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In this paper, the effects of non-Maxwellian distribution of electrons on the characteristics of magnetized plasma sheath with secondary electron emission are investigated by using a magnetic fluid model of one-dimensional velocity and three-dimensional space. The velocity of electrons follows the non-extensive distribution, and the ions are magnetized in a magnetic field with a certain tilt angle relative to the wall. The effects of the non-extensive electron distribution parameter q and the magnetic field strength and angle on the Bohm criterion, the floating wall potential, the secondary electron number density at the sheath edge, the sheath thickness and the ion velocity are studied by establishing the self-consistent equations. When the electron velocity distribution deviates from the Maxwellian distribution, the results show that as the q-parameter increases, the value of the Bohm criterion decreases, the floating wall potential increases, the number of secondary electrons at the sheath increases, the sheath thickness decreases, the number density of ions and electrons decline faster, the number density of ions near the wall is higher, and the velocities of the ions in the three directions are all reduced. In addition, as the magnetic field strength increases, the sheath thickness decreases, and the number density of ions and electrons in the sheath area decrease rapidly; the larger the magnetic field angle, the more significant the influences of the parameter q on the wall potential and the sheath thickness are, while the velocity component of the ion in the x-direction decreases with the increase of the magnetic field angle, but in the case of super-extensive distribution (q < 1), the velocity change near the wall presents an opposite trend, the increase of magnetic field angle causes wall velocity to increase; when it is close to Maxwellian distribution (q → 1), the velocity near the wall does not depend on the change of the magnetic field angle and basically tends to be identical; in the case of sub-extensive distribution (q > 1), the velocity near the wall decreases with the magnetic field angle increasing.
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Keywords:
- non-extensive distribution /
- secondary electron emission /
- magnetized sheath /
- Bohm criterion
[1] Franklin R N 2000 J. Phys. D: Appl. Phys. 33 3186
Google Scholar
[2] Tskhakaya D D, Shukla P K, Eliasson B, Kuhn S 2005 Phys. Plasmas 12 103503
Google Scholar
[3] Shibata K, Ito H, Yugami N, Miyazaki T, Nishida Y 2001 Thin Solid Films 386 291
Google Scholar
[4] Hatami M M 2015 Phys. Plasmas 22 023506
Google Scholar
[5] Aanesland A, Rafalskyi D, Bredin J, Grondein P, Oudini N, Chabert P, Levko D, Garrigues L, Hagelaar G 2015 IEEE Trans. Plasma Sci. 43 321
Google Scholar
[6] Adhikari S, Moulick R, Goswami K S 2017 Phys. Plasmas 224 083501
Google Scholar
[7] Liu J Y, Zhang Q, Zou X, Wang Z X, Liu Y, Wang X G, Gong Y 2004 Vacuum 73 687
Google Scholar
[8] Driouch I, Chatei H, Bojaddaini M E 2015 Phys. Plasmas 81 905810104
Google Scholar
[9] Benlemdjaldi D, Tahraoui A, Hugon R, Bougdira J 2013 Phys. Plasmas 20 043508
Google Scholar
[10] 王正汹, 刘金远, 邹秀, 刘悦, 王晓刚 2004 物理学报 53 793
Google Scholar
Wang Z X, Liu J Y, Zou X, Liu Y, Wang X G 2004 Acta Phys. Sin. 53 793
Google Scholar
[11] Saslaw W C, Arp H 1986 Phys. Today 39 61
Google Scholar
[12] Cáceres M O 1999 Braz. J. Phys. 29 125
Google Scholar
[13] Tsallis C, Mendes R, Plastino A R 1998 Physica A 261 534
Google Scholar
[14] Hatami M M, Tribeche M 2018 IEEE Trans. Plasma Sci. 46 868
Google Scholar
[15] Meige A, Boswell R W 2006 Phys. Plasmas 13 92104
Google Scholar
[16] Singh H, Graves D B 2000 J. Appl. Phys. 88 3889
Google Scholar
[17] Tsallis C 1988 J. Stat. Phys. 52 479
Google Scholar
[18] Borgohain D R, Saharia K, Goswami K S 2016 Phys. Plasmas 23 122113
Google Scholar
[19] Driouch I, Chatei H 2017 Eur. Phys. J. D. 71 1
Google Scholar
[20] Hatami M M, Tribeche M, Mamun A A 2018 Phys. Plasmas 25 094502
Google Scholar
[21] Hatami M M, Tribeche M, Mamun A A 2019 Astrophysics Space Sci. 364 1
Google Scholar
[22] Zhao X Y, Zhang B K, Wang C X 2020 Phys. Plasmas 27 113705
Google Scholar
[23] Safa N N, Ghomi H, Niknam A R 2014 Phys. Plasmas 21 082111
Google Scholar
[24] Basnet S, Khanal R 2019 Phys. Plasmas 26 043516
Google Scholar
[25] Zou X, Liu H P, Zhu Y Z, Zang X N, Qiu M H 2020 Plasmas Sci. Technol. 22 125001
Google Scholar
[26] Dhawan R, Kumar M, Malik H K 2020 Phys. Plasmas 27 063515
Google Scholar
[27] Basnet S, Patel A, Khanal R 2020 Plasma Phys. Control. Fusion 62 115011
Google Scholar
[28] 赵晓云, 张丙开, 王春晓, 唐义甲 2019 物理学报 68 185204
Google Scholar
Zhao X Y, Zhang B K, Wang C X, Tang Y J 2019 Acta Phys. Sin. 68 185204
Google Scholar
[29] Ghani O E, Driouch I, Chatei H 2020 Phys. Plasmas 27 083701
Google Scholar
[30] 邹秀, 刘惠平, 张小楠, 邱明辉 2021 物理学报 70 015201
Google Scholar
Zou X, Liu H P, Zhang X N, Qiu M H 2021 Acta Phys. Sin. 70 015201
Google Scholar
[31] Dhawan R, Malik H K 2021 Plasmas Sci. Technol. 23 045402
Google Scholar
[32] Bezerra J R, Silva R, Lima J A S 2003 Physica A 322 256
Google Scholar
[33] Safa N N, Ghomi H, Niknam A R 2015 Phys. Plasmas 81 905810303
Google Scholar
[34] Borgohain D R, Saharia K 2018 Phys. Plasmas 25 032122
Google Scholar
[35] Tribeche M, Djebarni L, Amour R 2010 Phys. Plasmas 17 042114
Google Scholar
[36] Hatami M M 2015 Phys. Plasmas 22 013508
Google Scholar
[37] Chen F F 1974 Introduction to Plasma Physics (New York: Springer Science)
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图 1 等离子体磁鞘示意图
Figure 1. Schematic diagram of the magnetized plasma sheath.
图 2 不同非广延参数q下的电子速度分布及其对电子数密度的影响 (a)电子速度非广延分布; (b)电子数密度分布
Figure 2. Electron velocity distribution and its influence on electron number density under different non-extensive parameter q: (a) Normalized velocity non-extensive distribution of electron; (b) electron number density distribution.
图 3 玻姆判据随非广延参数q的变化 (a) q < 1; (b) q > 1
Figure 3. Bohm criterion versus non-extensive parameter q: (a) q < 1; (b) q > 1.
图 4 壁面电势随非广延参数q的变化 (a)不同
$ \gamma, \theta $ 对$ {\varPhi }_{\mathrm{w}} $ 的影响; (b)不同等离子体种类对$ {\varPhi }_{\mathrm{w}}$ 的影响Figure 4. Wall potential versus non-extensive parameter q: (a) The
$ {\varPhi }_{\mathrm{w}}$ for different values of$ \gamma, \theta $ ; (b) the$ {\varPhi }_{\mathrm{w}}$ for different kinds of plasma.
图 5 鞘边二次电子数密度随非广延参数q的变化 (a)不同
$ \gamma, \theta $ 下鞘边二次电子数密度分布; (b)不同等离子体种类下鞘边二次电子数密度分布Figure 5. Normalized density of secondary electrons at the sheath edge versus non-extensive parameter q: (a) Density of secondary electrons at the sheath edge for different values of
$ \gamma \; \mathrm{a}\mathrm{n}\mathrm{d} \; \theta $ ; (b) density of secondary electrons at the sheath edge for different kinds of plasma.
图 6 不同非广延参数q值下鞘层电势分布(
$B=0.04 \; \mathrm{T}, $ $ \theta ={20}\text{°},\; \gamma =0.5$ )Figure 6. Sheath potential distribution with different values of q (
$B=0.04 \; \mathrm{T},\; \theta ={20}\text{°},\; \gamma =0.5$ ).
图 7 非广延参数q值及磁场角度对鞘层厚度的影响(B = 0.04 T,
$\gamma =0.5$ )Figure 7. Influence of non-extensive parameter q value and magnetic field angle on sheath thickness (
$B=0.04 \; \mathrm{T}, $ $ \gamma =0.5$ ).
图 8 非广延参数q值对鞘层粒子数密度分布的影响 (
$B=0.04 \; \mathrm{T}, \;\theta ={20}\text{°}, \;\gamma =0.5$ ) (a)离子、电子数密度分布; (b)二次电子数密度分布Figure 8. Influence of the non-extensive parameter q on the particle number density distribution of the sheath (B = 0.04 T,
$ \theta ={20}\text{°},\; \gamma =0.5$ ): (a) Normalized density of ions and electrons distribution; (b) normalized density of secondary electrons distribution.
图 9 磁场对鞘层离子、电子数密度分布的影响(
$ q=0.8, \gamma =0.5 $ ) (a) 磁场大小的影响; (b)磁场角度的影响Figure 9. Influence of magnetic field on the number density distribution of sheath ions and electrons (
$ q=0.8, \gamma =0.5 $ ): (a) Variation of normalized particle density for different values of magnetic field strength; (b) variation of normalized particle density for different values of magnetic field angle.
图 10 (a)−(c)非广延参数q值和(d)−(f)磁场角度对离子速度的影响(
$ B=0.04 \; \mathrm{T}, \gamma =0.5 $ ) (a) x方向速度($\theta ={20}\text{°}$ ); (b) y方向速度($\theta ={20}\text{°}$ ); (c) z方向速度($\theta ={20}\text{°}$ ); (d) x方向速度, q = 0.5; (e) x方向速度, q = 1; (f) x方向速度, q = 1.5Figure 10. Influence of (a)−(c) non-extensive parameter q value and (d)−(f) magnetic field angle on ion velocity (
$ B=0.04 \; \mathrm{T}, \gamma =0.5 $ ): (a) x-component ($\theta ={20}\text{°}$ ); (b) y-component ($\theta ={20}\text{°}$ ); (c) z-component ($\theta ={20}\text{°}$ ); (d) x-component, q = 0.5; (e) x-component, q = 1; (f) x-component, q = 1.5. -
[1] Franklin R N 2000 J. Phys. D: Appl. Phys. 33 3186
Google Scholar
[2] Tskhakaya D D, Shukla P K, Eliasson B, Kuhn S 2005 Phys. Plasmas 12 103503
Google Scholar
[3] Shibata K, Ito H, Yugami N, Miyazaki T, Nishida Y 2001 Thin Solid Films 386 291
Google Scholar
[4] Hatami M M 2015 Phys. Plasmas 22 023506
Google Scholar
[5] Aanesland A, Rafalskyi D, Bredin J, Grondein P, Oudini N, Chabert P, Levko D, Garrigues L, Hagelaar G 2015 IEEE Trans. Plasma Sci. 43 321
Google Scholar
[6] Adhikari S, Moulick R, Goswami K S 2017 Phys. Plasmas 224 083501
Google Scholar
[7] Liu J Y, Zhang Q, Zou X, Wang Z X, Liu Y, Wang X G, Gong Y 2004 Vacuum 73 687
Google Scholar
[8] Driouch I, Chatei H, Bojaddaini M E 2015 Phys. Plasmas 81 905810104
Google Scholar
[9] Benlemdjaldi D, Tahraoui A, Hugon R, Bougdira J 2013 Phys. Plasmas 20 043508
Google Scholar
[10] 王正汹, 刘金远, 邹秀, 刘悦, 王晓刚 2004 物理学报 53 793
Google Scholar
Wang Z X, Liu J Y, Zou X, Liu Y, Wang X G 2004 Acta Phys. Sin. 53 793
Google Scholar
[11] Saslaw W C, Arp H 1986 Phys. Today 39 61
Google Scholar
[12] Cáceres M O 1999 Braz. J. Phys. 29 125
Google Scholar
[13] Tsallis C, Mendes R, Plastino A R 1998 Physica A 261 534
Google Scholar
[14] Hatami M M, Tribeche M 2018 IEEE Trans. Plasma Sci. 46 868
Google Scholar
[15] Meige A, Boswell R W 2006 Phys. Plasmas 13 92104
Google Scholar
[16] Singh H, Graves D B 2000 J. Appl. Phys. 88 3889
Google Scholar
[17] Tsallis C 1988 J. Stat. Phys. 52 479
Google Scholar
[18] Borgohain D R, Saharia K, Goswami K S 2016 Phys. Plasmas 23 122113
Google Scholar
[19] Driouch I, Chatei H 2017 Eur. Phys. J. D. 71 1
Google Scholar
[20] Hatami M M, Tribeche M, Mamun A A 2018 Phys. Plasmas 25 094502
Google Scholar
[21] Hatami M M, Tribeche M, Mamun A A 2019 Astrophysics Space Sci. 364 1
Google Scholar
[22] Zhao X Y, Zhang B K, Wang C X 2020 Phys. Plasmas 27 113705
Google Scholar
[23] Safa N N, Ghomi H, Niknam A R 2014 Phys. Plasmas 21 082111
Google Scholar
[24] Basnet S, Khanal R 2019 Phys. Plasmas 26 043516
Google Scholar
[25] Zou X, Liu H P, Zhu Y Z, Zang X N, Qiu M H 2020 Plasmas Sci. Technol. 22 125001
Google Scholar
[26] Dhawan R, Kumar M, Malik H K 2020 Phys. Plasmas 27 063515
Google Scholar
[27] Basnet S, Patel A, Khanal R 2020 Plasma Phys. Control. Fusion 62 115011
Google Scholar
[28] 赵晓云, 张丙开, 王春晓, 唐义甲 2019 物理学报 68 185204
Google Scholar
Zhao X Y, Zhang B K, Wang C X, Tang Y J 2019 Acta Phys. Sin. 68 185204
Google Scholar
[29] Ghani O E, Driouch I, Chatei H 2020 Phys. Plasmas 27 083701
Google Scholar
[30] 邹秀, 刘惠平, 张小楠, 邱明辉 2021 物理学报 70 015201
Google Scholar
Zou X, Liu H P, Zhang X N, Qiu M H 2021 Acta Phys. Sin. 70 015201
Google Scholar
[31] Dhawan R, Malik H K 2021 Plasmas Sci. Technol. 23 045402
Google Scholar
[32] Bezerra J R, Silva R, Lima J A S 2003 Physica A 322 256
Google Scholar
[33] Safa N N, Ghomi H, Niknam A R 2015 Phys. Plasmas 81 905810303
Google Scholar
[34] Borgohain D R, Saharia K 2018 Phys. Plasmas 25 032122
Google Scholar
[35] Tribeche M, Djebarni L, Amour R 2010 Phys. Plasmas 17 042114
Google Scholar
[36] Hatami M M 2015 Phys. Plasmas 22 013508
Google Scholar
[37] Chen F F 1974 Introduction to Plasma Physics (New York: Springer Science)
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