Energy loss in the ECPSSR theory and its calculation with exact integration limits

ECPSSR cross sections for inner-shell ionization are compared with numerical calculations that use correct and wrong expressions for exact limits of momentum transfer. For very adiabatic collisions, the employment of the wrong expression, as done in several popularly utilized computer codes, overestimates the cross sections up to two orders of magnitude. By contrast, even in such collisions the original ECPSSR approach of Brandt and Lapicki that approximates the effect of projectile energy loss by a multiplicative correction function fS and the relativistic effects by a relativistically corrected electron mass yields cross sections that are in agreement within a factor of two with respect to the calculation with hydrogenic Dirac wavefunctions and exact limits of integration with no relativistic correction. For plane-wave Born approximation calculations, the relativistic corrected expressions of the exact limits of integration are presented. The cross section calculated with such limits differs from the ECPSSR calculation using the fS-function by less than 2% in the region of lowest collision velocities covered by experiments. A straightforward modification of the existing computer codes with the correct expression for exact momentum transfer is suggested. (paper)