# Thermodynamics of Liquid Uranium from Atomistic and Ab Initio Modeling
## 4 Results
There is some scattering in the experimental data on the density of liquid uranium at the experimental melting point: 16.95 g/cm\({}^{3}\)[42], 17.23 g/cm\({}^{3}\)[41], 17.27 g/cm\({}^{3}\)[16], 17.60 g/cm\({}^{3}\)[43], 17.90 g/cm\({}^{3}\)[44]. Using the expression \(C_{p}=\left(\frac{\partial H}{\partial T}\right)_{p,N}=\left(\frac{\partial E }{\partial T}\right)_{p,N}\), we calculated the ionic contribution to the specific heat \(C_{p}^{ion}(T)=27.24\) J/(mole K) that does not change with temperature. The electronic contribution was calculated according to Equation (2). The fundamental input to Equation (2) from the electronic structure is the density of states at the highest occupied energy state (the Fermi level). For the liquid, we calculated this number assuming the atomic density and positions obtained from the MD simulation. In Figure 4, we show one example of the electronic density of states at 1400 K (liquid uranium) and compare it with the perfect BCC solid (\(\gamma\)-U) at the same atomic density. Figure 4: The electronic density of states for liquid uranium (solid black line) and \(\gamma\)-U at 1400 K and the same atomic density (red dashed lines). The vertical dashed line at zero energy indicates the Fermi level. Figure 3: The density of liquid uranium as a function of temperature. The experimental data are taken from Ref. [41]. We notice, unsurprisingly, that the density of states at the Fermi level, \(D(E_{F})\), is about 25% higher for the BCC solid relative to the liquid. This is due to a lowering of the symmetry in the liquid that is higher for the crystal BCC phase, leading to the symmetry-related degeneracies of some states and a higher \(D(E_{F})\). This phenomenon also takes place at low temperatures for uranium where the orthorhombic \(\alpha\)-U phase is stable due to this symmetry-breaking mechanism, which has been referred to as a Jahn-Teller or Peierls distortion [45]. As a result, the electronic contribution to the specific heat of liquid uranium is about 20% lower than the similar contribution to the specific heat of the BCC solid \(\gamma\)-U. The calculated ionic (LAMMPS) and total specific heat of liquid uranium, calculated by Equation (1), are shown in Figure 5 together with experimental data from Refs. [15; 46]. As was in the case for \(\gamma\)-U [1], the calculated heat capacity of liquid uranium has a significant thermal dependence due to electronic contribution. In addition to Refs. [15; 46], there are few measurements of the specific heat of liquid uranium. According to Ref. [47], the specific heat of liquid uranium at \(T\leq 6000\) K is described by the following equation:  
\[Cp(\text{J}/(\text{mole-K}))=42.144+3.232x+2.07/x^{2},\,\text{where}\,\,x=T/1000, \tag{3}\]  
which reveals \(Cp=47.72\) J/(mole K) at \(T=1400\) K and \(Cp=49.13\) J/(mole K) at \(T=2000\) K. [PERSON] [13] reported \(Cp=47.93\) J/(mole K) in the temperature interval 1415 K \(\leq T\leq 1579\) K. [PERSON] and [PERSON] [16] reported \(Cp=47.3\) J/(mole K) in the temperature interval 2786 K \(\leq T\leq 5408\) K, and [PERSON] [40] reported \(Cp=48.66\) J/(mole K) in the temperature range 1348 K \(\leq T\leq 2348\) K. So, according to Refs. [13; 16; 40], the specific heat of liquid uranium does not change with temperature at all, and Refs. [46; 47] indicate a very weak dependence of the specific heat of liquid uranium from temperature. The picture is different in the case presented by [PERSON] and [PERSON] [15]. Although their measurements were restricted to a very narrow temperature interval of 1405 K \(\leq T\leq 1500\) K, the slope of the measured heat capacity (the increase rate with the temperature change) slightly exceeds the similar characteristic derived by our present calculations. We do not make any conclusion on this matter, understanding the difficulties of any measurements in liquid uranium. Figure 5: The total specific heat of liquid uranium. The experimental data are taken from Refs. [15; 46].