## 4 SCG Model for Cerium
### Shear Modulus  
Our second step in the development of a SCG model for cerium is directed toward the shear modulus. Here, basing our model on that of [9] is a bit convoluted since, in [9], \(G=G(\rho,T)\), whereas under the SCG framework \(G=G(\rho,p,T)\) and pressure plays a prominant role in the model (see Equation (28)). [PERSON] et al. expressed \(G=G(\rho,T)\) as:  
\[G(\rho,T)=G_{0}(\rho)\left(1-\alpha\,\frac{T}{T_{m}(\rho)}\right) \tag{43}\]  
For the SCG model of shear modulus, we require five parameters (see Equation (28)): \(A\), \(B\), \(G_{0}\), \(\rho_{0}\), and \(T_{r}\)). We shall set \(\rho_{0}=6.7605\) g/cm\({}^{3}\) as in the previous section, and set \(T_{r}\) to 300 K. We shall set \(G_{0}\) to the value calculated using Equation (36), which turns out to be 10.94 GPa. Note that, in the calculation of \(G_{0}\), we set the values of \(G_{0}(\rho_{0})\), \(\rho_{0}\), \(\tau_{1}\), \(\tau_{2}\), and \(q_{2}\) to the values used in [9]; namely, 11.0 GPa, 6.775 g/cm\({}^{3}\), 2.62215, \(-5.0\), and 1.0, respectively. Parameter \(B\) can be calculated directly from Equation (43) as follows:  
\[B = -\frac{1}{G_{0}(\rho)}\frac{\partial G}{\partial T}\] \[= \left(-\frac{1}{G_{0}(\rho)}\right)\left(-\frac{G_{0}(\rho) \alpha}{T_{M}(\rho)}\right)\] \[= \frac{\alpha}{T_{m}(\rho)}\] \[B = 3.435\times 10^{-4}\;\text{K}^{-1}\]  
The determination of parameter \(A\) is somewhat less straightforward. Recall that \(A=1/G_{0}\;\;\partial G/\partial p\) (see Equation (30)). Since \(G\) in [9] is not a function of pressure, we are unable to calculate its value directly as was done above for parameter \(B\). We need \(G=G(\rho,p,T)\) from which we can determine \(\partial G/\partial p\). To accomplish this, we set \(T=300\) K and we use SESAME 90601 to obtain \(p=p(\rho)\). Then for any value of \(\rho\) we know the corresponding value of \(p\). Using this information and Equation (36) we can construct \(G=G(p)\) at \(\rho=\rho_{0}\) and \(T=300\) K. We can then fit a straight line to that construct, from which \(\partial G/\partial p\) is easily determined. That construct so determined along with a linear fit to that construct is shown in Figure 4. The value of \(A\) (i.e, the slope of the linear fit shown in Figure 4) is 0.0676322 GPa\({}^{-1}\), which we shall treat as 0.0676 GPa\({}^{-1}\) in subsequent calculations. Figure 4: Shear modulus as a function of pressure. The SCG model of the shear modulus determined as described above is compared to the model of [PERSON] et al. (2019) (Equation (43)) in Figure 5. The fact that \(G=G(\rho)\) (for constant \(T\)) differs significantly between the two models shown in Figure 5 is not of particular concern. Recall that, in the SCG model, \(G=G(\rho,p,T)\), whereas in the model of ([PERSON] et al., 2019), \(G=G(\rho,T)\). Consequently, it is not surprising that there are some differences in this regard. Note that it is quite possible that, with some other relationship between density and pressure (we chose to base that relationship on SESAME 90601),we would see a smaller difference between the curves shown in Figure 5. It is also possible that, if a five-term Taylor-Maclaurin power series had been used in the development of Equation (36) instead of a three-term series, this might also have served to result in a smaller difference between the two curves. At any rate, our objective is not to match the curve generated by Equation (43), but rather to develop a model based in large part upon SESAME 90601 that is logical and therefore justifiable. For given values of density and pressure (e.g., 6.6705 g/cm\({}^{3}\) and 1.01325 \(\times\) 10\({}^{-4}\) GPa, respectively), there is very little difference between plots of \(G\) versus \(T\) derived from \(G=G(\rho,T)\) as expressed in Equation (43) and from \(G=G(\rho,p,T)\) as expressed in our SCG model, as is shown in Figure 6. Again, the difference shown is Figure 6 is of little concern.