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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. -
Keywords:
- nonequilibrium /
- plasmas /
- energy level populations
[1] Rogers F J, Iglesias C A 1994 Science 263 50
Google 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 35 233
Google Scholar
[7] Bel'Kov S A, Gasparian P D, Dolgolyova G V 1997 J. Quant. Spectrosc. Radiat. Transfer 58 471
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[8] Novikov V G, Zakharov S V 2003 J. Quant. Spectrosc. Radiat. Transfer 81 339
Google Scholar
[9] 王民盛, 刘凌涛, 韩小英, 李家明 2006 物理学报 55 2371
Google Scholar
Wang M S, Liu L T, Han X Y, Li J M 2006 Acta Phys. Sin. 55 2371
Google Scholar
[10] Peyrusse O, Bauche-Arnoult C, Bauche J 2005 J. Phys. B: At. Mol. Opt. Phys. 38 L137
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[11] Bar-Shalom A, Oreg J, Klapisch M 2000 J. Quant. Spectrosc. Radiat. Transfer 65 43
Google Scholar
[12] Peyrusse O 2001 J. Quant. Spectrosc. Radiat. Transfer 71 571
Google Scholar
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Google Scholar
[14] Peyrusse O 2000 J. Phys. B: At. Mol. Opt. Phys. 33 4303
Google Scholar
[15] Hauschildt P H 1993 J. Quant. Spectrosc. Radiat. Transfer 50 301
Google Scholar
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Google Scholar
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Google Scholar
[18] 唐京武, 黄笃之, 易有根 2010 物理学报 59 7769
Google Scholar
Tang J W, Hang D Z, Yi Y G 2010 Acta Phys. Sin. 59 7769
Google Scholar
[19] Lee Y T 1987 J. Quant. Spectrosc. Radiat. Transfer 38 131
Google Scholar
[20] Abdallah Jr J, Sherrill M E 2008 High Energy Density Phys. 4 124
Google Scholar
[21] Fontes C J, Abdallah Jr J, Clark R E H, Kilcrease D P 2000 J. Quant. Spectrosc. Radiat. Transfer 65 223
Google Scholar
[22] Gao C, Zeng J, Li Y, Jin F, Yuan J 2013 High Energy Density Phys. 9 583
Google 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 144014
Google 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 38
Google Scholar
[26] Hansen S B, Chung H K, Fontes C J, Ralchenko Y, Scott H A, Stambulchik E 2020 High Energy Density Phys. 35 100693
Google Scholar
[27] Hansen S B, Bauche J, Bauche-Arnoult C, Gu M F 2007 High Energy Density Phys. 3 109
Google Scholar
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[30] Park C 1990 Nonequilibrium Hypersonic Aerothermodynamics (New York: Wiley Press) p5
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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 $k{T_{\rm{e}}}$/eV 5 15 40 15 40 Ne核总数密度/${\rm{c}}{{\rm{m}}^{ - 3}}$ 1018 1018 1018 1020 1020 ${\rm{Ne}}$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/% 88.6853 0.0015 5.0406 ${\rm{N}}{{\rm{e}}^{3 + }}$粒子数含量/% 6.0863 0.4202 38.9713 0.0002 ${\rm{N}}{{\rm{e}}^{4 + }}$粒子数含量/% 0.0008 19.4618 48.9205 0.0299 ${\rm{N}}{{\rm{e}}^{5 + }}$粒子数含量/% 67.8883 0.0037 6.9899 1.0996 ${\rm{N}}{{\rm{e}}^{6 + }}$粒子数含量/% 12.1407 0.6767 0.0777 15.7779 ${\rm{N}}{{\rm{e}}^{7 + }}$粒子数含量/% 0.0874 20.1261 48.7610 ${\rm{N}}{{\rm{e}}^{8 + }}$粒子数含量/% 79.1935 34.3313 ${\rm{N}}{{\rm{e}}^{9 + }}$粒子数含量/% ${\rm{N}}{{\rm{e}}^{10 + }}$粒子数含量/% e–数含量/(1018 cm–3) 2.01 4.92 7.79 351.00 712.00 
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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
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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 Fortran IBM服务器 Intel Xeon E5649: 6核2.53 GHz 约24 h 本文方法 Matlab Thinkpad笔记本电脑 Intel Core i5-3320 M: 2核2.60 GHz 约28 s
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[1] Rogers F J, Iglesias C A 1994 Science 263 50
Google 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 35 233
Google Scholar
[7] Bel'Kov S A, Gasparian P D, Dolgolyova G V 1997 J. Quant. Spectrosc. Radiat. Transfer 58 471
Google Scholar
[8] Novikov V G, Zakharov S V 2003 J. Quant. Spectrosc. Radiat. Transfer 81 339
Google Scholar
[9] 王民盛, 刘凌涛, 韩小英, 李家明 2006 物理学报 55 2371
Google Scholar
Wang M S, Liu L T, Han X Y, Li J M 2006 Acta Phys. Sin. 55 2371
Google Scholar
[10] Peyrusse O, Bauche-Arnoult C, Bauche J 2005 J. Phys. B: At. Mol. Opt. Phys. 38 L137
Google Scholar
[11] Bar-Shalom A, Oreg J, Klapisch M 2000 J. Quant. Spectrosc. Radiat. Transfer 65 43
Google Scholar
[12] Peyrusse O 2001 J. Quant. Spectrosc. Radiat. Transfer 71 571
Google Scholar
[13] Bauche J, Bauche-Arnoult C, Peyrusse O, Bachelier A, Gauthier J C 2003 J. Quant. Spectrosc. Radiat. Transfer 81 47
Google Scholar
[14] Peyrusse O 2000 J. Phys. B: At. Mol. Opt. Phys. 33 4303
Google Scholar
[15] Hauschildt P H 1993 J. Quant. Spectrosc. Radiat. Transfer 50 301
Google Scholar
[16] Duston D, Clark R W, Davis J, Apruzese J P 1983 Phys. Rev. A 27 1441
Google Scholar
[17] Lee R W, Whiten B L, Stout R E 1984 J. Quant. Spectrosc. Radiat. Transfer 32 91
Google Scholar
[18] 唐京武, 黄笃之, 易有根 2010 物理学报 59 7769
Google Scholar
Tang J W, Hang D Z, Yi Y G 2010 Acta Phys. Sin. 59 7769
Google Scholar
[19] Lee Y T 1987 J. Quant. Spectrosc. Radiat. Transfer 38 131
Google Scholar
[20] Abdallah Jr J, Sherrill M E 2008 High Energy Density Phys. 4 124
Google Scholar
[21] Fontes C J, Abdallah Jr J, Clark R E H, Kilcrease D P 2000 J. Quant. Spectrosc. Radiat. Transfer 65 223
Google Scholar
[22] Gao C, Zeng J, Li Y, Jin F, Yuan J 2013 High Energy Density Phys. 9 583
Google 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 144014
Google 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 38
Google Scholar
[26] Hansen S B, Chung H K, Fontes C J, Ralchenko Y, Scott H A, Stambulchik E 2020 High Energy Density Phys. 35 100693
Google Scholar
[27] Hansen S B, Bauche J, Bauche-Arnoult C, Gu M F 2007 High Energy Density Phys. 3 109
Google 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 026403
Google Scholar
[30] Park C 1990 Nonequilibrium Hypersonic Aerothermodynamics (New York: Wiley Press) p5
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