When neutron reduced widths, ${\Gamma }_n^0$, are evaluated for $N$ adjacent "single-population" resonances for an isotope, it is customary to express the fractional uncertainty in the $s$ strength function, $S_0$, as $\pm {\left(\frac{2.27}{N}\right)}^{\frac{1}{2}}$ or $\pm {\left(\frac{2}{N}\right)}^{\frac{1}{2}}$. It is assumed that the ${\Gamma }_n^0$ follow a Porter-Thomas (P.T.) single-channel distribution with a common $〈{\Gamma }_n^0〉$ for the interval, with no correlation between the different ${\Gamma }_n^0$. If the spacing distribution follows the Wigner formula for nearest neighbor spacings, but with no correlations, the ${\left(\frac{2.27}{N}\right)}^{\frac{1}{2}}$ fractional uncertainty applies for large $N$. For spacings following a statistical "orthogonal ensemble" (O.E.) behavior, the fractional uncertainty in $〈D〉$ is $\approx N^{-1\mathrm{}}$, so the fractional uncertainty in $S_0$ is $\approx {\left(\frac{2}{N}\right)}^{\frac{1}{2}}$ for large $N$. Experimentalists need easy to use rules for smaller $N$. We have used Monte Carlo methods with a P.T. form for ${\Gamma }_n^0$, and O.E. for spacings to establish the upper- and lower-bound values for $S_0$, divided by ${\overline{S}}_0\equiv \frac{{\overline{\Gamma }}_n^0}{\overline{D}}$ (the ratio of the measured averages of ${\Gamma }_n^0$ and $D$). The method of confidence intervals was used. We also suggest a "best choice" ratio for true $S_0$ to measured ${\overline{S}}_0$, all for a range of small $N$. The results should also apply for "single populations" of $p$ levels. The behaviors for ${\Gamma }_n^0$ and $D$ were also studied separately.

• Received 31 March 1972

DOI:https://doi.org/10.1103/PhysRevC.6.435