# High-Temperature Thermodynamics of Uranium from Ab Initio Modeling
## 4 Results
In principle, the AIMD treatment is more complete than SCAILD for lattice dynamics, but the results of Figure 1 suggest that SCAILD is more accurate in practice for \(\gamma\)-U. Next, we will discuss our results for the Gibbs free energy. Our calculated \(\gamma\)-U equilibrium volumes in Figure 1 are determined as the volumes that define the minimum of the Gibbs free energy, Equation (1), as a function of temperature. In our approach, we include three contributions to the free energy. The terms \(F_{el}\) and \(F_{mag}\) are calculated for a one-atom bcc cell, utilizing the all-electron FFLMTO method. The lattice contribution, \(F_{lat}\), is obtained from SCAILD where the atomic forces are calculated with the VASP-PAW method. Both electronic-structure methods represent the DFT + OP approach. In Table 2, we show our calculated magnetic moments and the separate contributions to the Gibbs free energy as functions of temperature. We see in Table 2 that the magnetic contribution to the free energy is of the same order as the electronic contribution at 750 K, but at higher temperatures it becomes rather small even though the magnetic moments are far from negligible. For actinides with less than half-full 5f-band occupations, such as uranium, contributions to the total magnetic moment, i.e., spin and orbital components, have antiparallel directions so that the net moment is largely suppressed due to the compensation. We note that the magnetic moments increase in magnitude with temperature, but the total moment, due to a relatively increasing fraction of the orbital component, decreases weakly above 1000 K. Overall, however, the thermal dependence of the net magnetism is rather small. It is not possible to measure the free energy directly for any material, and therefore an explicit comparison between our model and experiment is not conceivable. Nonetheless, the CALPHAD approach allows for a free-energy representation of available experimental thermodynamic data, and here, we compare our ab initio model with CALPHAD in Figure 2. We are not surprised to find that DFT + OP + SCAILD is very close to the CALPHAD results for \(\gamma\)-U. In our recent focus on actinide systems, we found very good agreement between the two approaches for Gibbs free energy [65]. Unfortunately, we were not able to find any other published modeling data on Gibbs free energy for \(\gamma\)-U that we could compare our predictions with. Next, we focus on the heat capacity that consists of terms (Equation (3)) analogous to the free energy (Equation (1)). In practice, we ignore the \(C_{p}{}^{mag}\) term, because, numerically, the total magnetic moment is nearly constant with temperature. The lattice-vibration term, \(C_{p}{}^{lat}\), is dominating, but the electronic contribution, \(C_{p}{}^{el}\), cannot be neglected at elevated temperatures. The reason is that the electronic density of states is quite high at the Fermi level and \(C_{p}{}^{el}\) is proportional to this quantity and temperature; see Equation (4). \begin{table}
\begin{tabular}{c c c c c c} \hline
**Temperature** & \(\mu_{spin}\) & \(\mu_{orbital}\) & \(F_{mag}\) & \(F_{el}\) & \(F_{lat}\) \\ \hline
750 & 0.1150 & \(-\)0.0870 & \(-\)0.0035 & \(-\)0.0053 & \(-\)0.4342 \\
1000 & 0.1383 & \(-\)0.1088 & \(-\)0.0049 & \(-\)0.0239 & \(-\)0.6572 \\
1250 & 0.1580 & \(-\)0.1290 & \(-\)0.0061 & \(-\)0.0470 & \(-\)0.9005 \\
1500 & 0.1740 & \(-\)0.1480 & \(-\)0.0066 & \(-\)0.0738 & \(-\)1.1571 \\
1750 & 0.1860 & \(-\)0.1650 & \(-\)0.0062 & \(-\)0.1041 & \(-\)1.4362 \\
2000 & 0.1950 & \(-\)0.1810 & \(-\)0.0048 & \(-\)0.1372 & \(-\)1.7264 \\ \hline \end{tabular}
\end{table} 
Table 2: Spin (\(\mu_{spin}\)) and orbital (\(\mu_{orbital}\)) magnetic moments (\(\upmu_{B}\)) and Gibbs free energy contributions (eV) as functions of temperature obtained from our DFT + OP + SCAILD model. \(F_{el}\) energies are shifted to zero at zero temperature.