# Thermodynamics Modeling for Actinide Monocarbides and Mononitrides from First Principles
## 3 Results
### PuC and PuN  
We now focus on the plutonium monocarbide and mononitride. For these compounds we are again able to use CALPHAD assessments [59; 60; 61; 62] of the available experimental data to help us better evaluate the validity and accuracy of our first-principles model. One difference we notice in the modeling between PuC, and PuN is that, although both are magnetic, the effect of magnetism on the thermodynamics is significantly stronger for PuN. The explanation is that the spin and orbital magnetic moments in PuC are very nearly of the same magnitude, see Table 2, and anti-parallel. Hence, the net magnetic moment is small for PuC. Actually, this situation is reminiscent of the magnetic cancellation that has been proposed to exist in \(\delta\)-phase plutonium [74]. For PuN, on the other hand, our ferromagnetic calculations suggest a substantially larger spin than orbital moment so that the resulting net magnetic moment is considerable. Figure 8: Specific heat at constant pressure, \(C_{p}\), for UN. Present theory is here indicated by the solid line. Experimental data are from [7; 67; 68; 69; 70; 71; 72; 73] and CALPHAD data are shown with a dashed line. Figure 7: Free energy for UN 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. We recently studied the thermodynamics for plutonium monocarbide [15] and the results from that investigation are included here with only minor modifications. We have here added the small magnetic contribution to the free energy. In Figure 9, we show the free energies from our full theory (lattice, electronic, and magnetic contributions) and the results from the CALPHAD computations. We notice that the agreement is not quite as good as it is for the uranium compounds shown above, especially at higher temperatures. This is also the case, as we shall see, for PuN. The reason for these discrepancies in the free energies, particularly at the highest temperatures, could be due to either the first-principles theory, the CALPHAD, or both. We will discuss some possible causes below. As is the case for the uranium compounds, there are experimental data available for the thermodynamical properties of plutonium monocarbide [75; 76; 77]. In Figure 10 we show the heat capacities from these experimental studies and also the results from two CALPHAD assessments for different stoichiometries of PuC. The calculated \(C_{p}\) shown here is the same as published [15] because the magnetic contribution to \(C_{p}\) is negligible. Our calculated \(C_{p}\) is relatively consistent with the older measurement [75] and the CALPHAD data for the ideal-stoichiometric PuC. We believe carbon deficiency plays a major role for these experiments but acknowledge that the present theoretical framework does not take any such effects into account. Next, we report our results for plutonium mononitride. As before, we scrutinize our theory by carefully comparing the free energy with the result from CALPHAD, in Figure 11. Analogous to PuC, our first-principles free energy for PuN is slightly greater than the CALPHAD free energy at higher temperatures. We have already touched on the issue of stoichiometry as a possible cause of the discrepancy, but there could also be other sources for the differences that we shall discuss below. Figure 9: Free energy for PuC 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. We conclude by discussing our results on the specific heat of plutonium mononitride. There are some published reports of measured data [13; 78; 79], in addition to a semi-empirical-model's results [9], that we can compare with, apart from our CALPHAD data. In Figure 12, we show these heat capacities and our present theory.