Statistical analysis of intermediate structure

We study several statistical tests which can be used to determine whether experimental data, typically a sequence of partial widths, are compatible with the statistical model of nuclear reactions or, on the contrary, imply the existence of intermediate structure. These tests are applicable to cases where the data can be ascribed to a definite partial wave. We give a brief, but critical, discussion of the few known methods, and develop several new tests. These new tests involve the study of the following quantities: (a) the length of the longest unbroken sequence (run) of partial widths lying either above or below their median, (b) the length of the longest run about a value that we call the value of optimal run length, (c) the distribution of runs up or down, i.e., of unbroken sequences composed of increasing or decreasing values, (d) the number of large adjacent partial widths, and (e) the ratio of the mean-square successive difference to the variance. We apply these new tests and the previously known ones, and discuss their merits and drawbacks, in the examples ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}(p,p^{\prime })$, ${}_{}{}^{56}{}_{}{}^{}\mathrm{Fe}\mathrm{}(n,n)\mathrm{}$, ${}_{}{}^{244}{}_{}{}^{}\mathrm{Cm}+n$, ${}_{}{}^{187}{}_{}{}^{}\mathrm{Re}(n,\gamma )$, ${}_{}{}^{115}{}_{}{}^{}\mathrm{In}(n,\gamma )$, ${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}+\gamma$, $\mathrm{Sn}+\gamma$, ${}_{}{}^{70}{}_{}{}^{}\mathrm{Ge}\mathrm{}(p,p)\mathrm{}$, ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}\mathrm{}(n,f)\mathrm{}$, and ${}_{}{}^{206}{}_{}{}^{}\mathrm{Pb}\mathrm{}(n,n)\mathrm{}$. We find reliable evidence for intermediate structure in all these cases, except in the reactions ${}_{}{}^{187}{}_{}{}^{}\mathrm{Re}(n,\gamma )$ and ${}_{}{}^{115}{}_{}{}^{}\mathrm{In}(n,\gamma )$.