Calculation of the detection limits with the least-squares method in gamma-ray spectrometry: A simple procedure

Abstract

A metrologically consistent procedure for assessing the detection limits of activity measurements for gamma-ray emitters with high-resolution spectrometers using the LSQ method is described and tested. As the input to the assessment, besides the measured contents of the spectral channels, the results of the peak analysis, i.e., the indication and its uncertainty, are used. The unfolding of the spectral region of interest into its components corresponding to the peak representing the indication and its background allows us to take into account the uncertainty budget, describing the uncertainty of the indication and the shape of the corresponding peak, making possible to include these sources of uncertainty in the calculation of the decision threshold. To assess the detection limit, the variance of the indication is calculated as a function of the indication itself, while considering the relative uncertainty of the conversion factor. The variance of the indication observed is approximated by a polynomial of the second order of the indication, thus making it possible to calculate the detection limit analytically. The method was tested on measured spectra using the empirically determined spectral shape of the peak representing the indication. It was shown how the empirically determined shape of an isolated and expressive peak close to the peak representing the indication can be used in the calculation of the decision threshold and how the presence of a peak overlapping with the peak representing the indication affects the decision threshold and the detection limit. It is explained that besides the counting statistics, the sources of uncertainty due to the shape of the peak representing indication also contribute to the decision threshold. However, to the increase of the detection limit over the decision threshold, besides the counting statistic, only the uncertainty of the conversion factor contributes. It is shown that in the presence of the indication, the decision threshold and the detection limit can be used to quantify the comparison between the observed value and the true value of the measurand with a predetermined quantity value in terms of the probabilities of making errors of the first and second kind. The application of the decision thresholds and detection limits to a conformity assessment is proposed.

Keywords: Conformity assessment; Decision threshold; Indication; Measurand; Uncertainty budget; Variance-covariance matrix.