Simple and Exact Method for Calculating the Nuclear Reaction Matrix

A new, simple, and exact method is given for calculating the reaction matrix $G$ in a two-particle harmonic-oscillator basis. The method makes use of an expansion of the Bethe-Goldstone wave function in terms of solutions of the Schrödinger equation for two interacting particles in a harmonic-oscillator well. Since a two-particle basis is used, the Pauli operator $Q$ is diagonal and can be treated exactly. Reaction matrix elements based on the Hamada-Johnston potential are used in a shell-model calculation of $A=18\mathrm{}$ nuclei. The results are compared with those of earlier calculations using approximate Pauli operators. The dependence of the reaction matrix on the starting energy is studied, and the relationship of this energy to the intermediate-state spectrum and to the Pauli operator $Q$ is discussed. In this same context the difference between using a Brueckner $Q$ and a shell-model $Q$ is also discussed.