Table of contents

The analogy between gravitation and electromagnetic theory is discussed and used to obtain, for linearized gravitation theory, an analogue of the magnetic monopole. This analogue results in a quantization of mass. If a similar attempt is made to use the analogy between spin and charge it is found that Einstein's theory must be substantially modified to obtain an analogue of a magnetic monopole.

It is argued that a solid consisting of fully entangled long chain molecules will show elastic properties similar to those of cross linked, i.e. vulcanized material, provided that elastic moduli are studied in periods short compared with creep times. An appropriate formula for the free energy is deduced, and calculated for a model in which each chain is confined to an environment formed by the others. Under deformation this environment changes and the elastic energy can be deduced. If ₁, ₂, ₃ are the principal strains, that part of the free energy due to the polymer entanglements is shown to be ΔF = - BκT(₁² + ₂² + ₃²)L(σl/m)^1/2 for small strains, where B is a numerical constant, L is the total length of polymer present, σ the density of polymer, and l, m the effective length and mass of one of the submolecules from which the polymer is made. An expression is also given for the case of a strain of arbitrary size.

A discussion is given of the best way to improve a wave function, for the ground state of a two-electron atom, when starting with either the full Hartree-Fock wave function or else some analytic approximation to it (as in the Hylleraas-Eckart function for helium). In both cases the Hartree-Fock energy is a saddle point, and it is possible to give formulae from which we determine the direction of steepest descent towards the best open-shell function. Numerical calculations are made for the case of helium.

The dipole polarizabilities and shielding factors of all members of the sodium isoelectronic sequence have been calculated as functions of the nuclear charge Z using an extension of a theory recently developed by Cohen. The results of calculations within the framework of the Hartree-Fock approximation are compared with those of more accurate approximations as well as with experiment.

A matrix method is employed to calculate the energy eigenvalues and the corresponding orbital wave functions for a number of ¹S states of the helium isoelectronic sequence. The `inner' 1s-electron orbital is represented by a hydrogenic wave function, while the valence electron orbital is expanded in a complete set of Laguerre functions.

Results are presented for the 1 ¹S, 2 ¹S and 3 ¹S states of He, Li⁺ and Be²⁺.

Variational methods are used to calculate the parallel and perpendicular dynamic dipole polarizabilities, the refractive index, the Verdet constant, the Rayleigh and Raman scattering cross sections, and the Rayleigh and Raman depolarization factors of molecular hydrogen as a function of frequency of the incident radiation. The results are satisfactory at long wavelengths but they fail for wavelengths less than about 2500 Šbecause of inadequacies in the representation of the wave function of the undisturbed molecule. By scaling the results to ensure they tend to the zero-frequency limits of Kolos and Wolniewicz we obtain, at the laser wavelength 6328 Šused by Bridge and Buckingham, 0.126 for the molecular anisotropy K, 0.818 ų for the mean polarizability and 0.309 ų for the polarizability anisotropy compared with the measured values of 0.128, 0.819 ų and 0.314 ų respectively.

Approximate analytical Hartree-Fock wave functions have been obtained for 1snp ¹P and ³P states of the He-like isoelectronic sequence up to N⁵⁺ for n = 2, 3, 4 and 5. Calculated energy eigenvalues compare well with previous results obtained by Davidson for n = 2, 3 and 4 for helium and Perrin and Stewart for n = 2. New results for higher members of the isoelectronic sequence are presented.

A variation principle for the expected values of Hermitian operators has previously been derived. This paper describes a calculation on the ground state of the helium atom, which uses this principle to obtain variational estimates of the expected values of a number of operators. Trial functions containing up to fifty parameters are used, and the results justify a cautious optimism over the practicability of the approach.

We show that the variational methods of Hulthén, Kohn and Malik are members of a larger group of methods, which can be applied in scattering theory. We illustrate and compare these methods, applying them to the elastic scattering of electrons by hydrogen atoms.

The impact parameter formalism is used to calculate the cross sections for the electron capture reaction H^τ + He(1s²) -> H(1s, 2s, 2p) + He⁺ (1s) over the energy range 1-1000 kev. It is concluded that the inclusion of the excited states of hydrogen does not improve the agreement of the theory with experiment for capture summed over all final states. The calculated cross sections for capture into the 2s and 2p states agree reasonably well with the few measurements available.

Excited atomic states are produced by electron impact with a pulsed beam of electrons of fixed energy and sharp cut-off. The spontaneous decay of these states is obtained as a function of time by a single-photon counting technique. Experimental values for the lifetimes of the 4 ³S, 5 ³S, 4 ¹S, 5 ¹S, 3 ³P, 4 ³P, 3 ¹P, 4 ¹D and 5 ¹D states of neutral helium are given.

The interaction energy between pairs of like atoms and ions has been calculated for different distributions of the atomic electrons. In the energy range which is of interest for radiation damage calculations, the interaction energies calculated in the case of neutral argon for Hartree electron densities are up to 30% higher than those calculated for Thomas-Fermi-Dirac densities. For Cu⁺, Al⁺ and neutral neon the difference is 15% or less.

The difference between the argon-argon interaction energies calculated by Firsov from the Thomas-Fermi model, and those calculated by Abrahamson from the Thomas-Fermi-Dirac model, is larger than might be expected from the approximations involved. The results of the present paper are in good agreement with those of Firsov.

Starting from the exact asymptotic expression of the energy separation of the symmetrical and antisymmetrical states of the system A₂²⁺ (where A is an inert gas atom), the effective cross sections for momentum transfer in A²⁺ + A collisions, as well as the mobilities of A²⁺ ions in their own gases, are calculated over a wide range of temperatures (100-20 000 °K). The results are compared with classical predictions at low temperatures and previous calculations for helium. At T = 300 °K the following values for the reduced mobilities of He²⁺, Ne²⁺, Ar²⁺, Kr²⁺ and Xe²⁺ are obtained: 25.2, 7.98, 2 75, 1.55 and 1 01 cm² v^-1 sec^-1, respectively.

Cross sections for the absorption of radiation by the lithium atom due to photoionization and inverse bremsstrahlung are calculated and depicted as a function of wavelength in the range 1800-9000 Å. The corresponding absorption and emission coefficients are deduced for wavelengths in the above range and temperatures in the range 2500-150 000 °K. A comparison is made with the measured continuum emitted by exploded lithium wire for low capacitor energies: the continuum in the visible and near ultra-violet regions is shown to be predominantly due to radiative recombination.

The first and second excited states of ²⁷Si have been populated by the reaction ²⁸Si(³He, α)²⁷Si. The particle-gamma-ray angular correlation associated with the decay of the first excited state is consistent with J = ½ for this state. The data on the second excited state are satisfied by J = 3/2, 7/2. The decay modes of these states are compared with those of the analogous states in ²⁷Al.

The direct production of electron pairs by cosmic-ray muons (mean energy 6 GeV and minimum energy 0.85 GeV) in carbon, aluminium and copper targets of thicknesses 0.06, 0.14 and 0 44 radiation units, respectively, has been studied with two multi-plate cloud chambers. Showers of two or more electrons associated with each outgoing muon from the targets were selected, and the energy transfer in each event was estimated. The results, after necessary corrections, show that the experimental cross sections for direct pair production events in the region of energy transfers, 12 to 1000 MeV, are in approximate agreement with the theory of Murota-Ueda-Tanaka. They also support the expected Z(Z + 1) variations of the cross sections and the required variation with the mean primary muon energy. The data also indicate that the most appropriate value of the arbitrary parameter α in the Murota-Ueda-Tanaka theory is 2.

The distribution of effective field at the ⁵⁹Co nucleus in cobalt-nickel alloys has been studied at 4.2 °K by the spin echo technique. Satellite lines of the ⁵⁹Co resonance have been observed in alloys of composition Co_cN_{11 - c} for c <or= 0.10 and c >or= 0.93 and broad distributions of effective field at intermediate compositions. The resonant frequency of cobalt nuclei with one and two nickel atoms in the nearest-neighbour shell is reduced by 7.5 Mc/s per nickel atom. The distributions of effective field in all the alloys can be reasonably well described by considering only the first-nearest-neighbour shell.

Measurements of the scattering from the ferromagnetic component of magnetic moment in MnCO₃ at 4.2 °K in applied fields of 1.57 and 7.0 kOe have been made using the polarized neutron method. An x-ray refinement of the structure was carried out to obtain accurate values for the positional parameter of the oxygen atom. From the results the distributions of the aligned spin density in the [1[1 over bar]10] projection have been calculated and these indicate that a significant proportion of the ferromagnetically aligned moment is not located within 1.2 Å of the manganese nucleus. The observed distributions have been separated into field-dependent and spontaneous parts, and the contribution of spin-orbit coupling to the spontaneous part has been estimated. Both moment distributions show considerable departures from spherical symmetry around the manganese atoms; the degree to which these can be explained by a simple theory of covalency is discussed.

Recently Syozi has shown how the specific heat of a model of a dilute ferromagnet may be expressed in terms of the specific heat of the Ising model. The effect of dilution is to reduce the specific heat singularity to a cusp, its value at the critical temperature being no longer infinite.

The index α', describing the critical behaviour of the specific heat of a ferromagnetic system as a function of temperature below the Curie point, is thought to be related to the corresponding indices β and γ' for the spontaneous magnetization and susceptibility by α' + 2β + γ' = 2. By extending the work of Syozi to the case when a finite external magnetic field is applied, we have shown that, if the above relation is satisfied by the Ising model indices, it is also satisfied by the indices for the diluted model. Specifically the indices for the diluted model are α_D' = -α'/1 - α', β_D = β/1 - α' and γ_D' = γ'/1 - α' where α', β and γ' are the Ising model indices. The indices above the critical temperature and the amplitudes of all the singularities are also given.

The dependence of the thermodynamic properties on the concentration of magnetic atoms at fixed temperature is also investigated and the critical indices are found to be the same as above.

With the aid of exact seriesexpansions the shape of the critical isotherm of an Ising ferromagnet (or fluid) is re-analysed and it is conjectured that δ = 5 for all three-dimensional lattices. (The analysis strengthens previous conjectures that δ = 15 for all two-dimensional lattices.) Estimates are given for the amplitude of the singularity for the various lattices.

Evidence is presented to support the conjecture that α' = α = 1/8, β = 5/16 and γ' = γ = 1¼ for all three-dimensional lattices.

A single band of exciton states is considered for a polymer rod consisting of identical monomers whose nuclei are able to vibrate. It is shown that the shift in the average energy of absorbed photons when the monomers are stacked is equal to the shift one would get in the absence of vibrations of the nuclei. The spread of energy of the absorbed photons (defined by the standard deviation) is equal to the corresponding spread induced by vibrations in the monomer. These results are shown to be true both at zero and at finite temperatures. The result for the average energy confirms the suggestion of McLachlan and Ball that the shift of the envelope of the absorption spectrum when the monomers are stacked should be independent of the vibrations of the nuclei, while avoiding some of the approximations on which that suggestion was based.

An earlier theory, given by Wexler in 1966, for electrons and phonons in disk geometry is shown to be consistent with existing approximations for neutrons. An electrode is found to be analogous to an absorbing disk with a blackness of two placed in an infinite medium containing neutrons. The large-radius régime for electrons would be analogous to an absorption of neutrons proportional to the radius, and independent of the blackness of the disk. It is shown that such behaviour would occur in the neutron problem for intermediate values of the radius but that it tends to be removed by absorption in the bulk medium. The earlier modified Skyrme formula, given by Ritchie and Eldridge in 1960, is shown to correspond approximately to the use of a different trial function in the variational formulation. Whereas the latter formula is appropriate for small disks of any blackness, and for large weak absorbers, the new formula is recommended for heavy absorbers, especially those of large radius. An extensive table for media of varying absorptions is provided. The basic parameter in the theory is approximately equal to the reflection coefficient of a disk-shaped region drawn in the bulk material.

Resistivity measurements have been made on a number of Gd-Yt alloys in the temperature range 4 2 °K to 1100 °K. The exchange coupling constant J has been calculated from the spin-disorder resistivity and found to vary from 2 1 ± 0.1 ev ų in pure gadolinium to 6.0 ± 0 04 ev ų in an alloy containing 10.1 at.% of gadolinium. To interpret this concentration dependence of J, covalent mixing between the localized 4f electrons and the conduction electrons at the Fermi surface has been assumed.

The calculation of transport coefficients in the Kirkwood theory of transport processes falls naturally into two parts: evaluation of force correlation functions and of reduced particle distribution functions. Both these quantities are calculated for the case of lattice thermal conductivity, the former for a face-centred cubic lattice with nearest-neighbour interactions and the latter for the Montroll model of a lattice. It is shown that the result for the high-temperature limit gives reasonable agreement with experiment for rare gas crystals. Generalizations to more realistic models are discussed.

The case of phonon interactions is considered to demonstrate that the equation-of-motion method is equivalent to the diagram technique for finding the self-energy occurring in the evaluation of Green functions. Precise approximations are stated to establish the equivalence.

Experiments are reported which measure the angular correlation of gamma rays from positron annihilation in solid and liquid mercury, solid and liquid bismuth and a liquid bismuth-mercury alloy. The positron wave functions and the angular correlations to be expected from positrons annihilating in the ion cores are calculated in the single-electron approximation. It is not possible to use these calculations to subtract core annihilation from the total observed annihilation because the shapes of the theoretical and experimental curves do not agree at large angles. This situation is discussed. Valence electron annihilation is shown to be more probable relative to core annihilation in the liquids than in the solids, and for liquid bismuth it is compared with theoretical calculations by Ballentine.

Different equivalent potentials variously proposed for muffin-tin ionic potentials in solids are described in a unified manner as obeying a conservation condition for the scattering amplitudes. This condition is the basis of a reformulation of the problem in which a projector operator method is used to generate different families of equivalent potentials. When the projectors are constructed with bound and scattering states, one obtains the orthogonalized plane wave pseudopotentials. When these are in and out projectors in real space, with respect to the muffintin spheres, one obtains the augmented plane wave effective potentials. It is shown that the plane wave representation of the generator of this family yields the Slater effective potential. A special choice consisting of eliminating the potential inside, while leaving the right surface value to ensure matching, yields the Ziman pseudopotential, whose precise relationship with Slater's is thus established. A quantitative theory of phase shift cancellation is proposed and applied to perform a phase shift analysis of the orthogonalized plane wave pseudopotentials. It is shown that the number of multiples of π deducted from the trace of the phase shift operator is equal to the number of pseudo-bound energy levels (of the pseudo-Hamiltonian) which lie above the energy level under study.

Interband and free-carrier Faraday rotation has been measured in a series of n-type InAs samples between 3 5 and 20 μm at 77 °K and 300 °K.

The carrier density ranged from 3 × 10¹⁶ to 1 1 × 10¹⁸ cm^-3 and the free-carrier results have been analysed by k.p theory using the Boltzmann method previously developed for InSb by Smith, Pidgeon and Prosser. The (E, k) shape for the conduction band has thus been obtained for energies up to 0 15 ev above the band minimum. The results give a mass at the band minima of 0 024m₀ at 77 °K and 0.022m₀ at 300 °K.

The interband rotation from the purest sample has been compared with the theories of Boswarva and Lidiard, and Roth, the latter giving the better frequency fit. In the more highly doped samples a significant change in the shape of the interband rotations was observed. At a doping of 9 × 10¹⁷ cm^-3 some thirty Landau levels are affected by the electron filling near k = 0, blocking transitions into these states.

The balance between positive and negative rotations is thereby disturbed. In particular, the negative rotation arising from the highest light-hole state is reduced and so the results are consistent with the competition between light- and heavy-hole states, a feature of the Boswarva and Lidiard theory The large free-carrier rotation prevented any definite conclusion on the presence of a low-frequency contribution to the interband rotation as suggested by Mitchell, Palik and Wallis.

The leading anharmonic contributions to the Helmholtz free energy of face-centred cubic crystals have been considered on the basis of the nearest-neighbour central-force model. Results of computer calculations of these contributions at T = 0 °K and at the high-temperature limit are presented for six values of a₁, essentially the ratio of the first to second radial derivative of an arbitrary interatomic potential energy. Our work depends on the potential parameters a₁, ϕ", a₃ and a₄, which are related to the first up to fourth potential derivatives. For a specified form of the interatomic potential energy our results yield the volume dependence of anharmonic effects.

Chiefly for illustrative purposes, results are given for the m-6 Mie-Lennard-Jones potentials and some of the inert gas solids. The high-temperature anharmonic effect is found to vary considerably over the range of volumes studied. It is pointed out that, perhaps for the inert gas solids, anharmonic effects to a higher order than the first are appreciable We have examined the work of Ludwig, Lloyd, Barron and Wallace and compared their results with ours. The leading-term approximation of Maradudin et al. is quite unreliable except for the calculation of the total groundstate energy and C_p, where we show that it is very good owing to an unexpected cancellation in the coefficients of the neglected terms when the force constants are defined with respect to the volume for which the static lattice energy is a minimum.

A new technique for obtaining the Sutherland-Wassiljewa coefficients A₁₂ and A₂₁ of binary gas mixtures from the experimental data is described and applied to the viscosities of the rare gas mixtures. For each gas pair a range of acceptable values of A₁₂ and A₂₁ is determined such that, if the viscosities of the mixtures are calculated from the viscosities of the pure components using any pair of these values, the mean absolute percentage difference between calculated and experimental viscosity values is less than a prescribed limit.

The results suggest that a direct theoretica prediction of the values of A₁₂ and A₂₁, from the viscosity ratio and molecular weight ratio of the components of a given gas pair, might be based upon the simultaneous use of two types of relationship involving, respectively, the ratio and the product of A₁₂ and A₂₁. It is shown that the product relationship would be the more critical for determining the coefficients, and therefore it is recommended that further theoretical studies should be directed to obtaining valid forms of this relationship.

Recently, interest has arisen in the thermal conductivity of the H₂-He gas system because of some views suggesting an abnormal composition dependence. We present some theoretical and experimental studies and conclude the system to have a normal composition variation.

The inertia tensor derivatives for a centre-of-mass principal axes system have been obtained for an XYZ₂ type molecule of symmetry C_s. From the force field for difluoramine, the centrifugal distortion constants D_J, D_JK, D_K, R₅, R₆ and δ_J have been calculated for NHF₂ and NDF₂ using a first-order perturbation theory. Except for R₆, the other constants have values comparable with D_JK, and thus a correction to the rigid-rotor energy levels should include all these terms.

A Coulomb wave perturbation theory is formulated to study the excitation and the ionization of atoms by electron impact. Some features of the theory and its immediate consequences, including threshold laws, are discussed.

The Bloch-like coherent-wave approximation of Phariseau and Ziman can be generalized by inclusion of an operator to take account of the average point-group symmetry of the nearest-neighbour environment in the liquid. It is concluded that liquids with fourfold average co-ordination and a high degree of angular correlation should show unusual electronic properties.