Chemistry Reference
In-Depth Information
d
y
d
x
2
m
p
2
E
=
+
U
(
y
)
=
U
(
y
max
)
(12.34)
where
y
max
is the maximum value of the “coordinate” (the maximum slope in the
bunch) that is attained when the total “energy” is
E
and the “kinetic energy” is zero.
To obtain information for the shape of the bunch we shall explore the solutions of
the differential equation (
12.34
) in the limits where the “potential” energy (
12.33
)
is dominated by the first or by the second term.
Inthefirstcase(
F
p
h
0
l
)
|
J
st
|
<<
we have
d
y
d
x
2
m
p
2
2
3
F
p
y
3
/
2
2
3
F
p
y
3
/
2
−
=−
(12.35)
max
which can be rewritten as
1
3
/
2
1
/
2
y
y
max
y
3
/
2
y
3
/
2
1
/
2
4
F
p
y
3
/
2
d
y
d
x
=
4
F
p
3
m
p
max
3
m
p
−
−
=
−
−
(12.36)
max
=
=
Integrating from
y
0to
y
y
max
and taking into account that the correspond-
ing values of
x
are
x
=
0 and
x
=
L
/
2 (where
L
is the width of the bunch of steps)
one obtains
L
1
/
2
1
4
F
p
y
3
/
2
ξ
d
L
2
2
Fd
s
h
0
9
g
max
3
m
p
y
3
/
4
=
−
=
−
y
max
(12.37)
1
2
1
/
2
max
−
ξ
3
/
0
which requires negative values of the electromigration force, i.e.
F
<
0.
z
/
2
2
where
H
is the height of the bunch
Taking approximately
y
max
=
≈
(
/
H
L
)
x
one finally gets
37
H
1
/
3
1
/
3
g
h
0
d
s
|
F
|
L
=
2
.
(12.38)
F
p
(
)
In the second limit
U
(
y
)
≈−
J
st
y
(with
|
J
st
|
>>
h
0
/
l
)
one can follow
the same procedure to arrive to
H
1
/
2
L
∼
(12.39)
As seen, expressions (
12.38
) and (
12.39
) predict rather different scaling of the
bunch shape. It is interesting to note that the numerical integration of the ordinary
differential equations (
12.25
) manifests [
12
] the scaling law (
12.38
).
