## 2 Computational Methods  
We leverage a couple of numerical methods to study the U-Nb alloy system. These methods rely on density-functional theory that was conceived more than 50 years ago [40; 41]. DFT is exact as a concept, but, in practice, approximations are necessary. For example, the electron exchange and correlation interactions, for which we use the generalized gradient approximation (GGA) [42; 43], are more accurate for actinides than the older local density approximation (LDA) [44]. For niobium, the solids-biased gradient approximation PBEsol [45] is better, but since we are focusing on the uranium-rich side of the U-Nb phase diagram we adopt the GGA either in its original [42] or similar but simplified form [43]. There are discussions in the literature on applying an intra-atomic Coulomb interaction with a Hubbard \(U\) parameter (DFT + \(U\)) for metallic uranium fuels [46], but we have previously proved that this ad hoc interaction does not improve the accuracy of the model, but rather worsens it [47; 48; 49]. Regarding numerical details of the computations, we apply two methodologies. First, we employ a reliable [50] all-electron full-potential linear muffin-tin orbital (FPLMTO) method [51]. Our FPLMTO method makes no numerical approximations, except for that of the GGA. Basis functions, charge densities, and potentials are represented in spherical harmonics inside non-overlapping spheres that surround each atom and in Fourier series in the region between these muffin-tin spheres. The electron potential includes all relativistic terms, including spin-orbit interaction [48]. Orbital polarization only operates on the d and f states. Generally, the f-electron systems are exceptionally represented with 6s and 6p pseudocore states and 7s, 7p, 5f, and 6d valence-band states. The number of k points that are included in the calculations depends on the crystal structure, but we apply \(\sim\)1000 k points for the simpler structures and fewer for the multi-atom phases. It should be mentioned that this method has been validated and proven accurate on actinide systems numerous times dating back to the early 1990's up until very recently [37; 38; 49; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78; 79; 80; 81; 82; 83; 84; 85; 86; 87; 88; 89; 90; 91; 92; 93; 94; 95; 96; 97; 98; 99; 100; 101; 102; 103; 104; 105; 106; 107; 108; 109; 111; 11; 110; 111; 112; 113; 114; 115; 116; 117; 118; 119; 120; 121; 122; 123; 124; 125; 126; 127; 128;  
The calculations we have denoted as the Exact Murffin-Tin Orbitals (EMTO) method are performed while using the Green's function formalism based on the improved screened Korringa-Kohn-Rostoker technique, where the one-electron potential is defined by optimized overlapping muffin-tin (OOMT) potential spheres [55; 56]. Inside the potential spheres the potential is spherically symmetric, and it is permanent between the spheres. The radius of the potential spheres, the spherical potential inside these spheres, and the permanent value in the interstitial region are resolved by minimizing (i) the deviation between the exact and overlapping potentials and (ii) the errors that are caused by the overlay between the spheres. Within the EMTO scheme, the one-electron states are calculated exactly for the OOMT potentials. As a product of the EMTO calculations, one can determine self-consistent Green's function of the system and the complete, non-spherically symmetric charge density. Lastly, the total energy is calculated while using the full charge-density approach [57]. We consider, as the valence states, the 7s, 6p, 6d, and 5f states for U, 5s, 4p, and 4d states for Nb and 4s, 3p, and 3d for Ti. The corresponding Kohn-Sham orbitals are expanded in terms of spdf exact muffin-tin orbitals, i.e., we selected an orbital momentum cutoff, \(l_{max}=3\).