Theory of cusped rainbows in elastic scattering: Uniform semiclassical calculations using Pearcey’s integral

Abstract

The first numerical application of the uniform Pearcey asymptotic approximation to a problem in semiclassical collision theory has been carried out. The problem investigated is the calculation of the differential cross section for elastic scattering by a potential energy curve that possesses a maximum and a minimum. The corresponding classical deflection function also possesses a maximum and a minimum, which means there are three contributing saddle points in the semiclassical analysis, two of which may be complex. The collision system investigated comes from a model curve crossing study of He++Ne scattering. In order to apply the uniform approximation, it is necessary to calculate Pearcey’s integral and its derivatives for many values of their arguments. Two methods for doing this have been developed. One method uses a series representation while the second method integrates a third order differential equation. It is also necessary to calculate the arguments of Pearcey’s integral from the classical trajectory data as well as the terms that multiply Pearcey’s integral and its derivatives. Iterative and algebraic methods for doing this are described. Both real and complex valued classical trajectories have been included in the semiclassical analysis. The uniform Pearcey approximation gives excellent agreement with results from the partial wave series. The accuracy of the transitional Pearcey, uniform Airy, and transitional Airy approximations is also investigated. The uniform Pearcey approximation can now be regarded as a practical tool for applications in semiclassical collision theory and other short wavelength phenomena.