On the direct inversion of total scattering cross sections in the glory region

The total scattering cross section as a function of velocity in the glory region can be accurately described semiclassically as the sum of an average part and an undulating part. In practice the velocity dependence of the average cross section is complicated because of contributions from different dispersion terms in the attractive part of the potential and eventually from repulsive terms as well. It is shown that the average cross section can be accurately retrieved from the data, without the assumption of a mathematical model for the potential, and that the retrieved values can be directly inverted to numerical values of the potential tail. The separated results for the undulating part yield numerical values for five of the usual glory parameters. The procedure is tested on simulated total cross section data showing several full glory oscillations, and the glory parameters and potential tail are reproduced to accuracies consistent with the graphical-numerical methods employed.