PLASMA ADIABATICITY IN
A DIVERGING MAGNETIC NOZZLE
J. P. Sheehan and Benjamin W. Longmier
University of Michigan
Edgar A. Bering
University of Houston
Christopher S. Olsen, Jared P. Squire, Mark D. Carter,
Franklin R. Chang Díaz, Timothy W. Glover,
Andrew V. Ilin, and Leonard D. Cassady
Ad Astra Rocket Company
ABSTRACT
We propose a fluid model for ambipolar ion acceleration in a magnetic
nozzle that preserves the adiabaticity of the plasma. This adiabatic theory
predicts that the change in average electron energy depends linearly on
the change in plasma potential, providing an important design metric for
electric propulsion devices which employ magnetic nozzles. The fluid
theory predictions were compared to measurements made in the VASIMR
VX-200 experiment which has conditions conducive to ambipolar ion
acceleration. A planar Langmuir probe was used to measure the plasma
potential, electron density, and electron temperature for a range of mass
flow rates (50 – 140 mg / s) and power levels (12 – 30 kW). The linear
relationship between electron temperature and plasma potential was
observed as predicted. The adiabatic theory relies on collisions to
rethermalize the electrons and establish a temperature gradient. Coulomb
collisions cannot account for the high collisionality but an ion acoustic
instability may enhance the collision frequency.
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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HELICONS AND MAGNETIC NOZZLES FOR
APPLICATIONS IN PROCESSING AND PROPULSION
 Helicons
 Radio frequency
 High ionizing efficiency
 Electron heating
 Magnetic nozzle
 Functions like physical
nozzle
 Accelerates ions
 Converts thermal energy
into directed kinetic
energy
 Applications
 Materials processing
 Electric propulsion
 CHI KUNG
 VASIMR
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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CURRENT FREE DOUBLE LAYERS
IN HELICON EXPERIMENTS
 Narrow layer (10s of λd) of large
potential jump (several Te/e)
 Isothermal
 Occurs downstream of nozzle
 Current free, expanding
 Accelerates ions
 May be thrust mechanism in helicon
thrusters
 Open question of how/why current
free double layers form
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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ION ACCELERATION IN VASIMR
 VASIMR: < 200 kW helicon + ICH
thruster
 Went looking for double layers, but
found none!
 Vp, ne, and Te derivatives coincide
 Long length scales: 10,000s of λd
 λd ~ 10 μm
 Corroborated with RPA
 B. W. Longmier et al., Plasma
Sources Sci. Technol. 20, 015007
(2011).
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Washington DC, May 28, 2014
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PLASMA IN NOZZLE IS ADIABATIC
 Assume plasma is adiabatic
 Energy loss to surfaces small
 Radiation loss small
 Geometry dictates degrees of freedom (N)
 Expanding plasma sphere: N = 3
 Magnetic nozzle: N = 2
 Electrons remain Maxwellian, though
temperature can change
 Relies on collisions
 Conservation of momentum
 Relationship between average electron energy
loss and potential (→ion energy gain)
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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HELICON EXPERIMENT
USING VASIMR HARDWARE
• Bmax = 20,000 G
• P = 15 – 30 kW
• ṁ = 50 – 140 mg/s
• Only helicon coupler used, no ICH
• Superconducting magnets generate converging/diverging
magnetic field
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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MEASUREMENTS IN PLUME WERE MADE
WITH PLANAR LANGMUIR PROBE
 Vp < 20 V
 Te < 15 eV
 ne = 1010 – 1012 cm-3
 Planar tungsten probe
 No RF compensation
needed
 Guard ring reduces
sheath expansion
effects
 Parameters extracted
from I-V traces
 Vp: knee
 Te: semilog fit
 ne: saturation
current
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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PARAMETRIC STUDY
OF OPERATING PARAMETERS
 Measured at fixed position
 50 cm from throat
 Highest density where probe
could survive
 Lower mass flow rate
 Higher Te, Vp
 Lower ne
 Power flow density ∝ input
energy per ion
 Optimize energy deposition
for given flow rate
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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Vp, Te, ne DECAY DOWNSTREAM
 Axial measurements
 Lowest and highest mass flow
rates are shown
 Low flow rates → high
temperature, larger
gradients
 Temperature decay: plasma is
not isothermal
 Length scale: 10,000s of λd
 No double layer, but
significant density and
temperature gradients
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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DATA WERE CONSISTENT
WITH ADIABATIC THEORY
 Plasma potential decays
proportionally to electron
temperature
 Only parallel electron temperature
was measured with planar probe
 Some electron energy may be lost
to other sinks
 Collisions
 Instabilities
 Radiation
 J. P. Sheehan et al., Plasma Sources
Sci. Technol., (submitted).
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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ALTERNATIVE THEORY:
RAREFACTION WAVE THEORY
 Nonlocal theory
 Far downstream,
rarefaction waves create
potential structures which
confine electrons
 Electrons lose energy on
wave, cooling in
downstream region
 Two regions
 Steady state: ambipolar
ion acceleration
 Rarefaction wave: further
acceleration
Steady-state part
Rarefaction wave part
Arefiev and Breizman, Phys. Plasmas 16, 055707 (2009).
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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COMPARISON BETWEEN ADIABATIC THEORY
AND RAREFACTION WAVE THEORY
 Fundamental difference: local
theory or non-local theory
 Both predict same relationship
between φ and Te
 Different relationship between φ
and ne
 Suggests some discrepancy in
number of degrees of freedom
 No observed acceleration in lowfield region
 Factor of 3 increase in ion
velocity from rarefaction
wave
 Importance of collisions
Rarefaction wave theory
Experiment
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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ADIABATIC THEORY RELIES ON
COLLISIONS, RETHERMALIZATION
 VASIMR has >95% ionization
fraction → Coulomb collisions
dominate
 Electron-electron collisions cause
rethermalization of EVDF
 Collisions necessary to reduce
thermal conductivity along field
lines
 Classical coulomb collision are
insufficient to explain
rethermalization
 Collisional mean free path only
becomes long enough after major
temperature gradient
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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PLASMA IS DETACHED FROM
MAGNETIC FIELD LINES
 In a fully attached nozzle, density
should follow field lines
 ne/B > 1
 Plasma beam more focused
 Detachment occurs
 Possible detachment mechanisms
 Particle collisions
 Anomalous transport
 Demagnetization
 Frozen flow
 Electron inertia
 C. S. Olsen et al., IEEE Trans. Plasma
Sci. (accepted, 2014).
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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INSTABILITIES INCREASE
THE COLLISION FREQUENCY
 In the kinetic equation, collisions represented by collision operator
 which has depends on the collisional kernel
 Collision frequency affected by both the stable and unstable parts
S. D. Baalrud, J. D. Callen and C. C. Hegna, Phys. Rev. Lett. 102, 245005 (2009).
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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AMBIPOLAR ION ACCELERATION SCALES
ARE SIMILAR TO PRESHEATH SCALES
 LE, LB, λcx >> λd
 vi ~ cs
Presheath and sheath
 Ti << Te
 Δφ ~ Te
 Quasineutral
 J=0
Ambipolar acceleration
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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CONVECTIVE ION ACOUSTIC INSTABILITY
INCREASES DOWNSTREAM COLLISION FREQUENCY
 In presheath:
 Te constant
 ne ~ constant
 In magnetic nozzle:
 dTe/dx ~ Te/LB
 dne/dx ~ ne/LB
 Instability grows as ions propagate
 Collisions rethermalize electrons
 Ignoring gradients (see figure)
overestimates collision frequency
 Complicating factors
 Ion temperature, i-n collisions
 Magnetic field divergence
 Electron-electron collision frequency
in magnetic nozzle without
considering parameter gradients
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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CONCLUSIONS
 Ambipolar ion acceleration
observed in VX-200 experiment
 Longer length scale than in
CFDLs
 Adiabatic theory describes
experimental results
 Potential drop depends on
temperature, not density
 Local vs. non-local theory—
unresolved questions
 Unknown cause of
rethermalization
 Proposed collision mechanism:
ion acoustic instability
41st IEE International Conference on Plasma Science
Washington DC, May 28, 2014
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FOR A COPY OF THE POSTER
41st IEE International Conference on Plasma Science
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