Electronic stopping power of protons in transition metals: Bragg peak

For the electronic stopping power of protons in period-6 transition metals, the Bragg peak positions measured in experiments are nearly independent of the element for lanthanides and suddenly show a linear growth for non-lanthanides, while the Bragg peak magnitudes monotonically decrease with increasing atomic number. These observations can be partially explained qualitatively by the velocity-matching criterion. In this study, we reproduce experimental results by using the dielectric response models. To interpret them, we develop approximate analytical models of the Bragg peak position and magnitude for specified shells, which are correlated to both the ionization energy and average electronic density. It is found that for inner shells, the element-independent Bragg peak position for lanthanides is due to a weak variation of ionization energies and average densities, and a strongly dependent relationship for non-lanthanides is attributed to a distinctly remarkable increment of ionization energies. In addition, it shows that the consideration of the electronic density, a factor neglected in the velocity-matching formula, is essential for reproducing the element-dependent peak positions for inner shells. For valence shells, a uniform density is the unique factor, which almost remains unchanged for lanthanides but can be increased by about an order of magnitude for non-lanthanides. An analytic simplification of the Bragg peak magnitude reveals that the maximal stopping cross section is inversely proportional to the Bragg peak position for inner shells, and it is determined by the atomic number density for valence shells, which well explains the experimental results in period 6.