Environmental Engineering Reference
In-Depth Information
where
D
is the diffusion coefficient of electrons in a plasma, and the minus sign in
the first term accounts for the direction of the electron current. Taking
E
and
N
e
to
exhibit harmonic behavior as expressed by (5.48), we can write
E
0
e
i
(
kx
ω
t
)
N
0
e
i
(
kx
ω
t
)
E
D
E
0
C
,
N
e
D
N
0
C
.
We then obtain the dispersion relation
N
0
e
dw
dE
iDk
2
ω
D
kw
i
4
π
(5.158)
on the basis of (5.156) and (5.157) using the connection
(
dE
/
dx
). Dispersion relation (5.158) describes drift waves (5.129) that move to-
gether with the electron current. The diffusion process removes electrons from the
perturbation zone, thus causing damping of these waves. But if
dw
/
dE
@
w
/
@
x
D
(
dw
/
dE
)
<
0, long
drift waves with
Dk
2
N
0
edw
/
dE
can develop.
We now study a nonlinear electric domain. In a nonsteady interval of the electric
field strengths in Figure 5.14, (5.157) has the form
<
i
4
π
D
@
N
e
@
j
0
D
N
0
w
(
E
2
)
D
N
e
w
(
E
)
.
x
We should add to this equation the Poisson equation (1.4) or (5.156). Eliminating
the electron number density between these equations, we obtain
D
d
2
E
dx
2
w
(
E
)
dE
D
dx
4
π
eN
0
[
w
(
E
)
w
(
E
2
)] .
(5.159)
Equation (5.159) describes the behavior of the electric field strength in the electric
domain.
When we assume that diffusion plays a secondary role, and ignore diffusion in
the first stage of the analysis, (5.159) takes the form
eN
0
w
(
E
2
)
1
.
dE
dx
D
4
π
w
(
E
)
(5.160)
We solve this equation with the boundary condition
E
0. The solution
is shown in Figure 5.15a. As follows from (5.160),
E
(
x
)increaseswith
x
until
E
2
D
E
2
at
x
D
<
E
<
E
3
,where
w
(
E
3
)
D
w
(
E
2
). At
E
D
E
3
we have
dE
/
dx
D
0, so for subsequent
values of
x
we obtain
E
E
3
. Thus, this solution describes the conversion of the
system from the unstable state
E
2
to the stable state with
E
D
E
3
.
This transition means that the change of the discharge regime as a result of
the perturbation leads to an increase of the electric field strength up to
E
3
.This
would require a variation of the discharge voltage, which is impossible because the
discharge voltage is maintained by the external voltage. Therefore, the variation of
the electric field strength from
E
2
to
E
3
is a perturbation that takes place in a limited
region of the plasma. A return to the initial value of the field occurs as a result of
diffusion, leading to decay of the perturbation. Hence, a typical size of the back
D
