A mathematical formalism for computing cross sections for the single ionization of atoms and simple molecules by fast proton impact: application to the H <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow/> <mml

Abstract In this paper, we present an analytical formalism for computing triple and fourfold differential cross sections for the single ionization by proton impact of atoms as well as oriented and randomly oriented molecules. To improve the accuracy of the standard one Coulomb wave model, we incorporate the Salin factor, which provides a better approximation for the electron’s transition to the continuum. Although demonstrated for proton impact, the formalism is generalizable to other light ions. Its mathematical structure is designed to be both simple and practical, facilitating straightforward implementation and high computational efficiency, making it ideal for large-scale simulations. We validate the formalism by comparing the double differential cross sections for the ionization of water molecules with existing experimental data, demonstrating close agreement.