Optical Potential in the R-Matrix Theory. I

Abstract

In conformity with the channel elimination method of the R-matrix theory, a theoretical development of the generalized optical potential is performed in the internal region of the configuration space where nuclear reactions take place and significance of the optical potential is, therefore, of particular importance. The concept of the boundary condition used therein generally yields the representation of this potential in terms of a purely discrete set of basis functions with specified quantum numbers. This discreteness serves to construct an effective scheme for treating the potential in a nonlocal form. derived by means of a space truncation procedure. As contrasted with a corresponding truncation of the space without boundary conditions at channel radii, it is in principle available irrespective of incident channel of low energy nucleon and choice of the Hamiltonian defining these functions. On the basis of this quality the nonlocal optical potential is formally designed to be obtained through a diagonaliza­ tion procedure employing the effective interaction resulting from the space truncation and a finite number of the wave functions satisfying the boundary conditions of the R-matrix type. It follows that the present method converts the nonlocal potential into a mathematically tracta­ ble one with a finite number of separable terms and thereby warrants the utilization of the potential of this type. Variations of the optical potential and corresponding scattering amplitude due to the change of the number of adopted levels are represented in terms of the basis functions and the effective interaction. The qualities and conditions of the convergence at specific energies are remarked by use of these representations. § I. Introduction Because of the unmanageable nonlocality of the generalized optical potential, the description of nuclear process utilizing this potential virtually demands the concepts and techniques which transform it into a mathematically tractable one. Since the wave function describing elastic scattering is basically determined by the nonlocal optical potential, a theoretical method is in some respects required to yield an available nonlocal form for this potential instead of a local one. This would allow further investigations on theoretical significance and aspects of the nonlocality in contrast with completely phenomenological approaches. The pur­ pose of the present paper is, therefore, to present a formal development on the generalized optical potential which employs the R-matrix method1l and leads to an effective scheme of calculation concerning this potential in a nonlocal form. The projection operator method2l straightforwardly derives the generalized optical potential which is defined over the whole space. In the asymptotic region it has to vanish or takes a simple or familiar shape o£ a real long-range potential. Accordingly, this part turns out to be unnecessary in the derivation and calcula