# High-Temperature Thermodynamics of Uranium from Ab Initio Modeling
## 4 Results  
First, we compile some results for \(\gamma\)-U at zero or room temperature in Table 1. Of course, \(\gamma\)-U does not exist at low temperatures, but it is still useful to compare different modeling approaches. Experimentally, an estimate of the room-temperature atomic volume from extrapolating molybdenum-stabilized \(\gamma\)-U to zero Mo content was suggested (20.89 A\({}^{3}\)) [94]. The experimental bulk modulus for \(\gamma\)-U was obtained from extrapolating diamond anvil cell compression data to zero pressure [14]. The MEAM potential [30] is developed by fitting model parameters to various physical properties of uranium metal obtained from experiments and theory. The MEAM results are relatively close to experimental data for \(\gamma\)-U, suggesting a reasonable parameterization of the MEAM, although the \(\gamma\)-U volume is overestimated by almost 3%. All pseudopotential results, on the other hand, underestimate the volume by about 3-9%, while overestimating the bulk modulus by about 16-56%. For DFT modeling, this level of error is noteworthy. The all-electron results show the best agreement with experiment, with more acceptable errors of <1% in volume and ~1-6% in the bulk modulus. There is a small difference between the 2002 all-electron data [15] and the present DFT + OP + SCAILD, and the reason is numerical. The earlier calculation [15] applied a 20 mRy Gaussian broadening to the energy eigenvalues, while in the present study it is replaced by a 300 K Fermi-Dirac distribution. Normally, one would not expect any meaningful volume dependence on the electronic-structure broadening, and it is also not large here. However, \(\gamma\)-U has very narrow 5f bands at the Fermi level, the reason for its dynamical instability, which causes this sensitivity. For \(\alpha\)-U, the corresponding density of states is much lower, resulting in less consequence of the broadening scheme; see further discussions in this section below. In Figure 1, we show our calculated volume expansion for \(\gamma\)-U together with results from AIMD [46; 66], MEAM [30], and experiments [97; 98]. Our theory (solid circles and dashed line) in Figure 1 agrees very well with the experimental data [97; 98]; only a very small shift is evident relative to one experimental set [97]. The small errors (<1%) are consistent with the findings for the low-temperature volume in Table 1. The MEAM results overestimate the volumes by about 4% and are also consistent with the low-temperature data in Table 1. The AIMD modeling, on the other hand, underestimates the volumes by about 3.5%, as is expected from the results in Table 1. The slopes of the curves (change of volume with temperature) suggest that \begin{table}
\begin{tabular}{l c c} \hline \hline
**Method** & **Atomic Volume (Å\({}^{3}\))** & **Bulk Modulus (GPa)** \\ \hline MEAM [30] & 21.49 & 115 \\ Pseudopotential [95] & 19.06 & 170 \\ Pseudopotential [19] & 20.18 & 176 \\ Pseudopotential [96] & 20.32 & 133 \\ Pseudopotential [32] & 20.12 & 132 \\ All-electron [15] & 20.76 & 120 \\ All-electron DFT + OP + SCAILD & 21.00 & 114 \\ \hline Experiment [14; 94] & 20.89 & 113 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Room-temperature atomic volume (Å\({}^{3}\)) and bulk modulus (GPa) for \(\gamma\)-U from present theory (DFT + OP + SCAILD), other modeling, and experiments. Figure 1: Atomic volumes (Å\({}^{3}\)) for \(\gamma\)-U as functions of temperature. Solid symbols represent our DFT + OP + SCAILD model, AIMD [46; 66], and MEAM results [30]. Open symbols indicate experimental data [97; 98]. The dashed line is provided only as a guide. \(\gamma\)-U melts at 1408 K and theoretical data points above that temperature are shown for a hypothetical \(\gamma\)-U solid phase. DFT + OP + SCAILD and the MEAM results are in better agreement with the experimental trend than AIMD, which indicates a too-small thermal expansion.