Influence of cusp-shaped magnetic fields on plasma density and thrust in an RF plasma thruster with a magnetic nozzle

Radiofrequency (RF) plasma thrusters offer a promising solution to overcome the limit of the operation lifetime of thrusters and have been widely studied^1–3 for future space propulsion. Their advantage lies in avoiding the erosion of acceleration grids owing to the direct plasma interaction with the grids and contamination of the discharge chamber. An RF plasma thruster system generally consists of an RF generator, a plasma-generation antenna wound outside an insulating tube, a propellant feeding system, and an external magnetic field source. The propellant gas is ionized with an antenna powered by an RF generator, and ion acceleration results from a spatial potential gradient in the presence of an expanding magnetic nozzle (MN). Some groups have identified a large ion momentum downstream of the expanding MN as an ion acceleration scenario under an ambipolar electric field formed by an electron static pressure gradient.^4–7 The motion of electrons significantly influences the total thrust of the RF thruster, including its cooling process and detachment from the MN, which has been thoroughly investigated.^5,7–9 Additionally, several numerical simulation codes have been developed to model the expanding plasma in the MN,^10–13 aiding the prediction of the thrust performance and efficiency. This contributes to the qualitative validation of experimental plasma physics.^14,15 The external magnetic field plays a vital role in enhancing thruster performance. Under typical MN conditions with a uniform magnetic field configuration, Takahashi et al. reported that the thrust increases with higher magnetic field strength,¹⁶ input RF power, and propellant gas flow rate. In contrast, a limitation of the acceleration effect using only a magnetic nozzle was identified.¹⁷ 

Concerning the field configuration in the vicinity of the plasma generation region, the use of a cusp-shaped magnetic field, which is a non-uniform magnetic field configuration with a spatial null field point, has shown promising results in the plasma generation realm. Shamrai et al. found efficient plasma generation using a cusp-shaped field based on theoretical analysis of a magnetized inductively coupled plasma source.¹⁸ Previous research has highlighted that the location of the cusp point relative to the RF antenna is critical for efficient plasma generation and reduction of plasma wall loss. Significantly improved thrust performance was observed when the generation antenna was positioned downstream of the cusp point, with thrust increment correlating with increased static pressure along the magnetic field lines emanating from the inside of the discharge tube.¹⁹ A numerical simulation employing a two-fluid plasma model for ions and electrons found that the decrease in the plasma wall loss was more dominant than the increase in the plasma number density, depending on the operational conditions.¹³ The ion energy distribution function was measured in the expanding MN region, and ambipolar ion acceleration was determined. Simultaneously, it was suggested that the dispersion of ion momentum is due to particle collisions, leading to momentum loss.²⁰ These findings underscore the critical nature of cusp-shaped conditions, including the positioning of the cusp point and MN strength, in optimizing thrust performance. In 2022, a thrust efficiency of 30% was achieved using a cusp-shaped field at an RF power of 5 kW.²¹ 

Inspired by these findings, which demonstrated the benefits of positioning the RF antenna downstream of the cusp point, we newly constructed an RF plasma thruster device capable of changing operational conditions of RF plasma thrusters, including gas-feeding methods, external magnetic field configurations, and plasma excitation power, etc. With this device, we aimed to clarify the dependences of the plasma parameters and thrust performance on the operational parameters and to study the physical phenomena of the expanding plasma in the presence of the MN. Among them, the detailed cusp-field condition is still an open question, although the efficacy of using the cusp-shaped magnetic field is to be evident. This paper focuses on the influence of the cusp field condition on plasma parameters from the inside of the discharge tube to the MN region. We compare the parameters between the cusp and uniform magnetic field cases as the field configuration of the MN region remains nearly the same. Based on the previous studies on the RF thrusters using cusp field conditions,^13,19 cusp points with respect to the RF antenna and field strength are varied to clarify a suitable cusp field condition in the developed RF plasma source. Measurement methods such as electrostatic probes and optical emission spectroscopy are applicable. This paper uses a probing technique to conduct an experimental investigation of the influence of cusp-shaped magnetic field conditions on plasma parameters, including electron temperature, density, and static pressure.

The remainder of this paper is organized in the following way: in Sec. II, we overview the experimental setup and condition, including the RF thruster device, external magnetic field conditions, and our probe measurement methodology and theoretical framework. Section III provides the experimental results and compares different magnetic field conditions: III A discusses cusp versus uniform magnetic field conditions, III B reviews dependence on the cusp points with respect to the RF antenna position while using the same MN configuration, and III C discusses dependence on the strength of the MN under cusp conditions. In Sec. IV, we discuss the thrust values derived from our measurements in the context of existing thrust models for RF thrusters with an MN and conclude with a summary of our findings.

Figure 1 shows the schematic of the RF thruster device. The apparatus consists of a quartz tube with 50 cm in length and 9 cm in diameter and a stainless vacuum chamber with 60 cm in length and 35 cm in inner diameter. The joint where the chamber flange meets the vacuum chamber is designated as z = 0, which situates the discharge tube region in the negative z-region. A single-turn, loop-type antenna spanning a width of 2 cm and crafted from an oxygen-free copper plate was wound outside the tube at z = −14.2 cm. A background pressure inside the chamber reaches 1.2 × 10⁻⁶Torr using a turbo molecular pump and a rotary pump. During the RF discharge, argon propellant gas was periodically introduced at the tube head flange located at z = −50 cm using a piezo valve (PEV-01). The valve opening duration was set to 20 ms, followed by an 80-ms pulsed discharge initiated 5 ms after valve closure. Pulse discharge was conducted to prevent heat damage. The chamber pressure typically reached 5.0 × 10⁻⁴Torr with a gas introduction. In the present experiment, the RF power was set to 0.5 and 1 kW at an RF frequency of 13.56 MHz, using an RF power source (Thamway, T161-6013HAJ) connected through an RF impedance matcher, which included two adjustable vacuum capacitors. The forward and reflected powers are detected using a directional coupler, and the reflected power is minimized to less than 5% with respect to the input power during the RF discharge. Present plasma measurement was conducted after thoroughly decreasing the reflected power component by adjusting the impedance condition by changing the capacitors.

The local plasma parameters, including the effective electron temperature T_e, electron density n_e, and spatial potential V_s, were measured using an L-shaped RF-compensated probe, as shown in Fig. 1. The probe consisted of a tungsten tip with a length and diameter of 3 and 0.8 mm, respectively. Although the probe shapes and sizes differ, the RF compensation circuit has been illustrated elsewhere.²² The probe was inserted from the rear flange of the chamber, as shown in Fig. 1, allowing the probe head position to be varied radially and axially. A ramp bias voltage V_p with a maximum amplitude of ±65 V at 70 Hz was applied to the probe tip to detect a probe current signal from the plasma. The current was monitored using an oscilloscope (Agilent Technologies, DSOX2004A) equipped with an isolation amplifier circuit. The probe data were averaged over four pulse discharges to remove random noise. The effective probe-tip surface changed depending on the plasma condition because a sheath was formed around the tip surface. The sheath length was several times the Debye length λ_D = (ε₀k_BT_e/(e²n_e))^0.5, where ε₀, k_B, and e represent the vacuum permittivity, Boltzmann constant, and elementary charge, respectively. Typical T_e and n_e values of 7 eV and 10¹¹ cm⁻³, respectively, were used to evaluate the λ_D, and an λ_D of approximately 50 μm was assessed. This λ_D value is much lower than the probe-tip radius, and the disturbance of the plasma flow owing to the presence of the sheath can be ignored. In addition, the influence of the presence of the probe on the plasma flow was negligible because the probe tip radius was smaller than the electron Larmor radius.²² A cylindrical probe tip direction should be perpendicular to magnetic field lines.^23,24 In the present experiment, the probe tip was almost perpendicular to the magnetic field lines at each measurement position. As an index of availability of a probe perpendicular to magnetic field lines, a diffusion parameter, Ψ_⊥ = a_pln(πl_p/4a_p)/(γR_Le), is utilized. a_p, l_p, and R_Le are a probe tip radius, length, and the electron Larmor radius, respectively. γ is given as 4/3 − 0.62exp[−λ_e/(2a_p)], where λ_e is electron mean-free path. Ψ_⊥ of 0.08 is calculated using the typical measured parameters, and therefore, the present application of the Druyvesteyn method is valid.²⁵ 

Figure 2 illustrates the spatial distribution of the static magnetic field strength and lines of the cusp and uniform conditions. The maps display each scenario's magnetic field lines (upper profiles) and corresponding field strengths (lower profiles). The magnetic field strength increased to approximately 400 G as DC was applied to the external electromagnets (EMs). Specifically, an EM coil, called an upside coil, is positioned 5 cm away from a pair of EM coils, called downside coils. All coils were equipped with water-cooling systems and shared identical specifications, including a coil turn number of 60, an inner radius of 12 cm, and an axial thickness of 5.6 cm. To establish the cusp configuration, the upside coil receives a DC I_up oriented in the opposite direction relative to the current applied to the downside coils I_down. This setup allows for an arbitrary magnetic field strength ratio adjustment between the upside and downside coils using two DC-stabilized current power supplies, specifically the Matsusada and ITEC models. The upside coil was not used when creating a uniform magnetic field configuration. I_down was set to 60 A to match the maximum field strength observed in the cusp condition and compare the plasma parameters between the different field conditions.

Figure 3 displays the z-axis profiles of T_e, n_e, and V_s measured on the z-axis (x = 0) from z = −13 to 30 cm for the cusp and uniform field conditions at an input RF power of 0.5 kW. The magnetic field conditions were consistent with those shown in Fig. 2. The cusp z-axis point was set at z = −20.4 cm, with the measurement point at z = −13 cm positioned nearest the loop antenna within the measurement regions. V_s was derived from V_p, where d²I_e/dV_p² is zero. The error bars in the figure represent the standard deviation of the measurements. Figure 3(a) depicts T_e profiles along the z-axis, derived from Eq. (3). Under the uniform field condition, T_e values increased around the thruster exit (z = 0), although these values remained lower than those under the cusp condition. This difference is attributed to the nature of the probe measurement and the effect of a low plasma density profile, which leads to such high-temperature values. Figure 3(b) shows measured n_e profiles along the z-axis. A decrease in n_e from z = −7 cm is observed as the plasma expands through the diverging magnetic field. The n_e profile in the cusp condition surpassed that in the uniform case, with the peak n_e value at z = −7 cm in the cusp case being approximately three times greater than that in the uniform case. In addition, the electron static pressure p_e = n_ek_BT_e was higher in the cusp condition, contributing to an enhanced exhaust plasma plume in the MN region and an axial thrust component derived from this static pressure. Thus, the cusp condition demonstrates superiority over the uniform condition regarding the static pressure profiles on the z-axis. Figure 3(c) shows V_s profiles along the z-axis. V_s values were higher in the uniform magnetic field case than in the cusp case. High V_s profiles in RF plasma source are found in a low-density plasma flow; thus, the uniform case having the lower n_e profiles shown in Fig. 3(b) results in the V_s profiles. Notably, a steep V_s gradient along the z-axis, known as a double layer,^26,27 was absent in both cases.

Subsection III A showed the advantage of a cusp-shaped magnetic field in augmenting the n_e profile within plasma flow. To further evaluate the effect of the cusp-shaped magnetic field on the plasma dynamics, plasma parameters on the z-axis were meticulously measured while adjusting the magnetic field ratio between the up and downside coils at an input RF power of 0.5 kW. The |B_ext| profiles in the nozzle region remained consistently similar across all cusp configurations.

Figure 5 shows the z-axis profiles of T_e, n_e, and V_s for the three cusp cases. The profiles are represented by blue dashed, thin-red solid, and gray dotted-dashed curves for B_up:B_down = 1:2, 2:3, and 6:7, respectively. The corresponding coil current combinations of I_up and I_down for these magnetic field ratios were set at (−80 A, 80 A) (−120 A, 90 A), and (−180 A, 105 A), respectively. The peak values of |B_ext| are located at approximately z = −7 cm. Additionally, three vertical dashed lines in blue, thin-red, and black indicate the cusp points at z = −23.8, −20.4, and −18.1 cm, respectively, correlating with the cusp-field conditions shown in Fig. 5. Figure 5(a) shows T_e profiles are relatively uniform in the z-direction for the three cusp cases. However, the highest T_e profile was found in the discharge tube for B_up:B_down = 2:3. The n_e profile peaked near the choke region of the nozzle and gradually decreased along the z-axis in all the cases, as shown in Fig. 5(b). The highest n_e profiles were present, as well as T_e profiles. Conversely, V_s profiles were nearly flat in the z-direction for the three cusp cases. As substantiated by previous research in a different device, the spatial relationship between the cusp point and the RF antenna has emerged as a critical factor for optimizing total plasma-generation efficiency. When the antenna position approaches the cusp point, the wall loss of the plasma increases, although the plasma generation inside the discharge tube does not largely differ between the different antenna positions.^13,19 The present comparison qualitatively clarified that a separation of approximately 6 cm between the cusp point and antenna position is suitable for the performance of our linear RF plasma device. Moreover, the magnitude of |B_ext| at the antenna position can influence plasma generation efficiency. B_up:B_down = 2:3 manifested the most favorable p_e profile among the cusp-shaped conditions assessed. To thoroughly assess the plasma generation efficiency or wall loss of plasma near the RF antenna region, future studies will endeavor to measure two-dimensional (2D) profiles of plasma parameters, offering a more detailed understanding of these phenomena.

For the cusp conditions, the dependence of the plasma parameters on the strength of |B_ext| was investigated by changing I_up and I_down, while the B_up:B_down was maintained at 2:3, in which the maximum density profile was found. Figure 6 shows the z-axis profiles of the parameters at an input RF power of 1 kW. The T_e profiles showed relative uniformity along the z-axis, indicating that |B_ext| did not significantly alter T_e profiles or induce anomalous heating within the plasma-generation region, as shown in Fig. 6(a). Figure 6(b) shows the n_e profiles on the z-axis, revealing a decrease in n_e with increasing the |B_ext| near the plasma-generation region. Conversely, n_e increased downstream of the MN. The magnetic field strength critically influenced plasma generation and flow within the nozzle region. This increase in n_e on the z-axis can result from the compression of electrons due to a radial component of the Lorentz force in the throat region of the MN. Figure 6(c) shows that the V_s profiles remained relatively flat for all three examined cases. This indicates that variations in |B_ext| predominantly affect n_e profiles with an increase in |B_ext|, which is beneficial for enhancing plasma density.

The axial profiles of p_e for the three cusp cases, maintaining the B_up:B_down = 2:3, are shown in Fig. 7. The profiles are represented by open blue triangles, red circles, and black squares for the (I_up, I_down) combinations of (−120 A, 160 A) (−90 A, 120 A), and (−45 A, 60 A), respectively, at an RF power of 1 kW. In addition, green crosses denote the cusp condition at (I_up, I_down) = (−120 A, 90 A) with an RF power of 0.5 kW, as discussed in Subsection III A. The p_e profiles increased with increasing RF power, as shown by comparing the open red circle and green cross plots at (I_up, I_down) = (−120 A, 90 A). The case of (I_up, I_down) = (−60 A, 45 A), in which the B-field strength is half of that at (I_up, I_down) = (−120 A, 90 A), is shown in Fig. 6. The highest p_e profile was observed at (I_up, I_down) = (−120 A, 160 A), and a larger |B_ext| increased the static force derived from p_e along the z-axis. This force plays a critical role in augmenting the total plasma thrust.

In summary, the axial profiles of plasma parameters on the z-axis of the RF plasma-thruster device were investigated by varying the external magnetic field conditions. It was found that the cusp magnetic field configurations affected the density profiles; the cusp conditions demonstrated an increase in the magnitude of n_e profiles on the z-axis compared to that for the uniform field condition, even under identical RF power and gas puffing settings, as illustrated in Fig. 3. Near the loop antenna, Maxwellian (Druyvesteyn)-like distributions were observed for the cusp (uniform) cases, as shown in Fig. 4. The Maxwellian EEDF suggests a larger plasma heating process, which affects the density profiles and plasma-generation efficiency through Maxwellianization. Further analysis of the plasma parameters for the several cusp conditions indicated that the cusp point relative to the antenna position significantly influenced the density profiles. At the present stage, the upside and downside magnetic field strength B_up:B_down = 2:3 ratio emerged as optimal, resulting in the highest n_e profiles along the z-axis for all cusp cases, as shown in Fig. 5. The distance between the cusp point and antenna position was identified as critical for thrust performance, as reported in previous studies.^13,19 Therefore, exploring this positional dependence is essential for future research to enhance performance improvement. It was also established that p_e, T_B, and T_s increased with MN strength. Notably, T_B was higher than T_s for all cusp cases, revealing the significance of the MN strength in thrust dynamics in our thruster device. In the present discussion of thrust, the radial profiles of p_e at each z position were approximated as nonlinear functions. Thus, future efforts will focus on measuring the 2D spatial profiles of the plasma parameters for a detailed understanding of the plasma flow within the MN and to facilitate an accurate quantitative evaluation. Note that probe anisotropy relative to magnetic field lines should be addressed, especially in the MN region. In addition, the increase in the RF power, the RF antenna position with respect to the thrust exit, gas feeding method, etc., will be optimized based on the present cusp-field condition.

This study was partially supported by JSPS KAKENHI (Grant No. JP22K14027) and NIFS Collaboration Research Programs (Grant Nos. NIFS23KUGM194 and NIFS22KIER002).

The authors have no conflicts to disclose.

Takeru Furukawa: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Funding acquisition (equal); Investigation (lead); Methodology (lead); Project administration (lead); Resources (equal); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Kento Shimasaki: Data curation (supporting); Formal analysis (equal); Investigation (equal); Visualization (supporting). Satoshi Nakamoto: Conceptualization (supporting); Investigation (equal); Supervision (supporting); Validation (supporting); Writing – review & editing (equal). Hiromasa Takeno: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

Data supporting the findings of this study are available from the corresponding author upon reasonable request.