Procedure for Computing Cross Sections for Single and Multiple Ionization of Atoms in the Binary-Encounter Approximation by the Impact of Heavy Charged Particles

A procedure is developed for computing cross sections for the multiple ionization of atoms by the impact of protons or other fully stripped nuclei. The ionization probability, as a function of energy and impact parameter, $P(E,b)\mathrm{}$, is computed at several beam energies in the binary-encounter approximation for a ground-state hydrogenic electron scattered by an incident proton. Scaling laws are given which may be used to extend these results to other projectiles, other targets, and other hydrogenlike filled atomic shells. It is shown that $P(E,O)=〈\frac{\sigma \mathrm{}(E,r)\mathrm{}}{2\pi r^2}〉$ for isotropic, but otherwise arbitrary, electron-density distributions. A formulation for multiple-ionization cross sections is developed in terms of the single-electron probabilities $P(E,b)\mathrm{}$ for each atomic shell, assuming that both the electrons and the shells are mutually independent. Numerical calculations are compared to recent predictions in the semiclassical Coulomb approximation and to recent satellite and hypersatellite x-ray data. The discrepancies are generally within those resulting from uncertainties of 30-200% in the single-ionization cross sections, when the ionization probability is much less than one. Then, approximating $P\left(\mathrm{Eb}\right)\mathrm{}$ vs $b$ as a step function, the multiple-ionization cross sections are reduced to simple combinations of single-ionization cross sections. These single-ionization cross sections may be evaluated in the binary-encounter approximation by applying scaling laws to the usual universal curve that we tabulate. Multiple-ionization cross sections may thus be estimated without the aid of a computer.