Geoscience Reference
In-Depth Information
3.5.4.2
General Considerations
Let us consider the ion flux toward a charge particle. The incident ion and the target
particle interact via the Coulomb and the image potentials:
qQe
2
r
q
2
e
2
2a
"
1
"
C
1
S.r=a/;
U.r/
D˙
(3.139)
where
Q
and
q
are the charges of the particles and the ion in units of
e
[
Q
and
q
are
nonnegative integers; the upper sign (here “
C
”) refers to the Coulomb repulsion].
Introducing
"
C
1
"
1
;
D
ˇq
2
e
2
2a
"
1
"
C
1
; and
r
a
D
;
D
Q
q
ˇE
u
D
(3.140)
allows Eq.
3.139
to be rewritten as follows:
ˇU ./
D
S./
;
2
˙
(3.141)
where S./ is introduced by Eq.
3.137
. We also introduce
1
1
"
C
1
S./:
u
./
D
1/
2
˙
(3.142)
.
2
Then
D
.
u
/ (see Eq.
3.111
) is the solution to the equation
u
D
u
./:
(3.143)
Equation
3.112
now looks as follows:
‰.
u
/
D
.
.
u
//;
(3.144)
where
./
D
2
u
./
C
S./
:
2
(3.145)
For calculating the ion flux we use Eq.
3.115
. Let us replace there the integration
variable
x
with :
Z
e
x
r
2
.x/dx
D
a
2
Z
e
u
./
2
d
u
d
d:
