# High-Temperature Thermodynamics of Uranium from Ab Initio Modeling

^{*}

## Abstract

**:**

## 1. Introduction

_{p}) of α-U in the temperature range up to 900 K. At low temperatures, up to 300 K, the calculated heat capacity has a reasonable agreement with the experimental data of Flotow and Lohr [67] and Jones et al. [68]. However, as temperature increases, there is a substantial underestimation of the quasi-harmonic heat capacity relative to the experimental data [69] with errors of the order of 50%.

_{e}(see definition in Section 2) for α-U have been shown to be microstructurally dependent, with an average value of the experimental results to be about 10.12 mJ/(molK

^{2}). To our knowledge, no measurements have been published for γ-U.

_{p}for γ-U, while Gathers [73] proposed a completely temperature-independent C

_{p}for the liquid. The results of the measurements by Marchidan and Ciopec [72] and Gathers [73] were questioned by Belashchenko et al. [74], who emphasized the importance of the electronic excitations for the heat capacity. This electronic contribution has a substantial temperature dependence, thus contradicting the conclusion of the experiments [72,73].

## 2. Ab Initio Computational Methodology

_{v}or C

_{p}, for γ-uranium using parameter-free first-principles density-functional theory for the electronic structure. The thermodynamic quantities are here divided into contributions from the lattice dynamics and electronic excitations. No explicit electron–phonon coupling term is included in the formulation of the free energy, simply because there is no straightforward way to compute this contribution. We introduce the most general expression for the Gibbs free energy at zero pressure that we consider for γ-U as a function of volume and temperature (V,T):

_{lat}is the free energy from lattice vibrations, F

_{el}is the electronic free energy, including entropy, and F

_{mag}is the magnetic contribution due to classic entropy of magnetic disorder, S

_{mag};

_{B}is the Boltzmann constant and μ is the total magnetic moment (spin and orbital contributions) [77,78]. In an analogous fashion to the free energy, we express and calculate the heat capacity:

_{F}), D (E

_{F}):

_{p}

^{el}.

_{p}

^{mag}, is acquired from the following thermodynamic equation at constant pressure [79]:

_{mag}, is expressed in Equation (2). Analogous to the electronic contribution, the magnetic term depends on temperature via the electronic structure and thermal expansion.

_{el}, F

_{mag}, C

_{p}

^{el}, and C

_{p}

^{mag}) are calculated using a very flexible and reliable all-electron full-potential linear muffin-tin orbital (FPLMTO) method [80]. The framework for this method is built on basis functions where the radial part is a linear superposition of atomic-like functions and their derivatives. What makes this implementation different and more flexible than most others is that the basis set can describe energy levels associated with atomic states with separate quantum numbers while having identical magnetic quantum numbers. In other words, for uranium, 6s and 6p states can be represented as well as 7s and 7p states. In some approaches, these states with the same quantum number are defined by a single set of energy parameters in separate energy panels. However, sometimes these panels can intersect, causing numerical problems. Contrarily, in the present FPLMTO approach bases belonging to the same quantum numbers are composed within one fully hybridizing basis set.

_{lat}and C

_{p}

^{lat}) are obtained from the SCAILD method that was described in [75]. The method requires calculations of forces on thermally perturbed atoms in a supercell representation of the crystal phase. In this case, we set up a 3

^{3}(27 atoms) supercell of bcc (γ) uranium and perform 400 SCAILD iterations to ensure very good convergency of the free energy and heat capacity. The supercell size is consistent with our previous SCAILD works [13,53,60,65] and is assumed to be sufficient. Three strategic volumes and six temperatures are simulated with this approach. The SCAILD free-energy volume dependence is obtained by interpolating a cubic polynomial between the three volumes. Atomic forces can be determined from applying the FPLMTO method [13], but it is computationally far easier to utilize a pseudopotential plane-wave-based method, namely, the Vienna ab initio simulation package (VASP) in conjunction with the projector-augmented-wave (PAW) method [87,88].

^{3}). The lattice contribution to the free energy changed with these more accurate calculations, but only by a small amount of about 0.24 meV.

## 3. CALPHAD Methodology

## 4. Results

^{3}) [94]. The experimental bulk modulus for γ-U was obtained from extrapolating diamond anvil cell compression data to zero pressure [14].

_{el}and F

_{mag}are calculated for a one-atom bcc cell, utilizing the all-electron FPLMTO 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.

_{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).

_{e}is about 10.6 mJ/(molK

^{2}), which is close to what has been measured for α-U [36]. The electronic heat capacity is thus predicted to be quite significant for γ-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.

_{p}at 298.15 K while also adjusting for the β–γ 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 Konings and Beneš [71]. Interestingly, some of the C

_{p}data obtained this way show a negative trend with temperature, particularly [102]. Because C

_{v}is bound by the law of Dulong and Petit (classical asymptotic limit), and C

_{p}for most solids has a positive temperature-dependent correction to C

_{v}[103], the negative thermal behavior of these data seems odd.

_{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. This behavior is compatible with the data from Moore [99], Ginnings [100], and the review compiled by Rand and Kubaschewski [102] on which the data of Dinsdale [93] is based. The abrupt change after 1050 K in the CALPHAD data is likely related to the β–γ transition in uranium.

_{p}shows an interesting S shape that does not correlate with experiment or our ab initio model or CALPHAD, but the general trend is an increase with temperature. In addition, the AIMD predicts too-small C

_{p}at the highest temperatures close to 1400 K, as Aly et al. point out in their report [66].

_{p}curve.

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Hofman, G.; Walters, L.; Bauer, T. Metallic fast reactor fuels. Prog. Nucl. Energy
**1997**, 31, 83–110. [Google Scholar] [CrossRef] - Meyer, M.K.; Hofman, G.L.; Hayes, S.L.; Clark, C.R.; Wiencek, T.C.; Snelgrove, J.L.; Strain, R.V.; Kim, K.-H. Low-temperature irradiation behavior of uranium–molybdenum alloy dispersion fuel. J. Nucl. Mater.
**2002**, 304, 221–236. [Google Scholar] [CrossRef] - Kim, Y.S.; Hofman, G.; Yacout, A. Migration of minor actinides and lanthanides in fast reactor metallic fuel. J. Nucl. Mater.
**2009**, 392, 164–170. [Google Scholar] [CrossRef] - Carmack, W.J.; Porter, D.L.; Chang, I.; Hayes, S.L.; Meyer, M.K.; Burkes, D.E.; Lee, C.B.; Mizuno, T.; Delage, F.; Somers, J.; et al. Metallic fuels for advanced reactors. J. Nucl. Mater.
**2009**, 392, 139–150. [Google Scholar] [CrossRef] - Todreas, N.E. Thermal-hydraulics challenges in fast reactor design. Nucl. Tech.
**2009**, 167, 127–144. [Google Scholar] [CrossRef] - Janney, D.E. Metallic Fuels Handbook, Part 1: Alloys Based on U-Zr, Pu-Zr, U-Pu, or U-Pu-Zr, Including Those with Minor Actinides (Np, Am, Cm) Rare-Earth Elements (La, Ce, Pr, Nd, Gd), and Y; Idaho National Laboratory: Idaho Falls, ID, USA, 2017. [Google Scholar]
- Capriotti, L.; Bremier, S.; Inagaki, K.; Poml, P.; Papaioannou, D.; Ohta, H.; Ogata, T.; Rondinella, V.V. Characterization of metallic fuel for minor actinides trans mutation in fast reactor. Prog. Nucl. Energy
**2017**, 94, 194–201. [Google Scholar] [CrossRef] - Imhoff, S.D. Uranium Density, Thermal Conductivity, Specific Heat, and Thermal Diffusivity; LA-UR-21-21810; OSTI.GOV: Idaho National Lab: Idaho Falls, ID, USA, 24 February 2021. [Google Scholar]
- Söderlind, P. Theory of the crystal structures of cerium and the light actinides. Adv. Phys.
**1998**, 47, 959–998. [Google Scholar] [CrossRef] - Söderlind, P.; Johansson, B.; Yongming, L.; Nordström, L. Calculated thermal expansion of the actinide elements. Int. J. Thermophys.
**1991**, 12, 611–615. [Google Scholar] [CrossRef] - Söderlind, P.; Eriksson, O.; Wills, J.M.; Boring, A.M. Elastic constants of cubic f-electron elements: Theory. Phys. Rev. B
**1993**, 48, 9306–9312. [Google Scholar] [CrossRef] - Söderlind, P.; Eriksson, O.; Wills, J.; Boring, A. A unified picture of the crystal structures of metals. Nature
**1995**, 374, 524–525. [Google Scholar] [CrossRef] - Söderlind, P.; Grabowski, B.; Yang, L.; Landa, A.; Björkman, T.; Souvatzis, P.; Eriksson, O. High-temperature phonon stabilization of γ-uranium from relativistic first-principles theory. Phys. Rev. B
**2012**, 85, 60301. [Google Scholar] [CrossRef] - Yoo, C.-S.; Cynn, H.; Söderlind, P. Phase diagram of uranium at high pressures and temperatures. Phys. Rev. B
**1998**, 57, 10359. [Google Scholar] [CrossRef] - Söderlind, P. First-principles elastic and structural properties of uranium metal. Phys. Rev. B
**2002**, 66, 85113. [Google Scholar] [CrossRef] - Wills, J.M.; Eriksson, O. Crystal-structure stabilities and electronic structure for the light actinides Th, Pa, and U. Phys. Rev. B
**1992**, 45, 13879. [Google Scholar] [CrossRef] - Richard, N.; Bernard, S.; Jollet, F.; Torrent, M. Plane-wave pseudopotential study of the light actinides. Phys. Rev. B
**2002**, 66, 235112. [Google Scholar] [CrossRef] - Hood, R.; Yang, L.H.; Moriarty, J. Quantum molecular dynamics simulations of uranium at high pressure and temperature. Phys. Rev. B
**2008**, 78, 24116. [Google Scholar] [CrossRef] - Taylor, C.D. Evaluation of first-principles techniques for obtaining materials parameters of alpha uranium and the (001) alpha uranium surface. Phys. Rev. B
**2008**, 77, 94119. [Google Scholar] [CrossRef] - Xiang, S.; Huang, H.; Hsiung, L.M. Quantum mechanical calculations of uranium phases and niobium defects in γ-uranium. J. Nucl. Mater.
**2008**, 375, 113–119. [Google Scholar] [CrossRef] - Bouchet, J. Lattice dynamics of α uranium. Phys. Rev. B
**2008**, 77, 24113. [Google Scholar] [CrossRef] - Beeler, B.; Good, B.; Rashkeev, S.; Deo, C.; Baskes, M.; Okuniewski, M. First principles calculations for defects in U. J. Phys. Condens. Matter.
**2010**, 22, 505703. [Google Scholar] [CrossRef] - Raymond, S.; Bouchet, J.; Lander, G.H.; Le Tacon, M.; Garbarino, G.; Hoesch, M.; Rueff, J.-P.; Krisch, M.; Lashley, J.C.; Schulze, R.K.; et al. Understanding the complex phase diagram of uranium: The role of electron-phonon coupling. Phys. Rev. Lett.
**2011**, 107, 136401. [Google Scholar] [CrossRef] - Bouchet, J.; Albers, R. Elastic properties of the light actinides at high pressure. J. Phys. Condens. Matter.
**2011**, 23, 215402. [Google Scholar] [CrossRef] [PubMed] - Huang, G.-Y.; Wirth, B.D. First-principles study of diffusion of interstitial and vacancy in α U-Zr. J. Phys. Condens. Matter
**2011**, 23, 205402. [Google Scholar] [CrossRef] [PubMed] - Huang, G.-Y.; Wirth, B.D. First-principles study of bubble nucleation and growth behaviors in α U-Zr. J. Phys. Condens. Matter
**2012**, 24, 415404. [Google Scholar] [CrossRef] - Akella, J.; Weir, S.; Wills, J.M.; Söderlind, P. Structural stability in uranium. J. Phys. Condens. Matter
**1997**, 9, L549–L555. [Google Scholar] [CrossRef] - Smirnova, D.E.; Starikov, S.V.; Stegailov, V.V. Interatomic potential for uranium in a wide range of pressures and temperatures. J. Phys. Condens. Matter.
**2012**, 24, 15702. [Google Scholar] [CrossRef] [PubMed] - Smirnova, D.E.; Starikov, S.V.; Stegailov, V.V. New interatomic potential for computation of mechanical and thermodynamic properties of uranium. Phys. Met. Metallogr.
**2012**, 113, 107–116. [Google Scholar] [CrossRef] - Beeler, B.; Deo, C.; Baskes, M.; Okuniewski, M. Atomistic properties of γ uranium. J. Phys. Condens. Matter
**2012**, 24, 75401. [Google Scholar] [CrossRef] - Pascuet, M.I.; Bonny, G.; Fernández, J.R. Many-body interatomic U and Al–U potentials. J. Nucl. Mater.
**2012**, 424, 158–163. [Google Scholar] [CrossRef] - Beeler, B.; Deo, C.; Baskes, M.; Okuniewski, M. First principles calculations of the structure and elastic constants of α, β and γ uranium. J. Nucl. Mater.
**2013**, 433, 143–151. [Google Scholar] [CrossRef] - Smirnova, D.E.; Kuksin, A.Y.; Starikov, S.V.; Stegailov, V.V.; Insepov, Z.; Rest, J.; Yacout, A.M. A ternary EAM interatomic potential for U–Mo alloys with xenon. Modell. Simul. Mater. Sci. Eng.
**2013**, 21, 35011. [Google Scholar] [CrossRef] - Smirnova, D.E.; Kuksin, A.Y.; Starikov, S.V. Investigation of point defects diffusion in bcc uranium and U–Mo alloys. J. Nucl. Mater.
**2015**, 458, 304–311. [Google Scholar] [CrossRef] - Smirnova, D.E.; Kuksin, A.Y.; Starikov, S.V.; Stegailov, V.V. Atomistic modeling of the self-diffusion in γ-U and γ-U–Mo. Phys. Met. Metallogr.
**2015**, 116, 445–455. [Google Scholar] [CrossRef] - Moore, A.P.; Beeler, B.; Deo, C.; Baskes, M.I.; Okuniewski, M.A. Atomistic modeling of high temperature uranium-zirconium alloy structure and thermodynamics. J. Nucl. Mater.
**2015**, 467, 802–819. [Google Scholar] [CrossRef] - Bouchet, J.; Bottin, F. Thermal evolution of vibrational properties of α-U. Phys. Rev. B
**2015**, 92, 174108. [Google Scholar] [CrossRef] - Tseplyaev, V.I.; Starikov, S.V. The atomistic simulation of pressure-induced phase transition in uranium mononitride. J. Phys. Conf. Ser.
**2015**, 653, 012092. [Google Scholar] [CrossRef] - Ren, Z.; Wu, J.; Ma, R.; Hu, G.; Luo, C. Thermodynamic properties of α-uranium. J. Nucl. Mater.
**2016**, 480, 80–87. [Google Scholar] [CrossRef] - Kuksin, A.Y.; Starikov, S.V.; Smirnova, D.E.; Tseplyaev, V.I. The diffusion of point defects in uranium mononitride: Combination of DFT and atomistic simulation with novel potential. J. Alloys Comp.
**2016**, 658, 385–394. [Google Scholar] [CrossRef] - Kolotova, L.N.; Kuksin, A.Y.; Smirnov, D.E.; Starikov, S.V.; Tseplyaev, V.I. Features of structure and phase transitions in pure uranium and U–Mo alloys: Atomistic simulation. J. Phys. Conf. Ser.
**2016**, 774, 12036. [Google Scholar] [CrossRef] - Kolotova, L.N.; Starikov, S.V. Anisotropy of the U–Mo alloy: Molecular-dynamics study. Phys. Met. Metallogr.
**2016**, 117, 487–493. [Google Scholar] [CrossRef] - Tseplyaev, V.I.; Starikov, S.V. The atomistic simulation of pressure-induced phase transition in uranium mononitride. J. Nucl. Mater.
**2016**, 480, 7–14. [Google Scholar] [CrossRef] - Starikov, S.V.; Kolotova, L.N. Features of cubic and tetragonal structures of U–Mo alloys: Atomistic simulation. Script. Mater.
**2016**, 113, 27–30. [Google Scholar] [CrossRef] - Huang, S.-Q.; Ju, X.-H. First-principles study of properties of alpha uranium crystal and seven alpha-uranium surfaces. J. Chem.
**2017**, 2017, 8618340. [Google Scholar] [CrossRef] - Bouchet, J.; Bottin, F. High-temperature and high-pressure phase transitions in uranium. Phys. Rev. B
**2017**, 95, 54113. [Google Scholar] [CrossRef] - Kolotova, L.N.; Starikov, S.V. Atomistic simulation of defect formation and structure transitions in U-Mo alloys in swift heavy ion irradiation. J. Nucl. Mater.
**2017**, 495, 111–117. [Google Scholar] [CrossRef] - Starikov, S.; Kuksin, A.; Smirnova, D.; Dolgodvorov, A.; Ozrin, V. Multiscale modeling of uranium mononitride: Point defects diffusion, self-diffusion, phase composition. Defect Diffus. Forum
**2017**, 375, 101–113. [Google Scholar] [CrossRef] - Starikov, S.V.; Kolotova, L.N.; Kuksin, A.Y.; Smirnova, D.E.; Tseplyaev, V.I. Atomistic simulation of cubic and tetragonal phases of U-Mo alloy: Structure and thermodynamic properties. J. Nucl. Mater.
**2018**, 499, 451–463. [Google Scholar] [CrossRef] - Starikov, S.; Korneva, M. Description of phase transitions through accumulation of point defects: UN, UO
_{2}and UC. J. Nucl. Mater.**2018**, 510, 373–381. [Google Scholar] [CrossRef] - Lunev, A.V.; Starikov, S.V.; Aliev, T.N.; Tseplyaev, V.I. Understanding thermally-activated glide of 1/2 110 {110} screw dislocations in UO
_{2}—A molecular dynamics analysis. Intern. J. Plast.**2018**, 110, 294–305. [Google Scholar] [CrossRef] - Torres, E.; Kaloni, T.P. Projector augmented-wave pseudopotentials for uranium-based compounds. Comp. Mater. Sci.
**2020**, 171, 109237. [Google Scholar] [CrossRef] - Söderlind, P.; Landa, A.; Perron, A.; Sadigh, B.; Heo, T.W. Ground-state and thermodynamical properties of uranium mononitride from anharmonic first-principles theory. Appl. Sci.
**2019**, 9, 3914. [Google Scholar] [CrossRef] - Kolotova, L.N.; Starikov, S.V.; Ozrin, V.D. Atomistic simulation of the fission-fragment-induced formation of defects in a uranium–molybdenum alloy. J. Exp. Theor. Phys.
**2019**, 129, 59–65. [Google Scholar] [CrossRef] - Castellano, A.; Bottin, F.; Dorado, B.; Bouchet, J. Thermodynamic stabilization of γ-U-Mo alloys: Effect of Mo content and temperature. Phys. Rev. B
**2020**, 101, 184111. [Google Scholar] [CrossRef] - Ladygin, V.V.; Korotaev, P.Y.; Yanilkin, A.V.; Shapeev, A.V. Lattice dynamics simulation using machine learning interatomic potentials. Comput. Mater. Sci.
**2020**, 172, 109333. [Google Scholar] [CrossRef] - Beeler, B.; Casagranda, A.; Aagesen, L.; Zhang, Y.; Novascone, S. Atomistic calculations of the surface energy as a function of composition and temperature in γ U-Zr to inform fuel performance modeling. J. Nucl. Mater.
**2020**, 540, 152271. [Google Scholar] [CrossRef] - Beeler, B.; Andersson, D.; Jiang, C.; Zhang, Y. Ab initio molecular dynamics investigation of point defects in γ-U. J. Nucl. Mater.
**2020**, 545, 152714. [Google Scholar] [CrossRef] - Kolotova, L.; Gordeev, I. Structure and phase transition features of monoclinic and tetragonal phases in U–Mo alloys. Crystals
**2020**, 10, 515. [Google Scholar] [CrossRef] - Söderlind, P.; Landa, A.; Perron, A.; Moore, E.E.; Wu, C. Thermodynamics of plutonium monocarbide from anharmonic and relativistic theory. Appl. Sci.
**2020**, 10, 6524. [Google Scholar] [CrossRef] - Beeler, B.; Mahbuba, K.; Wang, Y.; Jokisaari, A. Determination of thermal expansion, defect formation energy, and defect-induced strain of α-U via ab initio molecular dynamics. Front. Mater.
**2021**, 8, 661387. [Google Scholar] [CrossRef] - Migdal, K.; Yanilkin, A. Cold and hot uranium in DFT calculations: Investigation by the GTH pseudopotential, PAW, and APW + lo methods. Comput. Mater. Sci.
**2021**, 199, 110665. [Google Scholar] [CrossRef] - Ouyang, W.; Lai, W.; Li, J.; Liu, J.; Liu, B. Atomic simulations of U-Mo under irradiation: A new angular dependent potential. Metals
**2021**, 11, 1018. [Google Scholar] [CrossRef] - Söderlind, P.; Yang, L.H.; Landa, A.; Wu, A. Mechanical and thermal properties for uranium and U–6Nb alloy from first-principles theory. Appl. Sci.
**2021**, 11, 5643. [Google Scholar] [CrossRef] - Söderlind, P.; Moore, E.E.; Wu, C.J. Thermodynamics modeling for actinide monocarbides and mononitrides from first principles. Appl. Sci.
**2022**, 12, 728. [Google Scholar] [CrossRef] - Aly, A.; Beeler, B.; Avramova, M. Ab initio molecular dynamics investigation of γ-(U,Zr) structural and thermal properties as a function of temperature and composition. J. Nucl. Mater.
**2022**, 561, 153523. [Google Scholar] [CrossRef] - Flotow, H.E.; Lohr, H.R. The heat capacity and thermodynamic functions of uranium from 5 to 350 K. J. Phys. Chem.
**1960**, 64, 904–906. [Google Scholar] [CrossRef] - Jones, W.; Gordon, J.; Long, E. The heat capacities of uranium, uranium trioxide, and uranium dioxide from 15 K to 300 K. J. Chem. Phys.
**1952**, 20, 695–699. [Google Scholar] [CrossRef] - Nakamura, J.-I.; Takashi, Y.; Izumi, S.-I.; Kanno, M. Heat capacity of metallic uranium and thorium from 80 to 1000 K. J. Nucl. Mater.
**1980**, 88, 64–72. [Google Scholar] [CrossRef] - Grimvall, G. Thermophysical Properties of Materials; Elsevier BV: Amsterdam, The Netherlands, 1999; pp. 157–172. [Google Scholar]
- Konings, R.J.M.; Beneš, O. The thermodynamic properties of the f-elements and their compounds. I. The lanthanide and actinide metals. J. Phys. Chem. Ref. Data
**2010**, 39, 43102. [Google Scholar] [CrossRef] - Marchidan, D.I.; Ciopec, M. Enthalpy of uranium to 1500 K by drop calorimetry. J. Chem. Therm.
**1976**, 8, 691–701. [Google Scholar] [CrossRef] - Gathers, G. Dynamic methods for investigating thermophysical properties of matter at very high temperatures and pressures. Rep. Prog. Phys.
**1986**, 86, 341–396. [Google Scholar] [CrossRef] - Belashchenko, D.K.; Smirnova, D.E.; Ostrovsky, O.I. Molecular-dynamic simulation of the thermophysical properties of liquid uranium. High Temp.
**2010**, 48, 363–375. [Google Scholar] [CrossRef] - Souvatzis, P.; Eriksson, O.; Katsnelson, M.I.; Rudin, S.P. Entropy driven stabilization of energetically unstable crystal structures explained from first principles theory. Phys. Rev. Lett.
**2008**, 100, 95901. [Google Scholar] [CrossRef] [PubMed] - Söderlind, P.; Landa, A.; Hood, R.Q.; Moore, E.E.; Perron, A.; McKeown, J.T. High-temperature thermodynamics modeling of graphite. Appl. Sci.
**2022**, 12, 7556. [Google Scholar] [CrossRef] - Grimvall, G. Spin disorder in paramagnetic fcc iron. Phys. Rev. B
**1989**, 39, 12300–12301. [Google Scholar] [CrossRef] [PubMed] - Wang, Y. Classical mean-field approach for thermodynamics: Ab initio thermophysical properties of cerium. Phys. Rev. B
**2000**, 61, R11863–R11866. [Google Scholar] [CrossRef] - Grimvall, G.; Häglund, J.; Fernández Guillermet, A. Spin fluctuations in paramagnetic chromium determined from entropy considerations. Phys. Rev. B
**1993**, 47, 15338–15341. [Google Scholar] [CrossRef] - Wills, J.M.; Eriksson, O.; Andersson, P.; Delin, A.; Grechnyev, O.; Alouani, M. Full-Potential Electronic Structure Method; Springer Series in Solid-State Science; Springer: Berlin/Heidelberg, Germany, 2010; Volume 167. [Google Scholar]
- Sadigh, B.; Kutepov, A.; Landa, A.; Söderlind, P. Assessing relativistic effects and electron correlation in the actinide metals Th-Pu. Appl. Sci.
**2019**, 9, 5020. [Google Scholar] [CrossRef] - Söderlind, P. First-principles phase stability, bonding, and electronic structure of actinide metals. J. Electron. Spectr. Rel. Phenom.
**2014**, 194, 2–7. [Google Scholar] [CrossRef] - Söderlind, P.; Moore, K.T. When magnetism can stabilize the crystal structure of metals. Scripta Mat.
**2008**, 59, 1259–1262. [Google Scholar] [CrossRef] - Söderlind, P.; Landa, A.; Sadigh, B. Density-functional theory for plutonium. Adv. Phys.
**2019**, 68, 1–47. [Google Scholar] [CrossRef] - Eriksson, O.; Brooks, M.S.S.; Johansson, B. Orbital polarization in narrow-band systems: Application to volume collapses in light lanthanides. Phys. Rev. B
**1990**, 41, 7311(R). [Google Scholar] [CrossRef] - Eschrig, H.; Sargolzaei, M.; Koepernik, K.; Richter, M. Orbital polarization in the Kohn-Sham-Dirac theory. EPL
**2005**, 72, 611–617. [Google Scholar] [CrossRef] - Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B
**1996**, 54, 11169–11186. [Google Scholar] [CrossRef] - Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B
**1999**, 59, 1758–1775. [Google Scholar] [CrossRef] - Saunders, N.; Miodownik, A. CALPHAD Calculation of Phase Diagrams: A Comprehensive Guide; Elsevier Science: Amsterdam, The Netherlands, 1998. [Google Scholar]
- Lukas, H.; Fries, S.; Sundman, B. Computational Thermodynamics: The CALPHAD Method; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Andersson, J.O.; Helander, T.; Höglund, L.; Shi, P.F.; Sundman, B. Thermo-Calc & DICTRA, computational tools for material science. Calphad
**2002**, 26, 273–312. [Google Scholar] - Thermo-Calc Software PURE5/Pure Substances Database Version 5. Available online: https://thermocalc.com/products/databases/general-alloys-and-pure-substances/ (accessed on 3 February 2023).
- Dinsdale, A.T. SGTE data for pure elements. Calphad
**1991**, 15, 317–425. [Google Scholar] [CrossRef] - Wilson, A.S.; Rundle, R.E. The structures of uranium metal. Acta Cryst.
**1949**, 2, 126–127. [Google Scholar] [CrossRef] - Crocombette, J.; Jollet, F.; Nga, L.; Petit, T. Plane-wave pseudopotential study of point defects in uranium dioxide. Phys. Rev. B
**2001**, 64, 104107. [Google Scholar] [CrossRef] - Shang, S.L.; Saengdeejing, A.; Mei, Z.G.; Kim, D.E.; Zhang, H.; Ganeshan, S.; Wang, Y.; Liu, Z.K. First-principles calculations of pure elements: Equations of state and elastic stiffness constants. Comp. Mater. Sci.
**2010**, 48, 813–826. [Google Scholar] [CrossRef] - Shapiro, A.B.; Summers, L.T.; Eckels, D.J.; Sahai, V. Modeling of Casting Microstructures and Defects; UCRL-ID-128519; LLNL Internal Report: Livermore, CA, USA, 1997. [Google Scholar]
- Rohr, W.G.; Wittenberg, L.J. Density of liquid uranium. J. Phys. Chem.
**1970**, 74, 1151–1152. [Google Scholar] [CrossRef] - Moore, G.E.; Kelley, K.K. High-temperature heat contents of uranium, uranium dioxide, and uranium trioxide. J. Am. Chem. Soc.
**1947**, 69, 2105–2107. [Google Scholar] [CrossRef] [PubMed] - Ginnings, D.C.; Corruccini, R.J. Heat capacities at high temperatures of uranium, uranium trichloride, and uranium tetrachloride. J. Res. N. B. S.
**1947**, 39, 309–316. [Google Scholar] [CrossRef] - Levinson, L.S. Heat content of molten uranium. J. Chem. Phys.
**1964**, 40, 3584–3585. [Google Scholar] [CrossRef] - Rand, H.; Kubaschewski, O. Thermochemical Properties of Uranium Compounds; Oliver and Boyd; Ltd.: Edinburgh, Scotland; London, UK, 1963; p. 36. [Google Scholar]
- Fei, Y.; Saxena, S.K. An equation for the heat capacity of solids. Geochim. Cosmochim. Acta
**1987**, 51, 251–254. [Google Scholar] [CrossRef] - Holley, C.E., Jr.; Storms, E.K. Thermodynamics of Nuclear Materials; Proc. IAEA.: Vienna, Austria, 1968; p. 411. [Google Scholar]

**Figure 1.**Atomic volumes (Å

^{3}) for γ-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. γ-U melts at 1408 K and theoretical data points above that temperature are shown for a hypothetical γ-U solid phase.

**Figure 2.**Gibbs free energy for γ-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 γ-U solid phase.

**Figure 3.**Electronic density of states (states/eV) for α-U (solid red) and γ-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.

**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 γ-U solid phase.

**Table 1.**Room-temperature atomic volume (Å

^{3}) and bulk modulus (GPa) for γ-U from present theory (DFT + OP + SCAILD), other modeling, and experiments.

Method | Atomic Volume (Å^{3}) | Bulk Modulus (GPa) |
---|---|---|

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 |

Experiment [14,94] | 20.89 | 113 |

**Table 2.**Spin (µ

_{spin}) and orbital (µ

_{orbital}) magnetic moments (µ

_{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.

Temperature | µ_{spin} | µ_{orbital} | F_{mag} | F_{el} | F_{lat} |
---|---|---|---|---|---|

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 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Söderlind, P.; Landa, A.; Moore, E.E.; Perron, A.; Roehling, J.; McKeown, J.T. High-Temperature Thermodynamics of Uranium from Ab Initio Modeling. *Appl. Sci.* **2023**, *13*, 2123.
https://doi.org/10.3390/app13042123

**AMA Style**

Söderlind P, Landa A, Moore EE, Perron A, Roehling J, McKeown JT. High-Temperature Thermodynamics of Uranium from Ab Initio Modeling. *Applied Sciences*. 2023; 13(4):2123.
https://doi.org/10.3390/app13042123

**Chicago/Turabian Style**

Söderlind, Per, Alexander Landa, Emily E. Moore, Aurélien Perron, John Roehling, and Joseph T. McKeown. 2023. "High-Temperature Thermodynamics of Uranium from Ab Initio Modeling" *Applied Sciences* 13, no. 4: 2123.
https://doi.org/10.3390/app13042123