Chemistry Reference
In-Depth Information
3.7.3.4 Stable Positive Column
At sufficient high distance between cathode and anode, the Faraday dark space
and the positive column is observed. Due to the dissipation of the kinetic energy
of the electrons as well as the loss of charge carriers due to recombination and
ambipolar diffusion in the negative glow the loss processes have to be balanced
by electron heating in the electric field and further gas ionization to satisfy the
continuity of the total discharge current. Therefore, the electric field strength is
slightly increasing in the Faraday dark space toward the positive column. The positive
column is characterized by a constant electric field strength in axial direction and
corresponding weak potential gradient, which is sufficiently high to heat the electrons
for further electron impact ionization. The electron energy distribution function in the
positivecolumnismostlyapproximatedbytheMaxwellianorDryvesteyndistribution
function. Thereby, sufficient electrons of the high energetic tail of the electron energy
distribution function have kinetic energies above the threshold for inelastic collisions
such as ionization, excitation, or dissociation of gas atoms/molecules.
In cylindrical discharge tubes (radius
R
) a simple model describes the diffusion-
dominated stationary positive column (λ
e
, λ
+
R
), taking into account
•
Constant axial electric field strength
E
z
•
Ionization by electron impact from ground state with α
=
const
.
•
No volume recombination
•
Axially constant plasma density
•
Radial ambipolar diffusion to the tube wall, (
n
=
n
(
r
)
,
T
e
T
+
,
T
n
)
•
Neglecting of plasma sheaths
•
Totally absorbing wall
With the stationary particle balance equation for electrons
div
→
j
=
ν
ion
·
n
and the
diffusion current density
→
j
=−
D
a
·
grad n
it follows for the radial part in cylinder
symmetry
d
2
n
dr
2
1
r
·
dn
dr
−
D
a
·
+
=
ν
ion
·
n
.
(3.295)
The solution of this equation with the ionization frequency ν
ion
=
v
eD
results in Bessel function of zero order (
J
0
) for the radial plasma density distribution.
By use of the boundary conditions,
n
α
·
b
e
·
E
z
=
α
·
(
0
)
=
n
0
and
n
(
R
)
=
0, the radial plasma density
profile
J
0
2.405
r
R
n
(
r
)
=
n
0
·
·
(3.296)
follows.
Additionally, a useful equation was derived to estimate the electron temperature
in the diffusion dominated positive column of a stable glow discharge, taking into
calculation the equality of the ionization collision frequency and the loss frequency
