## 2 Review of Cryogenic-Cooling Benefits
### Thermal Parameters
#### 2.2.2 Thermal Conductivity Standard Debye Model  
Using the kinetics theory of gases, applied to a phonon gas, and the Debye specific heat theory [39], the thermal conductivity \(k\) in its simplest form can be written as  
\[k=\frac{1}{3}Cv_{s}\ \Lambda=\ \frac{1}{3}Cv_{s}^{2}\ \tau \tag{26}\]  
where \(C\) is the specific heat at constant volume, per unit volume. \(C\) is obtained from the relationship  
\[C=C_{V}\ \rho \tag{27}\]  
Here \(C_{V}\) is the specific heat per unit volume per unit mass and \(\rho\) the mass density. Note that for dielectric crystals, the specific heats for constant volume and pressure are normally assumed to be equal (\(C_{V}=C_{P}\)). \(v_{s}\) is the mean phonon speed in the crystal, and \(\Lambda\) is the phonon mean-free path. \(v_{s}\) can be calculated for any crystal using the sound speeds for the acoustic transverse and longitudinal waves, \(v_{t}\) and \(v_{l}\) respectively, and accounting for the fact that there are twice as many transverse as longitudinal waves, from  
\[v_{s}=\left[\frac{1}{3}\left(\frac{2}{v_{t}^{3}}+\frac{1}{v_{l}^{3}}\right) \right]^{-1/3} \tag{28}\]  
The transverse and longitudinal sound speeds may be calculate using the elastic coefficients, or equivalently for anaxial crystals, the bulk and shear moduli and the mass density. For YAG (Y\({}_{3}\)Al\({}_{5}\)O\({}_{12}\)) and YbAG (Yb\({}_{3}\)Al\({}_{5}\)O\({}_{12}\)), which we will discuss later in this paper, a recent publication [40] provides values for \(v_{t}\) and \(v_{l}\), as well as for the bulk and shear moduli, Poisson's ratio, Pugh's ratio, and Young's modulus. \(\tau\) is the phonon relaxation time, which for undoped crystals is the Umklapp relaxation time. In the Debye approximation, the total phonon energy can be expressed as  
\[S\left(\omega\right)=\left[\frac{9 nN_{A}\ \rho}{M}\right]\cdot\left[\frac{ \hbar}{\omega_{D}^{3}}\right]\cdot\int_{0}^{\omega_{D}}\frac{\omega^{3}}{ \left[\frac{\hbar\omega}{e^{\frac{\omega^{3}}{k_{B}T}}-1}\right]}d\omega \tag{29}\]  
\(n\) is the number of atoms in the crystal chemical formula, \(N_{A}\) Avogadro's number, \(M\) the molecular mass/mole, \(\hbar\) Planck's constant divided by \(2\pi\), \(\omega_{D}\) the Debye frequency, \(k_{B}\) the Boltzmann constant, \(\omega\) the angular phonon frequency, \(T\) is the absolute temperature and \(\rho\) the mass density. The derivative of Equation (29) with respect to temperature yields the specific heat per unit volume:  
\[C=\frac{dS\left(\omega\right)}{dT}=\left[\frac{9 nN_{A}\rho}{M}\right]\cdot \left[\frac{\hbar^{2}}{{k_{B}}^{2}\omega_{D}^{3}T^{2}}\right]\cdot\int_{0}^{ \omega_{D}}\frac{\frac{\hbar\omega}{\omega^{4}e^{\frac{\hbar\omega}{k_{B}T}} }}{\left[\frac{\hbar\omega}{e^{\frac{\hbar\omega}{k_{B}T}}-1}\right]^{2}}d\omega \tag{30}\]  
The Debye frequency is related to the Debye temperature \(\theta_{D}\) through the relationship  
\[\theta_{D}=\frac{\hbar\omega_{D}}{k_{B}} \tag{31}\]If we define the ratios \(x=\hbar\omega/k_{B}T\) and \(x_{D}=\hbar\omega_{D}/k_{B}T\), divide by \(\rho\), and paramaterize Equation (29), we find the standard Debye expression for the specific heat [39]:  
\[C_{V}=\left[\frac{9 nN_{A}}{M}\right].\,\cdot\,\left(\frac{T}{\theta_{D}}\right) ^{3}\cdot\,\int_{0}^{x_{D}}\frac{x^{4}e^{x}}{\left[e^{x}-1\right]^{2}}dx \tag{32}\]  
where \(x\) is integrated from \(0\) to \(x_{D}\). There are two often-used limits to Equation (32). For large temperature values (\(T>>\theta_{D}\)), the exponential function may be expanded in a Taylor series, with the result being the constant  
\[C_{V}=\frac{3 nN_{A}k_{B}}{M} \tag{33}\]  
This result is known as the Dulong-Petit law.