## 2 Materials and Methods
\[\mathrm{QDE}=\ \frac{\int_{0}^{\mathrm{E_{0}}}\Phi_{0}\ (\mathrm{E}) \Big{(}1-\mathrm{e}^{-(\mathrm{\mu_{att}}\ (\mathrm{E})/\mathrm{\rho})\mathrm{\rho}T}\Big{)} \mathrm{dE}}{\int_{0}^{\mathrm{E_{0}}}\Phi_{0}\ (\mathrm{E})\mathrm{dE}} \tag{2}\]2 EAE of LaBr\({}_{3}\):Ce, LaCl\({}_{3}\):Ce, BGO, LSO and CdWO\({}_{4}\) single crystals. While \(\mu_{\rm att}\) expresses the abovementioned probability, \(\mu_{\rm{\it{cn}}}\) expresses the probability of transfer and absorption of this energy within the crystal. Their calculated ratio with respect to energy is represented graphically in \[\mathrm{QDE}=\ \frac{\int_{0}^{\mathrm{E_{0}}}\Phi_{0}\ (\mathrm{E}) \Big{(}1-\mathrm{e}^{-(\mathrm{\mu_{att}}\ (\mathrm{E})/\mathrm{\rho})\mathrm{\rho}T}\Big{)} \mathrm{dE}}{\int_{0}^{\mathrm{E_{0}}}\Phi_{0}\ (\mathrm{E})\mathrm{dE}} \tag{2}\]3 A sharp drop of this ratio at the absorption-edge positions of the crystals is observed, resulting in the corresponding reduction of EAE in the graph of \[\mathrm{QDE}=\ \frac{\int_{0}^{\mathrm{E_{0}}}\Phi_{0}\ (\mathrm{E}) \Big{(}1-\mathrm{e}^{-(\mathrm{\mu_{att}}\ (\mathrm{E})/\mathrm{\rho})\mathrm{\rho}T}\Big{)} \mathrm{dE}}{\int_{0}^{\mathrm{E_{0}}}\Phi_{0}\ (\mathrm{E})\mathrm{dE}} \tag{2}\]4 The crystals of LaCl\({}_{3}\):Ce and LaBr\({}_{3}\):Ce show a common K edge at 39 keV, due to La, while LaBr\({}_{3}\):Ce shows a second one, due to bromide at 14 keV, which, however, does not affect the EAE in the range of energies we are studying. The EAE of both crystals decreases after 40 kVp, and in fact the decrease in LaCl\({}_{3}\):Ce is greater, due to the greater drop in the \(\mu_{\rm{\it{cn}}}/\mu_{\rm att}\) ratio, as shown in the graph. After 60 kVp, the EAE increases slightly, and stabilizes, due to the gradual increase in the \(\mu_{\rm{\it{cn}}}/\mu_{\rm att}\) ratio after 39 keV. Similarly, in BGO, LSO and CdWO\({}_{4}\), with K-edges at 91 keV, 64 keV and 70 keV, respectively, the EAE decreases after 91 kVp, 64 kVp and 70 kVp, respectively. After 110 kVp for LSO and CdWO\({}_{4}\) and 130 kVp for BGO, the EAE almost stabilizes, due to the partial recovery of the \(\mu_{\rm{\it{cn}}}/\mu_{\rm att}\) ratio values. Figure 5 shows the effect of tube voltage on QDE. LaBr\({}_{3}\):Ce demonstrates almost perfect efficiency in detecting the incident photons, as shown by the QDE values, which range from 0.998 to 1. The QDE is maximum at X-ray tube voltages up to 120 KVp. This is mainly due to the thickness of the crystal, since the attenuation coefficients drop after the K-edge of LaBr\({}_{3}\):Ce, above 39 keV. For this energy range, the radiation absorption is maximum, and all the incident radiation may contribute to signal generation. For higher energies, however, the continuous drop of the attenuation coefficient leads to a smaller number of X photons absorbed in the crystal. Figure 6 shows the variation of AE of the LaBr\({}_{3}\):Ce crystal in the X-ray tube-voltage range from 50 to 150 kVp, compared with previously published data for the LaCl\({}_{3}\):Ce, BGO, LSO, and CdWO\({}_{4}\) crystals of similar dimensions [25; 26; 27]. The experimental error is 4.25%. As can be seen, the AE of all crystals increases continuously with increasing tube voltage; however, the LaBr\({}_{3}\):Ce crystal's values are the highest at all energies. Specifically, at 50 kVp, the values were 23.5 EU for LaBr\({}_{3}\):Ce, 15.1 EU for LaCl\({}_{3}\):Ce, 1.2 EU for BGO, 7.6 EU for LSO, and 12 EU for CdWO\({}_{4}\). At 130 kVp, the difference was even greater, measuring 60.1 EU, 38.7 EU, 3.7 EU, 17.7 EU and 26.9 EU, respectively [25; 26; 27]. The very good absolute-efficiency values of LaBr\({}_{3}\)Ce, compared to the other crystals reported for comparison in this paper (even though it has lower EAE values, as shown in Figure 2), can be explained by its higher LY [27].