Table of contents

In response to a recent paper by Jakeman and Pike, it is pointed out that the classical and quantum-mechanical treatments of the fluctuations of the integrated light intensity are equivalent. The choice of the observable is dictated not by theoretical conditions, but by reference to what is usually measured.

The theory of gravitation of a previous paper is presented in a deductive and more rigorous form. The assumptions made about the space-time metric, the scalar gravitational potential and the special (Newtonian) charts are summarized. An action principle is stated, and the conservation laws of energy-momentum and angular momentum are derived. Lagrangian densities for the gravitational field are found by assuming that weak gravitational waves propagate at the speed of light. The assumption that gravitational energy is not itself a source of the gravitational field leads, as in a previous paper, to a theory that is at present observationally indistinguishable from Einstein's; the opposite assumption leads to a distinguishable theory. The interactions of the gravitational field with the electromagnetic field and with an ideal fluid are discussed. The simplicity of the theory (space-time formally flat and one scalar potential to describe the gravitational field) is emphasized.

Solutions of the Einstein-Maxwell equations corresponding to cylindrically symmetric distributions of stressless conducting matter with an axial magnetic field have been found, which can be matched with an outside pure magnetic field solution originally due to Bonnor. It is also shown that Melvin's magnetic universe cannot be fitted with dust distributions in this way.

A (2j+1)-spinor formalism is used to discuss the Bel-Petrov-Penrose classification of the Weyl conformal tensor. A convenient pictorial representation of this classification is presented in the form of a series of intersecting manifolds nested in a four-dimensional projective space. The relation to other formalisms is considered briefly.

A method recently introduced for obtaining rigorous lower bounds to the true quantum-mechanical expectation value <ψ|F|ψ> of a positive operator F >or= 0 is here extended and strengthened. The new formula always improves the previous result, but requires the more difficult integrals of FH and H². As a numerical illustration, lower bounds are calculated for various powers of r₁ and r₁₂ in the normal helium atom. Using, particularly, the quantum-mechanical virial theorem and an improved lower bound for the overlap integral <ϕ|ψ>, it is shown that rigorous lower bounds accurate to 5-30% can be obtained even from the simple screened hydrogenic approximation, and the nuclear diamagnetic shielding is given correct to 1%.

A simple inequality, expressed in terms of two arbitrary distribution functions of the same normalization, is shown to be useful. By choosing various different forms for the distribution functions one can derive important results, such as the upper and lower bounds of the configurational free energy.

Fast neutron time-of-flight studies of (³He, n) reactions have been used to obtain information on four proton-rich nuclei. The ground-state reaction Q values, and hence atomic masses, and mass excesses have been determined together with the energies of several of the low-lying levels of these nuclei. The results are consistent with other recent determinations, but the values presented here are more precise. In all cases there is disagreement with the 1964 Mass Tables.

The photodisintegration of the deuteron has been studied for photon laboratory energies from 100 MeV to 320 MeV. Recoil protons were detected in a counter telescope and the effects of more complex reactions investigated by varying the bremsstrahlung end-point energy.

We calculate the total cross section σ(γ, p) for the two-particle photodisintegration of the alpha particle. We describe the ground states of the alpha particle and the triton by modified Irving wave functions, the parameters of which have been determined from variational calculations of the binding energies of these systems, using a central velocity-dependent potential. We neglect the final-state interaction. We compare our results with the recent experiments of Gorbunov and with other similar calculations of Gunn and Irving and Bransden et al. We find that, though the velocity-dependent forces cause the total cross section σ(γ, p) to change in the right direction, the discrepancy of fitting simultaneously the binding energy, r.m.s. radius and the maximum cross section at the correct energy is not removed altogether.

The angular correlation of the cascade in ⁹⁹Mo decay was studied with a fast-slow scintillation spectrometer in the beta energy region 450-700 kev. A small negative beta-gamma anisotropy is observed and the resulting angular correlation function (W) is found to lie in between -0·023 and -0·034 with an uncertainty of about 25%. It is concluded that the results are in accordance with the ξ approximation applicable to certain non-unique first-forbidden beta transitions.

A scintillation counter telescope of aperture 0·47 m² sr and incorporating layers of crossed neon flash tubes has been used to search for relativistic e/3 and 2e/3 quarks in the near-vertical cosmic radiation at sea level. Events were selected where the scintillator pulse heights were as expected for quarks and the corresponding flash tube records were studied.

No events satisfied all the rigorous acceptance criteria and only upper limits can be given for the quark flux. These are, at the 90% confidence limits: <1·15 × 10⁻¹⁰ cm⁻² s⁻¹ sr⁻¹ for relativistic e/3 quarks and <8·0 × 10⁻¹¹ cm⁻² s⁻¹ sr⁻¹ for relativistic 2e/3 quarks.

Using a large directional water Cerenkov counter and a near horizontal magnetic spectrograph, the Cerenkov energy loss of relativistic muons has been studied. The experimental results, which cover the range of muon momentum 0.3-120 GeV/c, are consistent with the classical theory of Frank and Tamm. They do not indicate a decrease in the Cerenkov loss, as expected according to Tsytovitch, nor do they confirm the large rise reported by Bassi et al.

A study has been made of the ionization energy loss of cosmic-ray muons in a plastic scintillator (Ne 102a) over the momentum range 0·3-120 GeV/c.

The results have been combined with those of Crispin and Hayman for a similar material, and together give no support for the decrease expected from the radiative correction theory of Tsytovitch, the measured change in energy loss above 10 GeV/c being +(1·2±0·7)%, compared with a predicted reduction of 4-8%.

A least-squares fit to the most probable energy loss values above 2 GeV/c gives a rate of rise of (2·7±3·3)%, a value not inconsistent with the constancy predicted by the theory of Sternheimer.

The net absorption coefficient allowing for stimulated emission is derived for intense light in a hot non-relativistic plasma, using a semi-classical approach, which takes into account the non-Maxwellian velocity distribution caused by the strong electric field of the radiation but considers only absorption and emission processes involving a single incident photon. The absorption coefficient is found to vary inversely as both the electric field and the frequency, a result intermediate between the usual weak-field coefficient and a strong-field coefficient due to Rand which is intended to include multi-photon processes. It appears that the net effect of multi-photon processes is to produce stimulated emission. Rand's result is shown to imply that, in order to heat plasma to a given temperature, a minimum time and (on a simple dynamic model) a minimum quantity of plasma are necessary.

The programme of reducing N-particle problems to one-particle problems is extended to include the Pauli principle. To calculate the ground-state energy of a system of N fermions interacting by pair forces, a rigorous lower-bound shell model is derived. This shell model follows a building-up principle, and tends to improve with increasing number of particles as well as with increasing strength of interaction. For twenty particles antisymmetric in ordinary space interacting by square-well interaction of strength V₀a² = 200 hslash ²/2m the shell model differs by less than 8% from the calculated upper bound, and hence a fortiori from the exact energy. In order to test the quality of the approximation for various interactions, it was necessary to calculate upper bounds.

The Gel'fand-Levitan equations for the solution of the inverse scattering problem are written in terms of the half-off-energy-shell matrix element. This element is given in terms of the phase shifts, by a double integral equation in which the kernel has simple poles. We also find an integral equation for the wave function. We include the effects of bound states, and find a relation between the large and small γ behaviour of the bound-state wave function.