Table of contents

A simple derivation is given of both the Einstein and Møller total energy-momentum conservation laws in general relativity from Komar's covariant identity. The Einstein and Møller theories are briefly compared. The main purpose of the paper is to use the Møller theory to obtain the energy distribution in the field associated with the Born-Infeld model of a charged particle incorporated in general relativity. It is shown that the Maxwell-Einstein point charge may be regarded as a limiting case of the Born-Infeld model provided an infinite negative mass of non-electromagnetic origin is introduced to compensate for the infinite Maxwell self-energy.

Results are reported of machine calculations of specific heat and Grüneisen's γ and their variation with temperature for a face-centred cubic solid. Estimates are also made for the elastic constants of a polycrystal at 0 °K. The harmonic approximation is employed and Houston's method, as developed by Horton and Schiff in 1959, is applied to a model assuming central forces derivable from a Mie-Lennard-Jones m-6 potential. Dispersion curves are obtained for six different lattice directions. A systematic investigation is made of the effects of varying the value of m and of the number of interacting neighbours for each of the inert gas solids Ne, Ar, Kr and Xe. The potential parameters are in each case determined from experimental values of lattice constant and sublimation energy and the often neglected importance of this part of the calculation is demonstrated. The result of neglecting zero point energy in expanding the lattice is also investigated and is found to lead to appreciable error. A limited assessment of the effect of anharmonicity, made by using the potential parameters suggested by Domb and Zucker in 1956, indicates that this may be small at low temperatures.

A comparison is made with experimental specific heat data in the case of Ar and Kr and best agreement found using a model involving only limited neighbour interactions. The implications of the results for the validity of the model are discussed.

The usual assumption that the specific heat of a paramagnetic crystal can be written as the sum of a lattice specific heat and a spin specific heat needs modification when there is strong spin-lattice coupling. The necessary theory is developed for two simple examples, and it is shown to lead to an increase in the maximum of the Schottky specific heat. It is suggested that the previously unexplained magnitudes of the specific heat maxima in cerium and praseodymium ethyl sulphates arise from strong spin-lattice couplings.

The interaction energy between two spins in a cavity is derived. It is assumed that each spin interacts with all the magnetic fields which exist at the point in the cavity where it resides. One spin is then made aware of the presence of the other by the changes in the mode excitations which are produced. This approach is thought to be more fundamentally based than that which takes, from classical magnetostatic theory, the standard dipolar form of the interaction. On deriving a spin Hamiltonian description, by second-order perturbation theory, it is found that the equivalent coupling energy between the spins consists of an infinite number of terms, the largest of which is the usual dipolar energy. The other terms occur because each spin has an infinity of images in the walls of the cavity and there is a dipolar interaction with all the images of the other. The newly developed formalism is then applied to a problem in cross relaxation, to see whether there are any appreciable differences in the two treatments. The algebraic differences which exist are shown to be of negligible importance.

An analysis is given of the interaction between the electronic configurations (nl)^N(ńl')^4l'+2 and (nl)^N(n'l') ^4l'+1(n"l"), and of the way it affects the matrix elements of a single-particle double-tensor operator T^(κk). The theory is applied to rare earth ions, for which nl ≡ 4f. It is shown that the infinite sums that occur in the calculation of the corrections to the spin-orbit coupling due to this kind of configuration interaction can be related to the corresponding sums that occur when the corrections to the orbital part of the magnetic hyperfine interaction are analysed. Deviations between theoretical and experimental values of the spin-orbit coupling constant for rare earth ions are used to estimate the adjustments that should be made to ⟨1/r³⟩ when calculating the magnetic hyperfine interaction. The effective values of this quantity thus obtained are found to be in excellent agreement with experiment. Some discussion is given of the calculation of the analogous adjustments to ⟨1/r³⟩ for the quadrupole interaction.

The possible establishment of a nuclear ferromagnetic state of a metal depends on the hyperfine interactions between the electron and the nuclear spins. Since this interaction is of very short range (δ-like) methods have been developed in this work which, in contrast to previous treatments, avoid the use of perturbation theory. An explicit condition has been established which shows whether or not a metal should exhibit nuclear ferromagnetism at very low temperatures. This condition depends on details of the electronic wave functions. It is satisfied in the case of sodium but not satisfied for metallic hydrogen.

Discrepancies between experimental and theoretical results for the shape of the Compton line in X-ray scattering from metallic lithium are examined. The effect of electron correlations on the Compton profile is first examined using the high density momentum distribution of an electron gas as given by Daniel and Vosko. The smallness of the effect leads to an examination of earlier work based on approximate Bloch wave functions. By using experimental binding energies in conjunction with the virial theorem it is shown that the mean kinetic energy of the conduction electrons in lithium is almost certainly overestimated by earlier work and is about 7 eV. Such a figure indicates an even greater discrepancy between theory and experiment.

The theory of averaging neutron transmission over an energy interval in which the nuclear cross section contains many resonances is outlined. Particular consideration is given to the validity of assuming the single-level Breit-Wigner formula for the resonances in carrying out the averaging. Hibdon's data on the average transmission of neutrons through slabs of ²³⁸U and ²³²Th are analysed in order to extract the p-wave neutron strength functions. The values for T_n⁽¹⁾/D are (2 5 ± 0 4) × 10^-4 and (2.40 ± 0 36) × 10^-4, respectively.

The resonant elastic scattering of γ rays by ¹⁴⁴Nd nuclei has been investigated and the small cross section observed indicates a lower limit of 2.0 ± 0.4 × 10^-14 seconds for the lifetime of the 2 18 MeV (1^-) level.

Using an internal source scintillation spectrometer, the L/K electron capture ratio in the decay of ¹³¹Ba to the 620 keV level in ¹³¹Cs has been found to be 0.135 ± 0.009. This result may be compared with the theoretical value of 0.131, which is obtained using a value of 1.7 MeV for the atomic mass difference between the ground states of ¹³¹Ba and ¹³¹Cs. Alternatively, assuming the electron wave functions to be correctly determined by theory, the upper limit for the L/K ratio may be used to set a lower limit to the transition energy available in the decay. The value obtained by this method for the ground state atomic mass difference is 1.06 MeV, which is consistent with the observed existence of an electron capture decay branch to the 1 04 MeV level in ¹³¹Cs.

The Hall mobility of electrons in n-type germanium at high electric fields is studied, taking into consideration the ellipsoidal many-valley band structure. The distribution function of carriers in a many-valley semiconductor is first obtained assuming acoustic phonon, optical phonon and intervalley scattering. Expressions for Hall mobility are then derived using this distribution function and numerical computations made with the parameter values of n-type germanium. It is found that (i) the Hall mobility is anisotropic, (ii) its value is a maximum when the electric field is applied in the <100> direction, and (iii) its value is a minimum when the electric field is in the <111> direction. It is also found that the anisotropy factor is independent of the value of the electric field for predominant acoustic phonon scattering, but depends on the field for predominant optical and intervalley scattering.

The complex refractive indices have been measured for the three axes of single crystals of gallium over the range 0.4 to 8.0 micrometres using a reflection method with polarized light.

In the lower energy regions if the absorption is ascribed to conduction electrons the numbers taking part according to the orientation of the electric vector relative to the axes are

Na	Nb	Nc
1.22	1.94	0.31 × 1022 cm-3

At higher energies there is an absorption band at about 2 1 eV (5700 Å) for all three axes, a and c being much alike but the peak is much weaker for the b axis. The transitions responsible for this absorption have not yet been conclusively identified.

Time-resolved measurements of infra-red emission from deuterium plasma in the theta pinch have been made at wavelengths between 70 and 250 micrometres using an indium antimonide photoconductive detector. At long wavelengths there is a sharp drop in emission near peak compression and this is attributed to an increase of the critical plasma frequency beyond that corresponding to the wave-length of observation. Thus the effect affords a means of determining the electron density. For plasma whose dimensions and impurity content are known, the absolute measurement of the radiation at short wavelengths leads to an estimate of the electron temperature, but as the emission depends only weakly on temperature the method is not of great precision.

Experimental values of density and temperature show reasonable agreement with those predicted by numerical solution of the one-dimensional two-fluid hydromagnetic equations, though the comparison suggests axial loss of plasma in the practical case. The measured electron temperature is in excess of 10⁶ °K for both trapped parallel and trapped reversed magnetic field. Bremsstrahlung from impurities makes an important contribution to the radiation from reversed field discharges and appropriate allowance for this increases the estimated temperature.

The number distribution of fast and slow electrons in the cathode region of low voltage arc discharges has been calculated on the basis of diffusion theory. The results agree well with measurements obtained in hydrogen by a dynamic probe technique. A winging effect has been found which may be connected with the production of electron oscillations in the plasma.

The inherent instability associated with a specific differential negative resistance in a solid is discussed for the two cases voltage-controlled and current-controlled differential negative resistance. In the case of voltage-controlled differential negative resistance, it is shown that domains of high electric field occur. These domains are generally mobile and their movement is discussed. In the case, of current-controlled differential negative resistance, high current filaments form. The sizes of the domains and filaments are governed by the size of the specimen and by the principle of least entropy production. The effect of an external circuit is to inhibit stable filament formation in the case of current-controlled differential negative resistance and impose conditions under which stable domain formation can be observed in the case of voltage-controlled differential negative resistance.

Electron attachment and ionization cross section measurements have been made under single collision conditions both by an absolute and a comparative method, and are compared with recent relevant data. The corrections employed in the latter are seen to be valid in view of the good agreement with the absolute measurements. In the absolute measurements a Tate and Smith type apparatus is used where all the ions are collected and the measurements do not involve any kinetic energy or angular discriminations. Ionization cross sections for 80 eV electrons in the three gases oxygen, carbon monoxide and carbon dioxide are 2 76 × 10^-16 cm², 2.93 × 10^-16 cm² and 4.23 × 10^-16 cm² respectively, while the peak resonance attachment cross sections resulting in the formation of O^- are 1.32 × 10^-18 cm², 2.18 × 10^-19 cm², 1.37 × 10^-19 cm² and 4.44 × 10^-19 cm² at 6 7, 10.1, 4 45 and 8.15 eV respectively. The value 1.37 × 10^-19 cm² at 4.45 eV corresponds to a second peak observed in carbon dioxide. Ionization potentials corresponding to the formation of positive ions and the onset potentials corresponding to the formation of negative ions in the dissociative process have been determined.

The growth of pre-breakdown currents in uniform field conditions in C₃F₈ at various pressures, for values of E/p in the range 90-300 V cm^-1 mmHg^-1 indicates large electron attachment processes. Assuming that the relevant mechanism of negative ion formation is dissociative attachment, values of the ionization and attachment coefficients (α and η) have been calculated from the semi-logarithmic plots of current against electrode separation employing the modified Townsend equation for the current growth and a process of curve fitting. Static breakdown potentials have also been measured up to a value of pd = 400 mmHg cm.

According to the Stokes-Wilson approximation, the tails of the line profile resulting from diffraction by a crystal containing a screw dislocation approach zero in accordance with an inverse-cube law. The variance of the line profile, therefore, would increase asymptotically as the logarithm of the range, instead of being independent of the range, as for strains that nowhere exceed a few per cent. The difference arises from the singularity at the dislocation axis, for which the usual models give strains of the order of unity. Very large strains are presumably relieved by some physical mechanism, but the logarithmic law might well apply over the measurable portion of a line profile.

The accuracy of the Stokes-Wilson approximation is difficult to assess, and the inverse-cube law has therefore been rederived without serious approximation from the asymptotic form of the Bessel-function expression for the diffraction pattern of a screw dislocation. The logarithmic term is shown to increase as the square of the index of reflection corresponding to the direction of the dislocation axis. Independent confirmation of the inverse-cube law has been obtained by Wilkens in the course of a discussion of the Warren and Averbach analysis of line profiles.

Numerical calculations, based on spin wave theory, have been carried out in connection with the magnetic properties of two simple ferromagnetic salts, EuO and EuS, and the antiferromagnetic compound MnF₂. The resulting predictions regarding magnetization and specific heat have been compared with experiment over as wide a temperature range as possible. The calculations envisage the possibility that more than one set of exchange interactions may be involved and in the case of MnF₂ allowance is made for a temperature dependent anisotropy field. Account is taken of two-magnon dynamical interactions and renormalized spin wave energies are used in evaluating the magnon populations. The effects of kinematical interactions between spin waves are, however, ignored. It is found that the theoretical estimates of magnetization appear to fit experiment over a temperature range extending to about three-quarters of the Curie or Néel temperature of the system concerned. Specific heat calculations, on the other hand, seem to depart from the experimentally observed values at a much lower temperature, of the order of one-third the critical temperature. The calculations on EuO and EuS confirm the general model for the exchange constants of these salts proposed by McGuire, Argyle, Shafer and Smart and by Gorter and his co-workers.

Range distributions of ²²Na positrons have been measured in argon and nitrogen, using a two-photon coincidence apparatus. The distributions are compared with theoretical positron path length distributions, and are used to calculate correction factors applicable to the study of positronium formation in gases.

The mean number of collisions suffered by electrons in diffusing from the inner to the outer radius of a hollow infinite cylinder is calculated using a development of a method previously outlined by McClure for spherically symmetrical problems. His approach appears to be superior to the standard Harries-Hertz treatment in that it takes variation of total diffusion time into account at an early stage; it also appears to be applicable at lower gas pressures. It is shown that utilization of the Harries-Hertz formula in the classical Maier-Leibnitz experiment leads to a 5% overestimate of the inelastic electron collision cross sections of the helium atom for electron energies near the metastable threshold, in so far as the present calculation and that of Harries and Hertz are applicable to Maier-Leibnitz's apparatus. The total error in his cross section values is probably not less than 30%.

The conditions governing impact ionization in semiconductors are investigated, and then applied to determine the position in energy and k space of the various impact ionization thresholds in silicon and germanium. A combination of these results with the band structure, and with the known mechanisms of optical absorption, enables the enhanced quantum yield threshold in silicon and germanium to be explained. In silicon the quantum yield curve above the threshold appears to be determined primarily by the number of carriers available for ionization.

The luminescence characteristics of some members of a new class of phosphors, the group III-VI compounds, are described and in particular the results for GaS, GaSe, Ga₂S₃, Ga₂Se₃ and In₂S₃ are presented. Luminescence bands are studied in self-activated crystals prepared under oxidizing conditions. At low temperatures (60 °K) bands are observed in GaS at 0.66 μm and 0 75 μm, in GaSe at 0.71 μm, in Ga₂Se₃ at 0.9 μm and in Ga₂S₃ at 0 425 μm, 0.62 μm, 0 72 μm and 0 87 μm. The analysis of thermoluminescence in Ga₂S₃ is seen, as in the case of In₂S₃, to indicate the existence of carrier trapping states with the abnormally high capture cross section of about 10^-12 cm². Independent studies of the temperature dependence of phosphorescence in In₂S₃ give the same result.

Together with the recent structural analysis of In₂S₃ and Ga₂S₃, these results suggest that the trapping states may be a characteristic of the defect semiconductor lattice. The role of impurities in the formation of an impurity band in other semiconductors is seen in the case of some defect semiconductors to be assumed by the ordered array of lattice defects (vacancies). Although at low temperatures the `defect band' is unoccupied in the unexcited crystal, a contribution to the conductivity following excitation at low temperatures is observed which is not accompanied by luminescence, and is suggestive of carrier motion in such a band.

Luminescence characteristics are presented for the ternary compounds CdIn₂S₄ and CdGa₂S₄ and for the mixed binary systems Ga₂S₃-In₂S₃ and Ga₂S₃-Ga₂Se₃. The phosphors were `self-activated' and prepared under sulphurizing conditions. At 65 °K two emission bands are observed in CdIn₂S₄ centred at 1.42 eV and 1.65 eV respectively, and in CdGa₂S₄ at 2.15 eV and 1.63 eV respectively. The progressive replacement of indium by cadmium at tetrahedral sites on introducing CdS into In₂S₃ to form the spinel CdIn₂S₄, and the associated shifts of the luminescence spectra, suggest a model in which luminescence transitions are associated with the region of a cation vacancy. The two observed emission bands in In₂S₃ are associated with complexes formed by neighbouring anions in the vicinity of the two types of cation vacancy site in this material, one tetrahedrally and one octahedrally coordinated. Correlation of the luminescence with change in the nature of either the cation or the anion in the mixed binary systems is an important feature in the study of group III-VI compounds.

The elastic scattering of electrons by ground state hydrogen atoms, for electron energies below the first threshold of excitation of the atom (10 2 eV), is investigated numerically making full allowance for exchange and for the virtual excitation of the 2s and 2p states of atomic hydrogen. Singlet and triplet phase shifts are obtained for values of the total orbital angular momentum of L = 0, 1 and 2. The results indicate the existence of very narrow resonances just below the n = 2 excitation threshold for several of the spin and angular momentum states. This work was done by an integral equation method, reported previously by Smith, McEachran and Fraser, which explicitly allows for the damped exponential behaviour of the wave functions in the asymptotic region of the closed scattering channels. The results of the present calculation are in very good agreement, except very close to the n = 2 threshold in the orbital angular momentum state L = 1, with the previously reported results of Burke and Schey who have treated this same problem by a different numerical method.

The viscosity of supercooled water has been measured, using a capillary flow technique, down to -23.8 °C. The large supercooling was obtained by removing nearly all suspended particles from the water by an ion exchange column. These results agree with those obtained by White and Twining in 1914, who obtained results down to -9.3 °C.

An asymptotic approximation to the effect of multiple elastic scattering in gas electron diffraction has been made using stationary phase techniques. The intensity due to multiple scattering is shown to be similar in structure to the usual kinematic intensity function and may constitute as much as ten per cent of the observed molecular structure.

A close examination of recent data on the x-ray quadrupole and non-quadrupole transitions reveals, for the first time, five pairs of lines, namely (L_IM_I, L_IIM_II), (L_IN_I, L_IIN_II), (L_IO_I, L_IIO_II), (L_IIO_III, L_IO_IV,V) and (L_IM_I, L_IM_II), which form screening doublets. The data include the values reported by the authors in 1962 and 1963 for iridium, gold, thallium and thorium.

A singlet system B ¹Σ⁺-X ¹Σ⁺ ascribed to BeS has been observed in absorption at 1900 °C. Values of the constants are

	T00	ωe	r0(Å)
B 1Σ+	25868	850.4	1.818
X 1Σ+	0	997.4	1.745

The lower state X ¹Σ⁺ appears to be the ground state of BeS.

The spin-lattice relaxation has been measured in the two lowest Kramers doublets of Dy³⁺ ions in yttrium ethyl sulphate. For each doublet the results appear to be consistent with relaxation via other thermally accessible levels by the two-phonon resonance process.

Calculations have been carried out in the 1s-2s-2p close-coupling approximation for cross sections in the vicinity of the 2s, 2p threshold. It is shown that the cross sections Q(1s-2s) and Q(1s-2p) in the vicinity of the threshold do not tend towards zero.

The specific intensity of entropy radiation of the whole beam resulting from a superposition of N mutually incoherent, arbitrarily polarized beams is calculated using the coherency matrix and the Stokes parameters representation.

The procedure of using rotating coordinates in scattering problems, which can be justified for exact calculations, breaks down in calculations based on the Born approximation.

Following a recent discussion of the crystalline field of the (N₁(H₂O)₆)²⁺ complex based upon conclusions drawn from resonance experiments on Tutton salts, it is pointed out that recent crystallographic data show that the symmetry of the crystal field is less than orthorhombic. Results obtained for other Tutton salts are also briefly discussed.