## 3 Results and Discussion
### YbGAGG: Cr\({}^{3+}\) Ratiometric Thermometer Based on the Emission of Yb\({}^{3+}\) (Mode 1)
Based on the above parameters, we can derive the Tanabe\(-\)Sugano diagram (as shown in Figure 8). Under excitation by a 660 nm light source, Cr\({}^{3+}\) will transition from the \({}^{4}\text{A}_{2}\) ground state to the \({}^{4}\text{T}_{2}\) state and then relax to the \({}^{2}\text{E}\) state (as shown in Figure 9). The lowest excited state \({}^{2}\text{E}\) and the ground state \({}^{4}\text{A}_{2}\) are weakly coupled [33], so the emission from \({}^{2}\text{E}\) to \({}^{4}\text{A}_{2}\) does not experience strong thermal quenching due to the cross-over effect. The extremely low PL intensity of Cr\({}^{3+}\) at room temperature is mainly due to the high phonon assistant energy transfer efficiency from Cr\({}^{3+}\) to Yb\({}^{3+}\)[34]. Figure 8: Tanabe-Sugano diagram for Cr\({}^{3+}\), in YbGAGG, the orange dotted line represents the crystal field strength of Cr\({}^{3+}\) in YbGAGG. The luminescence intensity of Cr\({}^{3+}\) decreases sharply with increasing temperature, as indicated by the blue square in Figure 7b. For Cr\({}^{3+}\), as mentioned in Section 3.3, the Cr\({}^{3+}\) luminescence is quenched at a higher Yb\({}^{3+}\) concentration due to the efficient Cr\({}^{3+}\)\(\rightarrow\) Yb\({}^{3+}\) energy transfer process. The opposite temperature dependence of Yb\({}^{3+}\) and Cr\({}^{3+}\) exhibits considerable thermometric sensitivity, especially at temperatures around 100 K.  
The LIR plots shown in Figure 7c were directly obtained from the integrated intensities of Yb\({}^{3+}\) and Cr\({}^{3+}\) in the temperature range of 100-320 K in Figure 7b, based on the formula of \(LIR=\frac{I_{\textrm{Cr}}^{3+}}{\textrm{Yb}^{3+}}\) (Mode 2), where \(I_{\textrm{Cr}}^{3+}\) and \(I_{\textrm{Yb}^{3+}}\) represent the integrated intensities of Cr\({}^{3+}\) and Yb\({}^{3+}\), respectively. A fitting function was employed to analyze these LIR plots further, as defined by Equation (3):  
\[S_{\rm a}=\frac{\mathrm{d}(LIR)}{\mathrm{d}T} \tag{4}\]0  
The calculation of the temperature dependence of \(S_{\textrm{a}}\) and \(S_{\textrm{r}}\) of the YbGAGG: Cr\({}^{3+}\) thermometer based on the luminescence intensity ratio \(LIR=\frac{I_{\textrm{Cr}}^{3+}}{\textrm{Yb}^{3+}}\) was also conducted using the method described in Section 3.4 and shown in Figure 7d. Notably, at 100 K, \(S_{\textrm{r}}\) reaches its maximum value of 2.69% K\({}^{-1}\). This indicates a high level of sensitivity at lower temperatures for Mode 2. \[S_{\rm a}=\frac{\mathrm{d}(LIR)}{\mathrm{d}T} \tag{4}\]1 Energy level diagram of YbGAGG: Cr\({}^{3+}\) sample under 660 nm excitation, the blue solid arrow represent the transition process of Cr\({}^{3+}\) excited from the ground state to the \({}^{4}\)T\({}_{2}\) state by the 660 nm excitation source, the red solid arrow represent the luminescence transition process of Cr\({}^{3+}\) from the \({}^{2}\)E state emission back to the ground state, black dashed arrows represent the Cr\({}^{3+}\)\(\rightarrow\)Yb\({}^{3+}\) energy transfer process, and the black solid arrow represent the luminescence transition process of Yb\({}^{3+}\).