Ground Plasma Tank Modeling and Comparison to ... - IEEE Xplore

IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 36, NO. 5, OCTOBER 2008

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Ground Plasma Tank Modeling and Comparison to Measurements Jean-Charles Matéo-Vélez, Jean-François Roussel, Daniel Sarrail, Françoise Boulay, Virginie Inguimbert, and Denis Payan

Abstract—ONERA plasma tank JONAS is populated of drifting Ar+ argon ions representative of Low Earth Orbit environment, produced by a Kaufman source, and slow ions created by charge exchange with the background pressure of argon. When testing a mock-up or space equipment in this tank, it is often important to determine the drifting and slow ion densities in various locations. The orbital and radial motion models are compared to experimental current–voltage measurements and to numerical results. In the range of parameters used in this paper, the radial motion model is more adapted than the orbital-motion-limited model to determine slow ion density. Nevertheless, the number of measurements is necessarily limited and discriminating between fast and slow ions is not easy. A complementary approach consists in performing a numerical simulation of the plasma dynamics in the tank. Provided the modeled physics is validated, and the modeling is calibrated through measurements, this approach supplies fast and slow ion densities everywhere with an acceptable factor of uncertainty. This approach has been followed by characterizing in detail a given plasma configuration in JONAS and modeling it with Spacecraft Plasma Interaction Software open source code. The mock-up was a simple plate. We measured the plasma characteristics through numerous current–voltage sweeps and current space profiles at given probe potential. The modeling was based on fast ion beam measurements close to the source and the residual pressure. Index Terms—Charge EXchange (CEX), plasma diagnostics, plasma simulation, wake effect.

I. I NTRODUCTION

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HEN TESTING a mock-up or a space equipment in a plasma tank, it is often important to determine the ion density. Simple plasma diagnostics, such as Langmuir Probes (LPs) or Retarding Potential Analyzers (RPAs), are usually used to characterize one-component plasma, either drifting or at rest. In ionospheric plasma simulation chambers, such as the ONERA plasma tank JONAS, a drifting plasma is emitted to simulate ambient plasma flowing at spacecraft speed. It is yet Manuscript received February 4, 2008; revised May 27, 2008. First published October 31, 2008; current version published November 14, 2008. This work was supported by a CNES R&T contract. SPIS software development has been supported by ESA Contracts 16806/02/NL/JA (TRP) and 19884/06/NL/JD (ARTES, French funded), and several CNES R&T contracts. J.-C. Matéo-Vélez, J.-F. Roussel, D. Sarrail, F. Boulay, and V. Inguimbert are with ONERA, 31055 Toulouse, France (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). D. Payan is with CNES, 31401 Toulouse, France (e-mail: denis.payan@ cnes.fr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPS.2008.2002822

impossible to avoid the generation of slow ions through Charge EXchange (CEX) reaction between fast drifting ions and slow neutrals [1]. In this context, the current–voltage characteristics are more difficult to interpret. One objective of this paper is to show that a proper analysis of LP characteristics allows distinguishing between the two components of the plasma. Three approaches have been adopted by various authors to model the current collected by the probe. These are the monoenergetic-ion model [2], the radial motion theory [3], [4] and the orbital-motion theory [5]–[8]. For collection of the hotter species (usually electrons), the theory of Langmuir is valid. For collection of the colder specie, usually ions, the choice between different theories depends on the range of various parameters. Let introduce the Debye ratio α = R0 /λD between the probe radius and the Debye length, β = Ti /Te the ratio between ion and electron temperatures. The orbitalmotion-limited (OML) theory has been proposed for α 1. However, Allen et al. [9], [10] demonstrated that even for α 1 the OML theory is never satisfied in Maxwellian plasmas. The case considered by Allen is β ≤ 1 (which is the majority of experimental plasmas). Nevertheless, even if this model is not fully representative, it gives a good approximation of current collection. The case α 1 is covered by the asymptotic analysis of Lam [11]. The case α ≈ 1 is divided in two subclasses. The case β = 0, related to cold ions, is covered by the radial motion model of Allen et al. [3], who extended Bohm’s theory [2]. The case 0 < β < 1 is more complicated and necessitates numerical computations. Bernstein and Rabinowitz [6] were the first to compute monoenergetic ion currents. Laframboise [7] adapted this orbital-motion method to Maxwellian distributions and obtained exact results. On a different way, assuming ions have a radial motion, Parker [12] fitted the numerical results of Langmuir and Blodgett [13] and obtained an expression of sheath thickness versus probe voltage. The next degree of realism consists in taking into account magnetic effects for electrons collected by a positive probe [14]. This paper aims to fulfill two objectives. The first objective is to propose a method for both fast and slow ion determination in JONAS tank. The fast monoenergetic ion density is easily obtained by the ion saturation current of planar or spherical probes. The slow ion density must be estimated by one of the theories cited above, depending on the parameters of the problem. Another difficulty inherent to most experimental characterizations is the necessarily limited number of measurements. A complementary approach consists in performing a numerical

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Fig. 1. Scheme of the plasma tank JONAS; the origin of (X, Y ) axes is the plasma source; the plate is located at X = 1.71 m.

simulation of the plasma dynamics in the tank. The second objective of this paper is to show how the proper “calibration” of the numerical modeling thanks to some measurements can lead to much wider and reliable predictions (over the whole tank, with densities possibly extended to energy distributions, etc.). In the following Section II-A, the experimental setup used is presented. We present a method to determine the fast ion density in Section II-B. In Section II-C, the slow ion density is estimated by using a spherical probe in a fast-ion-free region, i.e., without fast ions. As α is shown to be of the order of unity (see next sections), two current–voltage characteristics interpretations are possible, depending on the slow ion temperature. We assume that ions have a radial motion and compare the results obtained by considering β ≈ 1 and β = 0 and following Parker’s [12] and Allen et al.’s [3] previous work, respectively. In Section II-D, a numerical simulation of the current collection by a spherical probe in nonflowing plasma is proposed. It shows that radial motion models are valid in the range of parameters used in this paper. In Section III, a global numerical simulation of the plasma dynamics in the JONAS tank is presented. Finally, Section IV is devoted to the comparison between numerical and experimental data, in particular in term of prediction of fast and slow ion densities. II. C URRENT C OLLECTION BY P ROBES A. Plasma Tank The experiments are conducted in a cylindrical vacuum chamber which is 3.4 m long and 2 m in diameter. The pressure in the chamber can reach 10−5 Pa and is typically 5 × 10−4 Pa when the plasma source is operated. The argon plasma is produced by a Kaufman source. The electrons are injected through a neutralizer filament. A typical plasma density is 1012 m−3 (depending on the location) with ion energy of about 22 eV with an argon gas flow rate of 2.5 × 10−3 Pa · m3 · s−1 . In these experiments, the mock-up is a simple plate as shown in Fig. 1. This plate is 50 × 50 cm. The drifting ions of the Kaufman source can exchange their charge with residual neutral argon to create slow ions. These CEX ions can populate the whole tank, even in the wake of the plate. Other ions

Fig. 2. LPs on the displacement system (the plate does not appear in this picture; the large radius spherical probe is not used in this paper). The planar probes are right-hand side.

Fig. 3.

Ion collection by guarded planar LPs.

are negligible. Some Ar++ ions are created at the source but RPA measurements show they are negligible. Thresholds for sputtering of tank surfaces are about 50 eV which is far beyond fast ion energy. CEX reaction between fast ions and desorbed gases (such as H2 0) is much less efficient than with neutral argon because the background pressure is 50 times lower when the source is turned off than when the source is turned on, and because resonant CEX is much more efficient. The pressures measured in the tank with Bayard–Alpert and Penning vacuum gauges are P = 4.3 × 10−4 and 9.2 × 10−4 Pa, respectively. The tank is equipped with Helmholtz coils to compensate the magnetic field, which becomes lower than 10 μT. In the range of parameters used in this paper, the Larmor radius for electrons is greater than 10 cm while the Debye length is lower than 1 cm. In consequence, magnetic field effects can be neglected. Numerous current–voltage sweeps (also called I–V sweeps) are taken with three LPs mounted on a controlled displacement system as shown in Fig. 2. Two of the probes are guarded planar probes with a diameter of 2 cm, located at the same place. The first one is directed toward the plasma source and the second one toward the bottom of the tank. A spherical probe of 1.5 cm in diameter is also used. B. Measurement Method for Fast Ions (M1) The ion densities are determined through the two guarded planar probes. That enables to discriminate fast and slow ion current collection as shown in Fig. 3. The plasma source directed LP (LP1) collects both fast and slow ions while the back directed one (LP2) collects only slow ions.

MATÉO-VÉLEZ et al.: GROUND PLASMA TANK MODELING AND COMPARISON TO MEASUREMENTS

Fig. 4.

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Ion branch extrapolation to near 0-V plasma potential.

Numerical results obtained by Roussel et al. [1] show as follows. 1) The fast drifting ions cannot easily be deflected by the probe potential usually smaller than their kinetic energy. 2) The slow ions are collected at the surface of the sheath whose radius rapidly increases with probe potential. It was also proven by Allen et al. [3]. Then, following Garnier [15] and assuming that the slow ion velocity distribution is isotropic, the current–voltage sweep of LP2 is subtracted from the one of LP1 to obtain the fast ion current Iif . Slow ion current Iis is directly obtained by LP2. A current density profile j(Y ) is then measured at a distance X near the plasma source for a given probe potential. Assuming that the ion branch method is valid for fast ion current (linear current variation with probe potential), j(Y ) profiles at −10 and −5 V probe potential φLP gives access to the undisturbed fast ion beam by extrapolation at plasma potential, close to 0 V (cf., Fig. 4). This method is directly inspired by OML theory. For a probe negatively biased with respect to the plasma potential φP , i.e., repulsive for electrons and attractive for ions, collected currents are −φP ) kTe e(φLP 2 Ie = − eNe πrpl e kTe (1) 2πme 2kTif e(φLP − φP ) f f 2 Ii = eNi πrpl 1− (2) mi kTif respectively, for electrons and fast ions. In these equations, Ne and Nif , respectively, denote the electron and fast ion densities, Te and Tif are their temperatures, me and mi their masses, rpl is the planar probe radius, e the elementary electric charge, and k the Boltzmann constant. During experiments, fast ion energy is always greater than φLP . It can then be assumed that their trajectories are not influenced by probe potential. A good estimation of the current is then given by the product of the monoenergetic ion current density multiplied by the probe surface (2). According to (2), the ion density depends on the plasma potential. Using the measured current–voltage sweeps, the plasma potential is known with an uncertainty of 0.5 V; its value stands between −1 and 1 V. Electrons are extracted from a heated filament and are used to neutralize the plasma. Their initial temperature is about 0.1 eV but they are submitted to potentials of the order of plasma potential. The electron temperature can be extrapolated from the exponential growth of I–V sweeps for probe potentials near the plasma potential. According to (1), the electron temperature is obtained by the slope of ln(Ie ). In the tank, the electron temperature is about 0.1 eV.

Fig. 5. Fast ion density Y -profiles measured at different X locations from the plasma source.

The fast ion beam profiles obtained at X = 230 mm and X = 800 mm are presented in Fig. 5 by assuming φP = 0 V. The maximal density is about 1013 m−3 . The density profiles obtained by assuming φP = 1 V or φP = −1 V are practically equal, the uncertainty being less than 10%. As a result, fast ion density can satisfactorily be deduced from the ion branch extrapolated to 0 V using (2). The aperture angle of the source is about 80◦ . The beam shape, clearly visible at X = 230 mm, is much flatter at X = 800 mm, where the beam is larger than the tank. In principle, slow ion current collection can be estimated thanks to LP2 measurements by the same linear interpolation of the ion branch. OML theory states that the slow ion current collected by a planar probe is given by kTis e(φLP − φP ) s s 2 1− . (3) Ii = eNi πrpl 2πmi kTis However, it has shown to be less accurate than for fast ions. The reason why this method gives good results for fast ions and not for slow ions is that fast ions are not very sensitive to −10-V LP potentials. Only fast ions whose initial direction intercepts the probe are collected. On the contrary, slow ions entering the plasma sheath around the probe are collected. OML theory is acceptable for electron collection but simply approximate for ions (see [9] and [10]). A planar probe, when immersed in a plasma, defines an ill-defined region even if guarded. As this sheath increases when the voltage becomes more negative, more slow ions are collected. As a consequence, the simple approach of the ion branch extrapolation heavily relies on the assumption of its linear behavior with potential, which is only approximate. The I–V sweeps measured in the plasma source axis with the planar probe directed toward the back of the tank are interpreted with the OML model (3). This equation must be provided with plasma potential and slow ion temperature. We assume that the temperature of slow ions, initially created at ambient temperature (0.025 eV), does not exceed electron’s one. At location X = 230 mm downstream the source, considering φP = 0 V and Tis = 0.05 eV, the slow ion density is 3.5 × 1012 m−3 . The maximal and minimal slow ion densities obtained for φP between −1 and 1 V, and Tis between 0.025 and 0.1 eV differs

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from a factor two (between 2.5 × 1012 and 5 × 1012 m−3 ). The Debye ratio α corresponding to these values is greater than two. Then, it should be noted that OML theory might not apply in the tank for slow ion determination. C. Slow Ion Measurements in Nonflowing Plasmas (Method M2) Slow ions are discriminated from fast ions by measuring I–V characteristics behind the plate. Fast ions do not enter this region. We assume that the plasma is composed of Maxwellian ions at rest, and Maxwellian electrons. These assumptions permit to consider different theories. Assuming moreover that ions are isotropic, we consider here the current collection by a spherical probe. OML theory can give a good approximation of plasma density only if α 1. Let us consider first the OML model and determine the plasma density obtained with the linear extrapolation approach of Fig. 4 (method M1). The I–V characteristics are interpreted considering the following electron and slow ion currents collected by a spherical probe of radius R0 : −φP ) kTe e(φLP 2 Ie = − eNe 4πR0 e kTe (4) 2πme kTis e(φLP − φP ) s s 2 Ii = eNi 4πR0 1− . (5) 2πmi kTis In the following, the probe used is 15 mm in diameter and is located 200 mm behind the plate. The electron temperature is estimated to 0.1 eV thanks to the slope of ln(Ie ). Ions are created at ambient temperature, i.e., 0.025 eV, and are submitted to potentials of the order of some tenths of a volt. Their temperature might not exceed electron’s one. Then, we assume that 0.025 eV ≤ Tis ≤ 0.10 eV. Maximal and minimal plasma densities obtained for plasma potential between −1 and 1 V, and ion temperature between 0.025 and 0.10 eV, are 4.0 × 1012 and 0.7 × 1012 m−3 , respectively. There is a factor six between the minimal and maximal estimates. It is worth noting that α is between 3 and 12. Then, OML theory is not applicable and the method M1 not reliable. For α ≈ 1, the ion collection is better defined by the radial motion approach (method M2). For the high Debye ratio case, a sheath appears close to the surface of the probe that collects the ion current. As said before, the current depends on the ion temperature. There is no easy way to determine the ion temperature and only physical qualitative arguments can be provided. Let us consider two extreme cases, namely β 1 and β = 1. In the first case, the thermal current collected at the sheath is increased by about 50% due to the effect of the presheath. The presheath is a quasi-neutral region in which ions are accelerated to match the Bohm condition [2]. Then, according to Allen et al. [3] who extended Bohm’s work by taking into account the sheath thickness, the slow ion current collected by a spherical probe is kTe , for β 1 (6) Iis = 0.61eNis 4πS 2 mi


in which one can see that the ion current depends on electron temperature and not on ion’s one. S is the sheath thickness. Following Cooke’s review [16], who refers to Langmuir and Blodgett [13] and Parker [12] previous works for nonflowing Maxwellian plasma, in the space charge limited regime (i.e., high Debye ratio), the plasma sheath S around a spherical probe is given by

1/2 1 D + 4 R0 D D D + 0.052 H − 0, 2 , for ≤ 19 R0 R0 R0 .753 .752 D S D = 1+ , for > 19 R0 R0 R0 1 S = + R0 2

(7) (8)

where R0 is the probe radius, H(x) is the unit step function (H(x < 0) = 0; H(x ≥ 0) = 1). D is the 1-D analytic sheath thickness obtained by Child [17] and Langmuir [18]

e(φLP − φP ) kTe

D = 1.26λD ε0 kB Te λD = N e2

3/4 (9) (10)

where λD and N are the Debye length and the plasma density, respectively. The model used for β = 1, so-called hot ion model, is the same as before except that ions have enough energy to enter the sheath, and that, no presheath is necessary. In that case, the current is due to ions entering the 3-D sheath thanks to their thermal velocity kTis s s 2 , for β ≈ 1. (11) Ii = eNi 4πS 2πmi The Langmuir–Blodgett model is valid only if the orbital motion of particles around the probe is negligible. A particle entering the probe sheath cannot miss it. This implies that the translation due to thermal velocity must not deflect the particle more than the probe radius R0 , leading to the necessary condition 2 S e(φLP − φP ) > . (12) kTis R0 The nonflowing plasma hypothesis of the model is fulfilled in the wake of the plate. I–V characteristics obtained with the 15 mm diameter spherical probe are used. We dispose of two models of ion current collection, depending on their temperature. For the sake of conciseness, we present the complete interpretation of I–V characteristics by assuming β = 1 and only give the results obtained when assuming β 1. An iterative method has been developed to determine ion density as follows. 1) Te (= Ti when β = 1) is set to a value Te [n] of some tenths of an electronvolt. 2) Calculation of the electron and ion currents with (4) and (11), respectively.


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Fig. 7. Simulation of the sheath around a spherical probe; φP = −10 V, N = 1.2 × 1011 m−3 , Ti = Te = 0.09 eV. Fig. 6. I–V characteristics of a spherical probe 200 mm behind the plate and comparison to the hot ion model.

3) Plasma density is adjusted to obtain a good comparison to the measured I–V characteristics. The plasma potential φP can be extracted from the measured curves with a precision of only about 0.5 V. For that reason, we adjust φP in the model to equal the measured and computed floating potentials (the measured one being very easy to obtain). 4) Estimation of the measured electron current by subtraction of the computed ion current (11) to the measured total current. 5) Calculation of the electron temperature Te [n + 1] with the slope of ln(Ie ). 6) If Te [n + 1] = Te [n], a new iteration is performed. By this way, it is possible to determine both electron temperature and plasma density. An example is given in Fig. 6 in which the I–V sweep is measured 200 mm behind the plate. The model conveniently fits the measurement for Te = Ti = 0.09 eV, N = 1.2 × 1011 m−3 , and φP = 0.35 V. We verify a posteriori that α ≈ 1. According to (12), the ion collection model (7)–(10) is valid only if φP < −0.9 V. For φP greater than −0.9 V, numerical computations must be carried out. Nevertheless, the domain of validity is large enough. In the cold ion assumption β 1, the ion current calculated at step 2) is replaced by (6). The model conveniently fits the measurements presented in Fig. 6 for Te = 0.09 eV, N = 5.0 × 1010 m−3 , and φP = 0.29 V. The radial motion models assume ion isotropy. Complementary measurements have been performed with the two planar probes placed behind the plate. The first one is directed toward the plate and the other one toward the opposite direction. There is a factor two between the two collected currents. Nevertheless, the probe used with the method M2 being spherical, the collected current is approximately an average of the two currents (upstream and downstream). The factor of uncertainty due to anisotropy is then about 50%, which is lower than uncertainties on ion temperature. The values of the slow ion density obtained by the two radial motion models define the limits within which the real value stands. It can then be deduced that a maximal factor of three exists between the estimated and real values of plasma density.

Fig. 8. Current collected by a spherical probe 15 mm in diameter, immersed in a Maxwellian plasma at rest. N = 1.2 × 1011 m−3 , Te = Ti = 0.09 eV; comparison between theories and SPIS simulation.

D. Simulation of Current Collection by Probes The Spacecraft Plasma Interaction Software (SPIS) [19]– [21] has been used to numerically simulate the physics of the current collection by a spherical probe immersed in a Maxwellian plasma at rest. The goal is to reproduce the hot ion model and more generally to show that the radial motion theory (including hot and cold ion models) is adapted to the slow ion current collection by probes in JONAS. The plasma dynamics consists in a particle-in-cell (PIC) method for ions. Their mean energy is set to 0.09 eV. The electrons follow Boltzmann distribution with a mean energy of 0.09 eV. The plasma density is 1.2 × 1011 m−3 . The spherical probe is 15 mm in diameter. The external boundary is a sphere of radius 200 mm, which is larger than the Debye length (λD = 6.4 mm) and than the sheath thickness estimated by (7) and (8) (S = 61 mm at φP = −10 V). The minimal mesh size is 2 mm. The number of tetrahedra is 62 000. The plasma potential is represented in Fig. 7 for φP = −10 V. The sheath thickness is about 65 mm which is in agreement with the Langmuir–Blodgett model (7) and (8). The computed current voltage characteristic is presented in comparison to the hot ion model (4) and (11) in Fig. 8. The theoretical OML current obtained by (4) and (5) is also represented. In this example, SPIS simulation is in better agreement with the radial motion model. For large negative values of probe potential, the difference between radial motion model and SPIS simulation is about 10%. The OML model is clearly not reproduced by this simulation (difference of 50%). It is due to the fact that the Debye ratio is close to unity.

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Fig. 9. Current collected by a spherical probe 15 mm in diameter, immersed in a Maxwellian plasma at rest. N = 1.0 × 1010 m−3 , Te = Ti = 1 eV; comparison between theories and SPIS simulation.

A complementary simulation has been performed for α = 0.1. The parameters used are the same, except N = 1010 m−3 and Te = T+ = 1 eV. The results presented in Fig. 9 show a good agreement between the computation and the OML model. The radial motion model is not satisfactory because the condition (12) is never satisfied. In conclusion, depending on the value of the Debye ratio, SPIS simulations of current collection by a spherical probe can give good approximations of the orbital or radial motion models. The domains of validity of these models are well reproduced by the computations. In the ionospheric plasma chamber JONAS equipped with probes of some millimeters of radii, the Debye ratio is close to unity and the radial motion theory applies. The slow ion density can be estimated in the wake of a plate with a factor of two or three, which is due to uncertainties on ion temperature. III. N UMERICAL S IMULATION OF THE P LASMA T ANK SPIS has numerically been used to simulate the physics in the plasma tank JONAS. The methodology consists in using experimental input data of the fast ion beam and to perform a simulation of the plasma dynamics taking into account the tank and the plate. A. Geometry The CAD model of the plasma tank done with gmsh mesh generator is represented in Fig. 10. It takes into account the 3.4 × 1.85 m cylindrical tank, the 50 × 50 cm pump orifice, the 3 cm diameter plasma source, and the 50 × 50 cm plate inside. The grid mesh is composed of 80 000 tetrahedra. B. Numerical Model The plasma source is derived from the experimental results obtained at X = 230 mm, as shown in Fig. 5. The fit is used as an input data for the velocity distribution function of fast ions of average energy 22 eV, with an energy dispersion estimated to 0.5 eV. The total current injected is 1.1 mA. The plasma dynamics consists in a PIC method for fast and slow ions. In the plasma tank JONAS, the electrons are emitted from a filament.


Fig. 10. CAD model of the plasma tank JONAS; (gmsh modeler and mesh generator, interfaced with SPIS, see http://www.spis.org and http://www.geuz.org/gmsh/).

It is not necessary to model this creation because electrons rapidly follow a thermal Boltzmann distribution. We use a reference density N0 = 1014 m−3 in Boltzmann distribution, which is slightly larger than the maximal ion density produced in the tank. This value is quasi-arbitrary since a rescaling of Boltzmann exponential is equivalent to a potential shift. The only volume reaction considered is the CEX between + fast ions and neutrals: Ar+ f + Ar → Ar + Ars since its cross section is typically around one order of magnitude larger than that of elastic collisions. The neutral dynamics is not calculated but their background density is estimated by the measured residual pressure. The variation of the slow ion density during the time step dt is proportional to neutral and fast ion densities N (Ar) and N (Ar+ f ) ΔN Ars+ = N Arf+ · N (Ar) · σ · Vrel · dt (13) where σ is the cross section of CEX reaction and Vrel the relative velocity of neutrals and fast ions. For 22-eV drifting argon ions, we have 0.35 × 10−18 m2 < σ < 0.5 × 10−18 m2 [22]. In this paper, σ = 0.4 × 10−18 m2 . The uncertainty on this parameter is then about 25%. The numerical model of CEX is of the simple Monte Carlo collisions type. Slow ions are generated from (13) by comparing ΔN/Nmin to a random number between 0 and 1, where Nmin is the smallest admissible weight of a superparticle (or generating several superparticles if this ratio is higher than 1). Fast ion and neutral are considered as not affected by the interaction (neutral background pressure). For N (Ar) = 1017 m−3 , which corresponds to the measured 4.3 × 10−4 Pa pressure, the fast ion mean-free path 1/N σ is 25 m. Hence, only a few percent of the fast ions undergo CEX in the tank, justifying the assumption of neglecting CEX for this population. The electric potential follows the nonlinear Poisson equation, which is solved with an implicit method: eφ e f s + N+ − N0 exp N+ . (14) −Δφ = ε0 kTe The numerical parameters used in this paper are presented in Table I. The fit of the fast ion beam measurement is used as an input data. The ion source in the model is obtained by


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TABLE I NUMERICAL PARAMETERS

Fig. 12. Slow ion density (logarithm scale); slow ions are attracted by the negative wake potential.

Fig. 11. Fast ion (upper cutting plane) and slow ion (lower cutting plane) densities; the same logarithm scale is used for both cutting planes.

generalizing this data by axisymmetry. The choice of N0 = 1014 m−3 , hence of potential origin, led us to set the plasma tank potential to a value of −2 V. The first justification of this value is that it gives approximate agreement with experimental measurements of plasma potential (slightly positive with respect to tank potential). Its exact value is indeed not very important since it is efficiently screened by this dense plasma, and it must indeed only be negative enough to avoid an unphysical reflection of the ions by the tank surfaces. Finally, the pump orifice is modeled as a sink for particles.

Fig. 13. (Top) Plasma potential and (bottom, logarithm scale) electron density; the wake is negative with respect to the plume.

attracts slow ions which are created at ambient temperature. As a consequence, there is an accumulation of CEX ions behind the plate. The electron density and the plasma potential are correlated because of the Boltzmann distribution assumption for electrons. IV. M ODEL D ATA C ROSS C OMPARISON

C. Results The spatial distributions of fast and slow ion densities are presented in Fig. 11. This figure represents cutting planes containing the source axis. They are represented with the same logarithmic scale. The fast ion beam clearly shows a spatial decay corresponding to the divergence of the plasma plume. The fast ion density decreases from 1014 to 1011 m−3 away from the source. The aperture angle of 80◦ is well represented, and the wake effect due to the plate is clearly visible. The slow ions are mostly created in the vicinity of the plasma source and are slightly accelerated by the potential gradient resulting from the fast (and slow) ion density gradient (through electron barometric law). That makes them drift slowly toward the tank walls. Their density is in the range of 1010 −1012 m−3 . A more precise view of their space distribution is given in Fig. 12. There is a more negative potential region behind the plate due to the lack of fast ions, as visible in Fig. 13. This region

The first comparison consists in verifying that the numerical fast ion density profile obtained in the plane X = 230 mm corresponds with the input data given by experiments. The two curves in Fig. 14 are in good agreement. The fast ion density profile along the Y = 0 axis is also shown in Fig. 15. The expected experimental 1/x2 decay is well reproduced by the numerical simulation. The slow ion density is represented in Fig. 16, which gives many information about both numerical and experimental results. In this graphic, the experimental data are represented for ion temperature Tis between 0.025 and 0.09 eV. Concerning the OML model, it also takes into account uncertainties on plasma potential (−1 < φP < 1 eV). The numerical slow ion density is also subject to uncertainties since, as introduced before, the pressure is known within a factor two (depending on the vacuum gauge used). The uncertainties on CEX cross section (< 25%) and on fast ion density are much lower (< 10%).

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Fig. 14. Experimental and numerical fast ion density Y -profiles at location X = 230 mm.


Fig. 16. CEX ion density profiles along the source axis obtained with methods M1 [OML model with a planar probe directed toward the back of the tank (3)] and M2 (cold ion (6) and hot ion (11) radial motion models with a spherical probe behind the plate) compared with numerical results (dashed line represents the error bar on numerical results).

modeling the inflow of neutrals at the source, the collisions with the wall, the ion-electron recombination, and the evacuation to the pumping system. The very interesting outcome of that would be a prediction of the pressure in the tank, giving another quantitative check of the simulation accuracy. V. C ONCLUSION

Fig. 15. Fast ion density profile along the source axis (logarithm scale); the slope of the curves are close to −2. The three points are located 230, 800, and 1230 mm away from the source.

Statistical noise of PIC method is less than 1%. Another possible source of error is the mesh size. A complementary calculation with a coarser mesh (12 000 tetrahedra instead of 80 000) shows a discrepancy lower than 10%. Finally, a factor two of uncertainty on SPIS calculation of JONAS tank can be roughly estimated, which is taken into account in the plot of Fig. 16. Numerical results are in better agreement with the experimental data treated with the method M2 than with the treatment following the method M1. 200 mm behind the plate, the computed slow ion density is 4 × 1010 m−3 . There is more than a factor 20 between this numerical simulation and the OML method M1. That suggests that, as announced earlier, the experimental method consisting in measuring the slow ion density in a wake region thanks to the radial motion model (method M2) is more adapted than the orbital-motion model (method M1) in the range of parameters used in this paper. The simulated slow ion density underestimates by a factor three the measurements treated with the Langmuir–Blodgett model. Nevertheless, the margins calculated from a factor two of uncertainty on background pressure can explain the differences between measurements and simulation. As a final remark, let us repeat that our numerical simulation considers the residual neutral pressure as a uniform background density, taken from measurements. An alternative would be to perform a Direct Simulation Monte Carlo of neutrals, explicitly

Two I–V measurement methods used in this paper provide fast and slow ion densities in plasma simulation tanks. The first one consists in linearly interpolating the ion branch of two guarded planar probes for fast ions. The second one relies on the radial motion model of Langmuir–Blodgett. In this model, the current collected by a spherical probe in nonflowing plasma gives access to the slow ion density with a factor of uncertainty of three. The OML model is not adapted to slow ions in the range of parameters used in the JONAS tank. The modeling of the plasma dynamics is convincing of the validity of the numerical approach. The spatial decrease in the fast ion density is close to the 1/x2 experimental law, and the wake effect is well modeled. Charge exchange ions fill the whole tank due to their low kinetic energy. That demonstrates the necessity to take them into account during experiments. This paper shows that radial motion models should be used in JONAS instead of the OML model in order to obtain the more realistic estimation of the slow ion density. Finally, this paper highlights the possibility to use SPIS open source code to determine the ion densities in the whole tank (with a factor of uncertainty of three for slow ions). A single fast ion profile measurement in the near field plume is sufficient to calibrate the plasma dynamics modeling. Future works may focus on neutral dynamics in JONAS, or the modeling of other ground experiments, e.g., ONERA SIRENE tank, devoted to GEO environment simulation. ACKNOWLEDGMENT The authors would like to thank the useful discussions with the participants to the Spacecraft Plasma Interaction Network in Europe workshops (cf., http://www.spis.org).


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Jean-Charles Matéo-Vélez, photograph and biography not available at the time of publication.

Jean-François Roussel, photograph and biography not available at the time of publication.

Daniel Sarrail, photograph and biography not available at the time of publication.

Françoise Boulay, photograph and biography not available at the time of publication.

Virginie Inguimbert, photograph and biography not available at the time of publication.

Denis Payan, photograph and biography not available at the time of publication.