# Thermodynamics Modeling for Actinide Monocarbides and Mononitrides from First Principles
## 4 Discussion and Conclusions  
We have studied the actinide monocarbides and mononitrides ThC, ThN, UC, UN, PuC, and PuN, focusing on the thermodynamics at higher temperatures. The approach is that of a fully relativistic density-functional theory that is extended to include orbital polarization for the electronic structure. The lattice dynamics is obtained from temperature-dependent self-consistent phonons that couple in a strongly anharmonic fashion. Our modeling approach is more accurate than most others we have found for these materials. In the literature, first-principles theory of lattice dynamics of these actinide compounds has been restricted to be quasi-harmonic, except our own studies for UN [8] and PuC [15]. In addition to our first-principles thermodynamics model, we performed CALPHAD computations for the uranium and plutonium compounds for which more experimental data are available than for the thorium compounds. The CALPHAD results satisfactorily reproduced the experimental data such as the heat capacity. Concentrating on the ambient-pressure, temperature dependence of the free energy and specific heat, we related our relativistic-anharmonic theory with CALPHAD and experiments, where available, or quasi-harmonic modeling. For ThC and ThN, our free energies were consistent with reported quasi-harmonic theory at temperatures up to about 1500 K but appeared to deviate above that. There are no high-temperature specific-heat data for ThC, and for ThN they are sparse and inconsistent at higher temperatures. For ThN, our predictions of \(C_{p}\) are centered between several experimental data sets, suggesting they are reasonable. However, more experimental work is required for a better understanding of the disparity between the various measured data sets of the heat capacity for ThN. In the case of the uranium compounds, UC and UN, more data are available, and we have available CALPHAD assessments for both. The first-principles free energies agree excellently with CALPHAD, and the calculated specific heats are consistent with some (not all) experimental data and CALPHAD. We found, however, that the inclusion of defects in the CALPHAD modeling sensitively impacts the specific heat, thus complicating the comparisons with our defect-free theory. For PuC and PuN, the free-energy discrepancy between theory and CALPHAD is greater than for their uranium counterparts. The difference is relatively small (less than 3%) but non-negligible. The reason could be inaccuracies in either the first-principles modeling or the CALPHAD approach, where the latter, semi-empirical models are reliant on the availability of experimental data. In terms of theory, it is plausible that electron-phonon coupling, that is not fully explored here, is underestimated for the plutonium  
Figure 12: Specific heat at constant pressure, \(C_{p}\), for PuN. The levels of theory are indicated by open circles. The connecting lines are guide for the eye only. Experimental data [13, 78, 79] are shown as solid black symbols, the semi-empirical molecular-dynamics (MD) simulations [9] as green triangles, and CALPHAD data with a dashed line. compounds. That would then explain why the theoretical free energy is slightly above that of CALPHAD. Another issue is the question of stoichiometry of the compound, which can be accounted for using the CALPHAD method with vacancies on the carbon or the nitrogen sublattice. We make no attempts to account for possible non-stoichiometric states in the theory, however. Lastly, it is conceivable that fundamental electron-correlation effects in the plutonium compounds are not sufficiently addressed in the present electronic structure model. However, for PuC we discussed this possibility at some length [15] and showed that the DFT + OP approach for the electronic structure is largely appropriate, and we believe this conclusion holds true also for PuN.