Unitary Models of Nuclear Resonance Reactions

While most reaction theories are formally flux conserving, the kinds of models and approximations used to specify the $S$ matrix for resonance reaction processes are often of doubtful unitarity, particularly in the overlapping resonance region. To investigate the consequences of unitarity in this domain, several classes of simple analytically specified unitary $S$ matrices are constructed by means of $R$-matrix models having various periodic arrangements of poles and residues. The resulting reaction amplitudes have a variety of fluctuating resonance spacings and widths as well as nonresonant direct terms and up to three competing channels. Relationships between resonance parameters, channel transmission coefficients, average cross sections, and cross-section fluctuations are discussed. It is found that contrary to common belief unitarity imposes no restriction on the average ratio of channel width to resonance spacing. In all models investigated having no direct coupling between channels, the transmission coefficients are given by $T_c=1-\exp (-\frac{2\pi {\overline{\Gamma }}_c}{D})$. Localized structure in the resonance parameters is investigated, and the effects of a single strong $R$-matrix pole are compared with those of a "giant-resonance" distribution of $R$-matrix pole strength. In this way, the shapes of both Robson's analog resonances and Feshbach's doorway state resonances are derived in a different dynamical context. Direct scattering and reaction amplitudes are found to be strongly correlated with resonance amplitudes. Thus, for example, two channels coupled by a direct reaction have correlated resonance width amplitudes, and the pole terms of the resonance reaction amplitude coupling them have a nonzero average. Evidence is found that unitarity imposes resonance-resonance correlations, and these in turn affect the relation between average resonance parameters and average cross sections and even more between these and cross-section fluctuations.