High-energy calculation of K -shell ejection cross sections as a function of projectile charge

We present a high-energy calculation of the $K$ -shell ejection cross section $\ensuremath{\sigma}({Z}_{p})$ as a function of projectile charge ${Z}_{p}$. Charge-transfer effects are not included. The wave function for the heavy projectile is written as ${e}^{i{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{0}\ifmmode\bullet\else\textbullet\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}\mathrm{F}(\ensuremath{-}in,1,i{k}_{e}|\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|\ensuremath{-}i{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{\mathrm{e}}\ifmmode\bullet\else\textbullet\fi{}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}))\ensuremath{\Gamma}(1+in){e}^{\frac{\ensuremath{-}n\ensuremath{\pi}}{2}}$, as opposed to the Glauber approximation ${e}^{i{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{0}\ifmmode\bullet\else\textbullet\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}\mathrm{exp}(\ensuremath{-}in\ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{Z}\mathrm{dZ}{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|}^{\ensuremath{-}1})$. These two approximations agree at large distances but differ as the electron and projectile approach closely to each other. A prediction is made that if ${T}_{m}$ (the maximum classical energy an electron at rest may receive) is much greater than ${I}_{K}$ then ${r}_{12}$ will be less than unity. Here ${r}_{12}=\frac{\ensuremath{\sigma}({Z}_{1}){Z}_{2}^{2}}{\ensuremath{\sigma}({Z}_{2}){Z}_{1}^{2}}$, ${Z}_{1}>{Z}_{2}$. Higher-energy experiments are needed to confirm this theoretical result.