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A wide-range semi-empirical equation of state is constructed for numerical simulation of high-energy density experiments, such as, wire-array Z-pinch etc. The equation of state consists of zero-temperature free energy term, and thermal contributions of electron and ion. Thomas-Fermi model, which was firstly put forward by Thomas and Fermi, is initially developed to study the electron distribution of multi-electron atoms. Since its advent, this model has been widely used in solid-state physics, atomic physics, astrophysics and equation of state computations. It is a particularly important model to describe the behavior of matter under extreme conditions of high temperature and high density. This model provides reasonably accurate results that are validated experimentally for some thermodynamic quantities, such as the pressure. However, the Thomas-Fermi model yields a pressure of a few GPa under normal density even at very low temperature, and the pressure is always positive, indicating an obvious limitation of this model. Kirzhnits has evaluated the influence of quantum effect and exchange effect on temperature-dependent Thomas-Fermi model and their contributions to the Thomas-Fermi equation of state. Basically, the Thomas-Fermi model with its quantum and exchange corrections which is called Thomas-Fermi-Kirzhnits model, can be applied to calculate the thermal contribution of electrons to the thermodynamic functions, which can lower the pressure given from the Thomas-Fermi model. The zero-temperature free energy term in the semi-empirical equation of state is described by a polynomial expression. The coefficients of the polynomial expression is calculated by using zero-temperature Thomas-Fermi-Kirzhnits model and the relation between thermodynamic quantities. A quasi-harmonic model is adopted to describe the behavior of ions. It is originally applied to calculate the contribution of ions in the condensed state. However, the quasi-harmonic model is close to an ideal equation of state in the high-temperature and low-density region. This model makes the description of the behavior of ions in the phase transition from the solid state to plasma state be approximated. Thomas-Fermi-Kirzhnits model is adopted to calculate the thermal contribution of electrons. The semi-empirical equation of state has the advantages of less calculation and clear physical concepts. Experimental data of isothermal compression at 300 K is fruitful and accurate. They can be used to verify the results of the semi-empirical equation of state. An isothermal compression curve is calculated by the present work and compared with experimental data. The pressures over a wide-range of temperature and density are derived and compared with corresponding data of SESAME database. The trajectory of the electrical explosion of aluminum is demonstrated from solid state to ideal plasma state.
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Keywords:
- equation of state /
- Thomas-Fermi-Kirzhnits model /
- Z-pinch
[1] Sheng L, Wang L P, Wu J, Li Y, Peng B D, Zhang M 2011 Chin. Phys. B 20 055202
[2] Zhang Y, Chen Q F, Gu Y J, Cai L C, Lu T C 2007 Acta Phys. Sin. 56 1318 (in Chinese) [张颖, 陈其峰, 顾云军, 蔡灵仓, 卢铁城 2007 物理学报 56 1318]
[3] Eliezer S, Ghatak A, Hora H 2002 Fundamentals of equations of state (London: World Scientific) p153
[4] Lin H L, Zhang R Q 1991 Chin. J. High Pressure Phys. 5 62 (in Chinses) [林华令, 张若棋 1991 高压物理学报 5 62]
[5] Tang W H, Zhang R Q 2008 Introduction to theory and computation of equation of state (Beijing: Higher Education Press) p254 (in Chinese) [汤文辉, 张若棋 物态方程理论及计算概论 (北京: 高等教育出版社) 第254页]
[6] Ji G F, Zhang Y L, Cui H L, Li X F, Zhao F, Meng C M, Song Z F 2009 Acta Phys. Sin. 58 4103 (in Chinese) [姬广富, 张艳丽, 崔红玲, 李晓凤, 赵峰, 孟川民, 宋振飞 2009 物理学报 58 4103]
[7] Meng C M, Ji G F, Huang H J 2005 Chin. J. High Pressure Phys. 19 253 (in Chinses) [孟川民, 姬广富, 黄海军 2005 高压物理学报 19 353]
[8] Yu J D, Li P, Wang W Q, Wu Q 2014 Acta Phys. Sin. 63 116401 (in Chinese) [于继东, 李平, 王文强, 吴强 2014 物理学报 63 116401]
[9] Shemyakin O P, Levashov P R, Khishchenko K V 2012 Contrib. Plasma Phys. 52 37
[10] Duan Y Y, Guo Y H, Qiu A C 2011 Nucl. Fusion Plasma Phys. 31 97 (in Chinese) [段耀勇, 郭永辉, 邱爱慈 2011 核聚变与等立体物理 31 97]
[11] Shemyakin O P, Levashov P R, Obruchkova L R, Khishchenko K V 2010 J. Phys. A: Math. Theor. 43 335003
[12] Kirzhnits D A 1957 Soviet Phys. JETP 5 64
[13] Chittenden J P, Lebedev S V, Ruiz-Camacho J, Beg F N, Bland S N, Jennings C A, Bell A R, Haines M G, Pikuz S A, Shelkovenko T A, Hammer D A 2000 Phys. Rev. E 61 4370
[14] Khishchenko K V 2004 Tech. Phys. Lett. 30 829
[15] Khishchenko K V 2008 J. Phys.: Conf. Ser. 121 022025
[16] McCarthy S L 1965 Lawrence Radiation Laboratory Report: UCRL-14365
[17] Latter R 1955 Phys. Rev. 99 1854
[18] Shi Z Q, Wang K, Li Y, Shi Y J, Wu J, Jia S L 2014 Phys. Plasmas 21 032702
[19] Nellis W J, Moriarty J A, Mitchell A C, Ross M, Dandrea R G, Ashcroft N W, Holmes N C, Gathers G R 1988 Phys. Rev. Lett. 60 1414
[20] Akahama Y, Nishimura M, Kinoshita K, Kawamura H, Ohishi Y 2006 Phys. Rev. Lett. 96 045505
[21] Cochrane K, Desjarlais M, Haill T, Lawrence J, Knudson M, Dunham G 2006 Sandia Report SAND2006-1739
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[1] Sheng L, Wang L P, Wu J, Li Y, Peng B D, Zhang M 2011 Chin. Phys. B 20 055202
[2] Zhang Y, Chen Q F, Gu Y J, Cai L C, Lu T C 2007 Acta Phys. Sin. 56 1318 (in Chinese) [张颖, 陈其峰, 顾云军, 蔡灵仓, 卢铁城 2007 物理学报 56 1318]
[3] Eliezer S, Ghatak A, Hora H 2002 Fundamentals of equations of state (London: World Scientific) p153
[4] Lin H L, Zhang R Q 1991 Chin. J. High Pressure Phys. 5 62 (in Chinses) [林华令, 张若棋 1991 高压物理学报 5 62]
[5] Tang W H, Zhang R Q 2008 Introduction to theory and computation of equation of state (Beijing: Higher Education Press) p254 (in Chinese) [汤文辉, 张若棋 物态方程理论及计算概论 (北京: 高等教育出版社) 第254页]
[6] Ji G F, Zhang Y L, Cui H L, Li X F, Zhao F, Meng C M, Song Z F 2009 Acta Phys. Sin. 58 4103 (in Chinese) [姬广富, 张艳丽, 崔红玲, 李晓凤, 赵峰, 孟川民, 宋振飞 2009 物理学报 58 4103]
[7] Meng C M, Ji G F, Huang H J 2005 Chin. J. High Pressure Phys. 19 253 (in Chinses) [孟川民, 姬广富, 黄海军 2005 高压物理学报 19 353]
[8] Yu J D, Li P, Wang W Q, Wu Q 2014 Acta Phys. Sin. 63 116401 (in Chinese) [于继东, 李平, 王文强, 吴强 2014 物理学报 63 116401]
[9] Shemyakin O P, Levashov P R, Khishchenko K V 2012 Contrib. Plasma Phys. 52 37
[10] Duan Y Y, Guo Y H, Qiu A C 2011 Nucl. Fusion Plasma Phys. 31 97 (in Chinese) [段耀勇, 郭永辉, 邱爱慈 2011 核聚变与等立体物理 31 97]
[11] Shemyakin O P, Levashov P R, Obruchkova L R, Khishchenko K V 2010 J. Phys. A: Math. Theor. 43 335003
[12] Kirzhnits D A 1957 Soviet Phys. JETP 5 64
[13] Chittenden J P, Lebedev S V, Ruiz-Camacho J, Beg F N, Bland S N, Jennings C A, Bell A R, Haines M G, Pikuz S A, Shelkovenko T A, Hammer D A 2000 Phys. Rev. E 61 4370
[14] Khishchenko K V 2004 Tech. Phys. Lett. 30 829
[15] Khishchenko K V 2008 J. Phys.: Conf. Ser. 121 022025
[16] McCarthy S L 1965 Lawrence Radiation Laboratory Report: UCRL-14365
[17] Latter R 1955 Phys. Rev. 99 1854
[18] Shi Z Q, Wang K, Li Y, Shi Y J, Wu J, Jia S L 2014 Phys. Plasmas 21 032702
[19] Nellis W J, Moriarty J A, Mitchell A C, Ross M, Dandrea R G, Ashcroft N W, Holmes N C, Gathers G R 1988 Phys. Rev. Lett. 60 1414
[20] Akahama Y, Nishimura M, Kinoshita K, Kawamura H, Ohishi Y 2006 Phys. Rev. Lett. 96 045505
[21] Cochrane K, Desjarlais M, Haill T, Lawrence J, Knudson M, Dunham G 2006 Sandia Report SAND2006-1739
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