Analytic Representation of the Generalized Oscillator Strength for Ionization

The energy distribution of secondary electrons resulting from ionizing collisions of fast charged particles with an atom or molecule is indispensable to microscopic modeling of radiation actions. In the Bethe theory, the distribution is determined from a more basic quantity, viz., the generalized oscillator strength for ionization. We examine analytic properties of the generalized oscillator strength for ionization from a fixed orbital, as a function of the squared momentum transfer K², or of the kinetic energy ε =k²/2 of an ejected electron. For fixed e, the generalized oscillator strength is analytic on the complex K² plane except at those points where Δ =[(K-k)²+2I][(K+k)²+2I] vanishes, I being the relevant ionization threshold energy (and quantities being measured in atomic units). On the basis of this observation and others, we show that the K² dependence of the generalized oscillator strength at fixed e should be well approximated by a polynomial in K² di...