Use of the Projection-Operator Method in Nuclear Reaction Theory

Abstract

Feshbach has shown how the introduction of the projection operator onto the bound states of a system of nucleons can facilitate the calculation of the transition amplitude for a nuclear scattering reaction. Feshbach's method is based on the relationship of the transition amplitude to the $T$ matrix. In this paper we show how the projection-operator method can be applied to the generalized $R$-matrix expression for the transition amplitude. In order to test the validity of the approximations customarily employed in applications of the projection-operator formalism, we have applied the method to a simple soluble model. We find that the method works quite well provided that all second-order terms are retained. The generalized $R$-matrix method is found to work better than the $T$-matrix method, but the $R$-matrix method depends sensitively on the choice of the value of the boundary radius.