Chemistry Reference
In-Depth Information
∂
z
1
x
i
−
d
x
i
d
t
t
=−
h
0
(12.28)
∂
x
i
−
1
which expresses the simple fact that the decrease of the crystal surface height during
sublimation is equal to the product of the single step height
h
0
, the step density
1
, and the velocity of the steps. Substituting expression (
12.25
)into
(
12.28
) and making use of expression (
12.18
) one obtains an equation in partial
derivatives ((12) in [
12
]). To simplify this equation we make use of the results from
the linear stability analysis. We notice that the most important role in stabilizing the
regular step distribution has the term which describes the relaxation due to the step-
step repulsion. On the other hand, the drift velocity is the most essential destabilizing
factor. Accounting only for these two terms into the equations for the motion of the
steps one obtains for the surface diffusion limited regime [
12
,
30
]
/(
x
i
−
x
i
−
1
)
∂
D
s
n
s
kT
2
2
z
2
z
2
h
0
n
s
(
∂
z
6
g
h
0
∂
z
x
∂
2
Fd
s
∂
−
α
x
)
t
=
−
+
(12.29)
∂
∂
x
2
∂
∂
x
2
∂
x
2
τ
s
It is instructive to consider (
12.29
) as a continuity equation
2
h
0
n
s
(
∂
t
+
∂
J
x
=−
α
x
)
z
(12.30)
∂
∂
τ
s
where
6
g
h
0
∂
∂
D
s
n
s
kT
2
z
z
x
∂
2
Fd
s
∂
z
J
=
−
(12.31)
x
2
∂
x
∂
∂
∂
x
The requirement
J
=
J
st
=
const
.
defines a steady-state shape of a bunch of
steps at the crystal surface. As seen
z
st
(
x
)
satisfies a non-linear differential equation
which can easily be transformed into
d
2
y
d
x
2
F
p
√
y
d
U
d
y
m
p
=
+
J
st
=−
(12.32)
which is an equation describing the motion of a classical “particle” with a “mass”
m
p
=
d
z
d
x
2
, and “time”
x
. This is the so-
called mechanical analog, which is very useful in obtaining the bunch shape [
32
].
According to (
12.32
) the classical “particle” oscillates into a potential well
D
s
n
s
kT
3
g
h
0
=
, “coordinate”
y
2
3
F
p
y
3
/
2
U
(
y
)
=−
−
J
st
y
(12.33)
2
D
s
n
s
kT
where
F
p
=
Fd
s
. The “energy” conservation law reads
