The Boltzmann Equation in the Theory of Electrical Conduction in Metals

Abstract

The motion of conduction electrons in a metal in an electric field, scattered by an irregular static potential, is considered; this model is applicable to the resistance due to lattice waves at high temperatures, and to imperfections at any temperature. In §2 the Boltzmann equation is re-derived without the customary perturbation theory, avoiding the usual necessity of averaging over phases of different electron states after repeated small time intervals. The assumption that the scattering centres are distributed at random in the crystal is alone sufficient. The theory is, however, still dependent on assuming that h /τ<< kT , where τ is the collision time, as is the usual perturbation theory. §3 gives a general formula for the conductivity of the model, not subject to any assumption. This helps to justify an argument by Landau according to which the usual theory is valid provided h /τ<<η, where η is the cut-off energy of the Fermi distribution. No direct evaluation of the formula has been achieved.