Uncertainty in the neutron-strength-function evaluation for a small number of measured resonances

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, S₀, 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 ≈N⁻¹, so the fractional uncertainty in S₀ 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 S₀, divided by ${\bar{S}}_0\equiv \frac{{\bar{\Gamma }}_n^0}{\bar{D}}$ (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 S₀ to measured S̄₀, 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.