# Analytic Model for U-Nb Liquidus and U-6 Nb Melting Curve
## 3 Application to the Uranium-Niobium System
### Melting Simulations of \(\gamma\)-U  
Here, we present the results of our QMD simulations of the melting of U, considered as its bcc modification \(\gamma\)-U, which were carried out using the Z method implemented with VASP. The Z method is a procedure used to calculate the melting curves of materials by modeling a solid system at different initial total energies. The idea of the method was put forward by [PERSON] et al. in Ref. [28]. A solid system is modeled in an NVE ensemble (\(N\) is the number ot atoms, and \(V\) and \(E\) are the volume and total energy of the system, respectively). In each simulation, \(E\) is controlled by the corresponding choice of the initial \(T\). Essentially, the Z method simulates a solid in a superheated limit. The solid will melt spontaneously if the simulation is long enough, and \(T\) will drop to \(T_{m}\) (at \(P=P_{m}\)). The shape of the \(P\)-\(T\) trajectory mapped out by the final states of simulations with different initial \(T\)s resembles the letter Z, which is the origin of the method's name. The Z method has been used in numerous studies of melting curves, e.g., of Zn [29], Sn [30], and Pt [17], just to name a few. The practical implementation of the Z method can be found in Refs. [28; 31; 32]. The typical time evolution of \(P\) and \(T\) in a QMD run is shown, with an example of Pa considered in what follows. In our simulations of the melting of U, its electronic structure was represented by [Xe 4f14 5d10] 5f3 6s2 6p6 6d1 7s2 (14 valence electrons). For the electron exchange and correlation functionals, the Perdew-Burke-Ernzerhof (PBE) implementation of generalized gradient approximation (GGA) was used, as suggested in Ref. [33]. A large 432-atom (\(6\times 6\times 6\)) supercell with a single \(\Gamma\)-point was used to avoid potential size effects associated with the mechanical instability of bcc at low \(T\) and its high-\(T\) stabilization. As our (unpublished) study of Ti, Zr, and Hf, which all melt from bcc and have bcc mechanically unstable at low \(T\) but stabilized at high \(T\), just like in U, shows, the critical supercell sizes needed to avoid the size effects related to bcc stabilization are 686 (\(7\times 7\times 7\)) for both Ti and Zr but 432 for Hf. Thus, the critical supercell size seems to decrease with the atomic number; hence, we assume that a 432-atom supercell is adequate for the simulations of U. We considered six different cases of the \(\gamma\)-U unit cells, which correspond to six different U densities and six different melting points on the U melting curve. In each of the six cases, the corresponding supercell was subject to a set of initial temperatures (\(T_{\rm init}\)) 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 15,000 time steps of 2 fs each, i.e., up to 30 ps, to determine the corresponding values of \(P_{m}\) and \(T_{m}\). Uncertainty of the value of \(T_{m}\) intrinsic to the Z method is therefore 75 K in the first case and 125 K in the remaining five cases, half of the corresponding increment in \(T_{\rm init}\)[32], which constitutes 3-9% of \(T_{m}\) (the largest uncertainty of 9% is for the first datapoint, and the lowest one of 3% for the last one). The uncertainties of \(P_{m}\) are negligibly small, within 0.5 GPa.