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As the core issue in helicon discharge, the physical mechanism behind the high ionization rate phenomenon is still not fully understood. Based on the warm plasma dielectric tensor model which contains both the particle drift velocity and temperature anisotropy effect, by employing the general dispersion relation of electromagnetic waves propagating in magnetized and uniform plasma with typical helicon discharge parameter conditions, wave mode propagation characteristic and collisional, cyclotron and Landua damping induced wave power deposition properties of azimuthally symmetric mode are theoretically investigated. Systematic analysis shows the following findings. 1) Under typical helicon plasma parameter conditions, i.e. wave frequency ω/(2π)=13.56 MHz, ion temperature is one tenth of the electron temperature, and for a given magnetic field B0 (or wave frequency ω), there exists a critical wave frequency ωcr (or magnetic field B0,cr), above which (or below B0,cr) the damping of the n = 1, 2, 3 cyclotron harmonics begins to increase sharply. 2) For the electron temperature isotropic case, the attenuation constants of different harmonics start to increase significantly and monotonically at different thresholds of magnetic field, while the phase constant abruptly increases monotonically from the beginning of the parameter interval. On the other hand, for the electron temperature anisotropic case, both the phase constant and attenuation constant have peaking phenomenon, i.e. the attenuation constant begins to increase sharply at a certain value of B0 and meanwhile the phase constant presents a maximum value near the same value of magnetic field, thus the phase constant starts to keep constant at a certain value of B0 and meanwhile the attenuation constant has a maximum value near this same value of magnetic field. 3) For the wave power deposition properties, under electron temperature anisotropy conditions, power deposition due to collisional damping of Trivelpiece-Gould (TG) wave plays a dominant role in a low field (B0 = 48 Gs) (1 Gs = 10–4 T); by considering the electron finite Larmor radius (FLR) effect, the power deposition of TG wave presents a maximum value at a certain point of parallel electron temperature Te,//; with the decrease of Te,⊥/Te,//, the maximum value of power deposition increases gradually. All these findings are very important in further revealing the physical mechanism behind the high ionization rate in helicon plasma.
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Keywords:
- helicon plasma /
- anisotropy /
- propagation characteristics /
- power deposition
[1] Varughese G, Kumari J, Pandey RS, et al. 2018 J. Mod. Appl. Phys. 2 13
[2] Omura Y, Summers D 2006 J. Geophys. Res. Space Phys. 111 A09222
[3] 倪彬彬, 赵正予, 顾旭东, 汪枫 2008 物理学报 57 7937
Google Scholar
Ni B B, Zhao Z Y, Gu X D, Wang F 2008 Acta Phys. Sin. 57 7937
Google Scholar
[4] 傅绥燕, 徐寄遥, 魏勇, 刘立波, 熊明, 曹晋滨, 宗秋刚, 王赤, 冯学尚, 史全岐, 师立勤, 任丽文 2019 中国科学: 地球科学 49 1641
Fu S Y, Xu J Y, Wei Y, Liu L B, Xiong M, Cao J B, Zong Q G, Wang C, Feng X S, Shi Q Q, Shi L Q, Ren L W 2019 Sci. Sin. Terrae 49 1641
[5] Caneses J F, Blackwell B D 2016 Plasma Sources Sci. Technol. 25 055027
Google Scholar
[6] Isayama S, Shinohara S, Hada T 2018 Plasma Fusion Res. 13 1101014
Google Scholar
[7] Shinohara S 2018 Adv. Phys. X 3 1420424
[8] Shinohara S, Hada T, Motomura T, et al. 2009 Phys. Plasmas 16 057104
Google Scholar
[9] Chen F F, Boswell R W 1997 IEEE Trans. Plasma Sci. 25 1245
Google Scholar
[10] Squire J P, Chang-Diaz F R, Jacobson V T, et al. 2003 AIP Conference Proceedings for the 15th Topical Conference on Radio Frequency Power in Plasmas Moran, May 19–21, 2003 p423
[11] Squire J P, Chang-Díaz F R, Glover T W, et al. 2006 Thin Solid Films 506 579
[12] Boswell R W, Sutherland O, Charles C, et al. 2004 Phys. Plasmas 11 5125
Google Scholar
[13] Bathgate S N, Bilek M M M, Mckenzie D R 2017 Plasma Sci. Technol. 19 083001
Google Scholar
[14] Furukawa T, Kuwahara D, Shinohara S 2020 AIAA Propulsion and Energy Forum New Orleans, August 24–26, 2020 p3630
[15] Liu X, Sun X, Guo N, et al. 2022 IEEE Trans. Plasma Sci. 7 2138
[16] Polzin K, Martin A, Little J, et al. 2020 Aerospace 7 105
Google Scholar
[17] Perry A J, Vender D, Boswell R W 1991 J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom. 9 310
Google Scholar
[18] Takahashi K, Motomura T, Ando A, et al. 2014 J. Phys. D Appl. Phys. 47 425201
Google Scholar
[19] Smyrnakis A, Dimitrakis P, Gogolides E 2018 J. Phys. D Appl. Phys. 51 455101
Google Scholar
[20] Goulding R H, Caughman J B O, Rapp J, et al. 2017 Fusion Sci. Technol. 72 588
Google Scholar
[21] Ivanov A A, Davydenko V I, Kotelnikov I A, et al. 2013 Fusion Sci. Technol. 63 217
Google Scholar
[22] Goulding R H, Biewer T M, Caughman J B O, et al. 2011 AIP Conference Proceedings for the 19th Topical Conference on Radio Frequency Power in Plasmas Rhode Island, June 1–3, 2011 p535
[23] Goulding R H, Chen G, Meitner S, et al. 2009 AIP Conference Proceedings for the 18th Topical Conference on Radio Frequency Power in Plasmas Belgium, June 24–26, 2009 p667
[24] Isayama S, Shinohara S, Hada T, et al. 2019 Phys. Plasmas 26 023517
Google Scholar
[25] Shinohara S 2002 J. Plasma Fusion Res. 78 5
Google Scholar
[26] Chen F F, Torreblanca H 2007 Plasma Sources Sci. Technol. 16 593
Google Scholar
[27] Tarey R D, Sahu B B, Ganguli A 2012 Phys. Plasmas 19 073520
Google Scholar
[28] Chen F F 1991 Plasma Phys. Controlled Fusion 33 339
Google Scholar
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Google Scholar
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Google Scholar
[32] Eom G S, Kim J, Choe W 2006 Phys. Plasmas 13 073505
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Google Scholar
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Google Scholar
Zhao G, Xiong Y Q, Ma C, Liu Z W, Chen Q 2014 Acta Phys. Sin. 63 235202
Google Scholar
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Google Scholar
Ping L L, Zhang X J, Yang H, Xu G S, Chang L, Wu D S, Lü H, Zheng C Y, Peng J H, Jin H H, He C, Gan G H 2019 Acta Phys. Sin. 68 205201
Google Scholar
[36] Guo X M, Scharer J, Mouzouris Y, et al. 1999 Phys. Plasmas 6 3400
Google Scholar
[37] Correyero Plaza S, Navarro J, Ahedo E 2016 52nd AIAA/SAE/ASEE Joint Propulsion Conference Salt Lake City, July 25–27, 2016 p5035
[38] Swanson D G 1989 Plasma Waves (New York: Academic Press) p155
[39] Huba J D 2016 NRL Plasma Formulary (Washington: Naval Research Laboratory) p34
[40] Mouzouris Y, Scharer J E 1998 Phys. Plasmas 5 4253
Google Scholar
[41] Fried B D, Conte S D 2015 The Plasma Dispersion Function: The Hilbert Transform of the Gaussian (New York: Academic Press) p1
[42] Sakawa Y, Kunimatsu H, Kikuchi H, et al. 2003 Phys. Rev. Lett. 90 105001
Google Scholar
[43] Shamrai K P, Taranov V B 1996 Plasma Sources Sci. Technol. 5 474
Google Scholar
[44] Chen F F, Arnush D 1997 Phys. Plasmas 4 3411
Google Scholar
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图 1 螺旋波等离子体柱示意图
Figure 1. Cross section of helicon plasma column.
图 2 Whistler波色散特性
Figure 2. Dispersion characteristic of whistler wave.
图 3 电子温度各向异性对Whistler波n = 1次回旋谐波传播常数的影响 (实线代表相位常数, 虚线代表衰减常数)
Figure 3. Effect of electron temperature anisotropy on the propagation characteristic of the n = 1 electron cyclotron harmonic (the solid lines represent the phase constant, and the dashed lines represent the attenuation constant).
图 4 电子温度各向同性情形下n=1, 2, 3 次回旋谐波传播常数对纵向静磁场的依赖关系 (实线代表相位常数, 虚线代表衰减常数)
Figure 4. Dependence of propagation characteristic of the n=1, 2, 3 electron cyclotron harmonics on magnetic field in the case of electron temperature isotropy (the solid lines represent the phase constant, and the dashed lines represent the attenuation constant).
图 5 电子温度各向异性情形下n = 1, 2, 3次回旋谐波传播常数对纵向静磁场的依赖关系 (实线代表相位常数, 虚线代表衰减常数)
Figure 5. Dependence of propagation characteristic of the n = 1, 2, 3 electron cyclotron harmonics on magnetic field in the case of electron temperature anisotropy (The solid lines represent the phase constant, and the dashed lines represent the attenuation constant).
图 6 n = 1, 2, 3次回旋谐波衰减常数随 (a) 电子温度各向异性和 (b) 电子纵向漂移速度的依赖关系
Figure 6. Dependence of attenuation constant of the n = 1, 2, 3 electron cyclotron harmonics on (a) the electron temperature anisotropy and (b) electron parallel drift velocity.
图 7 螺旋波与TG波有限拉莫尔半径效应因子随归一化静磁场的变化关系
Figure 7. Dependence of the FLR effect parameter of helicon and TG waves on the normalized static magnetic field.
图 8 波功率沉积随纵向电子温度的变化
Figure 8. Wave power deposition versus parallel electron temperature.
图 9 螺旋波碰撞阻尼产生的功率沉积径向分布 (a) Te,⊥/Te, // = 0.1; (b) Te,⊥/Te,// = 1; (c) Te,⊥/Te,// = 10
Figure 9. Collisional damping induced radial power deposition distribution of the helicon wave: (a) Te, ⊥/Te, // = 0.1; (b) Te,⊥/Te,// = 1; (c) Te,⊥/ Te,// = 10.
图 10 TG波碰撞阻尼产生的功率沉积径向分布 (a) Te,⊥/Te, // = 0.1; (b) Te,⊥/Te,// = 1; (c) Te,⊥/Te,// = 10
Figure 10. Collisional damping induced radial power deposition distribution of the TG wave: (a) Te,⊥/Te,// = 0.1; (b) Te,⊥/Te, // = 1; (c) Te,⊥/Te, // = 10.
图 11 TG 波功率沉积在
$({T_{{\text{e}}, //}}, {\text{ }}{T_{{\text{e}}, \bot }}/{T_{{\text{e}}, //}})$ 空间的分布 (a) 三维分布; (b) 二维分布Figure 11.
$({T_{{\text{e}}, //}}, {\text{ }}{T_{{\text{e}}, \bot }}/{T_{{\text{e}}, //}})$ space power deposition distribution of TG wave: (a) 3D; (b) 2D.表 1 色散关系元素
Table 1. Elements of dispersion relation.
${\varPi _{su} }$ u = 1 u = 2 u = 3 s = 1 $ {{\text{J}}_m}({k_{ \bot m, {\text{H}}}}a) $ $ {{\text{J}}_m}({k_{ \bot m, {\text{TG}}}}a) $ $ - {\text{j}}{k_{ \bot m, v}}{\text{H}}_m^{(1)}({k_{ \bot m, v}}a) $ s = 2 $\begin{gathered} k_{ \bot m, {\text{TG} } }^2\left[ {m{k_{//, m} }{ {\text{J} }_m}({k_{ \bot m, {\text{H} } } }a) } \right. \\ \left. +{ {k_{\text{H} } }{k_{ \bot m, {\text{H} } } }a{ {\text{J} }_m'} ({k_{ \bot m, {\text{H} } } }a)} \right] \\ \end{gathered}$ $\begin{gathered} k_{ \bot m, {\text{H} } }^2\left[ {m{k_{//, m} }{ {\text{J} }_m}({k_{ \bot m, {\text{TG} } } }a) } \right. \\ \left. +{ {k_{\text{H} } }{k_{ \bot m, {\text{TG} } } }a{ {\text{J} }_m'} ({k_{ \bot m, {\text{TG} } } }a)} \right] \\ \end{gathered}$ $ {\text{j}}k_{ \bot m, {\text{H}}}^2 k_{ \bot m, {\text{TG}}}^2 m{\text{H}}_m^{(1)}({k_{ \bot m, v}}a) $ s = 3 $\begin{gathered} k_{ \bot m, {\text{TG} } }^2\left[ {m{k_{\text{H} } }{ {\text{J} }_m}({k_{ \bot m, {\text{H} } } }a) } \right. \\ \left. +{ {k_{//, m} }{k_{ \bot m, {\text{H} } } }a{ {\text{J} }_m'} ({k_{ \bot m, {\text{H} } } }a)} \right] \\ \end{gathered}$ $\begin{gathered} k_{ \bot m, {\text{H} } }^2\left[ {m{k_{ {\text{TG} } } }{ {\text{J} }_m}({k_{ \bot m, {\text{TG} } } }a) } \right. \\ \left. +{ {k_{//, m} }{k_{ \bot m, {\text{TG} } } }a{ {\text{J} }_m'} ({k_{ \bot m, {\text{TG} } } }a)} \right] \\ \end{gathered}$ ${\text{j} }k_{ \bot {m}, {\text{H} } }^2 k_{ \bot {m}, {\text{TG} } }^2 {k_{ \bot {m}, v} }a{\text{H}_m^{(1)' }}({k_{ \bot {m}, v} }a)$ 
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[1] Varughese G, Kumari J, Pandey RS, et al. 2018 J. Mod. Appl. Phys. 2 13
[2] Omura Y, Summers D 2006 J. Geophys. Res. Space Phys. 111 A09222
[3] 倪彬彬, 赵正予, 顾旭东, 汪枫 2008 物理学报 57 7937
Google Scholar
Ni B B, Zhao Z Y, Gu X D, Wang F 2008 Acta Phys. Sin. 57 7937
Google Scholar
[4] 傅绥燕, 徐寄遥, 魏勇, 刘立波, 熊明, 曹晋滨, 宗秋刚, 王赤, 冯学尚, 史全岐, 师立勤, 任丽文 2019 中国科学: 地球科学 49 1641
Fu S Y, Xu J Y, Wei Y, Liu L B, Xiong M, Cao J B, Zong Q G, Wang C, Feng X S, Shi Q Q, Shi L Q, Ren L W 2019 Sci. Sin. Terrae 49 1641
[5] Caneses J F, Blackwell B D 2016 Plasma Sources Sci. Technol. 25 055027
Google Scholar
[6] Isayama S, Shinohara S, Hada T 2018 Plasma Fusion Res. 13 1101014
Google Scholar
[7] Shinohara S 2018 Adv. Phys. X 3 1420424
[8] Shinohara S, Hada T, Motomura T, et al. 2009 Phys. Plasmas 16 057104
Google Scholar
[9] Chen F F, Boswell R W 1997 IEEE Trans. Plasma Sci. 25 1245
Google Scholar
[10] Squire J P, Chang-Diaz F R, Jacobson V T, et al. 2003 AIP Conference Proceedings for the 15th Topical Conference on Radio Frequency Power in Plasmas Moran, May 19–21, 2003 p423
[11] Squire J P, Chang-Díaz F R, Glover T W, et al. 2006 Thin Solid Films 506 579
[12] Boswell R W, Sutherland O, Charles C, et al. 2004 Phys. Plasmas 11 5125
Google Scholar
[13] Bathgate S N, Bilek M M M, Mckenzie D R 2017 Plasma Sci. Technol. 19 083001
Google Scholar
[14] Furukawa T, Kuwahara D, Shinohara S 2020 AIAA Propulsion and Energy Forum New Orleans, August 24–26, 2020 p3630
[15] Liu X, Sun X, Guo N, et al. 2022 IEEE Trans. Plasma Sci. 7 2138
[16] Polzin K, Martin A, Little J, et al. 2020 Aerospace 7 105
Google Scholar
[17] Perry A J, Vender D, Boswell R W 1991 J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom. 9 310
Google Scholar
[18] Takahashi K, Motomura T, Ando A, et al. 2014 J. Phys. D Appl. Phys. 47 425201
Google Scholar
[19] Smyrnakis A, Dimitrakis P, Gogolides E 2018 J. Phys. D Appl. Phys. 51 455101
Google Scholar
[20] Goulding R H, Caughman J B O, Rapp J, et al. 2017 Fusion Sci. Technol. 72 588
Google Scholar
[21] Ivanov A A, Davydenko V I, Kotelnikov I A, et al. 2013 Fusion Sci. Technol. 63 217
Google Scholar
[22] Goulding R H, Biewer T M, Caughman J B O, et al. 2011 AIP Conference Proceedings for the 19th Topical Conference on Radio Frequency Power in Plasmas Rhode Island, June 1–3, 2011 p535
[23] Goulding R H, Chen G, Meitner S, et al. 2009 AIP Conference Proceedings for the 18th Topical Conference on Radio Frequency Power in Plasmas Belgium, June 24–26, 2009 p667
[24] Isayama S, Shinohara S, Hada T, et al. 2019 Phys. Plasmas 26 023517
Google Scholar
[25] Shinohara S 2002 J. Plasma Fusion Res. 78 5
Google Scholar
[26] Chen F F, Torreblanca H 2007 Plasma Sources Sci. Technol. 16 593
Google Scholar
[27] Tarey R D, Sahu B B, Ganguli A 2012 Phys. Plasmas 19 073520
Google Scholar
[28] Chen F F 1991 Plasma Phys. Controlled Fusion 33 339
Google Scholar
[29] Chen F F, Blackwell D D 1999 Phys. Rev. Lett. 82 2677
Google Scholar
[30] Blackwell D D, Chen F F 2001 Plasma Sources Sci. Technol. 10 226
Google Scholar
[31] Kline J L, Scime E E, Boivin R F, et al. 2002 Phys. Rev. Lett. 88 195002
Google Scholar
[32] Eom G S, Kim J, Choe W 2006 Phys. Plasmas 13 073505
Google Scholar
[33] Cho S 2020 Plasma Sources Sci. Technol. 29 095023
Google Scholar
[34] 赵高, 熊玉卿, 马超, 刘忠伟, 陈强 2014 物理学报 63 235202
Google Scholar
Zhao G, Xiong Y Q, Ma C, Liu Z W, Chen Q 2014 Acta Phys. Sin. 63 235202
Google Scholar
[35] 平兰兰, 张新军, 杨桦, 徐国盛, 苌磊, 吴东升, 吕虹, 郑长勇, 彭金花, 金海红, 何超, 甘桂华 2019 物理学报 68 205201
Google Scholar
Ping L L, Zhang X J, Yang H, Xu G S, Chang L, Wu D S, Lü H, Zheng C Y, Peng J H, Jin H H, He C, Gan G H 2019 Acta Phys. Sin. 68 205201
Google Scholar
[36] Guo X M, Scharer J, Mouzouris Y, et al. 1999 Phys. Plasmas 6 3400
Google Scholar
[37] Correyero Plaza S, Navarro J, Ahedo E 2016 52nd AIAA/SAE/ASEE Joint Propulsion Conference Salt Lake City, July 25–27, 2016 p5035
[38] Swanson D G 1989 Plasma Waves (New York: Academic Press) p155
[39] Huba J D 2016 NRL Plasma Formulary (Washington: Naval Research Laboratory) p34
[40] Mouzouris Y, Scharer J E 1998 Phys. Plasmas 5 4253
Google Scholar
[41] Fried B D, Conte S D 2015 The Plasma Dispersion Function: The Hilbert Transform of the Gaussian (New York: Academic Press) p1
[42] Sakawa Y, Kunimatsu H, Kikuchi H, et al. 2003 Phys. Rev. Lett. 90 105001
Google Scholar
[43] Shamrai K P, Taranov V B 1996 Plasma Sources Sci. Technol. 5 474
Google Scholar
[44] Chen F F, Arnush D 1997 Phys. Plasmas 4 3411
Google Scholar
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