Analytical formulas for Coulomb integrals involved in scattering problems

Radial matrix elements of multipole type involving the product of two Coulomb functions appear in many problems of theoretical atomic physics. Here we investigate some of their properties, with a generalization which includes the possibility of having the two functions related to two different charges. These would typically appear in calculations of ionization cross sections, going beyond the well studied case where the two charges are equal (elastic scattering or discrete excitation). We provide analytical formulas for the evaluation of matrix elements and recursion relations connecting them, both in the exact quantal formulation and in the WKB approximation. These theoretical results are illustrated by considering, within the Coulomb projected Born model, the electron impact ionization of ${\mathrm{Mg}}^{+}$ and ${\mathrm{Ar}}^{7+}$ with a relatively moderate incident energy. The radial matrix elements are evaluated with our exact quantal formulas and in the WKB approximation. The agreement is impressive, and is reflected in the values of triple, double, and single differential cross sections. Finally, a further study shows how the semiclassical approximation yields a very good estimate of single cross sections.