Using the $\Delta_3$ statistic to test for missed levels in mixed sequence neutron resonance data

Abstract

The $\Delta_3(L)$ statistic is studied as a tool to detect missing levels in the neutron resonance data where 2 sequences are present. These systems are problematic because there is no level repulsion, and the resonances can be too close to resolve. $\Delta_3(L)$ is a measure of the fluctuations in the number of levels in an interval of length $L$ on the energy axis. The method used is tested on ensembles of mixed Gaussian Orthogonal Ensemble (GOE) spectra, with a known fraction of levels ($x%$) randomly depleted, and can accurately return $x$. The accuracy of the method as a function of spectrum size is established. The method is used on neutron resonance data for 11 isotopes with either s-wave neutrons on odd-A, or p-wave neutrons on even-A. The method compares favorably with a maximum likelihood method applied to the level spacing distribution. Nuclear Data Ensembles were made from 20 isotopes in total, and their $\Delta_3(L)$ statistic are discussed in the context of Random Matrix Theory.