The Band Theory of Solids

Abstract

In a solid one deals with a large number of interacting particles, and consequently the problem of calculating the electronic wave functions and energy levels is extremely complicated. It is thus necessary to introduce a number of simplifying assumptions. In the first place we shall assume that the nuclei in the crystalline solid are at rest. In an actual crystal this is of course never the case, but the influence of nuclear motion on the behavior of electrons may be treated as a perturbation for the case in which they are assumed, to be at rest. As we shall see in the next chapter, the lattice vibrations play an important role in the interpretation of electrical resistivity and other transport phenomena. Even with the above assumption, however, we are still left with a many-electron problem which can be solved only by approximative methods. In the case of solids, the most important approximative method which has been applied extensively is the so-called one-electron approximation. In this approximation the total wave function for the system is given by a combination of wave functions, each of which involves the coordinates of only one electron. In other words, the field seen by a given electron is assumed to be that of the fixed nuclei plus some average field produced by the charge distribution of all other electrons.