# Analytic Model for U-Nb Liquidus and U-6 Nb Melting Curve
## 5 U-18 Nb as a Proxy for Protactinium  
Another much less known application of the U-Nb system is the ability of U-18 wt.% Nb (U-18 Nb) to serve as a proxy for protactinium. Protactinium (Pa), element 91, falls between thorium and uranium and is one of the more intractable natural radioactive elements. Protactinium is one of the rarest and most expensive naturally occurring elements. The average concentrations of Pa in the Earth's crust is typically on the order of a few parts per trillion, but may reach up to a few parts per million in some urannite ore deposits. It is only comparatively recently that a reasonably clear understanding of its physical and chemical properties has begun to emerge. Pa is typically obtained through isolation from spent nuclear fuel, typically uranium materials such as U-Nb, and subsequent purification. However, separating Pa from U-Nb alloy, which, as mentioned above, is a common material in the nuclear fuel cycle, is challenging due to the chemical similarity of Pa and Nb, as the two, along with vanadium and tantalum, share the same vertical group in the periodic table. Because of all this, Pa is extremely difficult to study in in situ experiments, and theoretical approaches should be invoked.  
It appears that having the values of density, bulk modulus, and ambient melting temperature all very close to those of Pa, the U-18 wt.% Nb (U-18 Nb) alloy is a natural candidate for being a proxy for protactinium. Indeed, the \(T=0\) density of Pa is \(\approx\)15.5 g/cm\({}^{3}\) (\(\approx\)15.4 at 300 K) [7], its (ambient) bulk modulus is in the interval 100-157 GPa [38], and the three experimental measurements cited in [21], 1848 \(\pm\) 20, 1833 \(\pm\) 30, and 1838 \(\pm\) 20 K, give the ambient \(T_{m}\) of Pa of \(1840.5\pm 22.5\) K, or 1818-1863 K. For U-18 Nb, the corresponding values are as follows: (i) a density of 15.69 g/cm\({}^{3}\), from Equation (3), with \(x=0.18\) and the literature values of \(\rho_{\rm U}=19.14\) and \(\rho_{\rm Nb}=8.620\); (ii) a bulk modulus of 132.9 GPa, from Equation (4), with \(x=0.18\) and the literature values of \(B_{\rm U}=120.0\) and \(B_{\rm Nb}=170.5\); and (iii) the ambient \(T_{m}\) of 1863.0 K from Equation (8). Thus, the two densities agree within 1%, the bulk modulus of U-18 Nb is right in the middle of the experimental interval of that of Pa, and the ambient \(T_{m}\) of U-18 Nb is at the upper end of the experimental interval of that of Pa. It is therefore quite plausible to consider U-18 Nb as a proxy for Pa, at least in regard to its melting curve, i.e., the behavior of \(T_{m}\) as a function of \(P\).  
We now consider the application of Equation (7) to U-18 Nb, where the two \(P_{\ell}\)s are determined by the corresponding SG forms (10) and (12), and \(C\) (\(\approx\)const) = \(-0.6\). With \(x=0.18\) (18 wt.% Nb), we numerically solve Equation (7) at the seven \(T\)s between 2000 (close to \(T_{\ell}^{U-6 Nb}(0)=1863.0\) K) and 4400 K with an increment of 400 K. The seven \((P_{\ell},T_{\ell})\) points are listed in Table 3 along with the ambient melting point.  
The best fit of the SG form to the eight datapoints in Table 4 results in the model melting curve of U-18 Nb:  
\[T_{\ell}^{\rm U-18 Nb}(P)\,=\,1863.0\,\bigg{(}1\,+\,\frac{P}{19.67}\bigg{)}^{0.4866}, \tag{19}\]  
for which \(T_{\ell}^{\rm dU-18 Nb}(0)=1863.0\cdot 0.4866/19.67=46.087\) K/GPa, which virtually coincides with 46.083 given by Equation (9). This melting curve is shown as a gray line in Figure 3.  
It is interesting to compare this melting curve of U-18 Nb with the results of QMD melting simulations on Pa, which U-18 Nb is presumably a proxy for.