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Virtually all terrestrial plasmas are contained inside solid vacuum vessels. So, an obvious question is: what happens to the plasma in the immediate vicinity of the vessel wall? Actually, to a first approximation, when ions and electrons hit a solid surface they recombine and are lost to the plasma. Hence, we can
treat the wall as a perfect sink of particles. Now, given that the electrons in a plasma generally
move much faster than the ions, the initial electron flux into the wall greatly
exceeds the ion flux, assuming that the wall starts off
unbiased with respect to the plasma. Of course, this flux imbalance causes the wall to charge up negatively, and so
generates a potential barrier which repels the electrons, and thereby
reduces the electron flux.
Debye shielding
confines this barrier to a thin layer of plasma, whose thickness is a few
Debye lengths, coating the inside surface of the wall. This layer
is known as a plasma sheath or a Langmuir sheath.
The height of the potential barrier continues to grow as long
as there is a net flux of negative charge into the wall.
This process presumably comes to an end, and a steadystate is attained, when the potential barrier becomes sufficiently large to make electron flux equal to the ion flux.
Let us construct a onedimensional model of an unmagnetized, steadystate, Langmuir sheath.
Suppose that the wall lies at , and that the plasma occupies the
region . Let us treat the ions and the electrons inside the sheath
as collisionless fluids.
The ion and electron equations of
motion are thus written
respectively. Here, is the electrostatic potential. Moreover, we have
assumed uniform ion and electron temperatures, and ,
respectively, for the sake of simplicity. We have also neglected any offdiagonal
terms in the ion and electron stresstensors, since these terms are
comparatively small.
Note that quasineutrality
does not apply inside the sheath, and so the ion and
electron number densities, and , respectively, are not
necessarily equal to one another.
Consider the ion fluid. Let us assume that the mean ion velocity, ,
is much greater than the ion thermal velocity,
. Since,
as will become apparent,
, this ordering necessarily implies
that : i.e., that the ions are
cold with respect to the electrons. It turns out that plasmas in the
immediate vicinity of solid walls often have comparatively cold ions, so our ordering assumption
is fairly reasonable. In the cold ion limit, the pressure term in Eq. (426)
is negligible, and the equation can be integrated to
give

(428) 
Here, and are the mean ion velocity and electrostatic
potential, respectively, at the edge of the sheath (i.e.,
).
Now, ion fluid continuity requires that

(429) 
where is the ion number density at the sheath boundary. Incidentally, since we
expect quasineutrality to hold in the plasma outside the sheath, the electron number density
at the edge of the sheath must also be (assuming singly charged
ions). The previous two equations can be combined to give
Consider the electron fluid. Let us assume that the mean electron
velocity, , is much less than the electron thermal
velocity,
. In fact, this must be the case, otherwise,
the electron flux to the wall would greatly exceed the ion flux.
Now, if the electron fluid is essentially stationary then the lefthand side of
Eq. (427) is negligible, and the equation can be integrated to
give

(432) 
Here, we have made use of the fact that at
the edge of the sheath.
Now, Poisson's equation is written

(433) 
It follows that

(434) 
Let
,
, and

(435) 
where
is the Debye length. Equation (434) transforms to

(436) 
subject to the boundary condition
as
.
Multiplying through by , integrating with respect to , and making use of the
boundary condition, we obtain

(437) 
Unfortunately, the above equation is highly nonlinear, and can only be solved numerically.
However, it is not necessary to attempt this to see that
a physical solution can only exist
if the righthand side of the equation is positive for all . Consider the the
limit
. It follows from the boundary condition that
. Expanding
the righthand side of Eq. (437) in powers of , we find that the zeroth and firstorder terms cancel, and we are left with

(438) 
Now, the purpose of the sheath is to shield the plasma from the wall
potential. It can be seen, from the above expression, that the physical
solution with
maximum possible shielding corresponds to , since this
choice eliminates the first term on the righthand side (thereby making
as small as possible at large ) leaving the much smaller,
but positive (note that is positive), second term.
Hence, we conclude that

(439) 
This result is known as the Bohm sheath criterion.
It is a somewhat surprising result, since it indicates that
ions at the edge of the sheath are already moving toward the wall at a
considerable velocity. Of course, the ions are further accelerated
as they pass through the sheath. Since the ions are presumably at
rest in the interior of the plasma, it is clear that there
must exist a region sandwiched between the sheath and the main plasma
in which the ions are accelerated from rest to the Bohm velocity,
. This region is called the presheath,
and is both quasineutral and much wider than the sheath (the actual
width depends on the nature of the ion source).
The ion current density at the wall is

(440) 
This current density is negative because the ions are moving in the negative direction. What about the electron current density? Well, the
number density of electrons at the wall is
, where
is the wall potential. Let us assume
that the electrons have a Maxwellian velocity distribution peaked at
zero velocity (since the electron fluid velocity is much less than the
electron thermal velocity). It follows that half of the electrons at are moving in
the negative direction, and half in the positive direction.
Of course, the former electrons hit the wall, and thereby constitute an electron
current to the wall. This current is
, where
the comes from averaging over solid angle, and
is the mean electron speed corresponding to a
Maxwellian velocity distribution. Thus, the electron current density at the wall
is

(441) 
Now, in order to replace the electrons lost to the wall, the electrons must have a mean velocity

(442) 
at the edge of the sheath. However, we previously assumed that any
electron fluid velocity was much less than the electron thermal
velocity,
. As is clear from the above equation,
this is only possible provided that

(443) 
i.e., provided that the wall potential is sufficiently negative to
strongly reduce the electron number density at the wall.
The net current density at the wall is

(444) 
Of course, we require in a steadystate sheath, in order to prevent wall charging, and so we obtain

(445) 
We conclude that, in a steadystate sheath, the wall is biased negatively with respect to the
sheath edge by an amount which is proportional to the electron temperature.
For a hydrogen plasma,
. Thus, hydrogen
ions enter the sheath with an initial energy
, fall through the sheath potential, and so impact the
wall with energy
.
A Langmuir probe is a device used to determine the electron temperature and electron number density of a plasma. It works by inserting an electrode which is biased with respect to the vacuum vessel into the plasma. Provided that the bias voltage is not too positive, we would
expect the probe current to vary as

(446) 
where is the surface area of the probe, and its bias with respect to the vacuum vesselsee Eq. (444). For strongly negative
biases, the probe current saturates in the ion (negative) direction. The
characteristic current which flows in this situation is called the ion
saturation current, and is of magnitude

(447) 
For less negative biases, the currentvoltage relation of the probe
has the general form

(448) 
where is a constant. Thus, a plot of versus gives
a straightline from whose slope the electron temperature can be deduced. Note, however, that if the bias voltage becomes too positive then
electrons cease to be effectively repelled from the probe surface, and the
currentvoltage relation (446) breaks down.
Given the electron temperature, a measurement of the ion saturation current allows the electron
number density at the sheath edge, , to be calculated from Eq. (447). Now, in order to accelerate ions to the Bohm velocity, the
potential drop across the presheath needs to be
,
where is the electric potential in the interior of the plasma. It follows
from Eq. (432) that the relationship between the electron number
density at the sheath boundary, , and the number density in the interior of the plasma, , is

(449) 
Thus, can also be determined from the probe.
Next: Waves in Cold Plasmas
Up: Plasma Fluid Theory
Previous: Closure in Collisionless Magnetized
Richard Fitzpatrick
20110331