Transition rate for impact ionization in the approximation of a parabolic band structure

Recently, Alig, Bloom, and Struck have reported a simple model of ionization scattering in semiconductors and insulators. Their model is based upon the random-$k$ approximation to the transition rate for impact ionization, and upon a generic band structure with only two free parameters to describe all materials. The present paper describes the first step in an attempt to understand in detail why such a simple model works so well. The random-$k$ approximation to the transition rate for impact ionization is tested on a highly symmetric band-structure model for which most of the dimensions of the twelve-dimensional transition-rate integral can be treated analytically. The difference near threshold between the random-$k$ approximation and the rigorous result can be much larger than indicated by Kene's Monte Carlo integration for the silicon band structure, but this difference seems to be unimportant in practical problems where impact ionization competes with phonon emission.