# Thermodynamics Modeling for Actinide Monocarbides and Mononitrides from First Principles
## 3 Results
### ThC and ThN
In a similar way as spin polarization, spin-orbit coupling and orbital polarization play roles for the electronic specific heat as well. The all-electron FPLMTO calculations we include orbital polarization for all six compounds, while OP is only included in the VASP-PAW force calculations (for lattice dynamics) for PuC and PuN and not for the thorium or uranium compounds (for them the OP effect is much smaller). The OP interaction is analogous to the spin-orbit coupling and helps to lower \(D(E_{F})\) because of the increased energy separation between electronic states with different magnetic quantum number (\(m_{l}\)). Next, we discuss our results for thorium mononitride, ThN. As in the case of ThC, we have no CALPHAD results to compare with but for the free energy we instead contrast our results with a quasi-harmonic model [11]. Again, the QH theory appears to be close to our anharmonic model up to about 1000 K but increasingly less accurate beyond that, assuming a simple extrapolation. Figure 2: Specific heat at constant pressure, \(C_{p}\), for ThC. Levels of theory are open circles, where the connecting line is a guide for the eye only. The dashed lines refer to quasi-harmonic (QH) modeling [3,5]. Solid symbols are from experiment [54]. Our predicted free energies (Figures 1 and 3) at higher temperatures can be helpful for future CALPHAD assessments or for constraining semi-empirical approaches for ThC and ThN. In regard to the specific heat for thorium mononitride, there are some older experimental data [56, 57, 58] but also some very recently published measurements [6, 7]. In addition, semi-empirical molecular-dynamics (MD) studies [9, 10] and quasi-harmonic modeling [11, 12] are reported. All these data are plotted in Figure 4 where we also show our calculated \(C_{p}\) from lattice vibrations only (\(C_{p}\)-_lat_) and with the electronic contribution added (\(C_{p}\)-_lat_-_el_). ThN is not magnetic so there is no magnetic contribution to \(C_{p}\) for ThN. Interestingly, our full theory (\(C_{p}\)-_lat_-_el_) is centered between the set of newer [6, 7] and older [56, 57, 58] experimental data. In this particular case, QH theory is not in reasonable agreement, not even below 1000 K, with our anharmonic model. The semi-empirical MD results are also not in agreement with either our theory or any of the experimental data sets. We have no good explanations for the drastic variations we find between either the different experiments or the different modeling approaches for the specific heats displayed in Figure 4. We believe, however, that our results are sensible as they are positioned in the middle between experiments and other modeling. Certainly, more experimental  
Figure 4: Specific heat at constant pressure, \(C_{p}\), for ThN. Levels of theory are open circles, where the connecting lines are guides for the eye only. The dashed lines refer to semi-empirical molecular-dynamics (MD) [9, 10] and quasi-harmonic (QH) modeling [11, 12]. Solid symbols are from experiments [6, 7, 56, 57, 58]. \[F_{mag}(V,T)=-k_{B}TS_{mag}=-k_{B}Tln(2\mu+1), \tag{2}\]0 Free energy for ThN from present theory together with results from a first-principles quasi-harmonic (QH) model [11] where the blue dashed curve (above 1500 K) is a cubic-spline extrapolation. Results are shown as solid symbols and dashed lines are guides for the eye only. and theoretical work may be required to better understand why these various data sets are so different.