Nuclear Scattering in the Random Phase Approximation

A simple extension of the random phase approximation (RPA) is described that includes scattering channels of a single nucleon. Coupled equations are derived for the single-nucleon amplitudes that describe the scattering states of the nucleon by an assembly of identical nucleons. These equations reveal that the exclusion principle is taken into account by antisymmetrized interaction matrix elements plus the appearance of the projection operator $1-{\rho }_0$ (${\rho }_0$ is the single-particle density) into unoccupied single-particle states in the compound nucleus. The use of the RPA also introduces the effects of long-range correlations, the well-known "backward-going graphs" of many-body perturbation theory, in the scattering problem. The system of equations thus obtained contains no parameters beyond those describing the two-body matrix elements that generate the Hartree-Fock field that serves as a zero-order approximation to the excitations considered within the RPA. The scattering solutions of these equations are illustrated with a soluble model. It is shown that the presence of correlations in the wave functions of resonance states can influence their particle decay widths considerably. The important practical problem of angular-momentum decomposition of the coupled-equation system is also discussed.