# Thermodynamics Modeling for Actinide Monocarbides and Mononitrides from First Principles
## 3 Results
### UC and UN  
For uranium monocarbide and mononitride there are considerably more experimental data available and consequently robust CALPHAD assessments can be done for these compounds [59; 60; 61; 62]. This is very helpful for the theoretical modeling because we can directly compare the free energies for the two approaches. Otherwise, the free energy is a quantity that is not directly measured. For the thorium systems, on the other hand, we correlated our approach only with lesser quasi-harmonic theory that tends to lose quality at higher temperatures. The only difference in our modeling of the uranium monocarbides and mononitrides, relative to their thorium counterparts, is the explicit inclusion of magnetic contributions to the free energy and the heat capacity. Admittedly, these contributions are rather small and in our previous study on UN [8] we did not consider them, but they do improve the agreement with CALPHAD as we shall see below. Otherwise, our theory for UN is the same as in our previous study [8]. First, we consider uranium monocarbide, UC. In Figure 5, we compare our present free energy with the results from the CALPHAD analysis. Apparently, the agreement is excellent between the two approaches, giving credence to both. For the theoretical model, the free energy has contributions from, in order of importance, lattice vibrations, electronic excitations, and magnetic entropy. Their individual relevance is reflected in the heat capacity, \(C_{p}\) (see below). Indeed, the magnetic contributions are rather small, we show its contribution to the heat capacity in Figure 6, and it slightly improves the comparison with the CALPHAD free energy. The specific heat has been measured numerous times for uranium monocarbide [63; 64; 65; 66]. Here, we are primarily concerned with the high-temperature \(C_{p}\) and focus our attention above 500 K. In Figure 6, we display experimental data [63; 64; 65; 66], two CALPHAD assessments, and our present theory. The cause of the difference between the two CALPHAD results is the inclusion of defects, such as vacancies, on the carbon sublattice in one of the assessments. The presence of defects results in non-stoichiometry, which sensitively affects the heat capacity at elevated temperatures [59]. We see from our theoretical treatment in Figure 6 that the main contribution to \(C_{p}\) is from lattice vibrations, but the electronic contribution is very important too. The full theory  
Figure 5: Free energy for UC from present theory together with results from our CALPHAD calculation (dashed line). Solid symbols show our present theory, and the solid line is guide for the eye only. (\(C_{p}\)-_lat-el-m_) also includes a small magnetic correction due to the magnetic-disorder entropy at high temperatures. It is rewarding to find in the figure that the CALPHAD approach, that assumes no defects (stoichiometric UC), is very close to our full theory. We do not expect the CALPHAD analysis that includes strong defect effects to agree with our theory because our model does not deal with any defect mechanisms but rather assumes a perfect, stoichiometric, crystal. Moving on to uranium mononitride, we acknowledge that there is a fair amount of high-temperature data on this compound [7; 67; 68; 69; 70; 71; 72; 73]. Before we reflect on the heat capacity, we once more compare the free energies from CALPHAD and our model, that includes a magnetic contribution. In Figure 7, we plot these energies and notice a very good consistency between them. This figure is very similar to a corresponding figure in [8], but in the present free energy for UN we also include the magnetic contribution. The magnetic entropy is increasingly important at higher temperatures, and its consideration significantly improves the agreement with CALPHAD in the high-temperature regime. Hence, for the two uranium compounds, our first-principles anharmonic treatment is in excellent agreement with CALPHAD for the free energies.