Debye expansion and JWKB approximation for the elastic scattering of heavy charged particles

By using the Debye expansion of the scatteringS-function and the JWKB solutions of the Schrodinger equation for the spherical-well nuclear potential in the presence of the Coulomb interaction (both for a point charge and for a uniformly charged sphere), it is shown that regions of the complex λ-plane exist in which the multireflection expansion of Knoll and Schaeffer coincides with the JWKB approximation of the Debye expansion. This expansion also provides the correct analytical contribution of all the terms of the multireflection expansion in the regions of the λ-plane where the Knoll and Schaeffer technique cannot be applied, because of the absence of the subdominant exponential. The JWKB approximation gives a reasonable estimate of the exactS-function except in the neighbourhood of the λ value corresponding to the classical grazing trajectory and allows one to interpret the scattering phenomenon in terms of classical trajectories. Furthermore, the Debye expansion allows one to overcome the mathematical difficulties one meets with in the evaluation of the scattering amplitude by the saddle point method in the cases in which the deflection function shows strong oscillations (false rainbows) produced by the superposition of contributions coming from the rays directly reflected by the nuclear surface on the multireflected ones.