Resonance Widths and Spacings: Their Averages and Distributions

Resonance-pole parameters of the collision matrix are calculated from models of the $K$ matrix that are specified by finite statistical samples of poles and residues. The results confirm that in the absence of direct reactions the relation between the average partial width to spacing ratio $\frac{{\overline{\Gamma }}_c}{D}$ and the transmission coefficient $T_c$ is given by $T_c=1-\exp (-\frac{2\pi {\overline{\Gamma }}_c}{D})$; that the average absolute values of the diagonal collision-matrix pole residues ${\overline{G}}_c$ satisfy the inequality $\frac{2\pi {\overline{G}}_c}{D}\ge \frac{T_c}{{(1-T_c)}^{\frac{1}{2}}}$; that the distribution of widths is broader than the usually assumed ${\chi }^2$ distribution, depending on transmission coefficients and the number of open channels; and that the repulsion of pole spacings disappears as transmission coefficients and the number of open channels increase. The edge effects which distort the results of such finite pole sample calculations are discussed.