# Thermodynamics of Liquid Uranium from Atomistic and Ab Initio Modeling
## 1 Introduction  
In a recent paper [1], we presented the results of first-principles calculations of the thermodynamic properties of \(\gamma\)-phase uranium (\(\gamma\)-U) within its temperature range of stability, 1049 K \(\leq T\leq T_{m}=\) 1408 K, where \(T_{m}\) is the melting temperature. The first-principles method was coupled to a lattice dynamics scheme which was used to model anharmonic lattice vibrations. The calculated heat capacity was predicted to have significant thermal dependence due to the electronic contribution. There are a significant number of calculations that have been performed to study thermal properties of the solid \(\alpha\)-, \(\beta\)-, and \(\gamma\)-phases of uranium metal, as seen in Ref. [1] for details. However, there are only a handful of theoretical studies of liquid uranium performed either by classical molecular dynamics (MD), i.e., the so-called atomistic modeling with empirical interatomic potential [2, 3, 4, 5, 6, 7], or by quantum molecular dynamics (QMD) [8, 9, 10, 11, 12]. The interatomic potential developed in Ref. [2] was fitted to the properties of \(\alpha\)-U and was not well suited for high temperature applications. Contrarily, the potential developed in Ref. [3] was fitted to data on the structure of liquid uranium in the vicinity of the melting temperature but was not appropriate to describe the properties of the low-temperature \(\alpha\)-uranium phase. By using the force-matching method, [PERSON] et al. [4] modified the embedded atom method (EAM) potential suggested in Ref. [3] in order to study the structure and thermodynamic properties of \(\alpha\)-U, \(\gamma\)-U, and liquid uranium. The same year, [PERSON] et al. [5] suggested the modified embedded atom method (MEAM) potential to calculate the atomistic properties of \(\gamma\)-U. [PERSON] et al. [6] expanded upon the MEAM approach for the case of the U-Zr system and reported the results of calculations of the numerous mechanical and thermodynamical properties of this system, including the enthalpy and heat capacity at the melting temperature. [PERSON] and [PERSON] [7] independently constructed an MEAM potential and applied it to study the melting of U-Al alloys. By using the ab initio plane-wave pseudopotential method and QMD, [PERSON] et al. [8] performed constant volume modeling of \(\alpha\)-U, \(\gamma\)-U, and liquid uranium. QMD based on the Vienna Ab initio Simulation Package (VASP) has been performed in Refs. [9; 10; 11] in order to study the thermodynamic properties of liquid uranium, e.g., the equation of state (EOS), the basic structural and thermodynamic properties, the bulk modulus, atomic self-diffusion, and viscosity. Finally, the fully relativistic calculations within VASP-QMD (the spin-orbit coupling (SOC) accounted for both core and valence electrons) [12] revealed the significant impact of the relativistic effects on the thermodynamic properties of liquid uranium at melting point. In our previous paper [1], we paid special attention to the temperature behavior of the heat capacity, \(C_{p}(T)\), of \(\gamma\)-U at high temperatures. There is significant scatter in the experimental data on the heat capacity of \(\gamma\)-U [1]. The experimental data on the heat capacity of liquid uranium show an even higher level of uncertainty due the chemical volatility of uranium metal at high temperatures [13; 14; 15; 16; 17; 18]. According to Equation (3) from Ref. [1], the following can be observed:  
\[C_{p}(T)=C_{p}^{lat}(T)+C_{p}^{el}(T)+C_{p}^{mag}(T), \tag{1}\]  
where \(C_{p}^{lat}(T)\), \(C_{p}^{el}(T)\), and \(C_{p}^{mag}(T)\) are the lattice, electronic, and magnetic contributions to the heat capacity, respectively. The lattice, or the ionic in the case of a liquid, contribution is calculated from MD as the change in total energy divided by the change in temperature.