Table of contents

Extended series for the spontaneous magnetization and reduced susceptibility of the triplet model on the triangular lattice are analysed using Pade approximants. Baxter and Wu's exact solution (1973) for the specific heat has been used as a guide in interpreting the approximants and it is concluded that beta =0.080+or-0.005 and gamma '=1.5+or-0.15.

A recurrence method is developed to derive a partial differential equation for the logarithm of the configurational integral of a hard-sphere gas. The free energy follows immediately, while comparison of analytical results to experimental data (on virtual hardsphere gases like argon and xenon) gives quite remarkable precision.

A preliminary analysis of irreducible unitary representations of SL(3,R) is given using O(3) shift operator techniques similar to those used for treating SU(3) in an O(3) basis. A full analysis is given of the Delta l=2 representations, for which minimum l values of 0, ¹/₂ and 1 are found, but not the l_min=³/₂ representations proposed by Biedenharn et al. (1972).

The transformation coefficients connecting the Stark and the angular momentum states belonging to the positive spectrum of the hydrogen atom are calculated by a group- theoretical method which makes use of the O(3,1) symmetry of the states and also by a purely analytical method. The results of the two calculations agree except for an undetermined factor not containing the orbital quantum number l. Complete agreement between the results is achieved by taking the normalization factors for the continuum states to be analytic continuations of those for the bound states. The transformation coefficients turn out to be SU(2) Clebsch-Gordan coefficients (j₁j₂l; m₁m₂m) with complex j₁j₂m₁m₂, and physical l,m. From the general theory the well known expansion of the Coulomb scattering function is obtained by giving the magnetic quantum number m and one of the electric quantum numbers n₂, the special values, 0 and -1, respectively.

Kolmogorov (1957), Arnol'd (1973) and Moser (1962) proved that invariant toroids of N dimensions occupy a finite volume of the 2N-dimensional phase space of nearly integrable bounded systems of N degrees of freedom. Variational principles are stated for such invariant toroids.

The laws of conservation of energy and momentum are used to eliminate Planck's constant from the usual relationship for Compton scattering. The resulting expression is identical to the relativistic Doppler equation and it is shown that, in the proper frame of the interaction Snell's law is obeyed and the scattering is specular. This leads to the possibility that the reflection occurs in a region rotating about the centre of the electron at almost the velocity of light.

Recently Biedenharn and Gamba (1972) described a simple group-theoretical method to calculate the quantitative splitting of a degenerate energy level under the action of a symmetry-breaking hamiltonian. The authors provide the general proofs for the rules of the method and clarify the group-theoretical background. Furthermore they discuss the various kinds of multiplicities which can arise and the difficulties which they entail.

For pt. III see abstr. A79584 of 1973. The case of materials which exhibit hereditary-or memory-effects is examined. By requiring that the general constitutive equations for a so-called 'simple' material obey different principles of formulation such as those of equipresence, material indifference, and fading memory, and studying the restrictions placed upon the response functionals by the second principle of thermodynamics, a complete set of functional constitutive equations is derived. In particular, it is shown that the heat flux is a functional with respect to the temperature gradient. This result allows the formulation of a heat conduction law which provides a possible answer to the paradox of infinite propagation velocity of thermal disturbances in relativity. By examining the limiting case of steady-state processes, the present formulation is shown to incorporate, in a general frame, the results obtained for nonlinear elastic solids in the preceding papers of this series.

It is shown that Einstein equations allow a special class of stationary solutions which correspond to spherically symmetric clusters of particles in circular motions, the total angular momentum of the cluster being zero, and all orbits being performed with the same period. The mass density of such clusters is everywhere regular and positive, decreasing with increasing radius.

The phase transitions of lattice fluids with molecules defined by first-neighbour exclusion, and interacting at short range, are considered using extended Kikuchi approximations. On the square lattice, a second-neighbour interaction does not produce a disordered low-density transition, but does make the packing transition first order below a critical temperature epsilon beta =-1.05. The resulting phase diagram therefore has only generalized fluid and solid regions. In a Kikuchi double- square calculation with third-neighbour interaction a disordered transition is produced, but this overlaps with the ordered state and again there is no stable intermediate liquid phase. Short-range interactions produce qualitatively the same effect on the triangular and simple cubic lattices. Particularly for the ordered state, the large set of non-linear equations produced by an analytical treatment of extended Kikuchi approximations have limited the range of possible calculations.

A large class of ideal fluids satisfies the equation pv=gU, where U is the internal energy and g is a characteristic constant of the system. The 'ideal quantum gases' are here defined thermodynamically by this equation. These systems then include weakly interacting fermions or bosons either in the non-relativistic or in the extreme relativistic limit. The relationship between different ideal quantum gases can be shown in a diagram with horizontal coordinate g. It is desirable to represent the classical ideal gas, which is here defined thermodynamically by pv=NkT, 'orthogonally' by a horizontal line and this is achieved by the choice of the coordinate y=(1-1/ gamma ) (1+1/x), where gamma is C_p/C_v. The ideal classical gas has then states which lie on y=1. Such a diagram has the additional remarkable property that important simple systems have states represented by one single point. The theory of this diagram and the associated equations of state are analysed in detail.

The radiation of a small system of harmonic oscillators is analysed. The exact solution of the problem in the dipole and rotating wave approximation is discussed. It is shown that only vibrations of the centre of charge are radiatively damped and that their damping constant is N times larger than for a single oscillator. Due to the collective emission, broadening and shift of the line, dependent on the number of oscillators, also occur.

The paper investigates the dispersion relation and the time and space damping rates of the high-frequency surface waves propagating on a warm homogeneous nonisothermal current plasma bounded by a vacuum. The description is done on the basis of a kinetic plasma theory under the specular reflection condition for particles at the plasma-vacuum boundary.