# Mechanical and Thermal Properties for Uranium and U-6 Nb Alloy from First-Principles Theory
## 2 Computational Methods  
All results presented in this article are derived from the calculated free energy of uranium or niobium atoms and their geometrical arrangement. The theoretical framework is established by DFT, which only fundamental approximation is that of the electron exchange and correlation. There are many different alternatives for this approximation that govern the quantum-mechanical interaction between the electrons. They make different assumptions that are better or worse depending on the material in question. Fortunately, the generalized gradient approximation (GGA) [34] is known [35] to be quite good for actinide metals, including uranium. There are better variants for niobium (a 4d transition metal) but the niobium content in the studied uranium-niobium alloy is relatively low (no more than 12.5% of the atoms) and therefore the GGA remains a good choice. Because the GGA depends on the spin-polarized electron density, the effects of magnetism and the relativistic spin-orbit interaction are included in the theory. However, in the conventional DFT formulation, there are no electron currents, so orbital-orbital interaction is ignored. Nonetheless, for the actinides and particularly plutonium [36], this interaction is important and there are approximate ways to amend DFT to include it. The \"orbital polarization\" (OP) scheme defines a parameter-free self-consistent procedure that is computationally tractable [37] and we employ it here (DFT+OP), even though its effect in uranium is less pronounced than in plutonium [38]. For best accuracy, it is essential to minimize numerical or other approximations beyond the GGA, and the all-electron technique is preferred over methods that rely on pseudopotential or other approximations for core electrons not included in the valence. Here, we utilize the all-electron full-potential linear muffin-tin orbitals (FPLMTO) method [39]. Basis functions, electron densities, and potentials are calculated with no geometrical approximation, and these are expanded in spherical harmonics (with a cutoff \(l_{\text{max}}=8\)) inside non-overlapping (muffin-tin) spheres around the atoms and in Fourier series in the space outside these muffin-tin spheres. The muffin-tin sphere radius is fixed and close to 0.8 of that of a sphere with a volume equal to the atomic volume. All calculations utilize semi-core states and valence states with two fixed energy parameters each for the s semi-core state, p semi-core state, and the valence states. The number of k points included in the electronic structure calculations depends on the crystal structure, but we generally use ~2000 k points or more for one atom/cell calculations, and less for structures with more atoms. Each energy eigenvalue is broadened with a Fermi-Dirac function at room temperature (300 K) unless free energies at higher temperatures are considered. We introduce a 16-atom supercell approximation for the U-6 Nb alloy by replacing 2 of the 16 uranium atoms with niobium in an ordered U14 Nb2 compound, either in the monoclinic \(\alpha^{\prime\prime}\) phase or the orthorhombic \(\alpha\) phase. This assumption is analogous to the study by [PERSON] et al. [32]. Geometrical energy optimization (structural relaxation) is performed for the genuine phases (\(\alpha\) for uranium and \(\alpha^{\prime\prime}\) for the U-6 Nb alloy). For \(\alpha\) uranium, the relaxation produces axial ratios b/a = 2.045, c/a = 1.755, an internal parameter, y = 0.1009, and an atomic volume 20.67 A\({}^{3}\). Correspondingly, the uranium-niobium alloy in the \(\alpha^{\prime\prime}\) phase has axial ratios b/a = 2.022, c/a = 1.753, monoclinic angle 91.3\({}^{\circ}\), internal parameters y\({}_{1}\) = 5/6 and y\({}_{2}\) = 1/3, and atomic volume 20.48 A\({}^{3}\). Specifically, the \(\alpha\) phase (Cmcm space group) has primitive vectors in cartesian coordinates (cc): (a/2, -b/2, 0); (a/2, b/2, 0); (0, 0, c) with the two atomic positions in cc: (0, by, c/4); (0, -by, -c/4).