Behavior of the Ionization Cross Section in Collisions of High-Energy Electrons With Hydrogenic Atoms

The ionization cross section per unit range of the energy $\epsilon$ of the ejected electrons for collision of an incident charged particle of high energy $E$ with an atomic system is given by $\frac{\mathrm{dQ}}{d\epsilon }=E^{-1\mathrm{}}\left[A\right(\epsilon )\ln E+B(\epsilon \left)\right]$, where $A$ and $B$ are functions of $\epsilon$, and the total cross section is given by $Q=E^{-1\mathrm{}}[A_{\tau }\ln E+B_{\tau }]$. The coefficients $A\left(\epsilon \right)$ and $B\left(\epsilon \right)$ for the particular case of the atomic hydrogen are derived analytically as an expansion with respect to $k=\surd \epsilon$. The expansion converges for all values of $k$. By integrating $A\left(\epsilon \right)$ and $B\left(\epsilon \right)$ with respect to $\epsilon$, ${\mathrm{A}}_{\tau }$ and $B_{\tau }$ are obtained and compared with the values given by Bethe using a different method. The coefficients of the first two leading terms of the expansion of $\frac{\mathrm{dQ}}{d\epsilon }$ with respect to $\epsilon$ or $\frac{1}{\epsilon }$ when $\epsilon$ is small or large are found. A relationship for the average energy of the ejected electrons is given. Different features of the high-energy impact ionization and a certain connection between the ionization and the discrete level excitation are discussed.