Abstract
The Bloch theory of electrical conductivity of metals assumes that the phonon distribution does not deviate from equilibrium. Peierls has criticized this assumption, showing that in the presence of an electric field a stationary state is achieved only through processes which do not conserve the total wave vector. These can be Umklapprozesse involving electrons and a phonon, or phonon-phonon interaction of the type responsible for thermal resistance in dielectric solids. He estimates that if only the latter were to act, strong deviations from Bloch's formula would occur below 50° K. He therefore concludes that the electrical resistance at low temperatures is determined by Umklapprozesse.
A more detailed treatment of the statistical equilibrium of phonons and electrons shows that the relaxation times of the important phonons is smaller than estimated by Peierls by a factor of about 100; consequently one would not, as a result of phonon-phonon interactions, expect important deviations from the Bloch theory above 10° K. It is thus unnecessary to consider Umklapprozesse. Since Umklapprozesse require a special shape of the Fermi surface in wave-vector space, Peierls has concluded from the apparent agreement with Bloch's theory that the Fermi surface has this special shape. Hence it is now shown that this shape is not necessary.