Geoscience Reference
In-Depth Information
where
x
2
ˆ
1
x
2
1
S.x/
D
(3.132)
and ˆ.x/ is the solution to the differential equation
1
"
C
1
ˆ
D
x
.1
x/
2
xˆ
0
C
(3.133)
with the initial condition ˆ.0/
D
0. This solution has the form:
Z
x
y
1=."C1/
.1
y/
2
dy
ˆ.x/
D
x
1=."C1/
(3.134)
0
or, on integrating twice by parts,
Z
x
x
x
1
1
"
C
1
ln
1
1
x
C
1
."
C
1/
2
x
1=."C1/
1
1
y
dy
y
"=."C1/
ln
ˆ.x/
D
0
(3.135)
The last term in this equation is nonsingular at the particle surface. The
approximation
x
x
1
1
"
C
1
ln
1
1
x
C
x
."
C
1/."
C
2/
ˆ
a
.x/
(3.136)
reproduces the asymptotic behavior of ˆ.x/ as x
!
0 and x
!
1.
The function S.x/ obeys the differential equation
xS
0
D
2
"
"
C
1
S
#
1
"
1/
2
C
(3.137)
.x
2
which follows from Eq.
3.133
.
At "
D1
the function ˆ.x/
D
x=.1
x/ and we come to the quite familiar
expression for the image potential of a metallic particle:
q
2
e
2
2a
a
4
r
2
.r
2
U
image
.r/
D
a
2
/
:
(3.138)
Equation
3.137
plays a key role in our further consideration.
