## 2 Materials and Methods  
The LaBr\({}_{3}\):Ce single cubic-shaped crystal examined in this study was supplied encapsulated in a thin aluminum protective layer from Advatech UK Limited, with dimensions of 10 \(\times\) 10 \(\times\) 10 mm\({}^{3}\) and with all its surfaces polished [18]. The part of the crystal used as the output was covered with a fused-silica-glass window [31]. The crystal was evaluated by determining, experimentally or with the use of mathematical formulas [22; 25; 26; 32; 33; 34], the following parameters: the energy-absorption efficiency (EAE), the quantum detection efficiency (QDE), the AE, the SMF, the effective efficiency (EE) and the DQE(0). The crystal was irradiated using an X-ray tube with 1.5 mm Al filter, coupled with a CPI, series CMP 200 DR 50 kW generator. The X-ray tube-voltages ranged from 50 to 150 kVp, keeping the current-time product constant at 63 mAs, with an irradiation time of 1s. An additional 20 mm Al filtration was added to the exit of the X-ray tube [25; 26; 32; 33]. QDE is defined as the fraction of the incident X-rays interacting with the scintillator [34], whereas the fraction of the incident X-ray energy absorbed in the crystal is described through EAE [34]. QDE and EAE can be calculated as [26; 34]:  
\[\mathrm{EAE(E)}=\ \frac{\int_{0}^{\mathrm{E_{0}}}\Phi_{0}\ (\mathrm{E}) \mathrm{E}\Big{(}\frac{\mathrm{\mu_{em}}\ (\mathrm{E})/\mathrm{\rho}}{\mathrm{\mu_{att}(E)/\mathrm{\rho}}}\Big{)} \Big{(}1-\mathrm{e}^{-(\mathrm{\mu_{att}}\ (\mathrm{E})/\mathrm{\rho})\mathrm{\rho}T}\Big{)} \mathrm{dE}}{\int_{0}^{\mathrm{E_{0}}}\Phi_{0}\ (\mathrm{E})\mathrm{EdE}} \tag{1}\]  
and  
\[\mathrm{QDE}=\ \frac{\int_{0}^{\mathrm{E_{0}}}\Phi_{0}\ (\mathrm{E}) \Big{(}1-\mathrm{e}^{-(\mathrm{\mu_{att}}\ (\mathrm{E})/\mathrm{\rho})\mathrm{\rho}T}\Big{)} \mathrm{dE}}{\int_{0}^{\mathrm{E_{0}}}\Phi_{0}\ (\mathrm{E})\mathrm{dE}} \tag{2}\]  
where \(\Phi_{0}\)(E) is the incident X-ray photon fluence on the scintillator, E is the photon energy, \(\mathrm{\mu_{att}(E)/\mathrm{\rho}}\) is the radiation-photon total mass-attenuation-coefficient, and \(\mathrm{\mu_{en}(E)/\mathrm{\rho}}\) is the corresponding total mass-energy-absorption-coefficient. T is the detector thickness and \(\mathrm{\rho}\) is its density expressed in g/cm\({}^{3}\)[26; 34]. The \(\mathrm{\mu_{att}(E)/\mathrm{\rho}}\) and \(\mathrm{\mu_{en}(E)/\mathrm{\rho}}\) were taken from XMudat software [35] and the X-ray fluence from TASMIP Spectra Calculator [36]. AE is defined as the ratio of the energy flux, \(\mathrm{\Psi_{\lambda}}\) (units \(\mathrm{\mu W\cdot m^{-2}}\)), of the light photons emitted by a stimulated crystal, over the incident exposure-rate, \(\mathrm{\dot{X}}\) (units mR\(\cdot\)s\({}^{-1}\)) [37]. The experimental setup for the energy-flux measurement comprised a light integrating-sphere (Oriel 70451) coupled to a photomultiplier tube (PMT) (EMI 9798B,) which was connected to a Cary 400 vibrating-reed electrometer. The exposure rate was measured separately, with an RTI Piranha P100B dosimeter [38]. The experimental setup is shown in Figure 1.  
According to its definition, AE is equal to [25; 33]:  
\[\mathrm{AE}=\mathrm{\dot{\Psi}_{\lambda}}/\ \mathrm{\dot{X}} \tag{3}\]  
Figure 1: Experimental measurement setup. The units of AE are known as efficiency units: (EU) where 1 EU = 1 uW\(\cdot\cdot\)m\({}^{-2}\)/(mR\(\cdot\)s\({}^{-1}\)). Crystal scintillators are always used in combination with optical photon sensors. Their compatibility is expressed by the spectral-matching factor, SMF, and it can be defined as [26]:  
\[\text{SMF}=\frac{\int\ S_{\text{p}}\ (\lambda)S_{\text{D}}(\lambda)\text{d} \lambda}{\int\ S_{\text{p}}\ (\lambda)\text{d}\lambda} \tag{4}\]  
where S\({}_{\text{p}}\)(\(\lambda\)) is the light-spectrum emitted by the scintillator, obtained manually from the website of the vendor [18], S\({}_{\text{D}}\)(\(\lambda\)) is the spectral sensitivity of the photodetector coupled to the scintillator, and \(\lambda\) is the wavelength of the light emitted [39].