complex phenomena in magnetized plasmas with an electron emission yevgeny raitses princeton plasma...
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Complex phenomena in magnetized plasmas with an electron emission
Yevgeny RaitsesPrinceton Plasma Physics Laboratory
Michigan Institute for Plasma Science and Engineering Ann Arbor, December 5, 2012
Plasma Science & Technology Research
at Princeton Plasma Physics Laboratory(PPPL)Heavy ion beam
MRI
Outline
• EB plasma devices:- Configurations- Electron rotating effects- Maximizing electric field applied in plasma
• Anomalous electron cross-field transport:- Secondary electron emission effects- Turbulent fluctuations and coherent structures- Suppression of anomalous electron transport
• Summary and concluding remarks
PPPL DC-RF EB discharge of Penning-type
DC E×B fields applied in a 20 cm × 50 cm st. steel chamber with ceramic side walls
Plasma cathode: 2 MHz, 50-200 W Ferromagnetic ICP
Operating parameters: Bkg. pressure: 0.1-1 mtorrRF-power: 50-60 WDC voltage/current: 0-100 V/0-3 AMagnetic field: up to 500 Gauss
Magnetically shieldedRF-plasma cathode
E
Coils Coils
B
Anode
Insulator
Axis
Plasma in E ×B region: weakly collisional, non-equilibrium, with magnetized electrons and non-magnetized ions
4/14
Neutral density ~ 1013 cm3
Plasma density ~ 0.5-3 1011 cm-3
Electron temperature ~ 3-5 eV
Magnetic field: 5-500 Gauss
ea/L ~ 1-2
ei/L ~ 10
ee/L ~ 20-50
ia/L~ 0.5-3
Energy relaxation length in inelastic range > *
*/L ~ 2
ce/coll ~ 150-200
For B = 35 GaussElectron cross-field displacement during time loss (inelastic or wall collisions)
X ~ 2RLe (scat /2loss )0.5
Examples of E ×B devices
Sputtering magnetron discharge
Large Plasma Device (LaPD) at UCLA20-meter long, 1 meter diameter
Penning Gauge
ee
Diam ~ 1 -100 cm
B ~ 100 Gauss
Working gases: Xe, Kr
Pressure ~ 10-1 mtorr
Vd ~ 0.2 – 1 kV
Power ~ 0.1- 50 kW
Thrust ~ 10-3 - 1N
Isp ~ 1000-3000 sec
Efficiency ~ 6-70%
Unlike ion thruster, HT is not space-charge limited Thrust density is limited by B2/2
e << L << i
Hall Thruster (HT) – fuel effective plasma propulsion device for space applications
E =-ve B
Neutral density ~ 1012-1013 cm3
Plasma density ~ 1011-1012 cm-3
Highly ionized flow: ion/n~ 80%
Electron temperature ~ 20-60 eV
Ion temperature ~ 1 eV
Ion kinetic energy ~ 102-103 eV
Collisionless, non-equilibrium plasma with magnetized electrons and non-magnetized ions
ea/h ~ 20 – 200
ei/h ~ 4103
ia/h ~ 10-100
Energy relaxation length in the inelastic range
*/h ~ 30 - 300
Parameters of HT plasma
Comparison of different E B plasma devicesDevice\Parameter R
cmL
cmT
eVB
GaussEmax
V/cmLAPD 50 1700 2-5 400 4-18Compact Auburn Torsatron 17 53 10 1000 5Blaamann 8 65 9 570 2-6Continuous Current Tokamak
40 150 150 3000 120
ALEXIS 10 170 5 100 2Reflex arc 2.5 300 5 4000 20Mistral 11.5 140 1.4 220
Maryland Centrifugal Experiment (MCX)
27 250 3 2000
CSDX 10 280 1.5-3 650 3-4
WVU Q-machine 4 300 0.2 1400 14State-of-the-art Hall thruster
2 2 20-60 100-300 700
PPPL Segmented Hall Thruster
2.5 2 100 115 1000
Cylindrical Hall thruster (CHT) – EB plasma in diverging magnetic field
Cathode-neutralizer
Electromagnets
N
N
Anode
Ceramic channel
S
S
Annular part
F = -B
F = -eEB
B
Cathode-neutralizer
Electromagnets
N
N
Anode
Ceramic channel
S
S
Annular part
F = -B
F = -eEB
B
• Similar to conventional HTs, the CHT operation is based on closed electron EB drift.
• Fundamentally differences from conventional HTs:
Electrons are confined in the magneto-electrostatic trap.
Ions are accelerated in a large volume-to-surface channelRelated concepts
DCF by MIT and HEMP by Thales, CHT by Osaka, etc.
Raitses and Fisch, Phys. Plasmas 8, 2579 (2001)
Unusual focusing of the plasma flow in diverging magnetic field of CHT
Cathode-neutralizer
Electromagnets
N
N
Anode
Ceramic channel
S
S
Annular part
F = -B
F = -eEB
B
Cathode-neutralizer
Electromagnets
N
N
Anode
Ceramic channel
S
S
Annular part
F = -B
F = -eEB
B
0
20
40
60
80
-90 -45 0 45 90
Angular position, deg
Ion
cu
rre
nt
de
ns
ity,
uA
/cm
^2
Self-sustained
Non-self-sustainedIkp=2.5 A
LIF measurements of ion velocity
Ion current in plume
Spektor et al., Phys. Plasmas 17, 093502 (2010)Raitses at al., Appl. Phys. Lett. 90, 221502 (2007)
Plasma with azimuthal symmetric magnetic field and E×B rotating electrons is commonin industrial and laboratory plasmas: non-neutral plasmas, solar physics, magnetic mirrors, magnetic fusion devices, plasma centrifuges and, most recently, plasma thrusters
Cathode-neutralizer
Electromagnets
N
N
Anode
Ceramic channel
S
S
Annular part
F = -B
F = -eEB
B
Cathode-neutralizer
Electromagnets
N
N
Anode
Ceramic channel
S
S
Annular part
F = -B
F = -eEB
B
Rotating electron effects
1211
rr
B
12 E
const
rB
EBE
12
12
Isorotation
For magnetized electrons and non-magnetized ions, common assumption is that magnetic surfaces are equipotential surfaces leads to a force field that is perpendicular to the magnetic surfaces, a good assumption for non-rotating cold magnetized plasma
Ion focusing due to rotating electron effects
)ˆ(ˆ
2 br
e
m
en
bPE e
e
eS
Pressure gradient Centrifugal force effect on electrons
2BEec rmF
Non-magnetized ions are not affected by the magnetic field, but the addition of the field Es results in focusing deflection of the original electric field En
Fisch et al., PPCF, 53, 124038 (2011)
sinsinsinthe
BELe
ec
BE u
r
Ion focusing should benefit from supersonic electrons
Challenging requirements for the generation of supersonically rotating electrons in a steady state
- Strong electric field and low magnetic field to get high EB speed
- Colder plasma
Common approach: Control of E-field with biased electrodes
HT is capable to generate supersonically rotating electrons Device\Parameter R cm
Lcm
TeV
B Gauss
Emax
V/cm
VE/B/Veth
LAPD 50 1700 2-5 400 4-18 < 210-2
Compact Auburn Torsatron
17 53 10 1000 5 < 410-3
Blaamann 8 65 9 570 2-6 < 810-3
Continuous Current Tokamak
40 150 150 3000 120 810-3
ALEXIS 10 170 5 100 2 210-2
Reflex arc 2.5 300 5 4000 20 510-3
Mistral 11.5 140 1.4 220 410-3
Maryland Centrifugal Experiment (MCX)
27 250 3 2000 710-2
CSDX 10 280 1.5-3 650 3-4 < 110-2
WVU Q-machine 4 300 0.2 1400 14 510-2
State-of-the-art Hall thruster
2 2 20-60 100-300 700 < 1
PPPL Segmented (No SEE) Hall Thruster
2.5 2 100 115 1000 1 - 2
Electric field and thruster performance are affected by anomalous electron cross-field transport
Thruster efficiency
With all other parameters held constant, HTs efficiency reduces with increasing electron current across the magnetic field
Classical collisional mechanism can not explain the discharge current measured for Hall thrusters: e-a~ 106 s-1 < eff ~ 107 s-1
Enhanced cross-field conductivity in HTs usually attributed to1) SEE induced near-wall conductivity
2) Anomalous (Bohm-type) diffusion induced by high frequency azimuthal plasma oscillations
3) A new route for electron transport across magnetic field - low frequency rotating spoke oscillations
ei
i
e
jet
II
I
P
TV
2
1
1.5
2
2.5
3
0 200 400 600 800Discharge voltage, V
Dis
ch
arg
e c
urr
en
t, A
Carbon segmented
Boron nitride
Effect of the channel wall material on the discharge characteristic
Carbon segments drastically change V-I characteristics
- Boron nitride - high SEE
- Carbon velvet - zero SEE
Raitses, Staack, Dunaevsky, Fisch, Phys. Plasmas (2006)
Wall material affects the maximum electron temperature in the thruster
0
30
60
90
120
100 200 300 400 500 600 700 800
Discharge voltage, V
Ma
xim
um
ele
ctr
on
te
mp
era
ture
, e
V
High SEE BN channel
Low SEE segmented
Raitses , Staack, Smirnov, Fisch Phys. Plasmas ,2005
Electron temperature from emissive probe measurements
PPPL Hall thruster setup
SEE from dielectrics reaches 1 at lower energies (< 50 eV) of primary electrons than for metals
0 20 40 60 80 1000.0
0.5
1.0
1.5
2.0
Eprimary
(eV)
Teflon
Boron Nitride
Pz26 -
Pz26 +
Note: for boron nitride, if primary electrons are Maxwellian(Te) 1 at Te = 18.3 eV
PPPL SEE setup
Dunaevsky, Raitses, Fisch, Phys. Plasmas (2003)
SEE can significantly enhance electron flux from plasma to the wall
e
ionew πm
M
e
kT
21ln
scs Te
w(x)
e
i
see
ecrcr
e TT scsw 1
SEE turns sheath to space-charge limited regime [Hobbs and Wesson, 1967]
When
Xenon)(for 275 ,0 w eT. When
Fluid Approach
SEE effect on plasma electrons: comparing experiment with predictions
Large quantitative disagreement with fluid theory!
0
30
60
90
120
100 200 300 400 500 600 700 800
Discharge voltage, V
Ma
xim
um
ele
ctr
on
te
mp
era
ture
, e
V
High SEE BN channel
Low SEE segmented
Fluid theory Temax 18.3 eV
According to fluid theories, the maximum electron temperature should not be above 18.3 eV (for BN and Xenon)
Hall thruster plasma, 2D-EVDF Isotropic Maxwellian plasma, 2D-EVDF
EVDF in HT is strongly anisotropic with beams of SEE electrons
Loss cones
and beams
Sydorenko et al, Kaganovich et al., Phys. Plasmas (2005, 2006, 2007 2009), Ahedo, Phys. Plasmas (2005)
(x)i
2
1b2
1p
i
1b
1p
1- primary2- secondary
SEE coefficients: p 2p / 1p - SEE due to plasma electrons
b 2b / 1b - SEE due to beam electrons
1b / 2 - Penetration of the SEE beams
Electron fluxes have several components, including counter-streaming SEE beams from opposite walls
)( bp
peff
1
Total emission coefficient:
Note, p can be > cr if eff < cr
PIC simulations predict:
• EVDF is decreasing f (vx)
• Beam penetration is high, 0.9
0)(2
vfv
f(vx)
f(vx)
unstable
stable
vx
vx
0)(2
vfv
Conditions for the existence of self-sustained counter-streaming SEE electron beams
Sydorenko et al., Phys. Plasmas 2007
1) Weak two-stream and plasma beam instabilities
Conditions for self-sustained counter-streaming SEE electron beams (Cont’d)
2) Sufficiently strong electric field
- SEE electrons gain additional energy
during the flight between the channel
walls due to EB motion
- This energy must be high enough to
induce strong SEE on opposite wall
The maximum additional energy is scaled as
For typical HT conditions: E = 100-200 V/cm, B ~ 100 Gauss
Bmax ~ 30-60 eV enough for strong SEE from any ceramic material
LeB eE 2max
gives average velocity
/c cv zd
x
Ev u
B
21p ex z
bz eb x
T EmJ n
H M B
The displacement , , during the flight time H/ubx
~ /z d bx cu u u H
0
1
-4 -2 0
z/c
x/H
E
B
Near-wall conductivity SEE-induced cross-field current
Wall collisionality - exchange of primary magnetized electrons by non-magnetized SEE electrons
and current
Kaganovich, Raitses, Sydorenko. Smolyakov, Phys. Plasmas (2007)
Two profiles for two regimes of SEE-induced electron cross-field current
Classical sheath with SEEE = 200 V/cm
Predicted profiles of the cross-field current density:
Inverse sheath at a very strong SEE > 1, E = 250 V/cm
Qualitative differences between the potential profile, relative to the wall, of a classical sheath (a), SCL sheath (b) and the new inverse sheath (c). Note that plasma electrons are still confined by the SCL sheath, but not confined by the inverse sheath.
Results of particle-in-cell simulations of Hall thruster discharge: a comparison of results with classical (Sim. A), E = 200 V/cm, and inverse sheath (Sim. B) E = 250 V/cm.
M. Campanell et al., Phys. Rev. Lett. 108, 255001 (2012)
Disappearance of near-wall sheath at a very strong SEE > 1
When plasma is bounded with non-emitting and zero-recycling (100% absorbing) walls
Low back flux of contamination:
•Ion grazing incidence
•Redep. is trapped in velvet texture
Low SEE because:
• Carbon has low SEE
• SEE electrons are trappedin inter-fiber micro cavities
Carbon fibers bonded to carbon substrate
Engineered materials to mitigate plasma-surface interaction effects, e.g. carbon velvet material
Raitses, Staack, Dunaevsky, Fisch, Phys. Plasmas (2006)
Without SEE, the magnetized plasma can withstand much stronger electric field
Probe path
0-4.6 cm
2.5 cm
High-SEE No-SEE
With No-SEE walls, the electric field at high voltages, 1 kV/cm, approaches a fundamental limit for a quasineutral plasma:
E ~ Te/D (Te ~ 100 eV, ne ~ 1011 cm-3)
Without SEE, the cross-field mobility reduces to almost classical collisional level
Experimental cross-field mobility estimated using measured data and 1-D Ohm’s law at the placement of Emax
Possibly EB shear effect?*
For No-SEE, the shearing frequency, d(Ez/Br)/dz, reaches 5-8 nsec-1 at 600 V
Such a large shear may affect the dynamics of all instabilities, which were previously predicted for Hall thrusters at moderate voltages
*Fernandez, Cappellli, et al., Phys. Plasmas 15, 2008
HT is capable to generate supersonically rotating electrons Device\Parameter R cm
Lcm
TeV
B Gauss
Emax
V/cm
VE/B/Veth
LAPD 50 1700 2-5 400 4-18 < 210-2
Compact Auburn Torsatron
17 53 10 1000 5 < 410-3
Blaamann 8 65 9 570 2-6 < 810-3
Continuous Current Tokamak
40 150 150 3000 120 810-3
ALEXIS 10 170 5 100 2 210-2
Reflex arc 2.5 300 5 4000 20 510-3
Mistral 11.5 140 1.4 220 410-3
Maryland Centrifugal Experiment (MCX)
27 250 3 2000 710-2
CSDX 10 280 1.5-3 650 3-4 < 110-2
WVU Q-machine 4 300 0.2 1400 14 510-2
State-of-the-art Hall thruster
2 2 20-60 100-300 700 < 1
PPPL Segmented (No SEE) Hall Thruster
2.5 2 100 115 1000 1 - 2
How azimuthal oscillations can cause cross-field transport?• In principle, HT discharge is azimuthallyy symmetric
• If there are azimuthal oscillations of ne and and they are correlated so that their time average over one period is nonzero, a wave-based azimuthal force appears:
• For
• Therefore, the F×B drift of that wave-based force could be responsible for collisionless cross-field transport
0 ezz e r e e e
ce
ueu n B m n u
u
' ' , 0e e z r e e eF en E F en u B m n u
2 :
B
BFJunvmF r
zeee
Hall Thruster OscillationsOscillations in Hall thruster plasma
Imaging of HT operation
Xenon operation of 12 cm diameter 2 kW PPPL Hall thruster
• Records 400,000 fps• Unfiltered emission• ~ 7.5 m away
PPPL Hall Thruster Experiment (HTX)
Phantom camera V7.3
High speed imaging of HT operation
12 cm diameter PPPL HT300 V, 20 sccm Xenon
100 Gauss700 W
Steady-state operation
Rotating spoke
Azimuthal non-uniformity of visible light emission and plasma density rotating in EB direction (~ 10 kHz) observed using fast cameras and electrostatic probes for different types of HTs
Low voltage operation (< 200 V), probesJanes and Lowder, Phys. of Fluids 9 (1966)Morozov, et al, Sov. Phys. Tech. Phys. 5 (1973)Meezan, Hargus, Cappelli, Phys. Rev E 63 (2001)
Modern HTs, > 200 V, fast imaging and probesParker, Raitses, Fisch, Appl. Phys. Lett., 97 (2010)McDonald and Gallimore, IEEE TPS, 11 (2011)Ellison, Raitses, Fisch, Phys. Plasmas 19 (2012)Griswold et al Phys. Plasmas 19, (2012)
Theory and simulations of low frequency azimuthal oscillationsEscobar and Ahedo, IEPC 2011Matyash , Schneider et al., IEPC 2011Vesselovszorov, IEPC 2011
Spoke is always ~ 10 times slower than local EB speed !
A possible mechanism of cross-field transport through the spoke
Br
E0z+- ++--
--- +
++
E0z×B
Eθ
Eθ×B
Possible transport mechanism through the spoke:
Initial density perturbation
Only electrons undergo azimuthal drift motion
Eθ generated across the perturbation
Eθ×B drift across the magnetic field, towards the anode
Correlated density and azimuthal electric field fluctuations would explain enhanced electron transport
Cross-field transport through coherent plasma structures in magnetically controlled plasmas
Serfanni et al, PPCF 49, 2007, Photo: Courtesy of S. Zweben
Evolution of turbulent structures at the edge of the NSTX tokamak
Non-diffusive transport - particles are not moving by a random walk (drift wave fluctuations), but rather form coherent structures (or blobs) that convect towards the walls
UCLA LAPD
Carter, Phys. Plasmas 13 (2006)
MISTRAL, Aix-Marseille Univ. (EB linear device)
Jaeger, Pierre, Rebont, Phys. Plasmas 16 (2000)
HIPIMSRU, Bohum
Cylindrical Hall thruster (CHT)
• Mirror-cusp magnetic field topology• Similar to conventional HTs, the operation involves closed EB electron drift• Electrons are confined in the hybrid magneto-electrostatic trap• Ions are accelerated in a large volume-to-surface area channel(potentially lower erosion)
Raitses and Fisch, Phys. Plasmas 8, (2001)
Cathode-neutralizer
Electromagnets
N
N
Anode
Ceramic channel
S
S
Annular part
F = -B
F = -eEB
B
Cathode-neutralizer
Electromagnets
N
N
Anode
Ceramic channel
S
S
Annular part
F = -B
F = -eEB
B
Cathode
100 W 2.6 cm CHT
Rotating spoke in CHT
Cusp: Enhanced Radial Field
Direction: ExBFrequency: 15-35 kHzVelocity: 1.2-2.8 km/sE/B: 10-30 km/secE/B frequency 100-500 kHzSize: 1.0-1.6 cm
Does spoke conduct current ?
• Rotating spoke can not be observed in the discharge current traces
• Segmented anode (4 isolated segments) allows to see the rotating spoke
• Synchronized measurements with the fast camera reveal spoke-induced current
More than 50% of the discharge current is conducted via the spoke
Similar results were obtained for cylindrical and annular Hall thrusters
Ellison, Raitses, Fisch, Phys. Plasmas 19 (2012)
Insight of spoke with probes
Plasma density oscillations by planar tungsten probes
Plasma potential oscillations by floating emissive probe
Inside the channel: probe tips are flush with channel wall
Outside the channel: probe tips are at radial position of channel wall
Stationary probe arrays
Slowmovableprobe
3 azimuthal probes, 90 degrees apart, per axial location
2 azimuthal probes, 30 degrees apart, on a movable positioner outside the channel
23 mm
13 mm
back middle front
Spoke is everywhere along the channel, but the coherent rotation is only near the anode
• An azimuthal mode does exist in all three regions. • Mode is strongest in the back, although also “noisy” and
extends over a large frequency range
Density fluctuations: S(kθ,ω)kθ>0 corresponds to E×B direction
Anode region Channel middle Cathode region
Parker, Raitses, Fisch, Appl. Phys. Lett., 97 (2010)
- Potential and density fluctuations- Cross-field current estimation
•The density oscillates in-phase with the spoke current
•The potential is ~45 out of phase
• The azimuthal electric field
• The current to the anode:where d =E/B
• The drift current is ~¼ the discharge current, explaining a large fraction of the electron cross-field current to the anode
Ellison, Raitses, Fisch, Phys. Plasmas 19 (2012)
Do we know how to explain the spoke instability in Hall thrusters?
•3-D Full PIC with MC collisions relate the spoke to neutral depletion
Matyash , Schneider et al., IEPC 2011
• A linear stability analysis of the ionization region in HT
An extension of Morozov’s linear analysis for collisionless instability Spoke appears when the ionization and E-field make it possible to have positive gradients of plasma density and ion velocity
Escobar and Ahedo, IEPC 2011
0)/( nB
Potential explanation was given 20 years ago
• Modified Simon-Hoh instability (MSHI) - electrostatic instability in a plasma with magnetized electrons and unmagnetized ions due to finite ion Larmor radius effect on azimuthal velocity difference between electrons and ions
Y. Sakawa, C. Josh, P. K. Kaw, F. F. Chen, V. K. Jain, Phys. Fluids B 5, 1993
F. C. Hoh, Phys. Fluids 6, 1963
neo Er0 > 0
1Simon-Hoh instability (SHI) for Penning discharge
Conditions for SHI
neo 0
Can MSHI be excited in CHT plasma ?
, / 00 rzion BE / 00 rz
ed BE
12
,2 2
0
2
0
0 L
Rb
bB
E Lion
r
zion
From the dispersion relation for MSHI, the instability is excited when
Y. Sakawa et al Phys. Fluids B 5, 1993
From probe measurements of plasma properties and spoke in near- anode region of the Xenon CHT thruster:
Br 900 Gauss, Ez 10-20 V/cm, k 1 cm-1, b 30
Azimuthal ion velocity at the location of instability
kHzkf ion 20-01~2/ Not far from our observations
Can spoke be suppressed and controlled?
• Resistors attached between each anode segment and the thruster power supply
• The feedback resistors, Rf, are either 1, 100 , 200 , or 300
Spoke increases the current through the segment leading to the increase the voltage drop across the resistor attached the segment. This results in the reduction of the voltage between the segment voltage and the cathode.
Spoke suppression with the feedback control
Feedback off Feedback on
The suppression of the spoke leads to a reduction in the total discharge current due to the anomalous current that is carried by the spoke.
Summary• EB electron rotation can be used to focus the ion flow in a weakly
collisional plasma with magnetized electrons and non-magnetized ions
• Ion focusing is due to centrifugal force on electrons
• To maximize ion focusing supersonically rotating electrons are needed
• To achieve supersonic rotation of electrons, plasma needs to withstand a strong electric field
• Off all steady-state EB plasma devices, Hall thruster can produce the strongest electric field
• Electric field in Hall thrusters is limited by anomalous electron cross-field transport: wall conductivity and spoke instability
• Need better understanding of spoke and near-wall conductivity: needed 3D PIC simulations, theory of instabilities, experiments
• Reduction of anomalous transport by minimizing SEE effects and suppression of spoke instability was demonstrated
Acknowledgement
Nathaniel J. Fisch and Igor Kaganovich (PPPL)
Alex Khrabrov, Michael Campanell, Lee Ellision and Martin Griswald (PPPL)
Konstantin Matyash and Ralf Schneider
(University of Greifswald, Germany)
Thiery Pierre (Aix-Marseille University, France)
Andrei Smolyakov (University of Saskatchewan, Canada)
Stephane Mazouffre (CNRS-ICARE, France)
Amnon Fruchtman (Holon Institute of Technology, Israel)
Can MSHI be excited in CHT plasma ?
ed
ionBEs
Iion
R
kck
)(
,22
For excitation of MSHI
, / 00 rzion BE / 00 rz
ed BE
12
,2 2
0
2
0
0 L
Rb
bB
E Lion
r
zion
From dispersion relation for MSHIY. Sakawa et al Phys. Fluids B 5, 1993
For Xenon CHT near the anode and spoke: Br 900 Gauss, Ez 10 V/cm, k 1 cm-1
Azimuthal component of ion velocity
kHzkfb ion 01~2/ ,30 Smirnov, Raitses, Fisch, Phys. Plasmas 14, 2007`