Analytical expression for the electronic stopping cross section of atomic gas targets for hydrogen projectiles - Cabrera-Trujillo, Remigio (Physical review, 2021)
The energy deposition of hydrogen projectiles in material targets is of great importance in material damage, as well as in radiotherapy and dosimetry, particularly for the treatment of cancer tumors. The energy-loss description has to take into account electronic excitation, electron transfer, and nuclear displacement for collisions with atomic targets. Each of these processes become relevant at different projectile collision energies, making the modeling of the full energy-loss curve difficult to accomplish. In this article, an analytical expression for the electronic stopping cross section of atomic gas targets at nonrelativistic velocities for hydrogen ions is reported. As the energy deposition process requires a correct description of the energy-loss and scattering process, our formulation uses theoretical results at low collision energies derived from ab initio electron-nuclear dynamics (END), which accounts for the nuclear displacement, electronic excitation, and charge-transfer process. At high collision energies, the expression is based on the independent particle orbital description [R. Cabrera-Trujillo, Phys. Rev. A 60, 3044 (1999)] combined with Bethe's analytical expression obtained through a harmonic oscillator representation of a target [L. Trujillo-Lopez et al., Radiat. Phys. Chem. 156, 150 (2019)]. The two approaches are matched at intermediate energies through the charge-exchange cross sections obtained from the END approach in a probabilistic interpretation of the processes [E. Montenegro et al. Phys. Lett. A 92, 195 (1982)]. The excellent agreement obtained when compared to experimental data for H to Ar gas targets (Z≤18) when hydrogen projectiles impinge gives support to the theoretical arguments implied in its derivation. The analytical expression is simple and has the correct asymptotic behavior at low and high collision energies. Furthermore, the formula explains properly the threshold effects at low collision energies which are the result of polarization effects and energy gaps in the excitation process of the target, as confirmed by our ab initio results.