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 ε of the ejected electrons for collision of an incident charged particle of high energy E with an atomic system is given by dQdε=E^-1[A(ε)lnE+B(ε)], where A and B are functions of ε, and the total cross section is given by Q=E^-1[A_τlnE+B_τ]. The coefficients A(ε) and B(ε) for the particular case of the atomic hydrogen are derived analytically as an expansion with respect to k=ε. The expansion converges for all values of k. By integrating A(ε) and B(ε) with respect to ε, A_τ and B_τ 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 dQdε with respect to ε or 1ε when ε 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.