where x is measured from the edge of the plasma sheath toward the probe and AD is a
parameter called the Debye length:
( 1 . 1 2)
In equation 1 . 12, k is the Boltzman constant, e0 is the permittivity of free space, and e is
the electronic charge.
I
The Debye length is an extremely useful quantity and is one of the defining length
scales in a plasma. A more precise definition of "quasi-neutrality" states that the ion and
electron density can be dissimilar only over distances on the order of the Debye length.
Since the ion and electron densities are dissimilar near probes and walls, the Debye length
will be intimately tied to any discussion of the sheath thickness. A more usable form of
the equation is given below:
An
=
7430!F, (m)
where Te is in eV, and ne is in m-3, giving AD in m.
Example: For a plasma with an electron temperature of 4eV and a density of I x 10 16
m-3 the Debye length is 1 .5 x 1 0-4m.
The approximations used to get equation ( 1 . 1 1) are not valid as the potential on a
prnbe drops to the floating potential or beyond, so the question of the plasma sheath
thickness is still unanswered. Poisson's equation can be solved with a different set of
approximations though. Assume (I) that the electron density in the sheath is zero, and (2)
that the derivative of the potential with respect to distance at the sheath-presheath
boundary is zero. These conditions are not true at the edge of the sheath, but are quite
good through the bulk of the sheath and near the probe. With some effort Poisson's
equation can be solved to give the thickness of the sheath, x, as a function of the Debye
length, AD, the electron temperature, Te, and the potential on the probe with respect to
plasma potential, Vo. (Note that Vo will be negative since there is only a sheath if the
probe is at a potential below plasma potential.) The answer is:
1/2
(1. 14) (D-7)
1/2