Environmental Engineering Reference
In-Depth Information
From this, because of the identity of
f
i
(
g
), we obtain
N
i
Z
exp(
i
E
i
g
)
D
X
i
1
d
r
i
,
ln
F
(
g
)
ln
f
i
(
g
)
D
n
ln
f
i
(
g
)
D
where
N
i
is the number density of ions. In accordance with (1.21), from
this we obtain for the distribution function
P
(
E
) for the electric field strengths in a
plasma
D
n
/
Ω
)
3
Z
exp
1
d
r
i
d
g
.
N
i
Z
exp(
i
E
i
g
)
1
P
(
E
)
D
i
Eg
C
(2
π
e
r
i
/
r
i
,wehave
Since the electric field strength from an individual ion is
E
i
D
Z
exp(
i
E
i
g
)
Z
exp
i
eg
cos
1
1
d
r
i
θ
i
i
r
i
dr
i
D
2
π
d
cos
θ
D
r
i
1
Z
1
(
eg
)
3/2
15
sin
8
π
(
eg
)
3/2
2
d
4
π
D
,
0
eg
/
r
i
. Hence, the distribution function for electric field strengths takes
the form [24]
where
D
Z
exp
(
eg
)
3/2
N
i
15
8
π
d
g
(2
1
E
0
H
(
z
),
P
(
E
)
D
C
i
Eg
D
π
)
3
e
4
N
i
15
2/3
E
E
0
z
D
,
E
0
D
2
π
.
(1.23)
As is seen, the typical electric field strength
E
0
corresponds to that of (1.19), and
the Holtzmark function
H
(
z
)isgivenby
Z
x
sin
x
exp
dx
.
1
x
z
2
π
3/2
H
(
z
)
D
(1.24)
z
0
The Holtzmark function has the following expressions in the limiting cases
r
2
π
4
3
15
8
z
2
z
5/2
,
H
(
z
)
D
,
z
1,
H
(
z
)
D
z
1
(1.25)
π
and the case of large electric field strengths
z
1 corresponds to (1.18). The
Holtzmark function is represented in Figure 1.5, and its maximum corresponds
to a typical electric field strength
E
max
1.6
E
0
that is created at average distances
between ions. The Holtzmark function may be constructed in a simple manner
from its limiting expressions as
4
z
2
h
(
z
)
D
1
z
0
9/2
,
z
0
D
1.32 .
(1.26)
3
π
C
