## 3 PTW Model for Cerium  
As stated earlier, [PERSON], [PERSON], and [PERSON] [9] recently developed a callibration of the PTW model specifically for gamma phase (fcc) cerium. Within this work they treat \(G_{0}\) and \(T_{m}\) not as simple constants but as functions of density: \(G_{0}=G_{0}(\rho)\), and \(T_{m}=T_{m}(\rho)\). Following the methodology presented in [PERSON], [PERSON], and [PERSON], [13], [PERSON] et al. initiate the development of equations for \(G_{0}(\rho)\) and \(T_{m}(\rho)\) with the following definitions:  
\[\frac{d\ln G_{0}(\rho)}{d\ln\rho}\equiv 2\gamma_{G}(\rho)+\frac{1}{3} \tag{32}\]  
\[\frac{d\ln T_{m}(\rho)}{d\ln\rho}\equiv 2\gamma_{T_{m}}(\rho)-\frac{2}{3} \tag{33}\]  
where \(\gamma_{G}(\rho)\) and \(\gamma_{T_{m}}(\rho)\) are the Gruneisen gammas corresponding to the shear modulus and melt, respectively. These two Gruneisen gammas can be calculated as follows:  
\[\gamma_{G}(\rho)=\frac{1}{2}+\frac{\gamma_{1}}{\rho^{1/3}}+\frac{\gamma_{2}}{ \rho^{q_{2}}} \tag{34}\]  
\[\gamma_{T_{m}}(\rho)=\frac{1}{2}+\frac{\gamma_{1}}{\rho^{1/3}}+\frac{\gamma_{ 3}}{\rho^{q_{3}}} \tag{35}\]  
where \(\gamma_{1}=2.62215\), \(\gamma_{2}=-5.0\), and \(\gamma_{3}=-6400\). The interested reader is directed to [13] for a discussion of the methodology used, in general, to determine \(\gamma_{1}\), \(\gamma_{2}\), and \(\gamma_{3}\). Solving Equations (32) and (33) yields the following for \(G_{0}(\rho)\) and \(T_{m}(\rho)\):  
\[G_{0}(\rho)=G_{0}(\rho_{0})\left(\frac{\rho}{\rho_{0}}\right)^{4/3}\exp\left\{ 6\gamma_{1}\left(\frac{1}{\rho_{0}^{1/3}}-\frac{1}{\rho^{1/3}}\right)+\frac{ 2\gamma_{2}}{q_{2}}\left(\frac{1}{\rho_{0}^{q_{2}}}-\frac{1}{\rho^{q_{2}}} \right)\right\} \tag{36}\]  
\[T_{m}(\rho)=T_{m}(\rho_{m})\left(\frac{\rho}{\rho_{m}}\right)^{1/3}\exp\left\{ 6\gamma_{1}\left(\frac{1}{\rho_{m}^{1/3}}-\frac{1}{\rho^{1/3}}\right)+\frac{ 2\gamma_{3}}{q_{3}}\left(\frac{1}{\rho_{m}^{q_{3}}}-\frac{1}{\rho^{q_{3}}} \right)\right\} \tag{37}\]  
where the values of \(\rho_{0}\) and \(\rho_{m}\) were taken from the open literature, and have values of 6.775 and 6.640 g/cm\({}^{3}\), respectively. The value of \(\rho_{m}\), for example, was taken from [PERSON] [14] (pp. 191-192). At the melting point \(\rho_{m}=6.640\) on the solid side and 6.687 on the liquid side, hence liquid is denser than the solid, which implies a negative slope on the melting curve. For the analytic models of [9], it is the value on the solid side that is required (i.e., 6.640 g/cm\({}^{3}\)). The values of \(G_{0}(\rho_{0})\) and \(T_{m}(\rho_{m})\), used in [9], were 11.0 GPa and 1071 K, respectively.  
Reasonable fits of the PTW model that bracket the experimental data available to the authors are given in Tables 1 and 2. Note that parameter set 1 matches parameter set 1 of [9], whereas parameter set 7 differs from parameter set 2 of [9]. Using parameter sets 1 and 7, comparisons are made to the empirical data in Figure 1 for \(T=333\) K, \(\rho=6.6877\) g/cm\({}^{3}\), and \(\dot{\epsilon}=4.2\)\(\times\)\(10^{3}\). The empirical data were provided to the authors by [PERSON] [15]. Two parameter sets have been developed since the empirical data are limited and there is uncertainty associated with any extrapolation beyond the data.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(p\) & 0.286 \\ \(\theta\) & 0.0544 \\ \(\kappa\) & 0.416 \\ \(\gamma\) & 0.00271 \\ \(s_{0}\) & 0.06885 \\ \(y_{0}\) & 0.000746 \\ \(y_{1}\) & 0.1 \\ \(y_{2}\) & 1.0 \\ \(\beta\) & 0.27 \\ \(\alpha\) & 0.36 \\ \hline \hline \end{tabular}
\end{table}
Table 1: PTW parameter set 1.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(p\) & 0.286 \\ \(\theta\) & 0.046 \\ \(\kappa\) & 0.416 \\ \(\gamma\) & 0.00271 \\ \(s_{0}\) & 0.096398 \\ \(s_{\infty}\) & 0.01006 \\ \(y_{0}\) & 0.01 \\ \(y_{\infty}\) & 0.001044 \\ \(y_{1}\) & 0.1 \\ \(y_{2}\) & 1.0 \\ \(\beta\) & 0.27 \\ \(\alpha\) & 0.36 \\ \hline \hline \end{tabular}
\end{table}
Table 2: PTW parameter set 7.