## 3 Materials and Methods
### Gamma Spectrometric Analysis  
The collected samples were crushed into a fine powder using a jaw crusher then sieved by a 200 \(\upmu\)m mesh screen. All samples were dried at a temperature of 110 \({}^{\circ}\)C. Each sample had been weighted and transferred into an airtight cylindrical plastic container (47.6 mm radius, 82 mm height, and 0.5 mm thickness). Finally, the samples were saved for four weeks towards a secular equilibrium action between parents and their short-lived progenies in natural decay chains.  
The gamma spectral analyses were performed using a coaxial Canberra HPGe detector (GR4020). The detector is characterized by a relative efficiency (40%) for 3\({}^{\prime\prime}\)\(\times\) 3\({}^{\prime\prime}\) NaI (Tl) crystal and energy resolution (2 keV) for the 1332 keV Cobalt-60 \(\upgamma\)-line. Moreover, it operates with a suitable lead shield (Model 747E, preventing more than 98% of the background noise). The radioisotopes concentration (terrestrial radioisotopes) for each sample were determined in Bq/kg using their counting spectrums, the latter were obtained and analyzed using Genie-2000 software.  
Before starting the measurement, the system was calibrated for energy and efficiency. The energy calibration was carried out by acquiring spectra from standards sources of known energies such as \({}^{60}\)Co (1.332 MeV and 1.172 MeV). For the efficiency calibration, ISOCS/Lab-SOCS Canberra's Geometry Composer software (as a part of Genie-2000 software and based on the Monte Carlo Simulation) was used instead of the standard source. This was done individually for each sample to insert the geometry dimensions optimally and improve the efficiency of the HPGe detector.  
The sample measuring time (counting spectrum) was approximately in the range between 8 to 24 h. The gamma-ray photopeaks corresponding to 1.4608 MeV (\({}^{40}\)K) were taken into account to compute \({}^{40}\)K activity in the samples. These gamma-ray photopeaks 0.6093, 0.1120, and 1.7645 MeV (\({}^{214}\)Bi) and 0.2952 and 0.3519 MeV (\({}^{214}\)Pb) were considered in reaching out the \({}^{238}\)U activity in the samples. \({}^{232}\)Th activity was reached through the gamma-ray photopeaks corresponding to 0.3383, 0.9112 and 0.9689 MeV (\({}^{228}\)Ac) and 0.5832 and 2.6145 MeV (\({}^{208}\)Tl) and 0.2386 MeV (\({}^{212}\)Pb).  
Figure 1: Sampling location map of EISibai-Abu EITiyur area (modified after [PERSON] et al. [17]).  
The activity concentration AC\({}_{E_{i}}\) for the radioactive daughter of the radioisotope of interest (\({}^{40}\)K,\({}^{238}\)U, and \({}^{232}\)Th) can be estimated from its corresponding energy peak \(E_{i}\) via the following equation:  
\[\text{AC}_{E_{i}}=\frac{N_{E_{i}}}{\gamma_{E_{i}}\times\varepsilon_{E_{i}} \times t\times M_{s}} \tag{1}\]  
where \(N_{Ei}\), \(\gamma_{Ei}\), and \(\varepsilon_{Ei}\) are the net peak count, the \(\gamma\)-decay transition probability, and the detector efficiency at energy \(E_{i}\), respectively, \(t\) is the sample measuring time and \(M_{s}\) is the sample mass in kg. Hence the specific activity AC\({}_{j}\) in Bq/kg of jth parent (\({}^{238}\)U, \({}^{232}\)Th, and \({}^{40}\)K) having a number n of detected daughters' photopeaks, is obtained by:  
\[\text{AC}_{j}=\frac{1}{n}\sum_{i}^{n}\text{AC}_{E_{i}} \tag{2}\]  
The AC\({}_{\text{U}}\), AC\({}_{\text{Th}}\), and AC\({}_{\text{K}}\) were used to express the specific activity concentrations of \({}^{238}\)U (\({}^{226}\)Ra), \({}^{232}\)Th, and \({}^{40}\)K radioisotopes, respectively. \({}^{226}\)Ra being the highest radiological significance in the disintegration chain of \({}^{238}\)U is considered an alternative for 238U [19; 20].