## 3 Results and Discussion
The thickness of polyepoxide-reinforced Bi\({}_{2}\)O\({}_{3}\) composites required to reduce the emitted photons by a factor of 50% is called the half-value thickness (\(\Delta_{0.5}\)). Figure 4 depicts the variation in the \(\Delta_{0.5}\) values versus E\({}_{\gamma}\). The \(\Delta_{0.5}\) values increase with rising E\({}_{\gamma}\) values due to the PE and CS interactions, as illustrated in the linear attenuation coefficient section. In the PE interaction interval (i.e., 15 keV \(<\) E\({}_{\gamma}\)\(<\) 122 keV), the \(\Delta_{0.5}\) rose from 0.20 to 4.01 cm (for Bi\({}_{2}\)O\({}_{3}\) 0%), from 0.07 to 1.94 cm (for Bi\({}_{2}\)O\({}_{3}\) 5%), and from 0.04 to 1.26 cm (for Bi\({}_{2}\)O\({}_{3}\) 10%). This rapid increase in \(\Delta_{0.5}\) values is attributed to the large decrease in the \(\mu\) values under the effect of the PE interaction. The results in Figure 4 also show that the \(\Delta_{0.5}\) values linearly increase when E\({}_{\gamma}\) values increase from 302 keV to 661 keV. For example, the increase in \(\Delta_{0.5}\) values from 5.54 to 7.66 cm (for Bi\({}_{2}\)O\({}_{3}\) 0%), from 4.71 to 7.17 (for Bi\({}_{2}\)O\({}_{3}\) 5%), and from 4.12 to 6.81 cm (for Bi\({}_{2}\)O\({}_{3}\) 10%) is accompanied by an increase in E\({}_{\gamma}\) from 302 keV to 661 keV. This moderate increase in the \(\Delta_{0.5}\) values is attributed to the CS cross-section, in which the \(\mu\) values of the fabricated composites decrease exponentially with rising E\({}_{\gamma}\) values. We calculated the relaxation length (\(\lambda\), cm), which is inversely proportional to \(\mu\) (\(\lambda\) = 1/\(\mu\)). The calculated values for \(\lambda\) show the same trend in variation with E\({}_{\gamma}\) as the \(\Delta_{0.5}\) values. In contrast, the \(\lambda\) values increased from 0.29 to 11.05 cm (for Bi\({}_{2}\)O\({}_{3}\) 0%), from 0.10 to 10.34 cm (for Bi\({}_{2}\)O\({}_{3}\) 5%), and from 0.06 to 9.83 cm (for Bi\({}_{2}\)O\({}_{3}\) 10%), by raising the E\({}_{\gamma}\) values from 15 to 661 keV. Figure 3: Influence of Bi\({}_{2}\)O\({}_{3}\) concentration on the LAC of the synthesized composite. The \(\Delta_{0.5}\) and \(\lambda\) values were strongly affected by the Bi\({}_{2}\)O\({}_{3}\) doping concentration, as illustrated in Figure 5. The obtained results show that the \(\Delta_{0.5}\) and \(\lambda\) decreased by approximately 75% with an increase in the Bi\({}_{2}\)O\({}_{3}\) doping ratio in the PE interval, whereas they decreased by approximately 16% in the CS interval when the Bi\({}_{2}\)O\({}_{3}\) concentration increased between 0 and 10 wt. %. For example, at an E\({}_{\gamma}\) value of 551 keV, the \(\Delta_{0.5}\) values decreased from 6.84 to 5.90 cm, whereas the \(\lambda\) values decreased from 9.86 to 8.51 cm as the Bi\({}_{2}\)O\({}_{3}\) concentration increased from 0 to 10 wt. %. This decrease in the \(\Delta_{0.5}\) and \(\lambda\) values with the addition of Bi\({}_{2}\)O\({}_{3}\) is attributed to the increase in the interaction cross-section with increasing \(Z_{\rm eff}\) of the fabricated composites. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Energy/keV**} & \multicolumn{8}{c}{\(\mu_{\rm m}\) **(cm\({}^{2}\)/g)**} \\ \cline{2-9}  & \multicolumn{2}{c}{**Bi\({}_{2}\)O\({}_{3}\) 0\%**} & \multicolumn{2}{c}{**Bi\({}_{2}\)O\({}_{3}\) 5\%**} & \multicolumn{2}{c}{**Bi\({}_{2}\)O\({}_{3}\) 10\%**} \\ \cline{2-9}  & **MCNP** & **Phy-X/PSD** & **EXP** & **MCNP** & **Phy-X/PSD** & **EXP** & **MCNP** & **Phy-X/PSD** & **EXP** \\ \hline
15 & 3.155 & 3.152 & 8.278 & 8.282 & 13.396 & 13.400 & \\ \hline
20 & 1.452 & 1.451 & 5.442 & 5.458 & 9.426 & 9.459 & \\ \hline
30 & 0.565 & 0.5  
The efficiency of the material can be also estimated through comparison of the number of transmitted photons (I\({}_{\rm t}\)) with the total number of emitted photons (I\({}_{\rm o}\)), where the transmission factor, TF, (%) = (I\({}_{\rm t}\)/I\({}_{\rm o}\)) \(\times\) 100.