Computational studies of first-born scattering cross sections I. spectral properties of bethe surfaces

Abstract

Computational techniques are reported for studies of atomic and molecular first-Born inelastic charged-particle scattering cross sections. Detailed illustrative results are presented for hydrogenic targets of appropriately defined momentum-transfer-dependent spectral moments, scattering cross sections differential in momentum transfer and in scattering angle, total inelastic scattering cross sections as functions of incident energy, and Van Hove autocorrelation functions describing target charge-density fluctuations. The variations with momentum transfer of the moments, the origins of the structures and asymptotic behaviors of the correlation functions, and the general characteristics of the scattering cross sections are described and clarified on the basis of the shape of the corresponding Bethe surface. Evaluations of the Born scattering cross sections differential in angle and in momentum transfer help to clarify the ranges of validity of various forms of corresponding static and binary-encounter approximations. The Bethe-Inokuti sum rule for the total inelastic scattering cross section is recovered in a transparent fashion from the static approximation by taking an appropriate high-energy limit, and its range of validity, as well as those of the static and binary-encounter approximations, is clarified by comparison with the correct Born result. Careful treatments of the scattering kinematics are seen to be important in these connections. It is noted throughout that spectral moments of the Bethe surface provide sufficient information for evaluation of Born cross sections, the consequences of which are investigated in a companion article in this issue.