Energy loss of protons in W using fully relativistic calculations and mean excitation energies of W, Au, Pb, and Bi

Abstract

We present a theoretical study on the energy loss of protons in wolframium by calculating target fully relativistic wave functions and binding energies. The HULLAC code is employed to obtain numerical solutions of the Dirac equation. We use the shellwise local plasma approximation (SLPA) to evaluate the different moments of the energy loss. The partial contribution of each subshell of target electrons is calculated separately, including the screening among the electrons of the same binding energy. We pay special attention to the role of the outer $4f$ shell and the screening between electrons of near subshells (i.e., the $5p$ and $4f$ electrons). Results for stopping and straggling cross sections are compared to the experimental data available. Our calculations describe rather well the stopping measurements around the maximum and for very high energies, but overestimate the data for impact energies around 1MeV. We find that the SLPA results tend clearly to Bethe limit, but show a systematic overestimation in the energy region of 1--2 MeV. This overestimation may indicate the presence of other mechanisms included neither in the SLPA nor in Bethe formulations. We also present results for the stopping number of W, Au, Pb, and Bi which follow quite well the Lindhard scaling. A theoretical mean excitation energy $I(W)=710\text{ }\text{eV}$ is obtained, in good agreement with the suggested value of $727\ifmmode\pm\else\textpm\fi{}30\text{ }\text{eV}$. Theoretical mean excitation energies for Au, Pb, and Bi are also presented, which are in good agreement with the experimental ones.