# Thermodynamics of Plutonium Monocarbide from Anharmonic and Relativistic Theory
## 3 Results
### Ground-State Properties and Thermodynamics
Moving on to thermodynamics and our lattice-dynamics predictions, we show in Figure 4 our phonon density of states (p-DOS) at 1200 K. The p-DOS clearly suggests that including spin-orbit coupling and orbital polarization (DFT+SO+OP) in the electronic structure stiffens the lattice dynamics resulting in weight of the phonon modes shifting to higher energies. The consequence of the results shown in Figures 1 and 4 is that when the temperature dependent part of the electronic structure (Fermi-Dirac distribution and electronic entropy) is added to the Gibbs energies of the two models, the \"DFT\" is below that of the full \"DFT+SO+OP\" theory because the magnitude of the lattice-vibration contribution is greater. In Figure 5 we show the Gibbs energies obtained from adding the electronic and lattice-vibration contributions at constant volumes (16.2 A\({}^{3}\) and 15.6 A\({}^{3}\)) together with our CALPHAD Gibbs energy for PuC. The CALPHAD data begin at 298 K, but an extrapolation to zero K and a shift so that the Gibbs energy vanish at zero temperature makes the comparison appropriate. The lesser (DFT) theory lies below the full (DFT+SO+OP) treatment and appears to be in slightly better agreement with the CALPHAD result. Figure 4: SCAILD phonon density of states (states/meV) at 1200 K from electronic structure that includes spin-orbit coupling and orbital polarization (DFT+SO+OP) and not (DFT). The atomic volumes are 15.6 Å\({}^{3}\) (DFT) and 16.2 Å\({}^{3}\) (DFT+SO+OP), respectively. Figure 5: Free energies from SCAILD lattice contribution and electronic structure at volumes of 15.6 (DFT) and 16.2 Å\({}^{3}\) (DFT+SO+OP), respectively. The full lines are guides to the eye only. The dashed line shows the calculation of phase diagram (CALPHAD) Gibbs energy. This is an accidental better agreement because important contributions to the Gibbs energy are still missing in the first-principles modeling. First, removing the constraint of a fixed atomic volume, i.e., by accounting for thermal volume expansion, lowers the Gibbs energies a substantial amount. It comes with a significant computational cost to compute this contribution, however, because SCAILD needs to be repeated for two or more volumes (for each temperature) so that the equilibrium volume can be determined. In addition, since we are studying magnetically disordered PuC (over 100 K [52]), there is a simple magnetic-entropy contribution due to magnetic disorder [62; 63] that we shall include:  
\[F_{mag}(V,T)=-k_{B}Tln(2\mu+1) \tag{4}\]  
in this equation, \(k_{B}\) is the Boltzmann constant and \(\mu\) the total (spin and orbital) magnetic moment. Because orbital polarization, that enhances the magnitude of the orbital moment [5], is addressed in the model the total magnetic moment is small (-0.1 u\({}_{B}\)) as is the \(F_{mag}\) contribution. As mentioned, the small total moment is due to a near complete compensation between the spin and orbital components. Lastly, there is an electron-phonon-coupling term that we are not considering for the free energy. Accounting for all energy excitations from electrons and phonons and their distributions in a universal fashion is difficult and to our knowledge there is no efficient procedure to accurately determine this quantity. When we add the missing terms (except the electron-phonon term) to the Gibbs energy, the scalar-relativistic (DFT) energy is significantly below and incompatible with the CALPHAD results (not shown). The full theoretical treatment (DFT+SO+OP), on the other hand, produces energies that are above but near that obtained from CALPHAD. In Figure 6 we show our best theory of the Gibbs energy together with CALPHAD for PuC. They are quite close, with the first-principles data slightly above CALPHAD, consistent with the fact that electron-phonon interaction is neglected in the ab initio model. Figure 6: Free energies from our best first-principles theory and CALPHAD (same as in Figure 5). The solid line is guide to the eye only.