Exact and approximate triangle amplitudes for (in-)elastic three-body processes with charged particles

Abstract

The triangle amplitudes, which within the framework of the multiple-scattering approach represent the leading contribution to the amplitude for three-body elastic and inelastic reactions, contain the off-shell Coulomb T-matrix describing the intermediate-state scattering of the projectile off each of the target particles. We present results of the exact numerical calculation of that amplitude in which the rescattering particles have charges of opposite sign (`attractive case'), for several atomic processes. This is facilitated by a `new' representation of the Coulomb T-matrix which turns out to be very effective for numerical purposes. One interesting result is that the charge sensitivity of the full triangle amplitude apparently disappears at the elastic threshold, for all scattering angles. Furthermore, we propose a new approximation for the triangle amplitudes which can be viewed as a `renormalization' by a simple analytic expression, of the well known approximation which consists in replacing by the potential . While the latter is known to be generally inadequate, this new approximation is shown to yield results in excellent agreement with the numerically calculated exact amplitude, for atomic elastic reactions, over a wide range of (medium to high) projectile energies and scattering angles (including the near-forward-scattering direction). An even simpler approximate amplitude is derived which contains no quadratures at all. It yields similarly good results provided the masses of the two particles experiencing intermediate-state rescattering are of the same order of magnitude but differ from that of the spectator particle. In addition, the explicit forms of the approximate amplitudes are used to derive a variety of interesting theoretical results.