Table of contents

Angular momentum operators transforming in the same way as spherical tensors are listed up to sixth order, and a general expression for their matrix elements is given.

Potential energy integrals between overlapping Gaussian charge distributions, with two-body potentials involving inverse integral powers of the interparticle distance, are reduced to computational form. All angular dependences are expressed in terms of spherical harmonics, and the analysis allows for the introduction of two-body correlations between all the particle pairs in a system. The formulae are applicable to both molecular and solid systems in which Gaussian wave functions are used.

An atomic integral involving three odd powers of the interelectronic separations, which occur in the variational solution of the atomic many-body problem, is reduced to an infinite series of one-dimensional integrals. Convergence of this series is discussed.

The impact parameter version of Born's approximation is used to calculate cross sections for proton impact excitation of the n¹D states of He (n = 3, 4) from 1 ¹S and 2 ¹P. The results for 1 ¹S -> 4 ¹D are in satisfactory accord with experiment.

A general formula has been developed for the fluctuations in the X-ray absorption coefficients observed on the high-energy side of an absorption edge, on the assumption that the ejected photoelectron undergoes inelastic collisions in the vicinity of the absorbing atom only. For the intermediate energy range 50-250 eV the directional dependence of the scattering event has been taken into account by replacing the atoms at a given distance from the absorbing atom by a continuous distribution on a spherical shell of the same radius. The positions of the absorption maxima and minima have been calculated for copper and compared with experiment. Reasonable agreement has been found between the experimental results and the present calculations.

On the basis of Lewis's theory of multiple scattering in an infinite medium, relatively simple expressions are obtained for the first and second moments of the electron distribution in depth, and for the second moment of the transverse distribution, as functions of atomic number and incident electron energy. The results are compared with some Monte Carlo calculations due to Bishop. These moments are then used to determine semi-empirical expressions for the distribution functions. These results are compared with the Monte Carlo distributions for copper, in which instance good agreement is found. Some discussion is given of the applicability of these results, and possible modifications, to the case of the semi-infinite medium.

The purpose of this paper is to make a study of the circular polarization of the bremsstrahlung emitted in the collision of a high-energy neutrino with a nucleon.

The energy distribution of neutrons scattered from polycrystalline RbMnF₃ at room temperature (T/T_N = 2.8) has been measured using time-of-flight analysis. Evidence is obtained for the existence of a cut-off in the energy distribution at large scattering vectors. The nearest-neighbour exchange interaction 2J₁(S_i. S_i) is derived both by direct evaluation of the second moment of the energy distribution and from the best-fitting Gaussian. The results suggest J₁/k = 3.3 ± 0.3 °K, in good agreement with other determinations. A general calculation is made of the effect of biquadratic exchange on the second moment. It is shown that this is unlikely to affect the above result.

The liquid systems H₂ in argon and H₂ in neon have been investigated by neutron scattering. Quasi-elastic and inelastic spectra have been clearly observed, the inelastic spectra being more intense than in pure argon or neon.

A diffusion constant and a frequency of vibration for each case have been extracted and compared with a calculation by Sears. The experimental and calculated vibrational spectra are in fair but not close agreement. However, the diffusion constant for hydrogen in argon is greater than that for pure argon while for hydrogen in neon it is probably smaller than for pure neon.

A simple approximation is described which may be applied to Ising and classical Heisenberg ferromagnetic systems containing forces of arbitrary range. (In this paper this approximation is loosely referred to as an `extension' of the usual BPW (Bethe-Peierls-Weiss) approximation although this is not strictly true.) In the case of the S = ½ Ising model in the absence of an external field, it is shown that the present results reduce correctly to the first two terms (i.e. the `point' and `bond') of the linked cluster expansion developed by Hiley and Joyce. Specific calculations for the second-neighbour problem (S = ½ Ising model) are compared with results obtained using exact series expansions, and a formal similarity (within this approximation) between the S = ½ Ising and the classical Heisenberg models is pointed out.

The aim of this series of papers is to help bridge the gap in our knowledge of the properties of the Ising and Heisenberg models between very short-range forces corresponding to nearest-neighbour interactions, and very long-range forces corresponding to the mean-field approximation. The present paper is concerned with the equivalent neighbour model in which the interactions between a spin and a certain finite number of its neighbours are equal, and the remaining interactions are all zero. Methods derived previously for obtaining series expansions at high temperatures are generalized to include second- and third-neighbour shells for standard two- and three-dimensional lattices. It is found that as the co-ordination number q increases the properties of the model approach an asymptotic value which depends on dimension, and on q in a given dimension, but not significantly on the type of lattice structure. An estimate is made of this asymptotic behaviour as a function of q.

A theoretical justification is sought for the asymptotic behaviour of the equivalent neighbour model described in the previous paper. An approximation to the lattice constants for this model is suggested in terms of the virial integrals of a hard-sphere gas. For large co-ordination numbers it is found that the dominant terms in series expansion coefficients correspond to simple polygons for which virial integrals can be evaluated exactly to any order. If we take account of these polygonal terms, satisfactory agreement is obtained with the empirical results of the previous paper.

In two previous papers the distribution of magnetization about a non-magnetic impurity in a Heisenberg ferromagnetic or antiferromagnetic host was considered in the molecular field approximation. The present paper is concerned with a magnetic impurity within essentially the same theory. The magnetic moments associated with the impurity and its first neighbours are calculated as functions of temperature and impurity-host exchange coupling for arbitrary impurity and host spins. Numerical results are presented for several systems. The moments calculated for a simple cubic lattice with a spin ½ impurity and host are compared with the corresponding results from a Green function calculation by Wolfram and Hall. The spatial extent of the moment disturbance on the host is investigated. The elastic diffuse neutron cross section for the system is fully discussed as a function of impurity-host exchange coupling, reduced temperature and scattering vector.

A number of spin resonance and zero phonon optical absorption lines are broadened by microscopic strains in the lattice containing the centres observed. In MgO, for example, the electron spin resonance line of Fe²⁺, the ENDOR line of Co²⁺ and the zero phonon lines of colour centres all show strain broadening. The strains appear to be due to dislocations and other crystal imperfections. It is shown that these line shapes can be calculated in terms of the statistical distribution of the imperfections and their individual strain fields. The approach can be generalized to cases where other broadening mechanisms are important, such as spin-spin interactions. The method may also be used to obtain information about the statistical distribution of the strain sources from the observed line shape. To illustrate the method the line shape of MgO: Fe²⁺ is calculated on the simple, if rather unrealistic, assumption of an isotropic homogeneous distribution of dislocations. The results are in fair agreement with experiment.

The nuclear magnetic resonance absorption spectrum of solid chloropentafluorobenzene has been studied at temperatures between 77 °K and the melting point. The experimental values of the second moment have been divided into two parts, one due to broadening by nuclear dipole-dipole interactions and the other due to broadening by the anisotropy of the fluorine chemical shift. The absorption line begins to narrow at 190 °K, close to the temperature of a specific heat anomaly. Consideration of the effects of molecular motion on both parts of the second moment show that the narrowing is due to rotation about an axis perpendicular to the plane of the molecule.

A variational approximation is obtained for the resistance (or total entropy production) of a circular orifice in a plane diaphragm at which the scattering is specular. It is found that the Ohm's law prediction must be multiplied by a correction factor of the order of unity and added to the `small orifice' Knudsen contribution. The results for the intermediate régime can be applied to the transport of electricity and heat by electrons and to phonons which are dominated by elastic scattering (e.g. by crystal imperfections). Results are also obtained for the case when the diaphragm separates materials of different mean free path. It is shown that the above problems are equivalent to the determination of the spreading resistance of a small disk-shaped electrode. The mathematical method is based on the formulation of an integral equation over the region of the disk with a kernel containing the Green function for the infinite medium Boltzmann equation.

The fine structure of the optical and microwave spectra for ruby is analysed in terms of trigonal crystal-field parameters, using the complete 120 × 120 d³ crystal-field matrix. It is shown that the crystal-field model is rather more successful than earlier work suggests.

A method of estimating the covalent contribution to the trigonal field parameters in ruby is presented. This avoids the very complicated task of evaluating the matrix elements directly. The method assumes that the cubic field can be accounted for by invoking covalency, as has been demonstrated for the simpler case of KN1F₃. Since the covalent parts of both trigonal and cubic fields originate from the same class of perturbations, a covalent contribution to the trigonal field can be deduced.

A covalent orbital bond model is used to calculate the overlap energy of an interstitial lithium positive ion in a crystal, and applied to silicon. An estimate of the polarization energy is also made. From these it is possible to derive force constants needed in studying the vibration of Li⁺ impurities in Si.

The dynamical equations describing local modes of interstitial Li⁺ in silicon are derived, using the force constants derived in the previous paper. They lead to a relationship between the isotopic mass of the interstitial and the frequency of the local mode, which is solved numerically. Local modes occur only if the interstitial mass is below a critical value. The deviation of the local mode frequency from the top of the silicon vibration spectrum is proportional to the square of the deviation of the mass from the critical value, for small deviations. The critical mass is calculated from the model to be 12 A.U. Experimental data on boron-doped, lithium-compensated silicon suggest 8 A.U.

Results are reported of calculations of the Lennard-Jones m-6 intermolecular potential parameters of solid neon and argon, including cubic and quartic contributions to the lattice free energy at 0 °K. The treatment of the cubic and quartic terms is due to Feldman and Horton, using a first-neighbour approach which is an extension of the procedure introduced by Flinn and Maradudin. This approach is used in conjunction with first-, second- and all-neighbour treatments of the static lattice and zero-point contributions to the free energy at 0 °K. The parameters are determined from the experimental sublimation energy and lattice spacing at T = 0 °K, by the procedure used by Horton and Leech and by Brown, who reported calculations on the potential parameters for the same potential without including the effects of the cubic and quartic terms. A comparison of the present parameters with those reported earlier is made for solid neon and argon for a number of models based on first-, second- and all-neighbour theories, with m = 11, 12 and 13. For neon the values of are typically increased by about 4%, and σ by about 0 5%, for argon is increased by about 0 4% and σ by less than 0 05%. Similar calculations for solid krypton and xenon show negligible differences from the earlier results, and are not included. A discussion is given of the use of the present parameters in the calculation of the equivalent Debye θ_o^c, and a brief comparison with earlier calculations is made.

Calculations are presented of the relative volume differences and sublimation energy differences of solid neon and argon isotopic lattices, assuming isotopic invariance of the intermolecular forces, which are chosen to be of the Lennard-Jones type. The calculations, performed at 0 °K on ²⁰Ne and ²²Ne, and ³⁶Ar and ⁴⁰Ar, make use of the anharmonic potential parameters presented by Brown in 1966. It is found that the relative volume and sublimation energy differences in solid neon and argon are well accounted for by the inclusion of cubic and quartic anharmonic terms at 0 °K. Comparison of our calculated sublimation energy differences with experiment is not possible owing to the lack of any measurements, and the only measured volume difference which is available is that of solid neon. It is suggested that the accuracy of the measurements has advanced to a point where non-adiabatic contributions may contribute slightly.

As the numerical value of the force between a representative pair of molecules decreases, the numerical value of the radial distortion u(r) of the equilibrium distribution function should also decrease, since the distortion is zero for a flowing gas. Application of the `strong' boundary condition of Eisenschitz (r<sup>3</sup>u(r) -> 0 as r -> [infinity]) to the equation for u(r) results in solutions which meet this requirement. In this paper it is demonstrated that the solutions obtained using the `weak' boundary condition of Kirkwood et al (r² e^-ψdu/dr -> 0 as r -> 0) do not satisfy the requirement. This suggests that the appropriate u(r) is not determined by the weak boundary condition.

It is generally accepted that the equations of thermodynamics have survived the discovery not only of quantum mechanics, but also of relativity theory. In this paper, however, departures are proposed in order to allow for a velocity w of the centre of mass of a system through the inertial frame of the observer. The earlier authors, for example Planck, Einstein, Pauli and Tolman, used conventional thermodynamics to describe such systems. If U is the internal energy, E the total energy of the system, it is here proposed that TdS = dU + pdV - μdN be amended to TdS = dU + β_wpdV - μdN (βw equivalent to (1-w²/c²)^-1/2) and dU = β_w(dE - w.dP) where P is the total linear momentum of the system. As a result many thermodynamic relations undergo a modification, for example (δU/δV)_N.S = -β_wp, (δS/δV)_T = β_w(δp/δT)_v. The latter is an amended Maxwell relation.

The results are justified by an argument based on a maximum entropy distribution. It follows that the thermodynamic variables T and U are Lorentz invariant. This result differs from previous ones, and its relation with some of the earlier work is discussed.

Analyses of the level-crossing effect have usually been given for broad-band excitation of resonance fluorescence. The fluorescence excited by monochromatic light from an assembly of static atoms would be expected to show a modified form of level-crossing effect, but, when motion of the atoms is taken into account, the same result is obtained as for broad-band excitation.

Intensity distributions in progressions in non-totally symmetric vibrations are discussed. It is shown that these distributions can be expressed in concise form as the terms in certain simple series of the form (1 - x)^-n/2, where n is a positive integer, x = {(ν" - ν')/(ν" + ν')}² and ν', ν" are the vibration frequencies in upper and lower states respectively. For the progressions v^l <- l^l v = l, l+2, l+4,... in electronically allowed systems, the intensity of the band v^l <- l^l relative to that of the band l^l <- l^l is given by the {½(v - l + 2)}th term in the series (1 - x)^{-(2[vertical line]l[vertical line]+d)/2}, where d is the degeneracy of the vibration, v is the vibrational quantum number and [vertical line]l[vertical line] = 0 (d = 1) or v, v - 2,..., 0 or 1 (d = 2, 3). It is also shown that (intensity of band l^l <- l^l)/(Σ_ν intensity of band v^l <- l^l) = q^{2[vertical line]l[vertical line]+d} where q = (ν'ν")^½/½(ν' + ν").

Systems which are electronically forbidden are discussed in terms of the Herzberg-Teller series expansion of the transition moment, and it is shown that for the forbidden progression v¹ <- 0⁰v = 1, 3, 5... the intensity of the band v¹ <- 0⁰ relative to that of the band 1¹ <- 0⁰ is given by the {½(v + 1)}th term in the series (1 - x)^-(d+2)/2. Also, the ratio (intensity of band 1¹ <- 0⁰)/(Σ_ν intensity of band ν¹ <- 0⁰) = q^d+2. The results are applied to the 2600 Å system of benzene; the agreement between experiment and theory is poor. The application of the results to the infra-red spectrum is indicated, and finally the effects of mechanical and electrical anharmonicity are briefly discussed.

A general expression is derived for the expectation value of the intensity in the Fraunhofer diffraction pattern of a plane screen of any shape containing any number of randomly positioned identical apertures. The special case of a rectangular screen is treated in more detail and the result is generalized for a screen of any shape. It is shown that the expectation value of the intensity has an intense narrow maximum in the centre, the shape of which resembles the Fraunhofer pattern of an aperture of the same size and shape as the screen. The effect of apertures overlapping is taken into account and it is shown that the effect can be corrected for by multiplying two terms in the expression for the expectation value of the intensity by different factors which depend on the number of apertures and the ratio of the area of one aperture to the area of the screen. Consideration is given to the failure of the scalar theory for apertures whose edges are separated by less than one wavelength and it is shown that in most cases the effects due to this cause are less serious than those due to overlapping except where the apertures are smaller than about twice the wavelength.

The dependence of the saturation magnetoresistance of a polycrystalline rod on the angle θ between rod and field is shown to involve cos² θ but no higher powers of cos θ. It is suggested that the occurrence, in practice, of a slight dependence on cos⁴ θ is spurious and is connected with the application of a demagnetizing-field correction.

The value of [partial differential]μ/[partial differential]α in the lower state of water is derived by infra-red spectra. This value, which represents the rate of change of the vector dipole moment relative to the bond angle α, is found to be approximately equal to the corresponding value derived by self-consistent field molecular orbital theory.

An analysis of 81 hammer tracks from interactions of antiprotons of momentum 5 GeV/c with Ag and Br nuclei is presented. The results are compared with evaporation theory using the values V = 5 8 MeV and T = 11 2 MeV.

Recent observations of helium-like spectra of carbon, nitrogen and neon are reported, consisting of lines due to the 2p ¹P-nd ¹D, 2s ³S-np ³P and 2p ³P-nd ³D transitions. Their wavelengths and further wavelengths calculated for these elements and oxygen are listed.