## 3 Results and Discussion
The mean free path (MFP) of the investigated alloys was calculated based on the simulated MAC values for a gamma-ray energy range of 0.015-15 MeV. The variation of the MFP versus the gamma photon energies for all selected alloys is shown in Figure 4. The variation of the MFP versus the gamma photon energies is similar to the variation of HVL discussed previously. The lowest MFP was obtained for the MAR-247 alloy and varied between 0.001 and 3.361 cm, while the highest MFP was achieved for the alloy encoded as Inconel-625 and varied between 0.002 and 3.936 cm. Moreover, the MFP of the studied alloys were compared with other common radiation shielding materials in Figure 4: ordinary concrete, commercial glass RS-253 G18, and lead at some fixed energies (0.5, 1, 5, and 15 MeV). The MFP of the studied alloys was lower than the MFP of ordinary concrete and commercial glass RS-253 G18. In contrast, the MFP of the studied alloys was higher than the MFP of Pb, with the MFP for lead varying between 0.0546 and 1.557 cm for gamma photon energies between 0.5 and 15 MeV, respectively. Figure 4: The mean free path (MFP) of the studied alloys versus incident gamma-ray energy. Figure 3: Variation of the half-value layer (HVL) of the studied alloys against the incident gamma-ray energy. The effective atomic number (\(Z_{\rm eff}\)) is used to describe a multi-element material in terms of its elements. The calculation of \(Z_{\rm eff}\) is based on the calculated MAC, electronic cross section, and atomic cross section. The variation of the \(Z_{\rm eff}\) versus the \(\gamma\)-ray energies is presented in Figure 5a for the investigated alloys. The \(Z_{\rm eff}\) trend can be explained through photon interaction phenomena. Values reach their maximum in the PE interaction region because of the dependence of the photoelectric cross-section with \(Z^{4-5}\)[34]. In the PE interaction region, two sudden increases appeared at around 0.02 and 0.07 MeV. The first increase appeared at around 0.02 MeV for the Inconel-625 and Inconel-718 alloys. This increase was due to the K-absorption edges (X-ray) of Nb and Mo, which appeared at 0.019 and 0.02 MeV, respectively. The second increase was detected at around 0.07 MeV for MAR-247, MAR-302, and WI-52 alloys. This second increase was due to the W K-edges, which appeared at 0.069 MeV. Then, the \(Z_{\rm eff}\) values sharply decreased with the increase in \(\gamma\)-ray energy because of the PE cross-section, which was proportional to E\({}^{-3.5}\). In the region where CS was dominant, \(Z_{\rm eff}\) demonstrated a moderate reduction with an increase of \(\gamma\)-ray energy, since the CS cross-section varied with Z. Above several MeV, the \(Z_{\rm eff}\) reversed its downward trend and began to increase with an increase in photon energy due to the PP cross-section, which varied with \(Z^{2}\). The chemical composition of a material plays an important role in its \(Z_{\rm eff}\) value and the value of other shielding parameters. The MAR-302 alloy, with tungsten (W), tantalum (Ta), and hafnium (Hf) contents, possessed the highest \(Z_{\rm eff}\), while Nimocast-75 had the lowest, which mainly consisted of nickel (Ni) and chromium (Cr). The equivalent atomic number \(Z_{\rm eq}\) is the first step to calculate the buildup factors of the studied materials and describes the material in term of its equivalent elements. Unlike the \(Z_{\rm eff}\), the \(Z_{\rm eq}\) of the selected alloys is calculated based on the ratio between the Compton scattering mass attenuation coefficient and the total mass attenuation coefficient of the sample at energies of 0.015-15 MeV. Figure 5b demonstrates that the \(Z_{\rm eq}\) tends to decrease in the energy region where the PE is prevalent. As energy keeps increasing, the \(Z_{\rm eq}\) values have an increasing trend with the increase of the CS interaction  
Figure 5: Variation of the atomic number of the investigated alloys versus incident gamma-ray energy: (**a**) effective atomic number (\(Z_{\rm eff}\)), (**b**) the equivalent atomic number (\(Z_{\rm eq}\)).