A Kinetic Theory of Planar
Plasma Sheaths Surrounding
Electron Emitting Surfaces
J. P. Sheehan1, I. Kaganovich2,
E. Barnat3, B. Weatherford3, H. Wang2,
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1
2
D. Sydorenko , N. Hershkowitz , and Y. Raitses
1
University of Wisconsin – Madison
2
Princeton Plasma Physics Laboratory
3
Sandia National Laboratories
4
University of Alberta
Outline
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Fluid theory of emissive sheaths
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Kinetic theory of emissive sheaths
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Electrons lost to surface
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Temperature of emitted electrons
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Particle in cell simulations
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Afterglow of capacitively coupled plasma
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Measurements of emissive sheath versus time
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Conclusions
DOE Plasma Science Center Teleseminar, December 7, 2012
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Emitted electrons reduce sheath
potential and electric field at surface
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Three species
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Plasma electrons
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Plasma ions
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Emitted electrons
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Plasma fills the -x̂ half-plane
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One dimensional
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Emitted electrons reduce:
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Collecting Sheath
The electric field at the
surface
The floating sheath potential
DOE Plasma Science Center Teleseminar, December 7, 2012
Emissive Sheath
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Fluid Theory: a SCL emitting surface
floats Tep below the plasma potential
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Collisionless
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Φ = 0 at the sheath edge (definition)
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Plasma electrons
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Maxwellian (temperature Tep)
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Boltzmann relation
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e ϕw
Φ w =−
=1.02
T ep
2
mu
E 0= i 0 =0.58
2T ep
Emitted Electrons
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Space-Charge Limited Solution
Zero energy at surface
Plasma Ions
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One species
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Cold (Ti = 0 eV)
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Singly ionized
Integrate Poisson's equation
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Bohm's criterion, E = 0 at sheath edge
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E = 0 at surface
DOE Plasma Science Center Teleseminar, December 7, 2012
G. D. Hobbs and J. A. Wesson, Plasma Physics 9 (1), 85 (1967).
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Emissive probes are used to
measure the plasma potential
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Emissive probes are Langmuir probes
that emit electrons
Usually Joule heated to emit
thermionically
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Allows good control over emission current
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Used to measure the plasma potential
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Electrons are emitted when probe bias is
above plasma potential, but not when
below
Can be used in plasmas where Langmuir
probe measurements fail
Smaller uncertainty than Langmuir probe
DOE Plasma Science Center Teleseminar, December 7, 2012
J. P. Sheehan and N. Hershkowitz, Plasma
Sources Science and Technology 20
(6), 063001 (2011).
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The floating point method is
often used in Hall thrusters
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DOE Plasma Science Center Teleseminar, December 7, 2012
Heat probe until floating
potential saturates
Potential at saturation is
measure of plasma potential
Heating voltage swept at
0.1Hz
Potential measured through
a high impedance op-amp
Potential saturates past peak
heating current because
probe continues to heat
Uncertainty ~0.1Te/e
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Inflection point in the limit of zero emission
attempts to reduce space-charge effects
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Typically 7 I-V traces were
taken
Emission current less than
electron saturation current
Inflection point versus
temperature limited emission
current approximately linear
Extrapolate inflection point to
zero emission current
Noise increases uncertainty, but
using multiple emission levels
reduces it
Uncertainty ~0.1Te/e
DOE Plasma Science Center Teleseminar, December 7, 2012
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The floating potential of a highly emitting
probe in a Hall thruster was ~2Tep below the
plasma potential
J. P. Sheehan, Y. Raitses, N. Hershkowitz, I. Kaganovich and N. J. Fisch, "A comparison of emissive probe
techniques for electric potential measurements in a complex plasma," Phys. Plasmas 18, 073501 (2011).
DOE Plasma Science Center Teleseminar, December 7, 2012
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Motivation
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Emissive probe measurement of plasma potential
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Floating potential of a highly emitting probe is near the
plasma potential
Knowledge of emissive sheath yields more accurate
measurements
Secondary electron emission in laboratory plasmas
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Significant in determining plasma potential and EVDF in
low temperature plasmas
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Increase electron loss to divertors in tokamaks
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Modify operation of Hall thrusters, etc....
DOE Plasma Science Center Teleseminar, December 7, 2012
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A fully kinetic model of the planar
emitted sheath was developed
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Plasma electron loss cone: modification of
EVDF due to electrons lost to the boundary
Kinetic emitted electrons: half-Maxwellian
distribution with temperature parameter Tee
Ions are assumed to be cold
Poisson's equation and the generalized Bohm
criterion solved simultaneously
Highly nonlinear equations were solved
numerically
DOE Plasma Science Center Teleseminar, December 7, 2012
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Kinetic Theory: plasma electrons do
not follow the Boltzmann relation
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Boltzmann relation
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DOE Plasma Science Center Teleseminar, December 7, 2012
Assumes fraction of electrons lost to surface
is small
Valid for collecting sheath, not for emissive
sheath
Full kinetic model
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n ep ( Φ)
=exp(−Φ)
n ep ( 0)
(
1+erf ( √ Φw −Φ )
n ep (Φ)
=exp(−Φ)
n ep ( 0)
1+erf ( √ Φw )
)
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Accounts for electrons lost to surface
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Close to surface, lost electrons are significant
Boltzmann relation over-estimates the
plasma electron density in the sheath
Considering electrons lost to the surface
reduces net charge in the sheath, reduces
the sheath potential
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Emitted Electrons: account for kinetic effects of
non-zero emitted electron temperature (Tee)
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DOE Plasma Science Center Teleseminar, December 7, 2012
Plasma to emitted electron
temperature ratio Tep/Tee = Θe
Fluid expression
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n ee (Φ)
= 1− Φ
Φw
n ee (0)
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Assumes Θe → ∞
(
−
)
1
2
Kinetic expression
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n ee (Φ) exp (Θe (Φw −Φ))erfc (√ Θe (Φ w−Φ))
=
n ee (0)
exp (Θe Φ w )erfc (√ Θe Φ w )
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Maxwellian emitted electrons (Tee)
Fluid equations over estimate emitted
electron density in the sheath
Higher emitted electron temperature
reduces emitted electron density in
sheath, reduces sheath potential
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EVDFs of emitted and plasma
electrons are modified Maxwellians
Plasma Electrons
DOE Plasma Science Center Teleseminar, December 7, 2012
Emitted Electrons
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Emitted electrons modify
the Bohm criterion
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For arbitrary electron
distribution
Required for positive
space-charge in sheath
Solved for E0
Since ions are cold in all
descriptions, defines
simple condition for ion
energy
DOE Plasma Science Center Teleseminar, December 7, 2012
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Higher temperature emitted electrons
reduce net electron density in sheath
DOE Plasma Science Center Teleseminar, December 7, 2012
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Emissive sheath potential is reduced
by the emitted electron temperature
DOE Plasma Science Center Teleseminar, December 7, 2012
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Emitted electrons only slightly
affect the Bohm criterion
DOE Plasma Science Center Teleseminar, December 7, 2012
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Emissive sheath was
simulated using EDIPIC
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(Performed by Hongyue
“Della” Wang)
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Argon
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System length of 5 mm
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Plasma source electron
temperature: 1 eV
Plasma source ion
temperature: 0.025 eV
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At source (x = 0 mm)
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No magnetic field
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Simulated time: 100 μs
DOE Plasma Science Center Teleseminar, December 7, 2012
Zero electric field
At emitter (x = 5 mm)
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Collisionless
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Escaping particles
thermalized and
reflected
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Fixed potential of 0 V
Constant emission
current
Emitted electron
temperatures of 0.2 –
0.001 eV
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Potential profiles of emissive sheath
calculated from PIC simulations
DOE Plasma Science Center Teleseminar, December 7, 2012
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Kinetic theory was confirmed using
particle in cell simulations
DOE Plasma Science Center Teleseminar, December 7, 2012
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Enhanced EEDF tail (>eΦw)
increases the sheath potential
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Bi-Maxwellian electron energy
distribution function (EEDF)
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Two electron temperatures
(Tep2/Tep = Θp)
Hot electron fraction
β=
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n ep2 (0)
n ep (0)+n ep2 (0)
Sheath potential normalized to
colder electron temperature Tep
5% hot electrons in figure
Hot electrons can significantly
affect the sheath potential even
at low concentrations
DOE Plasma Science Center Teleseminar, December 7, 2012
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Emissive sheath potential depends
nonlinearly on the hot electron fraction
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DOE Plasma Science Center Teleseminar, December 7, 2012
Above a certain fraction of
hot electrons, the
temperature of the hot
species begins to dominate
This break point depends on
the plasma electron
temperature ratio Θp = Tep2/Tep
In figure, Θe = Tee/Tep = 10
In laboratory plasmas,
secondary electrons can be
source of hot electrons and
constitute a significant
fraction of the plasma
electrons
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Sheath potential has a non-monotonic
dependance on the hot electron fraction
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For data shown
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Θe = Tee/Tep = 10
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Θp = Tep2/Tep = 10
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Sheath potential normalized to
colder plasma electron
temperature
The colder electrons define the
ion flux via Bohm's criterion
The hotter electrons dictate the
electron flux through the sheath
Sheath must be large to reduce
electron current to maintain
current balance through the
sheath
DOE Plasma Science Center Teleseminar, December 7, 2012
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Planar dispenser cathode was
installed in GEC reference cell
DOE Plasma Science Center Teleseminar, December 7, 2012
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Working gas: Helium
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Neutral pressure: 25 mTorr
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Electron density: ~109 cm-3
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RF frequency: 10 MHz
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Pulse frequency: 60 Hz
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Afterglow time: 2.5 ms
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Barium tungsten dispenser cathode
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Dispenser cathode floating potential vs.
time at various heating currents
DOE Plasma Science Center Teleseminar, December 7, 2012
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Langmuir Probe
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1 cm long, 250 μm diameter
Positioned 3 cm above the edge of the
dispenser cathode
Aluminum tube protected against displacement
currents
I-V traces to measure electron temperature
DOE Plasma Science Center Teleseminar, December 7, 2012
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Emissive Probe
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1 cm long, 76 μm
Thoriated tungsten wire was secured by crushing the
ends of copper tubes around it
Aluminum tube reduced displacement currents for
emissive probe, as well
I-V traces to measure plasma potential using inflection
point in the limit of zero emission
DOE Plasma Science Center Teleseminar, December 7, 2012
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Slow-sweep emissive probe method
measured Vp versus time in afterglow
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DOE Plasma Science Center Teleseminar, December 7, 2012
Measured current vs time at
many probe biases
Transpose to determine I-V
trace vs time
Easy, inexpensive to execute
Used for both Langmuir
probe and emissive probe I-V
traces
First time inflection point in
the limit of zero emission
technique was used to
measure temporally varying
plasma potential
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Measuring Te
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Slope of semilog
Langmuir probe I-V trace
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Requires good signal to
noise ratio
Number averaging and
smoothing may be
necessary
Approximated by sheath
potential of floating
Langmuir probe
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Many assumptions for
this method
DOE Plasma Science Center Teleseminar, December 7, 2012
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Electron temperature decay
measured versus time
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RF ring down affected
measurements tens of μs
into the afterglow
Langmuir probe could not be
used for Te measurements
later than ~250 μs into
afterglow
Collecting sheath potential
was used to approximate the
electron temperature
Remarkable agreement
between these two
measurements
DOE Plasma Science Center Teleseminar, December 7, 2012
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Floating potential of heated electron
falls, then rises in afterglow
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DOE Plasma Science Center Teleseminar, December 7, 2012
Afterglow: 0 – 2.5 ms
Floating potential initially
drops as plasma cools
and loses density
Increases as emitted
electrons begin to
dominate the discharge
Only data before the
minimum (870 μs) is
relevant to compare to
theory
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Plasma potential decreases
monotonically in afterglow
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DOE Plasma Science Center Teleseminar, December 7, 2012
Plasma potential
drops to a few volts in
the first 100 μs
Decays slowly
through afterglow
Becomes negative at
1150 μs, after emitted
electrons begin to
dominate discharge
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Electron temperature
decays in afterglow
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DOE Plasma Science Center Teleseminar, December 7, 2012
Electron temperature
decays rapidly once RF
heating is turned off
Monotonic decay in
afterglow
Measurement become
negative after 1240 μs
when floating potential
exceeds plasma potential
“Negative temperature”
measurements excluded
since it is in the emission
dominated discharge
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Normalized emissive sheath potential is
greatly reduced at low electron temperatures
DOE Plasma Science Center Teleseminar, December 7, 2012
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Data qualitatively follows trend
predicted by theory
0.1
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1
10
100
1000
10 4
Cannot directly compare: experimental measurements include
presheath
Sheath disappears when plasma electron temperature equals emitted
electron temperature
For intermediate temperatures, measured sheath is larger than
expected from kinetic theory
DOE Plasma Science Center Teleseminar, December 7, 2012
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Conclusions
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Kinetic theory of emissive sheaths
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Considering the plasma electrons lost to the surface reduces
the emissive sheath potential by 10%
Considering the non-zero emitted electron temperature
reduces the emissive sheath potential by up to 50% for
some low temperature plasmas
Validated with particle in cell simulations
Measurements of emissive sheath in afterglow
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Confirms that as plasma electron temperature approaches
emitted electron temperature emissive sheath disappears
Emissive sheath was larger than expected for intermediate
electron temperatures
DOE Plasma Science Center Teleseminar, December 7, 2012
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Acknowledgments
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This work was supported by US Department of Energy
grants No. DE-AC02-09CH11466 and No. DE-FG0297ER54437, the DOE Office of Fusion Energy Science
Contract DE-SC0001939, and the Fusion Energy
Sciences Fellowship Program administered by Oak
Ridge Institute for Science and Education under a
contract between the US Department of Energy and
the Oak Ridge Associated Universities
DOE Plasma Science Center Teleseminar, December 7, 2012
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