Saddles and trajectories, and poles and surface waves in heavy-ion elastic scattering

Abstract

The terms I m ±( θ) of the Poisson summation formula are used to describe and classify the oscillatory structures of the heavy-ion elastic cross sections coming from a strong-absorption model. The method of stationary phase, which relates these terms to branches of a deflection function, is shown to be inadequate and is replaced by the saddle-point method in the plane of complex angular momenta. The positions of the saddle points are found to be fairly independent of the parametrisation and define certain “active” regions which contribute to the scattering amplitude. The complex saddle points may be thought of as corresponding to complex trajectories in the nuclear potential. If the strong-absorption parametrisation possesses poles many saddle-point contributions can be simply expressed as the residues of the poles nearest the real axis. One of these leading parametric poles has an energy dependence similar to that of the dominant Regge pole generated by a complex optical potential. Poles lying above the real axis give rise to terms which may be thought of as surface waves, i.e. they are damped as they progress around the nuclear surface. These terms possess classically equivalent paths in the sense that they correspond to deflections less than π. Terms coming from poles below the real axis have no classical equivalent, increase as they progress and can, then, only be thought of as being diffractive. Some useful formulae for the cross section are given and the limiting case of a sharp cut-off is discussed.