High-speed charge and isotope scaling of electronic stopping ($\delta E\propto Z^2/v^2\simeq Z^{2}M/(2E)$)

For a fast ion traversing a fixed material thickness with only a small fractional energy loss, the dipole excitation probability per atomic encounter scales as $Z^2/v^2$ at fixed impact parameter. Summing transferred excitation energies and integrating the encounter measure $2\pi bdb$ gives the leading electronic stopping power scaling, with slower logarithmic dependence from the allowed excitation energies and impact-parameter range.

At fixed ion kinetic energy $E=M{v}^2/2$, a heavier isotope is slower, so the leading loss scales as $Z^{2}M/E$. At fixed velocity, isotopes of the same element instead have approximately the same electronic loss; plotting against energy per nucleon removes the leading isotope mass dependence. The charge should be the effective projectile charge if its bound electrons screen the nucleus. At lower speeds, excitation thresholds, slow-passage suppression, charge exchange and nuclear energy transfer invalidate a simple extrapolation of the high-speed law.