The electrical and thermal conductivities of monovalent metals

Abstract

The scattering of electrons by the thermal lattice vibrations is considered. The original Bloch theory is qualitatively in agreement with experiment, but has been shown by various authors to be unrealistic in its detailed assumptions. In particular, it ignores Umklapp processes, neglects the difference between longitudinal and transverse phonon velocities, and treats electron-phonon interaction as if it were independent of the scattering angle. The Bloch theory is generalized to take account of these effects, and formulae are obtained for the transport properties. Numerical calculations for the case of sodium, using the Bardeen (1937) formula for the scattering cross-section and Blackman’s (1951) value for a ‘longitudinal Debye temperature’, agree better with observation than do the simple Bloch expressions, but there still remain discrepancies. An alternative description of these is to treat the differential scattering cross-section, which is not really known exactly, as if it were an unknown function to be determined. The observed electrical and thermal conductivities of sodium are consistent with a cross-section which has a narrower forward lobe than the Bardeen formula, but does not become so small for scattering in the backward direction.