ECCPA: Calculation of classical and quantum cross sections for elastic collisions of charged particles with atoms

Abstract

The Fortran program eccpa calculates differential and integrated cross sections for elastic collisions of charged particles with atoms by using the classical-trajectory method and several quantum methods and approximations. The collisions are described within the framework of the static-field approximation, with the interaction between the projectile and the target atom represented by the Coulomb potential of the atomic nucleus screened by the atomic electrons. To allow the use of fast and robust calculation methods, the interaction is assumed to be the same in the center-of-mass frame and in the laboratory frame. Although this assumption neglects the effect of relativity on the interaction, it allows using strict relativistic kinematics. The equation of the relative motion in the center-of-mass frame is shown to have the same form as in the non-relativistic theory, with a relativistic reduced mass and an effective potential. The wave equation for the relative motion, as obtained from the correspondence principle, is formally identical to the non-relativistic Schrödinger equation with the reduced mass and the effective potential, and it reduces to the familiar Klein-Gordon equation when the mass of the target atom is much larger than that of the projectile. Collisions of spin 1/2 projectiles are also described by solving the Dirac wave equation. Various approximate solution methods are described and applied to a generic potential represented as a sum of Yukawa terms, which allows a good part of the calculations to be performed analytically. The program eccpa is useful for assessing the validity and the relative accuracy of the various approximations, and as a pedagogical tool.