Stopping power of hydrogen in hafnium and the importance of relativistic 4f electrons

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The stopping power of protons through Hf foil has been studied both experimentally and theoretically. The measurements were performed at the Laboratory of Accelerators and X-Ray Diffraction in Lisbon by using the transmission method on self-supporting stopping material. The overall uncertainty of around $5%$ was established over the protons energy range (0.6--2.5) MeV. The theoretical developments involved fully relativistic atomic structure calculations for Hf, which required the solution of the Dirac equation. The shell-wise local plasma approximation was used to describe the energy transferred to the bound $1s\text{\ensuremath{-}}4f$ electrons, and the outer four electrons were considered as a free electron gas. We found the relativistic description of the $4f$ shell and the screening between $4f$ and $5p$ electrons to be decisive around the stopping maximum. Present theoretical and experimental results are in very good agreement in the energy region of the new measurements. However, our theoretical stopping cross sections show substantial differences with the most used semiempirical models (SRIM2013 and ICRU-49) at intermediate to low energies. Our calculations suggest the stopping maximum to be higher and shifted to lower energies than these previous predictions. Future measurements around the maximum and below would be necessary for a better understanding of the stopping power of hafnium.