The Angular Distribution of Scattering and Reaction Cross Sections

TLDR

Racah coefficients, originally introduced for complex atomic spectra, underpin the resulting expressions for angular distributions. The authors simplify the general angular‑distribution formula by explicitly summing over magnetic quantum numbers, express cross sections as real Legendre‑polynomial series, and specialize the theory to nuclear reactions involving a single isolated resonance level. They derive explicit formulas for differential reaction cross sections and for scattering with or without channel‑spin change, and provide detailed interference terms between resonance and potential scattering for both neutral and charged particles.

Abstract

The general formula for the angular distribution in collisions between pairs of particles is simplified by performing explicitly all sums over magnetic quantum numbers. The resulting expressions involve coefficients introduced by Racah for the study of complex atomic spectra. The cross sections are expressed as series in Legendre polynomials, each coefficient in the series being manifestly real.The general theory is then specialized for the case of nuclear reactions and scattering associated with one isolated resonance level of the compound nucleus. Formulas are derived for the various differential reaction cross sections and for scattering with and without change of channel spin. The interference terms between resonance and potential scattering are written explicitly, both for neutral and for charged particles.