Threshold Effects in Average Cross Sections According toR-Matrix Theory

Abstract

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 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 ievel 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. It is shown that certain formulas for energy-averaged total and elastic cross sections, which were made plausible in a previous paper by the author, follow from the above-mentioned general expressions by performing suitable energy averages. Consequences of the random-sign approximation of the value quantities'' gamma lambda and of a partial breakdown of this assumption are examined and related to the assumptions of the optical model. It is shown that under the assumption of random signs, the total cross section should showmore » no threshold effects, whereas if this assumption is relaxed threshold effects appear. Hence it is, in principle, possible to decide by experiment which situation obtains. Finally, cross section threshold effects under the single-level approximation are given; with a slight generalization of the phase shift, these are identical to expressions derived by Baz and by Newton. (auth)« less