## 2 Review of Cryogenic-Cooling Benefits
### Thermal Parameters
#### 2.2.5 Modeling Thermal Conductivity  
A number of models have been developed to describe the variation in thermal conductivity as dopant concentration increases. One of the most successful is due to Klemens [42]. Other contributions  
Figure 7: Thermal conductivity of undoped YAG (red), 4.0 (green) and 15.0 (blue) at. % Yb-doped YAG as a function of absolute temperature. Data are from [29]; also shown are fits to the data showing that the undoped case scales as \(b=1.315\) (pink), the 4.0 at. % Yb case (black) scales as \(b=1\), and the 15 at. % Yb case (orange) scales as \(b=0.8\). were made by Slack and Oliver [45] and Holland [46]. A similar model was discussed in [47]. We follow the treatment of Klemens [42], who defines two relaxation times \(\tau_{U}\) and \(\tau_{D}\) as  
\[\frac{1}{\tau_{U}}=B\omega^{2}\,\ \frac{1}{\tau_{D}}=A\omega^{4} \tag{41}\]  
The relaxation time \(\tau_{U}\) is associated with Umklapp anharmonic three-phonon scattering, while \(\tau_{D}\) is the relaxation time for acoustic phonon scattering from dopants or impurities. The equation for \(\tau_{D}\) is analagous to the Rayleigh scattering of optical photons, which has the same frequency dependence. The quantity \(A\) is a constant that depends on the mass difference between donor and acceptor ions, the unit cell volume, and the phonon speed. \(A\) can formally be written  
\[A=\frac{a^{3}}{4\pi v_{s}^{3}}\cdot\delta \tag{42}\]  
In Equation (42), a is the unit cell lattice parameter, \(v_{s}\) the speed of sound in the crystal, and \(\delta\) is the mass-mismatch variance, given by  
\[\delta=\sum_{i}x_{i}\left[\frac{M_{i}-M}{M}\right]^{2} \tag{43}\]  
and \(M\) is  
\[M=\sum_{i}x_{i}M_{i} \tag{44}\]  
The quantity \(x_{i}\) is the concentration parameter of the donor atoms, and satisfies the inequality \(0\leqslant x_{i}\leqslant\ 1\). B in Equation (39) is a constant that is proportional to the temperature. For Umklapp and dopant scattering together, the effective phonon relaxation time \(\tau_{e}\) is calculated using Matthiessen's rule:  
\[\frac{1}{\tau_{e}}=\frac{1}{\tau_{U}}+\frac{1}{\tau_{D}} \tag{45}\]  
Substituting in Equation (39), and using Equation (41), we find  
\[k=\frac{k_{B}}{2\pi^{2}v_{s}}\cdot\int_{0}^{\omega_{D}}\frac{\omega^{2}}{ \frac{1}{\tau_{U}}+\frac{1}{\tau_{D}}}d\omega=\frac{k_{B}}{2\pi^{2}v_{s}}\cdot \int_{0}^{\omega_{D}}\frac{\omega^{2}}{B\omega^{2}+A\omega^{4}}d\omega \tag{46}\]  
We can define a characteristic frequency \(\omega_{0}\) using the condition \(\tau_{U}\left(\omega_{0}\right)=\tau_{D}\left(\omega_{0}\right)\) where the impurity scattering time constant is equal to the intrinsic Umklapp time constant. This condition can be expressed as, using Equation (41):  
\[\frac{1}{B\omega_{0}^{2}}=\frac{1}{A\omega_{0}^{4}} \tag{47}\]  
This may be re-arranged to yield  
\[\omega_{0}=\sqrt{\frac{B}{A}} \tag{48}\]  
Equation (46) may be simplified to give  
\[k=\frac{k_{B}}{2\pi^{2}v_{s}B}\cdot\int_{0}^{\omega_{D}}\frac{1}{1+\left( \frac{A}{B}\right)\omega^{2}}d\omega \tag{49}\]The integral is Equation (49) may be solved analytically, yielding  
\[\frac{1}{\tau_{U}}=B\omega^{2}\,\ \frac{1}{\tau_{D}}=A\omega^{4} \tag{41}\]0  
We now define the intrinsic thermal conductivity \(k_{0}\) by setting \(A=0\) in Equation (47), and taking \(k=k_{0}\), the following results:  
\[\frac{1}{\tau_{U}}=B\omega^{2}\,\ \frac{1}{\tau_{D}}=A\omega^{4} \tag{41}\]1  
Dividing Equation (50) by Equation (51), we obtain the result  
\[\frac{1}{\tau_{U}}=B\omega^{2}\,\ \frac{1}{\tau_{D}}=A\omega^{4} \tag{41}\]2  
By use of Equation (39) and with \(B\) being proportional to the temperature \(T\), we can use Equation (48) to get  
\[\frac{1}{\tau_{U}}=B\omega^{2}\,\ \frac{1}{\tau_{D}}=A\omega^{4} \tag{41}\]3  
where \(K\) is a proportionality constant.