# Development of a CALPHAD Thermodynamic Database for Pu-U-Fe-Ga Alloys
## 2 Thermodynamic Modeling
### Electronic Structure Calculations
The prototype structure is the same as U\({}_{6}\)Fe, but the exact crystallographic parameters are not known. Therefore, it is necessary to calculate these parameters, or the energy will be too high for the compound. Hence, we have relaxed the crystal structure (i.e., found the structural parameters that correspond to the lowest total energy) for Pu\({}_{6}\)Fe. The obtained parameters are significantly different than those for U\({}_{6}\)Fe. For completeness, structural relaxation is performed for all FPLMTO calculations, but we find that for U\({}_{6}\)Fe the optimized structure is very close to the experimentally reported one. The calculations referred to as exact muffin-tin orbital method (EMTO), are performed using the Green's-function technique based on the improved screened Korringa-Kohn-Rostoker method, where the one-electron potential is represented by optimized overlapping muffin-tin (OOMT) potential spheres [24; 25]. Inside the potential spheres the potential is spherically symmetric, and it is constant between the spheres. The radius of the potential spheres, the spherical potential inside these spheres, and the constant value in the interstitial region are determined by minimizing (i) the deviation between the exact and overlapping potentials, and (ii) the errors caused by the overlap between the spheres. Within the EMTO formalism, the one-electron states are calculated exactly for the OOMT potentials. As an output of the EMTO calculations, one can determine self-consistent Green's function of the system and the complete, non-spherically symmetric charge density. Finally, the total energy is calculated using the full charge-density technique [26]. The valence states are treated as the \(7s\), \(6p\), \(6d\), and \(5f\) states for U and Pu and \(4s\) and \(3d\) states for Fe. The corresponding Kohn-Sham orbitals are expanded in terms of \(spdf\) exact muffin-tin orbitals, i.e., an orbital momentum cutoff is applied, where \(\mathrm{l_{max}}=3\). The EMTO orbitals, in turn, consist of the \(spdf\) partial waves (solutions of the radial Schrodinger equation for the spherical OOMT potential wells) and the \(spdf\) screened spherical waves (solutions of the Helmholtz equation for the OOMT muffin-tin zero potential). The completeness of the muffin-tin basis was discussed in detail in Ref. [25] and it was shown that for metals crystallizing in close-packed lattices \(\mathrm{l_{max}}=3\) (\(spdf\) orbitals) leads to the well converged charge density and total energy. For the electron exchange and correlation energy functional, GGA (PBE96) is considered [27]. Integration over the Brillouin zone is performed using \(25\times 25\times 25\) k-points grid generated according to the Monkhorst-Pack scheme [28]. The moments of the density of states, needed for the kinetic energy and valence charge density, are calculated by integrating the Green's function over a complex energy contour (with 1.9-2.4 Ry diameter) using a Gaussian integration technique with 40 points on a semi-circle enclosing the occupied states. When treating compositional disorder, the EMTO method is combined with the coherent potential approximation (CPA) [24]. For instance, the EMTO-CPA formalism has been successfully used to describe thermodynamic properties of the metallic nuclear fuels, including: U-Zr, U-Mo, Pu-Zr, Pu-Mo, Pu-U, Pu-Np, Pu-Am, U-Ti, U-Nb, Np-Mo, Pu-Mo, for fast breeder reactors [29; 30; 31; 32; 33; 34; 35; 36]. Relaxing (or optimizing for lowest energy) the crystal structures are not currently feasible and for this reason all calculations applying the EMTO assumes the experimental structure. For Pu\({}_{6}\)Fe the experimental structure is not known in detail and in this case the parameters for U\({}_{6}\)Fe have also been applied for Pu\({}_{6}\)Fe.