On the τ‐Approximation in the Quantum Theory of the Conductivity Tensor

In the framework of the random phase approximation a relaxation time is introduced into the quantum mechanical calculation of the conductivity tensor for a degenerate system of free electrons being elastically scattered by randomly distributed impurities. The result satisfies the requirements of gauge invariance and particle conservation and accounts for the modification of the level density by the scattering processes. In the special case of a weakly varying field the result obtained by Mertsching [6] is retained under certain conditions. Mertsching's semiphenomenological treatment is based on a heuristic formulation of the equation of motion for the statistical operator. The result is retained here if the Landau‐condition for the level width is met (i.e. the level width is much less than the Fermi energy). In the optical limit of zero wave number the Drude expression for the conductivity is obtained only for energies considerably less than the Fermi energy, otherwise a frequency‐dependent optical mass can be introduced.