## 3 Results and Discussions
### Time-Integrated Photoluminescence Measurements
Actually, the averaged value of the inter-site distance of Er\({}^{3+}\), \(R_{C_{2}-C_{3i}}\) extends from 2.66 A to 7.03 (11.6) A as the Er-concentration is reduced from \(x\) = 1.000 to 0.054 (0.012) in Sample A to B (C) as shown in Table 1.  
Other important information about the inter-site ET process is found from the photoluminescence excitation (PLE) measurement as shown in Figure 4d where the excitation wavelength was swept around \({}^{4}I_{13/2}\) manifold (1520-1555 nm). At the \(Y_{1}^{\prime}\) resonant excitation (\(\lambda_{\rm exc}\) = 1548 nm), the PL signal of Y\({}_{1}-Z_{1}\) transition (1535 nm) was observed. Accordingly, the \(Y_{1}^{\prime}\rightarrow\)Y\({}_{1}\) transition seems to be a dominant process relevant to the \(C_{3i}\to C_{2}\) ET rather than the \(Y_{3}^{\prime}\rightarrow\)Y\({}_{1}\) transition, and the inter-site ET occurs after the rapid internal energy relaxation in \(C_{3i}\) sites (i.e., \(Y_{3}^{\prime}\rightarrow\)Y\({}_{1}^{\prime}\rightarrow\)Y\({}_{1}\)). Further, it should be notedthat the strong PL of the \(Y_{1}^{\prime}-Z_{1}^{\prime}\) transition (1548 nm) appeared at the \(Y_{1}\) and \(Y_{2}\) resonant excitation; it denotes the presence of the inter-site ET in the reverse direction (\(C_{2}\to C_{3l}\)). For the energy transfer mechanisms between Er ions, the Forster type (dipole-dipole interaction) and Dexter type (wave-function overlapping) are well known from the previous studies [32; 33], and the former and the latter are proportional to \(R^{-6}\) and \(\exp(-2R/L)\), respectively (\(R\): the inter-ionic distance, \(L\): the van der Waals radius). In our case, the remarkable suppression by extending \(R_{C_{2}-C_{3l}}\) and the energy mismatch between \(Y_{1}^{\prime}\) and \(Y_{1}\) levels of \(\sim\)7 meV suggest that the Dexter-type mechanism, which is more sensitive to the change in \(R\) than the Forster-type mechanism, seems to be dominant in the inter-site ET process. Figure 4e shows the temperature dependence of the TI-PL intensities of the \(Y_{1}^{\prime}-Z_{1}^{\prime}\) transition in Figure 4a-c for the three samples. In Sample A, a significant thermal quenching of the PL intensity (\(I_{\rm PL}\)) occurs till around 20 K, and the \(I_{\rm PL}\) at 20 K is reduced approximately to the one-quarter of \(I_{\rm PL}\) at 4 K. In contrast, the decrease in PL intensities in Samples B and C are more moderate; the \(I_{\rm PL}\) at \(\sim\)20 K is about a half of \(I_{\rm PL}\) at 4 K. These behaviors appear clearly in the values of non-radiative recombination rate \(W_{\rm nr}^{0}\) (or time \(\tau_{\rm nr}^{0}\)) in Table 2, which is deduced from the time-resolved PL measurements in the next subsection. The data suggests that the extension of the inter-ionic distance serves as a suppression of the thermal quenching as well as the reduction in the inter-site ET. The suppression of the thermal quenching in Samples B and C may come from the reduction of the possibility to reach non-radiative centers due to the reduction in the inter-site ET. The dilution of Er ions replacing by Sc ions induces  
Figure 4: Time-integrated photoluminescence (TI-PL) spectra at different temperatures for the (Er\({}_{x}\)Sc\({}_{1-x}\))\({}_{2}\)O\({}_{3}\) crystals (**a**) Sample A (\(x=1.000\)), (**b**) Sample B (\(x=0.054\)), and (**c**) Sample C (\(x=0.012\)) under the \(Y_{3}^{\prime}-Z_{1}^{\prime}\) resonant excitation with \(P_{\rm exc}=1\) mW. (**d**) The 2D plot of PLE spectra where the excitation laser was swept around the \({}^{4}I_{13/2}\) manifold in Sample A at 4 K. In the excitation region of 1520–1540 nm, the PLs below 1540 nm as depicted by a gray square are blocked by a low-pass filter. (**e**) The temperature dependence of the TI-PL intensities of the \(Y_{1}^{\prime}\)-\(Z_{1}^{\prime}\) transitions shown in (**a–c**) is plotted: Samples A (circles), B (triangles), and C (squares). the increase of the lattice mismatch since the lattice constant of Sc\({}_{2}\)O\({}_{3}\) is 9.85 A, and thus the increase of the defect density may be expected.