Geoscience Reference
In-Depth Information
3.6.2.1
Ion Density Profile
Let us find the expression for the ion concentration profile n.r/ in the free molecule
zone.
We begin by considering the attractive potentials. Equation
3.28
gives
L
2
m
Z
Z
dL
2
p
L
2
m
.r/
L
2
Œ1
C
.L
L
a
/
C.R/
m
2
r
dE
e
ˇ
.
EU.R/
/
n.r/
D
U.R/
0
Z
dE
e
ˇ
.
ECjU.R/j
/
L
m
C
q
L
2
m
.r/
L
a
2C.R/
m
2
r
D
j
U.R/
j
2
p
2mC.R/
m
2
Z
dE
e
ˇ.ECjU.R/j/
D
jU.R/j
p
E
C j
U.r/
jC
.r
r
.E// .E
C
E
0
.E//
r
E
Cj
U.r/
j
!
L
a
2mr
2
In what follows we omit .r
r
.E//. It can be easily restored where it is
needed. We thus have
dx
e
x
"
p
x
C
ˇ
j
U.r/
j
2
mˇ
3=2
Z
1
p
C.R/e
ˇjU.R/j
n.r/
D
ˇ
j
U.R/
j
#
;
C
.E
C j
E
0
.E/
j
/
r
x
C
ˇ
j
U.r/
j
.x
C
ˇ
j
U.r
/
j
/
r
2
r
2
(3.164)
where
C.R/
D
n.R/
mˇ
2
3=2
2
1
C
e
ˇjU.R/j
F.R/
;
(3.165)
with
dx
e
x
.x
C
x
0
/
r
x
C
ˇ
j
U.R/
j
.x
C
ˇ
j
U.r
/
j
/
r
2
Z
2
p
R
2
;
F.R/
D
ˇ
j
U.R/
j
(3.166)
where x
0
D
ˇ
j
E
0
j
.
