Stopping Powers and Ranges of 5-90-MeV ${\mathrm{S}}^{32}$, ${\mathrm{Cl}}^{35}$, ${\mathrm{Br}}^{79}$, and ${\mathrm{I}}^{127}$ Ions in ${\mathrm{H}}_2$, He, ${\mathrm{N}}_2$, Ar, and Kr: A Semiempirical Stopping Power Theory for Heavy Ions in Gases and Solids

Stopping powers and range curves for approximately 5—90-MeV beams of ${\mathrm{S}}^{32}$, ${\mathrm{Cl}}^{35}$, ${\mathrm{Br}}^{79}$, and ${\mathrm{I}}^{127}$ in ${\mathrm{H}}_2$, He, ${\mathrm{N}}_2$, Ar, and Kr are presented. Polynomial fits to the range curves followed by differentiation showed the $\frac{\Delta E}{\Delta X}$ measurements to be equivalent to $\frac{\mathrm{dE}}{\mathrm{dX}}$ to within experimental uncertainties. The experimental stopping powers are estimated to have errors no greater than ±3% for ${\mathrm{S}}^{32}$ and ${\mathrm{Cl}}^{35}$, ±4% for ${\mathrm{Br}}^{79}$, and ±5% for ${\mathrm{I}}^{127}$. Integral ranges should be accurate to ±1% or better. Small corrections for nuclear stopping were applied to the stopping powers, and the resulting stopping powers were analyzed in terms of fractional effective charge, defined as the quotient of the charge giving the correct stopping power in the Bethe equation to the nuclear charge, using experimental proton stopping powers. The same was done for published heavy-ion stopping powers in solids (Be, C, Al, Ni, Ag, and Au); and the fractional effective charges $\gamma$ were compared against a reduced velocity parameter $v_r=\frac{v}{\left(v_0{Z}^{\frac{2}{3}}\right)}$, where $v$ is the ion laboratory velocity and $v_0{Z}^{\frac{2}{3}}$ is the Thomas-Fermi electron velocity. All data so analyzed fell on a single smooth curve of $\gamma$ versus $v_r$ to ±3%, with the exception of some of the data taken in ${\mathrm{H}}_2$. The curve may be parametrized as $\gamma =[1-\exp (-0.95\mathrm{}{v}_r\left)\right]$. For the reduced velocity region $v_r\ge 0.1$, electronic stopping powers may be computed from the semiempirical relationship