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基于Thomas-Fermi-Kirzhnits模型的物态方程研究

王坤 史宗谦 石元杰 吴坚 贾申利 邱爱慈

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基于Thomas-Fermi-Kirzhnits模型的物态方程研究

王坤, 史宗谦, 石元杰, 吴坚, 贾申利, 邱爱慈

Study on equation of state based on Thomas-Fermi-Kirzhnits model

Wang Kun, Shi Zong-Qian, Shi Yuan-Jie, Wu Jian, Jia Shen-Li, Qiu Ai-Ci
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  • 本文针对丝阵Z箍缩等高能量密度物理实验的数值模拟研究, 建立了一种适用温度、密度范围宽的三项式半经验物态方程. 三项式半经验物态方程包括零温自由能项, 电子热贡献项和离子热贡献项. 零温自由能项采用多项式拟合的方法确定. 多项式系数通过多项式计算的结果与高压缩比区域和压缩比为1时零温Thomas-Fermi-Kirzhnits模型计算的结果对应相等得到. 离子对物态方程的热贡献采用一种准谐振模型, 此谐振模型可以描述离子在固态相中的行为, 并且在高温度、低密度区域趋近于理想气体物态方程. 电子对物态方程的热贡献采用含温Thomas-Fermi-Kirzhnits模型计算. 利用所建立的三项式半经验物态方程计算了铝的等温压缩曲线, 并与实验数据做了对比. 给出了很宽温度、密度范围内铝的压强, 其数据与相应的SESAME数据库数据做了对比.
    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.
    • 基金项目: 国家自然科学基金(批准号: 51322706, 51237006, 51325705), 教育部新世纪优秀人才支持计划(批准号: NCET-11-0428)和中央高校基本科研业务费专项资金资助的课题.
    • Funds: Project supported in part by the National Science Foundation of China (Grant Nos. 51322706, 51237006, 51325705), in part by the Program for New Century Excellent Talents in University, China (Grant No. NCET-11-0428), and in part by the Fundamental Research Funds for the Central Universities.
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    Shemyakin O P, Levashov P R, Khishchenko K V 2012 Contrib. Plasma Phys. 52 37

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    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]

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    Shemyakin O P, Levashov P R, Obruchkova L R, Khishchenko K V 2010 J. Phys. A: Math. Theor. 43 335003

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    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

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    Khishchenko K V 2004 Tech. Phys. Lett. 30 829

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    Khishchenko K V 2008 J. Phys.: Conf. Ser. 121 022025

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    McCarthy S L 1965 Lawrence Radiation Laboratory Report: UCRL-14365

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    Latter R 1955 Phys. Rev. 99 1854

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    Shi Z Q, Wang K, Li Y, Shi Y J, Wu J, Jia S L 2014 Phys. Plasmas 21 032702

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    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

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    Akahama Y, Nishimura M, Kinoshita K, Kawamura H, Ohishi Y 2006 Phys. Rev. Lett. 96 045505

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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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  • 收稿日期:  2015-01-03
  • 修回日期:  2015-03-18
  • 刊出日期:  2015-08-05

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