## 2 Simulations and Theoretical Aspect
### Radiation Shielding Parameter Calculation  
The simulated track length was used to calculate the linear attenuation coefficient (LAC) of the studied alloys in the mentioned energy range. The mass attenuation coefficient (MAC) is calculated based on the simulated average track length and LAC, as described in Equation (1). The MAC is used to describe the capacity of a given material to attenuate incident radiation. Aside from the evaluation of the MAC using the Monte Carlo simulation, it was calculated theoretically using XCOM software.  
Figure 1: The geometry used for the gamma-ray shielding simulation.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Alloy**} & \multicolumn{10}{c}{**Fractional Abundance (wt.\%)**} & \multicolumn{1}{c}{**Density**} \\ \cline{2-13}  & **C** & **Cr** & **Co** & **Mo** & **W** & **Ta** & **Al** & **Ti** & **Hf** & **Zr** & **B** & **Ni** & **Fe** & **Nb** & **(g-cm-3)** \\ \hline MAR-247 & 0.02 & 8.50 & 10.00 & 0.65 & 10.00 & 3.00 & 5.60 & 1.00 & 1.40 & 0.04 & 0.02 & 59.63 & 0.00 & 0.00 & 8.53 \\ \hline MAR-302 & 0.85 & 21.50 & 56.94 & 0.00 & 10.00 & 9.00 & 0.00 & 0.20 & 0.00 & 0.00 & 0.01 & 0.00 & 1.50 & 0.00 & 9.21 \\ \hline Inconel-625 & 0.00 & 21.50 & 0.00 & 9.00 & 0.00 & 0.00 & 0.20 & 0.20 & 0.00 & 0.00 & 0.00 & 61.00 & 2.00 & 3.60 & 8.44 \\ \hline Inconel-718 & 0.00 & 19.00 & 0.00 & 3.00 & 0.00 & 0.00 & 0.50 & 0.90 & 0.00 & 0.00 & 0.00 & 52.50 & 18.50 & 5.10 & 8.22 \\ \hline Nimocast-75 & 0.12 & 20.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 79.88 & 0.00 & 0.00 & 8.44 \\ \hline WI-52 & 0.85 & 21.50 & 63.65 & 0.00 & 11.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 1.00 & 2.00 & 0.00 & 8.88 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The chemical composition of the selected alloys in (wt.%).  
The MAC for any material consists of the ith constituent element and can be calculated according to Equation (2) [1].  
\[\left(MAC\right)_{alloy}\ =\ \frac{LAC}{\rho\ (g\ \mathrm{cm^{-3}})} \tag{1}\]  
\[\left(MAC\right)_{alloy}\ =\ \sum_{i}\omega_{i}\star\left(MAC\right)_{i} \tag{2}\]  
where \(\left(MAC\right)_{alloy}\), \(\omega_{i}\), \(\rho\), and \(\left(MAC\right)_{i}\) refer to the MAC of the studied alloys, the weight fraction, and MAC for the ith constituent element, respectively.  
The half value layer (HVL) is calculated based on the results of the linear attenuation coefficient (LAC, or \(\upmu\)) as presented in Equation (3). The HVL describes the thickness of the material required to diminish the incident intensity to a half of its initial value.  
\[HVL=\frac{\ln(2)}{\mu\ (\mathrm{cm^{-1}})} \tag{3}\]  
The MFP refers to the distance that the radiation travels in the studied shielding materials between two subsequent collisions and can be calculated according to Equation (4).  
\[MFP=\frac{1}{\mu\ (\mathrm{cm^{-1}})} \tag{4}\]  
Furthermore, other shielding parameters such as \(Z_{\mathrm{eff}}\), equivalent atomic number (\(Z_{\mathrm{eq}}\)), geometric progression fitting parameters exposure buildup factor (EBF), and energy absorption buildup factor (EABF) were calculated using a newly developed software, Phy-X/PSD, in the energy range of 0.015-15 MeV [30]. The SRIM program was used to calculate the stopping power and projected range of the protons and alpha particles [31].