# Thermodynamics Modeling for Actinide Monocarbides and Mononitrides from First Principles
## 3 Results  
### ThC and ThN  
For thorium monocarbide and mononitride, as well as their uranium and plutonium counterparts, we adopted an approach combining electronic-structure calculations with lattice dynamics to obtain free energies and specific heats as functions of temperature. We focused on higher temperatures where these materials have not previously been studied to any greater extent. The free energy is not explicitly measured in any experiment, but the CALPHAD technique allows for a determination of this quantity that we can compare with. However, due to the scarcity of experimental data, we are not able to make a comparison to CALPHAD data of either ThC or ThN. Although the free energy can be straightforwardly calculated as a function of volume compression or pressure condition, we are only concerned about the free energy at constant, ambient (zero), pressure in this investigation. We introduce the most general expression of the Gibbs free energy at zero pressure (or Helmholtz free energy) that we consider:  
\[F(V,T)=F_{lat}(V,T)+F_{el}(V,T)+F_{mag}(V,T). \tag{1}\]  
Here, \(F_{lat}\) is the free energy from lattice vibrations, \(F_{el}\) the electronic free energy, including entropy, and \(F_{mag}\) is the magnetic contribution due to entropy of magnetic disorder, \(S_{mag}\). \[F_{mag}(V,T)=-k_{B}TS_{mag}=-k_{B}Tln(2\mu+1), \tag{2}\]  
where \(k_{B}\) is the Boltzmann constant and \(\mu\) the total magnetic moment (spin and orbital contributions) [52, 53]. The lattice free-energy contribution, \(F_{lat}\), is derived from the VASP-PAW-calculated atomic forces that are coupled with SCAILD, while the other two contributions are obtained from the FPLMTO electronic structure. Density-functional theory predicts no magnetism in the thorium compounds, so the magnetic contribution is only relevant for the uranium and plutonium systems. Table 2 lists the calculated magnetic moments for UC, UN, PuC, and PuN that we are using for the free-energy contributions. In principle, there is also an electron-phonon coupling term in the free energy expression in Equation (1) but there are, to our knowledge, no well-defined or straightforward procedures to calculate it accurately. Nonetheless, electron-phonon coupling is implicitly included in our strategy because all free-energy contributions are evaluated at the equilibrium volume and this volume, in turn, depends strongly on the calculated lattice dynamics. Lacking CALPHAD free energy for ThC (and ThN, see below), we have chosen to compare our result with published theory that is constrained to quasi-harmonic lattice vibrations [4]. In Figure 1, we show our calculated free energy together with results from quasi-harmonic (QH) modeling up to 900 K. Above 900 K we have added a cubic-spline extrapolation of the QH data as an approximate comparison at the highest temperatures. In our experience, a quasi-harmonic calculation for an actinide compound is often relatively good up to about 1000 K [8], so we are not surprised to find the same here for ThC. Certainly, at lower temperatures there is no significant difference between our present anharmonic approach and the QH treatment [4]. The [PERSON] and [PERSON] model [4] does not go above 900 K but the extrapolation to higher temperatures suggests that beyond about 1500 K their QH model is becoming increasingly inaccurate. Perhaps a more interesting evaluation of the present theory for the thermodynamics of ThC is to compare the specific heat at constant pressure, \(C_{p}\), with available experimental data. However, as already alluded to, there are very limited thermodynamical data on ThC. Actually, we cannot find any measured \(C_{p}\) at all for higher temperatures, only up to room temperature by [PERSON] [54].