Effective stopping-power charges of swift ions in condensed matter

The effective charge of energetic ions as it pertains to the stopping power of solids is calculated in a dielectric-response approximation. The density distribution of $N$ electrons bound in an ion of atomic number ${Z}_{1}$ is given by a variational statistical approximation. The effective charge $Z_{1}^{}{}_{}{}^{*}e$ is always larger than the ionic charge ${Q}_{1}=({Z}_{1}\ensuremath{-}N)e$, because of close collisions. A comprehensive low-velocity formula predicts $Z_{1}^{}{}_{}{}^{*}e$ for given $Q$ as a function of the ratio between the ion size and the mean electron spacing in the medium. At high velocities one obtains a partition rule of stopping powers for the effective charge of ionic projectiles. The results are compared with new precision stopping-power measurements on C, Al, and Au with $_{7}\mathrm{N}$ ions.