Energy loss of protons in Au, Pb, and Bi using relativistic wave functions

Abstract

We present a theoretical study on proton energy loss in solid targets of atomic number greater than 54. Fully relativistic wave functions and binding energies are obtained by solving numerically the Dirac equation. Ab initio calculations are developed for the first ͑stopping͒ and second ͑straggling͒ moments of the energy transferred from the ion to the target electrons. The shellwise local plasma approximation is employed for the inner shells, and the Mermin dielectric function is employed for the valence electrons. The dielectric response of each subshell is calculated separately, including in this way the screening among the electrons of the same binding energy. Results for stopping and straggling cross sections of protons in Au, Pb, and Bi are compared with the available experimental data. The theoretical stopping results are very good in the case of Au, reproducing the experimental data in an extensive energy region ͑10 keV-100 MeV͒. For Pb and Bi, the stopping results agree with the measurements for energies above 300 keV, for which the inner shells play a major role. However, we found some difficulties around the stopping maximum. For the energy-loss straggling, we obtained reasonably good agreement with the experiments for the three targets studied.

Low- and intermediate-energy stopping power of protons and antiprotons in solid targets

“…At sufficiently high impact energies the impinging projectile will be able to remove sub-valence electrons. To extend the theoretical description to intermediate and high velocities we include the inner-shell contribution by resorting to the SLPA [11,59,60]. During the last years we have developed this model based on the dielectric formalism and the local plasma approximation by Lindhard and Scharff [9].…”

“…6 we display the SLPA results for the stopping power due to the inner-shells of Cr, C, Be, Ti, Al, Si, Ge, Li and Pb. For Pb (Z=82, relativistic target) we used the results obtained by employing the GRASP code in [60]. For the rest, we used the atomic wave functions by Bunge [65].…”

“…By combining our non-perturbative and perturbative calculations in different energy regions, we managed to cover an extended range of (0.25 − 500) keV. The extension to higher impact energies by using the dielectric formalism and the SLPA has already been demonstrated [11,59,60].…”

“…This case is special because we are dealing with a relativistic target, 82 bound electrons, including the K-L-M-N-O shells and the 6s 2 -6p 2 electrons as free electron gas. We follow [60] to calculate the inner-shell contribution to the energy loss by using the SLPA together with the relativistic densities of electrons of each subshell (spin-orbit split) obtained with the GrasP code. In the present calculations, the experimental binding energies were used [68].…”

“…[color online] Inner-shell contribution to the stopping power of protons in Cr, Pb, Ti, Ge, Al, Si, Li, Be and C, as function of the impact velocity. Curves, results obtained with the SLPA considering from the K-shell to the sub-valence one, according to each target [11,59,60]…”