# New Inorganic Scintillators' Application in the Electromagnetic Calorimetry in High-Energy Physics
## 2 Simulations in Geant4
### Results of the Simulations for Combined Calorimeters
However, for the energies below 1 MeV, the situation is inverse: the spatial resolution of GAGG + PWO improves by 2-4 times. Figure 11: Comparison of spatial resolution in PWO-, GAGG-, and GAGG + PWO-calorimeters, excluding noise [20]. Experimental data are taken as a reference (PHOS). Figure 10: The fraction of gammas and hadrons accepted for varying upper limit on the shower dispersion as a function of this cut with fixed \(\lambda_{short}\) = 2.2 cm\({}^{2}\). In the case of a GAGG calorimeter or a combined GAGG + PWO calorimeter, the energy distributions are essentially non-Gaussian, so it seems more correct to approximate them with the Crystal Ball function [24] (which better describes the decaying \"tails\" in the distributions):  
\[f(x)=\left\{ \begin{array}{ll}e^{-\frac{(E-E_{mean})^{2}}{2\sigma^{2}}}&\frac{ E-E_{mean}}{\sigma}>-\alpha\\ A\cdot(B-\frac{E-E_{mean}}{\sigma})^{-n}+C,&\frac{E-E_{mean}}{\sigma}\leq- \alpha\end{array} \right. \tag{1}\]  
From here, by obtaining parameters \(\sigma\) and \(E_{mean}\), one can construct the energy resolution. Figure 13 shows such a resolution for the case including light collection and excluding the noise in the PWO-calorimeter, as well as in the GAGG- and GAGG + PWO-calorimeters with 3 cm long GAGG crystals. Experimental data obtained from radioactive sources are also included as a reference. The simulated resolution and experimental resolution of the GAGG-calorimeter are nearly the same; thus, once again, the correctness of the simulations is proven. This difference can be explained by the fact that, in a simulation for the light yield, the mean value of 420 phe/MeV is used but is expected to vary for crystals of different lengths due to the light absorption. Furthermore, for the GAGG-calorimeter, it is not possible to obtain the energy resolution in the whole range of photon energies (from 100 keV to 100 GeV) as, with the energy increase, the peak in energy distribution disappears or becomes non-fittable with the Gaussian-type distribution. The first sharp increase in the energy resolution in the range of 60-100 MeV can be explained by the increase in the energy deposited in PWO-section; the second one occurs due to the limited length of GAGG crystals. On the other side, at low energies,, the resolution of the GAGG-calorimeter is an order of magnitude better than that of the PWO-calorimeter: \(\sim\)5% instead of \(\sim\)45% at \(E\) = 1 MeV. In the case of the compound calorimeter (GAGG + PWO), the energy resolution is better than that in the PWO calorimeter across the entire energy range (from 100 keV to 100 GeV). Figure 12: The ratio of the spatial resolution of PWO-calorimeter to the one of GAGG + PWO-calorimeters with different GAGG-section lengths (excluding noise). A comparison of the energy resolution obtained for GAGG + PWO-calorimeter with different GAGG-section length is depicted in the Figure 14. Below 10 MeV, the energy resolution of the compound calorimeter becomes 1.5-10 times better than the resolution of the PWO-calorimeter. This improvement is also seen for energies above 20 GeV. As one can see from the Figure 15, the optimal length of the GAGG-section is about 3 cm because further GAGG-section length increases do not result in the energy resolution becoming significantly better, although the cost of the crystals rises. Figure 14: The ratio of the energy resolution of PWO-calorimeter to that of GAGG + PWO-calorimeters with different GAGG-section lengths (excluding noise). Figure 13: Comparison of the energy resolution in PWO-, GAGG-, and GAGG + PWO-calorimeters excluding noise with a GAGG-section length of 3.0 cm.