Stopping Powers and Ranges of 5-90-MeVS32,Cl35,Br79, andI127Ions inH2, He,N2, Ar, and Kr: A Semiempirical Stopping Power Theory for Heavy Ions in Gases and Solids

Abstract

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 $\frac{\ensuremath{\Delta}E}{\ensuremath{\Delta}X}$ 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 \ifmmode\pm\else\textpm\fi{}3% for ${\mathrm{S}}^{32}$ and ${\mathrm{Cl}}^{35}$, \ifmmode\pm\else\textpm\fi{}4% for ${\mathrm{Br}}^{79}$, and \ifmmode\pm\else\textpm\fi{}5% for ${\mathrm{I}}^{127}$. Integral ranges should be accurate to \ifmmode\pm\else\textpm\fi{}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 $\ensuremath{\gamma}$ were compared against a reduced velocity parameter ${v}_{r}=\frac{v}{({v}_{0}{Z}^{\frac{2}{3}})}$, 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 $\ensuremath{\gamma}$ versus ${v}_{r}$ to \ifmmode\pm\else\textpm\fi{}3%, with the exception of some of the data taken in ${\mathrm{H}}_{2}$. The curve may be parametrized as $\ensuremath{\gamma}=[1\ensuremath{-}\mathrm{exp}(\ensuremath{-}0.95{v}_{r})]$. For the reduced velocity region ${v}_{r}\ensuremath{\ge}0.1$, electronic stopping powers may be computed from the semiempirical relationship ${(\frac{\mathrm{dE}}{\mathrm{dX}})}_{Z,A,E}={(\frac{\mathrm{dE}}{\mathrm{dX}})}_{p,\frac{E}{A}}{Z}^{2}{[1\ensuremath{-}\mathrm{exp}(\ensuremath{-}0.95{v}_{r})]}^{2}{[1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2.5{v}_{p})]}^{\ensuremath{-}2},$ where $Z$, $A$, and $E$ represent the atomic number, mass number, and kinetic energy of the heavy ion; the $\frac{\mathrm{dE}}{\mathrm{dX}}$ factor subscripted $p$, $\frac{E}{A}$ denotes the experimental proton stopping power for protons of energy $\frac{E}{A}$; ${v}_{r}$ is the reduced velocity of the heavy ion; and ${v}_{p}$ is the proton velocity in units of Me${\mathrm{V}}^{1/2}$. To the extent that the extrapolations of the correlations found in this work are valid, stopping powers so calculated should be accurate to \ifmmode\pm\else\textpm\fi{}8%. The effective charge parameter for gases and solids is approximately equivalent to the root-mean-square (rms) charge for the same ions in gas. The effective charges for heavy ions in solids are thus shown to differ markedly from the rms charges measured for solid stripping foils. These data suggest that the extra electrons are bound while the ion is in the solid and are lost when the ion leaves the stripping foil.