Table of contents

The traditional unit tensor operators are shown to be orthonormal vectors in a trace metric approach. The operator equivalents of crystal field theory are used as models of the tensor operators which act as generators in Lie group theory.

The path integral describing a particle in a randomly distributed potential is examined. A heuristic discussion is given, which should elucidate the qualitative features as well as the mathematical problems of density-of-states calculations.

Resonant light scattering is considered in the Heisenberg picture by making use of operator radiation reaction theory. A crucial commutator between matter and free-field variables is shown to vanish, and the validity of the quantum fluctuation-regression theorem, like that of the optical Bloch equations which govern the reduced atomic density matrix, is firmly established, apart from renormalization difficulties, within the rotating wave approximation.

Kimball's work (1975) on the momentum distribution of electrons is utilized to establish the form of the expansion of the first-order density matrix for jellium around the diagonal. There is non-analytic behaviour and the connection between the present result and the analytic form for non-interacting electrons in established.

The eigenvalues of the homogeneous Fredholm integral equation, whose kernel is a sin² x/x² function, are shown to be non-degenerate. An upper bound for the largest eigenvalue is established and an approximate formula for evaluating the eigenvalues is suggested. Applications of these results to speckle and photocounting statistics are presented.

A numerical study of the time-like geodesics in the first axially symmetric solution of the Einstein vacuum field equations given by Tomimatsu and Sato is made. The study is made on the equatorial plane, following the method employed by Bardeen, Press and Teukolsky (1972) in the Kerr field.

The results are reported of a search for short pulses of gravitational radiation, using two split-bar detectors 30 km apart. The detectors have masses of 625 kg each and are capable of millisecond resolution over a broad band around 1200 Hz. Pulses imparting energy of ¹/₂ to 1kT to each bar, and coincident within +or-1 ms, were indistinguishable in trace characteristics from pulses due to thermal noise; and the number in the bin registering coincidences within +or-2 ms (ie 308 in total), was indistinguishable from the numbers in bins of the same timewidth but registering coincidences with time differences of up to +or-10 ms between the two detectors.

A method is given of evaluating Madelung constants for invariant cubic lattice complexes. It is shown that a linear combination of at most 8 lattice sums is sufficient to give the Madelung constant of all such lattices. In many cases this reduces to just 3 sums. The 8 sums suffice also to determine the Madelung constant for tetragonal lattices with axial ratio square root 2 or 2 and for orthorhombic lattices with lattice parameters in the ratio 1: square root 2:2. The method developed here is also applied to the evaluation of lattice sums arising from particles interacting with a general r^-s potential.

A new method for the identification and ordering of linear graphs is described. The method pays particular attention to the group properties of graphs and is easy to adapt for use with electronic computers.

A method is described for the efficient generation of lists of graphs which can be embedded on very loose-packed lattices.

The effectiveness of Chisholm's new rational approximation scheme for functions of two variables is examined for applications to problems in the field of critical phenomena. The examples chosen are double power series expansions for the high-temperature susceptibility chi ₀( alpha ,T) where the additional variable alpha is (i) the degree of anisotropy in the anisotropic Heisenberg model, (ii) the relative strength of second-neighbour interactions in Ising model systems and (iii) the relative strength of pair and three-spin interactions in a two-dimensional pair-triplet Ising model. In case (i) additional support is obtained for the principle of universality in relation to the discontinuous increase in the value of the exponent gamma at the suspected symmetry breaking point eta =1 for all quantal cases.

The solutions of the site percolation problem for some Bethe lattices of interacting spins are shown. For zero external field some arguments are given which predict for three-dimensional models p_c<¹/₂ and for planar models p_c=¹/₂ where p_c is the critical density of overturned spins for the percolation problem.

The 32-vertex model on a triangular lattice is solved exactly in two special cases. In one case the model is reducible to a three-spin Ising model and has the exponents alpha = alpha '=²/₃. The other soluble case is the free-fermion model which exhibits the Ising behaviour but may possess multiple transitions.

A compact parametrization is derived for the high-temperature series expansions of the free energy and two-spin correlation function of classical n-vector models with completely general anisotropic pair interactions. The number of independent coefficients needed for general anisotropy is much less than required in conventional expansions. The series for the free energy of the Heisenberg model (n=3) on three cubic lattices is presented to tenth order as an example.

Considers the usual Ising model with pair interactions between nearest-neighbour spins sigma on the triangular lattice. The author evaluates the correlation M₃=( sigma ₁ sigma ₂ sigma ₃), where sigma ₁, sigma ₂, sigma ₃ are the three spins at the vertices of a triangular face. This can be thought of as an order parameter associated with a three-spin 'field'. It vanishes at and above T_c, and has a critical exponent of ¹/₈.

The elastic and inelastic scattering of electrons by a standing wave of intense and coherent light are investigated by means of the Helmholtz-Kirchhoff diffraction theory. The time-averaged theories, used previously for the description of Kapitza-Dirac scattering, are shown to be inadequate.

Asymptotically free models with massive vector fields are constructed within the Su(2) group. Physical consequences of such models are briefly discussed.

The supersymmetry of Wess and Zumino is generalized to square roots of a class of ordinary Lie groups. The case of O_2,3 is studied in detail. The authors obtain supersymmetric equations of motion for fields on a variant of de-Sitter space. The only dimensional parameter is the radius of space-time.

Gives a theory of non-retarded dispersion interaction energy of molecules with charge distributions on them that readily allows inclusion of higher-order multipole interactions. The two-particle dispersion energy for a pair of harmonic oscillators is evaluated for dipole-dipole, dipole-quadrupole and quadrupole-quadrupole interactions. It also is found that the dispersion energy remains finite for all separations between the oscillators, supporting the earlier result of Mahanty and Ninham (1975) for dipole-dipole interactions with distributed dipole moments.

The electrostatic force on a conducting sphere positioned symmetrically above a uniformly charged, insulating, circular disc is calculated using the method of images. Results for spheres which are either earthed, uncharged or carry a known charge are given in terms of infinite series which converge rapidly under certain stated conditions. The dependence of the force on sphere radius, sphere-plane separation and the radius of the charged area is presented graphically.

It is shown how Brysk's expansion for the function F can be obtained directly by analytic continuation of its series representation.