# Analytic Model for U-Nb Liquidus and U-6 Nb Melting Curve
## 7 Concluding Remarks  
Here, we briefly summarize the main results of this work. In this study, we have presented a theoretical model for the melting curve (liquidus) of a mixture and applied it to the uranium-niobium system being considered as a mixture of pure U and pure Nb. The model is based on several assumptions listed in Section 2, the use of which seems fully justified by the reliability of the results produced by the model. The advantage of our formulation over alternative theoretical approaches that may be found in the literature is that it only requires the knowledge of the melting curves of the constituents of the  
Figure 5: The same as in Figure 4 for pressure. During the melting run with \(T_{0}\) = 11,250 K pressure increases from \(\sim\)96.8 GPa for a superheated solid up to \(\sim\)97.3 K for a liquid. mixture but not of their EOSs or other thermodynamic parameters, and it only uses one free parameter, \(C\), the value of which can be easily determined based on the available experimental data. As the examples of both the Mo-W system considered in this work and the uranium-plutonium mixed oxide (MOX) system considered in our previous work [15] clearly demonstrate, large variations in the value of \(C\), by as much as 60% in either case, cause a very minor shift in the liquidus of the system within \(\sim\)50 K. Thus, as already mentioned above, the exact knowledge of the value of \(C\) may not be really necessary to obtain the liquidus of a mixture in good agreement with experiment. The examples of the application of the model to real mixtures, Mo-W and U-Nb, considered in this work, as well as Si-Ge and MOX, considered in our previous work, clearly demonstrate that the model is reliable and relatively easy to apply in practice, in contrast to more complicated and more time-consuming Calphad calculations. A comparison of the U-Nb liquidus given by our analytic model to the available experimental results is shown in Figure 2, which demonstrates very good agreement between the model and the available experimental data. Figure 2 represents the current knowledge of the ambient phase diagram of the U-Nb system; this knowledge may be advanced further in subsequent studies on the subject. The example of U-6 Nb of the generalization of our analytic model to nonzero \(P\) clearly demonstrates that keeping the value of \(C\) a constant over some range of \(P\) is a good approximation. In the case of U-6 Nb, keeping \(C=-0.6\) fixed works well for pressures up to \(\sim\)50 GPa, as Figure 3 clearly shows. At higher \(P\), the model curve and QMD results start to deviate from each other, but the deviation is small and so agreement between the two remains good even with further increases in \(P\). In fact, below 100 GPa, the deviation of the model from QMD remains within 10%. Therefore, within the pressure range 0-100 GPa, the model should be considered as quite reliable in its ability to predict the melting curve of U-6 Nb, and perhaps the melting curves of U\({}_{1-x}\)Nb\({}_{x}\) with different alloy compositions. Further improvement in agreement between the model and experiment and/or independent theoretical calculations can be achieved via the determination of the more exact physical nature of the coupling parameter \(C\), particularly its possible \(x\) dependence and pressure (density) dependence. Below, we present a formula for such a possible \(x\) dependence of \(C\), which ensures an almost exact match between the model liquidus and the ambient data. It is very likely that the use of such an \(x\)-dependent \(C\) will lead to better agreement between the model melting curve and the QMD (as well as the future experimental) results. We have shown that the U-18 Nb alloy can be considered as a proxy for protactinium, an element which is extremely difficult to study experimentally. Specifically, the melting curves of U-18 Nb and Pa are very close to each other, as our QMD simulations on the melting point of Pa at \(\sim\)100 GPa clearly demonstrate.