## 3 Results  
The primary objective of this study is to explain why the addition of a small amount of titanium (at the expense of uranium) in the U-6 Nb alloy promotes the stabilization of the \(\gamma^{0}\)-tetragonal phase at the expense of the \(\alpha^{\prime\prime}\)-monoclinic phase [29]. However, we demonstrate the applicability of the DFT approach more generally for unalloyed uranium metal before addressing this particular alloy. For elemental metals, the most exact results are obtained from the FPLMTO method because of its lack of any structure-geometrical approximations. In Figure 5, we show total energies for \(\alpha\), \(\alpha^{\prime\prime}\), \(\gamma^{0}\), and \(\gamma\) uranium, as functions of the atomic volume for both FPLMTO and EMTO methods. In these calculations, the axial ratios, and internal parameters for \(\alpha\), \(\alpha^{\prime\prime}\), and \(\gamma^{0}\) have been optimized (relaxed) in order to produce the lowest energy state utilizing the FPLMTO approach. The same structure (no further relaxation) is adopted for the EMTO computations. The details of these structures can be found in Refs. [7; 13; 16; 18; 27; 31] (see Appendix A). Importantly, we find that the ranking of the phases in terms of energy is the same for both methods with the ground-state \(\alpha\) phase lowest and the BCC (\(\gamma\)) phase highest. The energy differences are also about the same, except for EMTO over-estimating \(\gamma^{0}\) relative to FPLMTO. There are also slight shifts in the equilibrium volumes between these two methods, where EMTO predicts smaller values than both FPLMTO and experiments. For example, FPLMTO predicts an \(\alpha\)-phase volume of 20.67 A3, while EMTO concludes 19.94 A3. The smaller volumes that were obtained from EMTO calculations originate from neglecting the spin-orbit coupling [49; 67] (in the present study only scalar relativistic EMTO calculations are performed). The experimental zero-Kelvin volume is very close to the FPLMTO value, 20.55 A\({}^{3}\)[70]. In Figure 6 we illustrate the FPLMTO energy landscape associated with \(\gamma\) (BCC) and \(\gamma^{0}\) (tetragonal) phases at a constant atomic volume 20.3 A\({}^{3}\). Interestingly, there are no local maxima or minima, except at the locations of \(\gamma^{0}\) and \(\gamma\) phases, on this two-dimensional energy surface. The optimized \(\gamma^{0}\) structure (c/a = 0.94 and z = 0.13) is adopted for the EMTO calculations of this phase. Uranium metal also has a complicated 30-atom tetragonal \(\beta\) phase that only exists at high temperatures [1] and it is less important for the U-Nb system. Therefore, we have not included \(\beta\)-U in our current modeling. Figure 5: Density-functional-theory (DFT) total energies; full-potential linear muffin-tin orbital (FPLMTO) (solid), and EMTO (dashed) total energies for \(\alpha\), \(\alpha^{\prime\prime}\), \(\gamma^{0}\), and \(\gamma\) uranium. Figure 6: DFT-FPLMTO energy landscape for uranium at a constant atomic volume of 20.3 Å\({}^{3}\). The “+” indicates the lowest (optimized or relaxed) energy for \(\gamma^{0}\) (shifted to zero energy) and the top right corner, “BCC”, that of the \(\gamma\) energy. For an explanation of the atomic displacement z, see Refs. [7; 16; 18; 23; 27; 31]. In Figure 7, we show the total energies for \(\alpha\)\", \(\gamma^{0}\), and \(\gamma\) structures, mentioned above for uranium, calculated for pure uranium and niobium metals within both FPLMTO and EMTO methods. The total energy of the \(\alpha\) (monoclinic) structure is assumed to be equal to zero (as the reference point). As in the case of uranium, for niobium metal, the axial ratios and internal parameters for \(\alpha\), \(\alpha\)\", and \(\gamma^{0}\) have been optimized (relaxed) in order to produce the lowest energy state utilizing the FPLMTO approach. The same structure (no further relaxation) is adopted for the EMTO computations for niobium.