Download Diffusion of Arc Plasmas across a Magnetic Field

P/366 USA
Diffusion of Arc Plasmas across a Magnetic Field
By Albert Simon*
The effect of a magnetic field В is to reduce the
coefficients of diffusion, £) p , across the magnetic field
to the values
£>+p = 2 , /
x2?
D T
-
=
l + U T_)2
equations for the ion and electron densities, щ and ne
are:
¿«i
та
d,
^)
where D+ denotes the coefficient for В = 0,
tü-t = eB¡ni±c and т± = mean free time between
collisions for ions or electrons. Diffusion in the
direction of the magnetic field, D\ is unchanged from
its field-free value. Hence
At 1 = D±.
тл -о d2%i
^1
dt
(2)
In many arc experiments, (OJ-T-) > {OJ+T+) > 1 and,
as a result, DJ > D+? > Z)_ p . In this case the plasma
has a highly anisotropic conductivity, since currents
may flow in the direction of the field far more easily
than in the perpendicular direction. In the analysis
which follows, it will be shown that this anisotropy
produces a diffusion across the magnetic field which is
not ambipolar. 1 The ions and electrons, instead,
diffuse at their own intrinsic rates. Space-charge
neutralization is maintained by slight adjustments of
the currents in the direction of the field lines. These
results will then be compared with the experiments of
Bohm, Neidigh, and Bickerton.
The analysis below will be valid only for the case of
a weakly ionized gas. This means that the dominant
mechanisms of diffusion are ion-neutral atom and
electron-neutral atom collisions. In typical arc
plasmas (ion and electron energies of the order of a
few electron volts) this requires the degree of ionization to be smaller than about 1%.
In the final section, the effects of providing magnetic
mirrors at the ends of the arc and the effect of increasing the tube length in the direction of the field
lines will be considered.
2
= T) v d n2e
dx
2
d ne
p d_
dx
d
Here /x denotes the mobility and Ex and Ey are electric
field components in the plasma. The relation between
the mobility across a magnetic field and that parallel
to the field is the same as that for the diffusion coefficients in Eq. (1). Hence /xp <^ /x. In addition, let us
consider a finite system bounded by conducting walls.
If L is the dimension of the system in the ^-direction
and R the dimension in the ^-direction, then one would
expect that Ey\Ex ~ R/L. The reason for this is that
the electric fields are the derivatives of the scalar
potential, which varies monotonically from a central
positive value to zero at the walls. Thus, except for
the case of a very long and thin container, one may
safely neglect the second term on the right side of
Eq. (3) compared to the last term. This is the formal
way of stating that the essential currents which adjust
for space charge neutrality are those which flow in the
direction of the magnetic field.
One may now proceed in the usual fashion to
eliminate the explicit dependence on Ey by the
assumption of approximate charge
neutrality
m = ne = n. By combining the ion and electron
expressions of Eq. (3), there results:
dn
dt
(L-D+-fL+D-\
(4)
\
Note that the effective diffusion coefficient in the
^-direction has the usual ambipolar value,
DIFFUSION IN A UNIFORM FIELD OF FINITE
LENGTH
Theory
Consider a two-dimensional problem in which the
magnetic field is in the jy-direction and the ^-direction
is perpendicular to the field. The time dependent
Deit =
(5)
while the coefficient in the ^-direction is that of the
ions since /x_ > fjL+ and D+ p > Z)_ p . Thus
* Oak Ridge National Laboratory, operated by Union
Carbide Corporation for the U.S. Atomic Energy Commission.
The report covers work by R. V. Neidigh and A. Simon,
Oak Ridge National Laboratory.
=
143
=
D
p
344
SESSION A-10
P/366
ALBERT SIMON
e-Folding Length
Suppose now that ions and electrons are being produced equally and uniformly along the^-axis (x = 0).
The steady state solution of Eqs. (4), (5) and (6), with
inclusion of the source term and boundary conditions,
will be of the form n(x}y) = N(x)G(y). With the
principal solution having the value G(y) = sin (тгу/L),
and with N(x) = Ае~х\1я. where A is a constant and
where
Г2П.Р
-»-
b V +
(10)
The corresponding result for the case of free streaming
to the end walls is
VACUUM MANIFOLD
T
Figure 1. Apparatus used by Neidigh
In many cases the dimensions of the system in the
field direction are small compared to the collision
mean free paths. In this case, the streaming of ions
and electrons in the ^-direction may not be represented by the diffusion terms. It may be shown that
the new equations corresponding to those of Eq. (3)
are now,
dni _
dt
+
P
^i__
dx2
щ\е\Еу
p
t
(7)
dt
where M is the ion mass, m the electron mass, and Vi
and ve are the average thermal velocities. Note that
there is now free streaming to the end walls as weH as
electric field accelerations of the ions and electrons
through a distance of the order of half the arc length.
Once again, it may be shown that the mobility term
in the ^-direction is usually negligible. In the same
fashion as before, one finds
dn
It
-*\d4
)dx2
\
n
L\
/
The effective streaming velocity to the end walls (for
comparable electron and ion energy) is 2v\ while the
effective diffusion coefficient across the field is D+p,
as before.
It should be noted that the effective diffusion coefficient across the magneticfield,D+p, is much larger
than the " ambipolar " coefficient, DPAMB> which
would have been obtained by requiring that the
electron and ion currents balance precisely to zero in
the x- and ^-directions separately. This coefficient, by
analogy to Eq. (5), has the value
P
-D AMB =
2P_g. (9)
Note that the ion density decreases exponentially
with an ¿-folding length q. Note also that the ¿-folding
length varies as Po/B in the diffusion case of Eq. (10)
and as PQ%/B in the free streaming case of Eq. (11).
Here Ро(~т-!) is the gas pressure. These results go
over exactly in the case of cylindrical symmetry
except that
N{x) -> r-ie-r/Q.
(12)
Experiment
A measurement of the ion diffusion coefficient across
a magnetic field was obtained by Bohm et al. at
Berkeley.2 A complete description of the experiments
may be found in Ref. 2. In essence, an arc was struck
parallel to the magnetic field and along the long axis
of a rectangular arc chamber made of graphite. The
resultant ion density in the median plane was measured with a cold shielded probe. In most measurements,
the chamber was filled with argon gas at a pressure of
about 10~3 mm Hg and the magnetic field strength
was 3700 gauss. These observations established that
the ion density decreased exponentially from the
center outwards with an ¿-folding length of about
0.3 cm.
If one assumes that the ion energy in the plasma is
about 2 ev, the resultant mean free path at a pressure
of 1.4xlO~3 mm is then Л# 5 cm. The thermal
velocity is t>i # 3 x 105 cm/sec and the arc length
L = 12 cm. From Eq. (11), using the measured value
of the ¿-folding length, one finds that D+p л 4.5 x 103
cm2/sec. This is to be compared with the theoretical
value of Z)+p obtained from Eq. (1). Since си+т+ = 14.4,
this value is 2.5 x 103 cm2/sec, which is in good agreement with the measured value.
Unfortunately, at that time, it was assumed that
the diffusion across the magnetic field was ambipolar,
and the experimental value was compared with the
theoretical value of DPAMB which is given in Eq. (9).
This value is D P AMB # 1 0 cm2/sec in complete disagreement with the experiment. As a result of this
apparent anomaly, Bohm postulated that the principal
mechanism of diffusion was by plasma oscillations
rather than by the classical collision mechanism. The
345
DIFFUSION OF ARC PLASMAS
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366
2
MAGNETIC
*
FIELD
i
STRENGTH
10
12
1<
( kilo - oersteds)
Figure 2. Radial variation of ion density
Figure 3. Variation of e-folding length with В
details of this theory have not been published. However, the resultant diffusion coefficient is stated to be 2
sec) is in good agreement with that determined in the
experiment of Ref. 2.
D
16B '
(13)
where kTe is in electron volts and В in kilogauss. Note
that this coefficient varies as B~l and is pressure
independent, whereas the classical coefficient varies as
B~2 and is directly proportional to the pressure.
In order to test these theories, Neidigh3 has carried
out a series of experiments at Oak Ridge. A sketch of
his apparatus is shown in Fig. 1. The ion chamber is
cylindrical and most experiments were carried out
with nitrogen gas at a pressure of about 10~3 mm. The
magnetic field could be varied from 2000 to 14,000
gauss. The carbon probe is biased 20 volts negative,
which is on the flat portion of its characteristic curve.
Measured ion currents varied from 10 /xa near the
cylinder wall to 10 ma near the arc. It is assumed that
ion density is proportional to probe current. Further
details of the apparatus may be found in Ref. 3.
Some typical measurements of the ¿-folding length
as a function of В are shown in Fig. 2. The resulting
values of the reciprocal of the ¿-folding length are then
plotted as a function of В in Fig. 3. The plot should be
linear if Z>+p varies as B~2 [see Eqs. (1) and (11)] and
should vary as the square root if <D+P ~ В~г. The
results clearly favor the B~2 behavior. Incidentally,
the experimental magnitude of Z)+p ( # 4 x l O 3 cm2/
In addition to a study of the dependence on B,
Neidigh has also investigated the variation with
pressure.4 For a case in which free streaming to the
end wall should prevail (Л # 6 cm, L — 6 cm), the
results of Fig. 4 indicate a square-root dependence, in
agreement with the discussion following Eq. (11).
Another case in which diffusion to the end walls has
set in (Л # 6 cm, L = 26 cm) is shown in Fig. 5. Here
a Hnear behavior is observed, again in agreement with
the theory. These last experiments on the pressure
dependence are only preliminary; more careful
investigations are planned.
Further confirmation of the B~2 dependence of the
diffusion coefficient is provided, for electrons, by the
experiments of Bickerton.5 In this case the spreading
parameter R was found to vary linearly with B2 in
agreement with the classical theory.
DIFFUSION IN A MAGNETIC MIRROR GEOMETRY
The theory presented above indicates that the
essential cause of the non-adiabatic diffusion across
the magneticfieldis the very large conductivity of the
system in the direction of the field lines. It is of
interest to inquire as to how this conductivity might
be reduced and ambipolar diffusion restored. One
SESSION A-10
346
10
P/366
ALBERT SIMON
s
• UNIFORM MA3NETIC FIELD: 58D 0 9 0 U S S W < ^ f
О UNIFORM MA(3NETIC FIELD.* 35(X) gaussJ
/
/
/
/
/
06
/
04
/
A
/
/
02
/
366
2
3
4
Pressure (mm Hg X 10 3 )
/
5
Figure 4. Variation of e-folding length with pressure. Free
streaming case
366
PRESSURE ( mm Hg x 1O~ 3 )
Figure 5. Variation of e-folding length with pressure. End diffusion
case
possible way is to put magnetic "mirrors" on the ends
of the arc. This effect is discussed below.
Theory
The theory of the confinement of charged particles
by a magnetic mirror 6 indicates that a particle will be
reflected from the mirror if its velocity angle 9 is
greater than 9C where
eG = sin-i (1/Й).
(14)
which has the value P # (2R)-1 for large R. It may
be shown that the effect of an electric field potential
ф is to increase the escape cone for the ions and
decrease it for the electrons in such a way that
P « P(l ±еф!Щ
(18)
where W is the average kinetic energy of the particles
Here 9 is the angle between the velocity vector of the
in the region between the mirrors. The combined
particle and the mirror axis. The ratio of the magnetic
effect of Eqs. (15) and (18) may be seen to be equivafield strength in the mirror region to that in the central
lent to changing the streaming terms in Eq. (7) by
region is denoted by R.
the factor LP/X, since ф/L & Ey. This in turn yields a
Let us assume that the mean free path Л is large
new expression for the e-folding length,
compared to the arc length L. Those ions with 9 < 9C
l
q2 = \D+*l2viP.
(19)
escape immediately. The remaining ions are confined
until a collision produces a new 9 which is less than
It should be noted that the addition of the mirrors
9C. If P is the probability that an ion will be scattered reduced the current in the direction of the field by
into the escape cone (9 > 9C) the rate of production of the factor LP/X. This will have no effect on the
such ions per unit volume per sec is then nvP/X and non-ambipolar diffusion, as the discussion in the final
the total current streaming to the end walls is
section will indicate. However, there is an immediate
2
J = nvPL/X.
(15) effect in the behavior of q with pressure.2 Equation
(19), together with Eq. (1), indicates that q should be
This is to be compared with the corresponding
pressure independent and that q # ro/2P% where ro
expression for free streaming without mirrors, which is
is the Larmor radius of the ions.
/ = nv.
(16)
The case of diffusion flow to the end walls is of no
If one assumed isotropic scattering, the probability interest for a mirror field, since the mirrors will then
P of scattering into the escape cone, by use of Eq. (14) have only a minor effect on the currentflowingin this
becomes:
direction.
DIFFUSION OF ARC PLASMAS
N2
PRESSURE (Ю
347
mm Hg)
r0 (cm)
Figure 6. Radial variation of ion density. Magnetic mirrors used
Experiment
Again there has been a preliminary set of measurements by Neidigh.4 The mirror field used has a 2:1
ratio. The field was 3500 gauss in the central region
and 7000 gauss in the mirror region. The results are
plotted in Fig. 6. The ¿-folding length seems independent of pressure over the range studied. However,
the magnitude of q is about a factor of 4 larger than
the Larmor radius. Further experiments are planned.
LONG TUBE LENGTH
It seems clear that if the arc chamber is made long
enough the effect of the electric fields parallel to the
tube axis will finally become comparable with the
radial electric fields; ambipolar diffusion will then be
restored. The necessary length may be estimated.
From the first part of Eq. (3) the order of magnitude
of the perpendicular current is
(<x)+T+)2q
(20)
where q is the ¿-folding length. The magnitude of the
parallel current is
It * р+щЕу/Ь.
(21)
Finally, since Ex « ф/R and Ey « ф/L, one has
(22)
Hence, for ambipolar diffusion to be restored, one
must have
LjR « œ+r+(q/R)i.
(23)
Finally, since it is electron current in the magnetic
field direction which neutralizes space charge, it is
necessary that this relation be satisfied for the
electrons as well. This is the worst case, hence the
condition is
L/R S* a>-T-{q¡R)K
(24)
This condition may be difficult to obtain in practice.
For example, in Neidigh's experiments а)+т+ # 14.4
and œ-r- » 3 x 103. Since q/R must be appreciably
larger than unity, this requires that LjR > 104. It
should be kept in mind that ео+т+ must be kept
appreciably larger than unity if the magnetic field is
to play a dominant role in the behavior of the gas.
In the case of free streaming to the end walls, Eq.
(7), the corresponding condition is easily shown to be
LjR > (a)T)2q/X. This condition is incompatible,
however, with the necessary requirement for free
streaming, A > L.
It might be thought that the use of insulating end
walls would require exact ambipolar diffusion in the
field direction and hence produce ambipolar diffusion
radially. However, there is no perfect insulator (for
example, surface collisions allow electrons to walk
radially along the wall) and this probably prevents
this scheme from compensating for the huge anisotropy-of the plasma conductivity. Some unpublished
experiments by Neidigh have shown that an insulating
end wall had only a small effect on the radial ion
diffusion.
348
SESSION A-10
P/366
SUMMARY
It is shown that experimental results on diffusion of
ions and electrons in a weakly ionized plasma across a
magnetic field are in agreement with the classical
collision-diffusion mechanism. Hence, no additional
mechanisms, such as plasma oscillations, need be
postulated. The resultant diffusion rate is not ambipolar, however. The ions and electrons diffuse at their
own intrinsic rates.
REFERENCES
1. A. Simon, Ambipolar Diffusion in a Magnetic Field, Phys.
Rev. 98, 317 (1955).
ALBERT SIMON
2. A. Guthrie and R. K. Wakerling, The Characteristics of
Electrical Discharges in Magnetic Fields, Vol. 5, Div. 1,
p. 197. McGraw-Hill Co., Inc., New York (1949).
3. A. Simon and R. V. Neidigh, Diffusion of Ions in a Plasma
Across a Magnetic Field, Oak Ridge National Laboratory
Report, ORNL-1890 (1955).
4. R. V. Neidigh, Some Experiments Relating Ion Diffusion In
a Plasma to the Neutral Gas Density in the Presence of a
Magnetic Field, Oak Ridge National Laboratory Report,
ORNL-2024 (1956).
5. R. J. Bickerton, The Lateral Diffusion of an Electron Swarm
in a Magnetic Field, Proc. Phys. Soc. (London), 70B, 305
(1957).
6. E. Fermi, On the Origin of the Cosmic Radiation, Phys. Rev.
75, 1169 (1949).