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在霍尔推力器中, 电子漂移、电子碰撞, 以及等离子体密度、温度、磁场梯度所蕴含的自由能会驱动各种频率和波长的不稳定性. 不稳定性的存在会破坏等离子体的稳定放电, 削弱推力器与电源处理单元的匹配度, 降低推力器的性能. 基于此, 本文利用基于流体模型推导的色散关系研究了霍尔推力器中由电子碰撞、等离子体密度和磁场梯度驱动的不稳定性. 结果表明: 1)在考虑电子惯性、电子与中性原子的碰撞, 以及电子${\boldsymbol{E}} \times {\boldsymbol{B}}$漂移时能够在推力器近阳极区到羽流区内的任一轴向位置处激发不稳定性. 随着角向波数${k_y}$的增大($k = 2{\text{π /}}\lambda $, $\lambda $为波长), 模式将从由碰撞激发的低杂波不稳定性转变为离子声波不稳定性. 当${k_y} = 10 {{\text{ m}}^{ - 1}}$时, 最大增长率${\gamma _{\max }}$对应频率${\omega _{\text{r}}}$随着碰撞频率${\nu _{{\text{en}}}}$的增大而轻微减小; 当$ {k_y} = 300 {{\text{ m}}^{ - 1}} $时, ${\gamma _{\max }}$对应的频率${\omega _{\text{r}}}$以及最大频率${\omega _{{\text{rmax}}}}$几乎不随碰撞频率变化. 不依赖于$ {k_y} $的大小, 对于碰撞激发的不稳定性, 模式的增长率随着碰撞频率的增大而增大. 2) 同时考虑电子惯性、电子碰撞效应, 以及密度梯度时, 密度梯度对驱动不稳定性占主导作用. 模式的动力学行为不会随${k_y}$的增大而变化, 但模式的本征值随${k_y}$的增大而增大. 在密度梯度${\kappa _{\text{N}}} = 0$的两侧, 由于密度梯度引起的抗磁性漂移频率${\omega _{\text{s}}}$的符号发生了变化, 模式的本征值在${\kappa _{\text{N}}} = 0$两侧有相反的变化趋势: 当${\omega _*}$与${\omega _{\text{r}}}$符号相反时, 密度梯度对不稳定性的激发有削弱作用(${\kappa _{\text{N}}} \gt 0$); 当${\omega _*}$与${\omega _{\text{r}}}$符号相同时, 密度梯度对不稳定性的激发有增强作用(${\kappa _{\text{N}}} \lt 0$). 3)在模型中同时考虑等离子体密度梯度、磁场梯度, 以及电子惯性和碰撞效应时, 模式本征值的变化依赖于电子的漂移频率, 以及密度和磁场梯度引起的抗磁性漂移频率的相对大小. 当仅包含密度梯度和磁场梯度时, 推力器放电通道内将出现稳定窗, 即增长率为0的区间; 包含电子惯性和碰撞效应后, 稳定窗消失.The free energy contained in electron drift, electron collision, and plasma density gradient, temperature, magnetic field gradient can trigger off the instabilities with different frequencies and wavelengths in hall thrusters. The instabilities will destroy the stable discharge of plasma, affecting the matching degree between the thruster and the power processing unit, and reducing the performance of the thruster. Based on this, the instabilities triggered off by electron collision, plasma density gradient, and magnetic field gradient in the hall thruster are studied by using dispersion relation derived from the fluid model. The results are shown below. 1) When in the model includes the effects of electron inertia, collision between electrons and neutral atoms, and electron drift, instability can be excited at any axial position from the near anode region to the plume region of the thruster. With the increase of azimuthal wavenumber ${k_y} = 2\pi /\lambda $, the lower-hybrid mode excited by electron collision transitions into the ion sound mode, where ${k_y} = 2{\text{π }}/\lambda $, $\lambda $being the wave length. The real frequency ${\omega _{\text{r}}}$ corresponding to the maximum growth rate ${\gamma _{\max }}$ slightly decreases with collision frequency increasing for ${k_y} = 10{\text{ }}{{\text{ m}}^{ - 1}}$. However, the maximum real frequency and real frequency ${\omega _{\text{r}}}$ corresponding to the maximum growth rate ${k_y} = 300{{\text{ m}}^{ - 1}}$ will not change with collision frequency for ${k_y} = 300{\text{ }}{{\text{ m}}^{ - 1}}$. Independent of the value of ${k_y}$, the growth rate of mode triggered off by electron collision increases with collision frequency increasing. 2) The plasma density gradient effect plays a dominant role in triggering off instabilities when the electron inertia, electron-neutral collisions and plasma density gradient are simultaneously included in the model. The dynamic behavior of the model does not change with the increase of ${k_y}$, but the eigenvalue of the model increases with the ${k_y}$ increasing. Since the sign of anti-drift frequency induced by the plasma density gradient is changed, the mode eigenvalues have the opposite change trend on both sides of point ${\kappa _{\text{N}}}$. When the sign of ${\omega _r}$ and ${\omega _r}$ are opposite, the density gradient effect has a stabilization effect on instability excitation (${\kappa _{\text{N}}} > 0$). When the sign of ${\omega _{\text{s}}}$ and ${\omega _{\text{r}}}$ are the same, the density gradient effect enhances the excitation of instability (${\kappa _{\text{N}}} < 0$). 3) If the plasma density gradient, magnetic field gradient, electron inertia and electron-neutral collisions are included in the dispersion, the mode eigenvalue relies on the electron drift frequency, and the diamagnetic drift frequency induced by the density gradient and magnetic field gradient. When the density gradient effect and the magnetic field gradient effect are considered, there is a stable window in the discharge channel. However, if the electron inertia and electron-neutral collisions are also included, the stable window will disappear.
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Keywords:
- Hall thruster /
- density gradient /
- magnetic gradient /
- electron collision /
- instability
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图 1 (a)归一化的电势$\phi $、等离子体密度$n$、 电场强度$E$、电子温度${T_{\text{e}}}$、磁场$B$、离子速度${\upsilon _{\text{i}}}$、${\boldsymbol{E}} \times {\boldsymbol{B}}$电子漂移速度, 密度梯度和磁场梯度引起的抗磁性漂移速度${\upsilon _*}$和${\upsilon _{\text{D}}}$, 图中灰色虚线表示推力器出口的位置; (b)密度梯度和磁场梯度与轴向位置的依赖关系
Fig. 1. (a) The normalized potential $\phi $, plasma density $n$, electric field $ E $, electron temperature ${T_{\text{e}}}$, magnetic field $B$, ion velocity ${\upsilon _{\text{i}}}$, ${\boldsymbol{E}} \times {\boldsymbol{B}}$drift velocity, electron diamagnetic drift velocity due to density gradient and magnetic gradient ${\upsilon _*}$ and ${\upsilon _{\text{D}}}$, respectively, the grey dashed line in the indicates the exit plane; (b) the density and magnetic gradient on axial position.
图 2 包含碰撞效应时, 不稳定性的频率(a)和增长率(b)与轴向位置的依赖关系
Fig. 2. Including collision effects, the dependence of the frequency (a) and growth rate (b) of the instability on the axial position.
图 3 模式频率和增长率在相空间中的变化
Fig. 3. The variation of mode frequency and growth rate in phase space.
图 4 ${k_y} = 10$(a)和${k_y} = 300$(b)时, 不同碰撞频率下模式频率和增长率在相空间中的变化
Fig. 4. The variation of mode frequency and growth rate in the phase space for different collision frequences: (a) ${k_y} = 10$; (b) ${k_y} = 300$.
图 5 包含密度梯度和碰撞效应时, 不稳定性的实频(a)和增长率(b)与轴向位置的依赖关系
Fig. 5. Dependence of frequency (a) and growth rate (b) of instability on the axial position, including the gradient of plasma density and electron collision effects.
图 6 包含密度和磁场梯度以及碰撞效应时, 不稳定性的实频(a)和增长率(b)与轴向位置的依赖关系
Fig. 6. Dependence of frequency (a) and growth rate (b) of instability on the axial position, including the gradient of plasma density and magnetic field as well as electron collision effects.
表 1 计算中的输入参数
Table 1. Input parameters for calculation.
参数 数值 参数 数值 ${\varPhi _{\text{m}}}$/V 270 ${l_1}$ $0.62 d$ ${B_{\text{m}}}$/T 0.018 ${l_2}$ $0.84 d$ ${n_{\text{m}}}$/m–3 5.0×1017 ${l_3}$ $0.92 d$ ${\vartheta _1}$ 2.5 ${\alpha _2}$ 1.5 ${\vartheta _2}$ 4 $T_{\text{e}}^{{\text{max}}}$/eV 24 ${\vartheta _3}$ 7.5 $T_{\text{e}}^{{\text{min}}}$/eV 3 ${\vartheta _4}$ 8 
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[1] Koo J W, Boyd I D 2006 Phys. Plasmas 13 033501
Google Scholar
[2] Lazurenko A, Coduti G, Mazouffre S, Bonhomme G 2008 Phys. Plasmas 15 034502
Google Scholar
[3] Appleton B R, Moak C D, Noggle T S, Barrett J H 1972 Phys. Rev. Lett. 28 1307
Google Scholar
[4] Anders A, Ni P, Rauch A 2012 J. Appl. Phys. 111 053304
Google Scholar
[5] Brenning N, Lundin D, Minea T, Costin C, Vitelaru C 2013 J. Phys. D: Appl. Phys. 46 084005
Google Scholar
[6] Smolyakov A I, Chapurin O, Frias W, Kosakarov O, Romadanov I, Tang T, Umansky M, Raitses Y, Kaganovich I D, Lakhin V P 2017 Plasma Phys. Control. Fusion 59 014041
Google Scholar
[7] Boeuf J P, Takahashi M 2020 Phys. Rev. Lett. 18 124
[8] Boeuf J P, Garrigues L 2018 Phys. Plasmas 25 061204
Google Scholar
[9] Morozov K N, Esipchuk Y V, Kapulkin A, Nevrovskii V, Smirnov V A 1972 Sov. Phys. Tech. Phys. 17 482
[10] Esipchuk Y V, Tilinin G N 1976 Sov. Phys. Tech. Phys. 21 417
[11] Gorshkov O A, Tomilin D A, Shagaida A A 2012 Plasma Phys. Rep. 38 271
Google Scholar
[12] Tomilin D 2013 Phys. Plasmas 20 042103
Google Scholar
[13] Romadanov I, Smolyakov A, Raitses Y, Kaganovich I D, Tang T, Ryzhkov S 2016 Phys. Plasmas 23 122111
Google Scholar
[14] Lakhin V P, Ilgisonis V I, Smolyakov A I, Sorokina E A, Marusov N A 2018 Phys. Plasmas 25 012106
Google Scholar
[15] Marusov N A, Sorokina E A, Lakhin V P, Ilgisonis A I, Smolyakov A I 2019 Plasma Sources Sci. Technol. 28 015002
Google Scholar
[16] Boeuf J P 2017 J. Plasma Phys. 121 011101
[17] Ducrocq A, Adam J C, Héron A, Laval G 2006 Phys. Plasmas 13 102111
Google Scholar
[18] Lafleur T, Baalrud S D, Chabert P 2016 Phys. Plasmas 23 053502
Google Scholar
[19] Tavant A, Croes V, Lucken R, Lafleur T, Bourdon A, Chabert P 2018 Plasma Sources Sci. Technol. 27 124001
Google Scholar
[20] Taccogna F, Minelli P, Asadi Z, Bogopolsky G 2019 Plasma Sources Sci. Technol. 28 064002
Google Scholar
[21] Mandal D, Elskens Y, Lemoine N, Doveil F 2020 Phys. Plasmas 27 032301
Google Scholar
[22] Chen L, Kan Z C, Gao W F, Duan P, Chen J Y, Tan C Q, Cui Z J 2024 Chin. Phys. B 33 015203
Google Scholar
[23] Artsimovich L A, Andronov I M, Esipchuk Y V, Bersukov I A, Kozubskii K N 1974 Kosm. Issled. 12 451
[24] Frias W, A I, Smolyakov, Kaganovich I D, Raitses Y 2012 Phys. Plasmas 19 072112
Google Scholar
[25] Koshkarov O 2018 Ph. D. Dissertation (Saskatoon: Saskatchewan University
[26] Kronhaus I, Kapulkin A, Balabanov V, Rubanovich M, Guelman M, Natan B 2012 J. Phys. D: Appl. Phys. 45 175023
[27] Litvak A A, Fisch N J 2001 Phys. Plasmas 8 648
Google Scholar
[28] Boeuf J P, Smolyakov A 2023 Phys. Plasmas 30 050901
Google Scholar
[29] Litvak A A, Fisch N J 2000 PPPL Reports posted on the U. S. Department of Energy’s Princeton Plasma Physics Laboratory Publications and Reports web site in Calendar Year 2000 The home page for PPPL Reports and Publications is: http://www.pppl.gov/pub_report/ PPPL-3521
[30] Boeuf J P 2014 Front. Phys. 2 74
[31] Lampe M, Manheimer W M, McBride J B, Orens J H, Shanny R, Sudan R N 1971 Phys. Rev. Lett. 26 1221
Google Scholar
[32] Lampe M, Manheimer W M, McBride J B, Orens J H, Papadopoulos K, Shanny R, Sudan R N, 1972 Phys. Fluids 15 662
Google Scholar
[33] McBride J B, Ott E, Boris J P, Orens J H 1972 Phys. Fluids 15 2367
Google Scholar
[34] Taccogna F, Garrigues 2019 Rev. Mod. Plasma Phys. 3 12
Google Scholar
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