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Penetration of a radio frequency electromagnetic field into a magnetized plasma: one-dimensional PIC simulation studies
- Fumiko Otsuka^{1}Email author,
- Tohru Hada^{1},
- Shunjiro Shinohara^{2} and
- Takao Tanikawa^{3}
- Received: 3 October 2014
- Accepted: 7 May 2015
- Published: 6 June 2015
Abstract
Penetration of a radio frequency (rf) electromagnetic field into a magnetized plasma is discussed by performing one-dimensional particle-in-cell (PIC) simulations. We consider two models: an electrostatic (ES) model using an external rf voltage and an electromagnetic (EM) model using an external rf current. The background magnetic field is perpendicular to the one-dimensional system. In the ES model, the external rf voltage across the plasma produces the rf electric field into the plasma. When ω _{ rf }=Ω _{ e }/2, where ω _{ rf } and Ω _{ e } are the externally applied field and electron gyrofrequencies, respectively, the ion-rich sheath is formed due to the electron wall loss caused by the electron polarization drift. When ω _{ rf }=ω _{ LH }/2, where ω _{ LH } is the lower hybrid frequency, the electron-rich sheath is formed due to the ion wall loss caused by its larger gyroradius than the electron. In the EM model, an induced electromagnetic field via the external rf current forms a standing wave in the plasma region bounded by the external antennas. When the diameter of the plasma is equal to the wave length, the spatial/temporal profiles of the rf standing waves are well explained by a cold collisionless plasma linear theory except for the existence of the sheath. When the rf magnetic field is increased to 10 % of the background magnetic field, the waveforms become irregular and complex because of the boundary effects.
Keywords
- Radio frequency electromagnetic field
- Extrenal electrode
- Magnetized plasma
- Electrodeless plasma thruster
Background
Wave penetration into a plasma is an important problem from the point of view of a fundamental physics and from numerous applications in various fields such as plasma production (Shinohara and Shamrai 2002; Shinohara et al. 2009) plasma heating, and confinement (Shoji 1980). In this paper, we consider the penetration of both electrostatic and electromagnetic waves into a magnetized plasma with a specific application in mind: the next-generation plasma thrusters. In many of the conventional electric (plasma) thrusters, such as the ion engine and Hall thruster, thrust is obtained by accelerating the plasma using a static electric field (Goebel and Katz 2008; Raitses and Fisch 2001). Such an electric field, however, is produced by electrodes situated inside the plasma, and thus, because of their contact with the plasma, electrode erosion cannot be avoided. To overcome this difficulty, we propose ‘electrodeless’ thrusters that have electrodes located outside the plasma so that there is no direct contact between the plasma and the electrodes (Helicon Electrodeless Advanced Thruster (HEAT) project (Shinohara et al. 2013)). In the electrodeless thrusters, either electrostatic or electromagnetic waves are emitted from antennas outside the plasma region, so they penetrate into the plasma and accelerate it. A very long thruster life span can be expected because it is not limited by the electrode erosion.
Three electrodeless plasma thrusters, in which the radio frequency (rf) electric field penetration is essential, are the Lissajous Helicon Plasma Accelerator (LHPA) (Matsuoka et al. 2012; Nakamura et al. 2012; Nishida et al. 2012), the Variable Specific Impulse Magnetoplasma Rocket (VASIMR) (Ando et al. 2005, 2006; Bering III et al. 2010), and the Ponderomotive Acceleration/Ion Cyclotron Resonance (PA/ICR) (Otsuka et al. 2013a, 2013b). First, the LHPA uses a Lorentz force which is a product of an azimuthal electron current and a radial magnetic field. In the LHPA, two sets of planar antennas are used to electrostatically excite a rotating electric field (REF), which produces the azimuthal electron current (Matsuoka et al. 2012). Second, the VASIMR uses the ion cyclotron resonance (ICR) for the ion acceleration (Bering III et al. 2010). In the VASIMR-type experiment, a helically wound antenna is used to electromagnetically excite an ion cyclotron wave propagating along the magnetic field line (Ando et al. 2005, 2006). Third, the ponderomotive acceleration (PA) uses a sign flipping of a ponderomotive potential near ion gyroresonance, leading to a unidirectional acceleration of ions (Dodin et al. 2004). We examined the PA combined with the ICR using the test particle simulations (Otsuka et al. 2013a, 2013b). In our test particle simulations, the rf electromagnetic fields are assumed to be present inside the plasma. This may well be an optimistic simplification because the plasma in general acts to shield the external waves from penetrating inside. Further, in the various experiments on the thrusters, it is difficult to observe the efficiency of the wave penetration, although it is directly related to the thruster’s performance. In this paper, for applications to various types of electrodeless plasma thrusters in general, we investigate the rf electric field penetration by performing one-dimensional particle-in-cell (PIC) simulations.
We study two models with different configurations of the external electrodes. The first is the electrostatic (ES) model in which an rf voltage difference is applied to a pair of planar electrodes to generate an rf electric field between the electrodes. The second is the electromagnetic (EM) model in which an rf current is applied to an external azimuthal coil which surrounds the plasma to generate the rf magnetic field and induced rf electric field inside the coil.
Our ES model is essentially the same as the one by Matsuoka et al. (2012) who developed a theoretical model in which the rf electric fields in a plasma and sheaths were derived in a high-frequency range of ω _{ LH }≪ω _{ rf }≪ Ω_{ e }. Here, ω _{ rf }, ω _{ LH }, and Ω_{ e } are an externally applied field frequency, lower hybrid frequency, and electron gyrofrequency, respectively. In this paper, we choose high and low frequencies for ω _{ rf }. The high frequency we use is ω _{ rf }=Ω _{ e }/2 to compare our numerical results with the theoretical model by Matsuoka et al. (2012). The low frequency we use is ω _{ rf }=ω _{ LH }/2 to apply the PA/ICR scheme. For the PA scheme to operate, ω _{ rf } should be near the ion gyrofrequency, Ω_{ i }, but we set ω _{ rf }=ω _{ LH }/2>Ω_{ i } as the low-frequency case, to reduce the computation time.
An ion-rich sheath, i.e., a positively charged boundary, and an electron-rich sheath, i.e., a negatively charged boundary, were studied using computer simulations and laboratory experiments. For example, the ion-rich sheaths were simulated around antennas immersed in space plasma (Miyake and Usui 2009; Usui et al. 2004). A transition from the ion-rich to the electron-rich sheath was experimentally observed, when a rapid rising voltage was applied to a plane gridded electrode in an unmagnetized discharge plasma (Stenzel et al. 2011). In our ES model, we show numerically that the electron-rich and ion-rich sheaths can be formed according to the value of ω _{ rf }.
Our EM model is similar to the one proposed by Ikeda et al. (2005) who focused on the evanescent rf wave with ω _{ rf }>ω _{ LH }; thus, the rf waves were highly damped near the plasma edge with a skin-depth scale and the spatial profile of the rf waves was independent of the plasma radius. Conversely, we focus on a propagating rf wave with ω _{ rf }<ω _{ LH }; thus, the rf wave excited on one side of the plasma boundary propagates toward the other side of the boundary as extraordinary waves. Accordingly, a standing wave is formed in the plasma region. In this paper, we compare spatial/temporal structures of the standing rf waves obtained from simulation and cold collisionless plasma linear theory (Hada et al. 2009; Stix 1992). Finally, we discuss which model should be adopted for application to the electrodeless plasma thrusters.
Methods
The plasma consists of either argon or helium ions and electrons. Neutral particles are absent, and the collisionless plasma is assumed. For numerical computation, at every time step, the ions and electrons are accelerated by the Lorentz force, the charge and current densities evaluated, and the electromagnetic fields time-advanced using Maxwell’s equations with a charge-conserving current deposition algorithm. The plasma is absorbed at the plasma boundaries of x=±a. We use VORPAL (Nieter and Cary, 2004) for the simulation code.
Plasma parameters
Plasma parameters in the ES and EM models are the same as follows. Initially, ions and electrons are uniformly distributed between x=−a and x=a, and their velocities are given by the Maxwellian distribution with temperatures, \(T_{i}=m_{i} v_{th,i}^{2}\) and \(T_{e}=m_{e} v_{th,e}^{2}\), where v _{ t h,i } and v _{ t h,e } are ion and electron thermal velocities, respectively, and m _{ i } and m _{ e } the ion and electron mass, respectively. There are 200 macro-particles per cell for both ions and electrons.
In this paper, we investigate the wave penetration process, assuming that the same physical process dominates other parameters with different scales. Here, to save the computation time, we choose the parameters as follows: m _{ i }/m _{ e }=200, T _{ e }=T _{ i }=0.0004 m _{ e } c ^{2}, and Ω_{ e }/ω _{ pe }=1, where c is the speed of light.
Under these parameters, the lower hybrid and ion gyrofrequencies normalized to the electron plasma frequency are ω _{ LH }/ω _{ pe }≃0.05, Ω_{ i }/ω _{ pe }=1/200, respectively; then, ω _{ LH }/Ω_{ i }≃10, where \(\omega ^{-2}_{\textit {LH}}=({\varOmega }_{i} {\varOmega }_{e})^{-1}+({{\varOmega }_{i}^{2}}+\omega _{\textit {pi}}^{2})^{-1}\). Alfvén and ion sound velocities normalized to the light speed are v _{ A }/c≃0.07 and v _{ S }/c≃1.4×10^{−3}, respectively, where \({v_{A}^{2}}/c^{2} \simeq {{\varOmega }_{i}^{2}}/\omega _{\textit {pi}}^{2}\) and \({v_{S}^{2}}/c^{2}=T_{e}/m_{i} c^{2}\). Thus, v _{ A }≫v _{ S } represents the validity of the cold plasma approximation.
Four runs are performed, where the system lengths, L=2a, are 1000Δ and 767Δ for run1 and runs 2–4, respectively
Method | ω _{ rf } | External field amplitude | |
---|---|---|---|
Run 1 | ES | 10 ω _{ LH } | V _{ rf }/c B _{0} L=1 |
Run 2 | ES | 0.5 ω _{ LH } | V _{ rf }/c B _{0} L=0.008 |
Run 3 | EM | 0.5 ω _{ LH } | μ _{0} j _{ rf } Δ/B _{0}=0.02 |
Run 4 | EM | 0.5 ω _{ LH } | μ _{0} j _{ rf } Δ/B _{0}=0.2 |
Electrostatic (ES) model
In the ES model, two planar electrodes at x=±a are given time-varying electrostatic potentials, ϕ(−a)=V _{ rf } cos(ω _{ rf } t)/2 and ϕ(a)=−V _{ rf } cos(ω _{ rf } t)/2. Here, an external voltage difference between the two electrodes is given by V(t)=ϕ(−a)−ϕ(a)=V _{ rf } cos(ω _{ rf } t). The rf electric field is then generated into the plasma. For the evolution of the electric field, the Poisson equation is solved at every time step based on the instantaneous charge density distribution by VORPAL (Nieter and Cary, 2004). Hence, in the ES model, the electric field is exactly parallel to the x direction, and there are no magnetic field fluctuations.
where \(\epsilon =1-\omega _{\textit {rf}}^{2}/{\Omega _{e}^{2}}\), q=8(ω _{ pe }/Ω_{ e })^{2}(δ _{ e }/L), and \(\delta _{e}={eV}_{\textit {rf}}/m_{e}{{\varOmega }_{e}^{2}} L=E_{1}/{\varOmega }_{e} B_{0}\), which represents an excursion length of the electron polarization drift for perfect penetration. Equation (1a) show that the amplitude of E _{ p } decreases with the increase in s _{0}, because of the shielding by the ion-rich sheath. In addition, (2) describes the maximal sheath width of 2s _{0}. The potentials corresponding to the electric fields (1) are shown in the ‘Appendix’. Under the simulation parameters for run 1, ε=3/4, q=8/20, δ _{ e }=c/ω _{ pe }; the sheath width is then 2s _{0}≃2.3c/ω _{ pe }, and the reduction rate in (1a) is \(2 \eta {s_{0}^{2}}/V_{\textit {rf}} \simeq 0.13\). In the ‘Results’ section, we compare the numerical results in run 1 with the theories (1) to (3).
Using the ES model, we perform run 1 and run 2 in Table 1. We set two different ω _{ rf } to examine the ω _{ rf } dependence. In run 1 for a high-frequency case, we set ω _{ rf }=Ω _{ e }/2≃10 ω _{ LH } and V _{ rf }=c B _{0} L. In run 2 for a low-frequency case, we set ω _{ rf }=ω _{ LH }/2≃5 Ω _{ i } and V _{ rf }≃0.008c B _{0} L. In run 1, the larger V _{ rf } than that in run 2 increases the sheath width, so that the sheath behavior is visible in detail, and can be compared with the theory. In run 2, the smaller V _{ rf } produces E _{1} which is equal to the amplitude of E _{ x } in run 3 based on the linear theory (6b), so that the results of the ES and EM models can be compared.
Electromagnetic (EM) model
where B _{1}=μ _{0} j _{ rf } Δ/2, and μ _{0} is the absolute permeability of a vacuum.
where ω _{ pe } and ω _{ pi } are the electron and the ion plasma frequencies, respectively.
The right panel of Fig. 2 shows the value of D/S as a function of ω/ω _{ LH } by inserting the plasma parameter into (8). When ω<ω _{ LH }, D/S is negative so that i E _{ x }/E _{ y } > 0. This means that the extraordinary wave for ω<ω _{ LH } is right-handed polarized. When ω=ω _{ LH }/2, we obtain D/S≃−6.6.
Using the EM model, we perform runs 3 and 4 in Table 1. We use two different external current density, j _{ rf }, to examine the amplitude dependence of the rf wave, with ω _{ rf } fixed at ω _{ rf }=ω _{ LH }/2. Run 3 uses μ _{0} j _{ rf } Δ/B _{0}=0.02 corresponding to B _{1}/B _{0}=0.01, whereas run 4 uses μ _{0} j _{ rf } Δ/B _{0}=0.2 corresponding to B _{1}/B _{0}=0.1.
From the equations above, E _{ x } and E _{ y } have nodes at x=0,±a, whereas δ B _{ z } has anti-nodes at x=0,±a. In this paper, we compare the numerical results in runs 3 and 4 with the theory (9), where v _{ w }/c=0.06 and D/S=−6.6 are used for the comparison. For run3 (B _{1}/B _{0}=0.01), the amplitude of E _{ x } in (9b) is 2B _{1} v _{ w }|D/S|≃0.008c B _{0}, which is equal to the vacuum amplitude of E _{1} in run 2, as described in the previous subsection.
Results
ES model
Figure 3b–d show the spatial profiles of E _{ x }; potential, ϕ; and charge density, σ, normalized to E _{1}, V _{ rf }, and e n _{0}, respectively. In these panels, the colored solid and dashed lines are obtained by the simulation and the theory from Matsuoka et al. (2012), respectively. In Fig. 3b–d, the measurement times are ω _{ rf } t=4π=12.6 and ω _{ rf } t=5π=15.7, which are displayed in Fig. 3a by the green and blue chained lines, respectively. At these times, E _{ x } in the vacuum are positively or negatively peaked in the time series of Fig. 3a. In Fig. 3b, the solutions (1) are depicted for two phases: (s _{1},s _{2})=(0,2s _{0}) for V(t)=V _{ rf } and (s _{1},s _{2})=(2s _{0},0) for V(t)=−V _{ rf }, indicated by the green and blue dashed lines, respectively. In Fig. 3c, the numerical potential profiles are obtained by the spatial integration of E _{ x } in Fig. 3b. The theoretical potential fields are plotted using (11). In addition to Fig. 3c, the vacuum solutions of ϕ(x)=∓(V _{ rf }/2)(x/a) are depicted by dotted lines. In Fig. 3d, the theoretical charge densities are σ=e n _{0} in the sheath of the cathode side because all electrons move toward the anode side. The theoretical sheath regions with the scale of 2s _{0}≃2.3c/ω _{ pe } are indicated by the gray shaded regions in Fig. 3b,c.
From the simulation result, we define the boundary between the plasma and sheath. When |x|<5c/ω _{ pe }, E _{ x } approximates to a constant in Fig. 3b, and the charge neutrality is conserved, i.e., σ is approximately 0 in Fig. 3d. Therefore, the plasma-sheath boundary is found to be |x|=5c/ω _{ pe }, indicated by the vertical chained lines in Fig. 3b–d. Hence, the scale of the sheath width is found to be 5c/ω _{ pe }≃4.4s _{0}, which is 2.2 times larger than 2s _{0}≃2.3c/ω _{ pe } obtained from the theory. In the plasma region of |x|<5c/ω _{ pe }, the spatial profiles of E _{ x } and ϕ match with the theory. In the sheath region of 5c/ω _{ pe }<|x|<a=10c/ω _{ pe }, the value of σ is locally positive or negative, and the value of ϕ is larger than the vacuum solution. These results show that the ion-rich sheath is formed in the whole region of the sheath, although the negative charge density exists locally in the sheath. Therefore, the simulation result is qualitatively consistent with the theory by Matsuoka et al. (2012). In addition, we predict that the amplitude of the electric field (solid line) in Fig. 3a is less than E _{1} (dashed line), because of the presence of the ion-rich sheath, which shield the applied field.
EM model
Discussion
In this section, four issues from our analysis will be examined. First, we discuss the plasma wall loss process in the ES model. When ω _{ rf }=Ω _{ e }/2≃10 ω _{ LH } (the high-frequency case), we found the ion-rich sheath, which was qualitatively consistent with the theory from Matsuoka et al. (2012). As the theory predicted, mainly, electrons were lost because of the polarization drift, resulting in the ion-rich sheath. However, the sheath width obtained from the simulation was 4.4 s _{0}, which was 2.2 times larger than from the theory, where s _{0} is the excursion length of the electron polarization drift. In addition, a negative-charged density was locally observed in the sheath. These results were inconsistent with the theory. In run 1, we used the external voltage as V _{ rf }=c B _{0} L, which was larger than that used in the PIC simulation by Matsuoka et al. (2012). Therefore, the larger V _{ rf } may break the assumptions of immobile ions and rigid box motion of electrons. Hence, we suggest, for a strong external voltage, the mobile ions and electron motion in detail should be considered for a proper description of the sheath dynamics.
Inserting the parameter of run 2 into E _{1}, we obtain δ _{ i }=5.6ρ _{ i }. If ions are lost as a result of the ion polarization drift, the sheath width should have the scale of 2δ _{ i }∼11ρ _{ i }, which is larger than the sheath width of 2ρ _{ i } obtained from the simulation. Therefore, we conclude that the ion wall loss in the low-frequency case is not the result of the ion polarization drift but of the thermal ion gyromotion.
Second, we discuss the validity of the cold collisionless plasma linear theory in the EM model. We fixed the plasma radius at a=π/k so that the rf electric fields were nodes at the boundaries within the linear theory. When the rf field amplitude was small, with B _{1}=0.01B _{0}, the simulation result matched the theory except for the existence of the sheath, and the rf standing waves were stably excited in the plasma. When the rf amplitude was increased to B _{1}=0.1B _{0}, the waveforms became complex, as well as time evolutions of the electromagnetic fields measured at fixed spatial locations (Fig. 6a–c). The complexity may be due to the superposition effect with a reflected wave at the boundary or due to the plasma compression from incident waves as seen in the plot of E _{ x } in Fig. 6e. A study of the boundary effects including a correct sheath physics is necessary for a proper description of the electromagnetic wave penetration into a plasma. In addition, a collisional effect, especially electron-neutral collisions, cannot be ignored in the high-density plasma produced in laboratories. The collisional effect may prevent the stable excitation of the rf standing waves into the plasma. For future work, we need to investigate whether the rf standing wave can survive in the plasma, without a strong damping from the collisions.
Third, we apply our results to the plasma acceleration in the electrodeless plasma thruster. For efficient acceleration of the plasma, the rf electric field should penetrate deeply into the plasma center because the plasma density is maximum at the center and low near the edge for typical cylindrical plasma produced in laboratories. In the present PIC simulation, when ω _{ rf }=ω _{ LH }/2, the rf electric fields penetrated into the plasma in both the ES and EM models. However, the spatial profiles were different. Namely, the ES model produced the nearly uniform rf electric field in the plasma, whereas the EM model produced the standing rf wave with a very small electric field at the center because of the symmetry. Therefore, we suggest that the ES model, which produces the rf field more deeply into the plasma, is more suitable for the external electrode configuration than the EM model. In addition, the electron-rich sheath in the low-frequency regime tends to cause an increase in the rf field amplitude in the plasma, whereas the ion-rich sheath in the high-frequency regime tends to cause a decrease. Therefore, the ES model in the low-frequency regime can produce larger amplitude of the rf electric field than in the high-frequency regime, leading to efficient plasma acceleration.
Finally, we discuss the validity of our one-dimensional model, in which the plasma is only bounded in the x axis. In the electrodeless plasma thrusters, the plasma is confined in a cylindrical boundary. This geometrical difference may lead to an essential difference for the rf field penetration process. For example, a one-dimensional fluid simulation for the electrostatic field penetration has indicated that the field penetration in the cylindrical geometry was reduced by about 6 % in comparison with a planar model (priv. comm. with K. P. Shamrai). Therefore, to make more realistic estimates, the wave penetration into a cylindrical plasma should be investigated for the development of the electrodeless plasma thrusters.
Conclusions
We investigated the external radio frequency (rf) electromagnetic field penetration into the magnetized collisionless plasma using one-dimensional particle-in-cell (PIC) code, which simulates the plasma and electromagnetic field evolution in a direction perpendicular to the background magnetic field. We considered two different electrode configurations: an electrostatic (ES) model using an external rf voltage and an electromagnetic (EM) model using an external rf current.
In the ES model, the external rf voltage across the plasma generates the electrostatic field in the plasma. The sheath behavior depends on the externally applied rf field frequency, ω _{ rf }. When ω _{ rf }=Ω _{ e }/2 (the high-frequency case), an ion-rich sheath appeared near both the anode and cathode sides of the electrodes with several scales of the electron polarization drift, because more electrons than ions are lost as a result of the electron polarization drift. The penetrated rf electric field was qualitatively consistent with the theoretical model (Matsuoka et al. 2012), but the details of the sheath did not match the model. When ω _{ rf }=ω _{ LH }/2 (the low-frequency case), an electron-rich sheath appeared near the cathode side of the electrode with a scale similar to the thermal ion gyroradius, because more ions were lost due to their larger gyroradii than electrons.
In the EM model, the standing electromagnetic rf waves were formed as a result of superposition of two anti-traveling extraordinary waves excited by the external rf current. When the plasma diameter was equal to the wave length, the rf electric field at the plasma boundaries were nodes in the cold collisionless plasma linear theory. For a small amplitude of the external rf field with 1 % of the background magnetic field amplitude, the spatial/temporal profiles of the rf standing waves, except for the sheath, approximately matched the linear theory. When the amplitude of the external rf field increased to 10 % of the background field amplitude, the waveforms were irregular and complex due to the boundary effects.
Appendix
where V(t) is the external rf voltage with V(t)=V _{ rf } cos(ω _{ rf } t) and L(=2a) system length.
Declarations
Acknowledgements
This work was partly supported by the Grants-in-Aid for Scientific Research under Contract No. (S) 21226019 from the Japan Society for the Promotion of Science.
Authors’ Affiliations
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