The Bloch integral equation and electrical conductivity

Abstract

An account is given of the solution, for effectively the whole temperature range, of the Bloch (1928) integral equation for the electron momentum distribution in a metal in an electric field. Solutions of this equation, from which the temperature variation of the electrical conductivity of the metal may be immediately calculated, have previously been obtained only in the limiting cases of ‘high- and low -temperatures’, corresponding to ( T /θ p )≫ 1 and ≫1, where θ p is the Debye characteristic temperature. As a preliminary to its solution by numerical methods the integral equation is expressed in a non-dimensional form (§2). Solutions are obtained by deriving a high-temperature approximation which is valid over a much wider temperature range than that previously known, and by means of a method of successive approximations (§3). The temperature variation of conductivity is calculated from these solutions, and it is shown that there are significant differences between the results and those obtained from the semi-empirical formula of Grüneisen (1930) (§4). A comparison is made between the calculated and observed temperature variation of conductivity for a number of metals. There are deviations in detail, and a brief discussion is given of secondary factors from which they may arise, but in general the agreement is good, and it is concluded that the theoretical treatment covers satisfactorily the main features of the observed variation (§5). In an appendix it is shown that the approximate relations obtainable by the variational method developed by Kohler (1949) are consistent with the more exact results obtained here.