Universal range-velocity and stopping-power equations for fission fragments and partially stripped heavy ions in solid media

Abstract

The applicability of Bohr's stopping-power formula for range and stopping-power calculations has been investigated by a detailed analysis of the available range data for fission products in various solid media (Be, C, Al, Si, Cu, Zr, W, Au, and U). In the case of heavy media ($Z\ge 45.5$), the result of the analysis is in agreement with the Thomas-Fermi statistical approach for obtaining the effective quantum number for the outer orbital electrons and also with Bohr's original estimate of the effective charge of a fast-moving heavy ion. In the case of light media, an empirical function $n\left(U_s\right)=\frac{0.28Z^{\frac{2}{3}}{U}_s}{V_0}$ is found necessary in place of $n\left(U_s\right)=\frac{Z^{\frac{1}{3}}{U}_s}{V_0}$, where $n\left(U_s\right)$ is the number of orbital electrons in a medium of atomic number $Z$ whose velocities are less than a given velocity $U_s$ and $V_0=\frac{e^2}{\hslash }$. The incorporation of these findings into Bohr's stopping-power formula leads to universal stopping-power and range-velocity equations valid for partially stripped ions ($Z>\mathrm{}30\mathrm{}$) in different solid media, including compound media of known molecular formula. Available experimental data on stopping powers and ranges of heavy ions in various solid media and also the equilibrium charges of very light ions in gaseous media are in excellent agreement with the corresponding calculated values.