Direct reactions and Hauser--Feshbach theory

In the presence of direct reactions, the calculation of energy-averaged cross sections, polarizations, analyzing powers etc. in terms of the energy-averaged S-matrix is possible with the help of a unitary transformation U, defined as the matrix which diagonalizes Satchler's transmission matrix. For a fixed number of channels and in the limit of a very large number N of compound nucleus resonances, we show analytically that the transition from the originally given statistical scattering matrix S to the matrix S' =USU/supT/ reduces the Hauser-Feshbach problem with direct reactions to the one without. The analytical proof is supported by numerical calculations which show that the results are valid on the 2 to 3% level of statistical accuracy of the numerical calculations for values of N as small as 100. We also show analytically that in the absence of direct reactions, energy-averaged cross sections depend only upon the transmission coefficients. The form of this dependence, as well as the dependence on the transmission coefficients of other quantities needed for the evaluation of cross sections in the presence of direct reactions, is parametrized by simple fit formulas which, while lacking theoretical support, give a very satisfactory overall representation of our numerical results