Environmental Engineering Reference
In-Depth Information
of ions
to be independent of the ion velocity, we represent the kinetic equation
for the ion distribution function in the form
λ
(
v
z
)
Z
eE
M
@
f
@
v
1
λ
δ
v
z
v
z
λ
v
z
f
(
v
z
)
d
z
D
f
(
v
z
),
where the
z
-axis is directed along the field. Using the reduced ion velocity
M
eE
1/2
u
D
v
z
,
λ
onecanrewritethisequationas
w
i
N
i
u
!
exp
r
2
π
(
u
)
Z
u
0
Φ
u
2
2
2
λ
d
du
C
(
u
0
)
du
0
,
η
(
u
)
D
u
Φ
δ
π
M
)
1/2
and the ion drift velocity is
w
D
(2
eE
λ
/
π
in accordance with (4.96). Solving
this equation, we obtain
u
2
2
C
!
r
2
π
C
w
N
i
exp
u
2
2
2
λ
Φ
D
η
(
u
).
π
The constant
C
in this expression follows from the normalization condition
R
Φ
1/2. For the longitudinal diffusion co-
efficient of atomic ions in a parent gas in a strong electric field this gives
(
u
)
du
D
0, which gives
C
D
2/
π
Z
(
v
z
w
i
2
1
N
i
1
2
D
k
D
w
i
)
Φ
d
D
λ
π
D
0.137
λ
w
i
.
(4.104)
v
z
The transverse diffusion coefficient of atomic ions in a parent gas can be found
because the distribution function in the direction perpendicular to the field coin-
cides with the Maxwell distribution function for atoms. The variables in the kinetic
equation are divided for motion along the field and for motion perpendicular to it,
and the diffusion coefficient in the direction transverse to the field is [67]
Mw
,
T
D
?
D
(4.105)
and the ratio of the longitudinal and transverse diffusion coefficients is equal in
this case:
0.137
Mw
i
T
D
k
D
?
D
0.087
eE
λ
D
,
(4.106)
T
which is large in a strong electric field.
