Environmental Engineering Reference
In-Depth Information
Figure 1.5
The Holzmark function
H
(
z
) according to (1.24) (1) and its approxima-
tion (1.26) (2).
The approximation
h
(
z
)iscomparedwiththeHoltzmarkfunctioninFigure1.5.
One can expand these results for a plasma containing multicharged ions. If the
ion charge is
Z
, it is necessary to change
e
to
Ze
, which for the typical electric
strength
E
0
in (1.23) gives
Ze
4
N
i
15
2/3
2.60
ZeN
2/3
i
E
0
D
2
π
D
.
(1.27)
Note that microfields in a plasma may be created by both electrons and ions. In
this consideration we assume electrons to be distributed uniformly in space. This
means that the observation time
for the ion distribution exceeds significantly a
typical time of approximately
r
/
v
e
forelectrondisplacementatadistance
r
that
determines this part of the distribution function (
v
e
is a typical electron velocity).
For the maximum of the distribution function (
r
τ
N
1/3
i
) this criterion gives
m
1/
e
T
1/2
N
1/3
i
τ
,
(1.28)
e
where
m
e
is the electron mass and
T
e
is a typical electron energy. Simultaneously,
the observation time is small compared with a typical time for establishment of a
uniform ion distribution, that is, for typical electric field strengths (
E
E
0
):
M
1/2
T
1/2
i
N
1/3
i
τ
,
(1.29)
where
M
is the ion mass and
T
i
is a typical electron energy. In particular, for a gas
discharge helium plasma with typical parameters
N
i
10
12
cm
3
,
T
e
1eV,and
T
i
10
12
s.
In addition, it is necessary to account for screening of the ion field in a plasma,
which requires us to change the electric field strength
E
in (1.23) to
E
exp(
D
400Kthisformulagives5
10
10
s
τ
r
/
r
D
)
