Average Resonance Parameters and the Optical Model

Averages over resonances of the elements of the collision matrix are derived from $R$-matrix theory for the general many-channel case. Of special interest are the average diagonal elements because of their relation to the complex phase shifts of optical models for the channels. In the absence of correlations among channel width amplitudes, the formula relating an average diagonal element to average resonance parameters is found to be the same as obtained by Thomas for the single-channel case, or when competing channels have small transmission factors. The evaluation of this formula with the aid of the single-particle level picture of Lane, Thomas, and Wigner is discussed, as well as the effect of correlations among channels which are indicative of direct reaction effects.