Effective-charge theory and the electronic stopping power of solids

Abstract

Electronic stopping powers $S$ of solids for penetrating ions are analyzed on the basis of effective-charge theory, and comprehensive comparisons are made to available data for random stopping. The effective projectile-ion charges ${Z}_{1}^{*}e$, extracted from the data agree to within experimental uncertainties with those calculated in a statistical model, for projectile atomic number ranging from ${Z}_{1}=1$ to ${Z}_{1}=92$ and for projectile velocity ${v}_{1}\ensuremath{\gtrsim}{v}_{0}\ensuremath{\equiv}\frac{{e}^{2}}{\ensuremath{\hbar}}$. The rise of ${Z}_{1}^{*}$ with ${v}_{1}$ fully accounts for an often-quoted apparent deviation of heavy-ion stopping power from the behavior expected in the limit ${v}_{1}\ensuremath{\rightarrow}0$. The Bloch and the ${Z}_{1}^{3}$ corrections to the usual formula for $S$ are calculated and found to make no appreciable contribution to presently available data, save possibly to one bromine datum. When ${v}_{0}\ensuremath{\lesssim}{v}_{1}\ensuremath{\lesssim}3{v}_{0}$ the analysis requires, and provides, an empirical velocity-dependent proton effective charge ${Z}_{p}^{*}e$. A theoretical account of ${Z}_{p}^{*}$ is given in terms of velocity and energy criteria for electron stripping. Thomas-Fermi densities for heavy ions are used to calculate ${Z}_{1}^{*}$. Our results lead to an interpolation linear in ${v}_{1}$ for the range $0\ensuremath{\le}{v}_{1}\ensuremath{\lesssim}{v}_{0}$ which gives satisfactory values for $S$ in this low-velocity regime.