## 3 Results and Discussion
### YbGAGG: Cr\({}^{3+}\) Ratiometric Thermometer Based on the Emission of Yb\({}^{3+}\) (Mode 1)
The integrated PL intensity of Yb\({}^{3+}\) above 900 nm, as discussed in Section 3.3, increases with rising temperature (as illustrated by the red triangle plot in Figure 7b). The Boltzmann distribution and different transition probabilities between \({}^{2}\)F\({}_{7/2}\)-\({}^{2}\)F\({}_{5/2}\) with Stark levels can result in a negative thermal quenching. Additionally, the temperature dependence of energy transfer efficiency from Cr\({}^{3+}\) to Yb\({}^{3+}\) can play a part in this phenomenon (a demonstration of the energy transfer process can be found in the earlier discussion about Figure 3b). The opposite tendency between temperature-dependent PL intensities of Yb\({}^{3+}\) and Cr\({}^{3+}\) implies this hypothesis. For an efficient energy transfer, the luminescence spectrum of the donor and the absorption spectrum of the acceptor should be overlapped. However, there is an energy mismatch between Cr\({}^{3+}\): \({}^{2}\)E \(\rightarrow\)\({}^{4}\)A\({}_{2}\) luminescence and Yb\({}^{3+}\): \({}^{2}\)F\({}_{7/2}\)-\({}^{2}\)F\({}_{5/2}\) absorption. This energy gap can be compensated efficiently at higher temperatures by two possibilities: Cr\({}^{3+}\) PL broadening and phonon assist. As the most common explanation for the temperature quenching of rare earth luminescence centers, the intensity of the weak multi-phonon relaxation effect for Yb\({}^{3+}\) depends on the relationship between the phonon cut-off energy of the host material and the energy difference between the excited state and the ground state. Although the phonon cut-off energy of the GAGG host is as high as 808 cm\({}^{-1}\)[27], the energy difference between Z\({}_{\text{low}}\) and the ground state \({}^{2}\)F\({}_{5/2}\) of Yb\({}^{3+}\) exceeds 9500 cm\({}^{-1}\), depending on the intensity of the Stark split effect. In this case, the possibility of the multi-phonon relaxation process is negligible. The shape of the PL signal of Cr\({}^{3+}\) in Figure 7a is sharp peaks, which is believed to be a phenomenon that occurs when Cr\({}^{3+}\) is in a strong crystal field. In GAGG, the octahedral sites are occupied by smaller Al\({}^{3+}\) ions (0.535 A) [28; 29], resulting in a strong crystal field for the Cr\({}^{3+}\) (0.615 A) dopants. This results in the \({}^{2}\)E energy level of Cr\({}^{3+}\) becoming the lowest excited state, leading to the sharp peak shape of the PL spectrum. Figure 7: **(a)** Temperature dependence of the photoluminescence spectra sample under 455 nm excitation; **(b)** corresponding integral intensity of Cr\({}^{3+}\) and Yb\({}^{3+}\) for YbGAGG: Cr\({}^{3+}\); **(c)** temperature dependence of the luminescence intensity ratio; **(d)** calculated temperature dependence of S\({}_{\text{a}}\) and S\({}_{\text{r}}\) for the YbGAGG: Cr\({}^{3+}\) thermometer based on the photoluminescence intensity ratio of Cr\({}^{3+}\) to Yb\({}^{3+}\). To investigate the influence of the crystal field on the PL of Cr\({}^{3+}\), Racah parameters were calculated to explore the effect of the crystal field strength on the d orbitals of Cr\({}^{3+}\) and the shape of the PL spectrum. The formulas for calculating the crystal field strength (10\(Dq\)) and the Racah parameters (B and C) are as follows [30; 31; 32]:  
\[E\Big{(}^{4}T_{2}\Big{)}=10 Dq \tag{6}\]  
\[\frac{Dq}{B}=\frac{15\left(\Delta E/Dq-8\right)}{\left(\Delta E/Dq\right)^{2}- 10\left(\Delta E/Dq\right)} \tag{7}\]  
\[\frac{E\big{(}^{2}E\big{)}}{B}=\frac{3.05C}{B}+7.90-\frac{1.80B}{Dq} \tag{8}\]  
where \(\Delta E=E(^{4}\text{T}_{1}(^{4}\text{F}))-E(^{4}\text{T}_{2})\). \(E(^{4}\text{T}_{1}(^{4}\text{F}))\), and \(E(^{4}\text{T}_{2})\) are, respectively derived from the positions of the \({}^{4}\text{T}_{1}\) and \({}^{4}\text{T}_{2}\) bands of Cr\({}^{3+}\) in Figure 3a, while the value of \(E(^{2}\text{E})\) is derived from the position of the R-line of Cr\({}^{3+}\) PL spectrum in Figure 7a (691.53 nm). The final calculations result in \(Dq=1658.9\) cm\({}^{-1}\), \(B=631.53\) cm\({}^{-1}\), and \(C=3244.1\) cm\({}^{-1}\).