Theory of (p, p′) and (p, n) reactions through the isobaric analogue resonances: (II). Calculation of elastic and inelastic cross sections

Abstract

Using previous results on the isospin purity of the doorway part of the proton scattering wave functions in the internal region, we use a statistical model for the complicated compound nuclear resonances to derive formulae for the energy-averaged S-matrix elements describing inelastic proton scattering and for the transmission coefficients. These expressions together with the Hauser-Feshbach theory of nuclear reactions provide a frame for the analysis of experimental data. The analysis suggested in this paper makes use of an optical-model description of the elastic background scattering cross section in each channel and aims at extracting the partial widths of the analogue resonance from the data. These widths are proportional to the spectroscopic factors. The influence of direct inelastic proton scattering upon the results is discussed. The relationship between our results and those found previously for the case of one open proton channel is established. The energy averaged S-matrix elements are sums of two terms, the first describing the purely elastic optical-model background. The second term has the form of a Breit-Wigner formula. The total width in this formula is given by the sum over all channels of the partial widths of the analogue resonance plus a spreading width. The latter contains terms arising both from internal and external mixing. The partial widths in the Breit-Wigner formula are proportional to the partial widths of the analogue resonance. The proportionality factors are channel-dependent and complex and contain the influence of optical-model corrections. The transmission coefficients are each sums of two terms. One has the usual form one obtains from a one-level approximation to the scattering matrix. It displays a Breit-Wigner resonance. The other term contains the asymmetry characteristic of the analogue resonances. Compared with the case of one open proton channel, this asymmetry is smoothed out both because of internal mixing and because several proton channels are open.