Inverse-energy stopping-power model ($S=k/E,R=E_0^2/(2k)$)

For an ideal particle energy-loss law $dE/dx=-k/E$, integration gives $E(x)=\sqrt{E_0^2-2kx}$ up to the formal range. Stopping power grows as the endpoint is approached but its integral remains finite. This simple law does not describe endpoint broadening, scattering or every low-energy stopping mechanism.