Application of Statistical Tests for Single-Level Populations to Neutron-Resonance-Spectroscopy Data

Dyson's new $F$-statistic test for spurious or missing levels has been applied to recent ${\mathrm{Er}}^{166}$ data and other of our results. The $F$ test was shown to give good agreement with the ${\mathrm{Er}}^{166}$ total observed level set and our $l=0\mathrm{}$ population choices for ${\mathrm{Er}}^{168}$ and ${\mathrm{Er}}^{170}$, which were made before this test was known to us. The $F$ test involves the evaluation, at each resonance position $E_i$, of a parameter $F_i$, defined as $F_i\equiv \Sigma {j\ne i}^{}f\left(x_{\mathrm{ji}}\right)$, where $x_{\mathrm{ji}}=\frac{(E_j-E_i)}{L}$, and $j$ is summed over all resonances between ($E_i-L$) and ($E_i+L$), and $f\left(x\right)=\frac{1}{2}\ln \left\{\frac{[1+{(1-x^2)}^{\frac{1}{2}}]}{[1-{(1-x^2)}^{\frac{1}{2}}]}\right\}$ for $x<\mathrm{}1\mathrm{}$. The interval $L=M〈D〉$ is characterized by a properly chosen integer, $M$. Dyson finds that $〈F_i〉=n-\ln \mathrm{}\left(n\right)-0.656\mathrm{}$, with a standard deviation of ${\left[\ln \mathrm{}\right(n\left)\right]}^{\frac{1}{2}}$, where $n=\pi M$. If a spurious level is present, $〈F_i〉=n$, evaluated at the spurious level. Thus, a large positive fluctuation occurs at a spurious level, while a low value of $F_i$ results near a missed level. We have used the $F$ test and other statistical tests to see if sets of $l=0\mathrm{}$ resonances could be chosen for ${\mathrm{Th}}^{232}$ and ${\mathrm{U}}^{238}$ which agree with the Porter-Thomas (PT) distribution for ${\left({\Gamma }_n^0\right)}^{\frac{1}{2}}$ and all statistical level-ordering tests. For the ${\mathrm{Th}}^{232}$ and ${\mathrm{U}}^{238}$ we used our older data for ${\Gamma }_n^0$ values and for the positions of the stronger $s$ levels. Our more recent 1970 data for these isotopes were used to better establish the position of the weak levels. It should be emphasized that ≥80% of the final choice $s$ population is made of stronger $s$ levels having ${\Gamma }_n^0$ values too large to be $p$ levels. The only degree of freedom is for the $s$-level selection from the weaker $s+p$ levels observed. The number of weak "$s$" levels chosen is determined by the PT distribution and we attempt to fit (simultaneously) the Wigner nearest-neighbor spacing distribution, $\rho (S_j,S_{j+1\mathrm{}})\cong -0.27\mathrm{}$, the Dyson-Mehta $\Delta$ statistic, and the Dyson $F$ test. For both isotopes, good simultaneous fits were obtained. This result is highly unlikely unless the "true" $s$ population satisfied the tests, since ≥80% of the final choice involves nonweak $s$ levels.