Table of contents

The minimum uncertainty relation which characterizes a coherent spin state is shown to be invariant under some special rotations and a model to interpret this result is given. 'Intelligence' of coherent states is discussed in the light of this interpretation.

It is shown that in an isometric flow the vorticity vector is invariant under the group of motions.

The weaknesses of the arguments used by Davies (1975) to invalidate an experiment proposed recently by Heron and Pegg (1974) to detect advanced radiation are pointed out.

Fourier expansions in modified Bessel functions are evaluated in terms of elementary functions. The results are obtained in the form of rapidly convergent series so that numerical computations are quite easy. This leads to the possibility of tabulating the modified Bessel functions in their critical domain. A physical application is also given: simple expansions giving the Madelung constant for cubic crystallographic structures are established; they involve only elementary functions and exhibit the greatest known accuracy.

An axisymmetric static system of charged point particles is investigated in the framework of general relativity. The particles are located on the axis of symmetry; the Einstein-Maxwell equations are solved in second approximation. Equilibrium is maintained by means of 'struts' connecting neighbouring particles. Assumptions are made which make the solution unique. The mass of the whole system and the gravitational mass of individual particles are defined.

Exact values are given for many two-dimensional lattice sums in terms of products of Dirichlet L-series. Madelung constants are included amongst the sums considered.

For pt.I, see ibid., vol.8, p.203 (1975). The theory of capture and release of an electron bubble from a quantized vortex line in rotating He II has been presented in part I, along with a derivation of the kinetic coefficients for capture and release to lowest order in the external field strength. This information is used to analyse the available experimental data on capture and release, and show that the field dependence of both the cross section and lifetime is accurately predicted by the theory of Brownian motion of an ion in the presence of a static vortex potential, as well as is the temperature dependence of the cross section for T>or approximately=1.4K.

High-temperature series for the free energy of the q-component Potts model in the presence of arbitrary external fields are generated to eight order for five lattices and subsequently analysed. In two dimensions the higher critical exponents gamma , gamma ₃ are estimated to be 1.42, 3.60 and 1.20, 2.62 for q=3 and 4 respectively. Evidence for the possible divergence of the order parameter fluctuation at the first-order transition point for q>or=5 is presented. In three dimensions the question of the order of the transition is directly investigated in the temperature-order parameter plane and analysis of the series indicates that the transition is of first order for q>or=3.

The spin 1 Schrodinger exchange model is shown to have the same symmetry in fields as the three-component Potts model so that the zero-field quadrupolar ordering can be simply related to the dipolar ordering. The question of the nature of the transition is investigated using previously derived high-temperature series for the free energy as a function of an extensive variable. For S=1, mean-field-like behaviour persists in our higher-order approximations on the face-centred cubic lattice, indicating a first-order transition in this model.

The Blume-Capel (1966) model in a transverse field is studied in the mean field approximation. It is found that a tricritical point exists in the transverse field-temperature plane for a range of values of anisotropy to exchange interaction ratio. This is in contrast with the original Blume-Capel model which shows the tricritical phenomena only at a single value of this ratio. The most interesting feature, however, is that the ordering field, which is the externally applied field along the longitudinal direction, is experimentally accessible and can be set at one's will. This allows an experimentalist to investigate all aspects of a system describable by this model. Equations of the lines of critical points including the wing boundaries are derived. Phase diagrams for the whole range of values of anisotropy are described, and the experimental situation is discussed.

A renormalization group method is applied to obtain a representation for the singular part of the free energy of systems described by scaling operators which exhibit a critical phase transition.

Recent publications on the configurational properties of polymer chains subject to excluded volume restraints, as deduced from studies of off-lattice model chains, are briefly reviewed. It is shown that the data for such models cast doubt on the validity of the frequently quoted equations (R_N²)=aN^6/5 and (S_N²)=a'N^6/5 where (R_N²) and (S_N²) are respectively the mean square end-to-end length and mean square radius of gyration of N-link chains, and a and a' are constants. The use of the technique of inversely restricted sampling to generate off-lattice model chains, with either freely varying or fixed bond angles, and free rotation about bonds, is described.

For pt.I see ibid., vol.8, p.929 (1975). Three-dimensional continuum model chains up to 1000 links in length have been generated, with both freely varying and fixed bond angles. The geometric expansion of intermediate length (<or=100 links) chains due to exclude volume effects increases monotonically with the volume of the spheres representing the individual monomer units of the chain. The expansion factor does not consistently follow any of the published two-parameter curves. The ratio of the mean square radius of gyration to the mean square end-to-end length of 100-link chains and the distribution function W(R) of the probability that the end-to-end length of a chain will be R are obtained.

The tube radius a is defined as the mean distance between nearest segments belonging to different chains. For a moderately concentrated system of sufficiently long chains, a is found to be inversely proportional to the segment density rho. By use of this result, several problems in the entanglement effect are discussed.

A theory of adhesive rubber friction is proposed where the final expression for the coefficient of friction obeys the Williams-Landel-Ferry (1955) transform and correlates with the viscoelastic shear loss modulus as shown by Grosch (1963) and others. The approach is based on the bead-spring model for polymer chains of Rouse (1953).

A model is proposed, no less realistic in most respects than the usual ones, which leads to physical conclusions which are the exact opposites of the results in the literature. In particular, with the model all normal modes have normalizable eigenfunctions, and the eigenvalues are analytic functions of the wavenumber k for all k. No so called 'Knudsen modes' exist. Qualitative arguments are given to suggest that the difference results from the fact that the model operator, like the true Boltzmann collision operator, is unbounded, whereas the conventional model operators are bounded.

Properties of electrostatically self-focused electron streams are calculated from stream injection parameters. The unneutralized ion density within the stream can be nearly an order of magnitude greater than that of stream particles alone. This effect is due to ionization by stream particles and collective behaviour of the stream and ambient plasma. Cold, uniform streams can occur only where stream linear density is sufficiently high. In certain cases ion cluster and molecular ion formation give rise to tight focusing. Periodic self-focusing provides axial electric fields which can resonantly drive stream particle oscillations and which can be used for acceleration of ambient ions.

The longitudinal structure of the air shower particles in the shower disc has been investigated using the Kiel air shower array with additional fast-timing equipment. From the time delay measurements the longitudinal density distribution of particles has been derived, which near to the shower core (r<10 m) can be described by d(t)=0.39t^0.9e^-0.6t. This distribution proves to be similar to the longitudinal dispersion calculated for purely electromagnetic cascades.

The time spread of muons within EAS has been determined by measuring the time of arrival of individual muons with respect to the extreme shower front with an accuracy of +or-10 ns. The integral time distributions averaged over a group of showers are presented as a function of shower parameters. For near vertical showers it is found that the 10%-90% time spread of muons increases from about 100 ns at an EAS core distance of 150 m to about 160 ns at 450 m. These results are shown to be in reasonable agreement with predictions based on model calculations.

The minimum sizes of four quasi-stellar objects, each of which exhibits an absorption redshift greater than its emission redshift, have been calculated by assuming that the absorption lines originate in clouds which are falling radially towards the quasi-stellar objects along the line of sight, and that the quasi-stellar objects are at a distance inferred from the cosmological interpretation of their redshifts. It has been found that the radii are greater than 10 pc and that the masses are of the order of 10¹² M(.).

The author gives an alternative derivation of the evaluation by Papadopoulos (see abstr. A11851 of 1974) of the path integral relating to an electron gas. The method employed is a simple application of the powerful technique of global integration on function space. The integration is direct and does not require coupling of the system to auxiliary external forces.