The theory of metals. —I

Abstract

1. The quantum theory of electrical conduction in a solid has two main problems to face, the number of “free electrons” and the “mean free path.” Of these the first is the simpler and has, to a certain extent, been solved. The evaluation of the mean free path, on the other hand, has given rise to some controversy and cannot be regarded as satisfactory. In his original paper on conduction Bloch gave a theory of the interaction of the electrons and the thermal vibrations in a metal which leaves much to be desired from the point of view of rigour, but which leads to results in good agreement with experiment. Peierls criticised this treatment and gave a new one, which, if correct, would considerably alter the theory.§ Peierls omitted most of the calculations, which are difficult, and based his treatment on physical arguments, which are by no means easy to follow, and which require justification. Recently L. Brillouin|| has given an extended mathematical treatment of the points in dispute, and obtains results which differ considerably from those of both Bloch and Peierls. None of these calculations is really satisfactory, the main objection being that the physical assumptions have not been made sufficiently precise. A method is given here which treats consistently the interaction of the electrons and the lattice, and which enables the assumptions to be clearly seen. It also has the advantage that it can be extended quite naturally to deal with the problems of the dispersion and absorption of light in metals, which will be treated in subsequent papers. In this paper the general theory will be developed, and applied to the discussion of the debatable points in the theories of Bloch and Peierls. Although the general opinion seems to be that Peierls’ criticisms are correct, the opposite view is arrived at here, and so, if the present theory is correct, the anomalous processes ” introduced by Peierls have little impor­tance for the electrical conductivity in a constant field.