## 4 Discussion  
Figure 7 demonstrates that there is a sublinear dependence of the scintillation yield on the accelerator current. To account for this effect, a model of saturation of the yield of scintillation in the GYAGG:Tb (Ce) crystal was developed. The saturation of the yield can have different origins: it can be due to the depletion of available radiative centers in the ground state or to the onset of the interaction of excited centers effectively limiting their output. Further, it will be shown that the former effect is not applicable to our results, but the latter one provides a reasonable agreement with experimental data. The model is applied to the results obtained with the Gd\({}_{1.120}\)Y\({}_{1.730}\)Tb\({}_{0.15}\)Al\({}_{2}\)Ga\({}_{3}\)O\({}_{12}\) and Gd\({}_{1.175}\)Y\({}_{1.777}\)Ce\({}_{0.03}\)Al\({}_{2}\)Ga\({}_{3}\)O\({}_{12}\) samples. The saturation of the scintillation light yield in the crystal might be described in terms of kinetic equations. As a first step, the average concentration of electronic excitations in the region excited in the sample by the electron beam is to be estimated. The average length of the track created by the incident 150 keV electron in the GYAGG crystal was estimated using the GEANT4 package [31] to be 98 \(\upmu\)m. The energy of about two values of the energy gap (i.e., \(\sim\)12 eV) is required to create one electron-hole pair. Thus, the incident 150 keV electron can create up to 12,500 electronic excitations. The mean-free path of the secondary electrons can vary from 10 to 100 nm, thus the volume of one track can be about 3 \(\times\) 10\({}^{-7}\) to 3 \(\times\) 10\({}^{-5}\) cm\({}^{3}\). The number of the 150 keV electrons for the maximal electron-beam current is equal to \(N_{e}^{i}=\frac{1.4.20\ \mu s}{1.602\cdot 10^{-19}\ \mathrm{C}}=1.24\times 10^{11}\). For the electron-beam spot size on the sample of 1 cm\({}^{2}\) and penetration depth of about 70 \(\upmu\)m (less than the track length since the track is not straight), the volume of the excited region of the sample is 0.007 cm\({}^{3}\). According to this estimate, the tracks are overlapping for the whole range of the available electron-beam current. For the current of 1 A, the number of electron-hole pairs created by the accelerator pulse is equal to 1.55 \(\times\) 10\({}^{15}\), and their initial concentration for overlapping tracks is \(n_{ex}=2\times 10^{17}\) cm\({}^{-3}\). The concentration of activator centers (Tb or Ce) in the GYAGG crystals is estimated to be \(n_{Tb}=6.7\cdot 10^{20}\) CM\({}^{-3}\) (for concentration equal to 0.15 Tb ions per formula unit, 5% sites are substituted by Tb) and \(n_{Ce}=1.34\times 10^{20}\) CM\({}^{-3}\) (for concentration equal to 0.03 Ce ions per formula unit, 1% sites are substituted by Ce). Thus, the concentration of activators (\(n_{Tb}\) or \(n_{Ce}\)) is substantially larger than the concentration of excitations \(n_{ex}\) produced by one electron-beam pulse. Therefore, we can neglect the saturation effects due to the exhausting of nonexcited activators. The decay time of the lowest radiative state of Tb ion in GYAGG crystal is known from the experimental results above. In addition, Tb can nonradiatively relax from the upper \({}^{5}D_{3}\) to the lower \({}^{5}D_{4}\) state with a decay time of 150 \(\upmu\)s. The decay time of an excited Ce ion in GYAGG crystal was set to be 50 ns [32]. The separation between the pulses of the beam of 0.1 s is substantially larger than the decay time of the Tb (Ce) activator, thus all excited activator ions relax between two pulses.