## 2 Review of Cryogenic-Cooling Benefits
### Thermal Parameters
#### 2.2.3 Thermal Conductivity: The Influence of Dopant Density
In addition to Lu\({}_{2}\)O\({}_{3}\), two other sesquioxides, Sc\({}_{2}\)O\({}_{3}\) and Y\({}_{2}\)O\({}_{3}\), have been intensely investigated as newer laser materials. No donor results in a good match for Y. For Sc, the rare-earths are a very poor mass match, and this is also reflected in the experimental data. Yb has the worst mis-match, and Er, Tm, and Ho are also poor. The poor mismatches of these donor-acceptor ions are reflected in the experimental data as well, as shown in Figure 6 where thermal conductivity is shown as a function of absolute temperature [43; 44]. Data are plotted for undoped hydrothermally-grown and Czochralski-grown undoped Sc\({}_{2}\)O\({}_{3}\), for 2.9 and 9 at. % Yb doping and for 3 at. % Er doping. As expected from Table 1, there is a strong drop in k for Er and an even worse drop for Yb, for all temperatures. Another interesting comparison is Yb:YAG with Yb:LuAG. The mass mismatch for Yb and Y is about \(-\)84.13 amu. For Yb and Lu it is only 1.93 amu. Based on this  
Figure 4: Thermal conductivity as a function of absolute temperature for undoped single-crystal Czochralski-grown YAG (red), and Yb:YAG with doping densities of 2 (blue), 4 (green), 10 (pink), 15 (black), and 25 (orange) at. %. The experimental data for the 2, 4 and 15 at. % values are taken from [29], while the 10 and 25 at. % values were determined from fits to the data of [29] for each temperature. comparison we would expect the mass scattering to be much smaller in Yb:LuAG than in Yb:YAG. We will see in Section 2.2.7 that as Yb doping increases, the falloff in thermal conductivity for Yb:LuAG is much smaller than for Yb:YAG. Figure 5: Thermal conductivity as a function of absolute temperature for hydrothermally-grown single-crystal undoped Lu\({}_{2}\)O\({}_{3}\) (green) and 11 at. % doped Yb:Lu\({}_{2}\)O\({}_{3}\) (blue) [43]. Figure 6: Thermal conductivity as a function of absolute temperature for the sesquioxide material Sc\({}_{2}\)O\({}_{3}\), for undoped hydrothermal (blue) [43] and Czochralski (green) [44] grown single-crystals, 3 at. % Er doping (light blue) [44], and 2.8 at. % Yb doping (orange) [44]. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Donor**} & \multirow{2}{*}{**At. Number**} & \multicolumn{2}{c}{**Acceptor**} & \multirow{2}{*}{**Li**} & \multirow{2}{*}{**Mg**} & \multirow{2}{*}{**Al**} & \multirow{2}{*}{**K**} & \multirow{2}{*}{**Ca**} & \multirow{2}{*}{**Sc**} & \multirow{2}{*}{**Y**} & \multirow{2}{*}{**La**} & \multirow{2}{*}{**Gd**} & \multirow{2}{*}{**Lu**} \\ \cline{3-3} \cline{5-12}  & & \multicolumn{2}{c}{**At. Number**} & \multicolumn{2}{c}{**3**} & \multicolumn{2}{c}{**12**} & \multicolumn{2}{c}{**13**} & \multicolumn{2}{c}{**19**} & \multicolumn{2}{c}{**20**} & \multicolumn{2}{c}{**21**} & \multicolumn{2}{c}{**39**} & \multicolumn{2}{c}{**57**} & \multicolumn{2}{c}{**64**} & \multicolumn{2}{c}{**71**} \\ \cline{3-12}  & & \multicolumn{2}{c}{**At. Weight (amu)**} & \multicolumn{2}{c}{**6.94**} & \multicolumn{2}{c}{**24.31**} & \multicolumn{2}{c}{**26.98**} & \multicolumn{2}{c}{**39.10**} & \multicolumn{2}{c}{**40.08**} & \multicolumn{2}{c}{**44.96**} & \multicolumn{2}{c}{**88.91**} & \multicolumn{2}{c}{**139.91**} & \multicolumn{2}{c}{**157.25**} & \multicolumn{2}{c}{**174.97**} \\ \hline Ti & 22 & 47.87 & \(-\)44.87 & 56.87 & \(-\)20.89 & \(-\)8.77 & \(-\)7.79 & \(-\)2.91 & 41.04 & 92.04 & 109.38 & 127.10 \\ V & 23 & 50.94 & \(-\)