Environmental Engineering Reference
In-Depth Information
We now consider a nonequilibrium plasma when the electron and ion tempera-
tures (
T
e
and
T
i
) are different. Moreover, the energy distribution function for elec-
trons differs from the Maxwell one, and its form is essential for the particle charge.
Nevertheless, we introduce the electron temperature such that a real energy distri-
bution of electrons provides the same flux of electrons on the particle surface and,
correspondingly, the same particle charge, as takes place for the Maxwell electron
distribution with a given electron temperature. Screening of the particle field is
determined by ions and therefore is independent of the electron distribution func-
tion. Next, by analogy with (6.21) and (6.22), the particle charge in this case is given
by [37, 41]
2
e
2
ln
T
e
m
i
.
r
0
T
e
j
Z
jD
(6.31)
T
i
m
e
Let us introduce the Coulomb field size
R
0
:
jD
j
Z
e
2
R
0
D
j
j
U
(
R
0
)
T
i
,
R
0
r
0
,
where
ln
T
e
m
i
T
i
m
e
,
e
2
T
i
D
D
j
Z
j
T
e
2
T
i
T
i
R
0
e
2
R
0
r
0
X
,
X
D
j
Z
jD
.
(6.32)
Let us apply this to a typical gas discharge plasma in argon with
T
e
D
1eVand
T
i
D
400 K and where the particle radius is
r
0
D
1
μ
m. According to the above
10
3
,
X
formulas, for this plasma we have
m.
The criterion for a rare plasma, if this plasma does not screen the particle field, is
N
0
R
0
j
Z
jD
5
D
210, and
R
0
D
210
μ
10
8
cm
3
.Asisseen,at
typical number densities of electrons and ions in gas discharge this criterion is not
fulfilled.
To determine the self-consistent field of the particle and ions in the region of ac-
tion of this field, we introduce the current charge
z
(
R
) inside a sphere of radius
R
,
and according to the Gauss theorem [74, 75], the electric field strength
E
(
R
)and
the current charge
z
(
R
)insideasphereare
1, which gives for these parameters
N
0
5
Z
R
z
(
R
)
e
R
2
dz
(
R
)
dR
D
r
2
dr
,
R
2
N
i
(
R
) .
E
(
R
)
D
,
z
(
R
)
Dj
Z
j
N
i
(
r
)
4
π
4
π
(6.33)
r
0
Using (6.30) for the ion number density, we have for the potential energy of the
particle field
Z
l
Z
l
z
(
r
)
e
2
r
2
z
(
R
)
e
2
R
2
z
(
r
)]
e
2
[
z
(
R
)
U
(
R
)
D
dr
D
C
Δ
U
,
Δ
U
D
dr
,
(6.34)
r
2
R
R
where
l
is a dimension for the region of action of the particle field that is defined as
z
(
l
)
D
0 .
(6.35)
