Table of contents

The energy derivative of the Dirac matrix for spherically symmetric potentials is shown to be a product of radial wavefunctions and an energy dependent term which is simply related to the phase shift.

Lattice sums required for evaluating the effect of the three-body triple dipole dispersion interaction on the elastic constants of face centred cubic crystals are evaluated. Previous calculations omitted a nonzero lattice sum. The present methods may have advantages in terms of rate of convergence.

Multiple scattering effects in cold neutron scattering by hydrogeneous liquids in an infinite flat sample are investigated. The intensity of twice scattered neutrons is calculated with no restrictions on angular dependence or energy transfer. Using a dynamical model for liquid methane numerical results are obtained. In the inelastic region, the intensity ratio of secondary to primary scattering is found to vary from about 0.50 to 0.30 as the scattering angle is increased from 30 degrees to 90 degrees , for a specimen thickness of 0.1 cm. Explicit multiple scattering corrections are made for the Trombay time of flight spectra. As a result, the agreement between experiment and theoretical cross section is improved significantly.

The contribution of the triple-dipole, three body force to the correlation energy of liquid ⁴He is calculated near 0 K. It is found to be less than 1% of the two body contribution, in contradiction to the previous estimate of Davison and Feenberg (1969).

The conductivity of pure NaI and NaI+CdI₂ and the ²²Na⁺ diffusion in pure NaI has been measured as a function of temperature. From a comparison of the diffusion results obtained by the two methods it is concluded that the point defects in NaI are of the Schottky type and that the cation selfdiffusion, which is the dominant contribution to the conductivity, occurs via isolated vacancies. The conductivity results for the Cd²⁺ doped crystals have been analysed in terms of the simple theory of ionic conductance. This analysis gives values of 2.27 eV, 0.47 eV and 0.51 eV for the enthalpies of Schottky defect formation, cation vacancy migration and Cd²⁺-cation vacancy association respectively.

Measurements of the nuclear magnetic resonance relaxation times T₁, T_1p, and T₂ made on pure and Cd²⁺ doped single crystals of NaI, over a temperature range 385 to 900 K, are reported. The process responsible for relaxation is Na⁺ selfdiffusion. The cation migration enthalpy is found to be 0.45+or-0.02 eV, and the activation energy for the intrinsic diffusion process is found to be 1.6+or-0.05 eV, in good agreement with ionic conductivity and tracer selfdiffusion studies. An expression for T_1p due to dipolar coupling between unlike spins is obtained, and is used to relate the T_1p measurements to the sodium ion jump frequency. A comparison of the sodium ion jump frequency obtained from the relaxation time measurements, and from ionic conductivity measurements is made. This indicates that at high temperatures diffusion takes place predominantly via cation single vacancies. At lower temperatures impurity-vacancy complexes contribute to the relaxation indicating that the free and complexed vacancies have similar diffusion rates. At the lowest temperatures impurity precipitation takes place and the total concentration of Cd²⁺ as a function of temperature is calculated from the T_1p measurements, and is in good agreement with that calculated from conductivity studies.

The electron charge density maps as determined by X-ray diffraction experiments lead to the conclusion that the charge transfer in lithium fluoride is about 0.88e rather than 1.0e usually accepted for all the alkali halides. A theoretical investigation of the lattice vibrations in LiH, LiF and LiCl has been carried out with a view to obtaining an independent check on this observation of partial charge transfer. The model used is a modified shell model which takes account of three body overlap forces arising in ionic solids due to nonorthogonality of free-ion wavefunctions and in which the valence Z of an ion enters as an independent parameter. It would appear that Z=0.88 is a more appropriate choice than Z=1.0.

An expression for the free energy of cubic crystals at finite temperatures is derived using an adaptation of the Bethe approximation. The harmonic and quartic anharmonic energies are in good agreement with the results of standard lattice dynamical methods at 0 K and in the high temperature limit. The cubic anharmonic energy is however, in error. The theory is applied to calculating the equilibrium lattice constant, specific heat at constant volume, expansivity and isothermal bulk modulus of solid krypton in the range 0 to 100 K. From 0 to about 40 K, where perturbation theory is valid for krypton, reasonable agreement is obtained with experiment considering the simple Lennard-Jones (12-6) potential used in the calculations. Beyond 40 K perturbation theory is no longer valid and increasing differences between theory and experiment are observed. Comparison with the results of a standard lattice dynamical calculation of Klein et al (1969-70) shows good agreement below 40 K and increasing disagreement with increasing temperature above that temperature.

The effective dynamical matrix for long wavelength acoustic phonons in a symmetry-induced zero-gap semiconductor is derived from microscopic considerations of the electron response to lattice displacement. A nonadiabatic theory is shown to be necessary in the very long wavelength limit for a pure system at T=0. Macroscopic correspondence is shown to require an effective charge-neutrality sum rule. This sum rule is used to examine acoustic phonons at somewhat shorter, but still long, wavelengths, and leads to the prediction of two regions of linear dispersion with different sound velocities separated in alpha -Sn at about a wavelength 10³ cm. The degenerately doped system is also considered at low temperatures where it is shown that for wavenumbers much less than the Fermi wavenumber the acoustic phonons have a linear dispersion determined primarily by intraband polarization but with a change of character above k_F.

The connection between the linearized phonon collision operator and the Callaway model for it is clarified. A derivation of this model is given for the case of large departure from equilibrium, together with a modification in its interpretation. The amended model is applied to thermal conduction with large temperature gradients and to the problem of relaxation to equilibrium.

The attenuation of 9.375 GHz longitudinal acoustic waves in germanium has been investigated for high photoexcited carrier densities, at temperatures between 1.6 K and 4.2 K. The attenuation was not measurable for light intensities less than about 10% of the maximum intensity and increased linearly with light intensity above that value. The relationship between light intensity and photoconductivity was investigated and found to have a discontinuity of slope at about 10% of maximum intensity. The observed hypersonic attenuation increased markedly with decreasing temperature and was about a thousand times larger than that predicted for free carrier absorption.

Mechanisms for the previously reported increase in attenuation of 9.375 GHz longitudinal hypersonic waves in germanium when irradiated by light are discussed. The experimental data are found to be consistent with attenuation via the exciton-phonon interaction and agree with theoretical considerations of Lipnik and Ruvinskii. The data suggests that the ground state of the exciton is split by 0.1 meV.

The thermodynamic theory of solids is generalized to take explicit account of internal strains in perfect non-Bravais crystals. Treating external and internal strains as independent leads to the definition of generalized thermodynamic properties, namely elastic constants, heat capacities, thermal expansion coefficients and Gruneisen parameters. Expressions are obtained relating these to the usual bulk properties, thus facilitating the calculation of the bulk properties from theoretical models and also providing a method of analysing experimental results obtainable from the direct measurement of internal strain. The theory can be applied both to the complete set of internal and external strains and to restricted sets appropriate to particular experimental conditions (eg those that preserve symmetry). By way of illustration, the theory is applied to the trigonal structure of selenium and tellurium, and a simple force constant model with this structure is used for the calculation of elastic stiffnesses for symmetry preserving strains. The limitations and advantages of the present theory are discussed and contrasted with those of methods previously used in treating internal strain.

The problem is formulated of the energy levels deriving from the degenerate valence bands of a semiconductor in the presence of an impurity potential and a magnetic field. An approximate solution is found using perturbation theory, the variational principle and the adiabatic approximation.

The problem of ac loss in the Meissner state of type II superconductors has been examined theoretically by considering the field enhancement and consequent flux penetration at surface irregularities. A rough estimate of the loss can be made with little difficulty, but to study the problem in more detail the author equates the Lorentz force, (- phi ₀*curl H(B,x)), with the pinning force on the fluxons. This gives the flux gradient as a function of flux density and position. Numerical integration is necessary to obtain the flux distribution and hysteresis loss, but the final results are in good agreement with experiment, and show that the loss is dependent upon the shape, the purity, and to a lesser extent the metallurgical state of the surface region. The low frequency motion of small quantities of flux trapped in the superconductor perpendicular to the surface has also been examined, and the contribution to ac loss determined.

Using the surface Green function method of Garcia-Moliner and Rubio (1969-71) the extra density of states of metal-Si(110) contacts is studied in detail. This is then used to calculate the barrier height for a wide range of Sommerfeld metals. The surface position of the Fermi level relative to the band edges of the semiconductor depends on both the work function and the Fermi energy of the metal. Some arguments are put forward to the effect that a similar behaviour is expected for crystallographic orientations of the Si surface other than the (110). The calculations are actually in fair agreement with experimental data on (111) surfaces.

A numerical study of field emission from tungsten covered with germanium layers of atomic thickness is given, based on a one dimensional pseudopotential model for the adsorbed atoms. As a result a theory is proposed which explains at a semiquantitative level both the reduction in the total current, and the appearance of a second peak in the total energy distribution of the field emitted electrons. The observed variation of the field emission characteristics with the thickness of the adsorbed germanium is also explained.

A theory is proposed for the evaluation of electron tunneling probabilities through a slowly varying potential barrier containing potential holes. In particular a method is proposed for the evaluation of the multiple scattering between strongly interacting holes. The method utilizes the two stage interaction formalism of the T scattering matrix. The series describing in all orders the scattering of the tunneling electron by the potential holes is summed using a technique of Beeby (1968) modified accordingly. The theory is developed for both the one dimensional and the three dimensional case. A discussion of the qualitative correctness of one dimensional calculations, with particular reference to field emission, is included. No numerical results on specific cases are reported.

For pt. I see ibid., vol. 4, 2097 (1971). The results of the previous paper with this title by the authors are applied to the electrical susceptibility and the specific heat of KDP and its deuterated isotope, in their paraelectric phase. It is found that the isotope effect and critical behaviour of these and other properties are explained in a selfconsistent way by our simple model if we allow for the expansion of the short hydrogen bonds on deuteration, an effect which has been discussed in connection with another model of KDP. The theory predicts the logarithmic behaviour of the specific heat above the critical point which has been observed recently by Reese. To explain the ferroelectric transition in full detail appears to require a more sophisticated model which would include short range forces.

A Green function method is developed to describe the excitations of a simple cubic Heisenberg ferromagnet which contains a small number of randomly distributed impurity spins. The method used demonstrates some novel features in the form taken by the averaged Green function describing the system and eliminates the need to work with spurious parameters depending on the square root of the impurity spin. A brief discussion is given of the eigenvalues and densities of states associated with the impurity spin wave modes and of the modification to the host spin wave band.

Measurements of the magnetic susceptibility of FeV₂O₄ between its Curie point of 109 K and 600 K, are reported. The experimental values are fitted against Neel's equation for the inverse susceptibility. A value for the Curie constant C_M=4.9+or-0.4 Bohr magnetons per molecule is found, and this, together with the low temperature saturation magnetization of 1.95 Bohr magnetons per molecule observed by Rogers et al., (1963), seems to imply that FeV₂O₄ is composed from a mixture of a normal and an inverse spinel.

The thermal expansion coefficients of CuCl₂.2H₂O have been measured in the temperature range 2-20 K using the three terminal capacitor technique. At the Neel temperature 4.37+or-0.01 K, lambda type behaviour was observed. From the results in the paramagnetic phase it has been possible to separate the lattice and magnetic components of the observed expansion. The magnetic thermal expansion coefficients are found to be proportional to the magnetic specific heat except for temperatures between 4 and 7 K. Qualitatively the form of the expansion is consistent with an exchange striction mechanism.

A Green function calculation for spin waves in a dilute antiferromagnet is presented. By generalizing a calculation of Kaneyoshi (1969) it is shown that in a second order Born approximation the spin wave modes become unstable at C*=0.296. Diagrammatic techniques are given from which selective summations may be performed. A possible candidate for experiments is indicated.

Dysprosium phosphate (DyPO₄) orders antiferromagnetically at a Neel temperature of 3.4 K; the direction of the sublattice alignment is the c axis of the tetragonal unit cell. This ordered state arises from a combination of dipolar and exchange interaction and is well described by the Ising model for a highly anisotropic system of magnetic ions with ground state g values of g_c=19.5 g_a=g_a' approximately=0. Analysis of the high field Zeeman spectra below 20 K on the transition Z₁(⁶H₁₅2/) to G₁, which displays a pronounced quintet structure, enables an evaluation of the strength of the exchange interaction to be made. The result is confirmed by a direct determination of the interaction field from high field moment measurements at 0.5 K.

The orientation dependence of W₂/W₁ has been measured for ²³Na in sodium nitrate at room temperature, using a steady state double resonance technique. The following ratios of the components were found: M₁₃₁₃/M₁₁₁₁=0.850+or-0.015, M₃₃₃₃/M₁₁₁₁=0.822+or-0.016, M₁₁₁₃/M₁₁₁₁=-0.120+or-0.018, M₁₁₂₃/M₁₁₁₁=0.142+or-0.017. Using symmetry arguments it is shown that M₁₁₂₃ should be zero in the coordinate system. The reason for the discrepancy between experiment and theory is not known. The values are compared with those obtained by Niemala (1967) at 77 K using a pulse technique. The results of a simple point charge calculation which takes into account only nearest neighbour atoms agree remarkably well with experiment. It is shown that the ratios of the M components are determined more by the configuration of nearest neighbour atoms than by the details of the lattice vibrations.

The authors discuss the origin of the photoluminescence of Er³⁺ in CdS and shows that the sensitizing behaviour of the group I co-dopants are rather different from those envisaged so far and that a judicious choice of preparative conditions leads to the observation of higher energy rare earth transitions. It is further shown that centres capable of producing broadband emissions in the same position as the emitting rare earth energy levels must be present in order for sensitized emission from the rare earth ions to be observed.

The electron spin resonance spectrum of Cu²⁺ in NH₄F was observed between 77 and 300 K. No evidence was found of a lambda transition in NH₄F. The spin Hamiltonian parameters at 77 K were g_x=2.0771, g_y=2.1040 and g_z=2.4702 with the hyperfine parameters all less than 4*10^-4 cm^-1. These parameters suggest that Cu²⁺ is in a tetrahedral site. Expressions obtained for the possible ground states suggest that the E"(B₂) state is lowest.

Approximate ground state wavefunctions and potentials for iron group transition metal impurity centres in semiconductors, for instance Cr, Fe, Cu, Ni, Co, Mn in GaAs are calculated using a model in which both the screened long range Coulomb potential and the short range core potential are taken into consideration. The magnitude and the wavelength dependence of the photoionization cross section are obtained via the wavefunction, in favourable agreement with experiment.

The magnetic resonance absorption at g=2 in irradiated MgF₂ is positively identified in this paper as arising from F-centres. EPR and ENDOR investigations have determined the hyperfine constants and tensor axis orientations for the first six fluorine shells. These agree in detail with the interactions expected from an electron trapped at a negative ion vacancy. The large nearest neighbour interaction (A/sub ///=382.83 MHz, A_{perpendicular to} =288.23 MHz) coupled with the smaller second and third neighbour interactions have been used to reconstruct in detail the resolved EPR spectra when H₀ is parallel to the crystal symmetry axes. A semiempirical wavefunction obtained from the observed contact interactions for all six shells predicts that neighbour cation ENDOR transitions for the 10% abundant ²⁵Mg nuclei should be observed near the first neighbour fluorine transitions.

For Pt. I see ibid. vol. 4, 2992 (1971). The 260 nm optical absorption band in irradiated MgF₂ has been identified as the F band. The irradiation dose dependence, irradiation temperature dependence, optical bleaching behaviour and thermal annealing properties of this band and the EPR F centre were compared using the same samples for both the optical and EPR work. A close correlation of all the properties measured with both techniques established that in irradiated MgF₂, the 260 nm band is indeed the F band with a oscillator strength of approximately one.

When hydrogen is diffused into calcium fluoride crystals which contain both rare earth (RE) and oxygen impurities, RE³⁺- H^- complexes of trigonal symmetry, labelled T₃, are formed. These centres have been detected by measurements of the infrared absorption from the localized vibrations of the paired H^- (or D^-) ions, and in the case of crystals containing Yb³⁺ ions these measurements have been correlated with EPR spectra and with optical electronic transitions. It is shown that the RE³⁺ and H^- ions occupy nearest neighbour substitutional sites and that T₃ is a simple modification of another trigonal centre, T₁, which is sometimes present in crystals grown by the Czochralski technique. It is suggested that models proposed previously for T₁ and for another centre of trigonal symmetry (T₂) may be incorrect, and alternative proposals are put forward.