Table of contents

The effect of the momentum dependence of the exchange integral on the Kondo temperature is discussed by considering two extreme limits of momentum dependence. A simple model in which there are some momentum effects illustrates the change in a qualitative fashion.

It is suggested that the exponent α for the specific heat of the classical Heisenberg model is -fraction one-sixteens. A closed-form expression is derived to represent the specific heat in the range T > T_c. Good agreement is obtained with experimental data on europium sulphide. It is concluded that the finite limitation of the specific heat observed experimentally is not due to the Heisenberg interaction.

Several a priori possible types of the antiferromagnetic ordering in GdAlO₃ are considered. It is shown that the actual structure can be deduced from symmetry considerations and measurement of a macroscopic property, for instance the magnetoelectric effect.

The phase transition in the CuAu system between the strictly ordered structure CuAu I and the superlattice structure CuAu II is accompanied by a small latent heat. This letter proposes a mechanism to account for the entropy of disorder in CuAu II which is compatible with the regular appearance of the micrographs of thin films of this alloy.

Evidence is presented which shows that the fine structure of a tunable sum frequency reported earlier by the authors can be attributed to the action of three pump frequencies, two of which are stimulated Raman emissions from the trapping liquid. The variation of sum frequency power with the length of the non-linear crystal tends to confirm the existence of three pump frequencies.

The dependence of the excitonic insulator transition temperature on the shift in chemical potential due to the presence of various types of impurities (donors, acceptors or normal impurities) and on the change of band masses due to small crystal deformations is investigated in terms of a mathematical parameter, η_p, which involves all these physical factors. It is shown that both in the semiconductor region and in the semi-metal region, the transition temperature is independent of the sign of this parameter but is a maximum when this parameter is zero.

A calculation is presented of the single-particle propagator for a free electron in a uniform electric field. In its time-dependent form this takes simple closed forms in both momentum and configuration space. Series expansions for the Fourier transform of the configuration-space propagator are given. Using the latter a calculation is made of the electric-field dependence of the (approximate) bound state of the delta-function potential. A variational method which is suitable for general potentials is also given.

A preliminary analysis of the absorption and fluorescence spectra of Er³⁺ (4f¹¹) in LiYF₄ has been made. With the aid of previous work on Er³⁺ in LaF₃, various states have been identified. The Racah parameters giving the best overall agreement between the predicted `free ion' spectrum and the spectrum of Er³⁺:LiYF₄ are identical with those for LaF₃, that is E¹ = 6747 cm^-1, E² = 31·63 cm^-1, E³ = 635·8 cm^-1 and ζ = 2345 cm^-1. An attempt has also been made to determine the nature of the crystal-field potential by considering the rare earth site as a tetragonally distorted cube. The most suitable crystal-field parameters were found to be B₂⁰ = -115 cm^-1, B₄⁰ = -171 cm^-1, B₄⁴ = -661 cm^-1, B₆⁰ = +25·5 cm^-1 and B₆⁴ = -168 cm^-1.

As part of a study of the aggregation of impurity ions in alkali halides, measurements of dielectric loss, of optical absorption and of emission spectra have been made on NaCl: Pb²⁺ and KCl: Pb²⁺ crystals. Changes in the optical absorption and the emission spectra with aggregation of the Pb²⁺-vacancy pairs are more pronounced in NaCl than in KCl. The main conclusions from the NaCl: Pb²⁺ measurements are:

1. changes in the absorption and emission spectra result from the same aggregation process studied previously in many alkali halide crystals by dielectric measurements;

2. the emission spectra, but not the absorption, change when the divalent ion-vacancy pairs (dipoles) form the initial aggregates (trimers); later in the aggregation process both absorption and emission change;

3. the optical measurements show that the aggregation takes place in several definite stages and the results are not inconsistent with the model proposed earlier, viz. the initial aggregates are trimers and these grow into pentamers, and then heptamers etc. by the addition of dipoles, two at a time.

The dielectric absorption measurements give activation energies for the diffusion of Pb²⁺-vacancy pairs through NaCl and KCl lattices of 0·80 ev and 0·85 ev respectively.

The propagation of transverse magneto-plasma waves in an absorptive plasma whose density is spatially periodic has been investigated. Numerical results are presented where the real and imaginary parts of the wave vector are studied as a function of the electronic collision frequency for a metal-semiconductor sandwich structure. These results show that scattering can play a major role in determining the dispersion relationship of plasma waves in a periodic structure. The source of the effects introduced by scattering is discussed, together with some comments concerning their implications for a proposed tunable microwave filter.

A detailed experimental investigation on the validity of the Bardeen-Rickayzen-Tewordt (BRT) theory for the ratio of the lattice thermal conductivity in the normal and superconducting states has been made for Nb-Mo and Nb-Ta alloys over a wide range of composition.

The measured lattice ratios for the nine samples investigated are in complete agreement with the BRT theory down to reduced temperatures t similar 0·4. Below this temperature agreement is still obtained for the thicker samples; in thin samples deviations appear which are attributable to boundary or dislocation scattering.

The variation of the lattice conductivity with composition has also been investigated. Addition of Mo to Nb noticeably increases the lattice conductivity, whereas addition of Ta to Nb has a much smaller effect. These results can be explained in terms of the variation of the electronic density of states N(0) with composition.

Non-equilibrium phenomena in crystal lattices determined by the long-time behaviour of the system are formulated in terms of the correlation function of quantities relaxing very slowly, such as the densities of the conserved variables in the system. Transport equations for such correlation functions can be derived and, depending explicitly on the conservation laws of the system, facilitate an analytic solution of the problem. As an example, we examine the autocorrelation function for the quasi-momentum density. The transport equation for it is similar in structure to the phonon Boltzmann equation with a modification of the collision term describing the relaxation of the quasi-momentum density rather than of the phonon density.

A theoretical investigation is made of the behaviour of soft X-ray emission spectra near the high-energy (Fermi) edge, where divergencies have been reported in previous many-body theoretic calculations, and a spike has been observed in the L-emission spectra of several metals. We use a simplified, but for our purposes sufficiently general, model Hamiltonian and introduce a calculational method which overcomes difficulties associated with previous treatments. The results of the calculation up to second order confirm general arguments leading to the conclusion that the divergence will prevail in the exact solution of the model problem. The effect will, however, be blurred by initial- and final-state effects neglected in the model.

New de Haas-van Alphen measurements in indium are reported, in which emphasis is placed on very-low-frequency oscillations and on the effective masses of the carriers. A Fermi surface model is deduced and four orbit sizes from this model are used in fitting the four parameters of an empirical pseudopotential calculation. The resulting potentials are: V(111) = -0·7617 eV, V(002) = -0·280 eV, V(200) = +0·0922 eV λ (a spin-orbit parameter) = +0·1756 eV. The band structure and Fermi surface computed with these parameters are shown to be consistent with the majority of experimental data.

Some of the features and characteristics of the general angular variation of the electron spin resonance spectra for impurity ions with spin 1, fraction three-twos and fraction five-twos in crystal fields of various strengths and symmetries are discussed and illustrated.

In metals at ultra-high compressions, pressure ionization of the cores takes place. The highest hydrostatic pressures so far attained (600 kbar) appear to be in or approaching this regime in the cases of K, Rb and Cs. A simple model for ionized cores leads us to predict order-of-magnitude increases in resistances above those at zero pressure. These are in agreement with the asymptotic behaviour of the observed curves for Rb and Cs and with that suggested by the observed curve for K.

Low-frequency helicon resonances have been observed in square plates made from single crystals of pure aluminium. The resonances have been interpreted in terms of the theory by Legendy, and the high field Hall effect and transverse magnetoresistance have been studied. Contrary to most direct current measurements the helicon measurements give a magnetoresistance which shows signs of saturation in high fields. The Hall coefficients are consistent with the predictions from theory for metals having closed Fermi surfaces. It is suggested that the linear increase of the magnetoresistance, which is well known from direct current studies, might be connected with the defect structure and is not a property of the ideal lattice.

Pauli operators are treated as Bose operators fulfilling q-number commutation relations. An analogue of Wick's theorem for Pauli operators is formulated suitable for perturbation theory. The lowest-order contributions to the perturbation series for the temperature dependent one-particle causal Green function are calculated, and a diagrammatic representation is introduced for them. It is not necessary to have a different meaning of the same graph depending on the order of temperatures contained in it. The connected graphs representing the perturbation series up to second order are drawn explicitly.

The diagrammatic representation proposed in the preceding paper is used to sum certain classes of graphs in the perturbation series for the one-particle Green function of a spin-½ Heisenberg ferromagnet. The excitation energy of spin waves and the low-temperature expansion of the magnetization are calculated in several approximations, and the results of the equation of motion method according to the decouplings proposed by Tyablikov, Callen, and Tahir-Kheli and ter Haar are rederived (with a slight alteration) in the low-temperature region. Due to the proposed diagrammatic representation no further approximations are necessary during the summation of graphs.

The band structure of a diamond-type semiconductor near the energy gap can be described simply in terms of the Jones zone. The energy gap is nearly constant over the Jones-zone faces except close to the edges and corners. This accounts for the success of the spherically symmetric Penn model of the dielectric constant . It also explains the qualitative shape of the optical absorption spectrum, in particular the occurrence of the first peak. Specifically covalent effects arise from the fact that the ν(111) Fourier component of the pseudopotential is relatively large and has to be treated beyond lowest order of perturbation theory. Simple algebraic formulae give the principal energy gap and the position of the minimum in the conduction band along ΓΔX, which agree with the large computer calculations to within about 25%. This picture accounts in a simple approximate way for the variation with pressure of the dielectric constant and of the position of the main peak in the optical absorption spectrum. Finally a microscopic calculation of the extra charge Q_b located in the covalent bond gives agreement with Phillips' conjecture that Q_b = 2^-1.

An investigation is made of the wave functions of electrons in the periodic field of a binary alloy possessing a long-period superlattice structure. The additional periodicity of a superlattice arises from the regular arrangement of antiphase boundaries, which requires a treatment different from conventional perturbation theory. Formulae are given for the magnitude of the energy gaps which arise as a result of the occurrence of the superlattice structure.

The basic physics of the processes producing flux flow resistance in superconductors is reconsidered. It is shown that in the dynamic intermediate state (in which at low fields flux tubes move through the material) there is no potential difference between the contacts; the measured voltage is produced by magnetic flux outside the superconductor which changes as each of the tubes move. On the other hand, in a type 2 superconductor near H_c2 there is a potential difference between the contacts and no effect of changing magnetic flux. We make the conjecture that in general in the mixed state there are contributions of both types to the voltage measured (V), the fraction V_p/V produced by a potential difference in the superconductor decreasing as the field is lowered below H_c2. In all cases the measured voltage V is the same as if there were an electric field (ν × B)/c in the superconductor, where ν is the average velocity of the flux `entities' and B the flux density in the material. The low-frequency noise in superconductors corresponds to fluctuations in the component of voltage V<sub>i</sub> produced by magnetic flux changes. This has the consequence that the noise spectrum should depend on the arrangement of the leads of the measuring circuit and also that the decrease in noise observed with increasing field in the mixed state corresponds to the decrease in the ratio V<sub>i</sub>/V and does not necessarily represent a decrease in the size of the flux `entities' with increasing field.

At distances greater than the rigid-core diameter σ, the soft interactions of the Rice-Allnatt theory are handled according to the Fokker-Planck approximation, and in this region the radial distortion u(r) of the equilibrium distribution satisfies the same equation as the radial distortion of the Kirkwood-Eisenschitz theory. In calculations based upon the Rice-Allnatt theory, Lowry, Rice and Gray have applied the boundary condition that r²e ^-ψ du/dr -> 0 as r -> σ to the equation for u(r). In this paper calculations and arguments are presented which suggest that the radial distortion selected by this condition is identical with that selected by the strong boundary condition of Eisenschitz, r³u(r) -> 0 as r -> infinity.

Neutron diffraction measurements confirm that electrolytic gamma manganese is antiferromagnetic, with a magnetic moment of 2μ_B per atom.

Durkan and March have found that the screening of ionized impurities by the conduction electrons in semiconductors depends on the magnitude of the external magnetic field and can readily be calculated for not too large values of the field. It is shown that this contributes to the longitudinal magnetoresistance even under conditions where this effect has hitherto been considered absent.

The band-structure contributions to the elastic shear constants of beryllium, magnesium, zinc and cadmium have been predicted qualitatively using a local model potential formulation. Quantitative agreement will require the consideration of non-local effects.