Table of contents

A solution representing a model of gravitational collapse in the background of an expanding universe is presented, there being no discontinuity in the density distribution anywhere and the outside universe being of negative curvature. It is found that the phenomenon of collapse would not be observable from the expanding region.

Equations are derived describing the scattering of electrons by hydrogen-like ions when electron-electron correlation terms are added to the close-coupling expansion of the total wave function. The 1s, 2s and 2p terms are retained in the latter expansion, while up to sixteen correlation terms are retained in the former expansion. Provided the incident electron energy is insufficient to excite the third quantum level of the target the resultant solution satisfies a minimum principle. The numerical solution of the equations is discussed and results are presented for elastic and inelastic S-wave transitions in e^--H and e^--He⁺. Above the energy where the minimum principle applies it is possible to obtain physically significant results by varying a non-linear parameter in the wave function in order to avoid singularities in the solution.

Calculations of the ionization cross section of hydrogen and of hydrogenic positive ions are described in which the initial state is either the ground or the excited 2s state. The first procedures used are the Born (ii) and Born-exchange approximations. These results are compared with other theoretical calculations and with experimental data. It is seen that, for the case of ionization of hydrogen from its ground state, none of the theoretical results is in good agreement with the experimental data. A certain defect of the theory is then corrected by adopting a third procedure for this case in which an angle-dependent Coulomb potential is used in the description of the final state for the process. It is then found that, despite the sounder theoretical footing of this latter calculation, no improved agreement with experimental data is obtained. In the near threshold region, however, our third procedure yields a linear behaviour, whereas the first two approximations do not. This is in accord with theoretical predictions though there exists some uncertainty in the experimental situation.

Convenient formulae are presented which represent the best data for the ionization cross sections, and the associated reaction rates for the case of an initial Maxwellian distribution of velocities.

The Born (ii) and Born-exchange approximations are applied to the calculation of ionization cross sections for the coronal ions Fe XV and Fe XVI by electron impact. Tables of ionization cross sections and reaction rates are presented.

The cross sections corresponding to the transitions Q(2s → np) and for 2 ⩽ n ⩽ 7 have been computed for Be II, N V, and Ne VIII, using the Coulomb-Born approximation. The cross sections Q(2s → 3p) and Q(2s → 7p) have also been computed for a hydrogenic ion with an infinite nuclear charge. The results obtained are compared with the semi-classical and the g treatments. Simple analytic forms are fitted to the calculated cross sections. The corresponding parameters are then extrapolated along the quantum number n and the charge Z. Tables and graphs are given which enable one to rapidly obtain any cross sections of the type Q(2s → np) in any ion of the lithium isoelectronic series.

At large energies E the cross section Q for the electron excitation from the n, l state of hydrogen to the n', l' state may be written in the form QE = A log E + B. The first Born approximation is used to calculate these A's and B's for all transitions with n = 1, 2, 3, 4, 5 and 6 and with n'= n+2. The A's and B's associated with the main optically allowed transitions (i.e. l' = l+1) are largest. In calculating the A's for the average cross section from one level to another we found that we need only consider the main optically allowed transitions, but in calculating B's for the average cross sections we could not ignore the other transitions, besides the main optically allowed transitions, since the sum total effect of these minor transitions is almost as large as that of the main optically allowed transitions. These minor transitions increased the B's for the average cross sections of most of the transitions considered by more than 50%.

The first-order exchange approximation, together with a separated-atoms approximation, is used to calculate the cross section for the dissociation of hydrogen molecules by electron impact assuming that the main process causing dissociation is the excitation of the repulsive 1 ³Σ_u state. It is found that the cross section obtained with the use of the first-order exchange approximation is considerably smaller than that calculated employing the Born-Oppenheimer approximation.

Comparison is made between the experimental data on the efficiency of dissociation of hydrogen molecules by electrons and the theoretical efficiency derived from the first-order exchange approximation to the dissociation cross section.

A complete distorted wave calculation is carried out of the rotational excitation of molecular hydrogen and of molecular deuterium in thermal collisions with atomic hydrogen. Total cross sections, angular distributions and rate coefficients are presented.

Cross sections for the vibrational excitation of H₂O by collision with H₂O and H₂ have been calculated by numerical solution of the appropriate scattering equations.

The angle-dependent interaction potentials generated by the dipole moment of the H₂O molecule have been replaced with approximate, spherically symmetric functions obtained by statistically averaging the potentials over all orientations of the colliding particles.

Coupling has been retained between five vibrational states variously distributed between the three normal vibrational modes of the non-linear H₂O molecule and cross sections are calculated for collision energies up to 2 5 eV.

The vibrational relaxation of H₂O and its dependence on the concentration of H₂ have been calculated in terms of these cross sections for temperatures from 300 °K to 1000 °K. It is found that the relaxation behaviour is dominated by transitions within the valence angle bending vibrational mode and can be well represented by a single relaxation time for all temperatures and hydrogen concentrations less than 5%.

Comparison with experimental results for the pure vapour indicates that the calculated inelastic cross sections for H₂O collisions may be low by less than an order of magnitude. The vibrational transition rate within the angle bending mode arising from collision with H₂ is found to be a factor of 40 greater than that produced by collision with H₂O.

Formulae are given for reducing certain atomic integrals between correlated 1s Gaussian functions to computational form.

Chisholm has shown that a particular change of nucleon field variable in neutral pseudoscalar-pseudovector theory yields an interaction which is exponential in the meson field. The S-matrix elements for neutral pseudoscalar-pseudovector and exponential theories are written down with the help of Feynman graphs and are shown to be identical. This result provides an example of the general theorem stated by Chisholm that the S matrix is invariant under any change of variables which preserves the free field part of the Lagrangian. It also emphasizes that the `equivalence' between pseudovector and pseudoscalar couplings is to first order only.

Proton transitions to the ground, 4.43, 7.66 and 9.64 MeV states of ¹²C arising from the ¹⁰B(³He, p) reaction have been investigated at bombarding energies between 1.2 and 12 MeV.

Excitation functions for the four particle groups were taken at 150°. Angular distributions of the p₀ and p₁ groups were measured at twelve energies and total cross sections were obtained. In addition angular distributions were obtained for the p₂ and p₃ groups at a bombarding energy of 5 4 MeV.

The experiment provides evidence for excited states of ¹³N at energies of 22, 24.5, 26 and 28 MeV.

Spin-lattice relaxation and cross relaxation have been studied for Cr³⁺ ions in emerald over a concentration range of 0.49 × 10²⁰ to 1.75 × 10²⁰ ions cm^-3. Temperature dependence of T₁ was measured in the range 1.5-20 °K and orientation dependence from θ = 0 to 90°.

Spin-lattice relaxation coupling rates have been calculated using the theory given by Van Vleck in 1940, the results being in fair agreement with experiment. The observed lengthening of T₁ at cross-relaxation points is explained in a simple manner.

Cross-relaxation linewidths and rates have been measured for a 2:1 harmonic point near θ = 29°. Linewidths are in the region of 1000 Mc/s and maximum rates about 1000 sec^-1 for Cr³⁺ concentrations close to 0 5 × 10²⁰ ions cm^-3.

Exact series expansions, valid for general spin and general lattice structure, are derived for the Ising and Heisenberg magnets in which first- and second-neighbour interactions are included. The expansions are developed to order six in reciprocal temperature for the zero field free energy and order five for the zero field susceptibility. Comprehensive tables of high-temperature lattice constants are also given which enable the appropriate series for the following lattices to be written down with ease: face-centred cubic, simple cubic, body-centred cubic, simple quadratic and triangular.

Microwave resonance measurements have been made on a number of paramagnetic, ferromagnetic and antiferromagnetic Gd-Lu, Gd-Y, Gd-Sc and Gd-La alloys. A paramagnetic value for g = 1.998 ± 0.005 has been obtained for those Gd-Y alloys containing small concentrations of gadolinium and close to the free-ion value; g = 1 995 ± 0.010 for dilute alloys of gadolinium in scandium or lanthanum. These values compare with a value for g = 2.010 ± 0 010 reported by Harris et al. in 1965 for Gd-Lu alloys of similar gadolinium content. In all alloys examined the g factor increases as the gadolinium content of the alloy decreases. The resonance line intensity of the Gd-Y and Gd-Lu alloys falls rapidly on cooling below the Néel temperature and observations have been made well into the spiral-spin region. As with ferromagnetic observations there is some uncertainty as to the factors influencing the absorption and no interpretation of these lines is made.

The low-temperature thermal conductivity of ferrimagnetic systems is calculated by using the thermal conductivity integral relating it to the phonon relaxation time. Different processes which contribute to the phonon relaxation, such as mass defect, external boundary and spin-wave scattering, are taken into account. The spin-wave contribution is calculated by using Boltzmann's equation for the relaxation time and by using the matrix elements of the phonon-magnon interaction term calculated earlier. The calculation is restricted to interactions involving `acoustic' modes of phonons and magnons, and the relaxation frequency is found to be proportional to the third power of the phonon wave vector. The theory is compared with experimental results on ferrites (e.g. MnFe₂O₄) and on magnetic garnets (e.g. Y₃Fe₂Fe₃O₁₂). The agreement in the case of the ferrite is good between 5 °K to 15 °K and is only satisfactory for the garnet. However, the agreement in the region where magnon scattering of phonons is important strongly supports the mechanism considered here. The agreement near 5 °K might improve by considering the magnon contribution to heat current which may not be entirely negligible.

A theoretical study is made of the thermal resistance of a paramagnetic spin system strongly coupled to the phonons of the lattice. Near the resonant frequency of the two systems dispersion occurs. Comparison with the work of Orbach on the thermal resistance of a non-Kramers salt suggests that in a strongly coupled material this dispersion will cause an increase in the height of the peak of the magnetic-field-dependent thermal resistance. An estimate of the effect to be expected in praseodymium double nitrate is made. It is found that the thermal resistance is increased by about thirteen per cent in the vicinity of its maximum above that given by Orbach's theory which neglects dispersion. The effect of boundary scattering is also considered.

The treatment of magnetic breakdown given by Pippard in 1964 is extended to apply to real metals, in which the lattice potential is not negligibly small. It is shown that for a hexagonal metal Pippard's results remain applicable, if suitably re-interpreted. An expression is given for the probability of breakdown, in terms of the geometry of the orbits at the breakdown point. The effect of breakdown on the de Haas-van Alphen effect is discussed, and a simple expression is given for estimating the signal amplitudes in the presence of breakdown.

A derivation of the Lindemann melting-point formula is given, for close-packed metals, by combining first-order electron theory with the Bragg-Williams theory of an order-disorder transition. The Lindemann constant cannot be predicted precisely from the theory, but it is shown that it can be estimated to better than order-of-magnitude accuracy.

It is also shown that, in this linear theory, the long wavelength limit of the structure factor S(K) just above the melting point should be approximately constant. The value should be less than or of the order of 1/30. This is in fair accord with the available data.

Finally, the concept of an `ideal' liquid metal is introduced, and a precise formulation of the Hamiltonian is given for this case.

We develop a general nth-order analytic continuation formalism for evaluating the nth-order response of a many-body system which is subject to an arbitrary time-dependent mechanical perturbation K(t) and which is initially in a state of thermal equilibrium (i.e. its initial density matrix is e^-βH/Z). The formalism is a generalization of the linear analytic continuation response formalism as presented by Luttinger and Nozières and similarly entails the analytic continuation of certain suitably defined correlation functions which may, in principle, be evaluated exactly by the techniques of temperature perturbation theory. In this way the time development of the system is, in principle, exactly prescribed. Finally, as an example of the use of the generalized formalism, we evaluate the Hall effect in a metal with impurity scattering by means of the second-order formalism and the result justifies the usual expression.

The existence of gaps in the energy density of states is investigated for some model systems in one and three dimensions. A sufficient condition for band gaps in a disordered `muffin-tin' potential is derived and is applied to a model of an array of attractive square-well potentials. The one-dimensional version of this condition predicts a band gap identical with that predicted by an extension of Borland's method.

Measurements have been made of the fundamental resonant frequency of helicon waves in thin rectangular samples of pure indium. The experimental results are compared with the predictions of the theory by Legendy in 1964 and a good agreement demonstrated. The high field Hall coefficient derived from the experiments is consistent with the theoretical value calculated on the assumption of one hole per atom.

The symmetry of the centre responsible for the 9570 cm^-1 (10 450 Å) line, which appears in the absorption spectrum of neutron-irradiated magnesium oxide, has been investigated at 77 °K by analysis of the polarization spectra of the line under uniaxial stress. The line is associated with a [110] linear electric dipole oscillator in a centre which has orthorhombic symmetry and [110] orientation.

The peaks in the phonon-assisted spectrum of the line can be explained by taking 280 cm^-1 and 380 cm^-1 as the energies of the fundamental phonon modes. There is some evidence for a second zero-phonon line at 10 070 cm^-1 having phonon energies of 380 cm^-1 and 490 cm^-1.

The spectral line-reversal method has been applied to determine population temperatures within an (argon-calcium) plasma, generated by a reflected aerodynamic shock. The measurements have been made simultaneously on (i) normal bound-bound Ca I lines, and (ii) lines involving strongly auto-ionizing levels. The agreement of the two sets of values identifies the electron temperature with the temperature characterizing the populations of the bound excited states, giving improved confidence in the existence of local thermodynamic equilibrium in the reflected shock.

A simple formula for the total Born cross section for the excitation of ground state H atoms to any excited state n is given and a table of cross sections is included for representative n and impact energies.

The Green function method is used to calculate an analytic expression for the excitation spectrum of a finite fermion system interacting through a pairing Hamiltonian. The number of particles is rigorously conserved and use is made of an approximation introduced by Flowers, Irvine and Johnstone in 1964 to simplify the equations and obtain results very similar to those of the Bardeen-Cooper-Schrieffer theory.

The polarization of the scattered electron in the collision of electrons with electrons, positrons and protons, and the cross section and μ^- polarization for the reaction e^- + e⁺ → μ^- + μ⁺ have been evaluated in first Born approximation with interactions that include the `anapole' moment for the leptons.