Evaluation of covariant behavior of experimental quantities through statistical analysis of matched pairs of pulse height data

Understanding and proper treatment of correlated information is very important in measurement science because it is a common problem. Covariance, if unrecognized or ignored without justification, can lead to errors. Correlations can be introduced in many ways. Measurement procedures, assumptions, algorithms, operators, nuclear data, model mismatch, and many other factors including physical variables such as environmental conditions can introduce co-dependences. In the current work, the correlation between a pair of measured data, namely, the full energy or photo-peak (FEP) and the background continuum under the peak, will be evaluated. Pulse height spectra were acquired using a high-purity germanium spectrometer and multiple uranium enrichment standards supplied by New Brunswick Laboratories. The uranium enrichment standards that were measured were Certified Reference Materials (CRM) with ²³⁵U fractional abundances ranging from 0.32 at.% to 93.23 at.%. The measurements were performed under near-ideal conditions, taking precautions to minimize sources of systematic uncertainties. Under fixed geometrical measurement conditions, the uranium enrichment, in atom fraction, of an optically thick chemical compound is directly proportional to the intensity of a gamma-ray of a given energy (e.g., 185.7 keV) emitted directly following decay of ²³⁵U and emerging from the sample. The objectives of the measurement were to acquire high-quality spectra that can be used to construct an “enrichment meter” calibration, and to utilize the FEP ratios to evaluate limitations in the knowledge of the branching ratios of gamma lines emitted by uranium isotopes. Propagation of uncertainties and evaluation of covariances were of utmost concern in this work. Using a three region of interest (ROI) peak analysis method, FEP areas and continuum counts under the FEP were estimated from the gamma-ray spectra. The FEP and continuum under the FEP are an associated pair of random variables because they are measured simultaneously (refer to the same measurement interval) and further are derived from the same pulse height spectrum. For a given measurement, the two random variables are expected to be correlated because of various factors. In addition, under well-shielded conditions and for pure compounds, a large part of the continuum under the FEP under the relatively lower energy gamma-ray peaks from ²³⁵U is expected to be due to down-scattered higher energy gamma rays emitted by ²³⁸U daughters. The ²³⁸U abundance goes down (up) as enrichment goes up (down), so it is natural to ask to what degree the FEP and the continuum under FEP are correlated. This work examines this question experimentally, for a given measurement item measured during a given measurement campaign, by acquiring a series of spectra close in time so that the sample linear correlation coefficient can be computed. For each of the eight uranium CRM standards, covariant behavior of FEP and continuum data from prominent gamma lines emitted by ²³⁵U (143.76, 185.715, and 205.315 keV) were evaluated directly through statistical analysis of the matched data pairs.