Theoretical investigation on extreme ultraviolet radiative opacity and emissivity of Sn plasmas at local-thermodynamic equilibrium

Gao Cheng; Liu Yan-Peng; Yan Guan-Peng; Yan Jie; Chen Xiao-Qi; Hou Yong; Jin Feng-Tao; Wu Jian-Hua; Zeng Jiao-Long; Yuan Jian-Min

doi:10.7498/aps.72.20230455

Abstract

Sn is the material for an extreme ultraviolet (EUV) light source working at 13.5 nm, therefore the radiative properties of Sn plasma are of great importance in designing light source. The radiative opacity and emissivity of Sn plasma at local thermodynamic equilibrium are investigated by using a detailed-level-accounting model. In order to obtain precise atomic data, a multi-configuration Dirac-Fock method is used to calculate energy levels and oscillator strengths of ${\rm{Sn}}^{6+}$-${\rm{Sn}}^{14+}$. The electronic correlation effects of $4{\rm d}^m\text{-}4{\rm f}^m$($m=1, 2, 3, 4$) and $ 4\mathrm{p}^n\text{-}4\mathrm{d}^n $($n=1, 2, 3$) are mainly considered, which dominate the radiation near 13.5 nm. The number of fine-structure levels reaches about 200000 for each ionization stage in the present large-scale configuration interaction calculations. For the large oscillator strengths (> 0.01), the length form is in accord with the velocity form and their relative difference is about 20%–30%. The calculated transmission spectra of Sn plasma at 30 eV and 0.01 g/cm³ are compared with the experimental result, respectively, showing that they have both good consistency. The radiative opacity and emissivity of Sn plasma at the temperature in a range of 16–30 eV and density in a scope of of 0.0001–0.1 g/cm³ are investigated systematically. The effects of the plasma temperature and plasma density on radiation characteristics are studied. The results show that the radiative properties near 13.5 nm are broadened with the increase of density at a specific temperature, while it is narrowed with the increase of temperature for a specific density. The present investigation should be helpful in designing and studying EUV light source in the future.

Keywords:

Authors and contacts

1.
College of Science, National University of Defense Technology, Changsha 410073, China
2.
State Key Laboratory of Laser Interaction with Matter, Northwest Institute of Nuclear Technology, Xi’an 710000, China
3.
College of Science, Zhejiang University of Technology, Hangzhou 310023, China
4.
Graduate School of China Academy of Engineering Physics, Beijing 100193, China

Corresponding author: Gao Cheng, gaocheng@nudt.edu.cn ; Wu Jian-Hua, wujh@nudt.edu.cn ;

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12074430, 11974423) and the Foundation Research of the State Key Laboratory of Laser Interaction with Matter, China (Grant No. SKLIM2008).

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References

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Cited By

图 1 (a) ${\rm{Sn}}^{5+}$, (b) ${\rm{Sn}}^{8+}$, (c) ${\rm{Sn}}^{10+}$和(d) ${\rm{Sn}}^{13+}$离子的4s, 4p, 4d和4f轨道波函数

Figure 1. Radial wavefunctions of 4s, 4p, 4d and 4f belonging to (a) ${\rm{Sn}}^{5+}$, (b) ${\rm{Sn}}^{8+}$, (c) ${\rm{Sn}}^{10+}$ and (d) ${\rm{Sn}}^{13+}$.

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图 2 ${\rm{Sn}}^{10+}$束缚组态, 其中长条表示相应组态分裂而成的精细能级的能量范围, 虚线表示电离阈值

Figure 2. Bound configurations of ${\rm{Sn}}^{10+}$. Each bar represents the energy range of fine-structure levels belonging to the corresponding configuration. The dashed line represents ionization threshold.

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图 3 ${\rm{Sn}}^{10+}$振子强度　(a) 振子强度的长度和速度表示, 红色虚线斜率为1; (b) 波长11—20 nm的振子强度长度和速度表示的比值, 上下两条红色虚线分别表示比值为1和0.8

Figure 3. Oscillator strengths of ${\rm{Sn}}^{10+}$: (a) Length and velocity forms of oscillator strengths; the slope of the red dashed line is 1; (b) ratio of length form to velocity form. The upper and lower red dashed lines represent 1 and 0.8, respectively.

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图 4 温度30 eV, 密度0.01 g/cm³条件下Sn等离子体中不同电荷态对辐射不透明度的贡献　(a) ${\rm{Sn}}^{9+}$; (b) ${\rm{Sn}}^{10+}$; (c) ${\rm{Sn}}^{11+}$; (d) ${\rm{Sn}}^{12+}$; (e)总不透明度, 红线为只包含基组态和单电子激发组态的结果, 绿线为包括了基组态、单电子和双电子激发组态的结果

Figure 4. Radiative opacity of Sn at a temperature of 30 eV and a density of 0.01 g/cm³ contributed by different ionization stages: (a) ${\rm{Sn}}^{9+}$; (b) ${\rm{Sn}}^{10+}$; (c) ${\rm{Sn}}^{11+}$; (d) ${\rm{Sn}}^{12+}$; (e) total opacity. The red line represents the result including only ground and singly excited configurations. The green line represents the result including ground, singly and doubly excited configurations

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图 5 Sn等离子体透射谱　(a)温度30 eV, 密度0.01 g/cm³时本文计算、ATOMIC^[28]与实验^[22]结果; (b) 本文计算的密度0.01 g/cm³, 温度为25, 27, 30和32 eV的Sn等离子体透射谱. 本文计算取仪器展宽为0.5 eV

Figure 5. Transmission spectra of Sn plasmas: (a) Present calculation, ATOMIC^[28] and experimental results^[22] of Sn at a temperature of 30 eV and a density of 0.01 g/cm³; (b) present calculated transmission spectra of Sn at a density of 0.01 g/cm³ and temperatures of 25, 27, 30 and 32 eV. The instrumental broadening in the present calculation is set to be 0.5 eV

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图 6 温度为20 eV, 密度为(a) 0.0001, (b) 0.001, (c) 0.01和(d) 0.1 g/cm³的Sn等离子体EUV辐射不透明度(左图)和发射系数(右图). 平均电离度分别为10.96, 9.63, 8.36和7.16, 自由电子密度分别为5.56 × 10¹⁸, 4.89 × 10¹⁹, 4.24 × 10²⁰和3.63 × 10²¹ cm^–3. 红色虚线表示中心波长13.5 nm, 带宽2%的波长范围

Figure 6. EUV Radiative opacity (left) and emissivity (right) of Sn plasmas at a temperature of 20 eV and densities of (a) 0.0001, (b) 0.001, (c) 0.01 and (d) 0.1 g/cm³. The average ionization is 10.96, 9.63, 8.36 and 7.16, respectively. The free electron density is 5.56 × 10¹⁸, 4.89 × 10¹⁹, 4.24 × 10²⁰ and 3.63 × 10²¹ cm^–3, respectively. The red-dashed lines represent the 2% wavelength region centered at 13.5 nm

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图 7 温度为23 eV, 密度为(a) 0.001, (b) 0.005, (c) 0.01和(d) 0.1 g/cm³的Sn等离子体EUV辐射不透明度(左图)和发射系数(右图)

Figure 7. EUV Radiative opacity (left panel) and emissivity (right panel) of Sn plasmas at a temperature of 23 eV and densities of (a) 0.001, (b) 0.005, (c) 0.01 and (d) 0.1 g/cm³

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图 8 温度为27 eV, 密度为(a) 0.001, (b) 0.005, (c) 0.01和(d) 0.1 g/cm³的Sn等离子体EUV辐射不透明度(左图)和发射系数(右图)

Figure 8. EUV Radiative opacity (left panel) and emissivity (right panel) of Sn plasmas at a temperature of 27 eV and densities of (a) 0.001, (b) 0.005, (c) 0.01 and (d) 0.1 g/cm³

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图 9 密度为0.0001 g/cm³, 温度为(a) 16, (b) 18, (c) 20和(d) 23 eV的Sn等离子体EUV辐射不透明度(左图)和发射系数(右图)

Figure 9. EUV radiative opacity (left panel) and emissivity (right panel) of Sn plasmas at a density of 0.0001 g/cm³ and temperatures of (a) 16, (b) 18, (c) 20 and (d) 23 eV

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图 10 密度为0.001 g/cm³, 温度为(a) 18, (b) 20, (c) 23, (d) 25和(e) 27 eV的Sn等离子体EUV辐射不透明度(左图)和发射系数(右图)

Figure 10. EUV radiative opacity (left panel) and emissivity (right panel) of Sn plasmas at a density of 0.001 g/cm³ and temperatures of (a) 18, (b) 20, (c) 23, (d) 25 and (e) 27 eV

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图 12 密度为0.1 g/cm³, 温度为(a) 20, (b) 23, (c) 25, (d) 27和(e) 30 eV的Sn等离子体EUV辐射不透明度(左图)和发射系数(右图)

Figure 12. EUV radiative opacity (left panel) and emissivity (right panel) of Sn plasmas at a density of 0.1 g/cm³ and temperatures of (a) 20, (b) 23, (c) 25, (d) 27 and (e) 30 eV

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图 11 密度为0.01 g/cm³, 温度为(a) 20, (b) 23, (c) 25, (d) 27和(e) 30 eV的Sn等离子体EUV辐射不透明度(左图)和发射系数(右图)

Figure 11. EUV radiative opacity (left panel) and emissivity (right panel) of Sn plasmas at a density of 0.01 g/cm³ and temperatures of (a) 20, (b) 23, (c) 25, (d) 27 and (e) 30 eV

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表 1 ${\rm{Sn}}^{10+}$基组态4s²4p⁶4d⁴的精细能级(单位: eV), 能级符号中省略了满壳层的4s和4p轨道.

Table 1. Fine-structure levels belonging to the ground configuration 4s²4p⁶4d⁴ of ${\rm{Sn}}^{10+}$ (Unit: eV), where the fully occupied 4s and 4p orbitals are omitted.

	能级	J	本文(MCDF)	实验^[14]	CI+MBPT ^[14]
1	${\rm{4d}}_{3/2}^4$	0	0.00	0.00	0.00
2	$(\text{4d}_{3/2}^3)_{3/2}\text{4d}_{5/2}$	1	0.36	0.38	0.39
3	$(\text{4d}_{3/2}^2)_{2}(\text{4d}_{5/2}^2)_4$	2	0.79	0.82	0.83
4	$(\text{4d}_{3/2}^2)_{2}(\text{4d}_{5/2}^2)_4$	3	1.22	1.25	1.27
5	$\text{4d}_{3/2}(\text{4d}_{5/2}^3)_{9/2}$	4	1.64	1.65	1.68
6	$(\text{4d}_{3/2}^2)_{2}(\text{4d}_{5/2}^2)_2$	0	3.48	3.32	3.28
7	$(\text{4d}_{3/2}^3)_{3/2}\text{4d}_{5/2}$	4	3.98	3.67	3.60
8	$(\text{4d}_{3/2}^2)_2(\text{4d}_{5/2}^2)_2$	3	4.58	4.33	4.29
9	$\text{4d}_{3/2}(\text{4d}_{5/2}^3)_{9/2}$	5	4.61	4.36	4.30
10	$(\text{4d}_{3/2}^2)_2(\text{4d}_{5/2}^2)_2$	1	4.68	4.39	4.39
11	$(\text{4d}_{3/2}^3)_{3/2}\text{4d}_{5/2}$	2	4.80	4.55	4.54
12	$(\text{4d}_{3/2}^2)_2(\text{4d}_{5/2}^2)_4$	6	5.09	4.74	4.67
13	$\text{4d}_{3/2}(\text{4d}_{5/2}^3)_{9/2}$	4	5.29	5.02	4.99
14	$(\text{4d}_{3/2}^2)_2(\text{4d}_{5/2}^2)_2$	2	5.61	5.38	5.40
15	$(\text{4d}_{3/2}^2)_2(\text{4d}_{5/2}^2)_2$	4	5.72	5.43	5.40
16	$\text{4d}_{3/2}(\text{4d}_{5/2}^3)_{3/2}$	3	5.87	5.60	5.60
17	$(\text{4d}_{3/2}^2)_2(\text{4d}_{5/2}^2)_2$	3	6.27	5.99	5.99
18	$(\text{4d}_{3/2}^2)_2(\text{4d}_{5/2}^2)_4$	5	6.36	6.06	6.03
19	$\text{4d}_{3/2}(\text{4d}_{5/2}^3)_{3/2}$	2	6.69	6.42	6.43
20	$\text{4d}_{3/2}(\text{4d}_{5/2}^3)_{3/2}$	1	6.81	6.55	6.56
21	$\text{4d}_{3/2}(\text{4d}_{5/2}^3)_{9/2}$	6	7.15	6.68	6.60
22	$\text{4d}_{5/2}^4$	4	7.72	7.40	7.38
23	$\text{4d}_{3/2}(\text{4d}_{5/2}^3)_{3/2}$	0	7.77	7.55	7.57
24	$\text{4d}_{5/2}^4$	2	8.41	8.01	8.00
25	$\text{4d}_{3/2}(\text{4d}_{5/2}^3)_{3/2}$	3	8.89	8.43	8.42
26	$(\text{4d}_{3/2}^2)_0(\text{4d}_{5/2}^2)_2$	2	9.86	—	9.40
27	$(\text{4d}_{3/2}^2)_0(\text{4d}_{5/2}^2)_4$	4	10.27	9.77	9.78
28	$(\text{4d}_{3/2}^2)_2(\text{4d}_{5/2}^2)_0$	2	10.48	9.97	9.98
29	$\text{4d}_{3/2}(\text{4d}_{5/2}^3)_{5/2}$	3	10.66	10.15	10.16
30	$\text{4d}_{3/2}(\text{4d}_{5/2}^3)_{5/2}$	1	10.77	—	10.30
31	$\text{4d}_{5/2}^4$	0	11.23	—	10.77
32	$\text{4d}_{3/2}(\text{4d}_{5/2}^3)_{5/2}$	4	11.79	—	11.11
33	$\text{4d}_{3/2}(\text{4d}_{5/2}^3)_{5/2}$	2	14.70	—	13.95
34	$(\text{4d}_{3/2}^2)_0(\text{4d}_{5/2}^2)_0$	0	18.87	—	17.98

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表 2 不同温度和密度条件下, Sn等离子体在(13.5 ± 2%) nm波长范围的总发射功率. a(b)表示a × $10^{b}$

Table 2. Total emissivity of Sn plasmas in (13.5 ± 2%) nm wavelength region at a variety of temperature and density. a(b) represents a × $10^{b}$

$T_{\rm{e}}$/eV	ρ/(g·cm^–3)	$j_{{\rm{tot}}}$/(W·cm^–3)
20	0.0001	4.52(10)
	0.001	7.86(11)
	0.01	6.68(12)
	0.1	8.36(13)
23	0.0001	5.50(10)
	0.001	1.01(12)
	0.01	1.42(13)
	0.1	1.69(14)
27	0.001	1.13(12)
	0.005	9.07(12)
	0.01	2.24(13)
	0.1	3.11(14)

DownLoad: CSV

[1]	Pirati A, Van Shoot J, Troost K, van Ballegoij R, Krabbendam P, Stoeldraijer J, Loopstra E, Benschop J, Finders J, Meiling H, van Setten E, Mika N, Driedonkx J, Stamm U 2017 Proc. SPIE 10143 101430G Google Scholar
[2]	Hutcheson G D 2018 Proc. SPIE 10583 1058303
[3]	David A 2007 Soft x-rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge: Cambridge University Press) pp17–20
[4]	宗楠, 胡蔚敏, 王志敏, 王小军, 张申金, 薄勇, 彭钦军, 许祖彦 2020 中国光学 13 28 Google Scholar Zong N, Hu W M, Wang Z M, Wang X J, Zhang S J, Bo Y, Peng Q J, Xu Z Y 2020 Chin. Opt. 13 28 Google Scholar
[5]	Versolato O O 2019 Plasma Sources Sci. Technol. 28 083001 Google Scholar
[6]	Kieft E R, Garloff K, van der Mullen J J A M, Banine V 2005 Phys. Rev. E 71 036402 Google Scholar
[7]	Ueno Y, Soumagne G, Sumitani A, Endo A, Higashiguchi T 2007 Appl. Phys. Lett. 91 231501 Google Scholar
[8]	Svendsen W, O’Sullivan G 1994 Phys. Rev. A 50 3710 Google Scholar
[9]	Churilov S S, Ryabtsev A N 2006 Opt. Spectrosc. 101 169 Google Scholar
[10]	D'Arcy R, Ohashi H, Suda S, Tanuma H, Fujioka S, Nishimura H, Nishihara K, Suzuki C, Kato T, Koike F, White J, O’Sullivan G 2009 Phys. Rev. A 79 042509 Google Scholar
[11]	Ohashi H, Suda S, Tanuma H, Fujioka S, Nishimura H, Sasaki A, Nishihara K 2010 J. Phys. B: At. Mol. Opt. Phys. 43 065204 Google Scholar
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