Transition Matrix for Nucleon-Nucleon Scattering

Abstract

As part of a study of the influence of off-the-energy-shell effects on the optical potential for nucleon-nucleus scattering, a method is presented for the calculation, via the reactance matrix, of the nucleon-nucleon transition matrix in terms of an internucleon potential and the scattering amplitude. The singular integral equations for the partial-wave amplitudes of the reactance matrix are reduced to a Fredholm form which contains the scattering amplitude parametrically. The iteration solution of these Fredholm equations is shown to be generally unreliable; however, the zeroth-order iteration approximates the exact solution quite well near the energy shell. The replacement of the kernels of these integral equations by separable functions is discussed; the validity of such an approximation is illustrated by a simple example. The requirement that the solutions of the (exact) Fredholm equations be consistent with the original singular integral equations yields a solution for the scattering amplitude in terms of the resolvent kernels of the Fredholm equations. The entire formalism is so constructed as to include the possibility of a hard core being present in the nucleon-nucleon interaction.