Excitation of atomic hydrogen by protons and helium ions

The Vainshtein-Presnyakov-Sobel'man approximation (VPSA) is extended for heavy projectiles and applied to calculating cross sections for direct excitation of atomic hydrogen by proton and ${\mathrm{He}}^{+}$ impact. The contribution of the projectile-target core interaction to the transition matrix element is included here for the first time. In addition an effective charge $\zeta$ is introduced to minimize the effect of neglecting certain terms in the Schrödinger equation of the collision system; this term-neglecting process is inherent to the VPSA. It is found that the projectile-target core interaction contribution is, as expected for heavy projectiles, small for calculating total cross sections but becomes significant for differential cross sections and increasing scattering angles. Inclusion of the effective charge raises the cross sections at lower-impact energies. The developed formulation is applied to excitation of atomic hydrogen to the $n=2,3,4,\mathrm{and} 5$ levels and comparison is made with the available experimental values for total and differential cross sections. The agreement is very good when the relative velocity is larger than $\sim 0.6v_0(v_0=\alpha c=2.188\mathrm{}\times {10}^{-8\mathrm{}} {\mathrm{cm}\mathrm{s}}^{-1\mathrm{}})$. The present results are also compared with the ones obtained by typical alternative theoretical approaches of varying sophistication.