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 Er¹⁶⁶ data and other of our results. The F test was shown to give good agreement with the Er¹⁶⁶ total observed level set and our $l=0$ population choices for Er¹⁶⁸ and Er¹⁷⁰, which were made before this test was known to us. The F test involves the evaluation, at each resonance position Eᵢ, of a parameter Fᵢ, defined as Fᵢ≡Σj≠i^f(xⱼᵢ), where xⱼᵢ=(Eⱼ-Eᵢ)L, and j is summed over all resonances between (Eᵢ-L) and (Eᵢ+L), and f(x)=1/2ln[1+(1-x²)1/2][1-(1-x²)1/2] for $x<1$. The interval $L=M〈D〉$ is characterized by a properly chosen integer, M. Dyson finds that 〈Fᵢ〉=n-ln(n)-0.656, with a standard deviation of [ln(n)]1/2, where n=πM. If a spurious level is present, 〈Fᵢ〉=n, evaluated at the spurious level. Thus, a large positive fluctuation occurs at a spurious level, while a low value of Fᵢ results near a missed level. We have used the F test and other statistical tests to see if sets of $l=0$ resonances could be chosen for Th²³² and U²³⁸ which agree with the Porter-Thomas (PT) distribution for (Γₙ⁰)1/2 and all statistical level-ordering tests. For the Th²³² and U²³⁸ we used our older data for Γₙ⁰ 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 Γₙ⁰ 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, ρ(Sⱼ,Sⱼ₊₁)-0.27, the Dyson-Mehta Δ 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.