# Thermodynamics Modeling for Actinide Monocarbides and Mononitrides from First Principles
## 3 Results
### ThC and ThN
Our general approach for calculating \(C_{p}\) is to add up the following terms, analogous to the free-energy expression shown in Equation (1):  
\[C_{p}(T)=C_{p}^{lat}(T)+C_{p}^{el}(T)+C_{p}^{mag}(T). \tag{3}\]  
Here, \(C_{p}^{lat}\) is obtained indirectly from SCAILD because this method is performed at constant volume (and temperature), not at constant pressure. Therefore, a conversion of the heat capacity to constant pressure is required and it can be done in several ways. One can do as we have done for the free energy, namely, calculate the total free energy, i.e., the lattice contribution, \(F_{lat}\), and the electronic contribution, \(F_{el}\), on a volume-temperature grid and establish the equilibrium properties (pressure is zero). Alternatively, one can extract the constant-volume specific heat, \(C_{p}^{lat}\) (from SCAILD) and convert to \(C_{p}^{lat}\) by means of this expression:  
\[C_{p}^{lat}(T)=C_{v}^{lat}(T)[1+\gamma\alpha T]. \tag{4}\]  
The temperature-dependent parameters \(\gamma\) and \(\alpha\), are the Gruneisen and volume-expansion coefficients, respectively. Both of these quantities are easily obtained from a quasi-harmonic model. The two approaches for determining \(C_{p}^{lat}\) give rather similar results and we are generally using Equation (4) for the conversion. The electronic contribution to the specific heat, \(C_{p}^{el}\), is derived from the electronic density of states, \(D(E)\), at the highest occupied energy state (Fermi level, \(E_{F}\)), \(D(E_{F})\):  
\[C_{p}^{el}(T)=\frac{\pi^{2}}{3}D(E_{F})k_{B}^{2}T. \tag{5}\]  
Note, \(D(E)\) depends on temperature because it is subject to a Fermi-Dirac temperature distribution. Thermal volume expansion, that depends on the lattice dynamics, influences \(D(E)\) as well. We account for both these effects when determining \(C_{p}^{el}\). Finally, the magnetic term in Equation (5), \(C_{p}^{mag}\), is acquired from the following thermodynamical equation at constant pressure [55]:  
\[C_{p}^{mag}(T)=k_{B}T\bigg{[}\frac{dS_{mag}(T)}{dT}\bigg{]}_{p}, \tag{6}\]  
Figure 1: Free energy for ThC from present theory together with results from a first-principles quasi-harmonic (QH) model [4] where the blue dashed curve (above 900 K) is a cubic-spline extrapolation. Results are solid symbols and dashed lines are guides for the eye only. where the magnetic entropy, \(S_{mag}\), is expressed in Equation (2). The temperature dependence of the total magnetic moment, \(\mu\), is derived from the DFT + OP electronic structure for the given temperature and the corresponding equilibrium volume. In Figure 2, we show our calculated specific heat for ThC with (\(C_{p}\)-_lat-el_) and without (\(C_{p}\)-_lat_) the electronic part, together with two quasi-harmonic models [3,5] and experimental data from Danan [54]. Unfortunately, the specific heat is measured at lower temperatures than we have studied, but at least we can compare our results with the two quasi-harmonic approaches [3,5]. It appears that one [3] of them is consistent with our present theory while the other [5] is not. The two sets of QH results are actually in disagreement in a wide temperature range, and that is indicative of the general uncertainty associated with the quasi-harmonic technique. It is clear from our results in Figure 2 that the electronic contribution to the specific heat is significant and cannot be ignored. This contribution is proportional to the electronic density of states at the Fermi level, \(D(E_{F})\), as expressed in Equation (5). This quantity directly reflects the electronic structure, and it is very sensitive to the fundamental theory as well as details of the calculation. Including the physics of the intra-atomic Coulomb repulsion, that has been suggested for actinide compounds, would likely (depending on the choice of Hubbard U parameter) lower \(D(E_{F})\) rather dramatically. Spin polarization also has a substantial effect of lowering \(D(E_{F})\) relative to a spin degenerate (non-magnetic) treatment of the electronic structure.