Fine Structure in Nuclear Resonances

A $K$-matrix formulation is given for the shell-model approach to a unified reaction theory. This $K$ matrix differs from the $R$ matrix of Wigner and Eisenbud in using distorted waves of a diffuse potential. The penetrabilities and channel phase shifts are therefore those for a diffuse potential rather than those for a square well with hard-sphere boundary conditions at some radius. This $K$ matrix is used to discuss the fine structure produced by a doorway-hallway system, and a result obtained earlier by Ferrell and MacDonald is rederived. A resonance expansion is then found which explicitly exhibits the widths and resonance energies in a multilevel formula. The distribution of the widths is found, a sum rule is derived, and the average cross section is derived. A general resonance formula is then derived for the case of any number of hallway and doorway states. This is used to generalize the previously obtained doorway-hallway results to the case of hallways coupled to the continuum. In certain cases, a characteristic asymmetry is shown to result.