Free-electron-gas approximation for the stopping power of ionic targets

The use of free-electron-gas (FEG) motivated theories, proposed by Lindhard and colleagues, has proved extremely successful in treating the stopping power of neutral targets, and their relatively modest requirements of computational effort has led to their recent application to the stopping by electrons bound to target ions in plasmas. The FEG approximation for the average-excitation-energy Bethe-Born parameter is shown, for hydrogenic targets with nuclear charge $Z$, to be $Z^{\frac{3}{2}}$ times the neutral hydrogen value instead of having the correct $Z^2$ proportionality constant. Numerical data are presented for ${\mathrm{Al}}^{q+}$ that show the FEG error obeys this rule and, while the error approaches zero for $q=0\mathrm{}$, it is substantial for all $q>\mathrm{}0\mathrm{}$. An empirical adjustment of the FEG theory is found that reproduces the accurate ${\mathrm{Al}}^{q+}$ average-excitation-energy data. Two empirical rules now in use are observed to differ markedly from each other and from the accurate ${\mathrm{Al}}^{q+}$ results. The use of the standard FEG theory for all incident projectile energies is questioned on the basis of these results and the augmented theory proposed here requires further testing.