Table of contents

A general study of a new system, known as a spin dipole, is made. Its properties are contrasted with those of a point dipole, a system which has recently received some attention. The Schrödinger equation for the eigenvalue problem associated with zero energy is solved. This makes it possible to give conditions for the existence of bound states for the finite (spin) dipole.

The structure of some low-lying autoionizing D states in helium has been investigated. Eigenvectors are presented which show that a plus-minus classification is appropriate. The relationship between structure and width is discussed.

The differential polarizabilities in the ground states of hydrogen and the alkalis are discussed. General angular momentum arguments are used to separate the contributions from the different hyperfine-structure interactions. Three separate polarizabilities are introduced and their contributions to the quadratic Stark effect are given. Explicit expressions for the polarizabilities are found for hydrogen.

It is shown how the one-electron exchange integral which arises in the theory of heavy particle collisions can be expressed as the solution of a first-order differential equation with respect to time. It is expected that this method will have a wide application in the calculation of heavy particle collision phenomena at high energies.

Total charge-transfer cross sections have been calculated for H⁺-H collisions taking account of nuclear symmetry at incident energies between 2 ev and 2 kev. It is shown that nuclear symmetry has negligible effect on the cross sections above 3 ev. The cross sections are above the Dalgarno-Yadav values but they are still below the experimental cross sections of Fite et al. It is suggested that this difference might be partly due to the geometry of the apparatus and partly due to scattering near the rainbow angles associated with the gerade potential.

Electron-impact ionization cross sections for Li, Na, K and Rb have been computed using empirical methods. These are in fair agreement with experiment in the moderate- to high-energy region. In the same energy region, however, the quantal calculations seriously underestimate the cross sections.

Ionization cross sections of He, Li, O and N by electrons and protons are calculated in a classical model, and compared, where possible, with quantal calculations and experiment.

The cross section for positronium formation by positron impact on hydrogen is calculated in the two-state approximation, taking account of the polarization of the hydrogen and positronium atoms in each channel. It is found that the polarization forces dominate the cross section near the threshold for positronium formation and evidence is presented for the existence of a positronium-proton virtual bound state that gives rise to a resonance in elastic e⁺-H scattering just below the formation threshold.

The authors have recently shown that the inclusion of hyperbolic perturber orbits considerably modifies the electron impact width and shift of an isolated ion line in the adiabatic limit. In this paper we extend these results in order to estimate the effect elastic collisions have upon the broadening parameters of several S II lines.

Approximate values for the inelastic contribution to the widths of these lines are also reported and the total calculated widths are found to be in satisfactory agreement with experimental measurements.

Our results show that elastic collisions constitute an important source of broadening for ion lines.

Previous calculations of neutral argon transition probabilities have been extended to higher energy levels. The transformation to intermediate coupling was carried out from the LS and the J_1J coupling schemes. The line strengths were put on an absolute scale using the Coulomb approximation

The relationship between the Coulomb and nuclear excitation of collective and one-particle levels in the inelastic scattering of complex nuclei has been estimated in the energy region above the Coulomb barrier. It has been shown that for heavy ions and medium and heavy target nuclei about 90% of the inelastic scattering cross section for E2 transitions is due to the Coulomb excitation. Estimates of the E3-transition inelastic scattering cross section for three different nuclei have also been made.

The inelastic scattering of cold neutrons by gaseous methane and ammonia has been investigated to test the accuracy of the theories of neutron scattering by molecules. The experimental results at small scattering angles, corresponding to small neutron momentum transfers, show separate peaks due to translational and rotational motions of the molecules, demonstrating the shortcomings of the Krieger-Nelkin theory. Agreement with Griffing's theory assuming discrete rotational levels is good at high scattering angles, but at the lowest angles there is a small discrepancy between the theoretical and measured heights of the translational and rotational peaks.

An approximate representation of the phase shifts of atomic potentials is derived making use of the weakness of the pseudopotential. By using this representation in conjunction with the Korringa-Kohn-Rostoker scheme for band structure calculation, various approximate schemes for band structure calculation are derived. In particular, when no virtual resonances are present one obtains the well-known pseudopotential method, and when a single virtual resonance is present one obtains a hybrid tight-binding plane wave scheme of the kind recently discussed by Heine and others.

A simple model in which various electron bands in a metal are represented by a set of electron gases is used to study the screening of charge and exchange perturbing potentials. Differences in the screening behaviour of para- and ferromagnetic metals are discussed. A three-electron gas version of the model is used to simulate an iron matrix containing transition element impurities. To understand the screening around elements to the right of iron in the periodic table, it is necessary to introduce a restriction limiting the possible increase in the population density of the gas representing the majority spin d electrons. Physically this restriction arises as a consequence of the corresponding d band in iron being almost full. With this addition the model leads to a qualitative understanding of the neutron scattering data and the experimental values of d[mu over bar]dc for dilute alloys of iron with other transition metals.

Measurement of the concentration dependence of the Knight shifts of the ¹⁹⁹Hg nucleus in the series of alloys Hg-Cd, Hg-In, Hg-Pb and Hg-Bi are reported together with measurements on the solute nucleus in the Hg-In and Hg-Bi systems. The results are discussed in terms of the Mott theory and in terms of a non-localized pseudopotential.

The optical constants of a single crystal of magnesium have been measured in the spectral range from 1.1 to 2.2 μm using a polarimetric reflection technique. The results show a strong polarization-dependent absorption peak, centred about 0.71 eV, arising from optical interband transitions.

An empirical pseudopotential model has been used to compute the interband absorption and correlate it with available de Haas-van Alphen data. The resulting pseudopotential from factors (V(10[1 over bar]0) = 0.160 eV, V(0002) = 0.310 eV, V(10[1 over bar]1) = 0.421 eV) are only slightly lower than those calculated by Animalu and Heine in 1965.

The model potential of Heine and Abarenkov may be used to calculate the band effective mass m* of a solid This is an important parameter in the calculation of many properties, of which some examples are given here. The values of m* given by Animalu and Heine have been found to be in error. The corrected values presented here are found to be in good agreement with roughly comparable quantities from detailed band calculations, and the mehtod therefore appears to be quite reliable, contrary to former opinion based on the results of Animalu and Heine.

Measurements were made of the variation of ultrasonic attenuation in indium between 4.2 °K and 1.3 °K for longitudinal waves at 50 Mc/s and 90 Mc/s and for transverse waves at 50 Mc/s. The results were analysed to determine the electronic contribution to the total attenuation. Low-amplitude dislocation attenuation was significant. Amplitude-dependent attenuation was investigated qualitatively.

The electronic attenuation and the electronic mean free path were very anisotropic. The transverse normal-state attenuation decreased with decreasing temperature, in spite of the increasing mean free path. The results are discussed in terms of a suggested mean free path reduction near the edges of the Fermi surface in the reduced-zone scheme. The anisotropies are attributed to the combination of a sharply peaked electronic distribution function under the action of the ultrasonic wave, a Fermi surface with sharp edges, and a high probability of small-momentum scattering by thermal phonons

For propagation along [110] the value of the average deformation parameter K_x is calculated to be equal to the free-electron value. For propagation normal to (111) K_x is 26% greater than this value. These results, together with the large observed transverse attenuation in the superconducting state, indicate that band structure effects on the deformation parameters for indium are appreciable.

A new method of investigating the problem of spectral gaps in the spectra of disordered lattices is presented, which is very general and not only free from any dimensional restriction but also does not depend on any special assumption concerning the dynamical structure of the lattice, provided only that the interactions are harmonic. By this `evaporation method', it is shown that for the vibrational frequency spectra of some mixed crystals of the alkali halide type, the Saxon-Hunter-type theorem is valid for all the expected spectral gaps, that is, each of the possible common gaps of the constituent regular lattices gives at the same time a gap of the mixed lattice.

It is shown that the Fourier coefficients in the expansion of the lattice Green function and the frequency spectrum can be calculated without evaluating the eigenvectors and eigenvalues of the dynamical matrix. With this simplification, the Fourier expansion method can now be conveniently applied to perturbed lattices.

Results of the dielectric loss studies of sodium chloride crystals doped with Co²⁺ are reported in the temperature range 90-200 °C and frequency range 30 c/s to 100 kc/s. Two peaks are observed in the tan δ-frequency isothermals. The plots of ln f_m (f_m being the peak frequency) against 1/T are found to be straight lines for both the peaks and yield 0.67 and 0.72 ev as the values for activation energies for the low (A peak) and the high (B peak) frequency peaks respectively The A and B peaks are attributed to the pairs of the substitutional impurity with the second-neighbour vacancy and with the first-neighbour (nearest cation neighbour) vacancy, respectively, as suggested by Dreyfus for the divalent impurity cations smaller in size than the cation of the host crystal. The height of the B peak changes from 0.24 × 10^-2 when quenched from 100 °C to 0.70 × 10^-2 when quenched from 400 °C or higher temperatures. The effect of the heat treatment on the height of the A peak is smaller. It is shown that, although the results strongly support the model of Dreyfus, the possibility of cobalt being in the interstitial positions cannot be ruled out on the basis of the present results. Experiments performed on crystals of different thicknesses, on the low-frequency variation of the capacitance of these crystals and on the effect of different heat treatments show that the observed loss peaks are not due to Maxwell-Wagner losses or interfacial polarization. The activation energies calculated from the dielectric loss data are found to be of the same order of magnitude as those obtained from the conductivity data, and are much smaller than those obtained from the diffusion data

Anharmonic contributions to the mean-square velocity of an atom in a crystal lattice have been obtained for all temperatures using the method of thermodynamic Green functions. The result is applied to the case of a linear chain.

A finite-temperature perturbation theory for the Heisenberg model of ferromagnetism is presented. A modified Wick theorem proved for S = ½ and finite temperature allows the terms in the expansion for the temperature-dependent Green function to be expressed by linked diagrams of the Feynman type. These differ from the normal boson and fermion diagrams in the appearance of time(temperature)-dependent broken lines between propagators, a modification caused by the spin statistics.

The finite temperature perturbation formalism, developed for the spin ½ Heisenberg model in the preceding paper, is applied under high-density conditions. The diagrams arising in the expansion for the temperature-dependent Green function are classified at finite temperatures with respect to the high-density parameter 1/z, where z is the number of spins interacting with a given spin. The results of molecular field theory are found in first order. The next two orders are considered in detail at low temperatures where they yield respectively spin-wave theory and Dyson's results in the first Born approximation to the scattering of spin waves. The high-density classification, previously justified only for T/T_c > 1/z is directly verified at low temperatures. Higher-order graphs are found to yield at low temperatures the expressions given originally by Tahir-Kheli and ter Haar for the lifetimes of the excitations.

In a concentrated paramagnetic salt at low temperatures it is possible to think of spin excitations with a well-defined wave vector k similar to spin waves. These are coupled to the phonons and result in mixed modes given by the Jacobsen-Stevens relation. These mixed modes are studied by the Green function method of Sears extended to higher order to obtain a finite lifetime of the modes which increases as T increases and the spins disorder. Paramagnetic relaxation and thermal conductivity are studied in this system. The main effect of the spin-phonon coupling at low temperatures is to modify the modes; lifetime effects occur in higher order. Comparison is made with some existing experimental data and more detailed experiments are suggested.

Antiferromagnetic resonance over a range of temperature, frequency and orientation has been observed at millimetre wavelengths in single crystals of orthorhombic Cr₂BeO₄ (T_N = 28 °K). Critical-field resonances occur along both b and c axes, at magnetic fields of 29.5 kG and 47.7 kG respectively, and a non-collinear spin array is inferred. The possibility of a spiral spin structure is considered.

We generalize the usual Dyson expansion to incorporate a time-dependent unperturbed Hamiltonian. The formalism is then used to perform a non-stochastic calculation of the free induction nuclear magnetic resonance decay of transverse magnetization (i.e. the Bloch decay) in a rotating single-crystal sample. We assume the magnetic nuclei to have spin one-half, to be located on a simple cubic lattice and to interact with each other by the direct dipolar interaction only. We have computed our analytic expression for a case of possible experimental interest namely a single-crystal sample of CaF₂ rotating about the [1, 1, 1] axis with the main field initially along the [1, 0, 0] axis of the crystal.

The de Boer model for the F centre in CaO has been confirmed by paramagnetic resonance measurements on polycrystalline samples of CaO enriched with ⁴³CaO, the isotropic and anisotropic hyperfine interactions were found to be 9.1 and 1 0 G respectively. The hyperfine lines were also measured for the F centre in neutron-irradiated BaO and isotropic interactions were obtained for ¹³⁵Ba of 75 5 G and for ¹³⁷Ba of 86.3 G, the anisotropic interactions were close to 5 and 6 G respectively. The F-centre hyperfine parameters are compared for the alkaline earth oxides using the point-ion-lattice model of Gourary and Adrian.

The allowed and forbidden hyperfine transitions of the S-state ion Mn²⁺ have been observed in a powdered sample of calcium zirconate. Since the resonance lines are narrow (similar 0.6-0.7 G) accurate measurement of the spin Hamiltonian parameters is possible. The analysis of the powder spectrum is based upon a spin Hamiltonian which includes a trigonal crystal field interaction with the ⁶S_5/2 state of Mn²⁺, magnetic hyperfine with the Mn²⁺ nucleus, and electronic and nuclear Zeeman interactions. The values obtained are g = 2.0022, A = (80.6 ± 0.1) × 10^-4 cm^-1 and D < (29 ± 2) × 10^-4 cm^-1. A spectrum due to Mn⁴⁺ ions is observed when the samples have been annealed in air at elevated temperatures.

The generation of frequency-modulated light is described. It has been used in a double-resonance experiment to stimulate intensity-modulated fluorescence from mercury vapour. Experimental results are compared with a theoretical expression.

The magnetic properties of three single crystals of dysprosium have been investigated along the principal directions of the hexagonal close-packed structure, over the temperature range 4 to 300 °K and in applied fields up to 18 kOe. The measurements along the a and b directions agree in general with the previous work. Along the c direction a small spontaneous moment is observed, appearing at (125 ± 2) °K, (89 ± 3) °K and (88 ± 4) °K for the three crystals studied. The corresponding values of the spontaneous moment at 0 °K are (1.9 ± 0.4) e.m.u. g^-1, (2.7 ± 0 3) e.m.u. g^-1 and 2.6 ± 0.3 e.m.u. g^-1. An analysis of the results indicates that the transition from antiferromagnetism to ferromagnetism is controlled by the strong dependence of the indirect exchange interaction with lattice dimensions.

The magnetic susceptibilities of Mo-y at.% Co and W-x at.% Co alloys, where y = 1.65, 1.085 and 0.535, and x = 1.05, 0.87 and 0.54, have been measured in the range 27-300 °K. The results can be fitted to a modified Curie-Weiss law of the form χ - χ₀ = C/(T - Θ), where the background susceptibility χ₀ is determined from the data. This represents an assumption that the alloying process modifies the background susceptibility to a small degree. A moment of about 1.8 μ_B then appears associated with Co in W, and about 1.5 μ_B in Mo. The Curie-Weiss behaviour is believed to represent increasing spin compensation of the moment as the temperature is lowered. The results are discussed in the light of ⁵⁹Co nuclear magnetic resonance results in four of the six alloys and in terms of current theories of s-d exchange in magnetically dilute alloys.

The complete set of third-order elastic constants of potassium chloride, sodium chloride and lithium fluoride have been measured at room temperature under conditions which eliminate the possibility that plastic deformation occurred during the experiments. A partial check on these values, provided by the pressure derivatives of the elastic constants, shows satisfactory agreement with experiment for potassium chloride and sodium chloride but shows some discrepancies for lithium fluoride With minor exceptions, the values agree with those predicted by Ghate using a Born-Mayer repulsive potential out to second-nearest neighbours

The decay time of luminescence emission from Mn²⁺ ion centres in zinc sulphide has been calculated. Tetrahedral crystal-field symmetry at these centres introduces configurational mixing of p and d states. The resulting values of decay time of between 1 and 6 msec are in good agreement with the experimental value of 3 msec. On the basis of further approximate crystal-field calculations a possible role for the so-called coactivators in zinc sulphide containing manganese is postulated, and this accounts for the observed effects on absorption and emission characteristics.

The theory of electric dipole transitions in ions situated in centrosymmetric and non-centrosymmetric environments is examined. The physical assumptions of one version of this theory are shown to be mathematically inconsistent and to lead to results, for a tetrahedral field, that disagree with experiment. Removal of this inconsistency leads to an interrelation between existing schemes. Further consideration of numerical examples, and the source of the errors involved, enables us to give a possible explanation of the reason why models assuming different odd-parity intermediate states give similar values for the calculated intensity.

Changes in the intensity of the lines of the spontaneous emission spectrum of a d.c. argon II laser tuned to 4880 Å are produced by chopping the laser radiation field in the cavity. Polarization ratios have been measured for several transitions in this perturbation spectrum. For the two transitions (4880 Å, 4228 Å) from the upper laser level (4p ²D_5/2) the polarization ratio (0.21) is independent of the discharge parameters and the laser intensity. Other transitions with levels collisionally coupled to the upper laser level are found to be unpolarized. A theoretical treatment is given which enables certain parameters (saturation effects, collisional lifetime of the upper laser level) to be evaluated. Magnetic depolarization effects are considered, but a satisfactory theoretical treatment has not been achieved.

In an earlier paper some determinations of the virial coefficients of carbon dioxide and ethylene, planned as part of a continuing investigation of the (p,v,t) properties of gases and gaseous mixtures, were described. These measurements utilized the `series-expansion' technique which it was intended to use consistently throughout the series. In view, however, of the aims of the work and the consequent demands on accuracy, it was decided, as a precaution against the possible intrusion of undetected systematic error, to carry out a limited series of measurements on one gas - carbon dioxide - by an independent method. For this the `piezometer' technique developed by Michels and his collaborators was chosen.

This paper compares the values of the second virial coefficients obtained over the temperature range -10 to 200°C in the two series. These proved to be in very precise agreement, the discrepancies at corresponding temperatures being nowhere greater than 0.03 × 10^-3 amagat (0 7 cm³ mole^-1) and on the average of the order of only 0 01 × 10^-3 amagat (0.2 cm³ mole^-1). The significance of these results is discussed.

Expressions for the second harmonic and sum and difference frequency components of the current density in a plasma (due to two alternating electric fields, an electron density gradient and a temperature gradient) derived by earlier workers have been modified to incorporate the effect of the higher-order asymmetrical term f₂ in the expansion of the distribution function. The normalization constant appearing in the time-independent part of the isotropic component of the distribution function has been assumed to be temperature dependent, in contrast with that of earlier workers. The results of the above analysis have been used for calculating the magnitudes of the second harmonic and sum and difference frequency components in the reflected waves from an inhomogeneous-plasma-free-space interface.

The absorption spectrum of AlO has been observed in the wavelength region 2000-3000 Å using a shock tube. Discrete band absorption and continuous absorption were observed. It is suggested that the continous absorption is a dissociation continuum of AlO and its long-wavelength edge gives a value of 4 54 ± 0.01 eV for the dissociation energy.

The influence of screen geometry on the diffracted field of a slit aperture is investigated, using measurements on a slit aperture formed by two inclined half-planes. An approximate solution of the integral equation for the problem is computed, and compared with experiment and with Kirchhoff's theory

An examination is made of the decoupling procedures which have previously been used in a Green function treatment of the spin-phonon interaction in paramagnetic crystals An exact equation connecting the Green function of the phonons with those of the spin system is derived.

The wave function used by Rudge as a basis for modifying the Ochkur approximation is generalized The results obtained with the new function cast doubt on the validity of the arguments used by Rudge and Crothers in support of Rudge's modification.

Using methods described by Saraph, Seaton and Shemming in 1966, collision strengths have been calculated for the isoelectronic sequences 2p³, 2p⁴, 3p² and 3p⁴.