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A simplified method of calculating electronic energy level populations in nonequilibrium plasmas

He Xin Jiang Tao Gao Cheng Zhang Zhen-Fu Yang Jun-Bo

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A simplified method of calculating electronic energy level populations in nonequilibrium plasmas

He Xin, Jiang Tao, Gao Cheng, Zhang Zhen-Fu, Yang Jun-Bo
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  • In order to investigate the radiative properties of plasma in non local thermodynamic equilibrium (NLTE), it is of great importance to determine energy level populations, which are often obtained by the so-called collisional-radiative (CR) model. As is well known, the CR model is accurate but computationally costly, and thus it is difficult to be applied to engineering calculations for such as complex three-dimensional plasmas. In this work, a bound-state characteristic temperature (BCT) method is proposed, which can be used to calculate quickly the energy level populations in non-equilibrium plasmas. In this method, we assume that for each kind of ionization stage, the bound-state population is Boltzmannian at a certain characteristic temperature. The assumed characteristic temperature is related to the degree of none-equilibrium and may be different from the electronic temperature of the plasma. Based on a modified Saha equation, the assumed characteristic temperature can be calculated easily, and then the energy level populations are obtained conveniently. Five cases of non-equilibrium neon plasma at variable electronic temperatures and densities are investigated and compared with the results from a CR model. Good agreement is found between them if the degree of non-equilibrium is not very large. It shows that the present method is effective and at least 3000 times faster in computation time than the CR model. The method is very useful in engineering applications.
      Corresponding author: Gao Cheng, gaocheng@nudt.edu.cn
    • Funds: Project supported by the National Numerical Windtunnel of China (Grant No. NNW2019ZT3-B07) and the National Natural Science Foundation of China (Grant No. 12074430)
    [1]

    Rogers F J, Iglesias C A 1994 Science 263 50Google Scholar

    [2]

    高城 2011 博士学位论文 (长沙: 国防科技大学)

    Gao C 2011 Ph. D. Dissertation (Changsha: National University of Defense Technology) (in Chinese)

    [3]

    Cowan R D 1981 The Theory of Atomic Structure and Spectra (Berkeley and Los Angeles: University of California Press) p2

    [4]

    吴泽清 2000 博士学位论文(北京: 中国工程物理研究院)

    Wu Z 2000 Ph. D. Dissertation (Beijing: China Academy of Engineering Physics) (in Chinese)

    [5]

    Surzhikov S T 2012 J. Heat Transfer 134 031002

    [6]

    Itoh M, Yabe T, Kiyokawa S 1987 Phys. Rev. A 35 233Google Scholar

    [7]

    Bel'Kov S A, Gasparian P D, Dolgolyova G V 1997 J. Quant. Spectrosc. Radiat. Transfer 58 471Google Scholar

    [8]

    Novikov V G, Zakharov S V 2003 J. Quant. Spectrosc. Radiat. Transfer 81 339Google Scholar

    [9]

    王民盛, 刘凌涛, 韩小英, 李家明 2006 物理学报 55 2371Google Scholar

    Wang M S, Liu L T, Han X Y, Li J M 2006 Acta Phys. Sin. 55 2371Google Scholar

    [10]

    Peyrusse O, Bauche-Arnoult C, Bauche J 2005 J. Phys. B: At. Mol. Opt. Phys. 38 L137Google Scholar

    [11]

    Bar-Shalom A, Oreg J, Klapisch M 2000 J. Quant. Spectrosc. Radiat. Transfer 65 43Google Scholar

    [12]

    Peyrusse O 2001 J. Quant. Spectrosc. Radiat. Transfer 71 571Google Scholar

    [13]

    Bauche J, Bauche-Arnoult C, Peyrusse O, Bachelier A, Gauthier J C 2003 J. Quant. Spectrosc. Radiat. Transfer 81 47Google Scholar

    [14]

    Peyrusse O 2000 J. Phys. B: At. Mol. Opt. Phys. 33 4303Google Scholar

    [15]

    Hauschildt P H 1993 J. Quant. Spectrosc. Radiat. Transfer 50 301Google Scholar

    [16]

    Duston D, Clark R W, Davis J, Apruzese J P 1983 Phys. Rev. A 27 1441Google Scholar

    [17]

    Lee R W, Whiten B L, Stout R E 1984 J. Quant. Spectrosc. Radiat. Transfer 32 91Google Scholar

    [18]

    唐京武, 黄笃之, 易有根 2010 物理学报 59 7769Google Scholar

    Tang J W, Hang D Z, Yi Y G 2010 Acta Phys. Sin. 59 7769Google Scholar

    [19]

    Lee Y T 1987 J. Quant. Spectrosc. Radiat. Transfer 38 131Google Scholar

    [20]

    Abdallah Jr J, Sherrill M E 2008 High Energy Density Phys. 4 124Google Scholar

    [21]

    Fontes C J, Abdallah Jr J, Clark R E H, Kilcrease D P 2000 J. Quant. Spectrosc. Radiat. Transfer 65 223Google Scholar

    [22]

    Gao C, Zeng J, Li Y, Jin F, Yuan J 2013 High Energy Density Phys. 9 583Google Scholar

    [23]

    Fontes C J, Zhang H L, Abdallah Jr J, Clark R E H, Kilcrease D P, Colgan J, Cunningham R T, Hakel P, Magee N H, Sherrill M E 2015 J. Phys. B: At. Mol. Opt. Phys. 48 144014Google Scholar

    [24]

    Ralchenko Y 2016 Modern Methods in Collisional-Radiative Modeling of Plasmas (Berlin: Springer International Publishing) p127

    [25]

    Piron R, Gilleron F, Aglitskiy Y, Chung H-K, Fontes C J, Hansen S B, Marchuk O, Scott H A, Stambulchik E, Ralchenko Y 2017 High Energy Density Phys. 23 38Google Scholar

    [26]

    Hansen S B, Chung H K, Fontes C J, Ralchenko Y, Scott H A, Stambulchik E 2020 High Energy Density Phys. 35 100693Google Scholar

    [27]

    Hansen S B, Bauche J, Bauche-Arnoult C, Gu M F 2007 High Energy Density Phys. 3 109Google Scholar

    [28]

    Pang J Q, Wu Z Q, Yan J 2007 Commun. Comput. Phys. 2 1085

    [29]

    Bauche J, Bauche-Arnoult C, Fournier K B 2004 Phys. Rev. E 69 026403Google Scholar

    [30]

    Park C 1990 Nonequilibrium Hypersonic Aerothermodynamics (New York: Wiley Press) p5

  • 图 1  ${A^{z + }}$的束缚态及连续态

    Figure 1.  Bound and continuum states of ${A^{z + }}$.

    图 2  计算流程图

    Figure 2.  Calculation flowchart.

    图 3  算例1的非简并能级布居

    Figure 3.  Non-degenerate electronic level populations for Case 1.

    图 4  算例2的非简并能级布居

    Figure 4.  Non-degenerate electronic level populations for Case 2.

    图 5  算例3的非简并能级布居

    Figure 5.  Non-degenerate electronic level populations for Case 3.

    图 6  算例4的非简并能级布居

    Figure 6.  Non-degenerate electronic level populations for Case 4.

    图 7  算例5的非简并能级布居

    Figure 7.  Non-degenerate electronic level populations for Case 5.

    表 1  算例参数

    Table 1.  Cell parameters.

    算例1算例2算例3算例4算例5
    $k{T_{\rm{e}}}$/eV515401540
    Ne核总数密度/${\rm{c}}{{\rm{m}}^{ - 3}}$10181018101810201020
    ${\rm{Ne}}$粒子数含量/%0.0258
    ${\rm{N}}{{\rm{e}}^ + }$粒子数含量/%5.2017
    ${\rm{N}}{{\rm{e}}^{2 + }}$粒子数含量/%88.68530.00155.0406
    ${\rm{N}}{{\rm{e}}^{3 + }}$粒子数含量/%6.08630.420238.97130.0002
    ${\rm{N}}{{\rm{e}}^{4 + }}$粒子数含量/%0.000819.461848.92050.0299
    ${\rm{N}}{{\rm{e}}^{5 + }}$粒子数含量/%67.88830.00376.98991.0996
    ${\rm{N}}{{\rm{e}}^{6 + }}$粒子数含量/%12.14070.67670.077715.7779
    ${\rm{N}}{{\rm{e}}^{7 + }}$粒子数含量/%0.087420.126148.7610
    ${\rm{N}}{{\rm{e}}^{8 + }}$粒子数含量/%79.193534.3313
    ${\rm{N}}{{\rm{e}}^{9 + }}$粒子数含量/%
    ${\rm{N}}{{\rm{e}}^{10 + }}$粒子数含量/%
    e数含量/(1018 cm–3)2.014.927.79351.00712.00
    DownLoad: CSV

    表 2  Ne原子及离子电离能

    Table 2.  Ionization energy of neon atom and ions.

    原子(离子)电离能/eV
    ${\rm{Ne}}$19.441
    ${\rm{N}}{{\rm{e}}^ + }$40.565
    ${\rm{N}}{{\rm{e}}^{2 + }}$63.007
    ${\rm{N}}{{\rm{e}}^{3 + }}$93.588
    ${\rm{N} }{ {\rm{e} }^{4 + } }$123.924
    ${\rm{N}}{{\rm{e}}^{5 + }}$157.561
    ${\rm{N}}{{\rm{e}}^{6 + }}$201.788
    ${\rm{N}}{{\rm{e}}^{7 + }}$230.156
    ${\rm{N}}{{\rm{e}}^{8 + }}$1183.642
    ${\rm{N}}{{\rm{e}}^{9 + }}$1345.217
    DownLoad: CSV

    表 3  计算耗费对比

    Table 3.  A comparison of calculation cost.

    程序语言计算平台CPU总耗时
    CR模型FortranIBM服务器Intel Xeon E5649: 6核2.53 GHz约24 h
    本文方法MatlabThinkpad笔记本电脑Intel Core i5-3320 M: 2核2.60 GHz约28 s
    DownLoad: CSV
  • [1]

    Rogers F J, Iglesias C A 1994 Science 263 50Google Scholar

    [2]

    高城 2011 博士学位论文 (长沙: 国防科技大学)

    Gao C 2011 Ph. D. Dissertation (Changsha: National University of Defense Technology) (in Chinese)

    [3]

    Cowan R D 1981 The Theory of Atomic Structure and Spectra (Berkeley and Los Angeles: University of California Press) p2

    [4]

    吴泽清 2000 博士学位论文(北京: 中国工程物理研究院)

    Wu Z 2000 Ph. D. Dissertation (Beijing: China Academy of Engineering Physics) (in Chinese)

    [5]

    Surzhikov S T 2012 J. Heat Transfer 134 031002

    [6]

    Itoh M, Yabe T, Kiyokawa S 1987 Phys. Rev. A 35 233Google Scholar

    [7]

    Bel'Kov S A, Gasparian P D, Dolgolyova G V 1997 J. Quant. Spectrosc. Radiat. Transfer 58 471Google Scholar

    [8]

    Novikov V G, Zakharov S V 2003 J. Quant. Spectrosc. Radiat. Transfer 81 339Google Scholar

    [9]

    王民盛, 刘凌涛, 韩小英, 李家明 2006 物理学报 55 2371Google Scholar

    Wang M S, Liu L T, Han X Y, Li J M 2006 Acta Phys. Sin. 55 2371Google Scholar

    [10]

    Peyrusse O, Bauche-Arnoult C, Bauche J 2005 J. Phys. B: At. Mol. Opt. Phys. 38 L137Google Scholar

    [11]

    Bar-Shalom A, Oreg J, Klapisch M 2000 J. Quant. Spectrosc. Radiat. Transfer 65 43Google Scholar

    [12]

    Peyrusse O 2001 J. Quant. Spectrosc. Radiat. Transfer 71 571Google Scholar

    [13]

    Bauche J, Bauche-Arnoult C, Peyrusse O, Bachelier A, Gauthier J C 2003 J. Quant. Spectrosc. Radiat. Transfer 81 47Google Scholar

    [14]

    Peyrusse O 2000 J. Phys. B: At. Mol. Opt. Phys. 33 4303Google Scholar

    [15]

    Hauschildt P H 1993 J. Quant. Spectrosc. Radiat. Transfer 50 301Google Scholar

    [16]

    Duston D, Clark R W, Davis J, Apruzese J P 1983 Phys. Rev. A 27 1441Google Scholar

    [17]

    Lee R W, Whiten B L, Stout R E 1984 J. Quant. Spectrosc. Radiat. Transfer 32 91Google Scholar

    [18]

    唐京武, 黄笃之, 易有根 2010 物理学报 59 7769Google Scholar

    Tang J W, Hang D Z, Yi Y G 2010 Acta Phys. Sin. 59 7769Google Scholar

    [19]

    Lee Y T 1987 J. Quant. Spectrosc. Radiat. Transfer 38 131Google Scholar

    [20]

    Abdallah Jr J, Sherrill M E 2008 High Energy Density Phys. 4 124Google Scholar

    [21]

    Fontes C J, Abdallah Jr J, Clark R E H, Kilcrease D P 2000 J. Quant. Spectrosc. Radiat. Transfer 65 223Google Scholar

    [22]

    Gao C, Zeng J, Li Y, Jin F, Yuan J 2013 High Energy Density Phys. 9 583Google Scholar

    [23]

    Fontes C J, Zhang H L, Abdallah Jr J, Clark R E H, Kilcrease D P, Colgan J, Cunningham R T, Hakel P, Magee N H, Sherrill M E 2015 J. Phys. B: At. Mol. Opt. Phys. 48 144014Google Scholar

    [24]

    Ralchenko Y 2016 Modern Methods in Collisional-Radiative Modeling of Plasmas (Berlin: Springer International Publishing) p127

    [25]

    Piron R, Gilleron F, Aglitskiy Y, Chung H-K, Fontes C J, Hansen S B, Marchuk O, Scott H A, Stambulchik E, Ralchenko Y 2017 High Energy Density Phys. 23 38Google Scholar

    [26]

    Hansen S B, Chung H K, Fontes C J, Ralchenko Y, Scott H A, Stambulchik E 2020 High Energy Density Phys. 35 100693Google Scholar

    [27]

    Hansen S B, Bauche J, Bauche-Arnoult C, Gu M F 2007 High Energy Density Phys. 3 109Google Scholar

    [28]

    Pang J Q, Wu Z Q, Yan J 2007 Commun. Comput. Phys. 2 1085

    [29]

    Bauche J, Bauche-Arnoult C, Fournier K B 2004 Phys. Rev. E 69 026403Google Scholar

    [30]

    Park C 1990 Nonequilibrium Hypersonic Aerothermodynamics (New York: Wiley Press) p5

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Publishing process
  • Received Date:  14 December 2020
  • Accepted Date:  07 February 2021
  • Available Online:  12 July 2021
  • Published Online:  20 July 2021

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