Theory of Average Neutron Reaction Cross Sections in the Resonance Region

Abstract

The scattering matrix for compound nucleus processes is studied in the $R$-matrix formalism, using a series expansion which is due to Thomas. It is shown that this series generally converges when (a) the average total resonance width is less than the average resonance spacing, (b) the number of important channels is not too large, and (c) the width amplitudes have random signs. The treatment also suggests strongly that the series does not converge in the continuum region. In the region of convergence the exact relationship between the channel transmission factor $T_c$ and the ratio of partial width to level spacing is found, in the absence of direct scattering reactions, to be $T_c=\frac{2\pi 〈{\Gamma }_{\lambda c}〉}{D}-\frac{{\pi }^2{〈{\Gamma }_{\lambda c}〉}^2}{D^2}$. The quadratic term is shown to be important in the vicinity of optical-model maxima. Correction terms to the Hauser-Feshbach relations for average reaction cross sections arising from the higher order terms of the series are obtained and are found to depend on the statistical properties of both resonance widths and resonance spacings. The effect on average neutron inelastic, compound elastic, and capture cross sections is discussed and an example of a calculation is presented.