## 2 Materials and Methods
The spectral sensitivities of various photodetectors were obtained from the manufacturers' data and the literature [33; 40; 41; 42; 43]. The overall efficiency corresponding to a specific scintillator-photodetector combination has been expressed by the effective efficiency (EE) [44], and is calculated as [33; 45]:  
\[\text{EE}=\text{AE}\cdot\text{SMF} \tag{5}\]  
The detective quantum efficiency expresses the efficiency of a system in transmitting the information it receives, and therefore can be used to evaluate and compare medical-imaging systems. DQE is defined as [22]:  
\[\text{DQE}=\frac{\text{SNR}_{\text{O}}^{2}}{\text{SNR}_{\text{I}}^{2}} \tag{6}\]  
where \(\text{SNR}_{\text{O}}\) and \(\text{SNR}_{\text{I}}\) refer to the system output and input, respectively. The detective quantum efficiency is a function of the scintillator's thickness (T) and spatial frequency (\(\mathbf{u}\)). Relation (6) transforms into the relation [22]:  
\[\text{DQE}(\mathbf{u},\text{E},\text{T})=\ \frac{\overset{-}{\mathbf{\Phi_{L}} (\mathbf{Q},0,\mathbf{T})\mathbf{M}_{\text{p}}^{2}}\ (\mathbf{u},\ \mathbf{T})}{\overset{-}{\mathbf{Q}} \left[\overset{-}{\mathbf{m_{0}}}\ \overline{\mathbf{g_{L}}}\ (\mathbf{T})\mathbf{M}_{\text{p}}^{2}\ (\mathbf{u},\ \mathbf{T})+1 \right]} \tag{7}\]  
where \(\overset{-}{\mathbf{\Phi_{L}}}\) is the mean photon fluence of the emitted light at zero spatial-frequency, and \(\overline{\mathbf{Q}}\) is the fluence of incident X-ray photons on the scintillator. \(\mathbf{M_{\text{p}}}\) (\(\mathbf{u}\),T) corresponds to the modulation transfer function of the scintillator, \(\overset{-}{\mathbf{m_{0}}}\) is the mean number of generated optical photons in the scintillator per absorbed X-ray quantum, \(\overline{\mathbf{g_{L}}}\) is the mean light-transmission-efficiency representing the probability that a photon exits from the crystal volume, and thus the product, \(\overset{-}{\mathbf{m_{0}}}\overline{\mathbf{g_{L}}}\), shows the number of the light photons escaping per absorbed X-ray photon. The detective quantum efficiency can also be defined for zero spatial-frequency DQE(0). In this case, M\({}_{\text{p}}\)(0,T) equals 1 and Relation (7) transforms into the following relation:  
\[\text{DQE}(0,\text{E},\text{T})=\ \frac{\overset{-}{\mathbf{\Phi_{L}} (\mathbf{Q},\mathbf{E},\mathbf{T})}}{\overset{-}{\mathbf{Q}}\cdot\overset{-}{ \mathbf{m_{0}}}\ \overline{\mathbf{g_{L}}}+\mathbf{Q}} \tag{8}\]  
The product, \(\overset{-}{\mathbf{m_{0}}}\overline{\mathbf{g_{L}}}\), can be calculated from the ratio, \(\overset{-}{\mathbf{\Phi_{L}}}\) to the absorbed fraction of the incident X-ray photon fluence. The number of the absorbed X-ray photons can be calculated from the product QDE\(\cdot\)Q:  
\[\mathrm{QDE}=\ \frac{\int_{0}^{\mathrm{E_{0}}}\Phi_{0}\ (\mathrm{E}) \Big{(}1-\mathrm{e}^{-(\mathrm{\mu_{att}}\ (\mathrm{E})/\mathrm{\rho})\mathrm{\rho}T}\Big{)} \mathrm{dE}}{\int_{0}^{\mathrm{E_{0}}}\Phi_{0}\ (\mathrm{E})\mathrm{dE}} \tag{2}\]0  
Figure 3 exhibits the graph of \(\mu_{\rm att}\) in terms of energy, as obtained from the XmuDAT software, featuring the characteristic saw-tooth shape in the energy range where absorption through the photoelectric effect dominates (up to approximately 100 keV). The discontinuities correspond to the absorption edges K, L, M or N, in which the energies of the photons become equal to the binding energies of the electrons of the atoms of the crystals in the shells K, L, M or N, and therefore the probability of absorption of the photons increases sharply, due to resonance [48]. \[\mathrm{QDE}=\ \frac{\int_{0}^{\mathrm{E_{0}}}\Phi_{0}\ (\mathrm{E}) \Big{(}1-\mathrm{e}^{-(\mathrm{\mu_{att}}\ (\mathrm{E})/\mathrm{\rho})\mathrm{\rho}T}\Big{)} \mathrm{dE}}{\int_{0}^{\mathrm{E_{0}}}\Phi_{0}\ (\mathrm{E})\mathrm{dE}} \tag{2}\]1 Attenuation coefficients of the LaBr\({}_{3}\)Ce, LaCl\({}_{3}\):Ce, BGO, LSO and CdWO\({}_{4}\) single crystals.