Table of contents

For pt.I see ibid., vol.19, p.L123, 1976. Radial moments of folding integrals for some non-spherical distributions that appear in problems relating to inelastic scattering off nuclei are evaluated in simple closed form in terms of the radial moments of the folded functions.

Using the variational block-spin approximation of Kadanoff (1975) a renormalization group recursion relation suitable for two-layered Ising systems is constructed. The critical behaviour of the Ashkin-Teller model is described in some detail.

The nature of the critical point in the triangular Ising model with pure triplet interactions is discussed. In particular, it is shown that the correction terms to the asymptotic power law behaviour of thermodynamic quantities require the existence of a third relevant scaling field in accord with recent renormalization group calculations. The exponent of this field is found to be y₃=7/8.

For pt.II see ibid., vol.9, p.1411, 1976. Using Poisson's summation formula of dimensionality less than or equal to three, a number of slowly convergent three-dimensional lattice sums, which appear in the theory of ionic crystals, have been reduced to rapidly convergent two-dimensional sums. As a result, some of the formulae reported recently by Hautot (1974, 1975) and by Zucker (1975, 1976) are reproduced and several new ones which exhibit a remarkably fast convergence are obtained.

Solutions of systems of linear equations or linear recurrence relations are obtained in the form of series of products of continued fractions. Simple rules are presented for obtaining the series of any given problem. The continued fraction expansion is superior to ordinary perturbation theory; each continued fraction represents a partial summation of the ordinary perturbation series. The use of the method in eigenvalue problems, and its possible application in numerical analysis are discussed.

The non-linear Schrodinger equations governing the evolution of the plane wave solutions of the sine-Gordon and the non-linear Klein-Gordon equations are derived. The former is unstable against modulational instability and thus evolves into envelope solitons. The latter however can be unstable or stable depending on the sign of the cubic term and can accordingly admit envelope soliton or hole solutions.

Using the Monte Carlo techniques single steps on Kossel-type crystal surfaces were simulated under equilibrium conditions up to the temperature region where statistical theories excluding overhanging step edge (ledge) configurations are questionable. The ledge length was varied between 40 and 960 units leading to average maximal step widths up to 40 units, so reaching dimensions observable with modern electron-optical techniques. Jump and pair frequency distributions of the ledge were investigated after an equilibrium state was reached by direct tracing of its successive paths over the surface. The distributions were compared with those of an analytical low temperature step model.

The relativistic equations of motion for a charged particle in interaction with a plane electromagnetic wave with arbitrary amplitude are solved rigorously both in terms of the co-moving time and the inertial time. The special case of a harmonic wave is considered in detail and a discussion is given concerning the radiated energy and the radiation reaction.

The exact penetrability for an inverted biharmonic oscillator potential is calculated and compared with that obtained from the WKB method.

The concept of fuzzy phase space Gamma _s, recently introduced in non-relativistic quantum mechanics, is extended to the relativistic case. The L²( Gamma _s) representation of the wave packet is used as a basis for discussing localizability in Gamma _s for non-zero mass particles as well as for the photon.

A generalization of the relativistic grad equations when time reversal symmetry for the scattering matrix does not hold is used in order to obtain a new set of transport equations for a gas in the vicinity of the Juttner-Synge equilibrium. In the particular 'Fourier situation' the heat equation very much resembles that of Cattaneo and Vernotte (1958). However, when PT invariance is introduced, this equation reduces to the usual Eckart-like one (1940). Nearly identical conclusions are obtained for a 'Bel-like' gas.

Following a method developed by Bel, Salas and Sanchez-Ron (see Phys. Rev. D, vol.8, p.1099, 1973) the equations of predictive relativistic mechanics are solved for the gravitational interaction of two structureless point particles, up to the second order in the coupling constant g identical to Gm₁m₂. The equations of motion for the 'static' case are explicitly given as well as the value for the perihelion shift of a planet in the corresponding Kepler problem.

A class of solutions of the Einstein-Maxwell field equations which satisfy the condition that the null tetrad, determined by the electromagnetic field, is parallelly transported along the curves tangent to the complex null vectors m_i and m_i is investigated. For the case when m_i has zero twist the general solution of the field equations is found in closed form. A symmetry of the Newman-Penrose equations (1963) is used to show the relationship between this class of solutions and those recently studied by Tariq and Tupper (1974, 1975).

The theory of light scattering from a solution of interacting spherical macroparticles as presented by Altenberger and Deutch (see J. Chem. Phys., vol.59, p.894, 1973) is re-examined. The Oseen model for the hydrodynamical interaction between the particles, is shown to omit a lowest order concentration term to the effective diffusion coefficient. This model is corrected to lowest order in the ratio of particle diameter to interparticle separation and the effective diffusion coefficient re-calculated. It is found that D_eff=D⁰(1+3.0 phi ), where D⁰ is the infinite-dilution self-diffusion coefficient for the particles and phi is the volume fraction. This is compared with earlier theoretical and experimental results.

Series data for the mean cluster size for site mixtures on a d-dimensional simple hypercubical lattice are presented. Numerical evidence for the existence of a critical dimension for the cluster growth function and for the mean cluster size is examined and it is concluded that d_c=6. Exact expansions for the mean number of clusters K(p) and the mean cluster size S(p) in powers of 1/ sigma where sigma =2d-1 and p<p_c are derived through fifth and third order, respectively. The zeroth-order terms are the Bethe approximations.

The critical exponents and scaling function are calculated with the Wilson-Feynman graph technique to second order in epsilon =4-d for an n-component system with anisotropic Brillouin zone, using a general cut-off for the bare propagator G(p,r₀, lambda )=(r₀+p²)^-1f(p/ lambda ,r₀/ lambda ²). Universality holds provided that f(v,x) is analytic in v and x around (O,O) and the first derivative f_v¹(O,x)=0.

The linear mono-energetic Boltzmann equation with isotropic scattering is considered for a semi-infinite medium in plane geometry and the spectrum of the corresponding operator under perfectly reflecting, vacuum, generalized or diffusely reflecting boundary conditions is explored in the frame of the 'initial-value problem'. By the Hille-Yosida theorem, the existence and uniqueness of the solutions of these problems are assured. As a common feature, one observes the absence of a true isolated asymptotic eigenmode, the solution displaying, due to the infinite extent of the medium, only 'transient' modes.

The electrostatic force on an earthed conducting sphere positioned symmetrically above a circular disc of charge placed on a dielectric half-space is calculated. Bispherical polar coordinates are used when the sphere does not touch the half-space and degenerate bipolar coordinates are used when the sphere touches the half-space. Results are given in terms of rapidly converging infinite series in the former and infinite integrals in the latter. The dependence of the force on sphere radius and sphere-half-space separation is presented graphically.

An analogue of Shirley's equation (see Phys. Rev., vol.128B, p.979 1965) for a spin-1 system subjected to a periodic perturbation is derived, and shown to be consistent with the results of Brossel and Bitter (1952).

Acausal interactions of a massive spin-1 field such as symmetric tensor and quadrupole moment couplings are not covariant at the basic c-number level. Curiously enough, certain causal derivative couplings are also non-covariant.

The problem of the construction of coherent states of relativistic particles (bosons and fermions) in electromagnetic fields has been solved. The states discovered are exact solutions to the Klein-Gordon and Dirac equations; the wavefunctions of the particles are found in the homogeneous magnetic and electric fields and the Redmond configuration field, as well as in the plane-wave field. The formulation of relativistic quantum mechanics is used on the 'null' plane.

An algebra providing causal minimal interaction with an external electromagnetic field is considered. It is incorrectly claimed in an earlier paper (see ibid., vol.9, p.1025, 1976) that this algebra was infinite. It is shown to be finite, but not large enough to support a theory with spin greater than one. A method is given for finding the size of some Young symmetrizer algebras, which may provide further examples of causal theories.

The interaction of radiation with matter in a high-temperature environment is studied. The radiation pressure is calculated carefully, including the coupling to the high-density electron plasma. The calculation yields a correction to the expression for radiation pressure given by Inman (1965). The results are applied to investigate whether radiation pressure can produce significant alterations of the electron density in atoms.

Continuum theory is employed to investigate the onset of a certain type of thermal instability when a sample of nematic liquid crystal confined between two horizontal flat plates is in the presence of a vertical temperature gradient. Two particularly simple experimental situations are examined, in one a parallel orientation is obtained at the plates and in the other the boundary orientation is perpendicular to the plates. Using a Fourier series method, an expression for determining the critical temperature gradient at which instability sets in is derived. In both cases, the analysis allows for the presence of an applied magnetic field.