Influence of the Applied Magnetic Field Strength on ...

Influence of the Applied Magnetic Field Strength on Flow Collimation in Magnetic Nozzles Justin M. Little∗ and Edgar Y. Choueiri† Electric Propulsion and Plasma Dynamics Laboratory, Princeton University, Princeton, NJ, 08544 The influence of the magnetic field strength on the collimation of the plasma flow in an electron-driven magnetic nozzle is investigated experimentally. A collimated plasma flow is required for efficient plasma propulsion to minimize plume divergence losses. Faraday probe measurements are used to estimate the ion streamlines in the diverging field of the magnetic nozzle. It is found that decreasing the strength of the applied magnetic field invokes a transition from a collimated plume to an under-collimated plume, where an under-collimated plume is defined such that the plume divergence is greater than the magnetic field divergence. Langmuir and emissive probe measurements reveal that the transition to an under-collimated plume is accompanied by anomalous deceleration of the ion beam along the nozzle centerline, broadening of the transverse density profile, and the disappearance of an ion-confining potential well at the plasma periphery. This transition offers a guideline for reducing the plume divergence of an electron-driven magnetic nozzle.

I.

Introduction

Magnetic nozzles (MNs) are the primary acceleration stage of electrodeless electric propulsion systems such as the helicon plasma thruster (HPT),1 electron cyclotron resonance thruster (ECRT),2 and Variable Specific Impulse Magnetoplasma Rocket (VASIMR).3 In addition, MN-like effects are thought to be present in applied-field magnetoplasmadynamic thrusters (AF-MPDTs),4 and cylindrical and cusped-field Hall thrusters.5 While the diverging magnetic field of a MN is ubiquitous to these system, the fundamental effect of the MN on the plasma dynamics of each device varies depending on the presence of applied currents and disparate plasma species temperatures. In this paper, we will concern ourselves primarily with MNs as used with HPTs and ECRTs, with the distinguishing quality being that the plasma source of these devices inherently heats electrons to much higher temperatures than ions (Te Ti ). As a result, thermal expansion in the MN is driven by the hot electron population.6 We refer to magnetic nozzles that rely on this process as electron-driven magnetic nozzles (ED-MNs). Ion acceleration in an ED-MN results from the ambipolar electric field left in the wake of the hot electrons outrunning the colder ions.7 The diamagnetic current thats results from magnetic confinement of the hot-electron plasma transfers momentum back to the magnetic circuit of the nozzle:8 an effect known as early as Seikel9 and recently demonstrated experimentally by Takahashi et al.10 Detailed experimental investigations of the fundamental processes of MNs have been limited to date, with recent efforts focused on the propulsion performance of thrusters that use MNs.11–15 Takahashi et al. showed12 that the measured thrust approached the theoretical value predicted from a simplified analytical model16 as the applied field strength of an ED-MN increased. They attribute this behavior to decreased cross-field diffusion in the downstream plasma for increased applied magnetic fields, however, they do not study the source of this diffusion in depth. Concern over the issue of plasma detachment from MNs has led to detailed investigations of the ion dynamics in the diverging field. Cox et al.17 and Deline et al.18 observed that the half-width at halfmaximum of the radial density profile diverges less than the magnetic field, from which they conclude that ions have begun to detach from the MN. A similar conclusion was reached by Terasaka et al.19 using Mach probes to spatially map the three-dimensional ion velocity vector. Direct measurements of ion detachment ∗ Graduate † Chief

Student, Research Assistant Scientist, EPPDyL, Professor, Applied Physics Group, AIAA Fellow

1 of 13 American Institute of Aeronautics and Astronautics

Coaxial Antenna Feedthrough

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were made using Faraday probe sweeps by Olsen et al.20 In contrast to the previous studies, Olsen et al. observe that the ion streamlines of an ED-MN detach outward with respect to the diverging magnetic field. Using a second stage to energize the ion population in addition to the electron population, they find that the ion beam detaches inward : a phonomenon they attribute to anomalous electron cross-field transport driven by the increased ion momentum. Many of these experiments have been performed at fixed or limited operational parameters. Notably absent from the literature is an experimental campaign addressing the influence of the applied magnetic field strength on both the ion dynamics of the magnetic nozzle and the density and potential structures in the flowing plasma. Therefore, the central goal of this study is to answer the question: How does the magnetic field strength influence the ion beam collimation in a MN exhaust? In the course of answering this question, we will remain focused on the relationship between the density and potential profiles and the ion beam divergence. Our experimental setup, diagnostics, and parameter space are described in Sec. II. Measurements of the ion streamlines, ion beam energy, and transverse density and plasma potential profiles are presented in Sec. III. Finally, in Sec. IV we discuss the implications of the observed plasma dynamics on the performance of MNs for plasma propulsion.

II.

Experimental Setup

In this section we provide a brief description of the plasma source and magnetic nozzle. We limit this description to the aspects of the experiment relevant to the present study, and encourage the reader to see Ref. [21] for a more detailed explanation of the system. II.A.

Plasma Source, Vacuum Chamber, and Magnetic Nozzle

The plasma source (Fig. 1) consists of a 7.5 cm inner diameter, 30.5-cm-long tube of borosilicate glass mounted concentrically inside two electromagnetic coils. A grounded copper mesh is wrapped around the outside of the glass tube to minimize stray electromagnetic radiation. Argon gas is injected through the center of a Macor backplate. The backplate is adjustable within a range of approximately 10 cm to change the effective length of the plasma source. All of the data presented in this paper was taken at an Argon mass flow rate of 0.5 mg · s−1 with the backplate at a fixed location 18.5 cm upstream from the MN throat. A two-turn spiral antenna positioned next to the backplate is used to excite the plasma using 13.56 MHz RF radiation. The antenna is connected to an L-type matching network via two rails that allow the antenna location to change along with the backplate. The power supply allows for approximately 750 W of delivered 2 of 13 American Institute of Aeronautics and Astronautics

power, however, the power is fixed at 500 W for the present study. The plasma source is mounted in the EPPDyLs large dielectric pulsed propulsion (LDPP) vacuum chamber. The cylindrical chamber is made out of fiberglass and measures 8 ft in diameter and 25 ft in length. Two roughing pumps, a roots blower, and a 48 inch diameter CVC diffusion pump are used to achieve a base pressure around 40 µTorr. We then use liquid nitrogen baffles to further reduce the background pressure, with measured values as low as 8 µTorr. The steady-state pressure with the plasma source operating at 0.5 mg · s−1 Ar was measured to be around 20 µTorr. A MN is formed by passing DC current through the electromagnets. The strength of the magnetic field, B, is controlled with the applied current, IB , such that B[G] ≈ 21IB [A]. The magnets are capable of operating at 50 A of current, however, we limited our measurements to applied currents less than 20 A due to the presence of an antenna-plasma coupling mode transition near this value. Although it has not been confirmed using magnetic probe measurements, Langmuir probe measurements of the density and electron temperature suggest that we are operating in a low-field helicon mode prior to this transition.22 The effective radius of each coil is located at rc = 7.51 cm, which roughly corresponds to the physical radius of the center of the coil windings. Although they were designed to move freely with respect to each other and the plasma source, in this study the magnets are rigidly connected with an axial separation of 4.5 cm between their effective centers. The downstream face of the magnet pair is mounted such that it is flush with the opening of the glass tube. II.B.

Plasma Diagnostics

Four different diagnostics are used throughout this study: an RF-compensated Langmuir probe, an emissive probe, a Faraday probe, and a retarding potential analyzer. A rotating translation stage allows accurate positioning of the probes over a large domain within the r-z plane. The locations at which we take measurements with each probe are depicted in Fig. 2. Here, “×”s and “+”s denote single-point measurements taken in the plume and radially along the MN throat plane, respectively, while solid lines indicate a measurement sweep along the line with a sampling rate determined by the specific probe. The RF-compensated Langmuir probe (RF-LP) is based on the design by Sudit and Chen23 that uses chip inductors and an auxiliary electrode to increase the probe impedance at the RF frequency and its first harmonic. Current-voltage (IV) traces are obtained for single-point measurements, which yield the plasma density and electron temperature. The ion saturation current is collected during the measurement sweeps to determine the plasma density profile along each arc. We find that the conventional methods for determining the plasma potential from the Langmuir probe IV curve exhibit significant errors; therefore, we measure the plasma potential using an emissive probe (EP).24 The emissive probe is made by friction fitting a tungsten filament between two copper wires separated from each other by the wall of a two-bore alumina tube.25 We heat the filament to electron emission using halfwave rectified AC current and measure the floating potential of the probe. Using the floating point method, the plasma potential is obtained from the rectified portion of the floating potential signal.26 Ion current densities in the exhaust plume are obtained using measurement sweeps of a Faraday probe (FP).27 The current collecting surface of the probe is mounted flush within a guard-ring, which is used to minimize the error due to sheath effects on the collection area. Current is collected through a shunt resistor for probe and guard-ring bias voltages of −30 V. Finally, a four-grid retarding potential analyzer (RPA)28 is used to measure the ion beam energy along the centerline of the plume. The IV trace of the RPA is obtained by applying a voltage sweep between 0 V and 80 V to the ion repelling grid and measuring the current to the grounded ion collection disc. A DC bias of -15 V is applied to each of two electron repelling grids to minimize the effect of secondary electrons. The ion energy distribution function is obtained from the derivative of the IV-curve, which can be combined with the plasma potential measurement to indicate the presence and energy of the ion beam.

III.

Magnetic Nozzle Plasma Dynamics

While theoretical and computational models have taught us much about the plasma dynamics of a magnetic nozzle, there is a limited database of experimental data to validate these models, test their predictions, and drive new insight. To fill this knowledge gap, we focus in this section on the prediction that the plasma dynamics of a MN are independent of the applied magnetic field strength, except for at very low29 or very


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Figure 2. Measurement locations for the (a) RF-compensated Langmuir probe, (b) emissive probe, (c) Faraday probe, and (d) retarding potential analyzer. “×”s and “+”s denote single-point measurements along the centerline and throat plane, respectively. Measurement sweeps are shown as red lines while the gray dashed lines show the magnetic flux surfaces. The purple dashed line is the magnetic flux surface that intersects the glass chamber of the plasma source, which is shown as a solid blue line.

high magnetic fields.8 Specifically, we will use experimental data to answer the question posed at the outset: how does the magnetic field strength influence the ion beam collimation in a MN exhaust? III.A.

Ion Dynamics

The performance of a MN depends strongly on the divergence of the exhaust beam, with a collimated beam required to minimize divergence losses. The beam divergence is intimately related to the ion dynamics in the plume because the ions carry most of the momentum by virtue of their large mass compared to the electron mass. Here, we characterize the influence of the applied magnetic field strength on the ion dynamics by measuring the ion streamlines and beam energy at different plume locations. Following the method of Olsen et al.,20 we can estimate the location of the ion streamlines by measuring the ion current density profile using the swept Faraday probe (see Fig. 2(c)). The streamlines may be found by solving for the location, θts = θχ , at which the area integral of the current density is a certain fraction, χ, of the total current, or Z θmax Z θχ χ= ji sin θts dθts ji sin θts dθts . (1) 0

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Here, ji is the ion current density measured using the Faraday probe, θts is the angle that the translation stage makes with the MN centerline and θmax = 95◦ is the maximum angle allowed by the translation stage


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Figure 3. Location of the ion streamlines containing (red) 30%, (green) 60%, and (blue) 90% of the total ion current for various (a)-(f ) applied field strengths. Dashed lines depict the magnetic flux surfaces, which are bounded by the magnetic flux surface that intersects the glass chamber of the plasma source.

for our system. The locations of the ion streamlines for different magnet currents are shown in Figs. 3(a)-(f) along with the magnetic flux surfaces (dashed lines) of the nozzle. Each figure contains three different data sets that represent the streamlines containing 30%, 60%, and 90% of the total ion current. Note that the error bar is indicative of the signal-to-noise ratio for each measurement. The signal-to-noise ratio scales with the density of the plasma, which is generally lower towards the plasma periphery and for small IB (due to decreased confinement in the plasma source).21 We immediately notice that the ion streamlines become more collimated as the applied magnetic field strength increases. Qualitatively, the streamlines for IB = 5.0 diverge faster than the magnetic field lines, while the streamline and field line divergence for IB = 7.5 A is comparable. As IB increases beyond 10 A the streamlines become less divergent that the field lines. For IB ≥ 15.0, this is accompanied by significant crossfield motion of the ions and linearization of the streamline in the far-field. We note that linear streamlines in the far-field were observed by Olsen et al.,20 and have been used as a criteria for ion detachment from the MN. Clearly a transition in the dynamics of the flow through the MN is occurring near IB = 5.0 A. We define this transition in terms of the collimation of the plume, which we will quantify momentarily. Specifically, we will refer to flow that diverges more than the applied magnetic field as under-collimated. Alternatively, flow that diverges less than the magnetic field will be regarded as collimated. The plume divergence may be quantified using the divergence half-angle of the ion beam, θdiv , such that Z θmax Z θmax ji cos θts sin θts dθts ji sin θts dθts . (2) cos θdiv = 0

0

Quantifying the magnetic field divergence is more obscure, however, because different magnetic field lines diverge at different rates. Furthermore, the ion current density is largest towards r = 0, which means that 5 of 13 American Institute of Aeronautics and Astronautics

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Figure 4. Exhaust plume divergence half-angle, θdiv , versus magnet current, IB , demonstrating the transition from an under-collimated flow to a collimated flow. The grey region represents the range of values corresponding to the magnetic field divergence, θB , for different IB . The dashed line is the approximate asymptotic value of the divergence ∗ half-angle, θdiv .

the plume divergence is weighted towards the MN centerline. To account for this, we calculate the magnetic field divergence, θB , as the hypothetical plume divergence if the ion current density profile remained constant (i.e. self-similar) throughout the plume, which would be true if the ion streamlines followed the magnetic field lines. We use RF-LP measurements to estimate the ion current density profile at the MN throat, j0 . We then substitute ji → j0 cos (∆θp ) and θdiv → θB into Eq. (2) to solve for θB , where we have defined ∆θp as the angle that the magnetic field vector makes with the probe. The results of this quantification may be seen in Fig. 4 where we plot the divergence half-angle as a function of the magnet current. Here, we use the furthest downstream FP sweep to determine θdiv . The shaded region represents the range of θB -values calculated using the RF-LP measurements at the MN throat for the different IB . In accordance with our qualitative analysis of Fig. 3, we observe a transition from an under-collimated flow to a collimated flow near IB = 5.0 A. As IB increases the flow becomes collimated as the beam diverges less than the magnetic field. Eventually, we observe the divergence half-angle to approach ∗ θB . an asymptote, θdiv We note that it is predicted that the plume divergence should increases at high magnetic fields due to the effect of the Lorentz force on the ion trajectories. The magnetic field strength at√which the Lorentz force has an appreciable effect on the ion motion can be approximated by B0∗ [kG] ≈ 0.1 µi Tev ,8 where µi is the ion atomic number. For our plasma (µi = 40, TeV ≈ 7 eV), the required magnetic field is B0∗ ≈ 1.7 kG. The maximum operating magnetic field of our magnets is Bmax ≈ 1.0 kG. Furthermore, for the operating parameters focused on in this study, the helicon mode is only stable for Bw ≤ 0.45 kG.21 We are therefore restricted in our experiments to operate under the threshold magnetic field at which ion magnetization becomes important. We further characterize the ion dynamics by measuring the energy of the ion beam at different locations along the MN centerline using the RPA (see Fig. 2(d)). IV curves are obtained by sweeping the ion repeller grid voltage and measuring the current at the ion collector.28 The ion energy distribution function (IEDF) follows from the derivative of this curve, 1 dI . (3) fi (V ) ∝ √ V dV The IEDF is expressed in Eq. (3) in terms of the ion repeller voltage, V , which is related to the ion velocity, vi , and the local plasma potential, Vp , through e (V − Vp ) =

1 mi vi2 . 2

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Plots of the axial evolution of the IEDF in the MN plume for different magnet currents are shown in Figs. 5(a)-(f) as gray lines. The black data points correspond to the plasma potential measurements obtained with the EP. The separation between the IEDF peak and the plasma potential is a clear indication 6 of 13 American Institute of Aeronautics and Astronautics

Potential HVL

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Potential HVL

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ã ã ãã ããã ãã ã ãã ãã ã ã ããã ããã ã ãã ãã ã ã ã ã ã ãã ã ããã ã ã ã ã ãã ã ã ã ã ã ã ã ã ã ã ã ããã ã ã ã ã ã ã ã ããã ã ãã ã ãã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ã ã ã ã ã ã ã ã ã ã ãã ãã ã ãã ã ã ã ã ãã ã ã ã ãã ã ããã ãã ã ã ã ããã ããã ã ã ããã ãã ãã ãã ãã ããã ãããã ãããã ãããã ããããã ããããã ãããã ããããããã ããããã ããããã ããããã ãããã ããã ããã ããã ãããã ããã ããã ããã ãããã ãã ããã ãããã ã ã ã ãããã ãããã ã ã ã ã ã ã ã ã ã ã ã ãããã ãããã ããã ããããã ãããã ããã ãããã ãã ãããã ããã ã ã ãã ãã ãã ããã ã ã ããã ããã ãã ãã ãããããããã ããã ããããã ãããããã ã ããã ãã ããããããã ããããããã ãããã ãã ãããã ãããã ãã ãã ããã ãã ãã ãã ãããããã ãã ã ãã ããã ã ãã ãã ããã ãããã ããããã ãããã ãã ãã ãã ã ããã ãã ãã ã ããã ãããããããããã ããã ããã ãã ã ããã ããããã ããããã ãã ããããã ãããããã ãã ããã ã ãã ã ãã ãããã ããããã ãããããã ãã ãã ããã ãããã ãããã ãã ããã ãããã ã ããã ã ã ã ã ã ã ã ãããããããã ããããã ãã ã ããã ãããã ãã ããããã ããããã ãããããã ãããããã ãããã ãã ã ãããã ãã ã ããããã ããã ã ããã ãããã ããã ãã ãããã ããã ããã ãã ãã ãã ã ãã ããã ãã ã ã ãããããã ãããã ããã ãããã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ã ãã ã ãã ããã ãã ãããã ãããã ããã ãã ãã ã ã ããã ãããã ã ãããã ã ãããã ã ãããã ããã ã ã ãã ããã ããã ã ãã ããããã ãã ã ã ã ãããã ããããã ããããã ãããã ãã ãã ã ãã ãã ã ãã ããã ã ãã ãã ã ãã ã ã ããã ãã ã ãã ã ããã ã ã ã ãã ããã ãããã ã ã ãã ãã ããã ãã ã ãããã ãã ã ã ã ã ã ã ã ã ã ã ãããã ãã ã ã ã ã ã ã ã ã ã ã ã ã ãã ã ãã ã ãã ããã ããã ã ã ã ã ã ã ãã ã ã ã ã ã ãã ããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ããã ããã ãã ã ã ã ã ã ã ã ããã ã ã ã ã ã ãã ã ã ã ãã ã ã ã ã ã ã ãã ã ã ã ã ã ã ããã ã ã ã ã ã ã ã ããã ã ã ãã ã ã ã ã ã ã ã ã ãã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ããã ããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ããã ãã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ã ã ã ã ã ã ã ã ã ã ããã ã ãã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã

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Potential HVL

Potential HVL Potential HVL Potential HVL

ãã

ããã ãã ãã ãããã ã ã ã ã ã ã ãã ã ã ãã ã ã ããã ããã ãã ã ãã ãã ã ã ãã ããã ã ãã ã ã ã ãã ã ãã ã ã ã ã ãã ã ã ãã ã ã ã ã ã ã ã ã ã ã ãã ã ã ã ã ã ã ã ãã ãã ããã ã ãã ã ã ã ãã ã ã ã ã ã ã ã ã ãã ã ã ã ã ã ã ã ã ãã ããã ã ããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ããã ããã ãããã ã ãã ã ã ã ã ããã ã ããã ã ã ã ã ã ã ãã ãã ã ãã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ã ãã ããã ãããã ã ã ã ããã ã ã ã ãã ã ã ã ããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ããã ããã ã ã ã ã ã ã ãã ã ã ã ã ã ããã ã ã ã ã ã ã ã ãã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ããã ã ãã ã ã ã ã ãã ã ã ã ã ã ã ã ã ã ã ã ããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ã ã ã ã ã ããã ãããã ãããã ã ã ãã ã ãã ã ãã ããã ã ã ã ã ã ã ããã ã ã ã ã ã ã ã ã ãã ã ããã ã ã ã ãã ãã ã ã ã ã ã ã ã ã ã ã ã ã ããã ããã ãããã ãããã ãã ã ãã ãã ã ãã ã ãã ãã ãããã ã ã ã ããã ããã ããã ããã ãããã ãããã ããããã ããããã ãããã ã ããã ãã ãã ãã ãã ã ã ããã ãããã ãããã ãããã ãããã ãããã ããã ãã ã ãã ããã ããã ãã ãã ãããã ãããã ããããã ããããã ããã ãããã ãããããã ããããã ã ã ã ãããã ããããã ãããããã ãããããã ãã ãããã ãããã ãããã ããããã ããããã ãããã ããã ã ãã ãã ãããã ããã ãããã ãã ãã ãã ã ãã ãã ãã ãã ã ã ã ã ã ã ã ããããã ãã ã ã ãã ã ã ã ã ã ã ã ãã ãã ããã ãã ãããã ããã ããã ã ã ã ãã ã ãã ã ãã ããã ããã ããã ãããã ãããã ããããã ãããããã ãããããã ãããããã ãããããããããã ãã ãããã ãã ããããã ãã ãããã ãã ãããã ãããã ããããã ããããã ãã ããã ãã ããã ãããã ãããã ãããã ããããã ãããã ãã ãã ãããã ãã ãã ããã ãããã ããããã ããããããã ãããããã ããããããããã ãã ã ã ãã ãã ãããã ãããã ãããã ããã ãã ãããã ããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ã ãã ãã ã ã ãã ãã ããããã ããã ãã ãããããã ãããããã ãããã ãã ãã ãããã ããããã ãããããã ãã ããã ããããããã ããããããã ããã ãã ããããã ã ã ãã ãã ãã ãã ããã ã ã ãã ããã ãã ã ãã ãããããã ããã ãã ãããã ãããã ã ãã ãã ããã ãããã ããã ã ãã ããã ã ãã ãããããã ã ã ã ã ãã ã ã ã ã ããã ãããã ãããããã ãããããã ãããããã ãã ãããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ãããã ãããããã ããããã ããããããã ããã ãããã ããããã ã ãã ãããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãããã ããã ãããã ãã ã ã ã ã ã ã ã ã ããã ã ã ã ã ã ã ã ããããã ã ãã ãã ãããã ãã ãããã ã ã ã ã ãã ã ã ã ã ã ã ã ã ã ãã

70 60 HaL IB =5.0A 50 40 30 æ æ æ æ æ æ 20 æ æ æ æ æ æ æ æ 10 æ æ æ æ 0 0 10 20 30 40 Axial Distance, z HcmL 70 60 HcL IB =10.0A 50 40 æ æ æ æ æ æ 30 æ æ æ æ æ æ 20 æ æ æ æ æ æ 10 0 0 10 20 30 40 Axial Distance, z HcmL 70 60 HeL IB =15.0A 50 æ æ æ æ 40 æ æ æ æ æ æ 30 æ æ æ æ æ æ æ æ 20 10 0 0 10 20 30 40 Axial Distance, z HcmL

50

ãã

ãã ããã

ã ã ãã ãã ã

ãã

ã

ããã ãã

ã ãã ãã ã ã ã ãã ã ãã ãã ã ã ã ã ãã ã ã ã ã ã ã ãã ããã ã ã ããã ã ã ã ãã ãã ã 70 ãã ã ã ã ã ã ãã ã ã ã ã ã ããã ã ã ã ã ãã ã ãã ã ã ãã ã ã ã ãã ãã ã ã ãã ããã ã ã ã ã ã ã ã ã ã ãã ã ããã ã ãã ãã ã ãã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã 60 HbL IB =7.5A ã ã ã ã ã ã ã ãã ãã ã ã ã ã ã ã ã ãã ã ãã ããã ã ã ã ãã ã ã ã ãã ãã ã ãã ã ã ã ã ã ã ã ãã ãã ã ã ãã ããã ã ã ã ãã ãããã ãããã ã ã ã ã ã ã 50 ãããã ããã ãããã ãããããã ããããããã ããããã ããã ã ã ããã ã ã ã ã ãããã ãããã ãããã ããããã ãããããããããã ãããã ããã ã ã ã ã ã ããã ããã ããããã ããããããããããããã ãããããããããããããããããããããããã ããã ããã ããããã ãããããã ãããã ãããã ããã ããããã ããããããããããããã ããã ããã 40 æ ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ãã ããããããã ããããã ããã ãããã ãã ãããã ããããããããããã ããããã ãããã ãã ã ããã ãã ããããã ã ã ãã ã ã ã ã ã ã ã ã ã ã ã ã ãã ã ã ããããã ãããããã ãããããã ããããã ããã ããã ãããã ããããã ããã ã ããããã ãããã ã ãã ãããã ã ãããã ããããããã ããããã ããããã ãã ãã ããã ããããã ããã 30 æ æ æ æ ããã ããã ãã ãã ãã ãããã ãããã ãããã ããããããã ãããã ãããã ããã ãããã ãã ããããã ãã ã ã ã æ ãããã ããããã ãã ãã ãããããã ãããã ã ã ãã ãã ã ãã ã ã ãããã ã ã ããã ãã ãããã ããããã ãããã ããããã ããã æ æ ã 20 ã ã ã ã ã ã ã ãã ã ã ã ã ãã ããã ãããããã ãããã ãããã æ æ æ ãã æ ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã æ ã ã ã ã ã ã ã ã ããã ããããã ããããããã ã ã æ ãã ã ã ã ã ã ã ã ã ã ã ã ã ã ã æ ãã æ ã æ ã æ ã æ ã æ ãã æ 10 ã ã ã ã ã ã ã ã ã ã ããã æ æ ã ã ã ã ã ã ã ã ãããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã 0 0 10 20 30 40 50 Axial Distance, z HcmL 70 60 HdL IB =12.5A 50 40 æ æ æ æ æ æ 30 æ æ æ æ æ æ æ æ æ æ 20 æ æ æ æ æ æ æ 10 0 0 10 20 30 40 50 Axial Distance, z HcmL 70 60 HfL IB =17.5A 50 æ æ æ æ 40 æ æ æ æ æ æ 30 æ æ æ æ æ æ æ æ æ æ æ 20 æ æ 10 0 0 10 20 30 40 50 Axial Distance, z HcmL ã ã ãã ããã ã ã ã ãã ã ããã ã ããã ã ãã ãã ã ãã ã ã ãã ã ã ã ã ã ã ã ãããã ã ã ã ã ã ãã ãã ã ã ããã ãã ã ã ã ãã ãã ãã ã ã ã ã ãã ãã ã ã ã ãã ã ã ã ã ã ã ã ã ãã ã ãã ã ã ã ããã ãã ã ã ã ã ãããã ã ã ã ã ã ã ããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ãã ã ã ã ããã ã ã ãã ããã ãã ã ã ã ã ã ãã ã ã ããã ãã ã ãã ã ã ã ãã ã ã ã ã ã ããã ãããã ãããã ãããã ãããããã ãããã ããããã ããããã ãããã ã ã ãããã ããã ããã ãããã ãããã ãããã ãããã ãããã ãããã ãããã ãããã ã ããã ããã ããã ãããã ããããã ãããã ãããã ãããã ããããã ãããããã ããããã ã ã ãã ã ãããããã ãããã ããã ããã ã ããããã ã ããããããããããããã ãããããã ããã ããã ãã ãããããããã ãããã ã ã ã ã ã ã ã ã ã ãã ããããã ããã ãã ãã ã ã ããããã ããã ããã ããããã ã ããããã ã ããããã ããã ã ããã ããã ã ã ã ã ã ãã ã ã ã ã ã ãã ã ã ã ãã ããã ããã ãã ãã ãããã ããããã ãããã ããã ãã ãããã ãã ãããã ããããã ããã ãã ã ããããã ãã ããã ãããã ããããã ãã ã ã ããã ã ã ãã ãã ãã ãã ããã ããããã ããããããã ã ãã ãã ãããããããã ãããã ã ããããã ãããã ãã ããããã ããã ã ãã ã ããã ãã ããããã ãã ã ãã ããããã ããã ããããããã ã ããã ããããã ãããããã ãããããã ãã ã ããããã ã ãã ã ããããã ã ãã ãã ãã ãããã ããã ãããã ããããã ã ã ã ãããã ãããã ããããã ã ããã ã ã ãããã ãããããããã ããããã ãã ã ã ã ã ã ã ã ããã ãã ããã ã ãã ã ãã ããã ãããã ãã ãã ã ããããã ãã ã ãã ã ãã ãããã ããããã ãããã ã ã ã ããã ã ãã ãããã ã ããã ããããã ãããã ã ãããã ã ãã ãã ããã ãããã ãã ã ã ã ã ã ã ã ã ã ããã ã ã ããã ã ã ãã ã ã ããã ããã ã ããã ããã ã ã ã ã ã ã ããã ã ã ã ãã ã ã ã ã ã ããã ããããã ãããã ã ã ã ã ã ã ããã ããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ãã ã ã ã ã ã ã ã ã ã ãããã ã ã ããã ããã ã ã ã ã ã ãã ã ã ãã ããã ãã ã ã ã ã ã ã ã ã ã ãã ã ã ã ã ã ãã ã ã ãã ã ãã ã ã ã ããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ã ãã ãã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ããã ã ããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãããã ãã ãããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ã ã ã ã ã ã ã ã ãã ã

ãã ã ã ããã ãã ãã ãã ã ã ã ãã ã ã ã ãã ãã ããã ã ã ãã ã ã ã ã ããã ãã ã ã ããã ãã ã ã ã ããã ã ã ã ã ã ã ã ã ãã ãã ãã ã ã ãã ããã ãã ã ã ã ã ã ã ã ã ããã ã ã ãã ã ããã ã ã ãã ã ããã ã ã ãããã ã ã ããã ããã ããã ããã ããã ã ã ããã ãããã ãããã ã ã ããã ããã ãããã ãããã ãããã ããããã ããããã ããããããã ããããã ããããã ã ããããã ãããã ããã ãããã ãããã ããã ããããã ãããã ãããã ãããã ãããããã ããã ããã ãããã ããã ãã ã ã ã ã ããããã ãã ã ã ããããã ããã ãããã ããããããããããããããã ãããã ããããã ããããããã ãããããããã ãããã ããããã ããã ãããã ãããã ãããã ãã ãã ããããã ãããã ãããã ãã ã ã ããããã ãã ã ã ããããããããããããããã ãã ãããã ãã ãã ãããããã ãã ã ã ãã ãã ããããã ã ã ã ã ã ã ã ã ã ãã ãã ãããããããã ã ãã ããã ããããããã ãããã ããã ã ããã ããã ãã ããããã ãã ã ããã ããã ãããã ãããã ã ãã ãã ããã ããããã ãããã ãããã ãã ãã ã ã ã ããã ãã ãããã ãããããã ããããã ãããã ãããããã ããããã ããã ããã ããã ããããã ãã ãã ãã ã ãã ãããã ããããã ã ã ãã ãã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ãããã ããã ã ããã ãããã ããããã ã ããããã ããããã ããããã ãããããããã ã ã ã ã ã ã ã ã ã ã ã ã ãã ãã ã ãã ã ã ããã ã ããã ããã ããã ã ããããã ããã ãããã ããããããã ããã ããã ããã ã ããã ããã ããã ãã ã ããã ã ã ã ãããã ãããã ãããã ããã ã ããã ããã ãã ããã ãã ããã ã ã ã ã ããã ã ã ã ãã ã ãã ãã ãããã ãã ã ã ã ãã ã ã ã ããã ããã ããã ãã ã ã ãããã ã ã ã ã ãããã ã ã ã ããããã ãããã ãããã ãããã ãã ã ã ã ã ã ã ã ã ã ã ã ã ãã ããã ãã ãã ãã ã ã ã ã ã ã ã ã ã ã ããã ã ããã ãã ã ã ãã ã ããã ãã ãã ã ã ã ãã ã ã ã ã ã ã ã ã ã ã ã ã ããã ãã ã ããã ã ã ã ã ã ã ã ãã ã ããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ã ã ã ã ã ã ã ã ã ãã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ããã ãããããã ããããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãããã ããã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ã ãã ã ã ã ã ã ã ã ã ã

Figure 5. Plasma potential (black) and ion energy distribution function (grey) measurements for various magnetic field strengths (a)-(f ). The units of the IEDF are arbitrary and normalized such that they possess a constant area. The dashed line corresponds to the expected beam voltage for a choked nozzle, with the shaded region representing the experimental uncertainty of this assumption.

that a supersonic ion beam has formed in the exhaust. Furthermore, in contrast to similar devices in the literature,14, 15, 30 a peak near the plasma potential is not typically found in our data, which indicates that the fraction of beam ions to background ions along the centerline remains large. 31 It is often assumed that the flow at the MN throat is choked, which implies that the average ion velocity p at this location is equal to the ion acoustic speed, cs = γe Te /mi . Under this assumption, the initial ion beam voltage is related to the plasma potential at the throat, Vp,t , the electron polytropic index, γe , and the electron temperature at the throat, Te,t in the following manner Vb,t = Vp,t +

γe Te,t , 2

(5)

where Te,t is expressed in terms of electron volts. Furthermore, it is possible to show that the beam voltage of the ions remains constant if the dominant force on the ions is the ambipolar electric field. Therefore, it should be expected that the peak of the IEDF approximately coincides with Vb,t in the downstream region, which is indeed the case. The voltage corresponding to Vb,t is labelled by a dashed line in Fig. 5, with the shaded region representing the error due to measurements of Vp,t and Te,t . It is clear that over large regions of the plume the IEDF peak occurs near the same voltage. Furthermore, this peak voltage is approximately Vb,t , which implies that the sonic condition is met at the MN throat. Interesting deviations from the expected behavior of a supersonic ion beam under the influence of ambipolar electric fields are observed as IB decreases. For IB ≤ 12.5 A, there exists a specific location where the beam begins to decelerate. This location depends on IB , and is pushed further upstream as IB decreases.


Surprisingly, this deceleration is not due to steady state electric fields as is indicated by the fact that the plasma potential continues to decrease in this region. It could be reasonably expected that the observed deceleration is due to collisions of the beam ions with background neutrals. Using conservative estimates for the total collisional cross section and background pressure near the exhaust, σAr ∼ 7 × 10−19 and Pbg ∼ 40 µTorr, the expected mean free path for collisions between the ion beam and background gas is λmfp ∼ 1 m. The beam begins to decelerate well before this location. Furthermore, a gradual deceleration would be expected if the beam were interacting with the background gas. Rather, the effective deceleration length scale is around λd ∼ 6 cm. Therefore, it would seem that the transition from a collimated flow to an under-collimated flow is coincident with anomalous deceleration of the ion beam, whose location approximately coincides with the region of increased beam divergence. III.B.

Electron Cooling, the Polytropic Law, and the Boltzmann Distribution

In the previous section we used the electron polytropic index, γe , to calculate the expected ion beam voltage assuming choked flow at the nozzle throat. We show here the method used to estimate γe and examine the influence of the applied magnetic field strength on its value. Theoretical models that assume isothermal expansion of the electrons are inherently problematic due to the fact that an infinite flux of heat from the PS to MN is required to maintain a constant electron temperature. To remove this inconsistency, a polytropic law32 may be introduced that assumes that the electron temperature decreases according to p e (6) B · ∇ γe = 0. n Here, B is the magnetic field vector and pe = nTe is the electron pressure. It has been shown using theoretical models that the value of the polytropic exponent strongly influences the resulting behavior and performance of the MN.33–35 Unfortunately, there is very little theoretical support for the polytropic law, and calculation of γe from existing experimental data provides little consensus with values ranging from γe ≤ 1 to γe > 5/3.18, 36, 37 RF-LP and EP measurements of the density, electron temperature, and plasma potential along the axis of the MN (see Fig. 2) allow us to accurately estimate γe for our experiment using two different methods. First, the definition of a polytropic process, Eq. (6), implies that d [log10 (n)] = γe − 1. (7) d [log10 (Te )] ψ Thus, measurements of Te and ne at different axial locations along the MN centerline (constant ψ) yield γe from the slope of the log10 (n) vs. log10 (Te ) curve. Furthermore, the electron momentum equation and Eq. (6) may be used to relate the plasma potential and electron temperature γe dVp = . (8) dTe ψ γe − 1 Here, we have made use of the fact that the electron centrifugal and Lorentz forces go to zero as r → 0. Furthermore, the small mass of electrons means that electron inertia is negligible. We plot in Figs. 6(a)-(f) the data used to calculate γe using Eqs. (7) and (8) for two different values of IB . The solid black lines correspond to the least-squares fit to the measured data, while the gray shaded region represents the 95% confidence interval. Also shown is the value of γe and its error calculated from the slope and slope error of the least-squares fit using Eqs. (7) and (8). We conclude from Fig. 6 that the polytropic law for electron cooling accurately describes the relationship between the electron temperature and plasma density over a wide region of the plume. We cannot rule out the possibility, however, that localized areas of increased or decreased cooling exist due to the limited fidelity of the RF-LP measurements. Rather, the polytropic law can be viewed as a global estimate of the electron energy flux from the MN. A deeper theoretical understanding of the electron thermal conductivity in the exhaust plume is required to understand the apparent validity of the polytropic law and explain the measured value of γe .


Log10 @Te HeVLD

Log10 @Te HeVLD

1.0 0.9 HaL IB =10.0A ææ 0.8 æ æ æ 0.7 æ æ æ ææ æ æ æ 0.6 æ æ æ æ 0.5 Γe =1.14 ±0.02 0.4 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5

1.0 0.9 HcL IB =15.0A æ æ æ æ 0.8 ææ 0.7 æ æ æ ææ æ æ æ ææææ æ 0.6 æ 0.5 Γe =1.15 ±0.02 0.4 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 Log10 @nHm-3 LD

60 50 40 30 20 10 0

Plasma Pot., VP HVL

Plasma Pot., VP HVL

Log10 @nHm-3 LD HbL IB =10.0A æ ææ æ ææ æ æ æææ

2

3

æ

æ æ æ

ææ

Γe =1.13 ±0.02

4 5 6 7 8 Electron Temperature, Te HeVL

9

60 50 40 30 20 10 0

HdL IB =15.0A ææ ææ ææ æ æ æ æ æ æ ææ

æ

ææ

æ

æ ææ

Γe =1.13 ±0.02 2

3

4 5 6 7 8 Electron Temperature, Te HeVL

9

Figure 6. Plots demonstrating the linearity between the measured quantities log10 (n) and log10 (Te ) for (a) IB = 10 A, (c) IB = 15 A, and (e) IB = 20 A, and the measured quantities Vp and Te for (b) IB = 10 A, (d) IB = 15 A, and (f ) IB = 20 A. The solid black lines and shaded regions show the resulting least-squares fit to the data along with the 95% confidence interval.

1.4

Polytropic Index, Γe

æ Log10 HnL vs. Log10 HTe L ç V p vs. Te

1.3

1.2

æ ç

æ ç

æ ç

æ ç

æ ç

æ ç

æ ç

1.1

1.0

5

10 15 Magnet Current, IB HAL

20

Figure 7. Magnetic field strength dependence of the electron polytropic index measured using the two methods indicated in the figure legend.

A comparison between the values of γe calculated using Eqs. (7) and (8) for different IB is shown in Fig. 7. The two methods of calculating γe exhibit remarkable agreement, with the largest discrepancy between measurements of about 3%, which is well within the measurement errors. The observed changes in γe with IB fall within the measurement errors of each technique, thus we cannot conclude if the applied field strength affects γe . However, we note that γe varies by only 6% for the data shown in Fig. 7. This variation is small compared to the range of values between isothermal (γe = 1) and isentropic (γe = 5/3) processes. Finally, it has been reported38 that electron expansion in a MN follows the Boltzmann distribution, or n = n0 exp (−∆Vp /Te ). This distribution results from the electron momentum equation projected along the magnetic field in the limit of negligible electron inertia and isothermal electrons. It is clear from Fig. 6 that the isothermal assumption is not technically valid in the exhaust plume. However, Eq. (8) implies that changes in Vp are much greater than changes in Te , meaning the Boltzmann distribution between n and Vp 9 of 13 American Institute of Aeronautics and Astronautics

will hold to within the experimental uncertainty of most diagnostics. III.C.

Exhaust Structure and Plasma Confinement

We observed in Sec. III.A a transition from an under-collimated to a collimated exhaust plume as the applied magnetic field strength increased above a certain value. Here, we use RF-LP and EP measurements to obtain the transverse density and plasma potential profiles at various locations in the exhaust to gain insight on the physical mechanism behind the collimation transition. Throughout our analysis we will refer to the probe location that corresponds to θts as z0 . Density profiles are obtained from the ion saturation current, Isat , of the RF-LP using a −27 V bias. An estimation for the electron temperature is required to determine the density from Isat . For this, we use the polytropic law (Eq. (6)) to relate the local electron temperature to the local density at the measurement location using measurements of the electron temperature and density at the MN throat, the value of γe determined from the axial RF-LP data (see Sec. III.B), and the known magnetic field topology. Therefore, this technique allows us to estimate the density anywhere in the plume using only Isat . The normalized plasma density profile for a centerline probe location of z0 = 30 cm can be seen for different applied magnetic field strengths in Fig. 8. Here, the density profile is normalized by the density at the MN centerline, ncl . Also shown is a dashed line that represents the location of the predicted plasmavacuum flux surface, θp , which coincides with the magnetic flux surface that intersects the glass tube at the plasma source exit. The plasma density is clearly peaked towards the MN centerline in all cases. The density profile is most narrow at high magnetic fields and seems to approach an asymptotic value. The density profile broadens significantly as the magnetic field is decreased, which agrees with the observation that the divergence of the ion streamlines increases as the magnetic fields strength decreases. Because the density and temperature profiles at the MN throat were measured to be similar regardless of the magnet current, we conclude that the observed density profile broadening is due to increased cross-field diffusion of the downstream plasma. Classical diffusion cannot be invoked to explain the broadening due to the low plasma densities in this region, thus we suspect that some form of anomalous transport is present. The influence of IB on the plasma potential profile is shown is Figs. 9(a)-(f) for six different z0 . The normalized plasma potential is given by φ = (Vp − Vp,0 ) /T0 , where Vp,0 and T0 are the plasma potential and electron temperature at the MN throat, respectively. Furthermore, φcl is the value of φ along the MN centerline. Again, the dashed line in each figure coincides with the predicted plasma boundary. Similar to the density profiles, it is observed that the plasma potential profile is peaked towards the MN centerline. This is due to the transverse electric field that forms to maintain quasineutrality at the plasma periphery.8 The most interesting feature, and one that is not predicted from existing theoretical models, is the potential structure that forms near the predicted plasma edge (see Fig.9(a)). Here, we observe the

1.0

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I = 5.0A

Density Profile, nn cl

I = 7.5A

0.8

I =10.0A I =12.5A

0.6

I =15.0A I =17.5A

0.4 0.2 0.0

-20

0 20 40 60 Translation Stage Angle, Θts HdegL

80

Figure 8. Normalized transverse plasma density profile for various magnet currents, IB . The dashed line represents the predicted plasma-vacuum boundary, which coincides with the magnetic flux surface that intersects the glass tube at the plasma source exit.


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-1.0

I = 5.0A I = 7.5A -1.5 IB =10.0A õ I =12.5A B -2.0 í IB =15.0A ¨ I =17.5A B 0

20

HaL z0 =25cm 40

60

-0.5 -1.0 -1.5 -2.0

0

= 5.0A = 7.5A =10.0A =12.5A =15.0A =17.5A

20

40

60

-1.0

I I I I I I

-1.5 -2.0

0

= 5.0A = 7.5A =10.0A =12.5A =15.0A =17.5A

40

60

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I I I I I I

-1.5 -2.0

= 5.0A = 7.5A =10.0A =12.5A =15.0A =17.5A

20

40

60

80

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0.0


80 HdL z =40cm

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20

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80

¨ ó ó ó óç ó ó ó ó í ¨ ó õ í ó ó ¨ í ó ó ó ¨ õ ¨ í õ ¨ í ¨ ¨ ó ¨ ó í í í õ õ ó õ í ¨ õá ¨ç õç ç óç õç ¨ õç õç ¨ íç í óç íç á á ¨ á óç õ í ¨ ¨ á í á óç á õç õ õç ¨ óç á ¨ óç ¨ ¨ õç á ç ¨ óç á óç á á óá ¨ õç ¨ í íç õ í óç á ç á ç õç óá á í á á ç ç ç íç õ í ¨ óç õ á á õ á óç í õç ¨ ¨ óç á á ¨ õ óç ¨ õ õ á óç õ õç á ¨ õç õ á õç á á óç í õç õ í óç õç ¨ ¨¨ç í óç õ¨ ¨ õ¨ ¨ í óá í á õ¨ õá óç õá ¨ íõ í¨ ç õá óá õ¨ õá á ¨ õ¨ óç íí óç á á ¨ óá õá í ¨ óá íí óá õõç ¨õ á ç ç ááááá ¨ óç á óç í¨ õõõç íõ ¨ 0 óç í¨ óç íí óç ááááááá ¨¨ óç õõõõóç ç óá óá í¨ óç í¨ áááá çç óç í¨ õ ç í¨ áç áááááá óóç ç ç õõõõ óóç í¨ í¨ í¨ áááááá ç ç ¨¨ õõ óóóç ç íí ó ç á á ç ó ç á ó õõõ óóóç á ááááááááá í¨ ç í¨ ç çç íí ¨¨ áááááááááá ç çç ¨¨¨õõõõ óóóóóóç íí ç á ááááááááá ç ç íí¨¨ õõ çç óóóóç áááááá ç ç çç íí¨ õõõ óóóóç ç ç ááááááááááááááááááááááááá í¨ ç ç ç ç ç õõ óóóó ç í¨ ç ç í¨ áááááá ç ç çç ç í¨ õ ó õ ó ó ¨ í¨ õ ¨ ó ç ííí¨¨ õõõ çç áááááááá óóóóóó ç çç ç ç ç íí¨¨¨ õõõ ç áááá ç ç óóóóóóó ç ç ç ç ç ç ííí¨ ç õõõ ç ó áááááááááááááááááááá ç ç ç ç ç ç ç ç ó ó ç ç ç ç ç ç ¨ í¨ ó ç ç ç ç ó ¨ ç ç ç ç ç ç ç ç ¨¨¨õõõ ç óóóóóóó ç áááááááááááááááá íííí ç õ ó ¨ õ ç ó íí¨ õ óóóóó õõõõ ç íí¨ ç ç ¨¨¨ ç ç í¨ ç ó õ ç ó ç í¨ ç ç õ õ ç ó ç ç ç íííí¨¨¨¨ õõõõõ óóóóóóóóóóóóóóóóóóóóó ç ç ç ç ç ç ç ç ç ç ç á íííí¨¨¨¨¨ õõõõ óóóóóóóóóó ç ç ç ç ç ç ç ç ç óóóóóóóóóó ç íííí¨¨¨¨ õõõ B ç ç ç ç ç ç óó íííí¨¨¨ õõõõõ ííí¨¨¨¨¨¨¨õ ç óóó óó õ õõõõõõ ííííí ¨õ ¨õ óóóóóóóóóó ¨¨ B õ íííí¨ õ óóóóóóó íííí¨¨¨¨¨¨¨õõõõ ó ó ó ó ó ó óó ííííí¨¨¨¨¨ õõõõõ¨ õ¨ õ¨ õ¨ B õ¨ õ¨ õ¨ õ¨ õ¨ ¨¨¨¨¨¨¨ õ¨ õ¨ ííí ¨¨¨¨ õ¨ õ¨ õ¨ õ¨ õ¨ õ¨ õ¨ õ¨ õ¨ õ¨ õ¨ õ¨ õ¨ õõõ ¨¨¨¨¨¨ ííííí õ ííííííííííííííííííííí õõõõõõõõõõ¨¨¨¨¨ B ííííííííííííí õõõõ¨¨¨¨ ííííí õõõ ¨¨¨¨ íííí õõõõ¨¨¨¨¨¨ í ííí õõõõõõ ¨¨¨¨ B ííííí õõõõ ¨¨ ííííí ¨ ííííí B ííííí ííí

0.0

I = 5.0A I = 7.5A I =10.0A I =12.5A -2.0 IB =15.0A ¨ I =17.5A B -1.5

0.0

HcL z =35cm

I I I I I I

-1.0

0

Potential Profile, Φ-Φcl


0.0

HbL z =30cm

-0.5

80

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-0.5

¨ ¨ õ ¨ ¨ õ ¨ õ ¨ ¨ á á ¨ õ õ á í ¨ ¨ ¨ á í ¨ ¨ í õ í õ õ í ç õ õ í í í í á á á ç á ¨ õ ç í í á ¨ õ óç á ¨ á í í á õ õ õ ¨ á óç óç óç ¨ í óç í á ¨ õ õ á óá óá ¨ õ á ¨ óç í õ óç á õ óç óç á ¨ õ í óá õ í í í á óç õ í ¨ ¨ óç óç óç í ¨ á ¨ ¨ óç óç á õ óç óç ¨ óç í óç ¨ á ¨ç á í í óç ¨ õ í óç õ õ ¨ í ¨ ¨ í õ á í á á óç ¨ óç á õ óç í ¨ á ¨ óç á óç õ ¨ óç á õ á ¨ óç õ á í õ óç á á óç óç á í ¨ á õ á óç í í íç õ ¨ óç ç í ó ¨ á ¨ õ í á ç õ ó á í í í ¨ õ á õ á õ õ óç õ í áç íç íç óç õ ¨ íç í áááç õ ç ¨ ¨ óç óç í ¨ óç õ ááááá í óç í õ ó óç ¨ í õ õ õ¨ óõ í ¨ õ ááááá óç ¨¨ç ¨ç ç í ç ó ç ó ¨ 0 ç í õ í ç áááá ó¨ õó ç í çá õ õí ó¨ õ ¨ áááá ó ç ç í ó çç ¨¨ç ó¨ í¨ á õí í õ ¨¨¨¨ç óó õí ¨ç ççááááá ó õí õó á á óóó¨¨¨ç ç á í á õí çç áááá óóó¨¨ç õí õí ç ç ááááá õí óó¨¨¨ç õí ç õí ç çç áááááá óóó¨¨¨ç ç õí õí áááááááá ç çç óó¨¨ ç õí õí ááááá ç çç óó¨¨¨ ç õí áááááá çç õí óóó¨¨ ç õí ¨¨¨¨¨ ç õí ç áááááá ç õí óóó¨¨¨¨ ç ç ¨ ¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨ ¨¨¨¨¨¨¨¨¨ ç õõ ç ç ç áááááááá óóó¨¨ ç íí ¨¨¨¨¨¨¨ ¨ ¨ õí ç õí ç á ááááááá¨¨¨á ç õõ ¨¨¨¨ ó ç ç ¨ ç ¨ ó ¨ ç ó íí á õí óóó¨¨¨ ç ¨¨¨¨¨¨¨ ç áááááááááá ç õí ¨ ç õõ ç ¨ ç ó ç ¨¨¨ õõ óóó¨ íí ç ç ç ç ó¨ íá íá ç íá íá á íí íá õõ íá íá ó¨ íá ¨¨¨¨ íá íá íá íá ç ç ó¨¨¨¨¨ ç íá íá íá íá íá íá ç ç íç íá á á íí íç íç íç íííííííííí õí íá íç á á íç ç íç íç ç ç á ç ç íç ç ç ¨ç õí ç íç íç ç á ç ç ç ç ç ç á í ¨ ç ç ç õõ óóó ç ç ç í ç í ç ç ç ó ç ç ç ç ç ç ç ç ç á ç ç ç í ¨ ó ¨ á ç ç á ¨ ç á ç ó í ç õõõõ óó¨ áááç íííí ¨¨¨¨ áç íí í áç ç B áá ¨¨¨ í íííõõ óó ç óóó¨¨ á í á ç ¨ áç íí áç õõõõõõõõõ ó¨ó¨ õõõ áááí ¨¨¨ ííí õõõ õóõõõõõõõç ç ó¨ õõõõ óó¨ ó¨ ó¨¨¨¨ õõ ííí íí õõõõõõ ííí ç óõõõõõõõõó óóóóóóóóóõõõõ õõç óõ ç B ç íííõõõõ óóóóóóó íí õõõ ç óóóóó óóóóóóóóóóóóóóóóóóóóóóõ í óóõ íííõõõõ ç õõ í óó óóóóóó óó óõ óõ ç ó óõ íííõõõõõ óóóóóóóó ç õõ óõ íóóóó óõ B õ íííí õõ óõ í õ óõ í õõ óõ íííííõõõõõõí í óõ í õ õ õ õ õõõõõ óõ ííííííí óõ B óõ óõ õ óó í

0.0



0.0

HfL z =50cm

-0.5 -1.0

I = 5.0A I = 7.5A I =10.0A I =12.5A I =15.0A IB =17.5A

-1.5 -2.0

0


Figure 9. Normalized transverse plasma potential profiles for various magnet currents, IB , and (a)-(f ) centerline probe locations, z0 . The dashed vertical line represents the predicted plasma-vacuum boundary, which coincides with the magnetic flux surface that intersects the glass tube at the plasma source exit.

formation of a potential well near θp . The potential well implies that the transverse electric field switches direction, from outward to inward, near the edge of the plasma. A similar potential structure was predicted using PIC simulations39 and verified experimentally40 using a helicon plasma expanding along magnetic field lines into a grounded diffusion chambers. In both of these studies the plasma density was observed to accumulate near the potential well, which is in contrast to our observations (see Fig. 8). Keeping in mind that the magnitude of the normalized transverse electric field scales with the gradient, or slope, of the potential profile, we observe the following from Figs. 9(a)-(f): (1) The magnitude of the inward electric field generally increases with increasing IB ; (2) The location of the maximum inward electric field moves closer to the MN centerline as IB increases; (3) The potential well ultimately disappears at a downstream location that increases with IB ; and (4) the disappearance of the potential well for the undercollimated plume approximately coincides with the location at which the ion streamlines being to diverge faster than the magnetic field lines, according to Fig. 3(a).


IV.

Conclusion

Using detailed measurements of the ion dynamics and plasma density and potential structure within an expanding magnetic nozzle plasma we are able to conclude the following: • There exists a transition from an under-collimated plume mode to a collimated plume mode with increasing applied magnetic field strengths, where an under-collimated plume is defined such that the plasma expands at a greater rate than the applied magnetic field. • The energy of the ion beam corresponds well with the value predicted by ambipolar acceleration through the electric field generated in the wake of an expanding population of hot electrons. Deceleration of the ion beam correlates with the outwards diffusion of the plasma at low magnetic fields, and cannot be explained by electrostatic forces or background effects. • Electron cooling in the expanding plasma approximately follows a polytropic law, with γe independent of the applied magnetic field strength to within experimental uncertainties. • The transverse plasma density profile is peaked towards the MN centerline and asymptotically approaches a similar shape for high magnetic fields. Significant broadening of the density profile occurs at low applied magnetic fields, which implies increased cross-field diffusion. • Increased plume expansion is correlated with the disappearance of an ion-confining potential well near the location of the bounding magnetic flux surface. The magnitude of the inward-directed normalized electric field that results from this potential well increases with the applied magnetic field strength. The above findings are in agreement with the increased MN performance at large applied magnetic fields recently observed by Takahashi et al.13 The physical mechanism that leads to this loss of confinement remains to be seen, however. One possibility is that the plasma detaches outwards with respect to the magnetic field for under-collimated plumes because the residual thermal energy of the electrons can no longer be magnetically confined.29 In a sense, the magnetic nozzle is under expanded. We will test this prediction, along with other theoretical detachment models, in a future study.

Acknowledgments The authors gratefully acknowledge Mark A. Hopkins and Prof. Lyon B. King of the ISP lab at Michigan Tech for allowing us to borrow their retarding potential analyzer.

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