Random-matrix approach to the statistical compound nuclear reaction at low energies using the Monte-Carlo technique

Using a random-matrix approach and Monte-Carlo simulations, we generate scattering matrices and cross sections for compound-nucleus reactions. In the absence of direct reactions we compare the average cross sections with the analytic solution given by the Gaussian Orthogonal Ensemble (GOE) triple integral, and with predictions of statistical approaches such as the ones due to Moldauer, to Hofmann, Richert, Tepel, and Weidenm\"{u}ller, and to Kawai, Kerman, and McVoy. We find perfect agreement with the GOE triple integral and display the limits of validity of the latter approaches. We establish a criterion for the width of the energy-averaging interval such that the relative difference between the ensemble-averaged and the energy-averaged scattering matrices lies below a given bound. Direct reactions are simulated in terms of an energy-independent background matrix. In that case, cross sections averaged over the ensemble of Monte-Carlo simulations fully agree with results from the Engelbrecht-Weidenm\"{u}ller transformation. The limits of other approximate approaches are displayed.