# Thermodynamics of Plutonium Monocarbide from Anharmonic and Relativistic Theory
## 4 Summary and Discussion  
We calculate thermodynamical properties; lattice dynamics, free energies, and heat capacities for ideal-stoichiometry plutonium monocarbide. The highest level of theory includes spin-orbit coupling and orbital polarization for the DFT-GGA electronic structure. This approach assumes delocalized (band) 5f electrons on plutonium and a direct comparison with the experimental photoelectron spectroscopy confirms that this assumption is appropriate for PuC. Furthermore, this interpretation of the 5f electrons is consistent with the calculated Pu-C and Pu-Pu distances (2.52 and 3.57 A, respectively). Namely, the Pu-Pu distance is close to the Hill limit (\(\sim\)3.4 A) [69] that is an approximate criteria for 5f-band formation in the actinides. The thermal properties are obtained from combining our advanced DFT to a temperature-dependent self-consistent phonon method that includes strong anharmonic lattice dynamics. We show that this level of electronic-structure theory and lattice dynamics are necessary to reproduce the CALPHAD Gibbs energy and produce consistent specific-heat data. The quasi-harmonic approaches, on the other hand, predict heat capacities that do not compare as well with CALPHAD and the most believable experimental \(C_{p}\) ([PERSON] and [PERSON] [64]). Still, the simplicity and efficiency of the Debye-Gruneisen technique are appealing for general thermodynamic modeling. However, we suggest that the best results are obtained if the inherent model-parameters are chosen so that experimental data (heat capacity) or the higher-order anharmonic theory (lattice-vibration energy) are reproduced as closely as possible. On the other hand, if the anharmonic effects are substantial, a good fit to either the heat capacity or the Gibbs energy may not be feasible. The consistency between our first-principles modeling, CALPHAD, and experiments is further validated by comparing PuC formation enthalpies. The zero-temperature formation enthalpy for PuC is calculated, with our best theory (DFT+SO+OP), as the total-energy difference between the compound and its solid-form constituents, i.e., carbon (\(\alpha\)-C, graphite) and \(\alpha\)-plutonium; E(PuC)-E(\(\alpha\)-Pu)-E(\(\alpha\)-C) = \(-\)21 kJmol\({}^{-1}\). Here, the structure of all phases is carefully relaxed (not a trivial task for \(\alpha\)-plutonium that is a complex monoclinic phase [5]). The first-principles formation enthalpy is in satisfactory agreement with CALPHAD data (\(-\)26.4 kJmol\({}^{-1}\)) and the experimental values, that range from \(-\)15.4 to \(-\)25.3 kJmol\({}^{-1}\)[16; 66]. Hence, there is favorable coherence in the thermodynamical properties between modeling and experiments. The high-temperature upturn in the specific heat found in two experiments [65; 66] could possibly be a magnetic effect but is more likely due to the defects of the material that are also affecting the comparisons with theory. Namely, the correlation between theoretical and experimental specific heats is obscured by the fact that plutonium monocarbide naturally forms with a deficiency of carbon atoms, i.e., PuC1-x, where x is \(\sim\)0.1. One can capture the effect of carbon deficiency in supercell calculations by removing carbon atoms and creating vacancies on the carbon sites. Specifically, by removing one carbon atom from a 16-atom supercell, we study the PuS\({}_{\text{C7}}\) compound, i.e., PuC1-x, where x = 0.125. The corresponding calculation is performed with VASP and the full DFT+SO+OP treatment, allowing for structural relaxation of the supercell with the vacancy. It turns out that the relaxation effects are small, but the atomic volume increases significantly accompanied by a softening of the bulk modulus and a small enhancement (\(\sim\)0.2 uB) of the total moment. Complementary to these supercell calculations, we conduct an EMTO-CPA investigation of the disordered PuC1-x system where the number-of-atoms ratio, C/Pu = 0.90, and 10% of the carbon atoms are replaced by vacancies but without structural relaxation.