Table of contents

The connection between Bopp's approximation to the energy of N-electron atoms and ions, and [infinity]-order perturbation theory is established. The approximation gives the first two terms in the perturbation series correctly, but contains only parts of all higher-order terms. Calculations show that Bopp's approximation is superior to first-order perturbation theory, but inferior both to second-order perturbation theory and to the Hartree-Fock method. For three-electron systems Bopp's approximation is a lower bound for the exact energy. Certain conditions on the second-order density matrix for atoms and atomic ions are noted, and a brief comment is made on the electron correlation problem in molecules.

Sum rules and hypervirial theorems are derived by using the simple technique of differentiating an eigenvalue equation with respect to parameters occurring in it. Various applications are discussed.

A comparison is made of the Kirkwood superposition approximation with the collective movement theory of liquids for the case of liquid metals. The correlations of density fluctuations of low wave vectors are compared. The results of the two methods do not agree and it is inferred that the Kirkwood superposition approximation should not be used for all particle configurations.

The theory of collective movement for liquids is extended to liquid binary alloys. A perturbation theory is then developed to evaluate correlations of particle coordinates. These correlations are used to evaluate properties of liquid binary alloys. The results are compared with experiment for Na-K alloys and agreement is good for the specific heat and electrical resistivity. These results do not explicitly use the interaction between the two types of atom. For further progress some knowledge of this interaction is required. A possible approach is suggested.

The calculation of Gor'kov for isotropic alloys is extended to anisotropic superconductors. The results are equations of the form proposed by Ginzburg. The mass tensor is given by the same expression as that found for a pure anisotropic material by Gor'kov and Melik-Barkhudarov, except that the factor relating to anisotropy of the energy gap is absent. It is pointed out that the mass tensor is proportional to the normal-state conductivity tensor. In an anisotropic material the parameter κ depends on the direction of the magnetic field; however, it changes with impurity content in exactly the same way as does κ for an isotropic material.

The formation of nuclear spin echoes in the presence of combined static or quasi-static magnetic and quadrupolar perturbations is analysed. The perturbations are assumed small in comparison with the Zeeman splitting by the applied field, and nuclear dipolar coupling is neglected. Echoes are formed by two pulses of radio-frequency excitation separated by an interval τ. Echoes appear only at t = 2τ. The quadrupolar echoes at t ≠ 2τ discussed by Solomon in 1958 are suppressed by the magnetic inhomogeneity. The condition for their suppression is roughly bτ > 1, where b is the magnetic inhomogeneity in nuclear frequency units. The dependence of the echo amplitude at t = 2τ on the nutation angle ϕ₂ induced by the second pulse has been calculated for I = 3/2 and 5/2, assuming an initial 90° pulse. The variation in amplitude with ϕ₂ of the echo from the combined perturbation is different from that of echoes arising out of either a purely magnetic or purely quadrupolar perturbation. The combined echo exhibits two main maxima as ϕ₂ increases from 0° to 180°. The echo shape at the first maximum is a hybrid of magnetic and quadrupolar parentage; the second at ϕ₂ = 180° is purely magnetic. The echo amplitude at the first maximum is substantially greater than that of the corresponding quadrupolar echo. Experimental confirmation of some of the analytical results has been obtained from observations of the ¹²⁷I signal from KI and the ⁶³Cu signal from Cu-4 4 at.% Zn and Cu-1 at.% Mn alloys.

The physical origin of the Azbel'-Kaner equations is briefly discussed, and these equations are used to calculate the line-shapes, allowing for the effect of non-resonant contributions to the surface impedance. The calculated line-shapes agree well with those observed experimentally.

The ⁹³Nb nuclear spin-lattice relaxation time T₁ has been measured at 77 °K in the body-centred cubic Nb-Mo alloy system from 0 to 60 at.% Mo. T₁T decreases from 0.32 sec degK for 100% Nb to 0.31 at 5 at.% Mo, and then increases to about 1.13 at 60 at.% Mo. These results are compared with existing measurements of electronic specific heat in the Nb-Mo system, and it is found that both can be related by a rigid two-band model only if quantities other than the densities of states are allowed to vary with composition.

The effect of dipolar and quadrupole interactions on the nuclear magnetic resonance line shape from a ferromagnet is calculated by taking an average over the domain wall weighted by the enhancement factor which varies through the wall. The results obtained disagree with the earlier work of Portis and Kanamori who used intuitive arguments to obtain the positions of the dipolar satellites. Detailed line shapes are calculated for face-centred cubic and body-centred cubic lattices which contain impurity atoms producing dipolar magnetic fields and electric quadrupole fields on their neighbours and also for hexagonal lattices where homogeneous electric field gradients are present. These are compared with experiment and used to reinterpret the available data and to show that the observed satellite lines generally come from different shells of neighbours and not from dipolar splitting as suggested by Portis and Kanamori.

The specific heat of antiferromagnetic α-manganese has been measured between 0.3 and 1.0 °K and is given by C = 12.0T + 0.51T^-2 + 0.032T³ mJ mole^-1 degk^-1. The contribution proportional to T^-2 demonstrates the existence of a hyperfine coupling interaction which may be expressed as an effective magnetic field H_eff of 90 kOe at the nucleus. Since nuclei at different lattice sites are likely to experience different interactions, this value for H_eff is the root mean square average of the fields over all nuclei.

The `Green's function' method for the calculation of electronic band structure is investigated. The general T-matrix theory of Beeby and Edwards is used to transform the formulae of Korringa and of Kohn and Rostoker to a `reciprocal lattice representation' which is equivalent to a `nearly free electron' formalism. The matrix elements between plane waves then correspond to a `pseudo-potential', which depends mainly on the phase shifts of the atomic potentials but which is also a function of the wave vector and energy of the Bloch state being considered. For small phase shifts it tends to the scattering amplitude or `quasi-potential' of an atomic potential.

But when a phase shift passes through π/2 this pseudo-potential goes through infinity, i.e. at a virtual bound state there is a singularity in the reaction matrix (K matrix) of the atomic potential. Such a `resonance' can give rise to a narrow band, such as a d band in a transition metal. At energies below the zero of the muffin-tin potential (where the phase shifts become imaginary), the formalism is still valid, and one gets further bands from the singularities of the T matrix - the bound states - of each potential well.

The discussion is purely formal, but suggests that the Green's function method is more than a complicated scheme of calculation capable of giving good numerical results over a wide range of energies; it can be physically interpreted and used empirically in the same way as the orthogonalized plane-wave, tight-binding and other apparently more direct techniques.

A variational principle for the calculation of free-free matrix elements is described and applied to the case of an electron moving in the field of a hydrogen atom. It is shown that, even using very simple trial functions, the method gives good agreement with the results of more elaborate calculations.

It is shown that for the scattering of zero-energy particles by a short range static potential whose sign never alters Kohn's variational method used in conjunction with `inappropriately normalized' trial functions provides an absolute lower bound to the reciprocal of the scattering length, as long as the potential does not possess more than a single bound state. Further, it is shown that the theorem still holds when the sign of the potential is permitted to alter if the Born approximation to the scattering length is negative.

The elastic scattering of positrons by ground state hydrogen atoms, for positron energies below the threshold for positronium formation (6.8 ev), is investigated numerically, full allowance being made for the virtual excitation of all bound states of atomic hydrogen up to and including the 3d state. The phase shifts are obtained for the S, P and D partial waves. The S-wave phases, when compared with those of Schwartz, point out the extremely slow convergence rate of the eigenfunction expansion approach to this problem. This slow rate of convergence also persists, especially at higher energies, for the P and D partial waves as well.

A semi-empirical procedure is used to calculate the mean excitation energies appropriate to stopping powers I and to straggling I₁ in neon, argon and krypton. The results are Ne: I = 150 ev; I₁ = 1200 ev; Ar: I = 220 ev, I₁ = 3200 ev; Kr: I = 360 ev, I₁ = 9600 ev. They are consistent with the experimental data.

The ranges and range distributions of nitrogen ions in nickel and silver have been determined for nitrogen ion energies of between 0.4 MeV and 2.5 MeV. Values of the stopping cross sections for nickel and silver for these ions were determined from the (range, energy) curves and compared with present theories of the stopping processes. The range straggles provide information on atomic collision processes, and the observed straggles were compared with a simple model based on present approaches to stopping in the low velocity region.

An expression for the magnetoresistance of many-valley semiconductors valid in the hot-carrier region is derived from the distribution of carriers obtained by solving Boltzmann's equation. Numerical results for small magnetic fields calculated with the parameter values of n-type germanium are also presented. The results indicate that the magnetoresistance coefficient increases with the strength of the electric field.

The bond orbital theory of the excited states of diamond, recently given by Rèdei, is extended to include charge transfer states, in which the electron and hole are associated with different bonds. All states up to and including those with second-neighbour electron-hole separations are considered. The results show that there is strong interaction between the charge transfer states and the excited bond states considered by Rèdei. The lowest energy exciton state, to which allowed transitions occur, is calculated to lie 4 ev above the direct band gap. But the lowest energy exciton state lies below the conduction band edge at X. These conclusions depend for their validity on the recent band structure calculation of Cohan, Pugh and Tredgold.

Values of the secondary ionization coefficient γ due to the Auger process of electron ejection caused by the incidence of atomic and molecular positive ions of helium on a silver cathode in low current discharges in the range 9 < E/p₀ < 41 v cm^-1 torr^-1 of electric field E to gas pressure p₀ have been deduced from measurements of ionization growth times. For He⁺ ions γ is found to be approximately 0.2 electrons per ion, but for He₂⁺ ions the electron ejection efficiency is about 0.05 electrons per ion. The rate of conversion of atomic to molecular helium ions is deduced from the measurements to be approximately 160 p₀² sec^-1.

Because of the p₀ and p₀² dependence of secondary processes at pressures in excess of about 20 torr the similarity principle for discharges is expected to fail in helium.

The polarization factor for the interference between Rayleigh scattering and Mössbauer scattering from ⁵⁷Fe is shown to be cos θ for both an unsplit scatterer and a non-aligned split scatterer, a result which differs from that recently published by Wong. Polarization factors have no effect on the ratios of the absorption cross sections for the hyperfine components of a non-aligned split scatterer.

It is shown that detailed balance is satisfied in the two-state and n-state impact parameter approximation to the atomic collision process even when approximate atomic or molecular eigenfunctions are used as a basis for expanding the total wave function.

A correction is made to the theoretical M1 transition probability predicted by the shell model calculation of Flowers and Wilmore. Agreement with experiment is thereby improved.

Surface barrier detectors have been used to measure the branching ratio in ²¹²Bi (ThC); the average value obtained for the α branch is 36.00 ± 0.03%, which is higher than the latest counter measurements (35.81%) but in better agreement with the value from magnetic spectrometers (36.08%). Some information is given on the change in value which results from different methods of analysis.

Corrections to 1965 Proc. Phys. Soc.85 625.