Please read the abstract in the section 00front of this document / Thesis (PhD (Mathematics))--University of Pretoria, 2005. / Mathematics and Applied Mathematics / unrestricted

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Magnetic space groups (MSGs) were first introduced (under a different name) by Heesch more than 30 years ago, and a list of all of them was published by Belov, Neronova and Srairnova in 1955. However, no mathematically rigorous derivation of MSGs can be found in the existing literature, although an outline of a method for obtaining a large class of MSGs was published by Zaraorzaev in 1957. In this thesis a systematic rigorous method for constructing MSGs is described in detail, and a proof that the method in fact gives all the MSGs is presented. The method also leads in a natural way to a classification of MSGs which is useful for a systematic study of the arrangements of spins in ferromagnetic, ferrimagnetic and antiferromagnetic crystals. The first and the last chapter of the thesis deal with the physical aspects of the problem, the remaining chapters with purely mathematical aspects of it. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

The theory of the phonon broadening of impurity spectral lines in homopolar semi-conductors is discussed within the framework of a Kubo-type formulation of the adiabatic dielectric susceptibility and the subsequent calculation of this using the double-time Green's function method. The basic assumption is the smallness of the interaction of the electrons (or holes) bound to impurity sites with the lattice vibrations. This interaction is then treated as a small perturbation of the independent systems of electron and vibrating lattice; the use of the adiabatic approximation is thereby avoided. The so-called decoupling of the infinite hierarchy of equations for the relevant Green's functions is discussed in detail and is given its justification in the present problem. In the case of nondegenerate electronic levels, the line-shape function is obtained explicitly in terms of the matrix elements of the electron-phonon interaction. It is found that the absorption line consists of a sharp peak with a width arising from a finite life-time of the unperturbed states due to the electron-phonon interaction and of a continuous background arising from the multi-phonon processes which accompany the optical absorption. In the degenerate case, a general method of obtaining the line-shape function is discussed and is illustrated in an example. The results are compared with those obtained by previous workers in the field. The general theory is applied to shallow impurity levels in silicon with the use of a modified hydrogenic model and a deformation potential description of the electron-phonon interaction; numerical estimates are made for typical contributions to the widths of the lines in both acceptor and donor cases. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

A simple method is presented for writing the matrix elements of transposition operators for discrete sets of quantum numbers. A proper product of these leads to easy computation of general permutation operators. It is shown how these operators may be constructed with operators defined in angular momentum space. Results agree with Dirac for transposition of two particles of spin ½ and with Kaempffer for spin 1. The calculations are performed to extend the results to spin 3/2 and 2 along with alternate representations. Special considerations are required for fermion creation and annihilation operators. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

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This thesis consists of two independent parts, both of which are applications of the quantum mechanical methods developed recently by R. P. Feynman. Part I is concerned with the non-relativistic theory, and applies Feynman’s formalism to the simple problem of the scattering of a particle by a potential field. The method and results are compared with those of the familiar Born-approximation. The two procedures are shown to be equivalent and to be valid under the same conditions. Feynman’s formulae are used to calculate the first and second order terms of the scattered particle wave function, with an arbitrary scattering potential. Part II uses the relativistic Feynman theory, and treats the scattering of positrons by electrons, and of two electrons. The calculation checks the work of H.J.Bhabha and C. Møller, who have obtained the same results by other methods. The differential cross-sections for the two scattering processes are calculated to first order, and an estimate is made of the feasibility of an experiment to determine whether the exchange effect described by Bhabha actually occurs in positron-electron scattering. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

In this thesis the problem of an electron in a crystal in the presence of a uniform magnetic field is investigated using group theory method. A group of operators [ symbol omitted ] commuting with the Hamiltonian of an electron in the presence of a uniform magnetic field and a crystal electric potential is constructed. This group is homomorphic to the group [ symbol omitted ] (a magnetic space group) of space time transformations that leave the magnetic field and the crystal electric potential invariant. The property of the subgroup [ symbol omitted ] of [ symbol omitted ] that under the above homomorphism is mapped onto the lattice [ symbol omitted ] of [ symbol omitted ] is studied in detail. It turns out the structure of [ symbol omitted ] depends on the magnitude and the orientation of the magnetic field, so that, in fact one has to deal with an infinite class of groups. In particular, it is useful to divide this infinite class of groups into two subclasses: one subclass is then referred to as corresponding to "rational" magnetic fields, the other as corresponding to "irrational" magnetic field. The group [ symbol omitted ] is a generalisation of the "magnetic translation group" recently introduced by Zak for the special case of a symmetric gauge. He also constructed "physical" irreducible representations of the "magnetic translation group" for the special case of a "rational" magnetic field. In this case a group [ symbol omitted ] always has a maximal Abelian subgroup with a finite index. (The term "physical" representation simply means a representation which can be generated by functions of spatial coordinates.) In this thesis no such restriction is introduced: the "physical" irreducible representations of [ symbol omitted ] are also constructed for the case of irrational magnetic field, in which case the index of a maximal Abelian subgroup is always infinite; the "physical" irreducible representations are then always infinite dimensional. Using a complete set of Landau functions the basis functions generating "physical" irreducible representations of [ symbol omitted ] are found for the special case when the crystal is simple cubic and the magnetic field is parallel to a lattice vector. It turns out when the field is "irrational" the basis functions are countably infinite sets of Landau functions, and the energy spectrum depends only on one of the parameters labelling the "physical" irreducible representations of [ symbol omitted ]. The problem of perturbation produced by a weak periodic potential on the Landau levels for a free electron in a magnetic field is also considered. In this connection we make plausible the validity of certain quite general selection rules for an arbitrary periodic potential. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

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