# Thermodynamics of Plutonium Monocarbide from Anharmonic and Relativistic Theory
## 3 Results
### Ground-State Properties and Thermodynamics
In order to make direct connection with experimental data we also calculate the specific heat at constant pressure, \(C_{p}\). We compare anharmonic theory (DFT+SO+OP combined with SCAILD) with lower-level quasi-harmonic theory within the Debye-Gruneisen model. This comparison allows us to evaluate the possible importance of anharmonicity at higher temperatures. In addition, we calculate CALPHAD \(C_{p}\) for ideal PuC and PuC\({}_{0.869}\) for a better understanding of the significance of sub-stoichiometry. Three sets of experimental data [64; 65; 66] are collected for comparison with theory. These data sets are shown in Figure 7 where we also plot the free-electron contribution obtained from the following simple Sommerfeld assumption [67]:  
\[C_{p}^{el}(T)=\frac{\pi^{2}}{3}D(E_{F})k_{B}^{2}T \tag{5}\]  
here, \(D(E_{F})\) is the e-DOS at the Fermi level that is predicted by the first-principles model (DFT+SO+OP). There is an implicit temperature dependence in \(D(E_{F})\) because the electronic structure is subject to a Fermi-Dirac thermal distribution that influences the e-DOS. From the slope of the free-electron specific heat we obtain the Sommerfeld coefficient 39.6 mJmol\({}^{-1}\)K\({}^{-2}\). We did not find any published experimental PuC Sommerfeld coefficient but a linear interpolation of the specific heat between 14 K and 19 K [68] gives a value of 40.6 mJmol\({}^{-1}\)K\({}^{-2}\) for PuC\({}_{0.8}\). The electronic contribution to the heat capacity is included in the anharmonic theory as well as the two examples of quasi-harmonic theory in Figure 7 and it is an essential component, particularly at higher temperatures. Figure 7: Measured and calculated specific heats at constant pressure, \(C_{p}\). The solid square (red), diamond (blue), and circle (black) are experimental data sets from references [64; 65; 66]. Dashed lines refer to results from CALPHAD. The solid black, blue, and red lines without symbols refer to our best anharmonic theory and two parameterizations of Debye–Grüneisen quasi-harmonic theory. The solid black line with triangles shows the calculated free-electron contribution to \(C_{p}\). Perhaps the most glaring aspect of Figure 7 is that the three experimental data sets deviate strongly above 1000 K. We mention that all points above 1300 K in the [PERSON] and [PERSON] [64] data set are extrapolated by them. To a degree, the differences in the data sets may be explained by different stoichiometry of the samples, but two of samples have the same stoichiometry. Contradictory experimental data complicates the CALPHAD assessments and the corresponding results become more uncertain. In the CALPHAD assessment of the PuC\({}_{1-\chi}\) phase [16], more weight was given to the experimental data of [PERSON] and [PERSON] [64] relative to that of references [65; 66] where the sharp increase of \(C_{p}\) above 1000 K leads to erroneous stability of the solid in the liquid regime. The CALPHAD data show, consistent with experiments, that the stoichiometry relatively weakly influences \(C_{p}\). At 700 K the CALPHAD \(C_{p}\) varies from 28.2 (PuC) to 26.8 (PuC\({}_{0.869}\)) Jmol\({}^{-1}\)K\({}^{-1}\), corresponding to a decrease of \(\sim\)5% (1.4 Jmol\({}^{-1}\)K\({}^{-1}\)) due to sub-stoichiometry. This result agrees with the data from [PERSON] and [PERSON] [64], which show a small difference in \(C_{p}\) of 1.25 Jmol\({}^{-1}\)K\({}^{-1}\) (\(\sim\)4.5%) between PuC\({}_{0.869}\) and PuC at 700 K.  
Our anharmonic theory agrees fairly well with all experimental and CALPHAD data up to 1000 K, but only with the Kruger and Savage [64] and the PuC-specific CALPHAD results above 1000 K. Not surprisingly, it agrees somewhat less favorably with CALPHAD for sub-stoichiometric PuC\({}_{0.869}\).