Analytical formulas for differential cross sections for ejection of electrons in ionization of water by protons in the PWBA and ECPSSR

Abstract

For comparison with measured ionization cross sections as a function of the ejected electrons from water by protons, analytical formulas of the plane-wave Born approximation (PWBA) and the ECPSSR theory – that accounting for the energy-loss (E) and Coulomb-deflection (C) protons and perturbed-stationary states (PSS) and relativistic (R) of electrons goes beyond the PWBA – are evaluated analytically. As opposed to the PWBA that overestimates the data with the increasing energy of the ejected electrons, the ECPSSR is in good agreement with the measured spectra of the electrons ejected in with keV to MeV energies.