Eur. Phys. J. D (2026) 80: 34
https://doi.org/10.1140/epjd/s10053-026-01147-6

Research - Atoms, Molecules, Ions, and Clusters

Trajectory effects on the energy loss cross section in collisions of H$_{}^{+}$ and He$_{}^{2+}$ ions with atomic hydrogen

¹ Department of Physics, University of Gothenburg, SE-412 96 , Gothenburg, Sweden
² Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad S/N, Col. Chamilpa, 62210, Cuernavaca, Morelos, México

^a This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 24 October 2025
Accepted: 16 March 2026
Published online: 6 April 2026

Abstract

Energy loss processes are fundamental for designing particle detectors, improving radiation therapies, and understanding fundamental processes in materials science. This property is often calculated by assuming straight-line trajectories. While this constant-velocity approximation is valid at high collision energies, it does not satisfy energy conservation. In this work, we investigate these trajectory effects by solving the time-dependent Schrödinger equation in a numerical lattice, using both coupled electron-nuclear dynamics and the straight-line trajectory approximation. We study H$_{}^{+}$ and He$_{}^{2+}$ projectiles incident on atomic hydrogen over the energy range 0.1–900 keV/u. We confirm that the total charge exchange cross sections are trajectory-independent within the energy range considered and show good agreement with experimental and theoretical data. However, clear trajectory-dependent differences emerge at intermediate to low energies for projectile energy loss with the straight-line approximation underestimating it at low collision energies. At high collision energies, the electronic stopping cross sections are in excellent agreement with available theoretical data. We also show that charge exchange and energy loss are correlated below the stopping cross section maximum, exemplified by the resonant charge exchange in H$_{}^{+}$+H collisions in contrast to the non resonant He$_{}^{2+}$+H. Overall, our work highlights the importance of nuclear trajectory considerations in collision dynamics and offers a foundation for further investigations of more complex systems.