A complex angular momentum theory of modified Coulomb scattering

Abstract

The authors develop an exact complex angular momentum (CAM) theory of elastic scattering for a complex optical potential with a Coulombic tail. The present CAM theory avoids complications due to the long range nature of the Coulombic potential in a straightforward way. This is in contrast to the conventional approach in which the partial wave series is divergent and the scattering amplitude f( theta ) is usually decomposed into a pure Coulomb amplitude and a modifying convergent partial wave series. After considering some general properties of the scattering matrix element S( lambda ), the Sommerfeld-Watson transformation together with a travelling wave (near-side far-side) decomposition is used to obtain an exact representation for f( theta ) in terms of a background integral fB( theta ) and a series of subamplitudes fn(+or-)( theta ). New exact representations are derived for fB( theta ) when S( lambda ) possesses local symmetries of the type S(- lambda )=S( lambda ) exp(+or-2i pi lambda ) and S(- lambda )=S( lambda ). The subamplitudes fn(+or-)( theta ) are contour integrals and are the exact equivalents of the saddle-point integrals that arise in semiclassical theories. The fn(+or-)( theta ) can also be evaluated in terms of the Regge pole positions and residues of S( lambda ), which allows a flexible representation for f( theta ) to be derived. The exact results obtained unify the CAM theory of scattering for Coulombic and short range potentials and are especially suitable for the introduction of semiclassical approximations.