## 2 Review of Cryogenic-Cooling Benefits
### Thermal Parameters
#### 2.2.6 Extended Thermal Conductivity Model  
Using the formalism shown in Section 2.2.5, we have developed an alternative model that splits the thermal conductivity into two components, \(k_{\mathrm{Y}}(x)\) and \(k_{\mathrm{Yb}}(x)\) whose properties are somewhat more physical, is symmetric with respect to the interchange of Y and Yb, and that has the reasonable boundary conditions that as \(x\to 0,k_{\mathrm{Yb}}(x)\to 0\), and for \(x\to 1,k_{\mathrm{Y}}(x)\to 0\). The thermal conductivity for any value of the doping parameter \(x\) can then be written as the equations  
\[k=\left(1-x\right)k_{\mathrm{Y}}+xk_{\mathrm{Yb}} \tag{59}\]  
and  
\[\begin{split} k\left(x\right)=\left(1-x\right)\cdot\left(\frac{k _{\mathrm{\sigma}Y}}{\xi_{Y}}\right)\cdot&\sqrt{\frac{T}{\xi_{Y }}}\cdot\tan^{-1}\left(\xi_{Y}\cdot\sqrt{\frac{\xi_{Y}}{T}}\right)+x\cdot \left(\frac{k_{\mathrm{\sigma}Yb}}{\xi_{Yb}}\right)\cdot\sqrt{\frac{T}{\xi_{ Yb}}}\\ &\cdot\tan^{-1}\left(\xi_{Yb}\cdot\sqrt{\frac{\xi_{Yb}}{T}}\right) \end{split} \tag{60}\]  
Figure 8: Thermal conductivity calculated using the Klemens model [42] as a function of concentration parameter \(x\) for the binary solid solution YAG-YbAG. Data points are taken from the comprehensive summary found in [49].  
These equations satisfy the aforementioned boundary conditions, are symmetric upon the interchange of Y and Yb, and are physically reasonable since now a crystal may be viewed as a solid solution of two separate scattering materials rather than a single crystal with impurity doping added, and there are now two constants \(\xi_{Y}\) and \(\xi_{Yb}\) that may be adjusted to fit the experimental data. The quantities \(k_{OY}\) and \(k_{OYb}\) are the experimental values of the thermal conductivity of YAG and YbAG. The mass scattering terms \(\delta_{Y}\) and \(\delta_{Yb}\) are now given by the expressions  
\[\delta_{Y}=(1-x)\cdot\left[\frac{(M_{Y}-M)}{M}\right]^{2};\ \delta_{Yb}=x \cdot\left[\frac{(M_{Yb}-M)}{M}\right]^{2} \tag{61}\]  
where \(M\) is given by Equation (44), and \(M_{Y}\) and \(M_{Yb}\) are the atomic masses of Y and Yb respectively.  
In Figure 9, we shown a fit to the experimental data summarized in [49]. The thermal conductivity \(k_{Y}\) goes to 0 for \(x=1\) and for \(x=0\), \(k_{Yb}=0\), as expected. The minimum appears to be closer to \(x=0.35\) than to \(x=0.5\) as found in the previous plot, Figure 8, but without a better more consistent data set, it is difficult to ascertain exactly where it is with any accuracy. The position of the minimum is determined by how quickly the thermal conductivities decrease as a function of \(x\). The equality of the two thermal conductivities occurs at \(x=0.4\).  
This new model seems to offer more flexibility with regard to fitting experimental data, however we view this as only a first step in developing a more comprehensive model that can be used to predict in advance the thermal conductivities of arbitrary laser crystals. Because the Klemens model that we started with is only valid for \(T>>\theta_{D}\), a reasonable next step is to combine the approach described here with Equation (36), and to solve for the thermal conductivity without approximation, using numerical methods.  
Figure 9: Thermal conductivities \(k_{Y}\) (blue), \(k_{Yb}\) (red), and \(k\) (green) calculated using our model as a function of concentration parameter \(x\) for the binary solid solution YAG-YbAG. Data points are taken from the comprehensive summary found in [49]. Constants used in this plot are \(\xi_{Y}=300\) and \(\xi_{Yb}=50\).