## 3 Materials and Methods
### Gamma Spectrometry Measurements  
Specific activity measurements of the investigated samples were performed by using a high-purity germanium (HPGe) detector (Ortec, Oak Ridge, TN, USA) for gamma spectrometry analysis. In order to reduce statistical uncertainty, samples were counted for 70,000 seconds and spectra were analyzed in order to obtain the activity concentration of \({}^{226}\)Ra, \({}^{232}\)Th and \({}^{40}\)K. The \({}^{226}\)Ra-specific activity was calculated through a weighted average of \({}^{214}\)Pb and \({}^{214}\)Bi (in secular equilibrium) specific activities. The \({}^{232}\)Th activity concentration was determined by using the 911.21 and 968.97 keV \({}^{228}\)Ac \(\gamma\)-ray lines; for \({}^{40}\)K, the evaluation was performed from its \(\gamma\)-line at 1460.8 keV. The experimental setup was composed by a positive biased Ortec HPGe detector (GEM) (Full Width at Half Maximum, FWHM) of 1:85 keV, peak to Compton ratio of 64:1, relative efficiency of 40% at the 1:33 MeV \({}^{60}\)Co-line), placed inside lead wells to shield the background radiation environment. Efficiency and energy calibrations were performed using a multipeak Marinelli geometry gamma source (AK-5901) of 1 L capacity, covering the energy range, 59.54-1836 keV, customized to reproduce the exact geometries of samples in a water-equivalent epoxy resin matrix [15]. The Gamma Vision (Ortec) software was used for data acquisition and analysis [16]. The activity concentration (Bq kg\({}^{-1}\)) of the investigated radionuclides was calculated using the following formula [17]:  
\[C=\frac{N_{E}}{\varepsilon_{E}t\gamma_{d}M} \tag{1}\]  
where \(N_{E}\) indicates the net area of a peak at energy \(E\), \(\varepsilon_{E}\) and \(\gamma_{d}\) are the efficiency and yield of the photopeak at energy \(E\) respectively, \(M\) is the mass of the sample (kg) and \(t\) is the live time (s). The measurement uncertainty is a combined standard one at coverage factor k = 2, taking into account the following components: uncertainty counting statistics, uncertainty in nuclear data library, uncertainty due to calibration efficiency and uncertainty about the quantity of the sample [18]. The quality of the gamma spectrometry experimental results was certified by the Italian Accreditation Body (ACCREDIA) [19]. This implies the continued verification (with annual periodicity) of the maintenance of the gamma spectrometry method performance characteristics. In particular, with regard to the blanks, they are generally determined by acquiring an empty lead well spectrum or an ultrapure water sample in the sample holder of the desired geometry (for determination of natural-emitting gamma radionuclides) or, in the case of samples requiring a material of support (e.g., atmospheric particulate aspirated on filter), by acquiring the support of clean sampling, under the same geometric conditions as the sample. The measurement of the blanks is performed on a quarterly basis. Regarding the precision and accuracy assessment, we proceed as follows:  
For the precision, with the double test method, the repeatability of the method is verified over time. A certified reference material (also containing the radionuclides K-40, Ra-226 and Th-232) is analyzed twice, with the activity concentration of the radionuclide of interest defined as X\({}_{1}\) (first measurement) and X\({}_{2}\) (second measurement). The probability level p = 0.95 is considered. The following formula is applied:  
\[|x_{1}-x_{2}|\leq\sqrt{2}\cdot s_{r}\cdot t \tag{2}\]  
where \(t\) is the Student variable and \(s_{r}\) is the standard deviation of repeatability obtained in the validation phase. For the accuracy, it is evaluated by comparing the reference (certified sample) and the measured values of the radionuclide of interest, taking into account the uncertainty.