## 2 Material and Methods
### Gamma Attenuation Parameters  
In this study, according to the formula Bi\({}_{1,64-x}\)Pb\({}_{0.36}\)Ag\({}_{x}\)Sr\({}_{2}\)Ca\({}_{2}\)Cu\({}_{3}\)O\({}_{4+2n}\) (x = 0.00, 0.04, 0.08, 0.12, 0.16), n = 3 was generalized and silver-doped superconducting ceramic samples were selected [25]. The samples were named BiPbSrCaCuO (x = 0.00), BiPbAgSrCaCuO-1  
Figure 1: Detector model for Monte Carlo (ESG4 code) calculations. (x = 0.04), BiPbAgSrCaCuO-2 (x = 0.08), BiPbAgSrCaCuO-3 (x = 0.12), and BiPbAgSrCaCuO-4 (x = 0.16), respectively. Mass attenuation coefficient (MAC) parameters were determined as prescribed in the following:  
The attenuation of gamma rays in a material can be described using the Beer-Lambert law, which states:  
\[I=I_{0}e^{-\mu x} \tag{1}\]  
where \(I\) is the intensity of the gamma-ray beam after it has passed through a thickness x of the material, \(I_{0}\) is the initial intensity of the beam, and \(\mu\) (cm\({}^{-1}\)) is the linear attenuation coefficient of the material for the specific energy of the gamma rays being considered. While the linear attenuation coefficient (\(\upmu\)) is commonly used to describe how strongly a material absorbs gamma rays, it has the disadvantage of depending on the density of the material. Therefore, it is often more useful to use the density-independent mass attenuation coefficient (\(\upmu/\rho\)), which is expressed in units of cm\({}^{2}\)/g. The density-independent mass attenuation coefficient can be calculated using the following formula \(\mu/\rho\) (cm\({}^{2}\)/g):  
\[I=I_{0}e^{-(\mu/\rho)\rho x}=I_{0}e^{-(\mu/\rho)d} \tag{2}\]  
According to the Beer-Lambert law equation, the thickness \"d\" is expressed in unit area mass of the material (g/cm\({}^{2}\)). When a superconductor sample contains more than one metal, the mass attenuation coefficient (\(\upmu/\rho\)) of the sample can be calculated using the mixture rule formula, which is given as:  
\[\mu/\rho=\sum_{i}w_{i}(\mu/\rho)_{i} \tag{3}\]  
where w\({}_{i}\) is the weight fraction of the ith component in the sample and (\(\upmu/\rho\))\({}_{i}\) is the density-independent mass attenuation coefficient of the ith component [26]. The weight fraction (w\({}_{i}\)) of a chemical component i in a mixture can be calculated using the following relation:  
\[w_{i}=\frac{a_{i}A_{i}}{\sum a_{i}A_{i}} \tag{4}\]  
where \(a_{i}\) is the number of formula units, \(A_{i}\) is the atomic weight of the ith element, and the summation is taken over all components in the mixture. Mass attenuation coefficients (\(\upmu/\rho\)) of the samples under study are computed by MC-EGS4 simulation code and compared to WinXCOM results in the energy range of 0.0595-1.332 MeV. The theoretical mass attenuation coefficients (\(\upmu/\rho\)) of the superconductor samples were determined using WinXCom software version 3.1 [6; 8; 23; 27]. The effective atomic number (Zeff) is a parameter that is often used to quantify the shielding efficiency of a material against radiation. Zeff is a weighted average of the atomic numbers of the constituent atoms in a material, taking into account both the number and energy of the incident photons or particles. The concept of Zeff is particularly useful in radiation shielding applications because it provides a convenient way to compare the radiation attenuation properties of different materials. The Zeff of the samples under investigation have been computed using the formula [28]:  
\[Z_{eff}=\frac{\sum f_{i}A_{i}\left(\frac{\mu}{\rho}\right)_{i}}{\sum\frac{f_{ i}A_{i}}{Z_{i}}\left(\frac{\mu}{\rho}\right)_{i}} \tag{5}\]  
where \(f_{i}\) is the weight fraction of the i-th element in the material, \(A_{i}\) is the atomic weight of the i-th element, and \(Z_{i}\) is the atomic number of the i-th element [29].