Chemistry Reference
In-Depth Information
Fig. 1.7
Elastic constants (in
units of
k
B
T
2
) for systems
/σ
at
k
B
T
0 for structured
walls (
top part
) and planar
walls (
bottom part
).
Horizontal straight lines
show the bulk values,
obtained for a system with
periodic boundary conditions
in both
x
-and
y
-directions
/ε
=
1
.
1.4 Insight Gained from the Harmonic Theory
of Two-Dimensional Crystals
In order to better understand the anomalous behavior of crystalline strips with planar
walls, it is useful to examine the mean square displacement correlation function in
the
y
-direction,
G
2
is the displacement
vector of a particle relative to its position in the perfect triangular lattice. Choosing
for simplicity a
D
(
y
)
=|
u
y
(
y
)
−
u
y
(
0
)
|
, where
u
=
(
u
x
,
u
y
)
×
L
geometry with periodic boundary conditions also in the
{
Q
x
-direction, one finds
=
(
Q
x
,
Q
y
)
}
Q
y
2
u
y
(
Q
u
y
(
−
Q
[
u
y
(
y
)
−
u
y
(
0
)
]
=
(
2
/
N
)
)
)
[
1
−
cos
(
yQ
y
)
]
(1.8)
Q
x
where
Q
x
,
Q
y
are “quantized” as follows:
2
D
,
−
4
D
,...,
+
2
D
Q
x
/π
=−
,
−
+
1
+
1
−
(1.9)
1
1
2
L
,
−
4
L
,...,
+
2
L
Q
y
/π
=−
1
,
−
1
+
1
+
1
−
(1.10)
