Environmental Engineering Reference
In-Depth Information
Hence, we expand the ion distribution function
f
(
v
) over this small parameter,
representing it in the form [67]
f
(
v
)
D
f
0
(
v
)
Φ
(
v
)
ln
N
i
,
r
and the nonperturbed distribution function
f
0
(
v
)satisfiestheequation
e
M
@
f
0
@
v
D
I
col
(
f
0
).
The normalization condition for the distribution function also gives
Z
Φ
(
v
)
d
v
D
0.
D
R
v
f
(
v
)
d
v
with the definition of
Comparing the expression for the ion current
j
i
the ion diffusion coefficient, we obtain
Z
(
1
N
i
D
z
D
w
z
)
Φ
(
v
)
d
v
,
(4.103)
v
z
where
z
is the direction of the ion density gradient. Substituting the above expan-
sion of the ion distribution function into the kinetic equation, we obtain the follow-
ing equation for the addition part of the distribution function
eE
z
M
d
Φ
(
v
w
z
)
f
0
D
v
z
C
I
col
(
Φ
).
z
d
We now obtain an integral relation for the distribution function addition
Φ
(
v
)
analogous to relation (4.18) by multiplying the equation for
Φ
by
v
z
and integrating
it over ion velocities. Using the normalization condition for
Φ
(
v
), we obtain
Z
g
z
g
w
z
N
i
m
a
σ
(
g
)
z
D
Φ
(
v
)
'
(
v
a
)
d
v
d
v
a
,
v
M
C
m
a
where we used the notation in (4.18). This equation allows us to find the diffusion
coefficient in the case
σ
(
g
)
1/
g
,whichgives
w
z
,
M
μν
D
z
D
z
v
σ
(
g
) is assumed to be independent of the colli-
sion velocity. One can see that for the isotropic distribution function the diffusion
coefficients of ions in different directions are identical, whereas they are different
for an anisotropic distribution function in the frame of reference where ions are
motionless.
We now determine the diffusion coefficient of atomic ions in a parent gas for the
direction along the electric field when the electric field strength is high. Accounting
for the character of the resonant charge process and assuming the mean free path
where the collision rate
ν
D
N
a
g
