Table of contents

A detailed investigation has been made of the application of the Hartree-Fock approximation to the stability and thermodynamic behaviour of spin-density waves in a model with a single electronic energy band and a delta-function-type interaction between electrons. It is shown that for a half-filled band the ground state at zero temperature is always a spin-density wave, provided only that the width of the band is sufficiently small. For other bands the conditions for the relative stability of the spin-density wave, ferromagnetic and paramagnetic states are more complicated, and are studied in detail for a special band form. For a half-filled band the energy per atom and specific heat have been calculated as functions of temperature for the spin-density wave, ferromagnetic and paramagnetic states. The temperature dependence of the amplitude for the spin-density wave and ferromagnetic states, and of the entropy for the spin-density wave and paramagnetic states have also been calculated.

The formulation of the problem has been made using a Green function method in order to allow the development in a later paper of a higher-order approximation to the correlation problem.

The Green function theory formulated in I is generalized to give a higher-order approximation to the correlation problem. The spin-density-wave stability in this approximation is compared in detail with the Hartree-Fock approximation results of I and are found to be in conflict in several respects.

Measurements of the specific heats of NaF and CsBr between 20 and 300 °K are reported. The results are believed to be accurate to within 0.5% down to 60 °K, falling to 2% at 25 °K.

The Grüneisen parameters γ = βV/C_Vχ_T = βV/C_Pχ_S of NaF and CsBr have been calculated from the present heat capacity measurements in conjunction with earlier thermal expansion and elasticity data, and in the case of NaF from the pressure derivatives of the elastic constants. In common with NaCl and NaI the Grüneisen parameter of NaF falls with diminishing temperature, but this parameter is independent of temperature for CsBr, in common with CsCl and CsI. The quasi-harmonic approximation has been applied to calculate the maximum frequencies v_D(n) of the Debye distributions having the same nth moments as the actual crystals for -3 <or= n <or= 6, interpolation being employed in evaluating those parts of the integrals occurring between 0 and 25 °K for -2 <or= n <or= 1. The variations display characteristic minima, and it has been observed that the dependence of v_D(n) upon n diminishes for the positive moments as the ratio of the masses of the constituent ions approaches unity for both face-centred and body-centred alkali halides.

It has been observed that Lytle's recent theory, which explains very well the extended fine structure occurring on the high-energy side of X-ray absorption edges of metals, is not satisfactory for compounds. The theory has been modified by considering a potential well of finite height around the absorbing atom. It has been shown that the modified model explains satisfactorily the extended fine structure in the vicinity of the Mn K absorption edge in manganese oxides.

Formulae giving the optical absorption due to second-order terms in the crystal electric moment expansion are obtained. These formulae are expressed as convolution integrals of matrix outer products of harmonic lattice Green functions.

The simplifications following from selection-rule requirements follow naturally from the results; the formulae are especially suitable for defect absorption calculations. The integrated absorption in both the high- and low-temperature limits is obtained and expressed in terms of the electric moment coefficients and the force constant matrix.

A Green function method is used to calculate the one-phonon infra-red absorption by uncharged point defects in a face-centred cubic Bravais host lattice. A nearest-neighbour central-force model of the host lattice is used. Changes in defect-host force constants are allowed for, but no correction is made for the relaxation of the host lattice around the defect. Choosing the distortion dipole moment parameters so as to fit the experimental results of Jones and Woodfine for xenon defects in argon indicates that a substantial distortion dipole moment results from non-radial displacements of the neighbouring atoms relative to the defect. The poor agreement with experiment obtained in previous shell-model calculations seems to be due to a neglect of this non-radial effect.

Two independent zero-phonon lines are seen at 6420 Å and 6490 Å in the absorption spectra of single crystals of magnesium oxide which have been neutron irradiated and thermally annealed. The 6490 Å line is split by 9 cm^-1 owing to spin-orbit interaction. The polarization spectra of the two colour centres were examined under uniaxial stresses of up to 50 kg mm^-2 and are in agreement with the theoretical splitting patterns for electric dipole transitions in trigonal centres. Models for the centres include the analogue of the R centre in the alkali halides and an R centre with a trigonally disposed cation vacancy.

Trivalent iron is formed in weakly doped crystals of AgCl:Fe²⁺ when these are irradiated at a temperature which allows silver ions to leave sites near to Fe²⁺ and diffuse away through the lattice. Fe²⁺ normally acts as a hole trap but below 168 °K it behaves as a recombination centre as no better electron-trapping centre exists. The stable configuration for Fe³⁺ is the tetrahedral (FeCl₄)^- complex identified by Hayes, Pilbrow and Slifkin in 1964, but a new spectrum which corresponds to a weak trigonal distortion of this complex by a neighbouring Ag⁺ ion is found during the formation of the more stable complex. Formation of the tetrahedral (FeBr₄)^- complex by optical irradiation in AgBr:Fe²⁺ is also reported.

Electron spin resonance measurements on sixteen undoped single crystals of CdS are reported. Three resonance lines have been observed which are not associated with impurities. Of the lines, which are designated as A, B and D, A and D are discussed in some detail. It is concluded that line D which has a g value near 1·8 is associated with the donor centre formed by a sulphur vacancy. It is suggested that line A is associated with a cluster of four nearest-neighbour cadmium vacancies which form an acceptor centre with a level approximately 0·7 ev above the valence band. The g value of line A varies slightly from sample to sample and is about 2·08.

In additively coloured cadmium-doped crystals an ultra-violet band at 310 nm is observed in addition to the F band at 565 nm. The absorption of the F band is found to decrease and the absorption at the 310 nm band is found to increase with the increase in the cadmium concentration. The F-aggregate centres and alkali metal colloids cannot be formed by optical and/or thermal bleaching of the F band in these crystals. On heating the crystals to 350-600 °C, the 310 nm band and the F band disappear and a new band at 275 nm with a half-width of 0·70 ev appears. The centres responsible for the 275 nm band give an electron spin resonance absorption line of Lorentzian shape and a g value of 1·999 and seem to be metallic colloidal particles of cadmium. The conductivity ratio σ_c/σ_n (σ_c is the conductivity of the coloured crystal and σ_n that of an identical uncoloured crystal) remains less than unity from room temperature to about 500 °C. In x-irradiated crystals the V₃ band observed at 218 nm in pure crystals shifts towards the longer-wavelength side and the band half-width increases with the increase of cadmium concentration. The growth of the F-band absorption is found to increase with the increase in the cadmium concentration up to about 1%. For larger cadmium concentrations the growth rate is found to decrease.

The spin-Hamiltonian coefficients for Fe³⁺ in (CH₃NH₃)Al(SO₄)₂.12H₂O have been determined at 77 °K by the classical paramagnetic resonance method. The value of g is 2·003±0·004 and is isotropic. The other coefficients in units of 10^-4 cm^-1 are D = 1892·4±4, a = 124·5±2, F = -8·6±2, E = 83·3±1. These values fit the resonance lines to within 0.1%. The zero-field splitting frequencies measured directly at 77 °K are v_{½-fraction three-twos} = (12 229±10) MHz and v_{fraction three-twos-fraction five-twos} = (22 080±10) MHz. These frequencies are consistent with the spin Hamiltonian using the measured coefficients. It is concluded that the spin Hamiltonian is valid for this salt in the magnetic field range 0-10 000 G.

The diffraction of electrons from the surface of a crystal is discussed using a matrix formulation of multiple scattering theory. Expressions are derived for the intensity of diffracted beams from a surface with an impurity superstructure and with a small group of impurity atoms on the surface. The behaviour of the expressions is qualitatively discussed.

Double-time temperature-dependent Green functions have been used to investigate the excitation spectra and magnetization of two kinds of magnetic system. The first example involves an induced-moment system in which the ions possess a singlet ground state and a singlet excited state, while in the second two limiting cases of a mixed system in which the ions possess a singlet ground state and three excited states are considered. The first example is also similar to a simple model of a hydrogen-bonded ferroelectric with the magnetization becoming the polarization. In both cases an approximation has been used to decouple the hierarchy of equations. Comparisons are made with previous work on this topic and it is found that the systems in which the magnetization is due only to an induced moment display first-order magnetic transitions at the critical point.

¹¹⁹Sn Mössbauer spectra have been measured for a dilute SnEr alloy (<0.03 at.% Sn). It is shown that the results obtained with the alloy in the conical ferromagnetic (low-temperature) and sinusoidal antiferromagnetic (high-temperature) phases are consistent with the known magnetic structures of these phases if it is assumed that the hyperfine field at each ¹¹⁹Sn nucleus is proportional to the host magnetization at its site. At 4.2 °K the magnitude of the single-valued hyperfine field acting on all the ¹¹⁹Sn nuclei was found to be (124±4) kG.

We consider the correlated motion of electrons in a conduction band hybridizing with completely localized non-degenerate atomic levels. Only intra-atomic Coulomb repulsion is included. In the limit of large Coulomb repulsion the quasi-particle bands are those of the non-interacting electrons but with hybridization matrix elements reduced by excluding atomic sites occupied by other electrons of opposite spin. Self-consistent solutions giving a ferromagnetic ground state exist if the hybridization is sufficiently large, the number of conduction band states lying below the atomic energy level T₀ is small and the Fermi level lies just below T₀. If the Fermi level lies above T₀ the ground state is non-magnetic.

In the ferromagnetic state the spin polarization of the conduction band is always parallel to the localized d moments. Possible implications for the ferromagnetism of the transition metals are discussed.

Magnetocrystalline anisotropy in gadolinium has been studied utilizing a spin-wave model in which a crystal field of hexagonal close-packed symmetry and magnetic dipole-dipole interactions are treated as a first-order perturbation to isotropic exchange. Corrections to the temperature dependence of K₁, due to normal and magnetostrictive thermal expansion, are shown to be small. The resulting agreement with the measurements of Graham and of Corner et al. for K₂(T), K₃(T), K₄(T) is good, but the temperature dependence of K₁ is anomalous. Crystal-field parameters, evaluated by comparison with experiment, are about 100 times larger than the corresponding parameters in gadolinium ethyl sulphate.

Several standard techniques for the calculation of electronic band structures in solids seem to require the crystal potential to be of `muffin-tin' form, but an elementary calculation shows that contributions from variations of the potential in the interstitial region may easily be included in the secular determinants of the augmented plane wave and Korringa-Kohn-Rostoker-Ziman methods. The corrections in the Korringa-Kohn-Rostoker formalism are more complex. A numerical example for germanium shows that these terms can be significant.

Band structures for the noble metals have been calculated at twelve points in 1/48th of the Brillouin zone by the Korringa-Kohn-Rostoker method. The bands were calculated at two values of the lattice constant for copper and silver and at one value for gold. The potentials used in this calculation were constructed by a method akin to that used by Abarenkov and Heine and by Animalu and Heine for the non-transition metals. The results are presented and compared with experiment.

Model Hamiltonians have been constructed to fit energy bands which have previously been calculated by Jacobs for the noble metals for two values of the lattice constant in copper, two values in silver and one value in gold. These model Hamiltonians are presented and the dependence of their parameters on the lattice constant is discussed and related to the work of Heine and of Jacobs.

The T-matrix formalism is employed to calculate the charge displaced by a localized perturbation acting on a many-channel system. The method is applied to the case of s-d mixing produced by an impurity in a noble metal.

The theory of the coupled Boltzmann equation was applied to describing the process of the energy exchange between the electron and phonon system of a metal. On the basis of experimental data the value of the relaxation time for this process has been found to be equal to 5 × 10^-7 s for copper at room temperature.

Measurements have been made on the statistical distribution of multiple emission of secondary electrons produced by bombarding Cu, Ag, Pt and Au targets with electrons having energies in the range 60-960 ev. A comparison between the experimental distributions and the corresponding Poissonian distributions, made by comparing their second-order moments, suggests that these distributions, although similar, are different. Some speculative comments are given on how the difference might arise. The possibility of describing it empirically with the use of combinatorial moments is suggested.

The results of a band-structure calculation for metallic lithium using the Korringa-Kohn-Rostoker method are presented briefly and a model for capital Sigma, Greek(small kappa, Greek) is obtained. This model is used to estimate the effect of the distortion of the Fermi surface on the values of the elastic constants c₁₁ - c₁₂ and c₄₄. The results obtained for c₁₁ - c₁₂ and c₄₄ were 0·224 and 1·11 dyn cm^-2 respectively.

The structure and pressure-induced phase changes of solid Ga are studied in terms of the pseudopotential and second-order perturbation theory. It is shown that the phase changes provide a very sensitive criterion for the pseudopotential form factor, the most important features of which are the position of the first zero and the height and position of the first maximum. The pressure-induced phase changes can be largely understood in these terms, and an explanation is suggested for the formation of a metastable phase when supercooled Ga freezes. The variation of the interionic potential with distance provides a qualitative interpretation of the results.

The transport of generated charge carriers in sulphur has been investigated by drift mobility techniques during the melting of single crystals (at similar 114 °c) and also in the liquid prepared from molten granules of ultra-pure (6N) or laboratory reagent sulphur. The discussion of the results leads to the conclusion that in the ultra-pure liquid the transport is predominantly electronic and that both electrons and holes propagate by an intermolecular hopping mechanism. For electrons the transport mechanism is the same as in the solid, but during melting the intermolecular overlap energy drops by about a factor of 5, leading to an electron mobility μ_e = 10^-4 cm² s^-1 v^-1 at 120 °c. The hole conduction, which takes place in a small-polaron band in the solid, changes to a hopping transport upon melting. The magnitude and temperature dependence of μ_h is then very similar to that of μ_e. These results are discussed in the light of Holstein's small-polaron theory. Experiments on less pure liquid specimens show additional features which appear to arise from the ionization of impurity molecules during the transit of the generated carriers. Above 160 °c a change in the temperature dependence of μ_e and μ_h occurs, connected with the break-up and polymerization of the S₈ ring molecules. The efficiency of photogeneration in the liquid is lower than in the solid, but its spectral dependence is substantially the same. The steady dark current in the ultra-pure liquid is ohmic, although on application of the field the initial current is injected under space-charge-limited conditions. The different steady dark current behaviour of the impure liquid has been attributed to the formation of heterocharge layers near the electrodes.

The electrical resistivity has been measured over the temperature range 1 °K to 120 °K in dilute alloys of nickel with Pd, Cu, Co, Fe, Mn, Cr, V and Ti. The impurity (residual) resistivity is very small for alloys of nickel with its near neighbours in the periodic table, Pd, Cu, Co, Fe and Mn, but is an order of magnitude greater for alloys containing Cr, V and Ti. The ice-point values of the deviations from Matthiessen's rule are unusually large in the Ni-Co, Ni-Fe, Ni-Mn, and Ni-Cr alloys, somewhat smaller in the Ni-Cu, Ni-V and Ni-Ti alloys, and negligible in Ni-Pd.

These results are interpreted on the assumption that the departures from Matthiessen's rule arise because of spin mixing between electrons in the spin downward arrow and spin upward arrow bands. On this basis it has proved possible to separate the two components of impurity resistivity for the two spin directions. The values obtained correlate well with the known band structure and impurity states of nickel and its alloys. In addition, the temperature dependence of the deviations in all the measured alloys can be explained on the basis of spin mixing, with the only adjustable parameter being capital Lambda, Greek(upward arrowdownward arrow), the mean free path between spin flips.

Photoemission measurements have been carried out on BaTiO₃ single crystals subjected to a heat treatment in an oxygen atmosphere. It is shown that the electron affinity of the surfaces is increased substantially by the presence of chemisorbed oxygen. The apparent conductivity of the crystals is enhanced by baking in an oxygen atmosphere prior to the deposition of electrodes. It is suggested that this enhancement is due to chemisorbed oxygen, and a model is put forward to explain the mechanism of carrier injection into BaTiO₃.

The Mössbauer effect in ⁵⁷Fe has been used to study the hyperfine fields at the iron sites in polycrystalline samples of the cubic Laves AFe₂ intermetallics where A is Ce, Sm, Gd, Tb, Dy, Ho, Er, Y or Zr. The ⁵⁷Fe spectra in the Tb, Er, Y and Zr intermetallics can be analysed in terms of two distinct six-line ⁵⁷Fe patterns, whereas those in Dy and Ho exhibit only one. This is shown to be consistent with [111] and [001] directions of magnetization respectively. The cases of Sm, Gd and Ce are found to be more complicated. An analysis of the results is attempted on the basis of a simple model in which the direction of magnetization is determined by the crystalline electric torques acting on the rare earth ions.

An estimate is made of the magnetic dipole contributions to the local internal fields. It is found that this leads to a magnetic interaction which is dependent on the iron site and on the direction of magnetization.

The hyperfine fields at the ⁵⁷Fe and ¹⁶¹Dy sites in DyFe₂ have also been followed as a function of temperature. This parallels other work done on TmFe₂ and allows a comparison of the rare-earth-iron exchange in these intermentallics.

A modified method for the preparation of narrow-line ⁵⁷Fe Mössbauer sources by electrodeposition has been developed and an investigation of the causes of line broadening has been carried out. The effects of the crystallographic structure and the state of perfection of the host have been studied. It has been observed that the linewidths are unaffected by the state of perfection of the host and that the narrowest lines, obtained when the host has face-centred cubic or body-centred cubic structure, have widths at least 30% greater than natural. Microprobe analysis showed that the impurity (source) atoms are homogeneously distributed in the host. The effect of increasing impurity concentration on the linewidth has been investigated. Annealing appears to lead to a rapid initial diffusion of electroplated material which is followed by a much slower penetration which is probably normal bulk diffusion.

Hot-electron galvanomagnetic effects in polar semiconductors (III-V and mixed) are studied considering the effects of different phonon scattering mechanisms. The effect of interelectronic collisions is taken into account by considering three different carrier concentrations - high, medium and low. In the high-concentration case a displaced Maxwellian distribution of carriers with a field-dependent effective temperature makes the ratio of Hall and conductivity mobilities equal to unity and the magnetoresistance coefficient zero, independent of electric and magnetic fields (for single-valley semiconductors only). In the medium- and low-carrier-concentration cases, on the other hand, the Hall mobility and magnetoresistance coefficient are highly dependent on the fields. Methods for calculation in two-valley semiconductors (GaAs, InP, etc.) are also discussed. Numerical results have been presented for acoustic and polar-optical phonon scattering and single-valley semiconductors with isotropic effective mass in the medium-concentration case. It is found that for the case of polar-optical scattering the magnetoresistance coefficient varies by several orders of magnitude for changes in the magnetic field. The dielectric breakdown field is also found to depend slightly on the magnetic field. The effect of these results on some recent experiments is discussed.

Pronounced minima separated by half the frequency of the longitudinal optical phonon have been observed in the frequency dependence of the low-temperature extrinsic photoconductivity of n-type CdTe. This observation is attributed to the occurrence of two separate and distinct series of minima, each series being spaced by the frequency of the longitudinal optical phonon, which was found to be 171·4 cm^-1. Both series amount to almost 100% modulation of the photoconductive response, and are therefore interpreted as arising from two alternative capture processes rather than from simultaneous excitation from two different impurity levels. The amplitude of the oscillatory response was found to be a strong function of the wavelength of the incident radiation, with the photoconductive response per unit power varying with frequency to the power of -4·9.

The optical absorption associated with charge carriers in NiO and CoO has been obtained from transmission measurements on pure and Li-doped crystals in the wavelength region 0·5 to 700 μm, at temperatures between 100 and 500°K. The absorption increases smoothly with frequency to a maximum near 1 ev and then shows a further increase extending to 2 ev and higher. In the region from 0·5 to about 4 μm the absorption increases slowly with temperature and correlates with the Li concentration. In the far infra-red the absorption σ_FI is independent of frequency and, for T <or= 300-400 °K, σ_FI > σ_dc. The activation energy associated with σ_FI is smaller than the d.c. value.

The behaviour between 1 and 700 μm is very similar to that predicted by the theories of Klinger, Reik et al. and Bogomolov et al. for non-diagonal (hopping) transitions by small polarons. Similar absorption in TiO₂ and other 3d oxides between 1 and 10 μm has been attributed to a hopping component in the free-carrier mobility. In NiO and CoO, however, we conclude that the absorption is primarily due to small-radius polarons hopping around Li centres with high-temperature activation energies E_a of 0·2-0·25 ev and 0·3-0·4 ev respectively.

Above 400°K in NiO, σ_FI -> σ_dc and electrical studies indicate that about 20-30% of the carriers are ionized. No new features appear in the absorption but the above E_a values give an upper limit for the hopping energy of any non-diagonal component in the d.c. mobility. We infer a relaxation time of less than 10^-13s for the diagonal component implying a bandwidth of at least 5 mev. Recent d.c. mobility data support a band picture, but a closer analysis indicates that an activated mobility with E_a less, similar 0·2 ev cannot be excluded.

Analysis of plasma edge reflectivity measurements can be extended to give more accurate determinations of carrier concentration or effective mass, and can also be used directly to obtain the scattering time and conductivity.

Using suitable geometries the technique can be used to study buried layers such as p-n junction interfaces.

When the cyclotron frequency relaxation time product is smaller than unity, the propagation coefficient of magnetoplasma waves increases with increasing magnetic field. At higher magnetic fields, where the cyclotron frequency relaxation time product is larger than unity, the magnetoplasma wave propagation coefficient decreases with increasing magnetic field. This inversion of the magnetic field dependence of the propagation coefficient of these waves allows them to be used to study the transport properties of a plasma.

The technique is applied to p-type germanium where light and heavy holes each produce structure in the propagation coefficient of the magnetoplasma wave. Analysis of the experimental results shows that, whilst the description of the heavy-hole system and the relative concentration of light and heavy holes is in close agreement with theory, the light holes appear to be only weakly sensitive to ionized impurity scattering.

Tables are presented which complete the theoretical analysis of the intensity patterns for transitions occurring in ions, or point defects, at sites of cubic symmetry.

Two formulae for calculating the Debye characteristic temperature at T = 0 for cubic elements are compared. One formula is due to de Launay and is derived from lattice dynamics. The other formula is the simple Debye formula expressed in terms of elastic constants from relations derived by Voigt and Reuss. Provided the anisotropy of the solid elements is not excessively large the two formulae give almost the same results. Even for the largest anisotropies which appear in nature the difference between the two formulae is less than 3%.