Some General Properties of Nuclear Reaction and Scattering Cross Sections

This paper deals with some general properties of nuclear reaction and scattering cross sections which can be derived from the dispersion-theoretic treatment of nuclear reactions as given, for example, by Wigner and Eisenbud. The form of the cross section near the resonances is found to satisfy the Breit-Wigner formula with corrections of the order $\frac{{\Gamma }^2}{D^2}$, $\Gamma =\mathrm{level}\mathrm{width}$, $D=\mathrm{level}\mathrm{spacing}$. In the region between the resonances, the behavior is found to be of three types. In the first two types, simple minima occur differing in order of magnitude in the two cases by a factor $\frac{{\Gamma }^2}{D^2}$. In the third type, a flat non-resonance maximum occurs between the resonances considered, accompanied by two minima. This type of behavior may be associated with a large fluctuation of $\Gamma$ or $D$ at some of the neighboring levels. The cross sections are explicitly calculated for some special choices of the $\Gamma$ and $D$, and the averages compared with those of the statistical theory. The effect of the potential scattering on the scattering cross sections is briefly discussed.