face-centred cubic. The results obtained are compared with those based on the Miller-Guggenheim approximation by considering the vapour pressure curves and it is shown that the present results are an improvement on these previous treatments.

The study of the equation of state of caesium bromide is based on the method adopted by Verma and Dayal for sodium chloride. It was observed that, corresponding to the volume ratio v_T/v₀ >or= 1.18 for some wave vectors, some of the frequencies obtained by solving the secular determinant were imaginary. As this ratio was smaller than the maximum volume ratio 1.27 for caesium bromide at the melting point, it implied that the usual form of the potential energy function leads to the instability of the lattice before it melts. All other available forms of the potential energy functions were also tried but did not improve the results. It was found that real frequencies could be obtained by augmenting Born and Mayer's potential by a term which is three times larger than the normal van der Waals and dipole-quadrupole terms.

The new potential energy function has been used further for evaluating the following four quantities. cohesive energy, melting point and the temperature variations of thermal expansion and compressibility. The agreement with experiment is good for the first and reasonable for the other quantities.

It is found that one of the six third-order constants of cubic crystals of sufficiently high symmetry and two combinations of the remaining five constants relate to strains in which the volume is unchanged. These shear constants are 1/8(C₁₁₁-3C₁₁₂+2C₁₂₃), C₄₅₆ and ½(C₁₄₄-C₁₅₅). The electrostatic and repulsive ion-ion contributions to these constants are obtained in general terms for face-centred cubic and body-centred cubic metals. The electrostatic contributions are positive for face-centred cubic metals and negative for body-centred cubic metals. The repulsive contribution to C₄₅₆ is relatively insignificant in face-centred cubic metals whereas it predominates in body-centred cubic metals.

These two contributions are not able to account for the observed third-order shear constants of noble metals, particularly C₄₅₆. It is shown that it is necessary to include the energy of the conduction electrons. Detailed calculations cannot be made, however, until the third-order constants have been determined at low temperatures.

It is shown that the diffracted field in certain parts of both the near and far field of a slit aperture formed by two non-planar screens satisfy the variational principle. Microwave measurements of the axial electric field intensity for a slit aperture with various angles of inclination of the non-planar screens are presented, and compared with the results of the variational principle, with Kirchhoff's approximation theory and with the results of an asymptotic solution of the integral equation.

The numerical evaluation for the van der Waals dipole-dipole interaction energy, using the recent data for the van der Waals coefficients, has been made for all the alkali halides The results are considerably different from those of earlier workers.

It is shown that the ionic radu due to Pauling and those derived by Fumi and Tosi could be different measures obtained from the same electron distributions rather than similar measures of different distributions, as suggested by Fumi and Tosi.

The rotational analysis of a new band at 7572 Å in the high-temperature thermal emission spectrum obtained with platinum in a King furnace has been carried out. The agreement of the ground-state combination differences with those obtained from the A ¹II, B ¹Σ⁺-X ¹Σ⁺ systems show that the band is the 0-0 band of a transition A' ¹II-X ¹Σ⁺ in PtC. Constants for A' (¹⁹⁵PtC) are as follows: T₀₀ = 13195.67 cm^-1, B₀ = 0.5038 ± 0.0002 cm^-1D₀ = (0.55 ± 0.03) × 10^-6 cm^-1, r₀ = 1.721 Å.

The infra-red absorption frequency has been computed using a simple and direct approach which connects it with other lattice properties. The results are, in general, in reasonable agreement with the experimental data that exist and also with the values obtained from the dispersion curves of frequency against wave vector.