Stopping power for hydrogenlike and heliumlike particles: Bethe theory

An analytical formula for the electronic stopping power S of atoms with atomic number ${\mathit{Z}}_2$ and mean excitation energy I can be obtained, on the basis of Bethe theory, for fast hydrogenlike (${\mathit{Z}}_{\mathit{p}}$=${\mathit{Z}}_1$, ${\mathit{N}}_{\mathit{e}}$=1) and heliumlike (${\mathit{Z}}_{\mathit{p}}$=${\mathit{Z}}_1$-5/16, ${\mathit{N}}_{\mathit{e}}$=2) projectiles with atomic number ${\mathit{Z}}_1$ and velocity v. The resultant expression is S=(4π${\mathit{e}}^4$/${\mathit{mv}}^2$)${\mathit{NZ}}_2$L(${\mathit{Z}}_1$,${\mathit{Z}}_2$,v), where L(${\mathit{Z}}_1$,${\mathit{Z}}_2$,v)=(${\mathit{Z}}_1$-${\mathit{N}}_{\mathit{e}}$ ${)}^2$ln(2${\mathit{mv}}^2$/I)+(2${\mathit{Z}}_1$ ${\mathit{N}}_{\mathit{e}}$ -${\mathit{N}}_{\mathit{e}}^2$)ln(v/${\mathit{Z}}_{\mathit{p}}$ ${\mathit{v}}_0$)+${\mathit{Z}}_1$ ${\mathit{N}}_{\mathit{e}}$-11/12${\mathit{N}}_{\mathit{e}}^2$.