Environmental Engineering Reference
In-Depth Information
Accounting for the equation of motion for the ion
M
d
v
x
dt
D
eE
allows us to connect the ion velocity with the time after the last collision by the
expression
v
x
eEt
/
M
,so
P
(
t
) is the velocity distribution function for ions whose
mass is
M
. Assuming the cross section of the resonant charge transfer process
D
σ
res
to be independent of the collision velocity, we find the the velocity distribution
function for ions
C
exp
,
M
x
v
f
(
v
x
)
D
>
0 ,
(4.95)
v
x
2
eE
λ
where
C
is a normalization factor, and the mean free path of ions i
n
a parent gas is
λ
D
1/(
N
a
σ
res
). The ion drift velocity
w
i
and the mean ion energy
ε
are
r
2
eE
λ
M
x
2
D
eE
λ
v
w
i
D
v
x
D
M
,
ε
D
.
(4.96)
π
2
The ion drift velocity exceeds significantly the thermal velocity of atoms.
The above operation allows one to take into account the energy dependence of the
cross section of resonant charge exchange. Indeed, approximating the velocity de-
pendence for the charge exchange cross section by the dependence given in (2.45),
we have
v
0
)
v
0
v
2
σ
res
(
)
D
σ
res
(
,
D
,
v
γ
R
0
where the argument gives the collision velocity, and
1. Repeating the above
operations in deriving the distribution function for ions, we obtain for the distribu-
tion function in this case instead of (4.95)
C
exp
,
2
x
N
a
M
σ
res
(
v
x
)
v
f
(
x
)
D
>
0.
v
v
x
(2
)
eE
Fr
om this we obtain for the ion drift velocity
w
i
and the average kinetic ion energy
ε
instead of (4.96)
Γ
2
2
1
(2
)
eE
2
w
i
D
Γ
1
2
,
res
(
v
0
)
v
0
MN
a
σ
Γ
3
2
2
M
2
(2
)
eE
2
Γ
1
2
ε
D
.
(4.97)
v
0
MN
a
σ
res
(
v
0
)
Formula (4.97) allows us to take into account a weak velocity dependence of the
charge exchange cross section in the expressions of the ion drift velocity and the
average ion energy as the limit
!
0. As a result, in the limit of high field strengths
(4.96) takes the form
t
2
eE
eE
res
1.3
q
eE
M
res
1.4
q
eE
M
w
i
D
,
ε
D
.
(4.98)
π
MN
a
σ
2
N
a
σ
