Table of contents

The equivalence of two published formulae (Power and Thirunamachandran 1974, Kielich 1968) for the isotropic averaging of fifth-order Cartesian tensors is demonstrated.

An analysis is undertaken of the mean surface s of clusters of size n from Monte Carlo data simulating a two-dimensional Ising model. At sufficiently high temperatures the data represent a percolation process and it is found that the clusters are completely ramified (tree- or sponge-like). At temperatures just below T_c the data do not correspond to circular droplets as assumed in standard nucleation theory and typical samples show a great deal of ramification. It is concluded that the parameters in Fisher's droplet model are empirical and should not be given a direct physical interpretation.

The recent scheme of approximants formed from double power series and proposed by Chisholm (1973) is examined in relation to known functions containing simple singularities of the form (x-x_s(y))^{- gamma} , in cases where x_s(y) is linear. Attention is focused on the usefulness of these approximants in determining sequences of approximations to the locus of singularities x_s(y), and the exponent gamma .

A special one-dimensional transport equation is solved by the variational technique of Djukic and Vujanovic (1971). Reasonable ansatz with one, two and three variational parameters are used. The variational solution is compared with the exact solution and its least mean-square approximation for the same ansatz.

It is shown that with certain restrictions on the trial functions the Schwinger variational method yields a convergent process for computing the scattering amplitude in scattering from short-range potentials.

A discussion is presented of the canonical averaging of time-independent, nearly multiple-periodic systems having Hamiltonians of the form H=H₀(J_alpha )+ lambda H₁(w_alpha ,J_alpha ;q_k,p_k)+ lambda ²H₂(w_alpha ,J_alpha ;q_k,p_k)+... where H₁, H_s,... are periodic functions of the angles w_alpha . A perturbation procedure is given for constructing a direct canonical transformation converting the Hamiltonian into a new one, independent of the proper angles w_alpha , irrespective of whether the system is non-degenerate or intrinsically degenerate. In each case constants of motion to all orders of the perturbation theory exist, corresponding to each proper angle w_alpha .

Two experiments are described in which mirrors were mounted on rotating turntables and measurements were made of the frequency of the reflected radiation. The order of accuracy was one part in 10¹⁷, which exceeded that required to detect second-order relativistic phenomena. It is shown that the mirrors are equivalent to transponder systems and that the application of logical arguments to the telemetered observations confirms earlier theoretical predictions of the radius and angular velocity of a rotating system measured entirely from a point within that system.

Einstein's field equations are solved for gravitational radiation propagating in a background metric given by the interior Schwarzschild solution. Expressions for the transmitted fraction as a function of frequency and propagation distance are found. Absorption is found to be present only for frequencies less than a critical frequency, the 'gravitational plasma frequency', which is roughly proportional to the square root of the mean density of the medium. The plasma frequency v_crit is found to be a useful concept for gravitational radiation, though the fraction transmitted does not fall as rapidly with distance of propagation below v_crit as it does in the analogous electromagnetic case and the transition to total transmission is not as abrupt as a function of frequency. For a neutron star v_crit=6700 Hz, well above the low frequencies of most current gravitational radiation detectors.

Solutions of Rosen's equations of gravitation for a homogeneous isotropic universe are obtained. The solutions resemble the Lemaitre universe. A universal repulsive force, introduced in general relativity by a positive cosmological constant, appears directly as a consequence of the (unmodified) field equations.

The exterior field of the Einstein-Rosen cylindrically symmetric metric in the Lyttleton-Bondi universe is considered and exact solutions are obtained in some specific situations. The solutions exhibit singularities on the z axis as well as at spatial infinity. The geodesic path of a test particle exhibits a spiral structure around the axis of symmetry.

The consistency of some assumptions made by Martin, Siggia and Rose (1973) in their operator formalism for the statistical dynamics of classical systems is demonstrated. This is achieved by the introduction of a simple representation for the operators involved.

The dynamics of Brownian particles in dilute dispersions interacting through a long-range repulsive potential are considered. Emphasis is on interpretation of recent light scattering photon correlation measurements on such a system. It is shown that the initial decay of the temporal correlation function F(K, tau ) of the Kth spatial Fourier component of particle number density is determined largely by free-particle Brownian motion. The initial decay rate of the normalized correlation function is given by D₀K²S(K)^-1 (D₀ is the free-particle diffusion constant and S(K) the static structure factor). The behaviour of F(K, tau ) at larger correlation delay time tau is discussed briefly.

For pt.VI see abstr. A72435 of 1973. It is shown how the principle of complete code balance can be exploited in a systematic way and the method is generalized to apply to partial codes. Euler's law of the edges is used to establish a latent symmetry property of the code system. The new results make possible the derivation of extended series.

For pt.VII see ibid., vol.8, no.9, p.1441 of 1975. The principle of partial code balance and the property of latent symmetry of the honeycomb-triangular code system are used to derive five new ferromagnetic polynomials psi ₁₇ up to psi ₂₁ for both the triangular and honeycomb lattices. For the honeycomb lattice the corresponding antiferromagnetic polynomials up to psi ₂₁^a are also derived.

For pt.VIII see ibid., vol.8, no.9, p.1448 of 1975. The derivation of high-field expansions for the honeycomb and triangular lattices is described briefly. New results are given for the high-field polynomials L₁₁ and L₁₂ on the triangular lattice and L₂₂, L₂₃, L₂₄, L₂₅ on the honeycomb lattice. The complete codes (partial generating functions) F₁₁ and F₁₂ which determine the corresponding sublattice polynomials are also derived.

The derivation of low-temperature (high-field) series expansions for the Ising model with pure triplet interactions on the plane triangular lattice is described. Euler's law of the edges is used to transform the linkage rule into a form convenient for the derivation of ferromagnetic polynomials. Explicit results are given for the ferromagnetic polynomials corresponding to the first twelve powers of the temperature variable (u₃) both as functions of the field variable ( mu ) and, in zero field, as functions of the temperature variable (u₂) of the simple Ising model.

For a particle with an anomalous magnetic moment the differential cross section for the emission of a photon due to interaction with an intense external wave field of circular polarization is calculated. The scattered frequencies appear in triplets of very small separation and there is in addition a 'zeroth' harmonic of very low frequency. In the cross section the contributions from the anomalous moment add to the relativistic terms in the corresponding formula for normal particles. The transition probabilities for all three numbers of the triplets differ from each other. For neutrons only the zeroth, first and second harmonics can appear. The intensity of the zeroth harmonic turns out to be very low both for electrons and nucleons.

From the Majumdar solution a simple analytical expression is obtained for the non-steady separation factor of a binary isotopic mixture in a thermal diffusion column closed at both ends. The result represents the exact solution very well, it is independent of the initial concentration and allows the value of the magnitude gamma =H²/ mu (K_c+K_d) to be easily obtained from experiment. The proposed formulation qualitatively interprets the principal characteristics of the experimental results of separation of ³⁶Ar-⁴⁰Ar at four temperatures. The influence of the non-active volumes present in the column is analysed and, by means of numerical calculus, an empirical expression is presented which gives a very good account of their influence on the determination of gamma .

The continuous n-component Potts model is studied in the framework of Wilson's multiplicative renormalization group. Apart from the isotropic phi ⁴ interaction, there is another one inherent in the model for n>2. For n>or=4, it is distinct from any phi ⁴ interaction that has been investigated in detail. Critical exponents are calculated. The n dependence of the fixed points shows new behaviour; in particular for n in the neighbourhood of five, the epsilon expansion must be reformulated as a power series in epsilon ¹2/.

It is shown that a branch point appears in the critical surface of the Ashkin-Teller model at the point where this model reduces to a Potts model, and its relation with the eight-vertex model is discussed. An estimate for the critical exponent alpha of the Potts model is given.

It is shown that for any system of non-relativistic electrons and nuclei the large-wavevector limit of the probability of finding an electron with momentum h(cross)k is proportional to 1/k⁸, and the electronic structure factor is proportional to 1/k⁴. Furthermore, the coefficient of the 1/k⁴ term in the structure factor is proportional to the zero separation electron-electron correlation. The coefficient of the 1/k⁸ term in the momentum distribution is proportional to the sum of zero-range electron-electron and electron-nuclear correlations.

It might be thought that if the energetic particles in the cosmic radiation are of extragalactic origin then they must be protons because heavy nuclei would be completely fragmented by interactions with the radiation fields in space. It is argued that this is not the case and that it is, in principle, possible to choose an origin model in which the very energetic particles commence their life as heavy nuclei; these nuclei then interact with extragalactic photons and give rise to the observed primary spectrum. Such a model would relate mainly to energies above about 10¹⁸ eV where there is the well known need to postulate extragalactic origin on the grounds of the observed near-isotropy of arrival directions.

A critical analysis is made of recent calculations of the sea level cosmic ray proton and pion spectra expected at ground level and comparison is made with two sets of experimental data measured at Durham. Particular attention is directed towards the theoretical approach advanced by the present author in a previous paper (1971) and some misconceptions by Hook and Turver (1974) are pointed out. The work of Jabs (1968, 1972) is also considered. It is concluded that all three sets of calculations represent the data equally well.

It is deduced, from the analysis of three previous papers on the Weaire model that topological disorder can only increase the band gap.