Properties of electrons in vibrating crystals

Abstract

A large number of the phenomena associated with solids can only be explained by the presence of the periodic potential arising from the lattice of ions. The starting point for most discussions -the so called Static Lattice Models -assumes that the ions constitute a fixed, rigid, immobile, periodic array within which the electrons move. However, although these models are impressively successful in accounting for a wealth of detailed metallic equilibrium and transport properties dominated by the behaviour of the conduction electrons and some of the equilibrium properties of ionic and molecular insulators, it is only an approximation to the actual ionic configuration since the ions are neither infinitely massive nor are they held in place by infinitely strong forces. While the theory of lattice vibrations explains more of the properties of solids e.g.melting, it is necessary to consider the interaction of the lattice and electron subsystems to account for all the properties of a solid and to get detailed agreement between theory and experiment. The interaction between the two subsystems is treated in the usual way: the vibration field is considered to be an ensemble of quanta -called phonons -and the interaction between the two subsystems is then considered as being between the electrons and phonons. However, under the conditions of high temperature -the so called Classical region -the problem of an electron in a vibrating crystal can be treated as a problem in time-dependent wave mechanics, since under such conditions the lattice vibrations behave in a classical way. The original section of this thesis considers the properties of an electron in a vibrating crystal from this point of view. For a special class of time-dependent potentials -those periodic in time -the states of an electron experiencing them can be explicitly constructed, and since such potentials are a good approximation to those that represent a vibrating crystal an attempt is made to calculate some measurable properties of the electron. The remaining sections of this thesis give a review of the standard theory of the electron in a vibrating crystal and of the theory required to understand it.