R-Matrix Shell-Model Calculations of Scattering and Reaction Cross Sections

The Wigner-Eisenbud $R$-matrix theory is applied to the calculation of neutron total and inelastic scattering cross sections for a system consisting of two neutrons interacting with an inert ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ core through a spherically symmetric Woods-Saxon potential and interacting with each other through a $\delta$-function force. The calculational method employed has the advantages that it includes the effects of shell-model configurations in which both neutrons are unbound, that it presents no obstacles to inelastic or reaction calculations, that it permits antisymmetrization of the compound space wave functions, and that it requires only one shell-model diagonalization for the computation of cross sections up to 5-MeV neutron energy. Use of antisymmetrized wave functions is shown to reduce substantially the number of compound-nucleus resonances and to reduce the magnitude of the inelastic cross section. By the correct calculation of the distant resonance contribution to the $R$ matrix, it is shown that the calculated cross sections are independent of the choice of channel radii. The application of the method to more complex systems with larger numbers of neutrons as well as protons and holes and also with direct coupling between channels is discussed. A selection rule encountered in the calculations suggests a possible $J^{\pi }$ dependence of the absorptive part of the optical-model potential.