Study of Thermodynamic Properties and Melting Curves of HP2 Structure Thallium Metal Under Extreme Conditions
Main Article Content
Abstract
Thermodynamic properties are important to understand and simulate thermodynamic processes, calculate and predict temperature, efficiency changes, improve energy efficiency in industrial systems. In this work, we studied the effect of pressure on several thermodynamic quantities including Debye frequency, Debye temperature, and Debye-Waller factor of HP2-structured Thallium (Tl) metal in which we considered the effect of non-ideal c/a axis ratio using a semi-empirical method. The correlation shift function in the X-ray absorption fine structure spectrum was determined on the basis of the Debye-Waller factor. The effective interaction potential between scattering and absorbing atoms with other neighboring atoms was approximately determined by the contribution of neighboring atoms to the fourth coordination sphere. Applying the Debye model, we obtain the volume (pressure) dependent analytical expressions of thermodynamic quantities including Debye frequency, Debye temperature, Debye - Waller factor. In addition, we have combined the Debye model and Lindemann melting law to determine the melting temperature of Tl metal. The previous experimental parameters are introduced by us to numerically calculate these thermodynamic quantities up to a pressure of 50 GPa. The melting curve of Tl according to our calculations is compared with compared with previous theoretical and experimental results and show good agreement. This study not only provides additional data on the thermodynamic properties of Tl metal but can also be applied to calculate other metals and alloys under extreme conditions.
Keywords:
Thallium, Debye model, Debye frequency, Debye–Waller factor, high pressure, Lindemann's law of melting.
References
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Vol. 43, No. 11, 1991, pp. 8850-8860, https://doi.org/10.1103/PhysRevB.43.8850.
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[34] H. K. Hieu, Melting of Solids Under High Pressure, Vacuum, Vol. 109, 2014, pp. 184-186, https://doi.org/10.1016/j.vacuum.2014.07.010.
[35] N. V. Hung, J. Rehr, Anharmonic Correlated Einstein Model Debye-Waller Factors, Phys. Rev. B - Condens. Matter Mater. Phys., Vol. 56, No. 1, 1997, pp. 43-46, https://doi.org/10.1103/PhysRevB.56.43.
[36] Y. Wang, R. Ahuja, B. Johansson, Melting of Iron and Other Metals at Earth’s Core Conditions: A Simplified Computational Approach, Phys. Rev. B - Condens. Matter Mater. Phys., Vol. 65, No. 1, 2001, pp. 1-3, https://doi.org/10.1103/PhysRevB.65.014104.
[37] N. Tsujino, Y. Nishihara, Y. Nakajima, E. Takahashi, K. Funakoshi, Y. Higo, Equation of State of γ-Fe: Reference Density for Planetary Cores, Earth Planet. Sci. Lett., Vol. 375, 2013, pp. 244-253, https://doi.org/10.1016/j.epsl.2013.05.040.
[38] P. Vinet, J. Ferrante, J. H. Rose, J. R. Smith, Compressibility of solids, J. Geophys. Res. Geophys Res, Vol. 92, No. B9, 1987, pp. 9319-9325, https://doi.org/10.1029/JB092iB09p09319.
[39] R. E. Cohen, O. Gülseren, R. J. Hemley, Accuracy of Equation of State Formulations, Am. Mineral., Vol. 85,
No. 2, 2000, pp. 338-344, https://doi.org/10.2138/am-2000-2-312.
[40] G. Singh, R. P. S. Rathore, Generalised Morse Potential for BCC Complex Metals, Vol.135, No. 2, 1986,
pp. 513-518, https://doi.org/10.1002/pssb.2221350208.
[41] N V. Hung, T. S. Tien, L. H. Hung, R. R. Frahm, Anharmonic Effective Potential, Local Force Constant and EXAFS of Crystial: Theory and Comparison to Experiment, Int. J. Mod. Phys. B, Vol. 22, No. 29, 2008,
pp. 5155–5156, https://doi.org/10.1142/S0217979208049285.
[42] E. Purushotham, N.G. Krishna, Mean Square Amplitudes of Vibration and Associated Debye Temperatures of Rhenium, Osmium and Thallium, Phys. B., Vol. 405, 2010, pp. 3308-3311, https://doi.org/10.1016/j.physb.2010.04.066.
[43] B. D. Singh, Y. P. Varshni, X-ray Debye Temperature for Hexagonal Crystals, Acta Cryst, Vol. 38, 1982,
pp. 854-858, https://doi.org/10.1107/S0567739482001740.