Poisson Statistical Methods for the Analysis of Low-Count Gamma Spectra

Although radioactive decay is well known to be a Poisson process, most of the gamma-ray spectral analysis and counting techniques in common use today have been developed in the “Gaussian limit”—that is, under the explicit assumption that the Poisson distribution can be well approximated by the Gaussian distribution. However the Gaussian approximation is not valid when the mean number of counts of the distribution is “small”, or when the behavior at the tails (i.e., “many” standard deviations away from the mean) of the distribution is of interest. We see increasing numbers of applications in disparate fields, from high energy astrophysics and particle physics to security screening and interdiction that fall into this regime. The blind application of Gaussian methods in these cases can yield erroneous and even non-physical results; in particular, reliance on the Gaussian notion of Critical Levels for detection decisions can have serious detrimental effects on detection probabilities and false alarm rates. In this paper, a set of rigorous Poisson-statistical tools have been developed, using a straightforward region-of–interest (ROI) approach, for the detection and quantification of signals in the analysis of low-count spectra. These tools provide improved accuracy over traditional Gaussian methods in both the quantitative evaluation and qualitative detection of small peaks. Formulae are derived for meaningfully estimating background and signal (net peak area) levels and their uncertainties, and for the evaluation of detection confidence. While these techniques are developed and presented here in the context of gamma spectroscopy, their applicability is quite general and can be extended to any radiation or particle detection scenario where Poisson statistics are expected to apply, including neutron, alpha and beta counting experiments.