## 2 Review of Cryogenic-Cooling Benefits
### Thermal Expansion Coefficient  
Thermal expansion of crystals can be understood by the realization that in the theory of lattice vibrations, thermal expansion only occurs when the potential energy of a classical harmonic oscillator is expanded to terms of higher order than two, since the quadratic dependence alone results in no net thermal expansion. For the thermodynamic volumetric thermal expansion coefficient \(\alpha_{V}\left(T\right)\), it can be shown that the Gruneisen relationship applies to cubic or isotropic materials. This equation relates the  
Figure 19: Thermal conductivity as a function of absolute temperature for the elemental metals copper (red [56]), titanium (green [56]), molybdenum (black [57]), and aluminum (blue [56]). Figure 18: Thermal conductivity as a function of absolute temperature for the uniaxial optical material \(\text{Al}_{2}\text{O}_{3}\)\(a\) (blue [55]) and \(c\) (green [56]) axes and for Type II a isotropic diamond (red) [56]. Gruneisen constant \(\gamma_{G}\), the bulk isothermal elastic modulus \(B_{T}\) and the crystal mass density \(\rho\) to the bulk thermal expansion coefficient according to  
\[\alpha_{V}=\frac{\rho C_{V}}{B_{T}}.\gamma_{G} \tag{62}\]  
For isotropic laser materials, the linear thermal expansion \(\alpha\left(T\right)\) coefficient is related to the bulk thermal expansion coefficient through:  
\[\alpha_{V}\left(T\right)=3\cdot\alpha\left(T\right) \tag{63}\]  
Using Equations (62) and (63), we can also write the following relationship for the linear thermal expansion coefficient:  
\[\alpha\left(T\right)=\frac{1}{3}K_{T}\left(T\right)\gamma_{G}\rho\left(T \right)C_{V}\left(T\right) \tag{64}\]  
where we have used the relationship \(K_{T}=1/B_{T}\). Finally we can also use the well-known relation for isotropic materials:  
\[K_{T}=\frac{E}{2\left(1-\
u\right)} \tag{65}\]  
where \(E\) is Young's modulus, and \(\
u\) Poisson's ratio, to arrive at  
\[\alpha\left(T\right)=\frac{1}{6}\gamma_{G}\rho\left(T\right)C_{V}\left(T \right)\cdot\frac{E}{\left(1-\
u\right)} \tag{66}\]  
Because \(E\), \(\
u\), and \(\rho\) are weak functions of \(T\), \(\alpha\left(T\right)\) is dominated by the strong functional dependence of \(C_{V}\left(T\right)\) on \(T\). This dependence can be seen in the thermal expansion coefficient data presented later in this section. Since theoretically \(C_{v}\to 0\) as \(T\to 0\), at absolute zero the thermal expansion coefficient is theoretically \(0\) also. While some data deviate from this simple rule, it is generally true that as temperature is lowered the thermal expansion coefficient decreases roughly with the temperature dependence of the specific heat. For simple cubic crystals like YAG, in the temperature range of interest in this paper, one may approximate the temperature dependence of \(C_{V}\) using an exponential function:  
\[C_{V}\left(T\right)=C_{0}\text{exp}\left(-\frac{\hbar\omega}{k_{B}T}\right) \tag{67}\]  
\(C_{0}\) is a proportionality constant equal to  
\[C_{0}=\frac{1}{6}\frac{\gamma_{G}\rho E}{\left(1-\
u\right)} \tag{68}\]  
This approach has been taken in [58], where very good fits to thermal expansion data for YAG were obtained by fitting the proportionality constant \(C_{0}\) to ceramic and crystalline data. Reported Thermal Expansion Coefficient Measurements  
In Figures 20-27 we display plots of thermo-optic coefficient as a function of absolute temperature for doped and undoped crystals. We have only included data sets available in the temperature region of interest in this paper, about 80-300 K, and have not included any single-point measurements, most of which are available at or near 300 K. Some data shown include ceramic laser materials since ceramic and single-crystal data are nearly indistinguishable.