PFC/JA-88-58
A Fluid Theory of Ion Collection By Probes in
Strong Magnetic Fields with Plasma Flow
I.H. Hutchinson
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139
October 1986
Submitted to: Physics of Fluids
This work was supported by the U. S. Department of Energy Contract No. DE-AC0278ET51013. Reproduction, translation, publication, use and disposal, in whole or in part
by or for the United States government is permitted.
By acceptance of this article, the publisher and/or recipient acknowledges the U. S. Government's right to retain a non-exclusive, royalty-free license in and to any copyright covering
this paper.
Abstract
A1T-diimensionalfluid theory of ~~Lanaiid
magnetic fields is presented.
prhe ~Oerai fri n
-n strong
Cross-field diffusion of ions both into and
out of the collection region is consistently accounted
differ from previous analyses,
by large factors,
present.
for.
The results
which did not account for outward diffusion,
especially when parallel flow of the external plasma is
These results provide a more reliable basis for interpretation of
recent probe measurements.
2
I. Introduction
The theory of Langmuir probe operation in strong magnetic fields is of
notorious difficulty
[1-5].
However,
the need for a reliable
theory of
probe operation in strong fields has become more urgent recently in view of
the
increased significance
attributed
to edge conditions
in magnetically
-confined fusion-research-plasmas and -the-accompanying proliferation of-probe
measurements
of these edge
plasmas
substantially smaller than the
field
is diffusive
effects.
As
a
even
if
result,
[6].
When the
probe radius,
the parallel
the
a,
ion collection
flow
quasi-neutral
is
until
the
cross-field
dominated
presheath
acceleration of the ions occurs into the sheath,
along the field,
ion gyroradius,
diffusion
pi,
is
across the
by
region,
inertial
in
which
becomes highly elongated
is able
to
balance
the
in this
process,
it
parallel collection flow.
Since the perpendicular
appears
attractive
presheath
as
to
momentum is unimportant
attempt
to
simplify
effectively one-dimensional.
the
problem
One
can
by
then
treating
seek
the
solutions
satisfying Poisson's equation and the equations of motion self-consistently
in the parallel direction, treating the perpendicular diffusion equation as
a source term in the parallel equations.
Stangeby [7,8] has championed this
approach recently in applications which have
adopted
either
a fluid or
a
particle description of the plasma.
The approximations adopted in the 1-dimensional model are that the ion
density, ni,
velocity,
vi, and plasma potential
*
at any parallel position,
x, can be regarded as given by single functions of x, representing some kind
of
mean
value
of
the
collection region.
the probe radius,
parameter
over
the
perpendicular
extent
of
the
The radius of the collection region is taken as equal to
a.
The cross-field diffusion of ions into the collection
3
may be represented
region
by a
source,
S,
in
(Without sources a
determine the parallel extent of the collection region.
one
dimensional
presheath
would
expand
to
which
ion equations
the
because
infinity
the
quasi
neutrality equation would then imply f - constant [9]).
Stangeby and others, in applying this model, have adopted forms of the
ion
~~thfi--the -ollection
sdf
source rate corresponding to 'birth --of
The resulting equations are then identical to those governing a 1-
region.
plasma discharge,
dimensional
sustained
within
by ionization
plasma
the
This latter problem has been studied from
region, between parallel plates.
a kinetic theory plasma viewpoint by Tonks and Langmuir [10] for zero birth
velocity and later by Harrison and Thompson [11] who demonstrated that the
and current density are independent of the spatial
sheath edge potential
et al,
More recently Emmert,
variation of S.
[12] have
extended
these
results to the case of finite temperature Maxwellian birth velocity.
These kinetic cases are not easily generalized to the experimentally
important
of
situation
a plasma
parallel
with
in which
flow,
presumably
Therefore, Stangeby
birth with an appropriate flow velocity should be used.
has given a fluid treatment [8] which proves (like the previously mentioned
cases)
to be analytically
formulae
providing compact
integrable,
for
ion
density and current.
It
of
is,
in part,
these analyses
theories
in
the purpose of the present work to point out that all
are
treating
fatally
the
flawed;
1-dimensional
validly applied to treating
that,
plasma
strong magnetic
only to diffuse
into the
i4
cannot
probe analysis.
field
As such,
region,
collection
they
discharge,
reason lies in the form of the source function adopted.
model the cross field diffusion of ions.
validity of
the
despite
it
but also
the
be
The
That source is to
must allow ions not
to
diffuse out.
In
other
words,
a diffusion
process
consists
of
the exchange
between the collection region and the outer plasma.
of
particles
This means that the
sources in the equations should model not only outer particles entering the
collection region but also
(a perhaps smaller number of)
particles,
which
may have spent some time in the collection region, leaving it.
From ~the point of vi
of ineAn-
isit
alrifnare--qua1, so
allowance for ion loss merely causes the source to vary according to the
difference between the inside and outside densities.
Of itself this would
make no difference to the collection
it
collection
region
extent)
because,
current
as
variation does not affect the current.
velocity,
all
ions are not equal.
(though
mentioned
However,
Loss
of
would change the
above,
the
source
rate
from the point of view of
ions which
have
already
been
accelerated in the presheath is not the same as gaining fewer ions at the
outer plasma velocity
(e.g.,
ions to be characteristic
only the
'birth'
It
incoming
at rest).
Therefore,
only of the external
particles not
only, not for 'death'
to allow the source
velocity,
models correctly
That
it
the outgoing.
is,
accounts
for
of particles in the collection region.
might be thought that this distinction is mere quibbling about the
details of a model which is already admittedly rather approximate.
in the
of
following
sections
we
will
see
that
the
quantitative
However,
differences
between the model we propose here, which does correctly account for particle
exchange,
and
the models
discussed
above,
are
in
especially when finite ion velocity in the outer
many cases
plasma
very
is accounted
It turns out that the simplest kinetic treatment mentioned above,
the zero ion birth velocity solution of Harrison
fairly well with the collection
results
based on finite
ion
and Thompson
current we shall calculate.
temperature
5
lead
to the
large,
for.
based on
Ell] agrees
However,
the
plainly unphysical
result that for Ti
(1/4 niv,),
Te the ions are collected at twice the free gas result
a problem which has been alluded to elsewhere
ion drift velocity is allowed,
[13].
And when
Stangeby's fluid model can be as much as a
factor of 4 wrong for Mach numbers up to 1.
It turns out, too, that the equations correctly accounting for particle
appear
exchange
Therefore,
in
to
not
order
be -suse-ptible~"to -exact
e
to minimize
obtaining applicable results,
to Stangeby's.
the
computational
anatytIc~solutTon
effort
and
focus
on
we shall treat a simple fluid model parallel
That will provide us with a direct comparison which will
illustrate the differences.
In Section II we briefly derive the equations;
then in Section III a
simple approximate analytic solution of the presheath
with
an
exact
numerical
integration
of
the
is given,
equations
for
together
outer
drift
velocities up to the sound speed.
These results provide the data with which
probe
ion
measurements
interpreted.
II.
to
Section
determine
IV
gives
density
a
brief
and
drift
velocity
discussion
and
can
be
conclusion.
The Model
The presheath is modeled as a 1-dimensional, two-fluid plasma, which is
quasi-neutral.
equation,
Thus, Poisson's equation is replaced by the quasi-neutrality
Zni-ne
(Z
is the
ion
charge).
Also,
we restrict
attention
to
cases in which the majority of electrons are reflected because the probe is
sufficiently negative.
Then the electron density can be taken as given by a
Boltzmann factor,
n
Subscript
collection
-
here
region,
- Zn, exp(e$/T )
denotes
where
quantities
we take
.
(1)
in the outer
the
6
potential
plasma,
0-0,
to
far
be
from
zero.
the
The
electron temperature,
Te,
is in energy units.
The diffusive exchange of ions between the collection region and the
outer plasma we suppose to take place at a rate 9.
the rate of
That is,
loss of particles per unit length is Qnfi(x) and the rate of gain is an.. We
can regard a as being approximated
by D1 /a2, the
proves
However , the collection current
to be
inverse
time
diffusion coefficient
perpendicular
.constant -of_ the-collection _regionfor
Ds.
diffusive
independent
D
of
and
Thus a determines only the length of
indeed of the spatial variation of D1 .
the collection region.
The 1-dimensional continuity equation in steady state is therefore
d (n v )Ui_x
(n-n)
.
(2)
The exchange of momentum between the collection region and the outer plasma
particles leaving with characteristic momentum mivi
is caused by the
entering with mivw.
Therefore,
the momentum equation is
dpi
dvi
n.
vi
v
nimiv. dx
and
+ M v u(n -n
)-n
ii
ZeE - di+
m n(n~v.-n v )(3)
where Ze and pi are the ion charge and pressure respectively,
By substituting for d /dx from Eq.
d$/dx is the electric field.
(3)
ifln-~~
dx
and E
(1)
=
-
this
becomes
2 dn.
dv.
niv
=
-
cs
In obtaining this equation we
dpi/dx.
Thus,
we
are
+
- v )
have ignored
adopting
corresponding to isothermal ions.
A n (v
a
()
a term
closure
of
dTi/dx
the
arising
fluid
from
equations
This is hardly justified a priori, and so
should be regarded as a simplifying approximation only.
c 5 , corresponding to this approximation is given by
7
The sound speed,
2
ZT +T
m
Ca
Equations (2)
solve.
and (4)
(5)
are now the plasma presheath equations we require to
It should be noted that they differ from those of Stangeby
in the final term of equation (2)
[8] only
( i.e. gni) which is absent in his model.
We now make the following non-dimensionalizing transformations
n - n /n.
M -v/c
(6)
fx
y
L
dx'
O s
These bring the equations into a form
Mdn + n-dM -1
M-+
dy
dy()
+ nM
-
-n
- M.
By elimination we then obtain
dn
dy
(1-n)M-(M,-M)
dM
(MH-M)M-(1-n)
M2 _1
(8)
n(M21)
and hence
dn
Notice that Eqs.
(1-n)M- (M-M)
=n
(8)
(M,-M)M-(1-n)
give singularities at M-±1.
sign to denote flow towards the probe,
Bohm
criterion
[1]
for
condition at the probe
sheath
(y-0).
M-+1
formation,
In view of
there is no regular solution satisfying M=-+1
8
Taking the
positive
is simply the equivalent of the
and
is
the other
the
required
boundary
singularity at M=-1,
at y-O with, M,<-1.
This is a
manifestation
of
supersonic flow.
the
fact
We,
therefore,
that
shocks
will
form
in
the
restrict our attention
this simple model cannot be expected to give an adequate
supersonic flow.
Il.Solutions
MI<1, since
description for
-
-
method of solution is to integrate
obtain n as a function of M.
Eq.
(9)
to
To do so requires attention to the condition
because both numerator and denominator tend
(corresponding to y--),
to zero there.
to
for
The boundary condition at infinity is n-1, M-M,.
The most convenient
at M-M,
presheath
A proper treatment of this limit shows that the required
derivative at M-M,, n-1 is
n
For
-
±1
(10)
.
1M.1<1 we expect M to be increasing and n to be decreasing towards the
probe; therefore, -1
is the appropriate choice.
An exact analytic solution seems not to be possible in closed form for
Eq.
(9).
However,
an approximate solution may be obtained by substituting
an appropriate value for n(M)
then integrating the equation.
into the fraction on the right hand side and
If we take n-1-(M-M,),
as indicated by the
boundary condition and perform this integration we get straightforwardly
n - exp(M,
- M)
(11)
A somewhat better approximation is obtained by seeking an expansion for n to
second order in (M-M.)
zero.
which makes the appropriate order
This gives n-1-(M-M.)+[(1+M.)/(3+M.)](M-M.)
2
terms in Eq.
(9)
, which when substituted
into the fraction in Eq. (9) leads to a solution
n - pa exp(-q)
(12)
where
9
1 (M3 3M +7M,+5) (M +1 ) (M,+5)
-
2(M-M)
p
- 1 + (MW+1)(M +3)
(M + 3M.
q-
(13)
+ 7M
+ 9)(M-M)
+ (M+1)(M-M
)2]
The obvious cumbersomeness of such solutions discourages one from pursuing
them further.
Instead,
for specified M..
numerical integration of Eq.
(9)
has been performed
The results are shown in Fig. 1.
Since ions
flow into
the
probe sheath at
the sound speed
the
ion
current drawn by the probe is proportional to the sheath edge density, i.e.,
the density at M-1.
Mach number M.
Figure 2 shows a plot of this density versus the flow
Also shown are the results obtained from the approximation
Eq. (12) and the corresponding results of Stangeby's model [8].
the approximate
solution is accurate
Notice
numbers.
also how
different
Notice that
for all but the most negative Mach
the
values
obtained
from Stangeby's
model are from our more physically appropriate model.
A
simple
downstream
measure
Langmuir
sides.
On
separately
Therefore,
for
probe
the
the
will
other
upstream
a particular
Mach
draw
hand
ion
a
and
number
current
divided
(or
downstream
of
flow
solution,
are plotted
These quantities,
(in terms of sheath-edge
both
upsteam
and
probe
can
'Janus')
collection
the
useful for diagnosis are the mean collection current
collection currents for ±M..
to
two
currents.
quantities
and the ratio
most
of the
obtained from the numerical
densities)
in Fig.
3.
The
flow Mach number may be deduced from the ratio and the density from the mean
of the ion saturation currents to either side.
Finally we may return to the spatial
integration to
obtain
(nondimensionalized)
y as
a function
equations
of M
(and
(8)
and perform the
hence n(y))
giving the
spatial variation for the presheath density.
10
This is
shown in Fig. 4.
The presheath potential is then given by Eq.
Although the Iresheath parallel length,
does
not
enter directly
into the
which is a few times cea2/D,
ion current deduced,
determining the applicability of the analysis.
greater
than the
parallel
(1).
distance to the
it
is important
in
If the presheath length is
plasma edge or
than
the
ion-
electron CQlliziQn mean free _path,_thenless . ion current will-be -collected
and our treatment will break down.
requires
an estimate
of D1 ,
but
To determine whether or not this occurs
provided
the
presheath
length
is
small
enough our results will be independent of D1 .
IV.
Discussion
The ratio of upstream to downstream ion current deduced from our model
has a value of about
12 at M.-1 and a slope of 2.1 at M=O.
compared with Stangeby's
These
differences
are
probe measurements
result
far
of
outside
3 at M,-1
the
typical
and so indicate that
and a slope
This should be
of
uncertainties
1
at
M,=O.
inherent
use of Stangeby's model will
in
give
deduced flow velocities which are in error by a large factor.
It
is interesting to note that for example Harbour
found ratios
up to
about
12 in their
measurements
and Proudfoot [14]
of
scrape-off
Using Stangeby's analysis these results indicate supersonic flow,
but
flows.
using
a naive particle model Harbour and Proudfoot offered an alternative subsonic
interpretation.
Our
present
results
indicate
that
their
velocities correspond within experimental uncertainties to Mach
The differences in mean ion saturation
that
of
Stangeby
are
less
dramatic
but
particle current density at the sheath edge
11
highest
flow
1.
current between our result and
still
significant.
(for M,=O)
We
obtain
a
ri - 0.35 nec3
whereas Stangeby gets a coefficient of 1/2.
It is often stated that the
Bohm formula for ion current is approximately
1/ nc,,
2
Stangeby's result appears more conventional.
Actually this is fallacious.
The correct (and original) formula is 1/
2 n.,/(ZTe/mi)
from which viewpoint
which Bohm showed [1]
had little dependence on ion energy when Ti<ZTe, for spherical probes in the
absence
of
magnetic
field
or
collisions.
Since
our
definition
of
c
includes an ion temperature term we must have some estimate of Ti before we
can relate our current to n.V(ZTe/mi).
It is clear physically that if the outer plasma ion temperature is much
smaller than Te it must nevertheless be a bad approximation to take Ti-0 in
the c5 definition.
The reason is that the most important place to obtain es
correctly is at the sheath edge. However, there the spread of ion particle
velocities is from zero to V(-Ze$/mi) corresponding to ions which enter the
collection region near or far from the sheath edge respectively.
sheath edge potential is (Te/e)ln(0.35) - -Te/e.
Now the
Therefore, the ion energy
spread at the sheath edge is -ZTe even when Ti=0 outside the collection
region.
external
This spread will be increased only a small amount
Ti.
Therefore,
the
most
appropriate
value
by non-zero
to
use
is
s f /(2ZTe/mi) for Ti < ZTe.
We conclude, therefore, that the ion current deduced from our model is
approximately
Bohm formula,
0.35x/2
whereas
- 0.49 times n,/(ZTe/mi),
Stangeby's
analysis would
for Ti<ZTe,
recovering a
give a value
about /2
higher.
The reason why Stangeby's formulation always gives a density (and hence
current) which is too high, as illustrated for example by Fig.
2, is that
loss of momentum from the accelerated presheath flow has been ignored.
12
In
our formulation this loss is properly accounted for,
with the result that
the sheath edge potential must be more negative in order for the mean ion
velocity to reach the sound speed.
Other qualitative differences exist between our solutions and those of
Stangeby.
We may mention first that our M(y) and n(y) tend smoothly to the
external values as ye., whereas Stangeby's have discontinuous derivatives at
the point where M-M,,
n-1,
(at finite y)
in an obviously unphysical way.
Another point is that our results give monotonic variation of n and M with
y, whereas Stangeby finds that there is a density
maximu
(and hence
potential)
on the downstream side, i.e. for M. < 0.
In conclusion,
collection
a 1-dimensional fluid model has been presented of ion
by probes in strong magnetic fields,
diffusion out of,
as well as into
correctly accounting for
, the presheath.
previous formulations are in error by large factors,
is parallel plasma flow velocity.
The results show that
particularly when there
The present results make it
diagnose these flows using divided (Janus)
possible to
probes.
Acknowledgements
I have been greatly assisted by stimulating and challenging discussions
with Dr.
Bruce Lipchultz.
This work was supported under U.S.
Energy Contract DE/AC02-78ET51 01 3.
13
Department of
References
1.
D. Bohm in Characteristics of Electrical Discharges in Magnetic Fields,
A. Guthrie and R.K. Wakerling (Eds.) (McGraw Hill, New York, 1949).
2.
M. Sugawara, Phys. Fluids, 9, 797 (1966).
3.
J.R. Sanmartin, Phys. Fluids, 13, 103 (1970).
4.
J.G. Laframboise and J. Rubinstein, Phys. Fluids, 19, 1900 (1976).
5.
S.A. Cohen, J. Nucl. Mater. 76 and 77, 68 (1978).
6.
See e.g., A. Miyahara, H. Tawara, N. Itoh, K. Kamada and G.M. McCraken
(Eds.)
Plasma Surface Interactions in Controlled Fusion Devices
(6th
Int. Conf.) J. Nucl. Mater. 128 and 129 (1984).
7.
P.C. Stangeby, J. Phys. D: Appl. Phys. 15, 1007 (1982).
8.
P.C. Stangeby, Phys. Fluids, 27, 2699 (1984).
9.
I.H.
Hutchinson,
Principles
of Plasma Diagnostics
(Cambridge
Press, to be published).
10.
L. Tonks and I. Langmuir, Phys. Rev. 34, 876 (1929).
11.
E.R. Harrison and W.B. Thompson, Proc. Phys. Soc. 74, 145 (1959).
14
Univ.
12.
G.A.
Emmert,
R.M.
Wieland,
A.T. Mense and J.N. Davidson,
Phys.
Fluids
23, 803 (1980).
13.
B.
Lipschultz,
Technol.
14.
I.H. Hutchinson,
B.
Labombard and A.
Wan,
J. Vac.
A4, 1810 (1986).
P.J. Harbour and G. Proudfoot, J. Nucl. Mater. 121, 222 (2984).
15
Sci.
Figure Captions
Fig. 1
Solutions for the normalized density as a function
number,
in the presheath.
external
Various cases are shown,
of ion Mach
for which the
plasma flow velocity is equal to the Mach number when
n-1.
Fig. 2
The normalized sheath-edge density as a function of external flow
Mach number.
Eq.
(9);
The curves are:
'Approx.',
'Exact'
numerical
integrations
the approximate analytical formula,
Eq.
of
(12);
'Stangeby' the result of using the equations of Ref. [8].
Fig. 3
The ratio and mean value of the sheath-edge density for the side
of the probe facing upstream and downstream.
The ion saturation
current density is equal to c. times the sheath-edge density.
Fig. 4
Variation
presheath.
of
density
with
non-dimensional
distance
in
the
This can be related to physical distance via Eq. (6).
16
0
cc
0
(U)
4l1suea
Fig. 1
9
I-
S
S
S
S
S
S
*S
0
0
0
0
0
0
0
*
0
S
0
0
0
S*
S
*
S
S
*
uc
*S
0
0
*0
S
S
S
*
c~o
S
S*
0
*
S
0
0
S
*S
*S
'4,
S
S
S
S
'B'S
0.
0
~
S
9
9
0*
44.
01
C1J
1~
AlISUeG q~eeq~
Fig.
2
xN
coc
cm~~l
o
0
V
0"
Fig.
C
3
0
Li.
'om
4)
8
0
0
Iy
I
-t
0
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CO)
I
a.
0
9z
Al !suea
Fig.
4