## 4 Discussion
The kinetic equations for Tb-doped GYAGG crystal can be written as follows:  
\[\frac{dn_{Tb^{5}D_{4}}}{dt}=-\frac{1}{\tau_{Tb^{5}D_{4}}}n_{Tb^{5}D_{4}}+ \frac{1}{\tau_{Tb^{5}D_{3}\to Tb^{5}D_{4}}}n_{Tb^{5}D_{3}}-\beta_{Tb^{ *}+Tb^{*}}n_{Tb^{5}D_{4}}n_{Tb^{*}}, \tag{1}\]  
\[\frac{dn_{Tb^{5}D_{3}}}{dt}=-\frac{1}{\tau_{Tb^{5}D_{3}}}n_{Tb^{5}D_{3}}-\frac {1}{\tau_{Tb^{5}D_{3}\to Tb^{5}D_{4}}}n_{Tb^{5}D_{3}}-\beta_{Tb^{*}+Tb^{*}}n _{Tb^{5}D_{3}}n_{Tb^{*}}, \tag{2}\]  
\[n_{Tb^{*}}=n_{Tb^{5}D_{4}}+n_{Tb^{5}D_{3}} \tag{3}\]  
Here, the radiative constants \(\tau_{Tb^{5}D_{4}}=3.0\ ms\). The radiative constant of \({}^{5}D_{3}\) is not known, so we chose for it to be equal to that of \({}^{5}D_{4}\): \(\tau_{Tb^{5}D_{3}}=3.0\ ms\). The effect of the exact value of this radiative constant is not critical for the solution of this set of equations since the characteristic time of nonradiative transfer from one state to the other is much smaller: \(\tau_{Tb^{5}D_{3}\to Tb^{5}D_{4}}=150\ \mu\)s. For the actual Tb concentration, the rate of Auger quenching for the pair of excited Tb ions \(\beta_{Tb^{*}+Tb^{*}}\) was set as 0.1\(\cdot 10^{-13}\frac{cm^{3}}{s}\), which corresponds to the best fit of the experimental result presented in Figure 8a. This value is noticeably smallerthan the value \(1.8\cdot 10^{-13}\frac{cm^{3}}{s}\) for YAG:Tb (10%) reported in [24]. The initial conditions for the system (1-3) can be written as \(n_{\frac{n_{C^{\prime}}}{Tb}_{3}}(t=0)=n_{ex}(I)=I(A)\cdot 4.055\cdot 10^{16}\)\(cm^{-3}\) and \(n_{\frac{n}{Tb}_{3}{}^{5_{D_{4}}}}(t=0)=0\). The substantial difference between the Auger rate obtained here and in [24] is well explained by the different Tb concentration in these two cases (5% in the present work and 10% in [24]). The interaction between the two excited terbium ions is efficient only if the distance between them is about one nanometer, which is a characteristic value for dipole-dipole and exchange interaction. Such a small distance can be achieved either by a high density of excitation (which is not the case here) or after the diffusion of excitations towards each other. A smaller terbium concentration corresponds to a larger mean distance between two terbium ions. The diffusion coefficient of the excitations over the terbium subsystem or the probability of excitation hopping between terbium ions is also concentration dependent. A smaller terbium concentration is characterized by a larger distance between terbium ions and has two consequences: a smaller diffusion coefficient of excitations over the terbium subsystem and a smaller portion of closely spaced Tb pairs. The kinetic equation for Ce-doped GYAGG crystal can be written as follows:  
\[\frac{dn_{Ce*}}{dt}=-\frac{1}{\tau_{Ce*}}n_{Ce*}-\beta_{Ce^{\prime}+Ce^{\prime }}n_{Ce*}^{2}+i(t) \tag{4}\]  
where \(\tau_{Ce*}=50\)\(ns\), \(i(t)=\frac{n_{ex}(I)}{\sqrt{2\pi}\
u}e^{-\frac{(t-\mu)^{2}}{2\sigma^{2}}}\) is the Gaussian source with \(\sigma=\frac{20\)\(ns}{2.355}=8.49\)\(ns\), and the rate of Auger quenching for pair of excited Ce ions \(\beta_{Ce^{\prime}+Ce^{\prime}}\) was set as \(0.7\cdot 10^{-13}\frac{cm^{3}}{s}\). Meanwhile, the \(\mu\) parameter of the Gaussian source \(i(t)\) should be set quite large to consider \(i(t=0)\approx 0\). In the current work, \(\mu=3\sigma\) was used. The initial condition can be set as \(n_{Ce*}(t=0)=0\). The solution to Equation (4) suggests a linear dependence of cerium luminescence yield on the beam current even for much higher values of the parameter \(\beta_{Ce^{\prime}+Ce^{\prime}}\) up to \(10^{-10}\frac{cm^{3}}{s}\), corresponding to nonrealistic values of the diffusion coefficient. We suppose that the Auger effect in the interaction of two excited cerium ions for 1% Ce concentration should be negligible. Figure 8 presents the comparison of the experimentally measured and simulated normalized yield of scintillation in GYAGG:Tb, which includes transitions both from \({}^{5}D_{3}\) and \({}^{5}D_{4}\) excited states and GYAGG:Ce (b) as a function of the beam power.