Effective-charge theory and the electronic stopping power of solids

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 $Z_1=1\mathrm{}$ to $Z_1=92\mathrm{}$ and for projectile velocity $v_1\gtrsim v_0\equiv \frac{e^2}{\hslash }$. 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 $v_1\to 0$. 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 $v_0\lesssim v_1\lesssim 3v_0$ 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 $0\le v_1\lesssim v_0$ which gives satisfactory values for $S$ in this low-velocity regime.