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Direct Vlasov simulations of electron-attracting cylindrical Langmuir probes in flowing plasmas


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Current collection by positively polarized cylindrical Langmuir probes immersed in flowing plasmas is analyzed using a non-stationary direct Vlasov-Poisson code. A detailed description of plasma density spatial structure as a function of the probe-to-plasma relative velocity U is presented. Within the considered parametric domain, the well-known electron density maximum close to the probe is weakly affected by U. However, in the probe wake side, the electron density minimum becomes deeper as U increases and a rarified plasma region appears. Sheath radius is larger at the wake than at the front side. Electron and ion distribution functions show specific features that are the signature of probe motion. In particular, the ion distribution function at the probe front side exhibits a filament with positive radial velocity. It corresponds to a population of rammed ions that were reflected by the electric field close to the positively biased probe. Numerical simulations reveal that two populations of trapped electrons exist: one orbiting around the probe and the other with trajectories confined at the probe front side. The latter helps to neutralize the reflected ions, thus explaining a paradox in past probe theory.
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Direct Vlasov simulations of electron-attracting cylindrical Langmuir probes in flowing
G. Sánchez-Arriaga and D. Pastor-Moreno
Citation: Physics of Plasmas (1994-present) 21, 073504 (2014); doi: 10.1063/1.4889732
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Direct Vlasov simulations of electron-attracting cylindrical Langmuir probes
in flowing plasmas
G. S
anchez-Arriaga and D. Pastor-Moreno
Departamento de F
ısica Aplicada, Escuela T
ecnica Superior de Ingenieros Aeron
Universidad Polit
ecnica de Madrid, Plaza de Cardenal Cisneros 3, 28040 Madrid, Spain
(Received 15 April 2014; accepted 24 June 2014; published online 10 July 2014)
Current collection by positively polarized cylindrical Langmuir probes immersed in flowing
plasmas is analyzed using a non-stationary direct Vlasov-Poisson code. A detailed description of
plasma density spatial structure as a function of the probe-to-plasma relative velocity Uis pre-
sented. Within the considered parametric domain, the well-known electron density maximum close
to the probe is weakly affected by U. However, in the probe wake side, the electron density mini-
mum becomes deeper as Uincreases and a rarified plasma region appears. Sheath radius is larger at
the wake than at the front side. Electron and ion distribution functions show specific features that
are the signature of probe motion. In particular, the ion distribution function at the probe front side
exhibits a filament with positive radial velocity. It corresponds to a population of rammed ions that
were reflected by the electric field close to the positively biased probe. Numerical simulations
reveal that two populations of trapped electrons exist: one orbiting around the probe and the other
with trajectories confined at the probe front side. The latter helps to neutralize the reflected ions,
thus explaining a paradox in past probe theory. V
C2014 AIP Publishing LLC.
Plasma structure in the neighborhood of material walls
is a fundamental problem in plasma physics. Theoretical
models were developed during almost one century and used
in relevant technological applications. Two examples are
Langmuir probes,
which are routinely used for plasma
diagnostic and bare electrodynamic tethers,
that can be
used to deorbit satellites at the end of life and mitigate the
space debris problem. The steady-state properties of plasma
sheaths in non-flowing plasma are now well understood.
In flowing plasmas, both analytical works in particular
and numerical simulations of the Vlasov-
Poisson system
were carried out. Some issues, however,
remain open.
An interesting paradox arises for cylindrical Langmuir
probes positively polarized within the so called mesother-
mal regime;
i.e., probe velocity small (large) compared
with electron (ion) thermal velocity. The faraway electron
population would still be (nearly) isotropic and an important
and very general result by Laframboise and Parker
electron density N
is less that the unperturbed
plasma density N
everywhere. This result is valid even if
the electric potential depends on the azimuthal angle h,asit
happens in the case of flowing plasmas. On the other hand,
since probe bias is positive and ions are repelled, the hyper-
sonic ion flow will result in an ion density N
exceeding N
in a broad region at the probe front side. This would break
the quasineutrality in a region much larger than the Debye
The explanation of the paradox may come from the vio-
lation of (at least) one of the hypothesis used in Ref. 16 to
find the condition N
<1. Besides the isotropic character
of the electron distribution function, Ref. 16 assumed that (i)
plasma reached an steady state and (ii) that there are no
trapped particles. Violation of hypothesis (i) was invoked in
Ref. 17, where plasma oscillations were detected in labora-
tory experiments. Possible electron trapping in energy
troughs due to collisional effects was discussed in Ref. 3.
However, since collisional trapping rate can be very small,
a non-stationary process called adiabatic trapping
has been
proposed in Ref. 15 to explain the paradox. As electron
potential wells develop in time, some electrons become
trapped adiabatically. The adiabatic term comes from the
fact that time variations in the electric potential, which are
controlled by the ions, are very slow as compared with the
motion of the electrons.
The confirmation of electron trapping at the front side
cannot be carried out with stationary Vlasov-Poisson solv-
because they explicitly neglect this particle popu-
lation. The computation of trapped particles requires a non-
stationary Vlasov-Poisson solver, like the particle-in-cell
(PIC) codes used to study Langmuir probes in the
However, PIC codes do not give an accurate rep-
resentation when the number of particles per cell is small
and they also introduce numerical noise. An alternative is
the implementation of Eulerian or direct Vlasov codes.
Unlike PIC codes, which use macro-particles to discretize
Vlasov equation, direct Vlasov codes make a discretization
of the distribution functions in both real and velocity space.
The result is a code free of numerical noise but expensive
from a computational point of view.
This work used numerical simulations to confirm that
electron trapping at the probe front side occurs. Section II
shows the modification carried out in the direct Vlasov code
named Kilaps (Kinetic Langmuir probe software)
1070-664X/2014/21(7)/073504/8/$30.00 V
C2014 AIP Publishing LLC21, 073504-1
PHYSICS OF PLASMAS 21, 073504 (2014)
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incorporate the probe-to-plasma relative motion. In Sec. III,
the time evolution of the macroscopic variables, including
collected current, is discussed. Sections IV and Vdescribe
the structure of the particle densities and distribution func-
tions. The signatures of the probe motion in plasma variables
are highlighted. In Sec. VI, two populations of trapped elec-
trons, one of them at the probe front side, are shown.
Discussion of results and comparisons with previous works
are presented in Sec. VII.
A perfectly absorbing cylindrical probe of radius Rand
bias U
(t) is moving with velocity U(t) inside a collisionless,
non-magnetized, Maxwellian plasma made of electrons an
single-charged ions. Faraway from the probe, plasma density
is equal to N
, and electrons’ and ions’ temperatures are T
and T
, respectively. Cylindrical coordinates (r,h), with ori-
gin at the center of the probe and angle h¼0(h¼p) corre-
sponding to the wake (front or ram) side, are used. Time,
position, and velocity vectors, electric potential, particles
distribution functions, and densities are normalized as
kDe !r;v
where the subscript a¼eand idenotes electrons and ions.
We also introduce the following dimensionless parameters
and variables:
Using this normalization, Vlasov-Poisson system in cy-
lindrical coordinates reads
¼1 and
The boundary conditions of the problem are
exp la
svrcos hvhsin h
Boundary condition (5a) imposes that no particle is emitted
by the probe. Faraway from the probe, a drifting Maxwellian
is assumed (see Eq. (5b)).
Given initial conditions, U(0, r,h) and f
(0, r,h,v
and laws /pðtÞand S(t), Sys. 3 is integrated with Kilaps,
which is a modification of the code presented in Ref. 21.It
implements a finite-difference method combined with an
explicit Runge-Kutta algorithm to carry out the time integra-
tion. Electrons and ions share the same spatial grid (r,h), but
they have different grids in velocity space (v
). The semi-
infinite spatial domain [qr<1] is truncated up to a maxi-
mum radius r
and discretized with N
points. These points
are not uniformly distributed, i.e., a fine grid is taken close to
the probe. Thanks to the symmetry of the problem, Kilaps
just simulates the range 0 hpusing a grid of N
uniformly distributed. Regarding velocity space, the infinite
domains are truncated to the intervals vrmax <vr<vrmax and
vhmax <vr<vhmax .Nvrand Nvhequispaced points are used
to created a grid in velocity space. For convenience, we take
Nvrand Nvhodd numbers. Due to the disparate masses, limits
of the velocity intervals are different for electrons and ions,
and we set velectrons
max ¼ffiffiffiffi
The total number of grid points for each specie is
where we took into account that no grid point is necessary at
r¼qand v
>0, because the distribution function is known
(see Eq. (5a)). The values of the distribution function of the
specie aat the grid points are organized in two vectors F
dimension N(a¼e,i).
After substituting the differential operator by appropri-
ate finite-differences formulae, Vlasov equation becomes a
system of 2Nordinary differential equation
dt ¼
NL Fe;Fi
that we integrate with a third order Runge-Kutta method.
L, which have a sparse structure and only depend
on the grid parameters, are computed at the beginning of the
073504-2 G. S
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simulation. Vectors C
, which take into account the bound-
ary conditions and the law S(t), and matrices
NL, involving
the non-linear term in Vlasov equation, are calculated each
time step. Potential Uis also found each time step by solving
Poisson equation with finite-difference methods. Boundary
conditions Uðq;h;tÞ¼/pðtÞand U(r
The above scheme has been parallelized in Kilaps. Each
processor computes the distribution function evolution
within a subdomain at certain hvalue and within a radial
range. In order to compute the radial and azimuthal deriva-
tives at the boundaries of the subdomains, communication of
adjacent processors each time step is needed. This strategy
ensures the scalability of the code and a similar computa-
tional load to each processor.
Parameter values of the simulation were R/k
¼1, and the unrealistic mass ratio m
¼100. This
value, which guarantees that ion motion is much slower
than electron motion, helps to save computational resour-
ces. From Ref. 6, we conclude that probe radius is small
enough to make the probe operate within the orbital-
motion-limited (OML) regime if no probe motion would
exist. The numerical parameters were N
¼153, N
¼65, N
¼65 and r
¼120 (normalized units). The
limits of the simulation box in velocity space for the elec-
trons were set to 7.5 <v
<7.5 and 7.5 <v
<7.5. The
ion box in velocity space was scaled by a factor ffiffiffiffiffiffiffiffiffiffiffiffi
and the equations were integrated with a fixed time step
equal to Dt¼0:001x1
pe .AsshownbySec.IVVI, the nu-
merical box and the resolution were large enough to impose
the boundary conditions appropriately and capture the main
physics of the problem. The simulation ran in 145 processor
during one month, approximately. The external laws /pðtÞ
and S(t) were varied as shown in Fig. 1(note the logarith-
mic scale in the horizontal axis).
Initially, the probe bias is zero (/pð0Þ¼0), the probe is
at rest (S(0) ¼0), and the electron and ions distribution
functions are given by Eqs. (11) and (12) in Ref. 21. The nor-
malized potential was increased until the value /p0¼10,
which is reached at x
t20. This part of the simulation is
similar to the one performed in Ref. 21, but with a lower
probe potential value. The current had an overshoot, and it
approached to the OML current (per unit length) I
wards. We recall that I
is given by
IOML ¼2eN0Rffiffiffiffiffi
rexp e/p
erfc e/p
and corresponds to the maximum current in steady condi-
tions of a long cylindrical Langmuir probe of radius Rand
bias /poperating at rest in unmagnetized, collisionless,
Maxwellian plasmas of electron temperature T
Probe potential was kept constant and equal to 10 in the
rest of the simulation. At about x
t1000, the probe-to-
plasma relative velocity was increased until it reached S¼0.5
at x
t1200. We remark that parameter Saffects the
boundary conditions 5b. In the interval 1200 <x
the (normalized) ram velocity is kept fixed and equal to 0.5.
This is enough time to reach an state close to equilibrium.
During the transient, the current dropped below I
then it reached an asymptotic value slightly higher than I
The black solid line in Fig. 1then looks thicker because the
current exhibits small oscillations with frequency close to x
(see inset in Fig. 1).
Within the interval 3500 <x
t<4000, Swas increased
from 0.5 to 1 and then kept fixed until the end of the simula-
tion at x
t7000. Plasma response was similar to the one
exhibited in the interval 1000 <x
t<3500; collected cur-
rent decreased for a moment and it then increased up to a
value slightly higher than I
. Very small oscillations in
the collected current are present at the end of the simulation.
A close look to the simulated variables at x
t3500 and
t7000, including collected current, electric potential,
and plasma distribution functions, indicates that the simula-
tions were long enough to reach an state close to equilibrium.
The small difference between the asymptotic collected
FIG. 1. Evolution of the normalized current I/I
, probe potential
/pðtÞ=/p0and probe-to-plasma relative velocity S(t). Here /p0¼10. The
inset shows the time evolution of I/I
in the interval 3450 <x
FIG. 2. Normalized electron density N
in log-polar coordinates at the
end of a simulation (/p¼10 and S¼1).
073504-3 G. S
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current value and I
is attributed to the slightly anisotropic
character of the electron distribution function as r!1
(note that in our simulation, we just have ffiffiffiffiffiffiffiffiffiffiffiffi
p¼10 in
Eq. (5b)).
For finite probe motion, electron density has a complex
spatial structure. Figure 2shows a map of the electron den-
sity at the end of the simulation. In order to magnify the
region close to the probe, a log-polar coordinate system,
with ð~
yÞ¼log10ðr=RÞðcos h;sin hÞ, is used. Very close to
the probe, there is a rarified plasma with normalized electron
density close to 0.5 (this is the value predicted by OML
theory if no probe motion exists). As radial distance is
increased, electron plasma density grows monotonically until
a maximum value is reached. The maximum, about
0.94, does not depend on the azimuthal coordinate
value (see Fig. 3). Its position, however, is closer to the
probe in the wake (r/k
1.17) than in the front side
1.6). For higher radial distances, the electron density
decreases until a minimum value, which depends strongly on
the probe side; at the front (wake), one has N
0.87 (N
0.57) at r/k
4.3 (r/k
4.8). The low electron
plasma region is especially important at the wake side,
within the range 2 <r/k
<10 and 0 <h<p/2. After the
minimum, the electron density grows monotonically and it
approaches to the faraway value (N
!1). These features
are in agreement with the results given by stationary-like
Figure 3shows electron density profiles at certain h
angles. We recall that h¼0(h¼180) corresponds to the
wake (front) side. There is a h-range in the front side where
the electron density is slightly above the background density
within certain radial domain. For h¼159, this radial range
extends from 40k
to the border of the simulation box. This
behavior, in apparent contradiction with Ref. 16, is due to a
population of trapped particles and it will be analyzed in
Sec. VI.
Probe-to-plasma velocity effect is illustrated in Fig. 4,
which shows density profiles at the front (top panel) and
wake (bottom panel) probe sides for different Svalues. The
thick solid black lines correspond to numerical results of a
stationary code (see for instance Ref. 5or Ref. 22). The three
thin lines are Kilaps results for S¼0, 0.5, and 1 at
t¼1000, 3500, and 7000, i.e., once the plasma reached
an equilibrium state. Clearly, the maximum in the electron
density is higher for Kilaps. This is due to particle trapping
(see Sec. VI), an effect not included in stationary codes. As S
is increased, the maximum in the electron density decreases,
and it approaches (moves away from) the probe in the wake
(front) sides. Regarding the density minimum, we observe
that it is less pronounced in the front side as Sis enhanced
but it becomes very deep in the wake side.
Figure 5shows an ion density map at x
t¼7000. Ion
density practically vanishes within the sheath, and it
increases monotonically until the background plasma density
. The sheath has a radius about 10k
, and, unlike the elec-
tron density, it exhibits a quasi-isotropic behavior. However,
there is small h-dependence, and the sheath radius is smaller
in the front than in the wake side. An analysis similar to
FIG. 3. Normalized electron density profiles at the end of the simulation
(/p¼10 and S ¼1).
FIG. 4. Electron density at the front (top panel) and the wake (bottom panel)
sides for different probe-to-plasma relative velocity. Results from Ref. 5,
which ignores particle trapping, are also shown.
FIG. 5. Normalized ion density N
in log-polar coordinates at the end of
the simulation.
073504-4 G. S
anchez-Arriaga and D. Pastor-Moreno Phys. Plasmas 21, 073504 (2014)
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Fig. 4applied to the ions (not shown) reveals that the sheath
radius decreases (increases) in the front (wake) side as the
parameter Sin enhanced. As compared with stationary simu-
lations, Kilaps shows that ion density in flowing plasmas is
higher at the front and lower at the wake; as expected,
rammed ions reach positions closer to the probe.
Electron and ion distribution functions in flowing plas-
mas exhibit a complex structure in real and velocity spaces.
Besides a dependence on the radial and azimuthal coordi-
nates, the distribution functions are not symmetric in velocity
space (except at the specific angles h¼0 and 180). Close to
the probe, they are far from Maxwellian functions and present
different features depending on the specie under considera-
tion. Some examples are given in Figs. 6and 7, which also
show that both the computational box and the resolution in
velocity space were large enough in the simulations.
Figure 6shows the normalized electron distribution
function at several positions. Panels (a) and (b) correspond
to r¼qat h¼0 and h¼p, respectively. Similarly, panels (c)
and (d) display f
at the same azimuthal angles but at
. Panels (a) and (b) show that, at the probe radius,
has a shape of half a ring with a radius in velocity space of
the order of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p. As compared with the no-flowing
case (see Fig. 3 in Ref. 21), there is a lack of electrons arriv-
ing at the probe with vanishing azimuthal velocity. This
effect is important in the wake [see panel (a)] and practically
negligible (but still present) in the front [panel (b)], where
the electron distribution function reaches its maximum value
for a normalized velocity magnitude of approximately 2. For
higher radial coordinate (bottom panels), the distribution
function resembles to a Maxwellian, but with vanishing
number of particles at a cone that extends for positive radial
velocity and has its axis at zero azimuthal velocity. The aper-
ture angle of this cone, which is a signature of the no-
emissive character of the probe, decreases as r!r
other azimuthal angles, the electron distribution functions is
not symmetric with respect to the line v
¼0; for instance, at
r¼Rand h¼32, there are much more particles with nega-
tive azimuthal velocity (not shown).
Panels (a)–(d) in Fig. 7display the ion distribution func-
tion at the same positions as in Fig. 6. Close to the probe,
inside the sheath, the ion distribution function almost van-
ishes. However, due to the probe velocity, there are more
ions in the front than in the wake side [see panels (a) and
(b)]. As rincreases, interesting and different features appear
in the wake and the front sides. Panel (c) in Fig. 7shows the
ions distribution function in velocity space at r¼20k
h¼0 (wake). There are two peaks with centers at vr
pand vh60:1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p. These peaks do not
appear for S¼0 simulations, where the ion distribution func-
tion has an isotropic behavior in velocity space (except for a
cone-like region with no particles due to the no emissive
character of the probe).
Panel (d) shows the ion distribution
function at the same radial distance but at the front side.
Even though the probe does not emit particles, there is a
prominent filament with positive radial velocity. Its azi-
muthal velocity range is very narrow, and it extends until
p. This population corresponds to rammed
ions that are reflected by the electric field (note that the probe
is biased positively).
Figure 8shows the structure of the filament in more
detail. For convenience, we plot with a thin solid line the
shifted Maxwellian distribution function given by Eq. (5b)
with S¼1. The thick lines are ion distribution sections at
¼0 and h¼180at the end of the simulation
t¼7000). For r¼2k
and 8k
, the ion distribution
function has a maximum at v
¼0; i.e., inside the sheath, the
influence of the ram motion in the ion distribution function is
weak. At r¼11k
we observe the appearance of two max-
ima that, as rincreases, separate apart. The left peak, with
<0, perfectly fits with the shifted Maxwellian, and it rep-
resents the incoming population of ions. The right peak,
which is the filament with v
>0 in Panel (d) of Fig. 7,isa
population of ions that have been reflected by the (positively
polarized) probe. Although, here, we just showed the ion dis-
tribution function at the specific value h¼p, ions are
reflected in a broad h-range. The consequence is a large
FIG. 6. Electron distribution function in velocity space at x
Panels (a) and (b) ((c) and (d)) correspond to sections at probe radius
) and h¼0and h¼180, respectively.
FIG. 7. Ion distribution function in velocity space at x
t¼7000. Panels
(a)–(d) correspond to the same radial and azimuthal values as in Fig. 6.
073504-5 G. S
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region at the probe front side with plasma density above N
As we will see, a trapped population of electrons is necessary
to recover quasi-neutrality faraway from the probe.
Figure 9shows a map of the normalized potential
(U=/p0) in log-polar coordinates at the end of the simulation.
The two insets correspond to potential profiles at h¼p(left)
and h¼0 (right). Clearly, the behavior of the potential is dif-
ferent at the front and wake sides of the probe. In particular,
the electrostatic potential reaches negative values at the
wake side, an effect already observed in stationary-like
As we will see, the potential asymmetry has conse-
quences in the electron trapped population, which happens at
the probe front side. We remark that the potential is not sta-
tionary at the end of the simulation but exhibits small oscilla-
tions at frequency close to x
Figures 3and 4seem to contradict results from Ref. 16;
electron density close to the probe is higher in Kilaps
simulations, where one also has N
>1 in the front. As
we will see, both differences are explained if trapped par-
ticles are considered. Trapped particles can be computed by
post-processing Kilaps simulations. At the end of the simula-
tion (x
t¼7000), once the plasma is in equilibrium, the
(almost periodic) potential profile U(t,r,h) is saved over one
electron period (2p/x
). Particle trajectories can then be
computed by integrating Vlasov characteristic equations
dt ¼vr;dvr
dt ¼a
dt ¼vh
dt ¼1
Neither the energy nor the angular momentum is conserved,
because the potential depends on time and azimuthal
angle h.
Examining the electron distribution function (for
instance Fig. 6), we can select initial conditions (r
) where electrons exist at the end of the simulation.
Systems 9 and 10 are integrated for a long time (typically
t50 000 in our calculations), and, if the particle does
not hit the probe and does not leave the simulation box, then
it is deemed to be trapped.
Figure 10 shows some examples of particle trajectories.
Trajectory (a) corresponds to a trapped electron orbiting
around the probe. This population of trapped particles is re-
sponsible for the high density (as compared with stationary
theory) exhibited by Kilaps simulations close to the probe
(see Fig. 4). A second population of trapped particles hap-
pens at the probe front side. An example is given by trajec-
tory (b) in Fig. 10. Neglecting this population yields to the
paradox explained in Sec. I. If it is included, one has
>1 at the front, and plasma can reach quasineutrality
(note that N
>1 at the front due to the reflected ions
showed in Fig. 8). Finally, electrons labeled by (c) and (d)
are captured by the probe and (e) goes around the probe and
leaves the simulation box.
FIG. 9. Normalized electric potential U/U
in log-polar coordinates at the
end of the simulation. Left (right) inset corresponds to a section at h¼180
FIG. 10. Electron trajectories: (a) trapped orbit around the probe, (b) trapped
orbit at the front side, (c) and (d) captured electrons and (e) trajectory leav-
ing the simulation box. Probe, with radius R/k
¼0.5, is plotted as a black
FIG. 8. The thick black line is the shifted Maxwellian given by Eq. (5b).
Thin lines correspond to sections of the ion distribution function (h¼180,
¼0) at x
073504-6 G. S
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Probe-to-plasma relative motion also produces a loss of
regularity in electrons trajectories. In non-flowing plasmas
and once it reached equilibrium and electric potential is
time-independent, both angular momentum and energy are
conserved. Systems 9 and 10 are integrable, and particle dy-
namics is regular. However, in flowing plasma, systems 9
and 10 are (in principle) non-integrable. This issue was
investigated by plotting Poincar
e sections of trapped elec-
trons orbiting around the probe and at the front; each time an
electron trajectory intersected the hypersurface h¼p,we
plotted v
versus r. The results are shown in panels (a) and
(b) of Fig. 11, where colors were used to denote trajectory
started with different initial conditions. Trapped electrons
trajectories are confined to certain bounded region of the
phase space and exhibit a complex behavior. In the case of
trapped electrons at the probe front side [panel (b)], the tra-
jectories seem to fill tori in phase space.
This paper presented numerical simulations of current
collection by a positively polarized infinite cylinder in flow-
ing plasmas, a problem with applications to Langmuir probes
and electrodynamic tethers. The non-stationary Vlasov-
Poisson solver named Kilaps was extended to incorporate the
probe-to-plasma relative velocity. Since there is no longer az-
imuthal symmetry in flowing plasmas, the problem is very
demanding from a computational point of view, and parallel
computing is required. Both normalized probe bias /pand
probe velocity Swere varied smoothly and then kept constant
during enough time to let the plasma reach an equilibrium
state. Repeating this strategy for several Svalues allowed to
make an analysis of plasma properties as a function of S.
The discretization algorithm implemented by Kilaps, an
Eulerian Vlasov code, provided results free of numerical
noise. However, due to the high computational cost, the
dimensionless parameters used in the simulations are far
from real space tether missions operating within the meso-
thermal regime. Since the code is parallelized, more realistic
dimensionless values will be achieved in the future thanks to
the development of computer clusters. This drawback did
not prevent to get insight into the paradox raised in Ref. 15.
The numerical simulations confirmed that particle densities
exceed the background density at the probe front side (see
Fig. 3), where electron trapping also occurs. This population
is crucial to recover quasineutrality at the probe front and
should not be neglected. A second population of electrons or-
biting around the probe was also detected. Unlike the no-
flowing plasma case, trapped particles trajectories do not
seem to be regular and exhibit a complex behavior. We
remark that neither particle energy nor angular momentum
are conserved because the potential depends (periodically)
on time and on the azimuthal coordinate. Another interesting
footprint of the probe motion is the presence of a filament in
the ion distribution function at the probe front side. This fila-
ment corresponds to rammed ions that were reflected by the
Besides electron trapping, most of the feature of the sim-
ulations are in agreement with results from steady-state solv-
ers. Electron density profiles (Figs. 24) are similar to the
one found in Sec. 4.6.1 of Ref. 22; in the ram side, electron
density minimum is less pronounced as Sincreases, but it
becomes deeper in the wake side. A second common feature
with Ref. 22 is the potential depression (below 0) on the
wake side (see right inset in Fig. 9). The main difference
between Kilaps simulations and the one reported in Ref. 22
is the collected current. It is slightly above I
in our simu-
lations but decays with Sin Ref. 22. Experimental measure-
ments indicated that plasma flow leads to a current
enhancement over that predicted by the OML theory.
Particle trapping occurs in the simulations during the
transient. It suggests that the final state of the plasma
depends on the specific S(t) and U
(t) temporal profiles. This
could be demonstrated by running two simulations with dif-
ferent S(t) and /pðtÞhistories but equal /pðt!1Þ and
S(t!1) values. Such a calculation is beyond the scope of
this paper due to computational limitations. However, the ex-
istence of multiple solutions for exactly de same value of the
parameters is not new. For instance, the solution found in
Ref. 21, which included particle trapping, was different to
the one found in Ref. 5, and they both correspond to the
same parameter values.
The parallelization scheme and the use of special sub-
routines to handle sparse matrices were essential to investi-
gate Langmuir probes with Eulerian Vlasov codes. Any
additional physical effect that would keep the problem
within a 2-dimensional geometry, could be incorporated to
the algorithm without increasing the computational cost
noticeably. Some examples are the analysis of emissive
Langmuir probes and the presence of an external magnetic
field component along the probe axis. Extensions of Kilaps
in these directions are currently in progress.
The authors wish to acknowledge very helpful
comments by the referee. D. Pastor-Moreno was supported
by a grant from the European Commission, FP-7 Space
FIG. 11. Poincare sections (with the hypersurface h¼p) of trapped elec-
trons. Top (bottom) panel corresponds to electron orbiting around the probe
(at the front).
073504-7 G. S
anchez-Arriaga and D. Pastor-Moreno Phys. Plasmas 21, 073504 (2014)
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project BETs (No. 262972). The authors thankfully
acknowledge the computer resources, technical expertise,
and assistance provided by the Supercomputing and
Visualization Center of Madrid (CeSViMa).
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... Langmuir and emissive probes, which are the oldest diagnostic devices for low-temperature plasmas, have also been investigated with Eulerian codes. The sheath formation around planar and cylindrical probe was studied in the framework of the non-stationary Vlasov-Poisson system with Eulerian solvers [17,18,19,20]. These solvers play an important role in the understanding of basic phenomena, like the particle trapping during the transient phase [19,20], but their use for constructing databases with current-voltage characteristics (I-V curves) in a broad range of physical parameters is beyond actual computational capabilities. ...
... The sheath formation around planar and cylindrical probe was studied in the framework of the non-stationary Vlasov-Poisson system with Eulerian solvers [17,18,19,20]. These solvers play an important role in the understanding of basic phenomena, like the particle trapping during the transient phase [19,20], but their use for constructing databases with current-voltage characteristics (I-V curves) in a broad range of physical parameters is beyond actual computational capabilities. For this reason, present models for the I-V curves in collisionless plasmas come from analytical and numerical analysis of the stationary Vlasov-Poisson system. ...
... Without trapping, the density of the electrons (attracted species) should be lower than N 0 [36] but the density of the hypersonic (repelled) ions should exceed N 0 in the front side, thus breaking the quasi-neutrality in a broad region. It was proposed [37], and later verified with non-stationary Vlasov simulations [20], that a population of trapped particles is indeed essential to recover the quasi-neutrality at the front. ...
Full-text available
Physical and numerical limitations of stationary Vlasov-Poisson solvers based on backward Liouville methods are investigated with five solvers that combine different meshes, numerical integrators, and electric field interpolation schemes. Since some of the limitations arise when moving from an inte-grable to a non-integrable configuration, an elliptical Langmuir probe immersed in a Maxwellian plasma was considered and the eccentricity (e p) of its cross-section used as integrability-breaking parameter. In the cylindrical case, e p = 0, the energy and angular momentum are both conserved. The trajectories of the charged particles are regular and the boundaries that separate trapped from non-trapped particles in phase space are smooth curves. However, their computation has to be done carefully because, albeit small, the intrinsic numerical errors of some solvers break these conservation laws. It is shown that an optimum exists for the number of loops around the probe that the solvers need to classify a particle trajectory as trapped. For e p 0, the angular momentum is not conserved and particle dynamics in phase space is a mix of regular and chaotic orbits. The distribution function is filamented and the boundaries that separate trapped from non-trapped particles in phase space have a fractal geometry. The results were used to make a list of recommendations for the practical implementation of stationary Vlasov-Poisson solvers in a wide range of physical scenarios.
... They cover a broad range of aspects, limited not only to mission analysis, such as tether dynamics and control, 10 risk assessment, and performance determination, but also to relevant scientific subjects such as EDT-plasma current exchange. For instance, particlein-cell simulations [4,5,6,7,8,9,10] and stationary [11,12,13] and nonstationary [14] Eulerian solvers enabled the study of current collection to and emission from EDT systems. With respect to simulation tools aimed at mis-EDT simulation lies in handling the disparate frequencies that naturally appear in tether dynamics, including the slow in-plane and out-of-plane oscillations, the moderate lateral string modes, and the fast longitudinal modes. ...
Full-text available
Five electrodynamic tether simulators (BETsMA v2.0, DYNATETHER, EDTSim, FLEX, and TeMPEST) have been cross-verified by running and analysing simulations of increasing complexity. A set of ten simulations without any tether was run to test the orbital propagators and the implementation of the perturbation force due to the non-sphericity of the Earth and its non-homogeneous mass distribution. The environmental models of the five codes and their implementation were then cross-verified by analysing the evolution of the magnetic field and the plasma and atmospheric densities. The electric modules of the simulators for electrodynamic tethers working in the passive mode, i.e., the routines in charge of computing the current and the voltage profiles along the tethers as well as the Lorentz force, were compared by running simulations with bare tethers. Configurations with an ideal electron emitter (zero potential drop), a real emitter, and a resistor and an ideal emitter were considered, as well as round and tape tethers. The electric models of BETsMA, EDTSim, FLEX, and TeMPEST, which assumed a straight tether, were also compared with the result of DYNATETHER for curved tethers. The consistency of the five simulators as a whole was tested by preparing performance maps with the deorbit time versus orbit inclination for a reference scenario and considering tape and round electrodynamic tethers. Although they implement different models and make different assumptions, the results of the five codes are consistent for the full simulation campaign. The simulation data and the software to visualize them are available in a public repository.
... Nonetheless, for many interesting scenarios such conservation laws do not hold. Relevant examples include objects with less regular geometries such as non-spherical dust grains [25] and EDTs with tape-like cross-sections [26,27], plasma probes with finite lengths [28], plasma-sheath lens [29][30][31] and objects with simple shapes but immersed in flowing plasmas [32][33][34]. ...
Full-text available
The structure of the sheath and the current exchange of two-dimensional electron-emitting objects with elliptic cross-section immersed at rest in Maxwellian plasmas are investigated with an energy-conserving stationary Vlasov-Poisson solver free of statistical noise. The parameter domains for current collection within the Orbital-Motion-Limited (OML) regime and current emission in Space-Charge-Limited (SCL) conditions were studied by varying the characteristic dimension of the ellipse, its eccentricity, and the emission level. The analysis reveals the correlations between the onset of the non-OML and SCL regimes and the local curvature of the ellipse. As compared to non-emitting ellipses, electron emission broadens the parameter domain for OML current collection for ions and reduces considerably the current drop for non-OML conditions. Under identical plasma environments, elliptic bodies are more prone to operate under non-OML and SCL conditions than cylinders. Their emitted current in SCL conditions can be computed accurately from well-known results for cylinders if appropriate dimensionless variables and an equivalent radius are used. The role of the eccentricity, which acts as an integrability-breaking parameter, on the filamentation of the distribution function of the attracted species is studied.
... Parameters kept constant across the following simulations are: T e = 4 eV, e mfp = 0.01 m, and i mfp = 0.04 m. As is often done in computationally expensive kinetic plasma simulations [40,41], a sub-amu ion mass m ion = 0.1 amu is used to reduce the time required for the plasma to evolve from its initial state to equilibrium. A key challenge for 2D simulation is capturing the proper separation of length scales and still being able to resolve a VC. ...
Full-text available
Recent one-dimensional simulations of planar sheaths with strong electron emission have shown that trapping of charge-exchange ions causes transitions from space-charge limited (SCL) to inverse sheaths. However, multidimensional emitting sheath phenomena with collisions remained unexplored, due in part to high computational cost. We developed a novel continuum kinetic code to study the sheath physics, current flow and potential distributions in two-dimensional unmagnetized configurations with emitting surfaces. For small negatively biased thermionic cathodes in a plasma, the cathode sheath can exist in an equilibrium SCL state. The SCL sheath carries an immense density of trapped ions, neutralized by thermoelectrons, within the potential well of the virtual cathode. For further increases of emitted flux, the trapped ion cloud expands in space. The trapped ion space charge causes an increase of thermionic current far beyond the saturation limit predicted by conventional collisionless SCL sheath models without ion trapping. For sufficiently strong emission, the trapped ion cloud consumes the entire 2D plasma domain, forming a mode with globally confined ions and an inverse sheath at the cathode. In situations where the emitted flux is fixed and the bias is swept (e.g. emissive probe), the trapped ions cause a large thermionic current to escape for all biases below the plasma potential. Strong suppression of the thermionic emission, required for the probe to float, only occurs when the probe is above the plasma potential.
... However, finding numerical solutions of the Vlasov-Poisson system, specially in non-stationary conditions, is computationally demanding. Particles-in-cell codes and Eulerian Vlasov solvers are the most popular methods and they both have been used for analyzing Langmuir probes (see for instance [29] and [30] and references therein). PIC codes are typically simpler but have lower precision and exhibit numerical noise [31]. ...
Conference Paper
Full-text available
Low Work-function Tethers (LWT) are a promising technology for propellantless space propulsion. The modelling of the charge exchange between the LWT and the ambient plasma, i.e. electron collection/emission, is a critical task, because the Lorentz force is strictly related to the electric current. Past works considered cylindrical tethers, but it is well-known that the tape geometry is advantageous from a performance point of view. Such a transition introduces important difficulties on the computation of the current because the angular momentum of the particles is not conserved. This work presents a 2-dimensional Vlasov-Poisson solver for the computation of the current exchange by LWTs with an arbitrary cross-section immersed in Maxwellian plasmas. After verifying the code with previous results on cylindrical emissive LWTs, the code has been used to compute the sheath structure around an electron-emitting LWT with an elliptical cross-section. Relevant magnitudes, including electrostatic potential, plasma densities, and distribution functions are presented.
... For instance, it was argued that a population of trapped electrons should exist at the ram side of an electronattracting Langmuir probe in flowing plasma, 12 and such a population has been recently detected by running non-stationary direct (eulerian) Vlasov simulations. 13 In the case of plasma expansion under the presence of magnetic field, adiabatic trapping in slowly varying time-dependent electric fields 14 were also considered. 15 Non-stationary particle-in-cell simulations, which have been also adapted to incorporate particle collisions, 16 constitute an alternative to study the particle trapping. ...
Conference Paper
Full-text available
The one-dimensional (paraxial approximation) transient expansion into vacuum of a collisionless electron-ion plasma guided by a magnetic nozzle is studied numerically. The simulation box, initially empty, has zero boundary conditions for the gyrocenter distribution functions of electrons and ions ¯ fe and ¯ fi, except at the entry of the nozzle, where particles with a positive axial velocity follow a Maxwellian. The time evolutions of ¯ fe and ¯ fi are computed with a parallelized direct Vlasov code, which solves a non-stationary guiding center equation for fully magnetized plasmas and discretizes the distribution functions in phase space. The latter involves the (conserved) magnetic moment, and the axial coordinate and velocity of the particles. The gyrocenter distribution functions of the electrons and the ions, affected by the axial components of the electrostatic electric field and the gradient of the magnetic field strength, are coupled through Poisson equation in the code. The evolution of macroscopic quantities, like particle density and electrostatic potential profiles , are discussed. Relevant kinetic features, such as the evolution of the ions towards a mono-energetic distribution function and the evolution of the plasma temperature profiles, are analyzed. The electron trapping, which the stationary models cannot determined self-consistently, and the transient trapping mechanism are captured by the code. This allows an assessment of the impact of the population of trapped electrons and a detailed analysis of their distribution function in terms of axial position, velocity and magnetic moment. Extensions of the code to two-dimensional configurations with axisymmetric geometry, but still fully magnetized plasmas, are discussed. Nomenclature B = magnetic field E = electric field f = gyrocenter distribution function H = particle energy I = total plasma current j = current density P = local plasma pressure R L = current loop radius B 0 = magnetic field at the nozzle throat T = local plasma temperature z = axial coordinate n = local plasma density N 0 = plasma density at the reservoir Q = local heat flux t = time u = average parallel velocity v = axial velocity z 0 = axial coordinate of the nozzle entrance z M = maximum axial distance of the computational domain Z = charge number β = particle-to-electron mass ratio δ = particle-to-electron temperature ratio φ = electrostatic potential λ De = electron Debye length µ = magnetic moment ω pe = electron plasma frequency Subscripts α = e, i = electrons, ions M = variables at z M T = variables at the throat of the nozzle = along the magnetic field lines ⊥ = normal to the magnetic field lines 2
Full-text available
We study qualitative and quantitative properties of the solutions of a Vlasov–Poisson system modeling the interaction between a plasma and a cylindrical Langmuir probe. In particular, we exhibit a class of radial sheath solutions for which the electrostatic potential is increasing and strictly concave with a strong variation in the vicinity of the probe which scales as the inverse of the Debye length. These solutions are proven to exist provided the incoming distributions of particles from the plasma verify the so‐called generalized Bohm condition of plasma physics. It extends our previous studies.
Methods for inferring the electron distribution function (EDF) from a probe current–voltage trace are reviewed for moderate-collisional plasma. The explicit accounting of particle collisions, modeling sheath depletion, and electron refilling processes are rarely performed when analyzing probe measurements, yet they are critical for regime validity, EDF-profile accuracy, and space potential quantification precision. The advantages, limitations, and utilization of large-area wall probes in moderate-collisional plasma, capable of highly resolved indications of energetic electrons in plasma, are outlined.
Full-text available
This article presents an overview of recent advances in the field of electron kinetics in low-temperature plasmas (LTPs). It also provides author's views on where the field is headed and suggests promising strategies for further development. The authors have selected several problems to illustrate multidisciplinary nature of the subject (space and laboratory plasma, collisionless and collisional plasmas, and low-pressure and high-pressure discharges) and to illustrate how cross-disciplinary research efforts could enable further progress. Nonlocal electron kinetics and nonlocal electrodynamics in low-pressure rf plasmas resemble collisionless effects in space plasma and hot plasma effects in fusion science, terahertz technology, and plasmonics. The formation of electron groups in dc and rf discharges has much in common with three groups of electrons (core, strahl, and halo) in solar wind. Runaway electrons in LTPs are responsible for a wide range of physical phenomena from nano- and picoscale breakdown of dielectrics to lightning initiation. Understanding electron kinetics of LTPs could promote scientific advances in a number of topics in plasma physics and accelerate modern plasma technologies.
Full-text available
The time-dependent current collection by a cylindrical Langmuir probe, whose bias is suddenly changed from zero to a positive or negative finite value, is studied with a novel direct Vlasov code. The numerical algorithm is based on finite-difference formulas to approximate spatial and velocity derivatives and the time integration is carried out with an explicit Runge-Kutta method, or in the case of probe radius small compared with the Debye length, by using the unconditionally stable backward Euler scheme. Both electrons and ions are treated kinetically by the code, which implements initial and boundary conditions that are consistent with the presence of the probe. Within the considered parameter range, the plasma sheath around the probe exhibited an overshoot and it later recovered a steady state. Phase space diagrams of the particle trajectories revealed the presence of a trapped population of particles. The dependence of this population as a function of the probe radius is presented as well as a comparison with the stationary theory. The performance of the code and a comparison with previously used particle-in-cell algorithms are discussed.
Full-text available
The current I to a cylindrical Langmuir probe with a bias {Phi}{sub p} satisfying {beta}{identical_to}e{Phi}{sub p}/m{sub e}c{sup 2}{approx}O(1) is discussed. The probe is considered at rest in an unmagnetized plasma composed of electrons and ions with temperatures kT{sub e}{approx}kT{sub i} Much-Less-Than m{sub e}c{sup 2}. For small enough radius, the probe collects the relativistic orbital-motion-limited (OML) current I{sub OML}, which is shown to be larger than the non-relativistic result; the OML current is proportional to {beta}{sup 1/2} and {beta}{sup 3/2} in the limits {beta} Much-Less-Than 1 and {beta} Much-Greater-Than 1, respectively. Unlike the non-relativistic case, the electron density can exceed the unperturbed density value. An asymptotic theory allowed to compute the maximum radius of the probe to collect OML current, the sheath radius for probe radius well below maximum and how the ratio I/I{sub OML} drops below unity when the maximum radius is exceeded. A numerical algorithm that solves the Vlasov-Poisson system was implemented and density and potential profiles presented. The results and their implications in a possible mission to Jupiter with electrodynamic bare tethers are discussed.
Full-text available
The present electron-collection concept for ionospheric electrodynamic tethers exposes a fraction of the tether length near its anodic end, so that electrons are collected in an orbital-motion-limited regime when a positive bias develops locally relative to the ambient plasma. The tether radius must be small compared with both the thermal gyroradius and the Debye length. Large currents can in this way be drawn with only moderate voltage drops, as is illustrated for the cases of generators and thrusters.
Full-text available
An asymptotic analysis is presented of the Langmuir-probe problem in a quiescent, collisionless plasma in the limit of large body dimension to Debye length ratio. The structures of the electric potential distribution about spheres and cylinders are analyzed and discussed in detail. It is shown that when the probe potential is smaller than a certain well defined value, there exists no sheath adjacent to the solid surface. At large body potentials, for which a sheath is present, the electric potential distribution is given in terms of several universal functions. Master current-voltage characteristic diagrams are given which exhibit clearly the effects of all the pertinent parameters in the problem. An explicit trapped-ion criterion is presented. The general problem with an arbitrary body dimension to Debye length ratio is qualitatively discussed.
BETs is a three-year project financed by the Space Program of the European Commission, aimed at developing an efficient deorbit system that could be carried on board any future satellite launched into Low Earth Orbit (LEO). The operational system involves a conductive tape-tether left bare to establish anodic contact with the ambient plasma as a giant Langmuir probe. As a part of this project, we are carrying out both numerical and experimental approaches to estimate the collected current by the positive part of the tether. This paper deals with experimental measurements performed in the IONospheric Atmosphere Simulator (JONAS) plasma chamber of the Onera-Space Environment Department. The JONAS facility is a 9-${rm m}^{3}$ vacuum chamber equipped with a plasma source providing drifting plasma simulating LEO conditions in terms of density and temperature. A thin metallic cylinder, simulating the tether, is set inside the chamber and polarized up to 1000 V. The Earth's magnetic field is neutralized inside the chamber. In a first time, tether collected current versus tether polarization is measured for different plasma source energies and densities. In complement, several types of Langmuir probes are used at the same location to allow the extraction of both ion densities and electron parameters by computer modeling (classical Langmuir probe characteristics are not accurate enough in the present situation). These two measurements permit estimation of the discrepancies between the theoretical collection laws, orbital motion limited law in particular, and the experimental data in LEO-like conditions without magnetic fields. In a second time, the spatial variations and the time evolutions of the plasma properties around the tether are investigated. Spherical and emissive Langmuir probes are also used for a more extensive characterization of the plasma in space and time dependent analysis. Results show the- ion depletion because of the wake effect and the accumulation of ions upstream of the tether. In some regimes (at large positive potential), oscillations are observed on the tether collected current and on Langmuir probe collected current in specific sites.
Results are presented of particle simulations of the time-dependent behavior of the plasma sheath surrounding an electrode whose potential is abruptly changed from 0 to some positive value. The surrounding plasma is assumed isotropic and collisionless, and the electrode is a sphere or infinite cylinder. Potential steps in the range 10 to 10,000 kT/e are considered. The transient behavior of the collected electron current is examined. For small (25kT/e or less) potential steps, the current quickly settles to a value consistent with existing static probe theory. Persistent Langmuir oscillations launched by larger potential steps significantly perturb the collected current. Trapping of electrons during all or part of an oscillation cycle is evident in many cases. In some cases, the oscillations are driven to large amplitudes by the ion-electron two-stream instability.
This thesis presents a numerical study of the ion current-collection behaviour of semi-infinite and infinite cylindrical probes, in a collisionless drifting Maxwellian plasma. The ion current collected, and the ion densities and the electric potential field surrounding a probe under a variety of plasma conditions typical of those encountered in the ionosphere are investigated. In the case of an infinite cylindrical probe, it has been found that the presence of plasma drift with respect to the probe strongly affects the presheath and sheath which surround the probe. The changes in the presheath and sheath induced by the plasma drift influence the ion current-collection behaviour of the probe and some of the structures that form in the ion density field around it. The changes in the presheath and sheath are shown to have a dependency upon the probe potential, probe size, and plasma temperature. The ion current has been found to have either a monotonic or non-monotonic dependency upon these parameters and the plasma drift speed. In the present work we show that this behaviour can be explained by the changes occurring in the presheath and sheath as a function of the plasma drift speed. In the case of a finite cylindrical probe, the calculation is not exact since Poisson's equation is not solved. The electric potential field around the finite probe has been modelled using hyper-ellipses to extrapolate from the potential field for an infinite probe. The ion current collected by a finite cylindrical probe with its axis at various angles with respect to the direction of plasma drift has been calculated. A peak in the ion current as reported by other researchers as the probe axis was brought into alignment with the plasma drift direction has been observed. The calculations were performed at a variety of electron-to-ion temperature ratios and the peak in ion current has been seen to depend upon the temperature ratio. The ion density field around a finite probe has also been calculated and is substantially different near the end of the probe than for an equivalent infinite probe. Ion density peaks at three times ambient were found in the wake of a probe. Comparisons of our results with the work of other investigators are made where applicable.
A new type of electrostatic probe design is proposed, which should permit temperature and density measurements for both ions and electrons from the accelerated current regions of the probe characteristics. Ion temperature measurements, in particular, have been difficult to obtain using standard probe techniques. The method is based on the use of one of a family of nonspherical probe geometries which can be operated as if they were spheres collecting orbit‐limited current. One such probe is a proposed multi‐electrode system having the advantage that the collector can be made much smaller than the usual spherical probe. The simultaneous use of this probe with a standard orbit‐limited cylindrical probe would then enable the above measurements to be made. The multi‐electrode probe may itself function as an orbit‐limited collector of both the spherical and cylindrical types, with different bias strategies on its electrodes for the two cases. The proposed probe operation is theoretically justified by rederiving the usual orbit‐limited current expressions for spheres and circular cylinders, without making any assumption regarding particle angular momentum conservation. The expressions are thereby shown to apply to a wider variety of probe shapes, including: (a) any convex cylinder, (b) any “sufficiently convex” three‐dimensional collector (for example prolate and oblate spheroids having major to minor axis ratios up to 1.653 and 2.537, respectively), and (c) the proposed multi‐electrode design. Some aspects of probe use in flowing plasmas and in magnetic fields are also discussed.
The theory of sperical and cylindrical probes immersed in plasmas of ; such low density that collisions can be neglected is formulated. The appropriate ; Boltzmann equation is solved, yielding the particle density and flux as ; functionais of the electrostatic potential, the situation in the body of the ; plasma, and the properties of the prohe. This infoimation whcn inserted in ; Poisson's equation serves to determine ihe potential, and hence the probe ; characteristic. No a priori separation into sheath and plasma regions is ; required. Though amenable to a deternmination of the full probe characteristic. ; The method is applied in detail and numerical resuits are presented only for the ; collection of monoenergetic ions. for the case of neglignble electron current. ; These resuits indicate that the potential is not so insensitive to ion energy as ; has been believed, and that if the probe radius is sufficiently small, there ; enters the possibility of a class of ions which are trapped near the probe in ; troughs of the effective radiai potential energy. The population of these ; trapped ions is determined by collisions, however infrequent. It is difficult to ; calculates and coneeivably cofin have a marked effeet on the local potential. ; (auth);