APPLIED PHYSICS LETTERS 96, 111502 共2010兲
Suppression of high-power microwave dielectric multipactor by resonant magnetic field C. Chang,1,a兲 G. Z. Liu,2 C. X. Tang,1 C. H. Chen,2 H. Shao,2 and W. H. Huang2 1
Department of Engineering Physics, Tsinghua University, Beijing 100084, People’s Republic of China Northwest Institute of Nuclear Technology, Xi’an, Shannxi 710024, People’s Republic of China
2
共Received 20 January 2010; accepted 23 February 2010; published online 16 March 2010兲 Through dynamic calculation and electromagnetic particle-in-cell simulation, high-power microwave dielectric multipactor is discovered to be suppressed by utilizing external dc magnetic field parallel to the surface, perpendicular to the rf field and satisfying the gyrofrequency close to the rf frequency ⍀ ⬃ . It is found that multipactor electrons emitted from the surface can be resonantly accelerated to obtain the impact energy e higher than the second crossover energy, leading to secondary emission yield lower than one. Besides, the corresponding flight time gets close to the rf period, also the period of the vector Erf ⫻ B, resulting in secondary electrons immediately pulled away without multipactoring along the surface. What is more, with the rf field increasing, suppression effect can be further enhanced due to e rising. © 2010 American Institute of Physics. 关doi:10.1063/1.3360853兴 Mechanisms of high-power microwave 共HPM兲 dielectric breakdown have been profoundly researched in Refs. 1–4. By applying external magnetic field, the dc, and pulsed vacuum flashover voltage can be enhanced when the vector E ⫻ B points away from the surface, and can be decreased when E ⫻ B into the surface.5,6 For HPM, the Erf ⫻ B drift of the external magnetic field periodically changes its direction compared with the directional Erf ⫻ Brf of the rf magnetic field.7,8 Magnetic insulation techniques for vacuum surface flashover are expected to be applicable to microwave frequencies as well, as long as the electron “hopping frequency” in a surface avalanche is small compared to the rf frequency.5,6 When B is oriented parallel to the surface and perpendicular to Erf, the strong magnetic force eu ⫻ B can significantly restrict the flight distance of the electrons and reduce their energy e to be lower than the first crossover energy 1 if and only if intense magnetic field and gyrofrequency ⍀ = eB/ m ⬎ 10 共e.g., B ⬎ 1 – 3 T for 2.85–9.3 GHz兲.9 However, with the rf field increasing, e can gradually rise to be higher than 1. Consequently, strong magnetic field is not appropriate to suppress HPM multipactor. In this paper, we propose to utilize the low-amplitude magnetic field and the resonant condition ⍀ ⬃ 共e.g., B ⬃ 0.1 T for 2.85 GHz兲 to accelerate the multipactor electrons with an impact energy higher than the second crossover energy 2 and to realize the flight time close to the rf period to eliminate multipactor. Assuming the rf field Erf = E0 sin共t + 兲ey, the normal positive field −Edcex, and the external magnetic field B = Bez, the schematic of an electron motion under vacuum is illustrated in Fig. 1, and its differential force equations are m
dux共t兲 = − euy共t兲B + eEdc , dt
共1兲
m
duy共t兲 = − eE0 sin共t + 兲 + eux共t兲B, dt
共2兲
m
共3兲
Focusing on strong magnetic force of high-energy electron, Edc which is weak at the initiation stage of multipactor can be neglected. The directional rf magnetic force Erf ⫻ Brf is toward the vacuum dielectric window in the upstream,7,8 chiefly playing a role of restoring force. The ratio of rf and dc magnetic force along the normal-x direction is approximately equal to Brf / Bdc ⬃ eE0 / 共mc兲 for ⍀ ⬃ . For E0 = 40 kV/ cm and f = 2.85 GHz, there is Brf / Bdc ⬃ 0.1. For different E0 and , if eE0 / 共m兲 Ⰶ c, Brf can be neglected compared with the external resonant Bdc. By eliminating ux共t兲 from Eqs. 共1兲 and 共2兲, the second-order differential equation of uy共t兲 yields ⍀E0 d2uy共t兲 cos共t + 兲. + ⍀2uy共t兲 = − dt2 B
共4兲
Since the initial electron energy e共0兲共⬃2 – 10 eV兲 is much smaller than the concerned energy 2共⬃5 – 10 keV兲,10 e共0兲 can be neglected, and uy共t兲 and ux共t兲 are written as
冤
冥
cos共t + 兲 E 0⍀ − 共⍀ sin2 + cos2 兲cos共⍀t + 兲 , uy共t兲 = B共2 − ⍀2兲 + 共⍀ − 兲sin cos sin共⍀t + 兲
a兲
Electronic mail:
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0003-6951/2010/96共11兲/111502/3/$30.00
duz共t兲 = 0. dt
共5兲
FIG. 1. Schematic of an electron motion under vacuum. 96, 111502-1
© 2010 American Institute of Physics
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FIG. 2. 共a兲 Analytical multipactor electron trajectories, 共b兲 analytical and simulative energy e共t兲 for ⍀ / = 1, E0 = 40 kV/ cm; dashed line, analytical = 0°; dashed-dotted line, = 90°; solid line, PIC simulation, = −180°.
冤
冥
− ⍀ sin共t + 兲 E 0⍀ + 共⍀ sin2 + cos2 兲sin共⍀t + 兲 . 共6兲 ux共t兲 = B共2 − ⍀2兲 + 共⍀ − 兲sin cos cos共⍀t + 兲 Setting uz共t兲 ⬅ 0 and using Eqs. 共5兲 and 共6兲, the energy e共t兲 = m关u2x 共t兲 + u2y 共t兲 + uz2共t兲兴 / 2 under the emission phase = 0° can be expressed as e2E20
e共t兲 =
冉
2m2 1 −
⫻
冢
⍀2 2
冊
2
1 + cos2共t兲 +
冉冊 ⍀
2
sin2共t兲
⍀ − 2 cos共t兲cos共⍀t兲 − 2 sin共t兲sin共⍀t兲
The flight time satisfying
兰0 ux共t兲dt = 0
⍀ 关1 − cos共⍀兲兴 + 关cos共兲 − 1兴 = 0. ⍀
冣
.
共7兲
for = 0° reads 共8兲
It can be found in Eq. 共8兲 that is independent of E0. The equations of the energy e共t兲 and flight time for different phases are not shown in this paper. The secondary emission yield ␦ for the impact energy e共兲 meets: ␦ = 1.125␦m / 0.35.10 Where, ␦m = ␦m0共1 + ks2 / 兲, = e共兲 / m ⬎ 3.6, and m = m0共1 + ks2 / 兲, ␦m0 is the peak secondary emission yield for normal incidence, m0 is the electron energy at the peak, ks is the parameter describing the surface roughness, and the angle to the normal meets = arctan关uy共兲 / ux共兲兴. The electron trajectory can be obtained by integrating uy共t兲 and ux共t兲 in Eqs. 共5兲 and 共6兲 for the flight time , the energy e共t兲 can be calculated by squaring Eqs. 共5兲 and 共6兲 and then summing up, and the special and e共t兲 for = 0° can be acquired from Eqs. 共8兲 and 共7兲. The parameters used in analytical calculation are: E0 = 40− 50 kV/ cm, = 0° and 90°, = 2ⴱ2.85 GHz, ⍀ / 苸 共0.5, 2兲, ␦m0 = 2.15, m0 = 400 eV, and ks = 1. The analytical trajectories and the corresponding energy e共t兲 for ⍀ / = 1, E0 = 40 kV/ cm, and = 0° and 90° are illustrated in Figs. 2共a兲 and 2共b兲. Besides, the statistical energy from the two-dimensional electromagnetic particle-in-cell 共PIC兲 simulation, whose model structure can be shown in Refs. 11 and 12, under the same value of ⍀, , and E0 except for the
FIG. 3. Statistical trajectories of the multipactor electrons by PIC simulation.
initial Edc = 1 kV/ cm is compared in Fig. 2共b兲, and the corresponding statistical trajectories of electrons are shown in Fig. 3. The trajectory components Sy and Sx, e共t兲 and time t have been respectively normalized by eE0 / 共m2兲, e2E20 / 共2m2兲 and rf-period T. Multipactor electron emitted at = 0° has a longer displacement Sy along the rf field and obtains a higher impact energy e共兲 compared with that at = 90° when ⍀ ⬃ , because the electron energy mainly comes from the rf field and longer Sy means higher e共兲. It can be discovered in Fig. 2共b兲 that the low energy in simulation for t ⬍ 0 represents multipactoring along the dielectric surface within the half period 关 苸 共−180° , 0°兲兴 of the vector Erf ⫻ B pointing to the surface. The low-amplitude trajectories of electrons multipactoring along the surface before Erf ⫻ B reversing can be also observed in Fig. 3. After that 共 ⱖ 0°兲, all of the electrons are pulled away from the surface with much higher and longer flights, and then impact the surface with e共兲 ⬎ 2 and ␦ ⬍ 1. Since the period of the vector Erf ⫻ B is T and the flight time gets close to T for ⍀ ⬃ 关shown in Fig. 4共a兲兴, electrons undergo two reverses of the vector during flights, the impact phase is approximately ⬃ 0° implying Erf ⫻ B pointing away from the surface, and the produced secondary electrons are pulled away at once without multipactoring along the surface. Consequently, multipactor can be suppressed quickly. Taking the simulation under ⍀ / = 0.7 for example, the initial condition is electron density ne ⬃ 1ⴱ104 cm−3 and emission phase ⬃ −180°. Consequently, electrons multipactor along the surface and ne increases in the initial half period T / 2 of Erf ⫻ B pointing to the surface, shown as the rising part of the curve in Fig. 4共b兲. After T / 2, Erf reverses, corresponding to ⬎ 0°, all of the electrons are pulled away from the surface under Erf ⫻ B, and ne keeps unchanged during the
FIG. 4. 共a兲 Analytical flight time vs ⍀ / and . 共b兲 Simulative electron density ne vs t for ⍀ / = 0.7 and E0 = 40 kV/ cm.
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FIG. 5. 共a兲 The analytical impact energy e共兲 and 共b兲 the corresponding yield ␦ with ⍀ / , , and E0; circle, 40 kV/cm, 0°; rectangle, 40 kV/cm, 90°; triangle, 50 kV/cm, 0°.
flight time , which for ⍀ / = 0.7 and ⬃ 0° attains 1.2T 关shown in Fig. 4共a兲兴, and the corresponding impact phase is ⬃ 90°. As a result, the flight time for secondary electrons is ⬃ 0.85T 关shown in Fig. 4共a兲兴, and the renewed impact phase is ⬃ 0°. For each impact, there are e共兲 ⬎ 2 and ␦ ⬍ 1, and the produced secondary electrons are immediately pulled away without multipactoring along the surface. Thereby, ne consecutively diminishes as going down the steps shown in Fig. 4共b兲. It can be deduced in Figs. 5共a兲 and 5共b兲 that, for ⍀ / 苸 共0.6, 1.7兲 and 苸 共0 ° , 90°兲, e共兲 / 关e2E20 / 共2m2兲兴 keeps higher than 2 and the corresponding ␦ maintains lower than 1. With the field E0 increasing from 40 to 50 kV/cm, e共兲 is increased, and ␦ is reduced under the same ⍀ / and , re-
sulting in the suppression effect of multipactor enhancing and the effective range of ⍀ / broadening. It should be mentioned that the relativistic effects can be neglected as long as the resonantly accelerated energy e共兲 ⬃ e2E20 / 共2m2兲 Ⰶ mc2, where  is the energy enhancement factor. It should be emphasized that, for the air pressure, multipactor electrons won’t be resonantly accelerated because of the collision frequency of the electrons with gas molecules c Ⰷ . The average electron energy for ⍀ ⬃ is approximately deduced as e ⬃ e2E20 / 关m共2c + 2 + ⍀2兲兴. 1
J. Krile, A. Neuber, H. Krompholz, and T. Gibson, Appl. Phys. Lett. 89, 201501 共2006兲. 2 Y. Hidaka, E. Choi, I. Mastovsky, M. Shapiro, J. Sirigiri, and R. Temkin, Phys. Rev. Lett. 100, 035003 共2008兲. 3 Y. Lau, J. Verboncoeur, and H. Kim, Appl. Phys. Lett. 89, 261501 共2006兲. 4 S. Nam and J. Verboncoeur, Appl. Phys. Lett. 92, 231502 共2008兲. 5 R. Korzekwa, M. Lehr, H. Krompholz, and M. Kristiansen, IEEE Trans. Electron Devices 38, 745 共1991兲. 6 M. Lehr, R. Korzekwa, H. Krompholz, and M. Kristiansen, J. Appl. Phys. 71, 389 共1992兲. 7 D. Hemmert, A. Neuber, J. Dickens, H. Krompholz, L. Hatfield, and M. Kristiansen, IEEE Trans. Plasma Sci. 28, 472 共2000兲. 8 A. Neuber, D. Hemmert, H. Krompholz, L. Hatfield, and M. Kristiansen, J. Appl. Phys. 86, 1724 共1999兲. 9 A. Valfells, L. Ang, Y. Lau, and R. Gilgenbach, Phys. Plasmas 7, 750 共2000兲. 10 J. R. M. Vaughan, IEEE Trans. Electron Devices 36, 1963 共1989兲. 11 C. Chang, H. Huang, G. Liu, C. Chen, Q. Hou, J. Fang, and X. Zhu, J. Appl. Phys. 105, 123305 共2009兲. 12 C. Chang, G. Liu, H. Huang, C. Chen, and J. Fang, Phys. Plasmas 16, 083501 共2009兲.
