# Analytic Model for U-Nb Liquidus and U-6 Nb Melting Curve
## 4 Generalization of the Theoretical Model to Nonzero \(P\): Melting Curve of U-6 Nb as an Example
### Melting Simulations of U-6 Nb  
As a consistency check for Equation (16), we carried out QMD melting simulations of U-6 Nb using the \(Z\) method implemented with VASP. As discussed above, the addition of Nb enhances the thermal stability of \(\gamma\)-U. Hence, a smaller bcc supercell can be used for the QMD simulations of the melting of U-6 Nb. Specifically, for these simulations, we used a 250-atom \((5\times 5\times 5)\) U-6 Nb bcc supercell. Since for Nb, 6 wt.% corresponds to \(\approx\)14 atomic % [3], the 35 atoms of U chosen at random were substituted with 35 atoms of Nb; thus, the supercell consisted of 215 atoms of U and 35 atoms of Nb. As before, the electronic structure of U was represented by [Xe 4f\({}^{14}\) 5d\({}^{10}\)] 5f\({}^{3}\) 6s\({}^{2}\) 6p\({}^{6}\) 6d\({}^{1}\) 7s\({}^{2}\) (14 valence electrons), while that of Nb by [Ar 3d\({}^{10}\)] 4s\({}^{2}\) 4p\({}^{6}\) 4d\({}^{4}\) 5s\({}^{1}\) (13 valence electrons). The PBE implementation of GGA was used for both U and Nb. Just as for \(\gamma\)-U, we considered six different cases of the U-6 Nb unit cells, which correspond to six different densities and six different melting points on the U-6 Nb melting curve. In each of the six cases, the corresponding supercell was subject to a set of initial 7s separated by an increment of 150 K in the first case and 250 K in the remaining five cases, and run with QMD in the \(NVE\) ensemble, for a total of up to 20,000 time steps of 1.5 fs each, i.e., up to 30 ps. The results of our melting simulations are summarized in Table 3.  
Just as in the case of pure \(\gamma\)-U considered above, the uncertainties of the values of \(T_{m}\) are within 3% for the last datapoint and 9% for the first one, and uncertainties of \(P_{m}\) are negligibly small, within 0.5 GPa. Fitting the SG form to the six melting datapoints in Table 2 results in  
\[T_{\ell}^{\rm U-6 Nb}(P)\,=\,1563.6\left(1\,+\,\frac{P}{15.0}\right)^{0.48}, \tag{17}\]  
which is the ab initio melting curve of U-6 Nb in the interval of \(P\) that corresponds to the six dataponts in Table 3, and for which \(T_{\ell}^{\rm U-6 Nb}(0)\approx 50.0\) K/GPa. If the value of \(T_{\ell}(0)\) is fixed at 1577.6 K, in agreement with (15), the fitting brings up  
\[T_{\ell}^{\rm U-6 Nb}(P)\,=\,1577.6\left(1\,+\,\frac{P}{15.4}\right)^{0.48}, \tag{18}\]  
which is only slightly different from Equation (17). The value of the initial slope of this melting curve (in K/GPa), \(1577.6\cdot 0.48/15.4=49.172\), virtually coincides with that in (15). The melting curve of U-6 Nb given by the new analytic model, Equation (7), with fixed \(C=-0.6\), is shown in Figure 3. The melting curves of both pure U and pure Nb are also shown in Figure 3 for comparison. As Figure 3 clearly demonstrates, the agreement between the new model generalized to nonzero \(P\) (with a fixed value of C) and the ab initio results is very good. This justifies the assumption that keeping the value of C fixed over some range of \(P\) is a good approximation. In our case, keeping \(C=-0.6\) fixed seems to work for pressures up to at least \(\sim\)50 GPa, and the model curve and QMD results start to deviate from each other at higher \(P\), but the deviation is rather small, \(\sim\)300 K at \(\sim\)90 GPa, which constitutes \(\sim\)7.5%. Possible ways to improve the accuracy of the model in regard to its melting curve predictions are discussed below in the concluding remarks. \begin{table}
\begin{tabular}{c c c c c} \hline
**Lattice Constant (Å)** & **Density (g/cm\({}^{3}\))** & \(P_{m}\) (GPa)** & \(T_{\ell}\) (K) & \(\Delta T_{\ell}\) (K) \\ \hline
3.70 & 14.269 & \(-\)11.0 & 860 & 75 \\
3.50 & 16.858 & 4.3 & 1780 & 125 \\
3.35 & 19.225 & 26.5 & 2550 & 125 \\
3.25 & 21.055 & 50.5 & 3170 & 125 \\
3.20 & 22.057 & 66.6 & 3520 & 125 \\
3.15 & 23.125 & 86.1 & 3900 & 125 \\ \hline \end{tabular}
\end{table}