Table of contents

In a dilute ferromagnet individual atoms can have a variety of different magnetic environments. The analogue of the Mayer theory for multicomponent systems is used to develop a cluster expansion for the Ising model from which both high and low temperature series expansions can be derived.

A new method for computing the exciton states is presented. Fredholm solutions to the integral equation for the expansion coefficients of the exciton wavefunction are obtained making use of Wannier functions. This method seems appropriate to give numerical solutions in the cases where neither the Frenkel-Peierls nor the Wannier-Mott schemes apply, such as in the low lying exciton levels of large gap insulators. The singlet-triplet splitting and the longitudinal transverse splitting are considered. This approach can be naturally extended to the case of resonant excitons and of core excitons.

A selfconsistent field theory is applied to Edwards' path integral model of an electron in the presence of a system of dense, weak, random scatterers. It is shown that for the electron's energy below a critical value, E_c, the electronic states are localized, (i.e. in the limit of infinite time the electron has nonzero probability of returning to its initial position), while for energies above E_c the states are extended.

From an empirical pseudopotential approach the charge distributions in real space have been calculated along various directions in ternary and quaternary alloy systems.

A comparison is made between the empirical pseudopotential and dielectric methods of calculating energy gaps in zincblende alloys. The details of the DM method are discussed with reference to charge density calculations in these systems.

The nature and possible origins of the structural distortions in the neighbourhood of the Cr³⁺ ion in MgCr₂O₄ are discussed in the light of experimental evidence from optical, X-ray diffraction, infrared absorption and other investigations. The results of the analysis may have implications for other compounds with cubic spinel structure.

Room temperature luminescence has been examined in nitrogen doped gallium phosphide grown by vapour epitaxy using the hydride system. Cathodoluminescence studies showed that as the nitrogen concentration was increased from 10¹⁸ to 1.5*5*10¹⁹ nitrogen atoms cm^-3 the green A line (2.225 eV) was joined by the A-0 line (2.175 eV). At higher concentrations (5*10¹⁹ atoms cm^-3) the NN₁ line (2.10 eV) became dominant with the NN₁-0 line (2.05 eV) becoming important at 1*10²⁰ nitrogen atoms cm^-3. The electroluminescence from p-n junction LED's made by zinc diffusion showed similar variations in spectral output. Diode efficiencies up to 0.1 per cent have been observed for diodes with emission peaking at 2.10 eV. Luminescence decay times of 40+or-4 ns have been observed in these diodes although these times were probably controlled by nonradiative transitions.

Electron spin resonance studies at room temperature (302 K) have been carried out on Cu²⁺ doped Na₂Zn(SO₄)₂.4H₂O. The electric field symmetry around Cu²⁺ is found to be rhombic. From the esr data it is found that the z axis of the rhombic field falls along Zn-O(H₂O₍₁₎) direction. The principal g values and the hyperfine constants are given.

For Pt.II see ibid., vol.4, 1921(1971). On the basis of an explicit determination of the singlet and pair atomic distribution functions at the surface of a simple liquid, expressions are obtained for the surface tension and surface energy in terms of the local atomic anisotropy. Numerical evaluation yields gamma (argon)=13.48 (dynes cm^-1) and U_s(argon)=35.35 (erg cm^-2) at the triple point. These values are in excellent agreement with the experimental values of 13.45 (dynes cm^-1) and 35.01 (erg cm^-2) for gamma and U_s respectively. Finally, the law of corresponding states is applied to obtain estimates for certain other systems.

For Pt. III see ibid., vol.4, 2433 (1971). A two dimensional molecular dynamics simulation of the liquid-vapour interface of argon at the triple point yields a strongly oscillatory density profile. It is found that the dynamical properties in the transition zone show quasicrystalline characteristics normal to the liquid surface. The velocity autocorrelation indicates phonon lifetimes of up to 5*10^-12 s, and the spectral density shows discrete peaks in the vibratory region indicative of a simple lattice structure. On the other hand, the existence of low frequency diffusive modes is evidenced by the nonzero value of the function in the limit omega to 0. Two diffusion constants for motions normal and parallel to the surface are obtained and these are discussed in terms of a jump-diffusion model.

The mobilities of positive and negative ions have been measured in liquid helium in the temperature range 0.9-1.0 K as a function of electric field. Within the limits of accuracy of 3%, the mobilities have been fitted to a hydrodynamic theory for the interaction of the ions with rotons in the normal fluid component of He II. A drop in mobility with velocity is predicted. Both the results and those of Schwartz (1970) can be fitted to the theory. The theory contains one adjustable constant. This constant links the variation of ion drift velocity to the velocity field of the superfluid which is responsible for producing an excess roton distribution around the ion, and it relates a macroscopically observed velocity to the microscopic processes of momentum loss by the ion. The results show also no clear evidence for periodic discontinuities in mobility and the theory does not require their existence.

High temperature series expansions are obtained for a lattice model of critical behaviour in multicomponent systems, for example a mixture of ³He-⁴He, or a disordered ferromagnet. The first nine terms of the series for the free energy are obtained for the face centred cubic lattice. The series is analysed to determine the critical behaviour and to obtain a phase diagram.

The authors discuss the metal-nonmetal transition in ferromagnetic as well as in antiferromagnetic systems having one electron per atom and described by the Hamiltonian which consists of one particle energies of electrons, intra-atomic Coulomb, and interatomic Coulomb and exchange interactions between electrons. It is found that the anomalous correlation functions corresponding to spin flip processes in the Hartree-Fock approximation give rise to the metal- nonmetal transition. The nature of phase transition in ferromagnetic and antiferromagnetic systems is compared and discussed.

The conventional group theoretical results of Landau's theory of continuous (i.e. second order) phase transitions are extended to cover the case of a transition between two magnetically ordered phases; this involves the use of the irreducible corepresentations of the magnetic space groups of the two phases between which the transition occurs. An example is given to illustrate the application of this theory. It is shown that while, for some systems, the correct conclusions can be obtained by neglecting the anti-unitary symmetry operations that contain theta , the operation of time reversal, this is not always the case.

The construction of accurate ion core scattering factors is discussed, and a Hartree-Fock treatment of exchange found to be important. A formalism is constructed for making accurate, fast, nonperturbative calculations of LEED spectra within the framework of the ion core scattering model.

For Pt. I see ibid., vol.4, 2501 (1971). Calculations of LEED spectra for a copper 100 surface in the ion core scattering model are presented and discussed in terms of band structure. The sensitivity of the excellent agreement with experiment to accuracy of the potential is thoroughly investigated with the conclusion that, provided an accurate Hartree-Fock potential is used, theoretical spectra will be in good agreement with experiment.

The Jahn-Teller Hamiltonian for an electronic p state equally coupled to E_g and T_2g vibrations in cubic symmetry, is diagonalized numerically. The invariance of the phonon Hamiltonian under the five dimensional rotation group, R₅, is used to get the matrix elements, and for this some fractional parentage coefficients for the reduction of R₅ to R₃ are calculated. The reduction factors K(E) and K(T₁) are calculated, lineshapes and zero phonon line intensities are given for s to p and p to s transitions. The wavefunction in strong coupling is worked out in a new coordinate system, and its properties match those derived from the numerical diagonalization. The s to p lineshapes are markedly different from those calculated by semiclassical methods, even in strong coupling.

The shape and first four moments of the F⁺ absorption band in CaO have been studied as a function of temperature between 4.2 K and 300 K. Analysis of these results shows that the properties of the band can be accounted for very well in terms of Jahn-Teller coupling of the excited ²T_1u state of the F⁺ centre to both E_g and T_2g vibrational modes, with weaker coupling to A_1g modes. The shape of the band agrees remarkably well with the calculations of O'Brien (1969). Values for the Huang-Rhys factors of S=3.3 for the E_g and T_2g modes and S_A=1.6 for the A_1g modes are derived from a model where all modes have the same effective frequency of h(cross) omega =264 cm^-1. An attempt to allow different frequencies for the various modes is only partially successful, due to the lack of detailed information on the frequency spectra of the coupled modes. Nevertheless such an analysis does suggest that the vibrational sidebands observed at frequencies of about 205 cm^-1 and 302 cm^-1 may be associated with A_1g and (E_g, T_2g) modes respectively. Measurements of the intensity, position and halfwidth of the zero phonon line as a function of temperature cannot be interpreted adequately in terms of conventional theories, unless the frequency of the interacting modes is made much smaller than the value used for the broad band.

The authors present an examination of the dominant sites of Yb³⁺ and Er³⁺ codoped with Li, Ag and Cu in ZnSe. Two sites have been analysed in detail and both have a tetrahedral arrangement of the monovalent group I ions around the rare earth ion. In one case the rare earth ion is an interstitial site while in the other case it substitutes onto a selenium site.

The nu ₁ and nu ₂ internal modes of the NO₂^- radical in NaNO₂ have been studied in detail at 4 K, 77 K and 300 K. Shifts in both these modes due to isotope substitution have been observed which have enabled a complete set of force constants to be determined at all three temperatures. The isotope shifts have been calculated using a general harmonic force field, and are shown to be in excellent agreement with experimental data. Using thin samples at low temperatures it has been possible to measure shifts as low as 5 cm^-1 from the fundamental with an accuracy of+or-0.3 cm^-1.

For Pt. I see ibid., vol.4, 1983 (1971). Magnetostriction measurements along the (110) and (112)axes of a NiO single crystal, which was compressed along the (111) direction in order to produce a twin free specimen, are reported. At fields up to 10-15 kOe the strain varies with H² and this is interpreted as due to the movement of spin domains. At higher fields a tendency towards saturation is found. The ordinary anisotropic magnetostriction was measured along the (110) and (112) axes by rotation of an applied field of 25 kOe in the (111) plane to the perpendicular directions. The temperature dependence of the resulting strain -¹/₃(A₁+A₂+²/₃A₄+¹/₆A₅), as described in terms of the equation of Birss (1959), is given and it takes values of 33*10^-6 and 23*10^-6 at room temperature as measured along respectively the (110) and (112) axes. An attempt towards comparison with the single ion theory of Callen and Callen (1963) was made, but this was hindered by the large temperature dependence of the elastic constants of NiO. The temperature dependence of the forced exchange magnetostriction in the (111) plane, which shows a discontinuity at the Neel point and which persists with a large and remaining constant value in the paramagnetic region, is also given.

A magnetostriction model for a second order phase transition in an insulating antiferromagnet is proposed and solved, treating the spin system in the two chain approximation with the distortion in the mean field approximation. The solution obtained agrees well with the experimental PrCl₃ data but badly with the corresponding NdCl₃ data.

The thermodynamic properties of a coupled spin-phonon system with S=1 are examined using a diagrammatic expansion of the free energy. The evaluation of the thermal averages of products of spin operators is facilitated by use of the drone- fermion representation. The specific heat contribution from the spin-phonon interaction is examined in detail for the case of ferrous ions in MgO and it is found to be negligible compared with the usual Schottky anomaly. This is in agreement with the experimental observation of Smith and Seidel (1970). The importance of using the full spin-phonon Hamiltonian as dictated by crystal symmetry is stressed because there is a large cancellation from the terms in the interaction that give 'resonances' at zero, omega ₀ and 2 omega ₀. If it was not for this cancellation the specific heat contribution from the spin-phonon interaction would be a factor omega _m/ omega ₀ times larger and hence observable experimentally.

An attempt has been made to analyse the effect of the plasmons on Compton scattering. Equations for the conservation of momentum and energy have been formulated and solved taking into consideration plasmon excitation in the scattering process. A comparison of the theoretical results with the experimental observations for beryllium is discussed. It has been pointed out that this formalism can be used to study the electron plasma in solids.

For Pt. I see ibid., vol.4., 916 (1971). The theory of tunnelling introduced in Pt. I is reformulated in the case of free electrons subjected to an arbitrary one-electron potential. An exact expression for the current is obtained, using Keldysh's generalization of perturbation theory. This result is particularly suited to an exact field theoretical treatment of many-body effects in the tunnelling junction.

For Pt. II see ibid., vol.4, 2598 (1971). The effect of an impurity level located in the barrier of a metal-insulator-metal tunnelling junction is investigated with the help of a new formalism. In agreement with Hurault's results (1971) it is found that a sharp resonance peak appears in the d²I/dV² characteristic, when the impurity state energy crosses either electrode Fermi level. The width of this resonance peak is proportional to the width of the virtual bound state arising from the interaction of the two electrode continuum with the impurity state. It is stressed that the resonance effect cannot be interpreted as the opening of a new tunnelling channel. The dependence of the effect on the location of the impurity state is briefly studied.

A method is presented for the calculation of the one-electron Green function of the periodic lattice. The method depends on knowledge of the pseudopotential matrix element of the periodic lattice potential in the momentum representation and essentially inverts the matrix (E-H)^-1. The matrix elements of G in the momentum representation are evaluated and graphs are presented for the case of germanium and silicon. They are found to have extrema at the symmetry points of the Brillouin zone. The matrix elements of G(r,r') are also evaluated numerically for the case of r=r=0 and enable the binding energy of a short range potential to be calculated. This can be regarded as an extension of Koster-Slater theory from a one band to a many band model.

The lower of the two threshold electron energies for binary compounds of the II-VI and III-V groups have been observed to represent the displacement of the lighter atoms of the compounds. This observation was used to predict atomic displacements in zinc sulphide.

Resonant effects in second harmonic generation (shg) in InSb near the energy gap have been studied by radiating with a high power Q-switched CO₂ laser at 10.6 mu m and tuning the energy gap by applying magnetic fields up to 55 kG in samples of carrier concentrations from 8*10¹⁴ to 5*10¹⁶ cm^-3 at a temperature of 12 K. Absolute, independent measurements of the coherence lengths l_c+- were made as a function of magnetic field, using circularly polarized radiation and wedge shaped samples, by a modification of the original 'Maker' experiment. These results, together with measurements of SHG _+- and absorption coefficients alpha _+- have yielded the magnetic field dependence of the second order nonlinear susceptibilities chi _+- at 5.3 mu m. Virtual transition processes have been used to formulate the theory for the resonant behaviour of chi and l_c and the effect of blocking by conduction band population is demonstrated experimentally.

The dielectric function epsilon (q) is calculated by the empirical pseudopotential method for fourteen semiconductors with the diamond and zincblende structures. The predictions of the static limit epsilon (0) are in excellent agreement with experiment. The form of the screening in real space is also given.

The magnetization and free energy of a ferromagnet with dipole-dipole coupling between spins are evaluated approximately using the drone-fermion perturbation method. The contributions are classified by means of a high density parameter 1/z, where z is the number of spins interacting with any given spin. The lowest order (1/z)⁰ results lead to molecular field theory, and fluctuation effects are introduced in higher orders. The calculations have been carried out up to and including order (1/z)². They are valid for all temperatures below T_c, and the results may also extend into the paramagnetic region if there is a suitable applied magnetic field. In a low temperature limit the (1/z)¹ results reduce to linear spin wave theory, and the (1/z)² contributions give a leading order description of spin wave interactions.

For Pt. I see ibid., vol.4, 2658 (1971). The method of Pt. I is used to derive approximate results for the spin-spin correlation functions of a ferromagnet with dipole-dipole coupling between spins. This formalism employs the drone-fermion representation for spin ¹/₂ and a high density 1/z classification of terms. Results are obtained for the spin wave energy and damping as functions of momentum and temperature. As in the case of the Heisenberg model, it is found that the damping is small for any temperature below T_c for sufficiently long wavelengths.

The validity of the smoothness postulate tested for the Ising antiferromagnet in two and three dimensions. The field dependence of the staggered susceptibility above the critical temperature is studied by means of series extrapolation, and the curvature of the phase boundary in zero field is deduced. The results are compared with those found from initial susceptibility and specific heat data.

The spin wave velocity in a one band itinerant electron model of an antiferromagnet is calculated using the equations of motion for the generalized transverse susceptibility. The formula is written down approximately, in terms of the ground state antiferromagnetic energy bands and is then evaluated in limits corresponding to wide and narrow antiferromagnetic energy gaps. A special case of the former yields an expression corresponding to the energy of a spin wave in a Heisenberg antiferromagnet. In the latter case it is found that the spin wave velocity tends to zero as the energy gap tends to zero, which is a new result. The problem of whether the spin wave velocity is real when the Fermi level is not in the gap is discussed.

The quadrupole interaction at the ⁵⁷Fe nucleus in FeF₂ has been measured at sixteen temperatures in the range 78-1023 K by means of Mossbauer spectroscopy. Least squares fits have been obtained between these results and values calculated on the basis of a model in which the following were to be determined: the ⁵⁷Fe quadrupole splitting, Delta E₀, which would be generated by a single 3d electron in FeF₂ in the absence of spin-orbit coupling; the temperature independent contributions to the quadrupole interaction; the crystal field splittings of the lowest orbital triplet; parameters designed to generate the effects of anisotropic thermal expansion of the lattice. Several models of thermal lattice expansion were considered. The most successful of these yielded excellent fits to the data. This work indicates that Delta E₀ is 3.8+or-0.3 mm s^-1 in FeF₂. This is close to the value 4.1+or-0.2 mm s^-1 found in FeSiF₆.6H₂O, indicating that, contrary to previous assertions, the 3d electrons of the metal are delocalized to similar extents in these two compounds. The value of Delta E₀ in FeF₂ agrees fairly well with the value 3.5+or-0.1 mm s^-1 predicted by other workers; better agreement would probably be obtained if the predicted value were to be corrected for the effects of sigma bonding.

Experimental data on 3d⁹ ions (Cu³⁺ and Ni⁺) at tetrahedral sites is reviewed and compared to the predictions of crystal field theory including the effects of covalency, 4p configurational mixing and the Jahn-Teller effect. Of the copper centres considered, only for the case of ZnO is this theory found acceptable and to make the optical and epr results consistent it is necessary to include second order Jahn-Teller effects. The first order reduction factor for angular momentum is found to be about 0.1. The hyperfine effect is explained using a reduction in the magnitude of the contact term caused by 4p mixing, as has previously been found in tetragonally distorted complexes.