Convergent series representation for the generalized oscillator strength of electron-impact ionization and an improved binary-encounter-dipole model

Abstract

The use of the Bethe cross section in the binary-encounter-dipole (BED) model for electron-impact ionization is studied. While the dipole contribution in the Born approximation accounts for the longest-range interaction in electron-neutral atom/molecule inelastic collisions at any incident energy, the Bethe formula is applicable only at high energies. To derive a suitable representation of the Born cross section for dipole-allowed transitions, a convergent series representation for the generalized oscillator strength (GOS) of electron- impact ionization is studied. It is shown that by transforming to a new variable determined by the location of the singularities of the GOS on the complex plane of momentum transfer K, a series representation for the GOS is obtained that is convergent at all physically attainable values of K. An approximate representation of the GOS that truncates the series representation to the first three terms is also given. The approximate GOS describes the interaction of the electron with a shielded dipole potential and satisfies both Lassettre's limit theorem at $K=0$ and the asymptotic behavior at large K derived by Rau and Fano [A. R. P. Rau and U. Fano, Phys. Rev. 162, 68 (1967)]. The dipole-Born cross section so obtained is applicable at all incident energies and goes to the Bethe cross section at the high-energy limit. It provides a more suitable representation of the dipole contribution in the BED model than the Bethe cross section and is valid over the entire energy range. A similar analysis of the optical-oscillator strength (OOS) as a function of the complex momentum for the ejected electron ${k}_{p},$ plus the requirement that the OOS satisfies both the low- and $\mathrm{high}{\ensuremath{-}k}_{p}$ limits produces an analogous series representation for the OOS. An approximate one-term representation of the OOS is also developed that can be used in modeling calculations. Numerical examples of total ionization cross sections of ${\mathrm{N}}_{2},$ ${\mathrm{H}}_{2}\mathrm{O},$ ${\mathrm{CO}}_{2},$ ${\mathrm{CH}}_{4},$ and ${\mathrm{CF}}_{4}$ using the new analytical representation are presented to illustrate the applicability of the improved BED model.