Threshold Effects in Average Cross Sections According toR-Matrix Theory

This paper attempts to place on a firm basis certain expressions for effects in elastic and total cross sections caused by—and in the neighborhood of—a reaction threshold. Explicit expressions are derived for the analytic behavior of the collision matrix near a reaction threshold. These expressions are based on the $R$-matrix theory of nuclear reactions and extend slightly work by Wigner and by Breit on threshold effects. The expressions are quite general, allowing for the presence of compound resonances. Both the channel matrix and the level matrix formulations of $R$-matrix theory are used. The former turns out to be convenient for formulating the general expressions of the collision matrix and the cross sections. The latter is more convenient for performing energy averages.