‘‘Soft threshold’’ result in impact ionization derived from ‘‘hard-sphere’’ model

It has previously been shown that a result similar to the Keldysh formula [L. V. Keldysh, Sov. Phys. JETP 37, 509 (1960)], for the probability of impact ionization in an avalanching semiconductor, can be derived using a purely classical model. In this letter an even simpler model, the hard-sphere model, is used to derive an analogous relation, showing that the ‘‘soft threshold’’ effect is due largely to the geometry of the interaction between the particles concerned, rather than the nature of the interaction itself.