# High-Temperature Thermodynamics of Uranium from Ab Initio Modeling
## 4 Results
In Figure 3, we show electronic density of states for \(\alpha\)- and \(\gamma\)-uranium, at atomic volumes corresponding to 300 K (\(\alpha\)) and 1250 K (\(\gamma\)), respectively. The density of states at the dashed vertical line (Fermi level) for \(\gamma\)-U is almost double that of \(\alpha\)-U. This explains why the \(\alpha\) phase is the ground state and not \(\gamma\), as we discussed in terms of a Peierls-type crystal distortion in the introduction. \(\gamma\)-U is, however, dynamically stable at high temperatures. For the \(\gamma\) phase, the high density of states contributes to a large electronic contribution to the heat capacity that is linear with temperature; see Equation (4). The calculated \(\gamma_{\rm e}\) is about 10.6 mJ/(molK\({}^{2}\)), which is close to what has been measured for \(\alpha\)-U [36]. The electronic heat capacity is thus predicted to be quite significant for \(\gamma\)-U. That is important, because the experimental data are rather scattered, as we shall see below, and the high-temperature behavior is not conclusively determined. In Figure 4, we present heat capacities from our DFT + OP + SCAILD approach, AIMD [66], together with CALPHAD and data from the experimental reports [71; 72]. [PERSON] and [PERSON] [71] used a polynomial equation for the heat capacity to fit experimental enthalpy-increment data from multiple authors [72; 99; 100; 101]. We conducted our own fit and constrained \(C_{p}\) at 298.15 K while also adjusting for the \(\beta\)-\(\gamma\) transition. The resulting \(C_{p}\) has decidedly less temperature dependence. Our fit is a more consistent representation of the heat capacity that clearly falls within the range of derived experimental value [72; 99; 100; 101]. Adapting an arbitrary polynomial function that can take any form of \(C_{p}=a+bT+cT^{2}+dT^{-2}\) to enthalpy-increment data may explain the scatter observed in Figure 4, whereas the measured data, (enthalpy-increment, not heat capacity) of [72; 99; 100; 101] are all consistent with each other, as described by [PERSON] and [PERSON] [71]. Interestingly, some of the \(C_{p}\) data obtained this way show a negative trend with temperature, particularly [102]. Because \(C_{p}\) is bound by the law of [PERSON] (classical asymptotic limit), and \(C_{p}\) for most solids has a positive temperature-dependent correction to \(C_{p}\)[103], the negative thermal behavior of these data seems odd. Figure 2: Gibbs free energy for \(\gamma\)-U from our DFT + OP + SCAILD and CALPHAD. Data below 300 K for CALPHAD are interpolated to zero temperature. Theoretical data points above the melting temperature (1408 K) are shown for a hypothetical \(\gamma\)-U solid phase. Figure 4: Specific heat from DFT + OP + SCAILD and AIMD [66] (solid lines), CALPHAD (dashed line), and data derived from experimental enthalpy results [71] (open circles), our own analysis of the data [71] (solid circles) (see main text), [99] (green triangles), [100] (red circles), [101] (solid diamonds), [102] (blue triangles), and [104] (red squares). \(C_{p}\) (lattice) is the DFT + OP + SCAILD result without the electronic contribution, and \(C_{p}\) (total) includes it. Theoretical data points above the melting temperature are shown for a hypothetical \(\gamma\)-U solid phase. Figure 3: Electronic density of states (states/eV) for \(\alpha\)-U (solid red) and \(\gamma\)-U (solid black) at volumes corresponding to 300 K and 1250 K, respectively. The energies are shifted so that the Fermi level is at zero energy, indicated with a dashed vertical line. For better readability, we apply a 2 mRy Gaussian broadening to the density of states. Our \"C\({}_{\rm{P}}\) (total)\" results in Figure 4 are in reasonable agreement with our adjusted values from [71] and fall within the bounds of the various experimentally derived heat capacity data [72; 99; 100; 101]. The CALPHAD results indicate a slope before about 1050 K, after which a constant value is assumed.