Eur. Phys. J. D (2015) 69: 145DOI: 10.1140/epjd/e2015-50571-4

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

Simulations on the influence of the spatial distributionof source electrons on the plasma in a cusped-field thruster

Tim Brandt1,2,a, Thomas Trottenberg2, Rodion Groll3, Frank Jansen1, Franz Georg Hey4, Ulrich Johann4,Holger Kersten2, and Claus Braxmaier1,3

1 DLR, Institute of Space Systems, Robert-Hooke-Str. 7, 28359 Bremen, Germany2 Institut fur Experimentelle und Angewandte Physik, Christian-Albrechts-Universitat zu Kiel, 24098 Kiel, Germany3 ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany4 Airbus Defence and Space, Claude-Dornierstraße 1, 88090 Immenstaad, Germany

Received 31 July 2014 / Received in final form 16 March 2015Published online 4 June 2015 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2015

Abstract. We present results from simulations on the influence of source electrons on the plasma propertiesin a magnetic cusps environment. Our simulations are based on the VSim/Vorpal particle-in-cell plasmasimulation package. Magnetic cusps are a typical feature of High Efficiency Multistage Plasma Thrusters(HEMPTs). This research is part of an effort to downscale a HEMPT to thrust levels in the µN and sub-µNregime. The aim is to fulfill the requirements of upcoming formation flight satellites and probes. Thosemissions demand very precise attitude control. In order to get the necessary insight, the plasma of a sectionof the HEMPT discharge chamber is simulated with idealized boundary conditions. The results for sucha section at two different distributions of source electrons are shown. A significant increase of the overallion number is recognized for one of the distributions. Comparisons with published similar simulations aremade. Factors that should be important for improvements of this thruster type are highlighted.

1 Introduction

Upcoming formation flying satellites and space probes likeNGO (new gravitational wave observatory) [1] or NGGM(next generation gravity mission) [2] require highly pre-cise attitude control. They put never before seen require-ments as to low thrust (0.1 μN. . . 150 μN) and thrust sta-bility (root of the noise spectral density ≤0.1 μN/

√Hz)

and long life expectancy. Due to the simplicity of its hard-ware components, low erosion and fuel efficiency, the HighEfficiency Multistage Plasma Thrusters (HEMPTs) seemto be an excellent candidate to fulfill those requirements.However, common HEMPTs like the Dm3a prototype [3]are more than an order of magnitude above the low thrustrequirement.

Therefore, an effort to downscale this thruster type isundertaken in cooperation of Airbus Defence and Space,the Center of Applied Space Technology and Micrograv-ity (ZARM) of the University Bremen and the GermanAerospace Center (DLR). This campaign consists of de-velopment on breadboard level [4], realization of a metrol-ogy system [5] and computer modeling [6] of this so-calledmicro-HEMPT. Currently, stable and reliable thrust levelsdown to 70 μN have been demonstrated [7], but furtherdownscaling to the 10 μN range still constitutes a con-siderable challenge. An improved understanding of this

a e-mail: tim.brandt@dlr.de

micro-HEMPT is desirable in order to reach the designgoals. The computer modeling part of the campaign aimsat a better understanding of the physics of a HEMPT,especially its downscaled version.

Several aspects of the HEMPT will be investigatedby computer modeling. One aspect is the influence fromthe spatial distribution of neutralizer electrons enteringthe thruster. Since a HEMPT is a kind of direct cur-rent discharge, the cathode (electron source) is crucialfor its behavior. Furthermore, the magnetic bi-conic cuspfield [8] has a strong influence on the mobility of the elec-trons, which makes the position and shape of the electronsource even more important. The cusps are zones wherekey thruster parameters like plasma confinement [9] andionization efficiency are determined. This paper describesresults from computer models with the focus on those as-pects, while the simulations are not intended to simulatea complete thruster.

The following section of this paper gives a quickoverview of the thruster concept, patented by ThalesElectron Devices GmbH in Ulm, Germany [10], followedby a description of the particle-in-cell (PIC) simulationmethod [11]. Due to the non-Maxwellian velocity distribu-tion and adiabatic invariant motion of the electrons nearthe zero magnetic field points [12], a kinetic particle modelwas indispensable. Therefore, PIC was selected over a fluidapproach. In consequence, a fully self-consistent kineticmodeling package is needed, as this enables to properly

Page 2 of 7 Eur. Phys. J. D (2015) 69: 145

Fig. 1. HEMP thruster principle.

represent non-local effects and accurate energy distribu-tion functions for all particles. As fully kinetic methodsare computationally expensive, it was desired to have acode with good parallel scaling for the PIC algorithm.Vorpal [13] provides an interface through the built-in li-braries to a number of models which have been used earlierfor thruster simulations [14,15]. Section 2 presents a de-scription of the setup of the models used in this work. Twoalmost identical models are created, which differ only inthe distribution of the source electrons. In Section 3 theresults of both models are compared. The possibility ofimprovements is discussed.

2 Model

2.1 The HEMPT principle

Like in a Hall effect thruster, the electric field in a HEMPTsupplies the energy for electron-neutral impact ionization,as well as it accelerates the ions. The discharge chamberconsists of a hollow ceramic tube with an anode at oneside and an open end at the other side, where magneticallytrapped neutralizer-electrons act as a virtual cathode. Aschematic view of a HEMP thruster is shown in Figure 1.The typical feature of the HEMPT is the periodic arrange-ment of opposing ring-shaped permanent magnets. Thesecreate so-called magnetic cusps with a radial magneticfield. Between the cusps, the field is mostly axial. There-fore, electrons can reach the discharge chamber wall onlynear the cusps, and even there, the magnetic mirror effectreduces the electron loss to the wall due to the convergingfield lines. Axial movement towards the anode is hinderedat the cusp, although not completely prohibited [16]. Botheffects result in an efficient plasma confinement.

2.2 Model theory

The presented simulations use a particle-in-cell (PIC) codewith integrated Monte Carlo algorithm [13]. A PIC simu-lation follows the kinetic movements of so-called superpar-ticles, each of which represents many real particles. Theseparticles with charge q and velocity vector v are movedaccording to the Lorentz force F = q(E + v × B) in a

(a)

(b)

(c)

Fig. 2. Geometries. Anode (red), grounded metal boundary(blue), Neumann boundary (violet), dielectric surface (yellow),electron source (orange), magnetic field lines (black). Electronsource current density and magnetic flux density (scaled) aregiven by color-schemes. (a) Position of the simulation domain(gray) within the cylindrical thruster geometry. (b) Simulationdomain of models 1 and 2. (c) Sketch of model 1 (left) andmodel 2 (right) with their respective electron source profiles.

leapfrog scheme by a Boris pusher [17]. While the mag-netic field generated by plasma currents is considered tobe negligible at the expected plasma densities and veloc-ities, the field B generated by the permanent magnets isan essential part of this simulation. For interactions onthe scale of the computational grid, the particles moveself-consistently in the electric field E, which is solved onthis grid according to the Poisson equation. Collisions aretreated by Monte Carlo algorithms [18]. This includes thecollisions that cause the ionization events.

2.3 Model setup

The aim of this model setup is to study in particular theplasma properties in the discharge channel near the cusps,and how they react to changes in the electron source. Thecylindrical simulation domain represents the last stage,i.e. the one at the exhaust, and covers the positions fromz = 7.5 mm to 17.5 mm, counted from the anode (seeFig. 2a). This way, the outermost magnetic cusp of thedischarge channel is included. The domain has a radius r

Eur. Phys. J. D (2015) 69: 145 Page 3 of 7

of 2.5 mm and includes not only the discharge channeland the inner floating dielectric wall, but also the dielec-tric material itself with its outer surface which is at groundpotential due to the adjacent grounded magnets and dis-tance rings.

To reduce the time needed by the computational pro-cess to get a stable result, the size of the system is scaleddown by a factor of four. In order to preserve the rela-tion of both the charged particles’ mean free paths andtheir gyration radii to the system length, the neutral gasdensity and the magnetic flux density are increased bythe same factor of four [19]. The magnetic field is im-ported from a stationary finite element simulation usingCOMSOL Multiphysics. Neutral gas distribution is spa-tially homogeneous at four times 4 × 1021 m−3, an aver-age value taken from a fluid simulation using OpenFOAMperformed on a model of the entire thruster and its nearexit region. Gas dynamics are considered negligible overthe time span of the PIC simulation which is 2.25×10−6 sor 1.8×106 time steps, but the depletion due to ionizationis accounted for.

The thruster has cylindrical symmetry, hence a cylin-drical coordinate system is chosen. Due to this symmetry,the model can be limited to a two-dimensional r-z-plane(with arbitrary angular position ϕ). The coordinates of thedownscaled system will be written with primes. For the co-ordinate system of the simulation, z′ = 0 mm correspondsto the real distance of 7.5 mm from the anode. The simu-lation domain covers the volume for 0 mm ≤ z′ ≤ 2.5 mmand 0 mm ≤ r′ ≤ 0.625 mm (see Fig. 2b). The volume for0 mm ≤ z′ ≤ 2.5 mm and 0.375 mm ≤ r′ ≤ 0.625 mmincludes the dielectric aluminum oxide (Al2O3) wall witha relative permittivity of εr = 9. The upper boundary ofthe simulation domain is the grounded potential of themagnetic and distance rings. The lower boundary (r′ = 0)is the symmetry axis. The two lateral boundaries of thisdomain are set to best approximate the environment (re-maining thruster channel and plume region). The rightboundary, at z′ = 2.5 mm, has a Dirichlet condition [20]for the Poisson solver with a fixed potential of 0 V as thereference potential. In this model, the boundary acts as acathode. At the opposing side, at z′ = 0 mm in the radialrange 0.375 mm ≤ r′ ≤ 0.625 mm, a Neumann boundarycondition [20] requires the electric field vectors to pointin radial direction, which takes into account that the realdischarge channel continues upstream. In the radial range0 mm ≤ r′ ≤ 0.375 mm, a potential drop is achieved by aDirichlet condition, i.e. a fixed potential. Previous simu-lations [21] and measurements [4] of others suggest a flatpotential inside the discharge channel. Therefore, this up-stream end of the model is assumed to lie on the anodepotential. This “anode” drives the plasma discharge ofthis model. In case of the lab version of this miniaturizedthruster, an anode current of 7 mA was measured at an an-ode potential of 400 V, resulting in a power consumptionof 2.8 W. In order to preserve the power to volume ratioof the 17.5 mm long discharge chamber for the 10.0 mmlong model (original size), the anode potential is reducedto 229 V.

At r′ ≥ 0.375 mm, particles are absorbed and chargeaccumulation is simulated by replacing each absorbed par-ticle with an unmovable one of the same charge. Thisboundary represents the dielectric surface of the thruster’sdischarge chamber wall. Both anode and cathode are ab-sorbing for all particles, with the exception of the cathodebeing reflecting for the electrons from the source, to avoida back-flow of those, in order to have a fixed input cur-rent. The electron source lies at the surface of the cathode(2.49 mm ≤ z′ ≤ 2.5 mm, 0 mm ≤ r′ ≤ 0.375 mm).

Only the charged particle dynamics, i.e. of singlycharged xenon ions and electrons, is followed in this sim-ulation. Particle positions are defined in r′ and z′ coor-dinates, but all three velocity vector components vr′ , vz′ ,and vϕ′ are calculated in order to account for the elec-tron gyration motion in the magnetic field. The gyrationmotion means that large scale motions of the electronsperpendicular to the magnetic field are prohibited. Dueto collisions with neutrals, a certain low mobility perpen-dicular to the magnetic field remains. The mass ratio ofthe electron and ion superparticles is the same as of thereal particles, which is especially important for the for-mation of realistic plasma sheaths [22]. The electrons aredistinguished by three species (with the same properties):the primary electrons from the source, the secondary elec-trons resulting from ionization, and the emitted electronsresulting from secondary electron emission at the dielec-tric surface [23,24]. Further distinction is not made, sothat a secondary electron can result from ionization bya primary or another secondary electron. The model in-cludes electron-neutral elastic, excitation and ionizationcollisions. For the collisions, the energy dependent colli-sion cross-sections are taken from reference [25]. Neutralgas density is used as an input parameter for the MonteCarlo code, which controls the collisions.

The cell size is Δz′ = Δr′ = 5×10−3 mm, resulting in acomputational grid of 500×125 cells. This size was chosento resolve the smallest Debye length resulting from theplasma density (1× 1018–1× 1019 m−3) and temperature(1–10 eV) expected for this type of thruster. The time stepsize was set to 1.25×10−12 s in order to resolve the electrongyration motion at the strongest magnetic field strength(2.8 T, corresponding to 0.7 T in the unscaled system).

Two models with the described setup are investigated,which differ only in the radial distribution of their re-spective electron source (virtual cathode). For the firstmodel, the current density is constant from r′ = 0 mm tor′ = 0.375 mm. For the second model, an r′-dependentGaussian current density distribution with a width ofσ = 0.0625 mm is applied, which concentrates most ofthe current at low r′-values (see Fig. 2c). The absolutecurrent remains the same for both models.

3 Results

The results of both models are presented in parallel. Fromnow on, the cusp will be seen as a separator of two re-gions, that will be denominated upstream and downstream(with regard to the ion movement) of the cusp. As in the

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(a)

(b)

Fig. 3. Source electron density at t = 2.25 × 10−6 s, (a) model 1, (b) model 2 (magnetic field lines in black).

(a) (b)

(c) (d)

Fig. 4. Electric potential for the two models at different times (magnetic field lines in black; dielectric surface in yellow).(a) Model 1 at t = 1.25 × 10−9 s. (b) Model 2 at t = 1.25 × 10−9 s. (c) Model 1 at t = 2.25 × 10−6 s. (d) Model 2 att = 2.25 × 10−6 s.

model description, source electrons will be called primariesand the ones generated by ionization secondaries. We be-gin with the final distribution of the primary electronsshown in Figure 3 (in all density plots the results aresmoothed over a width of 0.025 mm to reduce the PICnoise). Downstream of the cusp, in both models, the elec-trons have strayed off from the radial distribution givenby their source (at z′ = 2.5 mm) despite the magneticfield being mostly axial in this region. This can be ex-plained by the electrons having a certain mobility per-pendicular to the magnetic field lines due to neutral col-lisions [26,27]. In the second model, where the source isfocused on smaller radii, the electrons remain to some de-gree concentrated near the z-axis. In the first model, theelectrons have their peak concentration closer to the dis-charge chamber wall. Consequently, more electrons willbe absorbed by the dielectric discharge chamber wall, es-pecially near the cusp. The primary electron current to-wards this wall is 2.492 mA for the first model and only1.564 mA for the second model. This, in turn, creates asurface charge that is negative relative to the plasma po-tential. The ions follow that potential and are absorbed.

Consequently, the ion current towards this wall is higheras well in case of the first model, 4.994 mA instead of3.886 mA, indicating that it has higher plasma losses atthis wall. In both models, upstream of the cusp one can seethat the electrons roughly follow the magnetic field lineswhich strongly converge from the cusps to the z′ = 0 mmend. Thus the electrons are concentrated to low radii inthis region. In both models, the plasma potential is a fewvolts above the anode potential. Early in the simulation,at t = 1.25×10−9 s, a potential step has formed at higherradii (Figs. 4a and 4b) at the axial position of the cusps.The high radial magnetic field strength there creates ahigh resisitvity in axial direction for the electrons. Nearthe z-axis, the anode potential extends almost to the cath-ode. This indicates the low resisitvity in axial direction dueto the low radial magnetic field near the z-axis. At latertimes the potential has equalized and shows a rather flatprofile (Figs. 4c and 4d). While the flat profile in the entiredischarge channel is consistent with a simulation of theDm3a prototype including the plume [21], another sim-ulation [16], which investigated in general a cusped-fielddicharge with boundaries similar to our simulation, shows

Eur. Phys. J. D (2015) 69: 145 Page 5 of 7

(a)

(b)

Fig. 5. Xenon ion density at t = 2.25 × 10−6 s, (a) model 1, (b) model 2 (magnetic field lines in black).

(a)

(b)

Fig. 6. Xenon ionization rate, (a) model 1, (b) model 2 (magnetic field lines in gray).

a potential step already at the cusp closest to the cathode.The assumed reason for the formation of such a potentialstep is the hindering of the movement of the electrons inz-direction at the cusps, which depletes them in the up-stream region. Both in the simulation presented here andin the Dm3a simulation, the ion density is of the orderof magnitude of 1018 to 1019 m−3, while for the simula-tion from reference [16] it is 1017 to 1018 m−3. Therefore,an explanation for the discrepancy might be that for thehigher plasma densities the generation of new (secondary)electrons compensates the lack of primary electrons in theupstream region. A simulation with lower source currentand thus lower plasma density could clarify the origin ofthe two kinds of potential profile.

One can see that the ion distributions in both modelsshow a great similarity to the electron distributions in Fig-ure 5. This can be explained by the ambipolar movementof the two species. While the ions have the higher inertia,the electrons have a high resistivity perpendicularly to themagnetic field. Additionally, the primary electrons mostlydrive the ion generation, which also contributes to thesimilarity in the disribution of both species.

The ionization in the first model is concentrated nearthe axis in the upstream region (Fig. 6). While the au-thors of reference [16] could attribute such a concentration

0 0.5 1 1.5 20

5

10

15

time (µs)tota

l num

ber

of io

ns (

108 )

model 1model 2

Fig. 7. Comparison of the total xenon ion numbers versustime.

to the higher neutral gas density upstream, our simula-tion, that uses a homogeneous neutral gas density, showsthat the bundling of electrons by the magnetic field al-ready causes such an effect. In the second model, wherethe primary electrons are emitted near the z-axis, the ion-ization is dominated by a zone in the downstream regionclose to the z-axis. This can be attributed to the combi-nation of the concentration of primary electrons towardsthe z-axis and their energy gain at the potential step nearthe cathode. The total ion number is greater in the sec-ond model than in the first one (1.50 × 109 instead of1.22 × 109), as shown in Figure 7.

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(a)

(b)

Fig. 8. Density of electrons generated by ionization at t = 2.25 × 10−6 s, (a) model 1, (b) model 2 (magnetic field lines inblack).

The spatial distribution of secondary electrons (Fig. 8.)is similar to both the primary electrons and the ionsin each model. The anode current for the first model is1.465 mA and 5.054 mA for the primaries and the secon-daries, respectively. For the second model it is 2.311 mAand 4.421 mA for the primaries and the secondaries. Thismeans, that more primary electrons reach the anode inthe second model, which is in agreement with their highlosses at the wall in the first model. In both cases, the pri-mary electrons constitute a significant fraction of the totalelectron number. The contribution of emitted electrons iswith 0.175 mA for the first and 0.094 mA for the secondmodel almost negligible. The currents result in a powerconsumption of 1.49 W for the first model, and 1.54 Wfor the second, not deviating much from the value for thereal micro HEMPT (which is 2.8 W, corresponding to ap-proximately 1.60 W in the simulated section).

4 Conclusions

The work presented in this article studied the influence ofthe kind of electron supply on the discharge in a minia-turized HEMP thruster by means of a particle-in-cell sim-ulation. Instead of a complete simulation of the entirethruster together with the space containing the plume, asection extending from the midplane between the two rear-most magnetic cusps to the exhaust plane of the dischargechannel was simulated. To this purpose, idealized bound-ary conditions at the front and rear ends were applied, inorder to understand the main features rather than focus-ing on device specific details. The kind of electron supplyin a HEMP thruster depends critically on shape and posi-tion of the neutralizer. Neutralizers are mounted outsidethe thruster channel, and the electrons enter the thrusterthrough the exhaust, which constitutes a complex and ingeneral asymmetric situation, so that two simplified situ-ations have been examined: In the first case, the electronsenter through the rear plane with a flat current profile,and in the second case the current density at that planeis concentrated in form of a narrow Gaussian distributionabout the symmetry axis of the thruster.

It is found that in case of the concentrated electronsource, significantly more primary electrons pass throughthe simulated discharge channel, while in case of the flatsource, the electron losses to the channel wall are bigger.The spatial distribution of the ions is affected, too. In caseof the peaked electron supply, the ion density is also moreconcentrated to the axis of the channel. As a consequence,the plasma density is also higher in case of the electronsupply concentrated about the axis, even though the elec-tric power is comparable to the power in case of the homo-geneous electron supply. Since higher plasma densities in-volve higher exhaust currents, i.e. higher thrust-to-powerratios, the positioning of the electron source (neutralizer)should not be underestimated.

The author acknowledges Tech-X staff member SudhakarMahalingam, who’s experience in particle-in-cell codes pro-vided a great resource in the development of our own models.The numerical plasma simulations were done by Tim Brandt,while Rodion Groll did the numerical neutral gas simulations.Franz Georg Hey performed the experiments to generate in-put data for the modelling. Claus Braxmaier defined the taskand the scope, while Holger Kersten supervised the work. Theothers supplied scientific consultancy. The text was written byTim Brandt and Thomas Trottenberg.

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