Table of contents

A particular lattice Green function for the face centred cubic lattice with anisotropic nearest neighbour interactions is evaluated exactly as a product of two complete elliptic integrals. Applications of this result in lattice statistics are briefly described.

A new solution for the narrow band Hubbard model is developed and the results for the one particle Green functions are presented. The spectral density has poles and resonance terms, and is exact for zero bandwidth. The antiferromagnetic state and transition temperature are studied.

The Bloch density matrix of a disordered system can be expressed, at high temperatures, as a product arising from individual scattering centres. Using a property analogous to the idempotency of the corresponding Dirac matrix, it is shown how this approximation can be extended systematically to lower temperatures. Explicit forms are derived for the partition function of a random assembly and a liquid array with short range order, in terms of the single centre scattering properties and the structure factor.

A preliminary lowest order calculation of Brillouin scattering in molecular crystals indicates that in contrast to a formula already in the literature, the long wavelength dependence corresponds to that generally given for Rayleigh scattering, namely, lambda ^-4.

The simultaneous observation of intense magnetic field tunable anti- Stokes, Stokes and double-Stokes spin-flip scattering of 10.6 mu m Q-switched CO₂ laser radiation in InSb has demonstrated that the potential tuning range of such a near infrared source extends from 9 to 16 mu m for magnetic fields up to 100 kG. This is the first report of double-Stokes scattering in this wavelength range. The origins of the double and anti-Stokes radiation emissions are discussed.

F centres have been produced in LiF by illumination with ultraviolet light, indicating that the photolysis mechanism of defect production is operative in this material.

The methods suggested by Herbert and Tao Yuin for calculating the thermodynamic properties of a Heisenberg antiferromagnet are outlined. The relationship between the two methods and the degree of rigour involved in each are discussed.

The diffuse X-ray maxima exhibited by single crystal diamonds, which had undergone a density decrease of about 6% on fast neutron irradiation, lay closer to the Bragg positions of the irradiated crystals than those of the unirradiated crystals. Annealing resulted in shift of both the diffuse maxima and the Bragg positions. Diamonds in which the macroscopic density decrease on irradiation was less than about 12% recovered towards the original single crystal form on annealing. However, diamonds in which the density decrease on irradiation was more than about 12% reverted towards a semi-amorphous state, the density of which was about half that of unirradiated diamond, on annealing at temperatures below that required to produce significant graphitization in unirradiated diamond. Type I X-ray behaviour was observed after annealing diamonds which had undergone density decreases on irradiation of up to 9%.

A new method is proposed for the calculation of the potential of an extended charge distribution due to a lattice of point charges. The charge distribution is assumed to arise from the product of two exponential functions (Slate type orbitals). This method, which is exact for a one-centre charge distribution within the point charge lattice model, differs from the conventional 'crystal field potential' treatment in that the integration is first carried out analytically before the summation. This results in two types of lattice sum, called the 'exponential' and the 'harmonic' lattice sums. The convergence behaviours of these lattice sums are discussed. The exponential lattice sum is absolutely convergent and may be evaluated directly. Lower members of the harmonic lattice sums are conditionally convergent and transformation techniques are desirable. Explicit formulae valid for the general triclinic lattice, are derived for their evaluation, using the planewise summation scheme of de Wette.

For pt. I, see ibid., vol. 4, 263 (1971). The expressions derived for the calculations of the lattice potential are applied to crystals of different symmetry. These formulae are found to be simple to use and seem to give results superior to other approximate methods. In particular a comparison is made with values obtained by using a point potential (PP) approximation, that is, regarding the extended charge distribution as a single point. In general it is found that the lower the crystal symmetry, the poorer the PP approximation as compared to the more accurate method proposed here. The relative importance of the two lattice sums is also examined. For large orbital exponents, the exponential lattice sum is negligibly small, and the contribution to the lattice potential comes predominantly from the harmonic lattice sum. A practical way of evaluating the auxiliary Incomplete Gamma function is also described.

The lowest-order term due to anharmonicity in the thermal diffuse scattering of X-rays by some cubic crystals has been investigated. This term can be regarded as an interference between harmonic one- phonon and two-phonon scattering. The resultant intensities have been evaluated along several lines in reciprocal space, including the three main symmetry axes, for a number of face centred cubic alkali halides and for aluminium. The calculations were based on an anharmonic interaction restricted to nearest neighbour short range terms for the alkali halides with next nearest neighbour interactions included for aluminium. Significant intensity changes are found for all materials investigated.

The energy of inert gas interstitials and the activation energy for gas diffusion have also been calculated. Values for the binding energy of the gas into vacancies suggest that there may be a slight modification of the diffusion due to gas trapping. All the calculations were performed by minimizing the energy with respect to the displacements and dipoles of ions in a relatively large region (15-20 shells) surrounding the defect. A new numerical method used in the calculation of the relaxation is ten or twenty times faster than conventional minimization methods.

Long range van der Waals interactions to higher orders of both multipole and perturbation expansions are examined. In particular the dipole- dipole-dipole, dipole-dipole-quadrupole, dipole-quadrupole-quadrupole, quadrupole-quadrupole-quadrupole and dipole-dipole-octupole three body terms to third order perturbation, and the dipole-dipole-dipole- dipole term to fourth order perturbations are considered. Apart from the dipole-dipole-octupole term the contributions of these factors to the lattice energy of crystalline Ne, A, Kr and Xe are evaluated. The relative stability of the FCC and HCP structures is examined but does not appear to be firmly established by these terms. However, the results indicate that if the octupole polarizabilities of the rare gas atoms prove to be large then there would be a distinct preference for the FCC structure with respect to the dipole-dipole-octupole contribution.

The thermodynamic properties of an interacting spin-phonon system are studied by means of a diagrammatic expansion of the free energy which makes use of the drone-fermion representation for spin-¹/₂ operators. Not only do the diagrams systematize the evaluation of the traces of products of spin operators but they also assist in the physical interpretation of the results. For a model in which the spins interact independently with phonons it is found that the major effects are due to virtual phonon energy shifts. When the spins can exchange phonons coherently the excitations are coupled spin-phonon modes and the results can be correlated with the energy shifts of the spin- wave-like modes. By examining the contribution of the phonons to the thermal energy the enhancement of the phonon occupation number is calculated. For phonons with frequencies close to the spin precession frequency it is found that no anomalous peak as proposed by Becker and Clover for the independent spin mode. In the case of coupled modes agreement is obtained with the theory of Van Eekelen and Stevens.

Solid argon was irradiated with neutrons at 4.6 K in the low temperature irradiation facility of the Munich nuclear reactor FRM to produce point defects. After irradiation the stored energy of the samples was measured with a differential heat flow calorimeter at temperatures increasing at a rate of 0.3 K min^-1. By estimating the defect concentration, one obtains the enthalpy of formation of Frenkel pairs E_f^F approximately=0.5 eV. From the observed stored energy recovery spectra and the assumption that the annealing reactions are of second order, one obtains two migration enthalpies as follows: E_m1=(0.058+or-0.006) eV, E_m2=(0.130+or-0.013) eV. Those enthalpies of migration are tentatively attributed to the migration of interstitials and vacancies respectively.

A method has been developed which allows the dependence of internal friction on strain amplitude for a crystal to be measured in a period of 30 s. This method was used to study the variation of internal friction with strain amplitude in sodium chloride single crystals as a function of neutron irradiation. The results were found to agree with the Granato- Lucke theory of strain-amplitude dependent internal friction and this theory was used to examine the variation of pinned dislocation loop length with irradiation time. It was found that the loop-length variation from amplitude-dependent results was different to that from amplitude- independent results. An explanation is offered for this difference.

A theory to account for the effect of the atomic cores on an electron tunnelling through dielectric layers is developed. The theory utilizes the final-stage interaction formalism of the T scattering matrix. It is assumed that the atomic core potentials have a very short range and that they in no way overlap with each other. It is assumed that the core potentials are superimposed on smooth potential barrier U(x) which varies slowly with position. The effect of atomic core scattering on the transparency of the barrier depends on a parameter corresponding to a single scatterer, defined by F identical to (1/4 pi )(2m/h(cross)²) integral t(r,r')dr dr' and the geometry of the atomic layers, namely the distance between successive layers as well as the distance between atoms in the same layer. It is assumed that the atomic layers have a periodic structure. A qualitative calculation in relation to field emission from tungsten covered with a few layers of inert-gas atoms is discussed.

The residual resistivity rho is calculated for a binary alloy in terms of the degree of short-range order, the alloy composition and the number of conduction electrons per atom. A model is considered in which only s-s scattering occurs and in which the s electrons are described by free electron wavefunctions. This leads to an expression of the form rho varies as Sigma _i=0c_i alpha _iY_i when alpha _i are the usual Cowley short-range order parameters of the c_i atoms in the ith shell at distance R_i.Y_i is an integral which may either be positive or negative depending on R_i and the number of conduction electrons per atom. The sign of d rho /dT calculated from the above equation will thus depend on the signs of the parameters alpha _i and also of the integrals Y_i. In the system for which both the resistivity and the short-range order parameters alpha _i have been measured, substitution of the values of alpha _i into the derived equation yields results for d rho /dT in good agreement with those experimentally observed. The argument is extended to include the effects of the magnetic order-disorder phenomenon on the residual resistivity.

Microwave power transmission for both the circular modes and the linear mode of polarization through p-type germanium has been studied as a function of impurity concentration and magnetic field up to 0.85 Wbm^-2 at room temperature. Theoretical calculations have been done for the power transmission as a function of magnetic field taking into account the mixed scattering of lattice and impurities. The results have been used to estimate the mobilities due to impurity scattering and the thermal scattering.

An account is given of an experimental investigation into the source of the spin-phonon coupling in an undiluted electron paramagnet (Cs₂Cu(SO₄)₂.6H₂O). The saturation by ultrasonic (9 GHz) waves of the resonance absorption lines of the spin system, and the spin-lattice relaxation of the Zeeman and magnetic-dipole parts of its energy, have been measured at temperatures T approximately 1.4 K and in fields H between 1 and 7 kG. The results suggest that the interaction which the phonons modulate is isotropic, presumably an exchange coupling. Further it is suggested that the relaxation occurs in 'exchange pockets' in which exchange interactions are anomalously strong.

The effect is considered of macroscopic rotation of a solid about the magic axis on its nmr spectrum in those cases where the spectrum is already partially narrowed by microscopic nuclear motion within the solid. Two cases are distinguished; (a) in which rapid but restricted microscopic motion reduces the spectral width to a smaller plateau value, and (b) in which isotropic motion incompletely narrows the spectral width because it is insufficiently fast. In case (a) it is shown that macroscopic rotation can accomplish further substantial narrowing, and the limitations on the ultimate linewidth are discussed; in case (b) it is shown that macroscopic rotation at attainable speeds often has little effect.