A formalism is developed for a statistical treatment of the energy variations of nuclear scattering and reaction cross sections. A statistical collision matrix ${\mathrm{U}}^S$ is defined which has the form of an energy-independent direct-transition matrix plus a fixed simple resonance-pole expansion, the matrix residues of which are products of complex channel-width amplitudes. By direct comparison with the Wigner-Eisenbud and Kapur-Peierls collision matrices it is found that under widely applicable conditions the statistical collision matrix may be used to calculate averages of observables over energy intervals containing many resonances and many total widths. The problem of determining the statistical properties of the parameters of ${\mathrm{U}}^S$ is defined and is solved for several special cases by relating it to the statistics of $R$-matrix parameters. Using these methods averages and mean-square fluctuations of total and reaction cross sections are calculated under general conditions admitting direct and compound processes and arbitrary average values of the total widths $\Gamma$ and the resonance spacings $D$. The results are expressed in terms of the direct-reaction matrix elements and the statistical properties of resonance parameters appropriate to the energy region under consideration and are related to the locally applicable optical-model phase shifts and transmission coefficients. Simplifications are obtained under special assumptions such as uncorrelated width amplitudes, small and large $\frac{\Gamma }{D}$, pure compound-nucleus reactions, many competing open channels, and many competing direct processes. In the limit of small $\frac{\Gamma }{D}$ one obtains the leading terms of an expansion of the average cross section which had previously been derived from $R$-matrix theory directly. In the limit of large $\frac{\Gamma }{D}$, many competing channels, but no direct reactions, the nonelastic fluctuation (or average compound nucleus) cross sections approach the Hauser-Feshbach formula. Except in this limit, corrections due to partial-width fluctuations and resonance-resonance interference are applicable. The former are sensitive to the magnitudes of direct reaction matrix elements, the latter to the correlations of resonance energies. Competing direct reactions are shown to require reductions of the transmission coefficients. The mean-square fluctuations of cross sections are found to approach Ericson's results in the limit of large $\frac{\Gamma }{D}$ and many competing channels, but are in general much larger for moderate $\frac{\Gamma }{D}$ and few channels. They are also sensitive to the details of resonance parameter statistics.

• Received 2 March 1964

DOI:https://doi.org/10.1103/PhysRev.135.B642