We present laser-induced fluorescence measurements of acceleration zone scaling with discharge voltage (Vd), magnetic field strength (B), and facility background pressure (PBG) in NASA’s 12.5 kW Hall Effect Rocket with Magnetic Shielding. At fixed discharge current, the plasma potential profiles at discharge voltages from 300 to 600 V approximately overlapped in the region with plasma potential less than 300 V; ion acceleration began further upstream at higher Vd because the region with a steep potential gradient was broader. The radial divergence of mean ion velocity vectors in the outer half of the channel and near plume increased with decreasing Vd. At fixed Vd, the acceleration zone was located further upstream at higher B and at higher PBG. Bimodal ion velocity distribution functions (IVDFs) were measured along the channel centerline in the acceleration zone at high discharge voltages; this effect was attributed to time-averaging over movement of the acceleration zone during large-amplitude discharge current oscillations. At lower discharge voltages, the broadening of the IVDFs in the near plume could not be fully explained by ionization within the acceleration region. These results have implications for understanding front pole erosion, which can be an important wear mechanism over the long lifetimes of magnetically shielded thrusters, and they provide baseline data for validating first principles models of cross-field electron transport.

Laser-induced fluorescence (LIF) is a valuable tool for minimally invasive measurements of ion velocities in electric propulsion devices1–5 and other low-pressure plasmas.6–8 As NASA Glenn Research Center (GRC) and Jet Propulsion Laboratory (JPL) develop and qualify the 12.5 kW Hall Effect Rocket with Magnetic Shielding (HERMeS) through a combination of laboratory characterization, wear tests, and computational modeling,9–13 LIF measurements are playing a critical role in furthering physics understanding while informing and validating simulations.

Modeling of the thruster’s performance and operational lifetime is being carried out with JPL’s Hall2De,14–16,13 a code that solves the electron and ion fluid equations in two dimensions on a magnetic field-aligned mesh and includes a particle-in-cell (PIC) module. Like all other existing Hall thruster codes, Hall2De cannot yet predict, based on first-principles, the non-classical (anomalous) electron transport across the magnetic field that has been known to exist in Hall thrusters for decades.17 However, accurate modeling is still possible if laboratory measurements of ion velocities or plasma potentials are used to inform an empirical anomalous collision frequency profile in the simulations.

In the past, fast-scanning emissive probes have been used to make these measurements, but their presence in the Hall thruster channel can cause the discharge to shift axially by as much as 20% of the channel length.18 Inaccuracies of this magnitude in the plasma location would lead to unacceptably large errors in the HERMeS service life predictions. In particular, the rate of front pole erosion, which can be an important wear mechanism in magnetically shielded thrusters over the 10kHr lifetimes enabled by the elimination of significant channel erosion,19–22 is strongly dependent on the axial location of the acceleration zone.16,13 Therefore, non-perturbing LIF measurements are the preferred method for anchoring the models.

Because the location of the acceleration region and other aspects of the plasma potential structure depend strongly on the cross-field electron transport, LIF data can also be used to validate stand-alone models of wave growth and saturation in Hall thruster plasmas,23–26 which may eventually enable the self-consistent treatment of anomalous transport in hydrodynamics codes such as Hall2De.12,27,28 The spatially resolved LIF measurements yield not only the mean velocity but also the entire ion velocity distribution function (IVDF), providing a wealth of detailed information about the plasma kinetics.

This paper describes time-averaged Xe II (singly ionized xenon) IVDF measurements carried out in HERMeS Technology Demonstration Unit 2 (TDU-2).29 This thruster is designed to operate at discharge voltages (Vd) ranging from 300 to 800 V and discharge currents (Id) as low as 6 A and as high as 31.3 A.9 Over a throttling range from Vd=300600V at Id=20.83A, the nominal magnetic field strength (Bnom.) is constant, and the thruster can operate stably (Id,RMS/Id<0.25) at fields up to 25% higher or lower than the nominal setting.10 These features allow for scaling studies in which one parameter such as Vd is varied while holding other settings constant. We will present IVDFs, mean velocity vectors, and derived potentials at locations on and off the channel centerline as a function of Vd, B, and the facility background pressure PBG. A particular focus will be placed on the discharge voltage scaling, which featured a transition between two distinct discharge oscillation regimes.

The tests were carried out in the JPL Owens facility, a 3m diameter, 10m long cryogenically-pumped vacuum chamber lined with graphite. The operating pressure measured by a xenon-calibrated Stabil ion gauge located at the thruster exit plane was below 1.3×105Torr during the collection of all data presented in this paper, except for cases when the pressure was intentionally increased to study facility effects. During these studies, xenon was injected toward the downstream graphite beam dump using an auxiliary gas feed located several meters downstream of the thruster.

The electron source in HERMeS TDU-2 was a center-mounted lanthanum hexaboride (LaB6) hollow cathode.30 The thruster was operated exclusively in a “cathode-tied” configuration,9 with the thruster body electrically connected to cathode common and isolated from the facility ground. An RLC filter was installed between the power supply and thruster.29 Discharge current oscillations were monitored on the anode and cathode lines with Pearson current transformers, and a high-speed differential voltage probe monitored the anode-to-cathode voltage.

LIF takes advantage of the Doppler shift of laser photons in a moving particle’s reference frame to selectively excite ions moving with a certain velocity; by monitoring the intensity of fluorescence emission as the laser is scanned over a range of wavelengths, the IVDF can be measured. We employed a common4 non-resonant LIF scheme driving the near infrared 834.953 nm transition from the Xe II 5p4(3P2)5d2[4]7/2 metastable state to the 5p4(3P2)6p2[3]5/2o state, with fluorescence collected at a vacuum wavelength of 542.06 nm (541.92 nm in air). Two-dimensional (rz) ion velocity measurements were obtained by injecting laser beams along orthogonal lines of sight offset 45° from the axial and radial directions, as illustrated in Fig. 1. A single collection optic was mounted above the thruster, with its line of sight angled approximately 20° from vertical, and the two beams were mechanically chopped at different frequencies in order to simultaneously extract data along both injection directions. The measurement spot remained fixed in space, while the thruster was translated on two-axis motion stages in order to sample the IVDF in different regions of the plasma.

FIG. 1.

Schematic of the LIF measurement setup. Reproduced with permission from Chaplin et al., in 35th International Electric Propulsion Conference (Electric Rocket Propulsion Society, Atlanta, GA, 2017), IEPC-2017-229. Copyright 2017 Electric Rocket Propulsion Society.

FIG. 1.

Schematic of the LIF measurement setup. Reproduced with permission from Chaplin et al., in 35th International Electric Propulsion Conference (Electric Rocket Propulsion Society, Atlanta, GA, 2017), IEPC-2017-229. Copyright 2017 Electric Rocket Propulsion Society.

Close modal

The spatial resolution of the LIF measurements was 2mm, and the time-averaged optical power delivered to the interrogation spot was 89mW for the “East” line of sight and 44mW for the “West” line of sight. The laser intensity implied that the LIF transition was moderately saturated, which enabled a high signal-to-noise ratio (SNR) while only slightly broadening the measured line profiles and negligibly affecting the mean ion velocity calculations.31 During testing, the laser wavelength was continuously monitored using a Toptica WS/7 self-calibrating wavemeter with ±60MHz accuracy.

Daily in situ optical alignment was carried out on a pre-heated thruster using low-power visible lasers. The alignment was reproducible to within better than ±0.2mm,31 and the absolute systematic uncertainty in the axial and radial position references was estimated to be 1mm. During testing, the motion stage positions were tracked by inductive linear encoders built into the stage assembly. More detailed descriptions of the experimental hardware and procedures are provided in Refs. 32 and 31.

The LIF data were analyzed by subtracting off the mean background signal recorded at wavelengths far from the line profile peak in order to get the raw IVDF and then applying single or bi-Maxwellian curve fits of the form,

fv=A1expvv1vTi12+A2expvv2vTi22.
(1)

This fitting function was chosen for convenience and because its 6 free parameters were generally sufficient to fit the data well everywhere except near the plasma edges; it should not be taken to imply that the IVDF always consisted of one or two thermal populations. The mean ion velocity along a given injection direction was calculated from the average of v1 and v2, weighted by the relative areas under the two Maxwellian curves. This fitting procedure was preferred over a direct numerical calculation of the first moment of the IVDF because it was less influenced by random scatter in the data in the line wings and therefore produced smoother plots of mean velocity vs. axial position. However, for IVDFs that could not be fit well by Eq. (1), the first moment was used instead.

No correction was applied for effects other than an actual ion velocity spread that can broaden the line profile measured by LIF.33–35 These were analyzed in Ref. 31 and shown to have a negligible impact on the measured mean velocities and the subsequently derived plasma potentials, which are the primary quantities of interest in this paper. For the narrowest LIF line profiles we will present (with an apparent ion temperature of 0.2eV), Zeeman splitting36 and saturation broadening of the line profile with hyperfine splittings37,31 may have caused the measurements to appreciably overestimate the true width of the IVDF.

The off-axis laser injection geometry shown in Fig. 1 was chosen in order to protect the optics from the main thruster beam and to enable 2D velocity measurements within the discharge channel. The downside of this arrangement was that the full IVDF along the axial direction was not measured. However, the mean axial velocity could be rigorously calculated from the mean velocities along the East and West injection paths by applying a 45° coordinate rotation. The  Appendix provides a proof of the general validity of this procedure and also shows in contrast that it is not necessarily correct to calculate the most probable axial velocity (i.e., the velocity at the peak of the distribution) using an analogous rotation procedure.

Once the axial mean velocity uiz was known at a number of positions along the channel centerline, the plasma potential profile ϕz could be estimated by assuming that ions traveling at the mean velocity were born at the anode potential and applying conservation of energy,

ϕz=ϕanodemiuiz22e.
(2)

The axial electric field was then calculated as Ez=ϕ/z.

Since the creation of new ions from slow neutral gas within the acceleration region lowers the mean ion velocity, Eq. (2) can underestimate the electric field, and it is considered preferable to apply energy conservation to the most probable axial velocity38 or utilize more sophisticated approaches that leverage the details of the measured IVDF in a Boltzmann equation analysis.39,40 We could not apply these techniques without a measurement of the full axial IVDF. However, we show in Sec. III A that these analyses would not be likely to give a better estimate of the potential and electric field than our simple approach because the overlap of the ionization and acceleration regions was not the dominant source of IVDF broadening in HERMeS. Furthermore, our Gaussian curve fits to the main peak of the measured IVDFs tended to minimize the influence that slow ions created by charge exchange or ionization at sub-anode potentials had on the calculated mean velocity.

Figure 2 shows a selection of measured IVDFs at the Vd=300V, Id=20.83A, B=Bnom. operating condition. The data are plotted as dots, and single or bi-Maxwellian fits are overlaid. “East” and “West” label the two laser injection directions (see Fig. 1). Here and throughout the paper, distances are normalized to the discharge channel length Lchannel, z=0 is defined to be the axial location of the anode, and r=0 is defined to be the thruster centerline. The channel exit is at z/Lchannel=1, and the bulk of the ion acceleration at this operating condition occurred between 1<z/Lchannel<1.2.

FIG. 2.

IVDFs measured on the channel centerline at Vd=300V, Id=20.83A, B=Bnom.

FIG. 2.

IVDFs measured on the channel centerline at Vd=300V, Id=20.83A, B=Bnom.

Close modal

Well upstream of the acceleration zone (e.g., z/Lchannel=0.67), the time-averaged IVDF was narrow, and the mean axial ion velocity was slightly negative (directed back toward the anode). The IVDF was broadened within the acceleration region and remained relatively broad further downstream, while also developing a high-velocity “tail” (i.e., an excess of ions on the right-hand side of the distribution), as evident in the data at z/Lchannel=1.65. Since the East and West IVDFs were distorted similarly, it is likely that the suprathermal contribution arose mostly from the axial velocity component. This feature has been previously observed in a number of Hall thruster plasmas,2,41 and Ref. 2 suggested that the very fast population could arise due to ions “surfing” on ion transit time oscillations. Charge exchange processes involving double ions (Xe2+) could not be the main source of fast Xe+ ions near the channel exit because the relevant collision mean free paths were greater than one meter in the HERMeS near plume,42,43 while the observed distortion of the IVDFs occurred over a distance of only a few centimeters.

It is illustrative to compare the measured IVDFs in Fig. 2 with the classical expectation for ions accelerated through a potential drop. Hall2De predicts that the ions become approximately collisionless after their energy increases by 10V. Thus, to calculate the evolution of the axial IVDF along the channel centerline, starting at the point where the potential is 10 V below the anode potential (z¯z/Lchannel1.03), we can use the 1D steady state Boltzmann equation:

vzfiz¯,vzz¯+azz¯fiz¯,vzvz=neKizTefnz¯,vz.
(3)

The ionization term on the right-hand side is proportional to the ionization rate coefficient KizTe and the neutral atom velocity distribution fnz¯,vz. If we neglect this term, the problem can be solved analytically. We assume that the IVDF is a drifting Maxwellian at z¯=1.03 and temporarily define the potential at this position to be V¯=0. Then, a solution of the Boltzmann (Vlasov) equation is

fz¯,vzexpvz2+2eV¯z¯/miv02vT2.
(4)

Note that in this simplified situation, the final IVDF downstream of the acceleration region at potential V¯=ΔV is independent of the shape of the potential profile.

Figure 3(a) plots the IVDF from Eq. (4) upstream and downstream of a 220 V potential drop, using initial conditions appropriate for HERMeS. The IVDF is narrowed dramatically and also develops an excess of ions on the high-velocity side—these effects are known as “kinematic compression” and “kinematic distortion,” respectively.34 

FIG. 3.

(a) Classical IVDF evolution after acceleration through ΔV=220V, from Eq. (4), with ionization neglected. Based on HERMeS LIF measurements and Hall2De simulations, the Maxwellian ion temperature and mean ion velocity upstream of the potential drop were set to 0.5 eV and 4 km/s, respectively. The final IVDF is magnified in the inset so that the kinematic distortion can be more easily seen. (b) Numerical solution for the same initial IVDF, accounting for ionization within the acceleration zone.

FIG. 3.

(a) Classical IVDF evolution after acceleration through ΔV=220V, from Eq. (4), with ionization neglected. Based on HERMeS LIF measurements and Hall2De simulations, the Maxwellian ion temperature and mean ion velocity upstream of the potential drop were set to 0.5 eV and 4 km/s, respectively. The final IVDF is magnified in the inset so that the kinematic distortion can be more easily seen. (b) Numerical solution for the same initial IVDF, accounting for ionization within the acceleration zone.

Close modal

In order to include the contribution from ionization within the acceleration zone, we numerically integrated Eq. (3) in space, starting from the position where the ions become collisionless. The electron temperature, neutral density, and electric field as a function of position were taken from Hall2De simulations of HERMeS, and the neutral velocity distribution was assumed to be fnz¯,vz=δv500m/s. The contribution of the azimuthal E×B drift velocity to Kiz was included. The results are shown in Fig. 3(b). The downstream IVDF has a low-velocity tail arising from the creation of new ions at a range of potentials, but the bulk distribution is still narrow, unlike the measured IVDFs at z¯>1.2 shown in Fig. 2. A similar near plume IVDF has been obtained in a more complex 2D kinetic simulation by other authors [Fig. 17(a) of Ref. 23].

It has previously been suggested2,4 that broadening of axial IVDFs within and downstream of the acceleration zone in Hall thrusters is primarily caused by ionization at sub-anode potentials. However, the above calculation shows that this was not the case for HERMeS. Since ion acceleration occurs further downstream in magnetically shielded thrusters compared to unshielded thrusters, the ionization and acceleration zones may overlap less in shielded thrusters. However, note that the IVDF at z¯=1.03 [the “before accel.” curves in Figs. 3(a) and 3(b), which are similar to the data from z/Lchannel=1.01 shown in Fig. 2] contains very few ions with v<1km/s, so even if the ionization rate in the acceleration region was much higher than we have assumed, the slow ions born from neutral gas would appear as a distinct low-energy ion population, rather than broadening the main distribution.

These results are important for the analysis that follows because they show that the method developed in Ref. 39, in which an equation functionally identical to Eq. (3) is used to derive formulas for the electric field and ionization rate in terms of moments of the measured IVDF, will not produce accurate results for HERMeS. This method assumes that the spatial trends in the IVDF can be fully explained by classical kinetic effects plus ionization, but in HERMeS, there must be other dominant mechanisms (probably velocity-space diffusion due to wave-particle interactions44 and/or time-averaging over discharge oscillations) that broaden the downstream IVDF and produce a high-energy tail. Therefore, instead of using the full axial IVDF to derive the plasma potential and electric field as a function of position, we employ the simple energy conservation approach described in Sec. II B.

Figure 4(a) shows the measured mean axial velocities as a function of position along the channel centerline for the Id=20.83A, Vd=300600V HERMeS operating conditions. The corresponding potentials calculated from Eq. (2) and the electric fields Ez=V/z are plotted in panels (b) and (c), respectively. Time-averaged Hall2De results for the singly ionized xenon mean velocity and the plasma potential at the 300 V and 600 V conditions, obtained from simulations in which the LIF velocity data were used to inform the anomalous collision frequency profile, are overlaid on Figs. 4(a) and 4(b). In the simulations shown, better agreement in the acceleration region was achieved at 300 V than at 600 V. A closer agreement at 600 V likely could have been obtained by further decreasing the anomalous transport in the acceleration region to reproduce the extremely steep measured potential gradient, but the simulations as shown already produced sufficiently accurate thruster performance and erosion predictions, so additional efforts to more closely reproduce the measurements were deemed unnecessary.

FIG. 4.

(a) Mean axial Xe II ion velocities along the channel centerline at Id=20.83A and nominal magnetic field and background pressure. Hall2De predictions for the 300 V and 600 V operating conditions are also shown. (b) Plasma potentials derived from the velocity profiles in (a). (c) Axial electric fields obtained from the derivative of the potential (note the zoomed-in x-axis scale). Modified with permission from Chaplin et al., in 35th International Electric Propulsion Conference (Electric Rocket Propulsion Society, Atlanta, GA, 2017), IEPC-2017-229. Copyright 2017 Electric Rocket Propulsion Society.

FIG. 4.

(a) Mean axial Xe II ion velocities along the channel centerline at Id=20.83A and nominal magnetic field and background pressure. Hall2De predictions for the 300 V and 600 V operating conditions are also shown. (b) Plasma potentials derived from the velocity profiles in (a). (c) Axial electric fields obtained from the derivative of the potential (note the zoomed-in x-axis scale). Modified with permission from Chaplin et al., in 35th International Electric Propulsion Conference (Electric Rocket Propulsion Society, Atlanta, GA, 2017), IEPC-2017-229. Copyright 2017 Electric Rocket Propulsion Society.

Close modal

Figure 4(a) suggests that the acceleration zone moved upstream at higher discharge voltages; however, Fig. 4(b) shows that in fact it is more accurate to state that the acceleration region broadened with increasing Vd. Downstream of the ϕ=300V point, the potential profiles for the four different discharge voltages lay approximately on top of one another, and ion acceleration began further upstream at higher Vd. There may have been a corresponding narrowing of the ionization zone, which begins upstream of the acceleration zone, because the higher electron temperature at high discharge voltages would have enabled most of the anode flow to be ionized in a narrower region. The electric field peak was located furthest downstream at Vd=300V.

Some previous LIF studies45,46,2 on lower power Hall thrusters have likewise observed that the peak axial electric field was located further upstream at higher discharge voltages, while others33,47,48 have found that the acceleration region location was approximately independent of Vd (note that the discharge current was not held fixed as Vd was varied in some of these studies). Whether the acceleration region width narrows,45,46 remains unchanged,33 or broadens with increasing Vd (our result) also appears to be thruster-dependent. Of the studies cited here, only Ref. 48 tested a magnetically shielded thruster.

Figure 5 shows how the plasma potential changed as the magnetic field strength was varied between 75% and 125% of its nominal value. At the Vd=300V operating condition, the peak axial electric field was lower at B=1.25Bnom. than at weaker magnetic fields, but this trend was not observed at higher discharge voltages, implying that the minimum axial conductivity across the radial magnetic field was approximately independent of B. At all discharge voltages except 400 V, the region with a steep potential gradient was located further upstream at higher magnetic field strengths. This trend is consistent with previous studies on smaller unshielded thrusters.46,47,45,49

FIG. 5.

Plasma potential along the channel centerline at Id=20.83A with three different magnetic field strengths. (a) Vd=300V. (b) Vd=400V. (c) Vd=500V. (d) Vd=600V.

FIG. 5.

Plasma potential along the channel centerline at Id=20.83A with three different magnetic field strengths. (a) Vd=300V. (b) Vd=400V. (c) Vd=500V. (d) Vd=600V.

Close modal

In general, we would expect front pole erosion rates to be higher when ion acceleration occurs further downstream, since this enables more ions born at relatively high potentials near the beam edge to reach the pole faces. However, surface layer activation experiments have revealed that the inner pole erosion in HERMeS at Vd=600V increases with B.50 The evidence from Hall2De simulations of this operating condition suggests that higher magnetic strengths lead to higher resistivity between the cathode plume and the thruster plume, which increases the plume potential and leads to higher energies for ions impacting the pole covers.13 This conclusion is consistent with the LIF data in Fig. 5, which show that the plume potential was generally lowest at B=0.75Bnom. and highest at B=1.25Bnom..

Studies of background pressure sensitivity are important to assess the relevance of Hall thruster ground tests (and also modeling anchored by ground test LIF data) for predicting the performance and lifetime of a thruster in flight. IVDFs were measured at the minimum facility background pressure achievable during 20.83A operation at each discharge voltage and also at twice the nominal pressure. Plasma potential profiles derived from these measurements are presented in Fig. 6. At elevated background pressure, the acceleration zone moved upstream slightly, but the shift was less than 5% of the channel length at all discharge voltages. The downstream shift of the acceleration zone at lower background pressures appears to be a universal result for Hall thrusters.51,49,52 The steepness of the potential gradient did not vary significantly with PBG, consistent with the expectation that anomalous collisionality from wave-particle interactions is more important than classical electron-neutral collisions throughout the acceleration region and near plume.28 

FIG. 6.

Plasma potential along the channel centerline at Id=20.83A and B=Bnom. during operation at the nominal facility background pressure and twice the nominal pressure. (a) Vd=300V. (b) Vd=400V. (c) Vd=500V. (d) Vd=600V.

FIG. 6.

Plasma potential along the channel centerline at Id=20.83A and B=Bnom. during operation at the nominal facility background pressure and twice the nominal pressure. (a) Vd=300V. (b) Vd=400V. (c) Vd=500V. (d) Vd=600V.

Close modal

Figure 7 shows a selection of measured IVDFs at the Vd=600V, Id=20.83A, B=Bnom. operating condition. The data in the acceleration region are fundamentally different from those shown in Fig. 2, with two broadly separated peaks in the velocity distribution. Noting that the LIF data were averaged over a long timescale on the order of the lock-in amplifier time constant (300 ms), we can interpret the data as evidence that the plasma potential profile was oscillating in space, so that, for example, the mean axial velocity at z/Lchannel=0.96 was 3km/s at one phase of the oscillation and >15km/s at another phase. The motion of the ionization and acceleration zones during Hall thruster breathing mode oscillations has previously been observed in several time-resolved LIF studies53–56 and also in numerical models.57 

FIG. 7.

IVDFs measured on the channel centerline at Vd=600V, Id=20.83A, B=Bnom.

FIG. 7.

IVDFs measured on the channel centerline at Vd=600V, Id=20.83A, B=Bnom.

Close modal

Between Vd=400V and 500V (at Id=20.83A and B=Bnom.), HERMeS transitions from a mode exhibiting low amplitude breathing oscillations with a broad frequency spectrum to a regime with much larger amplitude, higher frequency coherent oscillations.10,31 The channel centerline LIF data in the acceleration region displayed a corresponding transition, from single-peaked IVDFs similar to those in Fig. 2 to bimodal distributions like those exemplified in Fig. 7. Figure 8 shows the discharge current oscillation amplitude and frequency as a function of mean discharge voltage Vd. Plots of the raw Idt and Vdt data and power spectra, as well as IVDFs measured at Vd=400500V, can be found in Ref. 31.

FIG. 8.

Discharge current oscillation peak frequency and RMS amplitude as a function of mean discharge voltage Vd, for Id=20.83A, and B=Bnom..

FIG. 8.

Discharge current oscillation peak frequency and RMS amplitude as a function of mean discharge voltage Vd, for Id=20.83A, and B=Bnom..

Close modal

In order to provide further support for the notion that the measured bimodal IVDFs arose due to time averaging over oscillations, simple simulations of this time-averaging at a single measurement location z0 were carried out for a drifting Maxwellian velocity distribution with an oscillating mean velocity v0t:

fiz0,v,t=Av0texpmivv0t22kBTi,
(5)
v0t=v0t=0+Δv0sinωt.
(6)

fiz0,v,t was assumed to be proportional to 1/v0t in order to account for the reduction in ion density at higher mean velocities due to particle flux conservation. Figure 9 shows two examples of simulated LIF line profiles, which bear qualitative resemblance to the acceleration zone data from Fig. 7 and other HERMeS IVDFs presented in Ref. 31. Peaks in the time-averaged IVDF at the extremes of the mean velocity oscillation arise due to the nearly flat slope of sinωt near ωt=π/2 and ωt=3π/2.

FIG. 9.

Simulated time-averaged LIF data for a drifting Maxwellian IVDF (Ti=1eV) with an oscillating mean velocity given by Eq. (6). (a) Δv0=2km/s. (b) Δv0=5km/s. Reproduced with permission from Chaplin et al., in 35th International Electric Propulsion Conference (Electric Rocket Propulsion Society, Atlanta, GA, 2017), IEPC-2017-229. Copyright 2017 Electric Rocket Propulsion Society.

FIG. 9.

Simulated time-averaged LIF data for a drifting Maxwellian IVDF (Ti=1eV) with an oscillating mean velocity given by Eq. (6). (a) Δv0=2km/s. (b) Δv0=5km/s. Reproduced with permission from Chaplin et al., in 35th International Electric Propulsion Conference (Electric Rocket Propulsion Society, Atlanta, GA, 2017), IEPC-2017-229. Copyright 2017 Electric Rocket Propulsion Society.

Close modal

To estimate how far the acceleration zone moved during the discharge oscillation, in Fig. 10, we have separately plotted potentials derived from the velocities of the “slow” and “fast” ion populations that appeared to exist in the time-averaged IVDFs in Fig. 7. The offset between the curves implies that the acceleration zone was oscillating 5% of the channel length in either direction from its mean location. Note that the ion transit time across the acceleration zone was 1μs, while the discharge oscillation period at Vd=600V was 18μs, so it is reasonable to assume that any single ion was accelerated through a nearly stationary potential profile.

FIG. 10.

Plasma potential profiles along the channel centerline derived from the mean velocities of the “slow” and “fast” ion populations that appeared to exist in Fig. 7 due to oscillations in the acceleration zone position at Vd=600V. At z/Lchannel>1.1, only the potential derived from the weighted average of the “slow” and “fast” velocities is plotted, since time-averaging over oscillations was not the primary driver of non-Maxwellian LIF profile features in this region.

FIG. 10.

Plasma potential profiles along the channel centerline derived from the mean velocities of the “slow” and “fast” ion populations that appeared to exist in Fig. 7 due to oscillations in the acceleration zone position at Vd=600V. At z/Lchannel>1.1, only the potential derived from the weighted average of the “slow” and “fast” velocities is plotted, since time-averaging over oscillations was not the primary driver of non-Maxwellian LIF profile features in this region.

Close modal

Oscillations in the acceleration zone position are of particular importance for HERMeS because they increase the pole erosion rates and thus can affect thruster lifetime. Hall2De simulations13 of the 600 V operating condition without discharge oscillations predicted an inner pole cover erosion rate that was approximately 5 times lower than the measured rate.50,58 When discharge current oscillations with frequency and amplitude matching the experiments were accounted for, the predicted inner pole cover erosion rate increased to approximately match the data.

It should be noted that there are other possible effects, unrelated to discharge oscillations, that could produce bimodal LIF profiles. For example, if a large population of metastable ions were excited directly from the neutral atom ground state—plausible in HERMeS at high discharge voltages because the peak electron temperature in the acceleration region is >50eV—there would be a low-velocity peak co-located with the high velocity peak corresponding to ions born near the anode potential. Time-resolved LIF measurements are underway to verify the interpretations put forth in this section and determine whether oscillations in the acceleration zone position are sufficient to fully explain the data.

LIF data were collected at the Vd= 300, 400, and 600 V operating conditions (Id=20.83A, B=Bnom.) across a grid of points spanning the outer half of the channel and near plume and near both the inner and outer chamfers at the downstream edge of the channel. The mean velocity vectors for the three discharge voltages are overlaid in Fig. 11. Some of the off-centerline IVDFs were complicated enough that Eq. (1) could not produce a good curve fit—in these cases, the mean velocity components along each injection direction were calculated directly from the first moment of the data.

FIG. 11.

Velocity vectors at Id=20.83A, B=Bnom., and Vd=300600V are overlaid to show the trend in the ion trajectory divergence versus discharge voltage. The vector lengths in panel (a) are proportional to the ion speed, while panel (b) shows unit vectors. Modified with permission from Chaplin et al., in 35th International Electric Propulsion Conference (Electric Rocket Propulsion Society, Atlanta, GA, 2017), IEPC-2017-229. Copyright 2017 Electric Rocket Propulsion Society.

FIG. 11.

Velocity vectors at Id=20.83A, B=Bnom., and Vd=300600V are overlaid to show the trend in the ion trajectory divergence versus discharge voltage. The vector lengths in panel (a) are proportional to the ion speed, while panel (b) shows unit vectors. Modified with permission from Chaplin et al., in 35th International Electric Propulsion Conference (Electric Rocket Propulsion Society, Atlanta, GA, 2017), IEPC-2017-229. Copyright 2017 Electric Rocket Propulsion Society.

Close modal

An important feature of the data in Fig. 11 is that the radial divergence of the ion trajectories increased as the discharge voltage was decreased. Ions born at high potentials near the edge of the beam are suspected to be the main source of pole erosion in magnetically shielded thrusters,32,59 and the electric fields that produced the observed ion velocity divergence at lower discharge voltages (particularly Vd=300V) would tend to drive a higher flux of ions toward the pole faces. Thus, the LIF results are consistent with the high pole erosion rate observed at the Vd=300V, Id=20.83A HERMeS operating condition,50,58 a phenomenon that was initially unexpected given that the 6.25 kW discharge power at this operating condition is only half of the thruster’s capability.

Figures 12–15 compare a selection of measured beam edge IVDFs at Vd= 300, 400, and 600 V (additional off-centerline IVDFs were presented in Ref. 31). Comparing the 300 V data in these figures with Fig. 2, we note that suprathermal ions were detected further upstream away from the channel centerline than they were along the centerline. Furthermore, bimodal IVDFs were detected at some off-centerline locations at all three discharge voltages, and some IVDFs had long low-velocity tails (for example, see the 300 V data in Fig. 15). The latter feature is expected to arise from the overlap of the ionization and acceleration regions (refer to the discussion in Sec. III A), but a similar low-energy ion population was not detected above the noise floor along the channel centerline.

FIG. 12.

Measured IVDFs at z/Lchannel=1.06, r/Lchannel=2.65.

FIG. 12.

Measured IVDFs at z/Lchannel=1.06, r/Lchannel=2.65.

Close modal
FIG. 13.

Measured IVDFs at z/Lchannel=1.06, r/Lchannel=2.77.

FIG. 13.

Measured IVDFs at z/Lchannel=1.06, r/Lchannel=2.77.

Close modal
FIG. 14.

Measured IVDFs at z/Lchannel=1.18, r/Lchannel=2.53.

FIG. 14.

Measured IVDFs at z/Lchannel=1.18, r/Lchannel=2.53.

Close modal
FIG. 15.

Measured IVDFs at z/Lchannel=1.18, r/Lchannel=2.77.

FIG. 15.

Measured IVDFs at z/Lchannel=1.18, r/Lchannel=2.77.

Close modal

It was not possible to conclusively determine which of the LIF profile features in Figs. 12–15 reflected the nature of the local instantaneous IVDFs and which were due to time averaging over oscillations. However, the general trend toward increasing time-averaged IVDF complexity at distances further from the channel centerline suggests that breathing mode oscillations or other waves may have had a greater impact on ion velocities at these locations, even at the 300–400 V operating conditions for which the global discharge current oscillation amplitude was low (see Fig. 8).

Examination of the full set of bimodal IVDFs at Vd=600V revealed that the maximum velocity separation of the two IVDF peaks occurred further downstream at positions further from the channel centerline; in light of the interpretation of the IVDF bifurcation in terms of acceleration zone movement (Sec. III E), this indicates that the peak axial electric field Ez was located further downstream at off-centerline positions. Comparing vz vs. z curves at different radial locations led to the same conclusion. On the other hand, at Vd=300V, the axial location of peak acceleration was approximately independent of radial position within the bulk of the channel.

We have presented LIF measurements of ion velocity distribution functions (IVDFs) along two orthogonal axes within the channel and near plume of the 12.5 kW magnetically shielded HERMeS Hall thruster. Variations in the plasma potential profile were observed as the discharge voltage Vd, magnetic field strength B, and background pressure PBG were varied individually while holding other operating parameters constant. At Vd=500600V, the measured IVDFs in the acceleration region displayed a broadly bimodal structure indicative of time-averaging over discharge movement during the large-amplitude discharge oscillations at these operating conditions. The time-averaged plasma potentials at Vd=300600V all approximately overlapped at locations downstream of the 300 V point, with the acceleration zone extending further upstream at higher Vd. The mean velocity vectors were more divergent at Vd=300V than at higher voltage operating conditions, consistent with a 2D potential structure that could push ions from the beam edge toward the front poles and drive erosion.

This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. The support of the joint NASA GRC and JPL development of HERMeS by NASA’s Space Technology Mission Directorate through the Solar Electric Propulsion Technology Demonstration Mission (SEP TDM) is gratefully acknowledged. The authors would also like to thank Ray Swindlehurst and Nowell Niblett for their assistance in setting up the LIF optics, installing the thruster for testing, and maintaining the vacuum facility.

Here, we prove that the mean ion velocity component along any direction may be calculated by a coordinate rotation of mean velocity components measured along two other orthogonal directions in the same plane. Consider a population of N ions, each with some velocity vj=vxjx^+vyjy^. The mean velocities in the x^ and y^ directions for the ensemble of particles are vx=j=1Nvxj/N and vy=j=1Nvyj/N. If we define a new coordinate system with axes x and y, related to the original coordinate system by a counterclockwise rotation through an angle θ, then the rotated velocity vectors of the individual particles are

vxjvyj=cosθsinθsinθcosθvxjvyj.
(A1)

The mean velocity components in the new coordinate system are

vx=1Nj=1Nvxj=1Nj=1Nvxjcosθ+vyjsinθ
(A2)
=cosθ1Nj=1Nvxj+sinθ1Nj=1Nvyj
(A3)
=cosθvx+sinθvy,
(A4)
vy=1Nj=1Nvyj=1Nj=1Nvxjsinθ+vyjcosθ
(A5)
=sinθvx+cosθvy.
(A6)

So, we can calculate the mean velocity components in the rotated coordinate system by applying a rotation matrix to the mean velocity vector measured in the original coordinate system.

This proof may seem intuitively obvious. However, we can show through a simple counterexample that an analogous rotation procedure cannot in general be used for the most probable ion velocity. Consider a plasma in which there are only three ions, with velocity components along the East and West directions (oriented at 45° angles with respect to the axial and radial directions as shown in Fig. 1) and along the axial and radial directions given in Table I. In this example, the most probable velocity along the East direction is 0. The most probable velocity along the West direction is also 0. Rotating this “most probable velocity vector,” we would find erroneously that the most probable axial velocity is 0. In fact, the most probable axial velocity is 1/2.

TABLE I.

Three-particle example demonstrating how applying a coordinate rotation to the most probable ion velocity can lead to incorrect results.

veastvwestvzvr
Ion #1 1/2 1/2 
Ion #2 1/2 1/2 
Ion #3 
veastvwestvzvr
Ion #1 1/2 1/2 
Ion #2 1/2 1/2 
Ion #3 
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