Table of contents

An explicit expression is given for the Dirac single-particle density matrix in terms of the particle density and its derivatives. For Thomas-Fermi type calculations a reduction by a factor 1/9 in the coefficient of the Weizsacker inhomogeneity correction to the kinetic energy is implied.

It is shown that the relative signs of the hyperfine coupling constants of two spin ½ nuclei with isotropic coupling to one unpaired electron can be experimentally determined using the ENDOR technique with two radio-frequency fields in addition to the microwave field.

Energy intervals in ⁷⁵As, ³¹P and ⁵³Cr ground state hyperfine multiplets have been measured using the techniques of atomic beam magnetic resonance. Values for the interaction constants have been found which are consistent with all measured intervals to within the experimental errors. The results are for ⁷⁵As: A = -66.204(1) mc/s, B = -0.535(3) mc/s, g_J = -1.9965(8); for ³¹P: A = +55.061(10) mc/s, g_J = -2.00165(40); and for ⁵³Cr: A = -82.5994(16) mc/s and B = +0.008(12) mc/s.

The magnetic dipole interaction constant A in ⁷⁵As has been calculated by estimating the atomic core polarization using superposition of configurations and by estimating the intermediate coupling in the ground configuration. The result is -89 Mc/s compared with the experimental value of -66.204(1) Mc/s. The agreement is regarded as reasonable within the limitations of the method used. An attempt to estimate the electric quadrupole interaction shows the use of the present theory of intermediate coupling in the ground configuration only to be inadequate.

Measurements have been made of the zero field transitions for Fe³⁺ ions in AlCl₃6H₂O. The splittings observed are in good agreement with the predicted values obtained from normal paramagnetic resonance experiments.

Evidence on lattice displacements due to nitrogen platelets in diamond is reviewed. It is shown that interplanar spacings of a diamond containing platelets may differ in either sense from that of pure diamond, depending upon the index of the Bragg reflection.

Structures previously proposed for substitutional nitrogen in diamonds are criticized. A platelet structure is proposed with six carbon and four nitrogen atoms per unit cell. With standard bond lengths it produces an expansion normal to the platelet of 1.18 Å, in accord with evidence from anomalous `spike' x-ray reflections. It also produces no density change in agreement with experiment.

A numerical method, previously developed for the calculation of partial cross sections for the collisional excitation of molecular vibrational states, has been extended to make proper allowance for the effect of the repulsive centrifugal potential term on the matrix transition elements.

This treatment of molecular collisions has been applied to a study of vibrational relaxation in carbon dioxide.

Cross sections have been calculated for collision energies up to 2.5 eV using the Lennard-Jones interaction constants tabulated by Hirschfelder et al. Coupling has been retained between up to five molecular states, variously distributed between the three vibrational modes. These cross sections are in qualitative agreement with the indications of earlier more approximate calculations and have been employed for the estimation of the vibrational relaxation time in carbon dioxide for temperatures up to 1000 °K. The results are consistent with the observations made by Witteman in 1962.

It is found that the vibrational level populations may diverge from a Boltzmann distribution by as much as 50% during the relaxation process.

The wave treatment of electron capture is used in calculations of differential cross sections for H⁺-H collisions at different scattering angles and at energies between 100 eV and 2 keV. It is shown that the impact parameter treatment is inaccurate for calculations of differential capture cross sections except at high energies and large scattering angles, that is above about 5 keV when the scattering angle of the incident proton is 0.5° and above about 0.5 keV when the scattering angle is 5°. At small angles and energies the two-state wave approximation shows considerable damping of the oscillations in the probability of electron capture, in qualitative agreement with experiment. The damping due to this wave effect is compared with that due to 2pσ_u - 2pπ_u coupling (based on calculations made by Bates and Williams in 1964), and it is shown that the wave effect causes more damping especially at low energies. The two-state wave treatment is corrected for 2pσ_u - 2pπ_u coupling in the case of 3° scattering and compared with the experimental results of Lockwood and Everhart obtained in 1962. Closer agreement with experiment is obtained than hitherto.

The effect of rainbow scattering, important at low energies and for small scattering angles, is also discussed.

All the available measurements of the energy spectra of electromagnetic cascades in the cosmic radiation have been combined to predict the muon spectrum at sea level on the basis of all pions or all kaons as the source of the cascades and the muons. Comparison has been made with the muon spectrum measured directly with magnetic spectrographs and, at higher energies, inferred from underground depth intensity measurements, and by interpolation the ratio of production of kaons and pions in high energy interactions has been derived. Under the assumption that the energy spectra of pions and kaons have the same form, the K/π ratio is found to vary from 20 ± 20% at 2 × 10⁴ GeV through 10⁺¹⁵_-10% at 7 × 10⁴ GeV to 40 ± 30% at the highest energy studied, 6 × 10⁵ GeV.

A variety of studies have been made of the energy spectrum of cosmic ray muons. For energies up to 1000 GeV magnetic spectrographs have been used and at higher energies indirect measurements have been made. Inconsistencies between the results of the various experiments have been reported and these are the subject of the present paper.

The differences are resolved and an estimate of the spectrum is made for the energy range 20-7000 GeV.

Using Bloch and hydrogen-like wave functions calculations have been made of the recombination rates by Auger processes in which one localized and three band states are involved. Four essentially different processes have to be considered, depending on how many of the states are in the conduction band and how many are in the valence band. In addition, each transition rate depends on the quantum numbers (n, l) of the localized state involved in the transition. This problem is solved in zero-order which is valid when the energy change of an electron making a transition is large compared with kT. The validity of this calculation is considered by extending the calculations for the ground state to higher order. The zero-order results for the excited states and the higher order results for the ground states, as well as the detailed consideration of exchange effects at various points in the paper, appear to be new. Zero-order results for the ground state have been obtained by previous workers. These results however, must be amended to ensure a self-consistent theory. Formulae are given which enable impact ionization coefficients to be obtained from the recombination coefficients. The calculated recombination coefficients are presented in graphical form, and show (i) at low temperatures a fluctuating variation with the quantum number l of the excited state into which an electron is captured, (ii) an expected variation with the energy level of the donor, (iii) a negligible effect due to exchange, (iv) that the higher order terms can be important.

An analytical and numerical treatment is given of the change in width of the Fermi soft x-ray emission edge produced by instrumental and physical factors. It is assumed that these can be represented by simple analytical functions and that their effect on the emission edge can be described by a convolution integral.

The problem of ultrasonic attenuation in the presence of magnetic breakdown is considered in a two-dimensional model of a metal with a rectangular lattice, and with a field applied in the third direction. Simple symmetry arguments are used, and the matrix elements for the processes are examined qualitatively. It appears that it is possible to have `normal' resonant absorptions, corresponding to the usual open-orbit absorptions, but if breakdown occurs in both directions perpendicular to the field there is also a possibility of having `anomalous absorptions which are repetitive in the reciprocal of the field, and whose repetition frequency depends on the lattice parameters and not on the detailed shape of the Fermi surface. It may be that both types of attenuation have an associated electroacoustic effect producing a voltage transverse to the field and to the wave vector of the sound, and this effect may be very strong for the anomalous absorptions.

The method of double-time Green functions is applied to the problem of the Schottky anomaly in the specific heat of a spin system which is strongly coupled to the lattice vibrations. The equilibrium properties of the system are investigated in a random phase approximation. The main effect of the spin-lattice interaction is found to be the anomalous dispersion of lattice waves which are on speaking terms with the spins. The dispersion law is essentially that found by Jacobsen and Stevens from classical equations of motion. The total entropy change associated with the Schottky anomaly is found to exceed the value R ln 2 which it would have in the absence of spin-lattice interaction. A comparison of the observed and calculated specific heats of cerium ethyl sulphate yields information about the crystalline field splitting of the ground level of the Ce³⁺ ion and about the strength of the orbit-lattice interaction.

In this paper the properties of athermal monomer-dimer liquid mixtures are considered on the basis of the quasi-crystalline model. The treatment is a development of the previous work of Rushbrooke, Scoins and Wakefield and the lattice considered is the face-centred cubic. The results obtained are compared with those of other workers by considering the molar entropy of mixing and it is shown that the present results are an improvement on these previous treatments.

By applying a small amplitude modulation to the mean current in a mercury positive column and observing the microwave reflection from the column in the region of resonance, the mercury ionization rate may be determined. The results show better agreement with theory than do probe measurements, probably because, unlike probes, the microwaves do not disturb the ion current flow to the walls. Electron temperatures may be calculated from the ionization rate using the `free-fall' theory proposed by Langmuir and Tonks in 1929.

The equilibrium shock wave parameters for xenon are calculated for several cases of fore-pressures between 0.5 and 760 torr and post-shock temperatures up to 20 000 °K. The problem of determining the electronic excitation partition function of neutral and ionized xenon is discussed with respect to both the presence of auto-ionized neutral xenon levels and the missing energy level data for both high principal quantum numbers and of intermediate azimuthal quantum numbers. The role of both the missing energy levels and also Debye-Hückel electrostatic effects on the shocked gas properties is discussed.

Expressions for the static dielectric constant of ice can be derived using either the Kirkwood-Fröhlich theory of dipolar dielectrics or the large-assembly method of Slater's ferroelectric theory. The two theories are here extended and brought into equivalence. The Slater theory is shown to be correct to the well-known first approximation in the theory of regular assemblies, and an alternative derivation of his combinatory formula is given which reveals more clearly the basis of the approximation. The Kirkwood correlation factor for ice is calculated on the same basis. Both methods indicate that the error due to the first approximation is of the order of 1%.

The problem of the local electric field in ice is critically examined, and reasons are given for thinking that Frohlich's cavity theory is not appropriate, and that the local field differs little from the externally applied field. The difficulty of estimating the dipole moment of a molecule in the crystal leaves any appeal to experiment inconclusive.

The electronic energy spectrum and degree of localization for a one-dimensional random alloy are calculated by numerically solving a modified form of Schmidt's functional equation. It is found that the spectrum is very complex. The results are similar to those obtained by Dean for the closely related vibrational problem.

The levels of ³⁶Ar have been investigated by means of the reaction ³⁹K(p, α)³⁶Ar. The existence of six excited states in ³⁶Ar has been confirmed.

The experimental and theoretical values of scattering cross sections of gamma radiation, reported by Lakshminarayana and Jnanananda in 1961, have been extended to cover a wider range of atomic numbers and photon energies.

A band system assigned to BaS has been observed in absorption at 2100 °C. The following constants have been derived from measurements of band heads.

State	T00	ωe	xeωe
A	25934	253	similar 0.25
X	0	377.1	1.22

The rotational structure appears to be simple and the system is probably the analogue of the A-X system of BaO.

An alternative interpretation of the axial Zeeman effect of the absorption line at 17 308 cm^-1 of CoCs₃Cl₅ is given, in which angular variations of the spectrum are related to polarization phenomena rather than to shifts in the energy levels. A number of theoretical difficulties that had arisen are thereby eliminated.