Total Cross Sections for Inelastic Scattering of Charged Particles by Atoms and Molecules. V. Evaluation to the Next Order beyond the Bethe Asymptote

The Bethe procedure to obtain an asymptotic expression for the total cross section for inelastic scattering is extended to include three constants instead of the usual two. Within the first Born approximation, the third constant is essentially related to the total number of electrons in the target. By subtracting the sum of the cross sections for discrete excitations from the total cross section, the ionization cross section can also be given in an asymptotic form with three constants. The Mott formula is used to estimate the correction for electron exchange in the ionization cross section. The resulting cross sections compare favorably with experiments on the ionization of H, ${\mathrm{He}}^{+}$, ${\mathrm{H}}^{-}$, He, and ${\mathrm{Li}}^{+}$ by fast charged particles.