Table of contents

Using a formalism set up by Bloch an effective Hamiltonian is introduced which, when expanded in powers of a perturbation V, generalizes the Rayleigh-Schrödinger perturbation theory. This effective Hamiltonian provides a simple way of handling degeneracies and the orthonormalization of the perturbed wave functions. As an illustration the spin Hamiltonian introduced by Pryce in paramagnetic resonance is derived up to third-order terms, and the electron-lattice interaction is discussed showing that the Born-Oppenheimer approximation follows from firstorder terms in the case of non-degenerate electronic ground state.

We use the time-dependent Ginzburg-Landau equation to calculate the effect of thermodynamic fluctuations in lowering the resistance below the normal state value for magnetic fields above a second-order critical field. We consider only dirty metals in the Ginzburg-Landau temperature region and evaluate the conductivity of a bulk specimen along and transverse to the applied field. There is a strong anisotropy in the effect: σ_zz is proportional to (H - H_c2)^{−fraction three-twos} but σ_xx is proportional to (H - H_c2)^−½, where the z axis is along the magnetic field. The results are in agreement with a recent microscopic calculation by Usadel.

The insulator model used is of greater generality than in most previous treatments in that the degree of retrapping is arbitrary and the dependence upon temperature of the lifetime of free carriers is controlled by the degree of occupation of deep trapping levels. The assumptions involved in the definition of this model are discussed and expressions are presented for the temperature variation of thermally stimulated conductivity and thermoluminescence, peak position and height, and the rate of change of peak position for variation of the heating rate. In particular the dependence upon the initial excitation level is discussed. The equations are illustrated with computed curves. The emphasis is on the use of these results in the interpretation of experimental data and to this end the basic parameters have been retained wherever practicable. Previous experimental results are discussed with respect to these extensions to the theory. The temperature dependences of parameters such as the capture cross section of the traps are explicitly considered and shown to cause a systematic error of the order of ±10% in the evaluation of activation energy from heating rate variation methods.

A perturbative approach to the calculation of the equilibrium lattice spacing of a solid at T = 0 °K is developed based upon an expansion of the lattice free energy and pressure in powers of the quantum parameter. Terms through order capital Lambda², representing cubic and quartic contributions to the free energy and pressure are included in the Feldman-Horton approximation, and the quasiharmonic zero-point energy is treated in the Domb-Salter approximation. A general procedure is proposed for calculating the coefficients in the perturbation expansion of the lattice spacing to arbitrary order, using a central Mie-Lennard-Jones potential to describe the interaction of the atoms. The method is then applied to the study of differences of volume in isotopic lattices of ²⁰Ne and ²²Ne. The results of the method give relative differences in lattice spacing which are then compared with the x-ray measurements of Batchelder and with the predictions of various other models in current use. These include the cell model of Hillier et al. and the variational calculation of Bernardes, and recent self-consistent phonon model calculations of Gillis, Werthamer, and Koehler. It is shown that the Feldman-Horton treatment of the anharmonic terms may lead to a certain inconsistency in the trend from first-neighbour to all-neighbour models, which may be overcome when an all-neighbour treatment of the anharmonic terms becomes available. In addition the effect of using quasiharmonic and anharmonic parameters in the present calculation is studied. It is shown explicitly that the relative shifts in both the quasiharmonic and anharmonic approximations can be written down as simple functions of the quantum parameter only, so that any other substance which obeys the Mie-Lennard-Jones interaction law can be easily treated simply by using its appropriate quantum parameter.

The dynamic Jahn-Teller effect is used in a theoretical study of the properties of the Cr²⁺ ion in corundum. The ion is produced in ruby crystals by X-irradiation. The trigonal distortion in the crystal is taken into account in setting up the Jahn-Teller interaction, in choosing zeroth-order orbital states for the ion and in the magnetic interactions. Using the cluster approximation, the effective Hamiltonian is thus derived. The effects of trigonal distortion are found to be small and thus the problem closely resembles that of Cr²⁺: MgO discussed previously in which there is a very strong orbit-lattice interaction. Theoretical values for the parameters needed to describe the low-lying energy levels of Cr²⁺: Al₂O₃ are deduced. They are in agreement with values calculated from preliminary acoustic resonance results when the applied magnetic field is along the trigonal axis. It is hoped that the theory given will serve as a satisfactory basis for the discussion of more detailed acoustic resonance results and measurements of the magneto-thermal resistance in irradiated rubies.

Using a dynamic Jahn-Teller model, it is found that the Cr²⁺ ion, produced in ruby crystals by x-irradiation, behaves as a fast relaxing impurity with a relaxation time of the order of 10^-9 s at very low temperatures. The problem of cross-relaxation between a Cr³⁺ ion and a Cr²⁺ ion in the presence of a magnetic field directed along the c axis is analysed in detail by computer using a dipole-dipole interaction Hamiltonian. The associated relaxation time T_x is found to be much shorter than the usual Cr³⁺-lattice relaxation time. T_x is found to vary with temperature T according to the relation T_x α T^-n where n is approximately 3·5 in agreement with the measurements of Mason and Thorpe. Possible relaxation processes in irradiated rubies are also briefly examined.

The hyperfine fields at the ⁵⁹Co nucleus have been observed in GdCo₂ (GdY)Co₂, (GdDy)Co₂ and Gd(CoNi)₂ using a spin echo technique. The field strength in the basic compound GdCo₂ was found to be 60·7 kOe.

The results can be interpreted in terms of contributions to the hyperfine field arising from the transition metal sublattice and from the rare earth sublattice. The former appears to be proportional to the magnetic moment associated with the transition metal ion while the rare earth contribution is taken to arise predominantly from conduction electron polarization. With yttrium substitution this second term is found to be directly proportional to the gadolinium concentration.

In the nickel substituted compounds the hyperfine field due to the rare earth sublattice is observed to decrease. This is believed to be the result of a change in conduction electron concentration.

The Mössbauer spectra of cytochrome c peroxidase (CCP) containing protohaem and mesohaem have been observed in a variety of physical conditions. At very low temperature intricate paramagnetic spectra are observed, while at 195°K magnetic effects are almost completely suppressed and two quadrupole pairs appear. All spectra indicate the presence of both high- and low-spin components in agreement with susceptibility results. For the proto and meso forms in 0·2M pH 7 buffer the high-spin fractions at 4·2 °K are 0·25 and 0·4 respectively, while corresponding values at 195°K are 0·63 and 0·67. In both cases the low-spin component is in rough agreement with known g values. The high-spin components are also consistent with electron spin resonance results, and the field dependence of the Mössbauer spectra indicate that the axial term in the spin Hamiltonian is 2D = 30 ± 6 cm^-1. The high-spin fraction increases with increasing buffer strength, and is freezing rate dependent at very low or zero buffer strength. Fluoride complexes of both enzymes also show low-temperature paramagnetic Mössbauer spectra. These appear to be entirely high spin with 2D = 14·6±0·6 cm^-1 and a quadrupole interaction ½QV_zz = 0·85 mm s^-1. The zero-field spectra at low temperature are diffuse and consistent with the assumption of a large neighbour-nucleus interaction. At 195°K paramagnetic effects remain large and spectra are diffuse both with and without applied fields.

The theory of the scattering of x-rays and neutrons by an anharmonic crystal is discussed, with particular reference to the interference between the one-and two-phonon scattering. This interference gives rise to an extra contribution to the intensity of the scattered x-rays and to a change in both the intensity and the shape of the one-phonon part of the neutron scattering cross section. Detailed numerical calculations are performed for alkali halides. The results show that the asymmetries found recently in the scattering of x-rays by alkali halides are of this origin. In the case of neutron scattering the change in shape of the one-phonon peaks is so small that it is probably unobservable under experimental conditions available at present, but the change in the intensity should be observable.

The second paper in this series casts the Boudreaux-Heine method of calculating low-energy electron diffraction (LEED) intensities into a matrix form suitable for extension to complicated surface structures. Calculations are made for the 00 and 01, macron beams reflected from a nickel 111 surface, for norrmal incidence, using the potential described in paper I. Predicted peak positions agree with experiment within ±2 ev and relative intensities agree qualitatively. We interpret our results in terms of band structure. A second calculation for the same face of a nickel crystal with the surface layer displaced by 5% of the bulk spacing between layers is presented, the results of which are interpreted in terms of the large ratio between forward and backward scattering cross section of atoms.

The consequences of absorptive processes for low-energy electrons are discussed from a band structure point of view. By introducing sufficient absorption as an imaginary component of the potential, calculated widths of peaks in the diffraction intensities are fitted to experiment, yielding an estimate for the inelastic intensity-extinction distance of 6 Å at E = 70 ev in nickel. Such strong absorption, together with the nature of the elastic scattering, tends to confine flux in the crystal to a narrow cone about the incident beam, making feasible an accurate approximate scheme of calculation which constitutes a considerable simplification on present, formally exact, schemes.

An assessment is given of the effect of inelastic electron scattering on the resolution of an electron microscope image. The calculations are primarily concerned with inelastic electron scattering in thin films of carbon, which is of interest from the biological point of view. It is shown that the energy loss distribution of electrons (incident energies in the range 20-100kev) scattered in carbon and carbonaceous materials (e.g. Formvar, polystyrene) obtained for an objective semi-angular aperture, α, may be predicted within a standard deviation of about 5%, for energy losses less than 100 ev. A detailed knowledge of the energy loss distribution of electrons scattered in organic and biological materials leads to the possibility of correcting an electron micrograph for chromatic aberration effects caused by inelastic electron scattering in the specimen.

The paramagnetic resonance spectrum of Tb⁴⁺ in zircon shows that Tb⁴⁺ ions enter the tetragonal lattice substitutionally for zirconium. The ⁸S_7/2 ground-state splittings are 23 times as large as those for Gd³⁺ in this mineral, totalling nearly 10 cm⁻¹.

Previous calculations of the field gradient in BeO on the basis of an ionic model have shown that it is doubtful whether such a model is adequate. Measurements of the temperature dependence of the quadrupole splitting in BeO are presented and it is shown that the very small observed temperature dependence is inconsistent with the ionic model.

An alternative covalent model is proposed, based on a description of the valence electron wave functions in terms of molecular orbitals. The difference between the variational parameters used in describing the molecular orbitals in the inequivalent bonds in BeO is unknown, and so it is not possible to deduce the field gradient uniquely. The covalent model, however, is consistent with the small temperature variation of the field gradient.

The Zeeman splittings of the lowest J = fraction five-twos, J = fraction seven-twos multiplet levels for Ce³⁺ in LaCl₃ have been investigated using pulsed magnetic fields up to 200 KG. The results are compared with static crystal field theory and it is suggested that, whilst the overall agreement is good, the discrepancies may indicate a significant contribution from phonon interaction. Crystal field parameters and wave functions are also reported for Ce³⁺ in LaBr₃. These have been obtained from measurements in zero magnetic field.

The ionized impurity limited electrical resistivity of ideal graphite is calculated. Details of the band structure are included in the screening. The results obtained are in fair agreement with experimental results.

By making use of the recently measured spin-wave dispersion relations we have calculated the spin-wave contributions to the low-temperature specific heats of antiferromagnetic MnF₂, FeF₂, CoF₂ and NiF₂ from 0 to 50 °K (0 to 35 °K for CoF₂) using two different numerical methods. The results are compared with the experimental measurements of Stout and Catalano for temperatures of 15 °K and above; it is suggested that further experimental measurements below 15 °K would be worth while.

The optimized model potential proposed by Shaw is applied to the problem of determining band-structure energies for simple metals. Expressions for the structure-dependent contributions to the total energy are derived in Hartree approximation. The derivation is similar to that given by Harrison for pseudopotentials, but has several features which are unique to model potentials. It is shown that the separation of structure-dependent contributions into electrostatic and band-structure energies is not unique and that the choice of effective valence is therefore arbitrary. An expression for the energy-wave-number characteristic is given in terms of the optimized model potential, and is evaluated as a function of scattering momentum for eight simple metals. A comparison is made between results obtained from the full nonlocal theory and those from various local approximations. The results for the energy-wave-number characteristic are used to calculate effective ion-ion interactions and to predict stable crystal structures.

The model (or pseudo) Schrödinger equation describing free-electron-like metals is solved by a modified perturbation method in which the diagonal matrix element of the potential is accounted for exactly by defining two κ-dependent effective masses which incorporate it from the outset. The new perturbation approach is used to derive mass-renormalized expressions for the energy of electronic states, the screened form factor, and the energy-wave-number characteristic. Numerical results for the effective masses are obtained using the optimized model potential, and a constant-mass approximation is used to estimate corrections to the form factors and energy-wave-number characteristics calculated from ordinary perturbation theory. The magnitude of these corrections suggests that mass-renormalized results may be required to predict metallic properties.

A triple-axis neutron spectrometer has been used to measure the frequencies of the normal modes in zinc with wave vectors along the direction [011, macron0] and polarization vectors perpendicular to the basal plane. Previous measurements of this polarization branch were in disagreement. The present measurements confirm and extend those of Borgonovi et al.

One of the band-structure calculations proposed by Deegan in a previous paper to explain the observed crystalline structures of the transition metals has now been completed. The results provide a quantitatively satisfactory explanation for the occurrence of the body-centred cubic phase near the centre of the three transition series. However, the calculation fails to favour the close-packed structure near the noble metal end of the series, suggesting that the effective hard-core repulsion between nearly closed d shells is not contained in the total band-structure energy, at least when the self-consistent dependence of the potential on structure is neglected.

The covalent charge density Fourier components are calculated as the difference between the Fourier components of the total and the linear screening charges. Comparison with previous estimates of the covalent charge is given.

While the total magnitude of the bond charge in silicon conforms to Phillips' conjecture (Q_b = 2₀⁻¹), it would appear to be less well localized than his model suggests.

The Mössbauer spectra of ¹¹⁹Sn in face-centred cubic copper manganese alloys containing from 3 at.% to 97 at.% Mn have been obtained at 300 °K, 77 °K and 4·2 °K, and the appearance of hyperfine fields at ¹¹⁹Sn nuclei as the temperature was reduced has been used to define ordering temperatures.

Long range magnetic ordering has been observed in all alloys, the distributions of hyperfine fields and the ordering temperatures reflecting the change in magnetic structure at 75% Mn. The ordering is discussed.

The isomer shifts for ¹¹⁹Sn in copper and gamma manganese with respect to white tin were determined as −0·83 ± 0·01 mm s⁻¹ and −0.91 ± 0.02 mm s⁻¹ respectively, and the hyperfine field at ¹¹⁹Sn in pure gamma manganese was found by extrapolation to be 255 ± 5 kOe.

The Seebeck and Nernst coefficients of several samples of Cd₃As₂ have been measured, in transverse magnetic fields of up to 2 tesla, between liquid nitrogen and room temperature. The saturation high-field Seebeck coefficient has been determined by extrapolation of the results of these measurements, taking into account inelastic scattering of the electrons. From the interdependence of the high-field Seebeck and Hall coefficients, it appears that the conduction band is linear, i.e. E is proportional to k. Thence, from the difference between the high- and low-field Seebeck coefficients, the scattering law has been determined. If the relaxation time τ is supposed to be proportional to k^2λ-1dE/dk a value of λ equal to 0·7 is indicated.

The high-field Hall mobility of n-type silicon has been studied by assuming Maxwellian distribution function of the carriers in the valleys. Different values of inter-valley coupling constants have been used. It has been found that a good fit to the experimental results for the <111> direction may be obtained by a proper choice of the inter-valley coupling constants. But even then there remain some discrepancies in the detailed characteristics.

Absorption due to boron vibrational modes in GaP has been measured. At 20 °K gap modes of ¹¹B and ¹⁰B occur at 284·2 and 293·8 cm^-1, local modes occur at 571·0 and 593·6 cm^-1 and local +gap modes occur at 849·0 and 882·4 cm^-1. An approximate calculation of the boron vibrational mode frequencies gives reasonable agreement with experiment.

The pair distribution function for isotropic molecules in a liquid is perturbed by the application of an external electric field. This anisotropy causes a pseudo-hyperpolarizability and results in an induced double refraction. Numerical results are given for carbon tetrachloride and liquid argon, the former showing reasonable agreement with experimental values.

The condensation coefficient (defined as the fraction of gas atoms striking the liquid-vapour interface which cross it to form part of the liquid) has been measured for liquid helium II from 1° K to the λ point, and was found to be not less than 0·90 and probably unity. The measurements, made by reflecting second sound pulses from the surface, are not in agreement with the macroscopic theories of Osborne and of Chernikova; the former theory gives agreement with experiment when modified to take account of conditions in the vapour a fraction of a mean free path above the surface. The problem of providing a microscopic interpretation of these results is briefly discussed.

The optical energy levels of Mn²⁺ ion in solids are shown to be modified by the lattice vibrations. The spectral shift is calculated within the framework of covalent model treating only the acoustic phonons and the low energy part of the Debye lattice spectrum. Significant shift in the optical spectrum due to lattice vibrations is found.

The magnetic susceptibilities of manganous propionate, butyrate, laurate and palmitate were measured between 2 °K and 300 °K. At the higher temperatures, all four compounds showed Curie-Weiss behaviour with c/χ_m = T - θ, θ being negative. Magnetic phase changes occurred, with the propionate becoming strongly magnetic below 45 °K and the butyrate below 25 °K. No phase changes were observed for the other two compounds although deviations from Curie-Weiss behaviour were observed below 10 °K for both of them.

The form of the spin pair correlation function in the critical region is investigated assuming that there is a single coherence parameter κ(t, H)(t = T/T_c−1). It is found that the equation satisfied is H² = κ^2Δ/νg (tκ^−1/ν) which is closely analogous to the equation of state for magnetization H = M^δh(tM^−1/β). The relationship between the functions g(u) and h(u) is established.

Recent measurements of the specific heats of single crystals of magnetic salts are compared with theoretical calculations on the high temperature side of the Curie point for the Ising and classical Heisenberg models. It is found that there is an anomalous rise of specific heat when (T/T_c−1) similar, equals 10⁻³. This is interpreted as arising from coupling between the magnetic interaction and compressibility of the lattice.

An error in a published expression for the relative intensity of forbidden hyperfine electron spin resonance transitions, from ions in crystal-field sites of cubic symmetry, is pointed out.