Table of contents

A procedure for improving the calculation of matrix elements when using approximate wave functions is described, and application is made to the expectation values of several operators for the ground states of the helium isoelectronic sequence.

A variational method is developed to find certain low-lying collective states of a closed-shell nucleus using the random-phase approximation with an effective particle-hole attraction of zero range. A numerical calculation is carried out to illustrate the use of this method for a very simple model nucleus and the significance of the results is discussed.

Exact solutions have been obtained for several problems in one-velocity neutron transport theory by means of the Wiener-Hopf technique. The basic problem concerns the calculation of the neutron flux and angular distribution at the surface of an infinite, purely absorbing foil or control plate embedded in a moderator. Solutions are obtained for the cases of isotropic and linear anisotropic scattering in the moderator. An approximate solution has been obtained for an infinite lattice of such absorbers by the `end-point method'.

Numerical results are compared with those from approximate theories such as P₁₃, double P₃ and variational methods. It is found that the double P₃ and variational methods are very accurate indeed, whereas the P₁₃ method, which although adequate, shows some unexpected errors for such a high order polynomial approximation.

An approximate formula has been developed which allows a certain amount of scattering in the foil; the limitations of this are discussed.

Cross sections for the ionization of neutral helium by electron impact are calculated in two versions of the Born approximation and in the Born exchange approximation, using accurate atomic wave functions. The effect of exchange in the Born exchange approximation is to remove more than half of the discrepancy between the conventional Born (Born (a)) cross sections and the experimental values of Smith. The Born (b) approximation, which is consistent with the Pauli principle, gives results intermediate between the other two. The calculated energy and velocity distributions conflict with previous Born (a) determinations.

Simultaneous measurements of small ionization currents I and formative time lags t_f in uniform electric fields E have been made for spectro-scopically pure hydrogen at pressures p₀ (< 15 mmHg) between plane parallel gold surfaces in the range 56 < E_s/p₀ < 170 v cm^-1 mmHg^-1 dagger over which the observed growth times lay between 10^-6 and 10^-4 sec for an over-voltage up to approximately 3% in excess of the static sparking potential V_s. In this time range the currents increased from about 10^-9 A to about 10^-4 A and the early stages of growth, approximately 10^-9 < I ≲ 10^-5 A, were found to be described by an expression of the form I = A + B exp(λt) in agreement with Davidson's analysis. The growth constant λ and hence the relative importance of the secondary ionization processes responsible for growth were evaluated directly from the current measurements and are tabulated for the whole range studied. Estimates of the photoelectric yield, k similar 10^-2 electrons per quantum, are deduced: these are in agreement with directly measured values.

Accurate numerical calculations are presented for the energy and angular dependence of the cross sections and polarizations of electrons scattered by mercury in the energy region 100 to 2000 eV. The results are compared with experiment and found to be in good agreement.

Positron lifetimes have been measured in Teflon (P.T.F.E.), polyethylene, nylon-6, polypropylene and paraffin wax. Using equipment of good resolution, W_1/2 = 0.7 nsec, wide range, 30 nsec, and a highly linear response, three components of the lifetimes are found in each case. These are about 0.3 nsec, 0 6-1.1 nsec and 1.7-4.1 nsec. Two components only have been reported so far.

A connection is found between the observed intensities of the longer components and the degree of crystallinity of the polymers, and an explanation is offered. The intensities of the long components vary according to the previous heat treatment of the polymer, especially in Teflon, in a way which depends on the physical structure of the solid. The identification of the three components should permit a useful re-examination of positron lifetimes in `molecular materials'.

The level-crossing technique in resonance fluorescence has been applied to determine the lifetime and hyperfine structure of the level (6s²6d)²D_3/2 in Tl 1. No distinction was made between the two stable isotopes 203 and 205 since their nuclear moments are identical within the accuracy of these experiments. The results (5 2 ± 0.8) × 10^-9 sec for the lifetime, and 42 2 ± 1 8 Mc/s for the hyperfine interaction constant, are in good agreement with the measurements of Gallagher and Lurio. The measured lifetime agrees with a value calculated from measurements of the spontaneous transition probabilities from the excited state, and the measured hyperfine interaction constant agrees with the value calculated from the Goudsmit-Fermi-Segrè theory. This latter agreement is probably fortuitous because the theory does not take into account the effects of configuration interaction, which might in this case be considerable.

Calculations are made of the magnitude of charge-dependent corrections to the polar vector matrix elements of a number of J = 0⁺, T = 1 → J = 0⁺, T = 1 beta decays. Both the Fermi gas model and jj coupled shell model are used, and with a most optimistic treatment it is not found possible to resolve the present difference between the experimentally determined polar vector β-decay coupling constant G_v and the μ-decay coupling constant G_μ. The role played by mesonic exchange corrections, should conserved vector current theory not hold, is also investigated and it is found that experiment is consistent with zero contribution from such effects. It is finally concluded that the measured ft values are in agreement with conserved vector current theory only if it is formulated along the lines suggested by Cabbibo in 1963.

The theory of closed-form approximations in crystal statistics developed by Yvon, Rushbrooke and Scoins, Fournet, and Domb and Hiley is extended to include long-range forces. This extension is used to obtain successive approximations for the entropy, specific heat and the susceptibility in zero field for the disordered state, for arbitrary long-range forces. The general forms of the approximations are studied to determine the shape dependence of the exact properties when the specimen is ellipsoidal and the interactions are dipolar. It is found that, in the disordered state, the susceptibility is shape-dependent while the specific heat and entropy are not. A numerical investigation of the susceptibility using the first two closed forms (a Bethe-type approximation) with an interaction potential of type r^-p shows that, as the range of the force increases, the critical point depends predominantly on the dimensionality and becomes insensitive to the detailed structure of the lattice. As p → d (the dimensionality of the lattice), it is found that this approximation gives the exact behaviour in the limit. This conclusion, however, is not valid for dipolar forces.

When indium antimonide is heated a Knight shift of (+0.920 ± 0.008)% of the ¹¹⁵In resonance is found to appear abruptly at the melting point and to remain constant in the temperature range 523 to 604 °C. This confirms the conclusion from the most recent magnetic susceptibility and electrical resistivity measurements that indium antimonide melts directly to a metallic state, but leaves unexplained the anomalous behaviour of the viscosity and specific heat above the melting point.

Measurements of the paramagnetic g factors have been made, using electron spin resonance at 8 mm wavelength, in solid solutions of gadolinium in yttrium, palladium and lutetium at temperatures between 60 °K and 400 °K.

The measurements show that the g factor for the trivalent gadolinium ion in the metallic lattice varies significantly from the free ion value and is concentration dependent. For alloys containing more than 30% Gd in the Gd-Y alloy system, 80% Gd in the Gd-Lu alloy system and for all concentrations in the Gd-Pd alloy system, the g factors are less than the free ion value. Lower concentrations of Gd in the Gd-Y and Gd-Lu alloy systems have g factors greater than the free ion value of 1 99.

A qualitative attempt has been made to explain this variation, using a model proposed by Kondo in which two interactions are considered.

For an unstressed solid a set of second-order elastic constants cαβστ can be chosen such that the strain-energy density, the first-order stress-strain relations and the equations of motion all have simple forms; the cαβστ are symmetric with respect to the interchanges (α, β), (σ, τ), (αβ, στ). In the present paper different ways of generalizing the cαβστ when the solid is under stress are discussed and compared, both for a general stress and for the special case of isotropic pressure. It is suggested that the symbols cαβστ (Voigt: c_ij) should be used for the elastic constants defined from the stress-strain relations and Cαβστ (Voigt: C_ij) for the elastic constants defined by expressing the strain-energy density as a function of the Lagrangian strain coordinates. The relation of these elastic constants to the dependence of strain energy on homogeneous strain is discussed in detail, and explicit expressions given for cubic crystals under isotropic pressure.

A method is described whereby the strain-dependence of the first few even moments of the vibrational frequency distribution of a crystal can be used to calculate the elastic constants of moderately anharmonic crystals. From the elastic constants at T = 0 the limiting Debye temperature Θ^C₀ can also be obtained, to the first approximation in the anharmonicity. Detailed calculations at T = 0 are carried out for cubic close-packed lattices with a number of different central force potentials.

For the 6-12 potential, contributions of the zero-point energy to the shear constants are less than those calculated by Salter and by Leibfried and Ludwig, who relied on the strain-dependence of the second moment only; and departures from the Cauchy relation are less severe. The bulk modulus is in fair agreement with earlier calculations, including the variational treatment of Bernardes. Anharmonic corrections to the quasi-harmonic value of Θ^C₀ are appreciable, but less than those estimated by Leibfried and Ludwig and by Wallace.

For a more general potential, ϕ(r), between nearest neighbours only, general expressions are derived for the c_ij and Θ^C₀ in terms of the derivatives ϕ⁽ⁿ⁾(r). A positive ϕ^(iv)(r) always tends to increase the c_ij above the values calculated from the static lattice, while the effect of ϕ⁽ⁱⁱⁱ⁾(r) is usually to decrease them. The full expression for Θ^C₀ includes not only the leading terms calculated by Flinn and Maradudin, but also additional terms which are of comparable magnitude.

The results are applied to the thermoelastic properties of the rare gas solids. With parameters chosen to fit the molar volume and cohesive energy, both all-neighbour and nearest-neighbour 6-12 models give bulk moduli at T = 0 in good agreement with experimental values for Xe and Ne; the predicted value for Kr is 3.21(±5%) × 10¹⁰ dyn cm^-2, and for Ar 2.61(±5%) dyn cm^-2. Close agreement is obtained with a recent measurement of the transverse wave velocity in polycrystalline argon. Anharmonic corrections to quasi-harmonic values for Θ^C₀ are found to be positive, being approximately 1.5%, 2%, 4% and 16% for Xe, Kr, Ar and Ne respectively. Roughly similar corrections will be needed for the exp-6 potential, but negative corrections are found for the Guggenheim and McGlashan argon potential, which turns out to be quite incompatible with the experimental Θ^C_O.

At high temperatures formal expressions are derived for the adiabatic and isothermal elastic constants, but their evaluation requires a rather uncertain extrapolation. For the 6-12 potential it appears that vibrational contributions to the shear elastic constants are small at high temperatures, and that C^ad₄₄ < C^ad₁₂. The model deviates quite strongly from the experimental equation of state of xenon, especially at the highest temperatures.

The method of double-time temperature-dependent Green's functions is applied to the problem of calculating the specific heat due to the spin-phonon interactions in paramagnetic crystals. It is shown that the coupling of the spin system to the lattice leads to a peaking of the Schottky anomaly in the vicinity of its maximum.

The behaviour of collinear signal, idler and pump waves in a non-linear medium without inversion symmetry is investigated under small-signal conditions when the frequency ω of either the signal or the idler approaches a transition frequency of the medium. The three waves are coupled together by the first-, second- and third-order susceptibility tensors of the medium. All three tensors exhibit a resonance in the case under consideration. The first-order resonance creates an absorption band and the third-order resonance produces stimulated Raman amplification. It is shown that the strength of the second-order resonance, which enters into the parametric effect, is determined by the product of the strengths of the other two. The growth constant of the signal and idler waves is a maximum when the mismatch of the propagation constants Δk in the presence of the pump field takes a non-zero value equal to the decay constant at frequency ω times the ratio of the imaginary and real parts of the complex parametric growth constant, a quantity which is independent of the pump field strength. Except in the immediate neighbourhood of resonance, the frequency dependence of the optimum growth constant is controlled by the ratio of the peak Raman growth constant to the off-resonance parametric growth constant. When this ratio is small the optimum growth constant is never significantly larger than the off-resonance parametric value; when it is large the optimum growth constant slightly exceeds the peak Raman growth constant over a frequency range of 10-10³ linewidths on one side of resonance. In the immediate neighbourhood of the resonance the optimum growth constant exhibits structure which is controlled by the anomalous dispersion and absorption resulting from the first-order resonance.

Some experiments are described in which the time dilatation at the tip of a high speed rotor is measured and compared with the expectations of relativity theory. Mössbauer sources and absorbers are attached to the centre and tip (or vice versa) of the rotor and the time dilatation deduced from the variation of gamma ray transmission with rotor speed. A technique is described whereby using a source absorber combination with an appropriate inherent frequency shift the experiment takes on the features of a null experiment. A result 2.1% ± 1.9% in excess of the expected dilatation is obtained.

A scalar gravitational theory is introduced which gives the correct rotation of the perihelion.

If the protons are treated as distinguishable particles classical mechanics gives the asymptotic electron capture cross section in H⁺-H(1s) collisions to be proportional to the inverse cube of the impact energy just as does quantum mechanics and in addition gives almost the correct constant of proportionality.

It is shown that the approximation of Ochkur is in a certain sense incorrect and an appropriate modification is indicated.

A corrected table of transformations of the operator equivalents O_n^m is given for m even and n = 2, 4, 6, for the rotation of coordinates where the new z axis is perpendicular to the old z axis. Results are also given for O₄³ and O₆³, which appear in a trigonal spin Hamiltonian.

The polytropic gas sphere model of a star has been suggested as a counter-example to the author's instability argument. It is shown that the assumptions of the instability argument do not hold for the star model.

The expressions for the partial vapour pressures of the two components in a monomer-dimer mixture involve certain cluster sums β_k. The values of β_k, as far as k = 5, were given in a previous paper. The evaluation of these quantities becomes more difficult as k increases and the method of obtaining β₅ for a face-centred cubic lattice is considered here in some detail.