An interesting type of 2-step reaction is that illustrated by A(d, n)B*, B* + C + c, i.e., a direct reaction to a resonant final state B*, which subsequently decays. We are especially interested in the situation in which the resonances B* are sufficiently dense in energy that they overlap, T> D, and we present a formalism for calculating the energy-averaged fluctuation cross section to any of the final channels (c + C). In particular, its application to the case in which the states B* are modulated by a doorway, (e.g., an isobaric analog state with fine structure) is worked out in some detail.
I. INTRODUCTION
An earlier publication [l] introduced an "optical background" representation for resonant multi-channel scattering amplitudes. Its use facilitated a generalization of the Hauser-Feshbach formula for fluctuation cross sections in 2-body reactions, to include the effects of direct-reaction coupling between these 2-body channels. Our purpose in this sequel is two-fold. First, by limiting ourselves to the important special case of strongly overlapping resonances (T > 0) decaying into many open channels, with only a "small number" of the channels strongly coupled by direct reactions, we show that the formulae of KKM [l] reduce to a form nearly as simple as the corresponding Hauser-Feshbach expressions. In particular, both the X-matrix and the fluctuation cross section are given directly in terms of the direct-reaction penetration matrix [2], by the approximate expressions