Uncertainty in relative energy resolution measurements

We suggest a new method for the determination of the detector relative energy resolution and its uncertainty based on spline approximation of experimental spectra and a statistical bootstrapping procedure. The proposed method is applied to the spectra obtained with NaI(Tl) scintillating detectors and 137Cs sources. The spectrum histogram with background subtracted channel-by-channel is modeled by cubic spline approximation. The relative energy resolution (which is also known as pulse height resolution and energy resolution), defined as the full-width at half-maximum (FWHM) divided by the value of peak centroid, is calculated using the intercepts of the spline curve with the line of the half peak height. The value of the peak height is determined as the point where the value of the derivative goes to zero. The residuals, which are normalized over the square root of counts in a given bin (y-coordinate), obey the standard Gaussian distribution. The values of these residuals are randomly re-assigned to a different set of y-coordinates where a new 'pseudo-experimental' data set is obtained after 'de-normalization' of the old values. For this new data set a new spline approximation is found and the whole procedure is repeated several hundred times, until the standard deviation of relative energy resolution becomes stabilized. The standard deviation of relative energy resolutions calculated for each 'pseudo-experimental' data set (bootstrap uncertainty) is considered to be an estimate for relative energy resolution uncertainty. It is also shown that the relative bootstrap uncertainty is proportional to, and generally only two to three times bigger than, 1/√(Ntot), which is the relative statistical count uncertainty (Ntot is the total number of counts under the peak). The newly suggested method is also applicable to other radiation and particle detectors, not only for relative energy resolution, but also for any of the other parameters in a measured spectrum, like peak position, shape parameters, etc