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# Time-evolution of the ion velocity distribution function in the discharge of a Hall effect thruster

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## Abstract and Figures

The temporal characteristics of the Xe$^+$ ion axial Velocity Distribution Function (VDF) were recorded in the course of low-frequency discharge current oscillations ($\sim$~14 kHz) of the 5 kW-class PPS$\circledR$X000 Hall thruster. The evolution in time of the ion axial velocity component is monitored by means of a laser induced fluorescence diagnostic tool with a time resolution of 100 ns. As the number of fluorescence photons is very low during such a short time period, a hom-made pulse-counting lock-in system was used to perform real-time discrimination between background photons and fluorescence photons. The evolution in time of the ion VDF was observed at three locations along the thruster channel axis after a fast shut down of the thruster power. The anode discharge current is switched off at 2 kHz during 5 $\mu$s without any synchronization with the current oscillation cycle. This approach allows to examine the temporal behavior of the ion VDF during decay and ignition of the discharge as well as during forced and natural plasma oscillations. Measurements show that the distribution function of the axial component of the Xe$^+$ ion does change periodically in time with a frequency close to the current oscillation frequency in both forced and natural cases. The ion density and the mean velocity are found to oscillate whereas the velocity dispersion stays constant, which indicates that ionization and acceleration layers have identical dynamics. Finally, variations over time of the electric field are for the first time experimentally evidenced in a crossed-field discharge.
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arXiv:0902.1432v1 [physics.plasm-ph] 9 Feb 2009
Time-evolution of the ion velocity distribution function in the
discharge of a Hall eﬀect thruster
S. Mazouﬀre, D. Gawron, N. Sadeghi,
February 9, 2009
ICARE, CNRS, 1C avenue de la Recherche Scientiﬁque, 45071 Orl´eans, France.
LSP, Joseph Fourier University - CNRS, 140 Av. de la Physique, 38402 St Martin d’H`eres, France.
Abstract
The temporal characteristics of the Xe+ion axial Velocity Distribution Function
(VDF) were recorded in the course of low-frequency discharge current oscillations (14 kHz)
of the 5 kW-class PPSrX000 Hall thruster. The evolution in time of the ion axial veloc-
ity component is monitored by means of a laser induced ﬂuorescence diagnostic tool with
a time resolution of 100 ns. As the number of ﬂuorescence photons is very low during
such a short time period, a hom-made pulse-counting lock-in system was used to perform
real-time discrimination between background photons and ﬂuorescence photons. The evo-
lution in time of the ion VDF was observed at three locations along the thruster channel
axis after a fast shut down of the thruster power. The anode discharge current is switched
oﬀ at 2 kHz during 5 µs without any synchronization with the current oscillation cycle.
This approach allows to examine the temporal behavior of the ion VDF during decay and
ignition of the discharge as well as during forced and natural plasma oscillations. Mea-
surements show that the distribution function of the axial component of the Xe+ion does
change periodically in time with a frequency close to the current oscillation frequency in
both forced and natural cases. The ion density and the mean velocity are found to oscil-
late whereas the velocity dispersion stays constant, which indicates that ionization and
acceleration layers have identical dynamics. Finally, variations over time of the electric
ﬁeld are for the ﬁrst time experimentally evidenced in a crossed-ﬁeld discharge.
Submitted to Physics of Plasmas
1
1 Introduction
A Hall Eﬀect Thruster (HET) is a gridless ion accelerator that ﬁnds applications in the ﬁeld
of spacecraft propulsion [1, 2]. Such a type of electric propulsive device is especially suited for
long duration missions and for maneuvers that require a large velocity increment. HETs are
at present mostly employed for geostationary communication satellite orbit correction and
station keeping. Other ﬁelds of application are envisaged for the near future. Low power
Hall thrusters seem suited for drag compensation of observation satellites that operate on a
low-altitude Earth orbit. The use of high power Hall thrusters for orbit raising and orbit
topping maneuvers of communication satellites would oﬀer signiﬁcant beneﬁts in terms of
launch mass, payload mass and operational life. In addition, large Hall thrusters appear as
good candidates to be used as the primary propulsion engine for robotic space probes during
interplanetary journeys towards far-oﬀ planets and asteroids.
A Hall eﬀect thruster is a low-pressure DC discharge in crossed electric and magnetic ﬁelds
conﬁguration [2, 3, 4]. Xenon is generally used as a propellant gas due to its high atomic mass
and low ionization energy. A schematic of a HET is depicted in Fig. 1. The anode is located
at the upstream end of a coaxial annular dielectric channel that conﬁnes the plasma. The
cathode is situated outside. A set of coils combined with magnetic parts provide a radially
directed magnetic ﬁeld Bof which the strength is maximum in the vicinity of the channel
exhaust. The magnetic ﬁeld is chosen strong enough to make the electron Larmor radius much
smaller than the channel characteristic sizes, but weak enough not to aﬀect ion trajectories.
The gas injected through the anode is ionized inside the channel by electron impacts. As the
magnetic ﬁeld considerably slows down the electron motion towards the anode, the applied
potential concentrates in a restricted area at the channel entrance. The corresponding axial
electric ﬁeld Ethen accelerates ions out of the channel, which generates thrust. The ion
beam is neutralized by a fraction of electrons emitted from the cathode. The crossed Eand
Bgeometry is at the origin of a large electron azimuthal drift – the Hall current – that is
responsible for the eﬃcient ionization of the supplied gas. When operating near 1.5 kW, a
HET ejects ions at 20 km s1and generates 100 mN of thrust with an overall eﬃciency of
It is well-established that the crossed-ﬁelds discharge of a Hall eﬀect thruster is strongly
non-stationary [3, 6]. This speciﬁc type of magnetized plasma displays numerous types of
oscillations, which encompass many kinds of physical phenomena, each with its own length and
time scales [6]. Current and plasma ﬂuctuations, of which the frequency range stretches from
10 kHz up to 100 MHz, play a major role in ionization, particle diﬀusion and acceleration
processes. Low-frequency plasma oscillations in the range 10-30 kHz, so-called breathing
oscillations, are especially of interest as they carry a large part of the power [7]. Breathing
oscillations ﬁnd their origin in a prey-predator type mechanism between atoms and ions
as shown by Boeuf and co-workers [8]. In short, these oscillations originate in a periodic
depletion and replenishment of the neutrals near the exhaust of the thruster channel due to
the eﬃcient ionization of the gas. The frequency is then linked to the time it takes for atoms
to ﬁll in the ionization region. With an atom thermal speed of 300 m/s and a region size of
20 mm, one ﬁnds a frequency of 15 kHz. The breathing phenomenon not only disturbs the
discharge current but also has a strong impact on several quantities. The plume shape and
the ion beam divergence change during an oscillation cycle of the current as was shown by
means of CCD imaging [9]. A time variation of the ion beam energy in a Hall thruster far
ﬁeld was evidenced using a retarding potential analyzer [11]. The electron density and the
2
plasma potential oscillate at low frequency [10]. Conversely, the electron temperature stays
unchanged. Oscillations of the aforementioned quantities are most likely connected with a
time variation of the potential distribution or, in other words, with the variation with time
of the accelerating electric ﬁeld. Therefore, it appears of great interest to investigate the
temporal behavior of the electric ﬁeld that certainly hides a rich and intricate dynamics.
Across the acceleration layer, the medium is collisionless, i.e. ion-ion and ion-atom collision
events are scarce. As a consequence, the electric ﬁeld can be directly inferred from the Xe+
velocity which then becomes the quantity to be examined.
Laser Induced Fluorescence (LIF) spectroscopy in the near infrared has often been used in
the past few years to measure the time-averaged velocity distribution function (VDF) of Xe+
ions in the plasma of a Hall eﬀect thruster [12, 13, 14, 15]. LIF spectroscopy is a non intrusive
diagnostic tool that enables an accurate determination of the local velocity of atoms along the
laser beam direction by measuring the Doppler shift of absorbed photons. The metastable
Xe+ion VDF is recorded by collecting ﬂuorescence radiation at 541.9 nm after excitation
of the 5d 2F7/26p 2Do
5/2transition at λ= 834.7233 nm [14]. A phase sensitive detection
method is often used to capture the ﬂuorescence signal. However, this method, which is
powerful enough for the extraction of a signal in an environment with a high background
noise level, oﬀers a poor time resolution. To achieve the measurement of the time-resolved
Xe+ion VDF in the plasma of a Hall eﬀect thruster, it is necessary to develop a bench able
to detect LIF photons with a time resolution around 1µs. In normal operating conditions
for a HET, the number of ﬂuorescence photons observed at 541.9 nm with a continuous laser
beam tuned at 834.7233 nm with about 1 mW/mm2power density is on the order of 102per
µs. Under identical experimental conditions, the number of background photons generated
by the plasma at 541.9 nm during 1 µs is typically 1, which means a ratio of 100 between the
two signal amplitudes. The laser system must therefore be able (i) to detect a tiny amount
of photons hidden in a strong background (ii) to determine with a high accuracy the exact
moment in time ﬂuorescence photons have been produced. One must therefore turn to a
photon-counting technique.
In this contribution, we present time-resolved measurements of the Xe+ion axial VDF in
the discharge of the 5 kW-class PPSrX000 Hall thruster ﬁred at 500 V discharge voltage and
6 mg/s xenon mass ﬂow rate. The evolution in time of the VDF was recorded at several loca-
tions during the transient regime that follows a fast anode discharge current ignition in order
to investigate the ion dynamics during forced and free low frequency current oscillations. The
current is switched oﬀ during 5 µs at a 2 kHz repetition rate without any synchronization with
the discharge current oscillation cycle. The outline of this paper is as follows. In Sec. 2, the
LIF optical assembly is described and the pulse-counting technique is introduced. Section 3
shows contour plots of the time-varying Xe+ion VDF as well as traces of various velocity
groups at the thruster outlet. In Sec.4, the temporal behavior of macroscopic quantities
like the density, the mean velocity and the velocity dispersion, is examined and discussed.
Section 5 reports on variation over time of the accelerating electric ﬁeld in the crossed-ﬁeld
discharge of a Hall thruster. Finally, concluding remarks will be presented in Sec. 6.
3
2 Diagnostic technique
2.1 Optical bench and collection branch
The LIF optical assembly is extensively described in Ref. [15]. The laser beam used to
excite Xe+metastable ions at 834.7 nm is produced by an ampliﬁed tunable single-mode
external cavity laser diode. The wavelength is accurately measured by means of a calibrated
wavemeter whose absolute accuracy is better than 100 MHz, which corresponds to 90 m/s.
A plane scanning Fabry-Prot interferometer with a 1.29 GHz free spectral range is used to
real-time check the quality of the laser mode and to detect mode hops. The primary laser
beam is modulated by a mechanical chopper at a low frequency 20 Hz before being coupled
into a 50 m long optical ﬁb er of 50 µm core diameter. The ﬁber output is located behind the
thruster. Collimation optics are used to form a narrow beam that passes through a small hole
located at the back of the PPSrX000 thruster. The laser beam propagates along the channel
axis in the direction of the ion ﬂow. Typically, the laser power density reaches 3mW/mm2,
which warrants a weak saturation eﬀect on the studied transition.
A collection branch made of a 40 mm focal length lens, which focuses the ﬂuorescence light
onto a 200 µm core diameter optical ﬁber, is mounted onto a travel stage perpendicular to
the channel axis. The magniﬁcation ratio is 1, meaning that the spatial resolution is 200µm
in axial direction. A 16 mm long slit was made in the channel dielectric outer wall in order to
carry out measurements inside the channel. The ﬂuorescence light transported by the 200 µm
ﬁber is focused onto the entrance slit of a 20 cm focal length monochromator that isolates the
541.9 nm line from the rest of the spectrum. A photomultiplier tube serves as a light detector.
2.2 Lock-in photon-counting device
The pulse (or photon) counting technique allows the detection of a very low level signal with
an excellent time resolution. When combined with a modulation of the laser light intensity,
the pulse counting technique can distinguish between LIF photons and spontaneous emission
photons. The technique is known as the time-resolved pulse counting lock-in detection tech-
nique. In this work, a customized pulse-counting system is used to measure time-dependent
ion VDF [16]. Here we brieﬂy outline the main characteristics and settings of the system.
A block diagram of the pulse counting system is shown in Fig. 2. Photons are detected by
means of a high gain and low dark noise PMT (R7518-P from Hamamatsu). A fast ampliﬁer
and discriminator module (9302 from Ortec - 100 MHz counting rate) is used to screen out
dark current from PMT dynodes, to limit the pulse rate thereby avoiding saturation of the
pulse counter, and to transform any single event – here the arrival of a photon – into either
a NIM or a TTL pulse. Pulses are subsequently treated by the lock-in pulse counter device,
which counts events as a function of time. A trigger starts the counter which segments photon
count data into sequential time bins. Notice that up to 32 kbins are available. The width of
the bins can be set from 10 ns to 655 s. The instrument records the number of photons that
arrive in each bin. In order to greatly improve the signal-to-noise ratio, the counter is able
to operate in real-time addition-subtraction mode. The laser beam intensity is modulated at
low frequency (20 Hz) by means of a mechanical chopper. Each pulse recorded when the
laser is propagating through the plasma (laser-on mode) is added to the time series; the signal
corresponds to LIF photons plus background photons. Each pulse recorded when the laser
is suspended (laser-oﬀ mode) is subtracted from the time series: in that case the signal is
solely composed of background photons. A 2 kHz trigger signal generated by the counter itself
4
was used to deﬁne the start of the measurement cycle. The time resolution, i.e. the width
of each bin, was set to 100 ns and 5000 bins were used. The duration of one measurement
cycle is therefore 500 µs, corresponding to about 6 times the period of low-frequency current
oscillations of the PPSrX000 thruster operating at 500 V and 6 mg/s. In order to obtain a
reasonable signal-to-noise ratio, light was accumulated over 1 million cycles.
The procedure to obtain the time-resolved ion VDF is the following:
the laser is ﬁxed at a given wavelength λcorresponding to a certain ion velocity group
δv. The extent (dispersion) of the velocity group results from the spectral width of the
laser beam and from the thermal expansion of the laser cavity. A feedback loop allows a
minimization of the shift of the laser wavelength. The dispersion is thus around 10 m/s.
The pulse counter is used to record the number of ﬂuorescence photons induced by
excitation of metastable ions at λ. That means we follow the temporal evolution of the
velocity group δv.
After about one million discharge current disconnection cycles the laser wavelength is
changed and a new measurement starts.
To obtain a smooth ion VDF about 15 to 20 diﬀerent wavelengths are used.
3 Temporal characteristics of the ion VDF
3.1 Experimental conditions - Ion emission light
All measurements were performed in the PIVOINE-2g test-bench. The PPSrX000 Hall eﬀect
thruster was equipped with BN-SiO2channel walls and with a carbon anode. All thruster
parameters were kept unchanged during the experiments: The applied voltage Udwas set to
500 V, the anode xenon mass ﬂow rate Φawas ﬁxed at 6 mg/s and the magnetic ﬁeld strength
Bwas 150 G. The mean discharge current is 5.4 A and the oscillation frequency is found to
be 13.7 kHz. The temporal characteristics of the ion VDF were recorded for three locations
along the channel centerline. Each position deﬁnes a distinct area in terms of electric ﬁeld
magnitude [15]. The position x= -15 mm corresponds to a zone through which the electric
ﬁeld is almost zero and the ionization is strong. At x= -2.5mm, the electric ﬁeld is large and
ionization is maintained. The channel exit plane, x= 0 mm, corresponds to a region where
ionization ceases and the electric ﬁeld strength is high with E300 V/cm.
In this study, the ion ﬂow dynamics was investigated before and after a fast shut-down of
the anode discharge current. It was therefore possible to examine the temporal characteristics
of the ion VDF during plasma breakdown and ignition as well as during forced and free current
oscillations. The anode current is switched oﬀ during 5 µs at 2 kHz by way of an optically
controlled fast power switch based on a MOSFET [9]. The power switch is directly driven by
the counter, as shown in Fig. 2. There is no synchronization between the power cut cycle and
the discharge current waveform. In other words, the anode current is switched oﬀ randomly
at any time with respect to the discharge current natural oscillations.
Figure 3 displays the intensity of the 541.9 nm ion line as a function of time observed
at x= -2.5 mm recorded with the photon-counting device operating in addition mode only.
A snapshot of the anode current waveform is also shown. Measurement reveals the time
5
evolution of natural plasma emission. The power is always switched oﬀ at t= 0 µs (reference
time). The plasma decays with a 1/etime of 2.1 µs. However, it does not fully vanish as
charged-particle recombination and diﬀusion processes timescales are not inﬁnitely shorter
compared to the 5 µs current-oﬀ time period. On the contrary, both the anode current and
the Hall current cancel in a few hundreds of ns [17]. This point is speciﬁcally addressed
in the next sections. A large amount of light is quickly produced at the re-ignition stage,
as can be seen in Fig. 3. In like manner, a discharge current burst always occurs at re-
ignition [9, 17]. This forced plasma oscillation originates in the sudden ionization of the great
amount of propellant atoms accumulated inside the channel when the discharge is oﬀ. The
plasma oscillates with a mean period of about 83 µs that corresponds to a 12 kHz frequency.
As can be observed in Fig. 3, the amplitude of light oscillations diminishes with time, and
the signal ﬁnally approaches a constant level. This phenomenon is due to the fact that, the
current disconnection cycle is not synchronized with the natural current oscillations, while
data acquisition is a cumulative process over thousand of cycles.
3.2 Contour map
Figure 4 shows contour plots of the time evolution of the Xe+ion velocity distribution func-
tions for three positions along the thruster channel axis, respectively x= -15 mm, x= -2.5 mm,
and x= 0 mm. All velocity groups vanish quickly after the current is switched oﬀ on a time
scale on the order of a few 100 ns. This property indicates the electric ﬁeld cancels almost
instantaneously when the current is stopped. As a consequence, the discharge current as
well as the Hall current disappear over an extremely short time period, as experimentally
observed [17]. Yet, very slow ions do not fully disapp ear in 5 µs as recombination, ambipolar
diﬀusion to walls and drift out of the acceleration zone are slow processes. In Fig. 4, the
ﬂuorescence signal is signiﬁcantly above zero after 5 µs for x= -15 mm. The remark holds
also true for the ion emission signal, see Fig. 3. Note that a Xe+ion travels 1.5 mm in 5 µs
at the thermal speed vth 300 m/s. Very slow ions, i.e ions with the atom speed, are always
produced ﬁrst at re-ignition as electrostatic acceleration is not an instantaneous process. In
Fig. 4, slowest ions are indeed observed ﬁrst and the mean velocity gradually increases up
to a limit. Ions moving with a velocity close to the thermal speed are not visible in Fig. 4
for x= -2.5 mm and x= 0 mm as the ion VDF is truncated due to a lack of data points.
The large production of ions at plasma ignition originates from the fact that the channel
is entirely ﬁlled up with xenon atoms, as previously explained. As can be seen in Fig. 4,
the Xe+ion VDF in axial direction changes in time, especially during the ﬁrst two plasma
oscillations. This phenomenon indicates that the acceleration potential, and thus the electric
ﬁeld, is likely to vary in time with a frequency on the order of the main discharge current
oscillation frequency.
3.3 Velocity groups
Examination of the temporal characteristics of the ion VDFs provides a general outline of
the ion dynamics in the discharge of a Hall thruster. In contrast, a critical analysis of the
temporal behavior of ion velocity groups reveals in great detail the intricate character of the
physics at work. The time evolution at the channel outlet of eight well-identiﬁed ionic velocity
groups δv is shown in Fig. 5. The temporal evolution is monitored at the thruster channel
exhaust. Graphs correspond to horizontal cross-sections of the lower contour plots in Fig. 4.
6
All velocity groups quickly vanish as soon as the power is switched-oﬀ: the mean 1/edecay
time for all δv is 1.5 µs. The discharge current as well as the Hall current decay in about
the same time period [17]. Nonetheless, fastest ions disappear out of the acceleration region
ﬁrst: the 1/edecay time is 1.1 µs at 12960 m/s. On the contrary, slowest ions are produced
ﬁrst when the discharge current is re-ignited. In Fig. 5, the group δv = 9515 m/s is detected
at 9 µs and it reaches its highest amplitude at 19 µs whereas the group δv = 13390 m/s is
detected at 20 µs and it goes through a maximum at 39 µs. All velocity families oscillate
nonetheless with the same period of time T73 µs, that means with a frequency around
14 kHz.
When the power is turned on again, ionization immediately takes place, see Fig. 3. Dis-
charge and Hall current are restored in less than 1 µs and ions are gradually accelerated as
shown in Fig. 5. All these facts indicate that the electric ﬁeld is established on a microsecond
timescale at re-ignition. However, it seems at ﬁrst sight in manifest contradiction with the
time it takes to detect fast ions inside the thruster channel. The time of ﬂight of a given
ion velocity group across the acceleration layer after plasma re-ignition can be assessed by
numerically solving the particle motion equation in an external electric ﬁeld E:
(1) dv =e
mE dt,
where eis the elementary charge and mis the mass of a xenon ion. For simplicity’s sake,
the electric ﬁeld is ﬁrst set constant in space and in time through the whole acceleration
region (from x= -10 mm to 20 mm) with a magnitude of 170 V/cm [15]. Moreover, the
ﬁeld is assumed to be instantaneously created and the ionization process is stationary and
homogeneous. The initial velocity v0is ﬁxed as the thermal speed vth. Computations reveal
that the group δv = 13390 m/s appears at the exit plane at t= 1.05 µs. Ions have then
travelled 7.2 mm, a distance compatible with the acceleration layer size L30 mm. However,
this velocity group is observed ﬁrst at t= 20 µs, see Fig. 5. Another approach consists of
taking a steady electric ﬁeld distribution similar to the measured one [15]. When ions are
created about 19 mm inside the channel, computations indicate that the group δv = 13390 m/s
is indeed observed at x= 0 mm for t20 µs. Yet ions moving at 9515 m/s are solely seen a
few hundreds of ns earlier, in contradiction with experimental outcomes.
One way to better duplicate reality consists for instance of considering that the ionization
proﬁle or the electric ﬁeld can evolve in the course of time. With a stationary electric ﬁeld
proﬁle similar to the measured one [15], numerical simulations using Eq. 1 show that ions
must be created ﬁrst in the vicinity of the channel outlet and the ionization front must move
towards the anode with a speed on the order of 500 m/s to reproduce trends that are ex-
perimentally observed in Fig. 5. Propagation of an ionization wave through the acceleration
layer was proposed to explain experimental results acquired by means of time-resolved optical
emission spectroscopy with a ﬁber comb [11] as well as CCD images of the plume behavior
with a sp eed around 1-2 km/s.
Figure 6 shows the time evolution of the velocity group δv = 9550 m/s for two locations,
respectively x= -2.5 mm and x= 0 mm. The ion family is ﬁrst observed at the channel
exit plane at t= 8 µs before being detected inside the channel at t25 µs. The highest
amplitude is attained for t= 19 µs and t= 45 µs at x= 0 mm and x= -2.5 mm, respectively.
Results indicate that ions are created within a broad region inside the channel: ions created
close to the exit plane are indeed detected ﬁrst. Nevertheless, experimental outcomes cannot
7
be correctly simulated when assuming that an ionization wave travels from the plume near
ﬁeld towards the anode while the electric ﬁeld distribution stays unchanged. On the contrary,
results suggest that the ionization front as well as the acceleration layer change in time to-
gether. Numerical simulations, though basic, and experimental data are therefore in favor of
complex plasma dynamics within the discharge of a Hall thruster. It is worth noticing that
the behavior of Xe+ion velocity groups during a forced oscillation that follows a power-oﬀ
period may not directly image the normal behavior during a natural plasma oscillation from
the viewpoint of amplitude of the observed phenomena.
4 Evolution in time of the density, mean velocity and disper-
sion
The time evolution of various averaged quantities is plotted in Fig. 7, 8 and 9 for three
positions along the channel axis, respectively x=15 mm, x=2.5 mm and x= 0 mm.
The relative metastable Xe+ion density is given by the area of the VDF. The mean velocity
and the velocity dispersion are computed from, respectively, the ﬁrst and the second order
moments of the velocity distribution. The velocity spread is in fact expressed in terms of p
parameter [14, 15]. The latter reads:
(2) p= 2 p2Ln(2) ×σ2.335 ×σ,
where σis the standard deviation. The quantity pis equal to the FWHM in the case of a
Gaussian proﬁle.
As can be seen in Fig. 7, the ion density is oscillating in time with a frequency of about
16 kHz whatever the position. Changes in the ion density are connected with temporal charac-
teristics of the ionization rate. The latter is driven by a prey-predator kind of process between
atoms and charged particles [6, 18, 19]. Besides, Langmuir probe measurements have shown
that the electron density periodically varies in time with the discharge current oscillation
frequency [10]. The fact that in Fig. 7, the ﬁrst maximum of the ion density waveform is
shifted in time with respect to the ﬁrst maximum of the 541.9 nm line intensity proﬁle for
x=2.5 mm and x= 0 mm is an experimental artefact due to the truncation of the measured
VDF.
The time evolution of the mean velocity is shown in Fig. 8. At x= -15 mm, the velocity
slightly oscillates around zero as the observation point is outside the acceleration layer. At
x=2.5 mm and x= 0 mm, the mean velocity oscillates around a constant value, which
corresponds with a reasonably good agreement to the value obtained by means of time-
averaged LIF spectroscopy at the same location [15]. The small diﬀerence is due to the
power-switch disturbance. As shown in Fig. 8, the velocity changes in time are however
small. In terms of kinetic energy the largest gap is 10 eV at -2.5 mm, respectively 25 eV at
the channel exit plane. Energy variation of a few tens of eV during a low-frequency current
oscillation cycle was recorded in the plume far-ﬁeld of a SPT100-ML Hall thruster using
a repulsing potential analyzer [11]. Hybrid ﬂuid/kinetic models also predict low-amplitude
periodical variation of the ion velocity [18, 19] with a strong correlation between discharge
current and ion velocity temporal behavior. The observed oscillations of the mean ion axial
velocity can solely be explained by a back and forth motion of the ionization and acceleration
layers and/or a change in the electric ﬁeld magnitude, as already suggested by the analysis
of ion velocity groups characteristics as a function of time.
8
As can be seen in Fig. 9, the velocity dispersion does not vary much in time other than
at current shut oﬀ and restart. At the thruster channel exhaust, the velocity dispersion (p
parameter) is almost constant with a value of 2950 m/s. In terms of energy spread it is 47 eV.
In previous investigations [14, 15], it was clearly demonstrated that in a Hall thruster environ-
ment the velocity dispersion originates mostly in the spatial overlap between the ionization
and the acceleration layers. As the velocity dispersion stays unchanged with time whereas
the ion density and velocity do vary, one can conclude the ionization front and the electric
ﬁeld distribution have a correlated dynamics. For instance, assuming the form of the two
proﬁles stay unchanged, the two layers would have to move along at once together in the
axial direction. Actually, computer simulations indicate that both the shape and the location
of the ionization and electric ﬁeld proﬁles change with time [18].
5 Low-frequency electric ﬁeld oscillations
The oscillation in time of the accelerating electric ﬁeld can be assessed from the time-
dependent proﬁle of the Xe+ion mean velocity ¯v, see Fig. 8, assuming a collisionless medium.
The electric ﬁeld is computed according to the formula:
(3) E(t) = mXe+
2e¯v(x1, t)2¯v(x2, t)2
dx1x2,
where xis the position and drefers to the distance. The time-dependent electric ﬁeld is plotted
in Fig. 10 for the area that ranges between -2.5 mm and the exit plane. The electric ﬁeld is
found to oscillate with a period T90 µs, i.e. f11 kHz, around a mean value of about
215 V/cm disregarding the power-oﬀ period. This value is close to the value of 245 V/cm
found by way of time-averaged laser spectroscopy [15]. The amplitude of the ﬁeld oscillations
is, however, relatively weak. Over the ﬁrst free oscillation that extends from t= 75 µs until
t= 165 µs the amplitude varies at most from 190 V/cm to 240 V/cm, therefore, the electric
ﬁeld variation is in the range ±10 % ahead of the thruster channel exhaust. Finally, in Fig. 10,
one can also notice the slow rise of the electric ﬁeld magnitude from 100 V/cm at 10 µs to
250 V/cm at 30 µs. This temporal evolution is likely to be connected with the time it takes
for the establishment of an equilibrium state for both the discharge and the Hall currents.
6 Conclusions
The examination of the temporal characteristics of the Xe+ion velocity distribution func-
tion in the magnetized discharge of a Hall eﬀect thruster reveal the complex dynamics of
the ionization and the acceleration processes. Three results are especially of great interest.
First, the ionization proﬁle and the electric ﬁeld distribution are unstationary. Second, the
ionization and acceleration layers vary in time in such a way that their spatial overlap stays
almost unchanged as the ion velocity spread does not change much after ignition. Third, the
amplitude of the low-frequency electric ﬁeld oscillations ahead of the channel exit plane is
weak.
Even though this study has brought new facts about the physics of a Hall eﬀect thruster,
it appears necessary to carry on this type of experiments aiming at building-up of a larger
set of data. The latter is actually necessary to improve our understanding of the time and
space evolution of the electric ﬁeld in a Hall thruster. Measurements of the time-dependent
9
ion VDF must be performed with a better signal-to-noise ratio at many locations along
the thruster channel centerline. Measurements must also be carried out for several thruster
operating conditions and various power levels. Finally, experimental results must be critically
compared with numerical outcomes of PIC models of Hall thruster behavior.
Acknowledgements
Works are performed in the frame of the joint-program CNRS/CNES/SNECMA/Universities
3161 entitled “Propulsion par plasma dans l’espace”. They are also ﬁnancially supported by
the French National Research Agency in the frame of the 06-BLAN-0171 TELIOPEH project.
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11
Figure 1: Cross-section view of a Hall eﬀect thruster. The symbol estands for electron, afor atom
and ifor ion. The channel exit plane is referred to as x= 0 in this work.
12
Figure 2: Block diagram of the lock-in pulse counting system used in this work to measure the time-
resolved Xe+ion VDF by means of LIF spectroscopy. The PMT is placed behind a 20 cm focal length
monochromator. The anode discharge current switch is externally driven by the counter.
13
Figure 3: (Top) Snapshot of the anode current waveform. The discharge current is switched for 5 µs
at t= 0 µs. (Bottom) Change in time of the ion emission at 541.9 nm observed using the photon
counting technique at x= -2.5 mm. The inset panel displays enlargement of the emission signal from
50 µs until 350 µs. The oscillation frequency is 12 kHz.
14
Figure 4: Contour map of the Xe+ion axial VDF as a function of time for three locations along the
channel axis of the PPSrX000 Hall thruster ﬁred at 500 V: x= -15 mm, x= -2.5 mm and x= 0 mm.
15
Figure 5: Trace of the time evolution of eight ionic velocity groups at the channel exit plane (x= 0
mm).
16
Figure 6: Time evolution of the velocity group δv 9 550 m/s for two lo cations: x=-2.5 mm and
δv = 9605 m/s (solid line) and x= 0 mm and δv= 9515 m/s (circle).
17
Figure 7: Time evolution of the metastable Xe+ion relative density for three positions along the
channel axis of the PPSrX000 Hall thruster. The density is given by the VDF area. Also shown is
the emission proﬁle at 541.9 nm (blue line).
18
Figure 8: Time evolution of the Xe+ion mean axial velocity for three on-axis positions. The time-
averaged velocity measured in a previous study at x=2.5 mm and 0 mm is 9300 m/s and 13100 m/s,
respectively [15]. Also shown is the proﬁle of the light emission at 541.9 nm (blue line).
19
Figure 9: Time evolution of the Xe+ion axial velocity dispersion (pparameter) for three positions
along the channel axis of the PPSrX000 Hall thruster. Also shown is the proﬁle of the light emission
at 541.9 nm (blue line).
20
Figure 10: Electric ﬁeld temporal characteristics ahead of the PPSrX000 thruster channel exhaust
(between 0 and -2.5 mm). The ﬁeld strength is determined from the mean Xe+ion axial velocity.
21
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