## 2 Review of Cryogenic-Cooling Benefits
### Thermal Parameters
#### 2.2.5 Modeling Thermal Conductivity
We then define a constant \(\xi\) as  
\[\frac{1}{\tau_{U}}=B\omega^{2}\,\ \frac{1}{\tau_{D}}=A\omega^{4} \tag{41}\]4  
so that \(\omega_{0}/\omega_{D}\) is expressed as  
\[\frac{1}{\tau_{U}}=B\omega^{2}\,\ \frac{1}{\tau_{D}}=A\omega^{4} \tag{41}\]5  
Substituting Equation (55) into Equation (52) gives the final result:  
\[\frac{1}{\tau_{U}}=B\omega^{2}\,\ \frac{1}{\tau_{D}}=A\omega^{4} \tag{41}\]6  
Knowing \(k_{0}\), \(T\), and \(\delta\), we can now calculate the thermal conductivity using \(\xi\), as a fitting constant. Turning our attention to the binary laser crystal system YAG and YbAG, we can use Vegard's Law for solid solution to write the intrinsic thermal conductivity as  
\[\frac{1}{\tau_{U}}=B\omega^{2}\,\ \frac{1}{\tau_{D}}=A\omega^{4} \tag{41}\]7  
whereas before \(x\) is the concentration parameter for Yb, and \(k_{0}^{Y}\) and \(k_{0}^{Yb}\) are the zero-doping intrinsic thermal conductivities of YAG and YbAG respectively. To obtain \(k\) as a function of Yb doping density, at any temperature, we then use the following relationship:  
\[\frac{1}{\tau_{U}}=B\omega^{2}\,\ \frac{1}{\tau_{D}}=A\omega^{4} \tag{41}\]8  
This model has been previously discussed in the Dissertations of Klopp and Fredrich-Thornton [48,49]. At a temperature of 300 K, and using the intrinsic thermal conductivities of YAG and YbAG, 0.111 W/(cm\(\cdot\) K) and 0.070 W/(cm\(\cdot\) K), we obtained the results shown in Figure 8, where the thermal conductivity is displayed as a function of \(x\), using the value \(\xi=100\), that visually best fits the experimental data, obtained from the summary provided in [49]. To simplify the presentation, we averaged the many values of [49] for YAG (\(x=0\)). Undoubtedly, a nonlinear least squares fit of Equation (56) to the experimental data would allow a more accurate determination of \(\xi\), however, for reasons discussed in the next paragraph we have reservations as to the wisdom of such an undertaking. While this model qualitatively explains the concentration dependence of \(k\), and of course because of Equation (57) duplicates the two end point thermal conductivities for YAG and LuAG, it is very likely that the appearance of accurately fitting the experimental data is illusory, because such a curve requires more than one constant to be theoretically duplicated. Another aspect of the Klemens model that is disquieting is that it treats the case where Y acoustic phonons are scattered as Yb ions are doped into the crystal, where the Yb concentration varies from 0%-100%, but ignores the reciprocal scattering of Yb ions from Y ions when Y is substituted into YbAG. Also, it is clear that beyond a Yb doping concentration of 50%, the crystal is predominantly YbAG rather than YAG. It should also be pointed out that YAG and YbAG have significantly different Debye temperatures of 760 and 575 K respectively [40]. Because of the interaction between the two crystal lattices, it is unlikely that a Vegard's law linear relationship exists that can be used to estimate the values of the Debye temperature for \(x\) values in the range \(0<x<~{}1\). The Klemens model uses only a single Debye temperature to describe the entire range of dopings, and for simplicity makes the assumption that one is operating at the high temperature limit where \(T\gg\theta_{D}\), well beyond the region of interest for room temperature and cryogenic lasers. We will discuss progress towards an alternative model in the following section.