Effects of Transition Metals (TM = Nb, Ta, and Zr) on Thermodynamic and Electronic Properties of Advanced (Th_1−xTM_x)C (x = 0.1 and 0.2) Fuels: A First-Principles Study

Abstract

In the development of advanced nuclear fuels, we investigated how transition metals (TM = Nb, Ta, Zr) affect the thermodynamic and electronic properties of thorium monocarbide (ThC) using first-principles calculations. We modeled six ternary carbide compositions (Th_1−xTM_x)C with x = 0.1 and 0.2 to predict key properties including heat capacity, entropy, Gibbs free energy, and bulk modulus across 0 K–1800 K. Results show that (Th_1−xTa_x)C has the lowest equilibrium energy and smallest volume, whereas (Th_1−xZr_x)C maintains the highest thermal conductivity and mechanical rigidity. Notably, (Th_0.8Ta_0.2)C exhibits significant phonon scattering and structural softening. Bonding charge density analysis reveals strong covalent Nb-C bonds and intensive Zr-C interactions, providing critical atomic-level insights to accelerate development of next-generation Th-based nuclear fuels.

Keywords

actinide carbides / nuclear fuel / structural ceramics / transition metal doping

References

[1] R. S. Kemp, E. S. Lyman, M. R. Deinert, R. L. Garwin, and F. N. von Hippel, “The Weapons Potential of High-Assay Low-Enriched Uranium,” Science 384, no. 6700 (2024): 1071–1073, https://doi.org/10.1126/science.ado8693.

[2] M. R. Abdussami and A. Verma, “Future Energy Landscapes: Analyzing the Cost-Effectiveness of Nuclear-Renewable Integrated Energy Systems in Retrofitting of Coal Power Plants,” Applied Energy 377 (2025): 124460, https://doi.org/10.1016/j.apenergy.2024.124460.

[3] Y. Gan, C. Ng, A. Elgowainy, and J. Marcinkoski, “Considering Embodied Greenhouse Emissions of Nuclear and Renewable Power Plants for Electrolytic Hydrogen and Its Use for Synthetic Ammonia, Methanol, Fischer–Tropsch Fuel Production,” Environmental Science & Technology 58, no. 42 (2024): 18654–18662, https://doi.org/10.1021/acs.est.3c06769.

[4] M. Hjelmeland, J. K. Nøland, S. Backe, and M. Korpås, “The Role of Nuclear Energy and Baseload Demand in Capacity Expansion Planning for low-carbon Power Systems,” Applied Energy 377 (2025): 124366, https://doi.org/10.1016/j.apenergy.2024.124366.

[5] B. Shirizadeh and P. Quirion, “Low-Carbon Options for the French Power Sector: What Role for Renewables, Nuclear Energy and Carbon Capture and Storage?,” Energy Economics 95 (2021): 105004, https://doi.org/10.1016/j.eneco.2020.105004.

[6] L. Liu, H. Guo, L. Dai, et al., “The Role of Nuclear Energy in the Carbon Neutrality Goal,” Progress in Nuclear Energy 162 (2023): 104772, https://doi.org/10.1016/j.pnucene.2023.104772.

[7] L. Cheng, N. Mahmood, and Y. Qiu, “Towards a Greener Economy: The Quest for Nuclear Energy Technology Budgeting,” Nuclear Engineering and Technology 57 (2024): 103356, https://doi.org/10.1016/j.net.2024.103356.

[8] M. Stan, “Discovery and Design of Nuclear Fuels,” Materials Today Physics 12, no. 11 (2009): 20–28, https://doi.org/10.1016/s1369-7021(09)70295-0.

[9] Y. Liu, Y. Lu, W. Y. Wang, et al., “Effects of Solutes on Thermodynamic Properties of (TMZrU)C (TM = Ta, Y) Medium-Entropy Carbides: A First-Principles Study,” Journal of Materials Informatics 3 (2023): 17, https://doi.org/10.20517/jmi.2023.19.

[10] R. Guzmán-Arriaga and G. Espinosa-Paredes, “Factoring-Based Method for the Design of a Nuclear Fuel,” Energy Conversion and Management 51 (2010): 918–927, https://doi.org/10.1016/j.enconman.2009.11.030.

[11] W. Y. Wang, J. Li, W. Liu, and Z.-K. Liu, “Integrated Computational Materials Engineering for Advanced Materials: A Brief Review,” Computational Materials Science 158 (2019): 42–48, https://doi.org/10.1016/j.commatsci.2018.11.001.

[12] W. Y. Wang, P. Li, D. Lin, et al., “DID Code: A Bridge Connecting the Materials Genome Engineering Database With Inheritable Integrated Intelligent Manufacturing,” Engineering 6 (2020): 612–620, https://doi.org/10.1016/j.eng.2020.05.001.

[13] S. M. Zamzamian, A. Zolfaghari, and Z. Kowsar, “Molecular Dynamics Investigation of Xenon, Uranium, and Oxygen Diffusion in UO₂ Nuclear Fuel,” Computational Materials Science 211 (2022): 111553, https://doi.org/10.1016/j.commatsci.2022.111553.

[14] W. Y. Wang, J. Yin, Z. Chai, et al., “Big Data-Assisted Digital Twins for the Smart Design and Manufacturing of Advanced Materials: From Atoms to Products,” Journal of Materials Informatics 2 (2022): 1, https://doi.org/10.20517/jmi.2021.11.

[15] C. Zou, J. Li, W. Y. Wang, et al., “Integrating Data Mining and Machine Learning to Discover High-Strength Ductile Titanium Alloys,” Acta Materialia 202 (2021): 211–221, https://doi.org/10.1016/j.actamat.2020.10.056.

[16] K. Anantharaman, V. Shivakumar, and D. Saha, “Utilisation of Thorium in Reactors,” Journal of Nuclear Materials 383, no. 1–2 (2008): 119–121, https://doi.org/10.1016/j.jnucmat.2008.08.042.

[17] S. S. Drera, K. Insulander Björk, and J. F. Kelly, “Thorium Fuel Production and Results From Beginning of Life Irradiation,” Progress in Nuclear Energy 72 (2014): 5–10, https://doi.org/10.1016/j.pnucene.2013.08.008.

[18] A. G. Croff, E. D. Collins, G. D. D. Cul, et al., “ORNL Experience and Perspectives Related to Processing of Thorium and 233U for Nuclear Fuel,” Nuclear Technology 194, no. 2 (2017): 252–270, https://doi.org/10.13182/nt15-59.

[19] K. Zhou, Y. Zhang, X. Chen, and S. Hu, “12-Membered Ring Carbides With Stabilization of Actinide Atoms,” Inorganic Chemistry 61, no. 4 (2022): 2119–2128, https://doi.org/10.1021/acs.inorgchem.1c03341.

[20] N. Bagheri, F. Nosratinia, F. Zahakifar, and T. Yousefi, “The Coprecipitation Method in the Production of Thorium Oxide and Uranium–Thorium Mixed Oxide Fuels-A Review,” Nuclear Engineering and Design 426 (2024): 113366, https://doi.org/10.1016/j.nucengdes.2024.113366.

[21] P. Martin, D. J. Cooke, and R. Cywinski, “A Molecular Dynamics Study of the Thermal Properties of Thorium Oxide,” Journal of Applied Physics 112, no. 7 (2012): 073507, https://doi.org/10.1063/1.4754430.

[22] D. Shi, J. Song, Y. Qin, et al., “Evolutionary Search for Novel Thorium Borides Toward Advanced Nuclear Fuels,” Journal of Physical Chemistry C 126, no. 41 (2022): 17759–17768, https://doi.org/10.1021/acs.jpcc.2c05753.

[23] A. Nuttin, D. Heuer, A. Billebaud, et al., “Potential of Thorium Molten Salt Reactorsdetailed Calculations and Concept Evolution With a View to Large Scale Energy Production,” Progress in Nuclear Energy 46, no. 1 (2005): 77–99, https://doi.org/10.1016/j.pnucene.2004.11.001.

[24] U. E. Humphrey and M. U. Khandaker, “Viability of Thorium-Based Nuclear Fuel Cycle for the Next Generation Nuclear Reactor: Issues and Prospects,” Renewable and Sustainable Energy Reviews 97 (2018): 259–275, https://doi.org/10.1016/j.rser.2018.08.019.

[25] J. Hou, Y. Xia, L. Ji, and G. Wang, “Theoretical Study on the Corrosion of Thorium Nuclear Fuel by Water: The Effect of Two-State Reaction Mechanism,” Chemical Physics Letters 786 (2022): 139167, https://doi.org/10.1016/j.cplett.2021.139167.

[26] N. A. Z. Kamarudin, A. F. Ismail, M. H. Rabir, and K. Kok Siong, “Neutronic Optimization of Thorium-Based Fuel Configurations for Minimizing Slightly Used Nuclear Fuel and Radiotoxicity in Small Modular Reactors,” Nuclear Engineering and Technology 56, no. 7 (2024): 2641–2649, https://doi.org/10.1016/j.net.2024.02.023.

[27] Q. Wei, L. Zhu, Y. Wu, Y. Liu, and B. Wang, “First-Principle Studies on Local Lattice Distortions and Thermodynamic Properties in Non-Stoichiometric Thorium Monocarbide,” Materials 16, no. 23 (2023): 7484, https://doi.org/10.3390/ma16237484.

[28] X. Kong, H. Kuang, A. Li, et al., “The Effect of Niobium on the Mechanical and Thermodynamic Properties of Zirconium Alloys,” Metals 14, no. 6 (2024): 646, https://doi.org/10.3390/met14060646.

[29] K. Mittag, “Kapitza Conductance and Thermal Conductivity of Copper Niobium and Aluminium in the Range From 1.3 to 2.1 K,” Cryogenics 13, no. 2 (1973): 94–99, https://doi.org/10.1016/0011-2275(73)90132-x.

[30] S. Gobert, F. Audubert, and J. Heintz, “Sintering Behaviour of Buffered UO₂ Fuel Doped With Niobium Oxides,” Journal of the European Ceramic Society 45, no. 4 (2025): 117049, https://doi.org/10.1016/j.jeurceramsoc.2024.117049.

[31] L. Tang, R. Xu, J. Tan, et al., “Modulating Electronic Structure of Monolayer Transition Metal Dichalcogenides by Substitutional Nb-Doping,” Advanced Functional Materials 31, no. 5 (2020): 2006941, https://doi.org/10.1002/adfm.202006941.

[32] R. A. Jat, D. Rawat, and A. Sharma, “Isotope Effect on Hydrogen Storage Properties of Ti and Nb Co-Substituted ZrCo Alloy,” International Journal of Hydrogen Energy 48, no. 44 (2023): 16802–16812, https://doi.org/10.1016/j.ijhydene.2023.01.161.

[33] B. Zhang, X. Ye, W. Luo, and G. Sang, “Influence of Niobium/Tantalum Doping on the Hydrogen Behavior of ZrCo(110) Surface,” International Journal of Hydrogen Energy 48, no. 46 (2023): 17577–17592, https://doi.org/10.1016/j.ijhydene.2022.11.110.

[34] B. Zhang, W. Luo, X. Ye, and G. Sang, “First-Principles Investigation on the Effect of Nb and Ta Doping on the Hydrogen Storage Performance of ZrCo,” Computational and Theoretical Chemistry 1238 (2024): 114688, https://doi.org/10.1016/j.comptc.2024.114688.

[35] X. Gao, W. Y. Wang, X. Chen, et al., “ProME: An Integrated Computational Platform for Material Properties at Extremes and Its Application in Multicomponent Alloy Design,” Materials Genome Engineering Advances 3, no. 3 (2025): e70029, https://doi.org/10.1002/mgea.70029.

[36] G. Yao, W. Y. Wang, P. X. Li, et al., “Electronic Structures and Strengthening Mechanisms of Superhard High-Entropy Diborides,” Rare Metals 42, no. 2 (2023): 614–628, https://doi.org/10.1007/s12598-022-02152-5.

[37] F. Tian, D. Y. Lin, X. Gao, Y. F. Zhao, and H. F. Song, “A Structural Modeling Approach to Solid Solutions Based on the Similar Atomic Environment,” Journal of Chemical Physics 153, no. 3 (2020): 034101, https://doi.org/10.1063/5.0014094.

[38] S. Wang, C. Xue, X. Yang, and J. Wang, “Structural Ordering Governs Stiffness and Ductile-to-Brittle Transition in Al–Li Alloys,” Journal of Chemical Physics 158, no. 10 (2023): 104303, https://doi.org/10.1063/5.0139566.

[39] C. Xue, S. Wang, J. Wang, et al., “An Ultrastrong Multilayer Core–Shell Nanostructure in Aluminum–Lithium Castings,” Nano Letters 25, no. 11 (2025): 4636–4643, https://doi.org/10.1021/acs.nanolett.4c06368.

[40] G. Sun, J. Kürti, P. Rajczy, M. Kertesz, J. Hafner, and G. Kresse, “Performance of the Vienna Ab Initio Simulation Package (VASP) in Chemical Applications,” Journal of Molecular Structure: THEOCHEM 624, no. 1–3 (2003): 37–45, https://doi.org/10.1016/s0166-1280(02)00733-9.

[41] I. Nieves-Pírez, A. Muñoz, F. Almeida, and V. Blanco, “Energy Efficiency and Performance Analysis of a Legacy Atomic Scale Materials Modeling Simulator (VASP),” Journal of Supercomputing 80 (2024): 16679–16702, https://doi.org/10.1007/s11227-024-06066-0.

[42] V. Eyert, M. Christensen, W. Wolf, et al., “Unravelling the Potential of Density Functional Theory Through Integrated Computational Environments: Recent Applications of the Vienna Ab Initio Simulation Package in the MedeA® Software,” Computation 6, no. 4 (2018): 63, https://doi.org/10.3390/computation6040063.

[43] R. Yongchao, L. Yongchao, Z. Lanting, et al., “A Metadata Schema for Lattice Thermal Conductivity From First-Principles Calculations,” Journal of Materials Informatics 2 (2022): 17, https://doi.org/10.20517/jmi.2022.20.

[44] S. Lv, L. Peng, W. Wu, Y. Yao, S. Jiao, and W. Hu, “Bridging Language Models and Computational Materials Science: A Prompt-Driven Framework for Material Property Prediction,” Materials Genome Engineering Advances 3, no. 2 (2025): e70013, https://doi.org/10.1002/mgea.70013.

[45] P. E. Blöchl, “Projector Augmented-Wave Method,” Physical Review B 50 (1994): 17953, https://doi.org/10.1103/PhysRevB.50.17953.

[46] G. Kresse and D. Joubert, “From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method,” Physical Review B 59, no. 3 (1999): 1758–1775, https://doi.org/10.1103/physrevb.59.1758.

[47] L. Zhang, J. Zhou, and X. Chen, “Data-Driven Exploration and First-Principles Analysis of Perovskite Material,” Journal of Materials Informatics 4 (2024): 13, https://doi.org/10.20517/jmi.2024.20.

[48] J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Physical Review Letters 77, no. 18 (1996): 3865–3868, https://doi.org/10.1103/physrevlett.77.3865.

[49] R. T. Liang, T. Bo, W. Q. Yin, et al., “Effects of Fission Product Doping on the Structure, Electronic Structure, Mechanical and Thermodynamic Properties of Uranium Monocarbide: A First-Principles Study,” Nuclear Engineering and Technology 55, no. 7 (2023): 2556–2566, https://doi.org/10.1016/j.net.2023.04.012.

[50] H. Wu, W. Shi, R. He, et al., “Atomistic Simulations of Thermodynamic Properties With Nuclear Quantum Effects of Liquid Gallium From First Principles,” Materials Genome Engineering Advances 3, no. 2 (2025): e70016, https://doi.org/10.1002/mgea.70016.

[51] M. Methfessel and A. Paxton, “High-Precision Sampling for Brillouin-Zone Integration in Metals,” Physical Review B 40, no. 6 (1989): 3616–3621, https://doi.org/10.1103/physrevb.40.3616.

[52] P. E. Blöchl, O. Jepsen, and O. K. Andersen, “Improved Tetrahedron Method for Brillouin-Zone Integrations,” Physical Review B 49 (1994): 16223, https://doi.org/10.1103/PhysRevB.49.16223.

[53] H. J. Monkhorst and J. D. Pack, “Special Points for Brillouin-Zone Integrations,” Physical Review B 13, no. 12 (1976): 5188–5192, https://doi.org/10.1103/physrevb.13.5188.

[54] Y. Wang and L. Li, “Mean-Field Potential Approach to Thermodynamic Properties of Metal: Al as a Prototype,” Physical Review B 62, no. 1 (2000): 196–202, https://doi.org/10.1103/physrevb.62.196.

[55] A. Otero-de-la-Roza, D. Abbasi-Pérez, and V. Luaña, “Gibbs2: A New Version of the Quasiharmonic Model Code.II.Models for Solid-State Thermodynamics, Features and Implementation,” Computer Physics Communications 182, no. 10 (2011): 2232–2248, https://doi.org/10.1016/j.cpc.2011.05.009.

[56] X. G. Lu, Y. He, and W. Zheng, “Design of Advanced Steels by Integrated Computational Materials Engineering,” Materials Genome Engineering Advances 2, no. 2 (2024): e36, https://doi.org/10.1002/mgea.36.

[57] J. Li, W. Hu, and J. Yang, “High-Throughput Screening of Rattling-Induced Ultralow Lattice Thermal Conductivity in Semiconductors,” Journal of the American Chemical Society 144, no. 10 (2022): 4448–4456, https://doi.org/10.1021/jacs.1c11887.

[58] L. Liu, M. Yao, Y. Wang, et al., “The MatHub-3d First-Principles Repository and the Applications on Thermoelectrics,” Materials Genome Engineering Advances 2, no. 1 (2024): e21, https://doi.org/10.1002/mgea.21.

[59] C. Yang, L. Shouhang, Y. Tao, B. Hua, and Z. Xiaoqin, “Research on the Thermal Conductivity of Metals Based on First Principles,” Acta Metallurgica Sinica 57 (2021): 375–384, https://www.jcscp.org/EN/10.11902/1005.4537.2025.158.

[60] D. Xu, Q. Zhang, X. Huo, Y. Wang, and M. Yang, “Advances in Data-Assisted High-Throughput Computations for Material Design,” Materials Genome Engineering Advances 1, no. 1 (2023): e11, https://doi.org/10.1002/mgea.11.

[61] H. Bai, Y. Duan, H. Qi, M. Peng, M. Li, and S. Zheng, “Anisotropic Elastic and Thermal Properties and Damage Tolerance of CrH: A First-Principles Calculation,” Vacuum 222 (2024): 112962, https://doi.org/10.1016/j.vacuum.2024.112962.

[62] Y. Zhang, K. Ren, W. Y. Wang, et al., “Discovering the Ultralow Thermal Conductive A₂B₂O₇-Type High-Entropy Oxides Through the Hybrid Knowledge-Assisted Data-Driven Machine Learning,” Journal of Materials Science & Technology 168 (2024): 131–142, https://doi.org/10.1016/j.jmst.2023.05.060.

[63] P. X. Li, W. Y. Wang, C. X. Zou, et al., “Lattice Distortion Optimized Hybridization and Superlubricity of MoS₂/MoSe₂ Heterointerfaces via Moiré Patterns,” Applied Surface Science 613 (2023): 155760, https://www.ams.org.cn/EN/10.11900/0412.1961.2020.00250.

[64] P. N. Nakashima, A. E. Smith, J. Etheridge, and B. C. Muddle, “The Bonding Electron Density in Aluminum,” Science 331, no. 6024 (2011): 1583–1586, https://doi.org/10.1126/science.1198543.

[65] K. Momma and F. Izumi, “VESTA 3 for Three-Dimensional Visualization of Crystal, Volumetric and Morphology Data,” Journal of Applied Crystallography 44, no. 6 (2011): 1272–1276, https://doi.org/10.1107/s0021889811038970.

[66] J. Raskind, T. Babakol, K. Mahmoud, and Y. D. Liu, “VESTA: Power Modeling Withlanguage Rruntime Events,” Proceedings of the ACM on Programming Languages 8, no. PLDI (2024): 621–646, https://doi.org/10.1145/3656402.

[67] T. Zhang, L. Zhu, H. Liu, J. Zhou, and Z. Sun, “Ordering Phenomena in Ternary Transition-Metal Dichalcogenides: Critical Role of Lattice Symmetry and vdW Interaction,” Materials Genome Engineering Advances 1, no. 2 (2023): e7, https://doi.org/10.1002/mgea.7.

[68] B. Y. Zheng, Y. D. Wu, D. Y. Lin, et al., “First-Principles Study on the Structure, Mechanical and Thermodynamic Properties of (Ti, Hf, Nb, Ta)C High-Entropy Carbide Ceramics,” Ceramics International 50, no. 13 (2024): 23097–23104, https://doi.org/10.1016/j.ceramint.2024.04.032.

[69] Y. Pan, “New Insight into the Effect of C Concentration on the Structural Stability, Elastic Modulus, Hardness and Thermodynamic Properties of NbC Carbides,” International Journal of Refractory Metals and Hard Materials 121 (2024): 106676, https://doi.org/10.1016/j.ijrmhm.2024.106676.

[70] S. Jiang, L. Shao, T. W. Fan, J. M. Duan, X. T. Chen, and B. Y. Tang, “Elastic and Thermodynamic Properties of High Entropy Carbide (Hf Ta Zr Ti)C and (Hf Ta Zr Nb)C From Ab Initio Investigation,” Ceramics International 46, no. 10 (2020): 15104–15112, https://doi.org/10.1016/j.ceramint.2020.03.045.

[71] K. Nakamura and M. Yashima, “Crystal Structure of NaCl-Type Transition Metal Monocarbides MC (M=V, Ti, Nb, Ta, Hf, Zr), a Neutron Powder Diffraction Study,” Materials Science and Engineering: B 148, no. 1–3 (2008): 69–72, https://doi.org/10.1016/j.mseb.2007.09.040.

[72] J. Duan, R. Ji, L. Ma, and L. Shao, “Ab Initio Study of Intrinsic Mechanical Properties of High-Entropy Carbides (Hf Ta Zr M)C (M = Ti and Nb),” Materials Today Communications 40 (2024): 109968, https://doi.org/10.1016/j.mtcomm.2024.109968.

[73] L. Xiong, B. Li, and Y. Hu, “Effect of Pressure on the Structural Properties of Tantalum Carbide,” International Journal of Modern Physics B 39, no. 9 (2024): 2550068, https://doi.org/10.1142/s0217979225500687.

[74] I. S. Lim and G. E. Scuseria, “A Screened Hybrid Density Functional Study of Metallic Thorium Carbide,” Chemical Physics Letters 460 (2008): 137–140, https://doi.org/10.1016/j.cplett.2008.06.008.

[75] L. Gerward, J. Staun Olsen, U. Benedict, J. P. Itíe, and J. C. Spirlet, “Structural Stability and Equation of State of Thorium Carbide for Pressures up to 36 Gpa,” Journal of Applied Crystallography 19, no. 5 (1986): 308–310, https://doi.org/10.1107/s0021889886089318.

[76] X. H. Lei, W. Liu, F. Luo, and X. G. Lu, “A Thermodynamic Database of the Ni-Mo-Re System,” Journal of Materials Informatics 2, no. 3 (2022): 11, https://doi.org/10.20517/jmi.2022.15.

[77] M. Yin, W. Liang, Q. Miao, et al., “Thermodynamic Properties, Interfacial Adhesion Energy and Tribological Properties Between MCN (M = Hf, Ta, Cr, Nb) Coating and Ti Alloy Substrates Investigated by Experiment and First-Principles Calculations,” Applied Surface Science 677 (2024): 161068, https://doi.org/10.1016/j.apsusc.2024.161068.

[78] Y. Pan and M. Wen, “Ab-Initio Calculations of Mechanical and Thermodynamic Properties of TM (Transition Metal: 3D and 4d)-doped Pt₃Al,” Vacuum 156 (2018): 419–426, https://doi.org/10.1016/j.vacuum.2018.08.010.

[79] L. Xu, Y. Pan, and J. Zhu, “Enhanced the Mechanical and Thermodynamic Properties of Mo₂AlC MAX Layered Carbide for High Temperature Applications,” Diamond and Related Materials 158 (2025): 112622, https://doi.org/10.1016/j.diamond.2025.112622.

[80] X. L. Liu, B. K. Van Leeuwen, S. L. Shang, Y. Du, and Z. K. Liu, “On the Scaling Factor in Debye–Grüneisen Model: A Case Study of the Mg–Zn Binary System,” Computational Materials Science 98 (2015): 34–41, https://doi.org/10.1016/j.commatsci.2014.10.056.

[81] J. Zhang, J. Cui, Z. Yang, and Y. Yu, “Heat Capacity and Thermal Expansion of Metal Crystalline Materials Based on Dynamic Thermal Vibration,” Computational Mechanics 63, no. 5 (2018): 971–984, https://doi.org/10.1007/s00466-018-1632-3.

[82] S. Aydin, A. Tatar, and Y. O. Ciftci, “A Theoretical Study for Thorium Monocarbide (ThC),” Journal of Nuclear Materials 429, no. 1–3 (2012): 55–69, https://doi.org/10.1016/j.jnucmat.2012.05.038.

[83] P. X. Zhang, L. Ye, F. H. Chen, W. J. Han, Y. H. Wu, and T. Zhao, “Stability, Mechanical, and Thermodynamic Behaviors of (TiZrHfTaM)C (M = Nb, Mo, W, V, Cr) High-Entropy Carbide Ceramics,” Journal of Alloys and Compounds 903 (2022): 163868, https://doi.org/10.1016/j.jallcom.2022.163868.

[84] E. C. Schwind, M. J. Reece, E. Castle, W. G. Fahrenholtz, and G. E. Hilmas, “Thermal and Electrical Properties of a High Entropy Carbide (Ta, Hf, Nb, Zr) at Elevated Temperatures,” Journal of the American Ceramic Society 105, no. 6 (2022): 4426–4434, https://doi.org/10.1111/jace.18400.

[85] J. Qian, C. Wu, and H. Gong, “Phase Transition, Thermodynamic and Elastic Properties of ZrC,” Transactions of Nonferrous Metals Society of China 28, no. 12 (2018): 2520–2527, https://doi.org/10.1016/s1003-6326(18)64898-8.

[86] E. J. Meitz, G. J. Wang, and A. J. H. McGaughey, “Phonon Mode Resolved Anharmonic Heat Capacity of Solids,” Physical Review B 111, no. 6 (2025): 064305, https://doi.org/10.1103/physrevb.111.064305.

[87] W. Y. Wang, S. L. Shang, H. Fang, Z. K. Liu, and L. Q. Chen, “First-Principles Calculations of Lattice Dynamics and Thermal Properties of Polar Solids,” npj Computational Materials 2, no. 1 (2016): 16006, https://doi.org/10.1038/npjcompumats.2016.6.

[88] N. K. Ravichandran and D. Broido, “Phonon-Phonon Interactions in Strongly Bonded Solids: Selection Rules and Higher-Order Processes,” Physical Review X 10, no. 2 (2020): 021063, https://doi.org/10.1103/physrevx.10.021063.

[89] D. Jiang, Q. Wang, W. Hu, Z. Wei, J. Tong, and H. Wan, “The Effect of Tantalum (Ta) Doping on Mechanical Properties of Tungsten (W): A First-Principles Study,” Journal of Materials Research 31, no. 21 (2016): 3401–3408, https://doi.org/10.1557/jmr.2016.358.

[90] Z. Wang, A. S. L. S. Pattamatta, J. Han, D. J. Srolovitz, and K. Zur, “Scaling Laws for Lattice Distortions: Application to High Entropy Alloys,” PNAS Nexus 3, no. 4 (2024): 117, https://doi.org/10.1093/pnasnexus/pgae117.

[91] K. Xiong, B. W. Wang, Z. P. Sun, et al., “Frist-Principles Prediction of Elastic, Electronic, and Thermodynamic Properties of High Entropy Carbide Ceramic (TiZrNbTa)C,” Rare Metals 41, no. 3 (2021): 1002–1014, https://doi.org/10.1007/s12598-021-01834-w.

[92] X. Y. Wang, G. Y. Shu, G. Zhu, et al., “An Interpretable Formula for Lattice Thermal Conductivity of Crystals,” Materials Today Physics 48 (2024): 101549, https://doi.org/10.1016/j.mtphys.2024.101549.

[93] A. J. Terricabras, J. O. Kiggans, L. Wang, and S. J. Zinkle, “Characterization of High Thermal Conductivity Fuel Surrogates Before and After Ion Irradiation,” Journal of Nuclear Materials 552 (2021): 153027, https://doi.org/10.1016/j.jnucmat.2021.153027.

[94] Y. Pan and W. M. Guan, “Origin of Enhanced Corrosion Resistance of Ag and Au Doped Anatase TiO₂,” International Journal of Hydrogen Energy 44, no. 21 (2019): 10407–10414, https://doi.org/10.1016/j.ijhydene.2019.02.131.

[95] Y. Yang, H. Lu, C. Yu, and J. M. Chen, “First-Principles Calculations of Mechanical Properties of TiC and TiN,” Journal of Alloys and Compounds 485, no. 1–2 (2009): 542–547, https://doi.org/10.1016/j.jallcom.2009.06.023.

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