Stopping power for partially stripped ions

The Bethe formula for the stopping power for bare ions is extended to the collisions between partially stripped ions and neutral atoms. The resulting formulas can be presented in the same form as the original Bethe formula by replacing the nuclear charge of the projectile $Z$ and the mean excitation energy $I$ by $Z_{\mathrm{eff}}$ and $I_{\mathrm{eff}}$, respectively. Both $Z_{\mathrm{eff}}$ and $I_{\mathrm{eff}}$ contain contributions from excitations (including ionizations) of the target by the projectile, excitations of the projectile by the target, and mutual excitations. They are defined in terms of projectile and target properties that can be calculated from first principles. Expressions for $Z_{\mathrm{eff}}$ and $I_{\mathrm{eff}}$ reduce to those in the Bethe formula for bare incident ions. Formulas for atom-atom and ion-ion collisions are also presented. Comparison with experimental data on Ar ions in Ar gas indicates that the new formula is reliable for projectile energies greater than a few MeV/amu.