Table of contents

The paper discusses the method proposed by Villars for describing the rotational states of nuclei. For simplicity only two-dimensional rotation is considered for the ground-state band of an even-even nucleus in which the last occupied sub-shell of the deformed potential is completely filled. It is shown how to avoid inconsistencies arising from the multi-valuedness of the angle variable which appears in the formalism. In the Villars method the moment of inertia appears in terms of a sum over the eigenstates of the `intrinsic' Hamiltonian. Since these are not calculable exactly, one needs an approximation scheme. A possible scheme based on the Hartree-Fock approximation is used, though for the present this is approximated further by the use of harmonic oscillator wave functions. The case of ²⁸Si is treated by way of illustration. The detailed model is too unrealistic to expect physically significant results; it does suggest, however, that the method is suitable for practical evaluation. Further steps for improving the accuracy are discussed.

Hypervirial theorems are treated from the point of view of observables and states. This technique yields an extension of the classical results and a new proof for the quantum case, and it makes explicit their intimate relationship to conservation theorems.

It is shown that the full specification of an assembly of long flexible molecules, needed for a statistical-mechanical study, requires an infinite set of topological invariants, and the first two of these are derived in detail. It is argued that these invariants provide a better description of the topology of the system than a more intuitively obvious one, for example, to state the condition that a molecule contains a single knot is very complicated requiring an infinite number of invariants, just as the specification of a function at a point requires an infinite number of Fourier coefficients. It is shown that the probability of molecules taking up configurations with given values for the invariants is a problem in quantum field theory, and that for example the first invariant leads to a formalism isomorphic with the electrodynamics of scalar bosons, and the governing differential equations for one and two molecules are derived. The transition from a real polymer to its representation by a continuous curve leads to divergences, but these can be absorbed by renormalizing the step length and entropy per monomer; within these two changes the topological properties are independent of monomer structure.

It is shown that a functional inequality may be used to establish variational principles for partition functions and distribution functions. The inequalities relate distribution functions of higher order to those of lower order. They can be used to suggest approximate integral equations for distribution functions, to test and modify approximate distribution functions, and to establish variational principles on which can be based the systematic computation of distribution functions by variation of parameters.

A self-consistent method, described by Bell and Fairbairn in 1967, is used to investigate Pca2₁ ordering in solid orthohydrogen. It is shown that when the orientations of the orthohydrogen molecules are ordered in accordance with the Pca2₁ structure a state of very low energy is produced. The zero-order statistical approximation is applied to the transition between this ordered state and the high-temperature disordered phase. It is found that a second-order transition occurs at a temperature of 3·6 °K.

The symmetrized cube and the antisymmetrized square of the irreducible representations of the point groups are reduced and tabulated; these are the products which are relevant to Landau's theory of second-order phase transitions.

The scintillator stack was operated for eight months in coincidence with two additional counters placed in the same tunnel at distances up to 6·3 m away. It is shown that this arrangement is satisfactory for studying muon showers and that the stack alone can be used to distinguish groups of muons. The measured rate of pairs of muons is compared with results from other experiments at various depths. The results are compatible with Greisen's expression for the muon density in air showers; in the extrapolated region close to the shower axis it is necessary to assume that the muon density rises more sharply than at greater distances, but there is no evidence for very strongly collimated groups of muons.

Recent measurements on cosmic rays deep underground have led to the necessity of studying the effect of fluctuations in range for muons of energy to some 10⁶ GeV and ranges down to 15 000 m.w.e. The results of such calculations are given here; in these calculations, which are more extensive than hitherto, an examination is made of the effect on the muon survival probabilities deep underground of different assumptions as to the form of the cross section for the contributions from nuclear and bremsstrahlung losses and different magnitudes of the energy loss coefficients.

Survival probability curves are given from which the enhancement in underground intensity can be calculated for any sea-level muon spectrum, and results are also given of the actual enhancement factors for sea-level spectra having constant exponents.

The energy spectra of atmospheric muons at sea level and underground have been calculated from assumed forms of the primary spectrum, the mass composition and the interaction model, and from various depth-intensity relations at great depths. The mean muon energies in the vertical direction at a depth of 7500 m.w.e. have been derived and are found to range from 286 GeV to 393 GeV according to the assumptions used.

The energy spectra of neutrino-induced muons underground have also been calculated. The mean energies are found to be much lower than for those of atmospheric origin, being typically in the region of some tens of GeV. It is shown that the rate of muons is sensitive to the mass of the (hypothetical) intermediate boson, and the mean muon energy is largely dependent on the saturation energy of an assumed linearly increasing cross section for the inelastic neutrino interaction. It is shown that more accurate measurements of both the rate and the mean energy of the neutrino-induced muons will provide useful information on the neutrino cross sections at very high energies.

The results reported form a continuation of the analysis given by de Beer et al. in 1966. In the present work an examination is made of the fluctuations expected in the electron and muon components from shower to shower, with particular regard to the number, lateral distribution and height of origin of the muons.

It is shown that fluctuations can affect considerably the picture of showers drawn from results with practical air shower arrays, largely through the correlations that exist between the various components.

The important problem of deriving results of significance to high-energy nuclear physics and to astrophysics is studied, and it is shown that the relative fluctuation of the muon number in showers of fixed size has some sensitivity to the characteristics of high-energy interactions, whereas the relative fluctuation of the shower size for constant muon number is sensitive to the mass composition of the primary particles.

The theoretical results derived in the previous paper by de Beer et al. in 1968 have been compared with experimental observations, with reference to the fluctuations in the total numbers of muons and electrons and to the variation of the mean number of muons with shower size.

It is found that the derivation of accurate parameters from experimental data is very difficult in view of the presence of a variety of bias effects. However, good agreement appears, in general, when comparison is made of the predictions of our preferred model (which assumes a CKP type of energy distribution for the secondary pions from high-energy interactions).

The nature of the mass composition of the primaries is still unclear in view of the rather low sensitivity of most parameters to the mass spectrum. What evidence there is, from the (N_μ, N_e) relation, gives only a very slight indication of a change in the primary composition above 10¹⁵ eV.

In this paper the authors have given a quantitative analytical investigation of the interesting concept of the modulation of an electromagnetic wave by its propagation along an alternating magnetic field in a semiconductor or a plasma. Numerical results, presented at the end, show that this phenomenon is appreciable.

Gravitational waves emitted from a spinning rod - taken as a rigid axially symmetric distribution of matter of uniform small cross section - are studied within the framework of general relativity, with special interest on transport of linear momentum from the rotating source. It is found that linear momentum is carried by the waves cyclically in such a way that the centre of rotation is a fixed point outside the axis of symmetry of the rod.

Two results for the rate of momentum flux, differing in numerical content, are derived from the solution of the linear approximation, one by the use of Synge's argument involving the energy-momentum tensor, and the other by means of the pseudo-tensor. The discrepancy between the results is explained.

The paper demonstrates the existence of a solution of the Einstein-Maxwell equations in general relativity where matter coexists with a cylindrically symmetric axial field.

The fields due to an accelerated dipole are calculated using straightforward electromagnetic theory. It is shown that the potentials and fields may be written so that the static field, its first correction and the radiation field are separated. The expression for the fields and potentials is therefore analogous to the expression for the field of a point charge discovered by Heaviside and Feynman.

The classical motion of an electron in the field of a finite electric-dipole is analysed in terms of Hamilton-Jacobi theory. It is shown that a bound state exists for an arbitrarily small dipole moment.

The classical motion of an electron in the field of a point electric dipole is analysed. It is shown that the only motion for which the distance γ from the dipole to the electron does not either increase without limit or decrease to and remain at zero is that for which γ is constant and the total energy E is zero. A necessary condition for such bound motion is D > 3 surd 3pϕ²/4me, where D is the dipole moment, pϕ is the component of angular momentum along the dipole axis, and m and e are the electronic mass and charge. It follows that any point dipole can bind an electron classically.

We discuss the quantum-mechanical problem of intensity fluctuations in samples of Gaussian light and its connection with the corresponding classical scalar problem involving fluctuations of random noise power. We obtain an expression for the intensity-fluctuation distribution of light with a Lorentzian spectrum, and investigate analytically its asymptotic behaviour. Numerical techniques are used to evaluate the distribution for various intermediate linewidths, and some examples of the corresponding theoretical photon-counting distributions are also given.

Expressions for the resistance to the motion of one gas through another at any relative drift speed are found, on the basis of a simple kinetic theory model. Approximate expressions giving a good degree of accuracy over all speeds are proposed, and it is suggested that the results will be of use in, for example, the study of gas streaming from stars.

An edge-current method is formulated for the problem of diffraction by a circular aperture formed by the envelope of a system of wedges. Some experimental results are presented for comparison. For various non-planar screens it is found that the aperture field is not changed appreciably by changes in the geometry of the screen on the image side, or by changes in the edge geometry. However, the geometry of the screen on the source side has a large influence on the aperture field.

A study has been made of the production of bursts by neutral cosmic-ray particles, presumably neutrons, in the energy range 20-4000 GeV. The energy spectrum is found to exhibit a deficiency above 100 GeV when compared with that derived from the primary nucleon spectrum, given by a number of authors, assuming a constant attenuation length. The discrepancy most likely represents a change of attenuation length with energy, but an alternative explanation involving quark production cannot be ruled out.

The effect of electron binding on the incoherent scattering cross sections of gamma rays in lead in the energy region 279-1330 kev is studied by subtracting theoretical values of all other partial cross sections from the total experimental cross sections. It is concluded that the Thomas-Fermi model underestimates the effect of electron binding in reducing this cross section in the energy region 279-662 kev.