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

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.

Influence of the solar activity on the background of a high-resolution gamma-ray spectrometer.

The count rate in the peak of a gamma-ray spectrum at 2223 keV was measured over a period of 25 years. The peak is produced by neutron capture on hydrogen, a constituent of the spectrometer's shield. Since the neutrons are produced by cosmic rays, the count rate in the peak is correlated with the solar activity via the interaction between the solar wind and the cosmic rays. The correlation between the total daily number of sunspots, as a measure of the solar activity, and the count rate in the peak was investigated as a function of the time shift between the time dependence of the count rate in the peak and the time dependence of the total daily number of sunspots. Variations of the correlation coefficient as a function of the shift are discussed in terms of phenomena occurring on the surface of the Sun. The variations indicate a long-term correlation, corresponding to the 11-year solar cycle, and a short-term correlation, corresponding to the sunspots.

Calculation of decision thresholds according to the standard ISO 11929-3 in case of presence of the peaked background.

The method of calculation of the decision threshold with the Least Squares Method, described in the standard ISO 11929, is presented for the case when the sources of peaked background contribute to the peak holding the indication. The decision threshold is calculated from spectral data corresponding to the indication zero; therefore, the observed indication must be removed from the spectrum. When the peaked background is present, the indication completely overlaps with the peaked background, so it can't be unfolded directly. Therefore, two steps are needed in the calculation: the unfolding of the peak, housing the indication, from the continuous background and the possible overlapping peaks, and separating of the indication from the peaked background using the background data obtained from separate calculations and measurements. In this article it is shown that the method of least squares is flexible enough to accommodate all sources of uncertainty into the uncertainty matrix of input quantities. Its derivation is presented in detail and the calculation of the indication corresponding to the decision threshold is described. As a proof of the concept an example of calculating the number of counts corresponding to the decision threshold as a function of the indication is presented. The method of calculation and the results of the calculation are briefly discussed.

Empirical determination of the correlation coefficient between the number of counts in a peak in a gamma-ray spectrum and the number of counts in the continuum where the peak is superimposed.

An account is given on the value of the correlation coefficient between the number of counts in a peak in a gamma-ray spectrum and the number of counts in the background, where the peak resides. It is supposed that the decomposition of the spectrum in the peak and in the background is performed by using the Least Squares method. The values of the correlation coefficient were determined empirically from measurements of gamma-ray spectra under repeatable conditions and from analyses of these spectra using four different kinds of peak-analysis software. These values were compared to the a-priori values, obtained from the Least Squares method.

Determining the probability of locating peaks using computerized peak-location methods in gamma-ray spectra as a function of the relative peak-area uncertainty.

The probabilities of locating peaks with a high relative peak-area uncertainty were determined empirically with nine types of peak-location software used in laboratories engaged in gamma-ray spectrometry measurements. It was found that it is not possible to locate peaks with a probability of 0.95, when they have a relative peak-area uncertainty in excess of 50%. Locating peaks at these relatively high peak-area uncertainties with a probability greater than 0.95 is only possible in the library-driven mode, where the peak positions are supposed a-priori. The deficiencies of the library-driven mode and the possibilities to improve the probabilities of locating peaks are briefly discussed.

An alternative approach to the decision threshold.

Two effects were identified that induce the dependence of the decision threshold on the uncertainty of the conversion factor. With the first effect, the conversion factor influences the decision threshold directly, through its variability. With the second effect, the variability of the conversion factor influences the decision threshold only when it is not evaluated for the null measurement. Then, the variability of the conversion factor influences the value of the measurand, where the decision threshold is evaluated. Both effects are explained and evaluated.

Calculation of the detection limits for radionuclides identified in gamma-ray spectra based on post-processing peak analysis results.

A new method for calculating the detection limits of gamma-ray spectrometry measurements is presented. The method is applicable for gamma-ray emitters, irrespective of the influences of the peaked background, the origin of the background and the overlap with other peaks. It offers the opportunity for multi-gamma-ray emitters to calculate the common detection limit, corresponding to more peaks. The detection limit is calculated by approximating the dependence of the uncertainty in the indication on its value with a second-order polynomial. In this approach the relation between the input quantities and the detection limit are described by an explicit expression and can be easy investigated. The detection limit is calculated from the data usually provided by the reports of peak-analyzing programs: the peak areas and their uncertainties. As a result, the need to use individual channel contents for calculating the detection limit is bypassed.

Systematic influences on the areas of peaks in gamma-ray spectra that have a large statistical uncertainty.

A method is presented for calculating the expected number of counts in peaks that have a large relative peak-area uncertainty and appear in measured gamma-ray spectra. The method was applied to calculations of the correction factors for peaks occurring in the spectra of radon daughters. It was shown that the factors used for correcting the calculated peak areas to their expected values decrease with an increasing relative peak-area uncertainty. The accuracy of taking the systematic influence inducing the correction factors into account is given by the dispersion of the correction factors corresponding to specific peaks. It was shown that the highest accuracy is obtained in the peak analyses with the GammaVision and Gamma-W software.

Calculation of the detection limits by explicit expressions.

A method for calculating the approximate value of the detection limit for measurements of ionizing radiation is presented. The method can be applied when the indication corresponding to the detection limit and its uncertainty are given as explicit functions. Then also the detection limit can be calculated explicitly, which means that the iteration procedure for its calculation can be avoided. The advantage of the method becomes apparent when the iteration process for calculating the detection limit is difficult to apply.

Determination of the measurement threshold in gamma-ray spectrometry.

In gamma-ray spectrometry the measurement threshold describes the lover boundary of the interval of peak areas originating in the response of the spectrometer to gamma-rays from the sample measured. In this sense it presents a generalization of the net indication corresponding to the decision threshold, which is the measurement threshold at the quantity value zero for a predetermined probability for making errors of the first kind. Measurement thresholds were determined for peaks appearing in the spectra of radon daughters 214Pb and 214Bi by measuring the spectrum 35 times under repeatable conditions. For the calculation of the measurement threshold the probability for detection of the peaks and the mean relative uncertainty of the peak area were used. The relative measurement thresholds, the ratios between the measurement threshold and the mean peak area uncertainty, were determined for 54 peaks where the probability for detection varied between some percent and about 95% and the relative peak area uncertainty between 30% and 80%. The relative measurement thresholds vary considerably from peak to peak, although the nominal value of the sensitivity parameter defining the sensitivity for locating peaks was equal for all peaks. At the value of the sensitivity parameter used, the peak analysis does not locate peaks corresponding to the decision threshold with the probability in excess of 50%. This implies that peaks in the spectrum may not be located, although the true value of the measurand exceeds the decision threshold.

Calculation of the correlation coefficients between the numbers of counts (peak areas and backgrounds) obtained from gamma-ray spectra.

Two simple methods for calculating the correlations between peaks appearing in gamma-ray spectra are described. We show how the areas are correlated when the peaks do not overlap, but the spectral regions used for the calculation of the background below the peaks do. When the peaks overlap, the correlation can be stronger than in the case of the non-overlapping peaks. The methods presented are simplified to the extent of allowing their implementation with manual calculations. They are intended for practitioners as additional tools to be used when the correlations between the areas of the peaks in the gamma-ray spectra are to be calculated. Also, the correlation coefficient between the number of counts in the peak and the number of counts in the continuous background below the peak is derived.

Evidence against solar influence on nuclear decay constants.

The hypothesis that proximity to the Sun causes variation of decay constants at permille level has been tested and disproved. Repeated activity measurements of mono-radionuclide sources were performed over periods from 200 days up to four decades at 14 laboratories across the globe. Residuals from the exponential nuclear decay curves were inspected for annual oscillations. Systematic deviations from a purely exponential decay curve differ from one data set to another and are attributable to instabilities in the instrumentation and measurement conditions. The most stable activity measurements of alpha, beta-minus, electron capture, and beta-plus decaying sources set an upper limit of 0.0006% to 0.008% to the amplitude of annual oscillations in the decay rate. Oscillations in phase with Earth's orbital distance to the Sun could not be observed within a 10-6 to 10-5 range of precision. There are also no apparent modulations over periods of weeks or months. Consequently, there is no indication of a natural impediment against sub-permille accuracy in half-life determinations, renormalisation of activity to a distant reference date, application of nuclear dating for archaeology, geo- and cosmochronology, nor in establishing the SI unit becquerel and seeking international equivalence of activity standards.

Calculation of the decision threshold in gamma-ray spectrometry using sum peaks.

In the presence of radon daughters, gamma rays from (88)Y with energies at 898.0keV or 1836.1keV appear on a high, continuous background or overlap with other peaks. Therefore a calculation of the decision threshold from the sum peak at 2734.1keV represents a useful alternative, because here the continuous background is low. The decision threshold calculated from this peak can attain a value being comparable to the decision threshold calculated from the gamma-ray peak at 898.0keV.

Measurement function for the activities of multi-gamma-ray emitters in gamma-ray spectrometric measurements.

In gamma-ray spectrometric measurements the background is usually taken into account by conducting measurements on a blank sample. However, this method is not appropriate if many different kinds of samples are measured and several types of detectors are used. In such cases it is easier to measure accurately the activities of the blank materials separately and then to subtract the activities of the blank sample from the total sample activities. In order to do this, the measurement function for the activity calculation must be modified.

Reliability of the peak-analysis results in gamma-ray spectrometry for high relative peak-area uncertainties.

When measurement results with values near the decision threshold are being considered, a relative uncertainty of 60% is expected. Since such measurement results can be reported, the performance of the peak-analysing software for gamma-ray spectra needs to be examined for peaks that have a large relative uncertainty. The investigation was performed on a series of spectra measured with a HPGe detector under identical counting conditions. It was found that under a limit value of the relative peak area uncertainty the peak-analysis results are reliable with respect to both the peak location and the peak area evaluation. At relative uncertainties exceeding this uncertainty, the probability of type-II errors increases and a systematic influence on the peak area occurs, which originates in fluctuations of the continuous background in the vicinity of the peak. For the counting conditions used in this investigation, the limit relative uncertainty is about 35%, and whereas a systematic influence can be taken into account by a correction factor, the frequency of the type-II errors can only be reduced at the expense of increasing the frequency of the type-I errors.

Calculation of the decision thresholds in gamma-ray spectrometry.

A method was developed for calculating the decision thresholds for gamma-ray spectrometric measurements. At the energies where gamma-ray emitters that are present in the nuclide library, but were not identified in the spectrum, radiate, peaks are supposed to appear. The peak areas are calculated by fitting, using the method of least squares, the spectral region of the supposed peaks with a continuous background and the spectrometer response function at the gamma-ray energies where the supposed peaks are positioned. The null measurement uncertainty of a gamma-ray emitter is obtained as the uncertainty of the weighted average of the activities calculated from the areas of the supposed peaks in a spectrum where the specified activity of the gamma-ray emitter is zero. For the calculation of the decision threshold the null measurement uncertainty is used. These decision thresholds overestimate the critical limits calculated with the Currie formula by about 10% in the case of single gamma-ray emitters. For multi-gamma-ray emitters the decision thresholds yield smaller values than the Currie formula. The presence of a peaked background or peaks that are near the supposed peaks increases the decision threshold considerably.

Determination of the shielding factors for gamma-ray spectrometers.

A method for determining the shielding factors for gamma-ray spectrometers is described. The shielding factors are expressed by decomposing the peaked background of the spectrometer into contributions of the detector, spectrometer shield and ambient radiation to the spectrometer background. The dimensions of the sample and its mass-attenuation coefficient are taken into account using a simple model. For six spectrometers, with contributions to the background quantified, the shielding factors were determined for the background based on the thorium decay series and the radon daughters. For a water sample with a diameter of 9 cm and a thickness of 4 cm and the nuclides of the thorium decay series that are in the spectrometer shields, the values of the shielding factors lie in the interval 0.95-1.00. For a spectrometer exhibiting the diffusion of radon into the shielding material, the values of the shielding factors for the same sample for gamma-rays from the radon daughters lie in the interval 0.88-1.00.

Probability of Type-I errors in the peak analyses of gamma-ray spectra.

The probability density of Type-I errors in the peak-locating step of the spectra-analyzing procedure was empirically determined at a low value of the sensitivity parameter for peak recognition as a function of the height of the continuous spectral background and peak width. On the basis of this probability density, the number of Type-I errors for any spectral shape and FWHM calibration can be estimated. A criterion that is based on the relative peak-area uncertainty and the relative difference between the peak width obtained from the FWHM calibration and the width reported by the peak-analyzing program, as a measure of the believability that a located peak presents a Type-I error, is established. In addition, we demonstrate that the peaks having the largest believability are most likely to originate from Type-I errors. Using this criterion, the number of peaks, not identified in the identification step of the spectra analyzing procedure, to be checked for their origin can be reduced in accordance with the number of estimated Type-I errors.

Evaluation of gamma-ray spectrometric results near the decision threshold.

A computerized procedure for analyzing high-resolution gamma-ray spectra was improved in three regards: the peak areas having large relative uncertainties were corrected for the possible contribution of statistical fluctuations in the continuous background, the peaks having a negative net peak area after background subtraction were included in the activity calculations and the primary measurement results were converted to the best estimate using an application of the Bayes theorem. It was proven empirically that the improvements that were introduced diminish the probability and severity of type-I errors and that they improve the consistency and accuracy of the measurement results near the decision threshold.

Uranium-induced background of germanium gamma-ray spectrometers.

The uranium-induced background in eight high-resolution gamma-ray spectrometers was analyzed in order to identify the locations of its sources. On the basis of the energy dependence of the peak count rates, normalized to units of emission probability and detection probability, the contributions of the molecular sieve and the beryllium window were extracted. Based on the uncertainties achieved the contribution of the detector shields could not be observed. However, some correlations between the construction of the detectors and the values of the parameters describing the energy dependence were observed and the uranium activity in the beryllium windows was assessed.