## 3 Results and Discussion
### Photoluminescence Analysis
For the case of the R\({}_{2}\)\({}^{\prime\prime}\)- and R\({}_{1}\)\({}^{\prime\prime}\)-lines, it is seen that the intensity of the R\({}_{2}\)\({}^{\prime\prime}\)-line decreases rapidly for temperatures above 70 K, while it is possible to observe the R\({}_{1}\)\({}^{\prime\prime}\)-line up to the RT, with a shift of its maximum toward lower energies. The R-lines show an analogous behavior, exhibiting a decrease in their PL intensity. This is more pronounced for temperatures above 90 K, accompanied by a redshift of the line peak position along with an enlargement of its width with increasing temperature (please note that each R-line is a multi-component line, as shown in Figure 6c). It was also observed that the intensity of the N\({}_{2}\)- and N\({}_{1}\)-lines dropped more rapidly than that of the R-lines for temperatures above 14 K. Their emission maxima also show a shift to lower energies, which is also accompanied by an increase in their FWHM. Figure 5: **(a)** The 14 K PL spectrum of \(\alpha\)-Al\({}_{2}\)O\({}_{3}\):Cr spheres obtained with 3.81 eV photon excitation; **(b)** enlarged spectrum in the region of the R- and N-lines. In the inset, the high-energy side of the R-lines is displayed. Figure 7 shows the temperature dependence of the peak position for the R\({}^{\prime\prime}\)-, R-, and N-lines and the dependence of the FWHM for the R- and N-lines. The linewidth and temperature shift of the R- and N-lines in dilute and heavily chromium-doped Al\({}_{2}\)O\({}_{3}\) can be explained based on the contributions of the microscopic strains and impurity-phonon interactions [8,46]. For the case of samples doped with 2.1 at.% and 0.94 at.% Cr, [PERSON] et al. [46] measured a redshift of about 2-2.5 meV for the peak position and a line broadening of about 1-2 meV for both the R- and N-lines when inspected between 13 K and RT. Considering a simplified model where the energy shift is proportional to the phonon population number, the temperature dependence of the peak position, \(E(T)\), can be expressed by [52],  
\[E(T)=E_{0}-\frac{A}{exp(\hbar\overline{\omega}/k_{B}T)-1} \tag{1}\]  
where \(E_{0}\) is the line energy at 0 K, \(A\) corresponds to a proportionality factor, \(\hbar\overline{\omega}\) is the energy of the phonons involved, \(\hbar\) is the reduced Planck constant, \(k_{B}\) is the Boltzmann constant, and \(T\) is the absolute temperature. The best fits to Equation (1) to the temperature dependence of the lines' peak position are shown by the solid lines in Figure 7. The \(E_{0}\) values of 1.809 eV, 1.805 eV, 1.793 eV, 1.789 eV, 1.769 eV, and 1.761 eV were found for the R\({}_{2}\)\({}^{\prime\prime}\), R\({}_{1}\)\({}^{\prime\prime}\), R\({}_{2}\), R\({}_{1}\), N\({}_{2}\), and N\({}_{1}\) lines, respectively. Constant factors of 2.7 \(\times\) 10\({}^{-3}\) eV, 0.7 \(\times\) 10\({}^{-3}\) eV, and 0.6 \(\times\) 10\({}^{-3}\) eV were determined for the (R\({}_{2}\)\({}^{\prime\prime}\), R\({}_{1}\)\({}^{\prime\prime}\)), (R\({}_{2}\), R\({}_{1}\)), and (N\({}_{2}\), N\({}_{1}\)) lines, and about 10 meV was obtained for the \(\hbar\overline{\omega}\) of the R\({}_{1}\)\({}^{\prime\prime}\), R\({}_{2}\), R\({}_{1}\), N\({}_{2}\), and N\({}_{1}\) lines. Figure 6: (**a**) Temperature dependence PL spectra of the \(\alpha\)-Al\({}_{2}\)O\({}_{3}\):Cr spheres obtained with 3.81 eV photon excitation. Enlarged spectra in the region of (**b**) the R\({}^{\prime}\)- and R\({}^{\prime\prime}\)-lines, (**c**) R\({}_{2}\)- and R\({}_{1}\)-lines, and (**d**) the N\({}_{2}\)- and N\({}_{1}\)-lines. Similarly, using the simplified model, the temperature dependence of the lines FWHM, \(\Gamma(T)\), can be determined by [52],  
\[\Gamma(T)=\Gamma_{0}+\frac{B}{exp(\hbar\overline{\omega}/k_{B}T)-1} \tag{2}\]  
where \(\Gamma_{0}\) is the lines' FWHM at 0 K and \(B\) corresponds to a proportionality factor.