Chemistry Reference
In-Depth Information
)
=
S
r
0
)
·
S
g
(
r
(
(
r
0
+
r
)
=
cos
[
φ(
r
+
r
0
)
−
φ(
r
0
)
]
exp
1
2
[
φ(
2
≈
−
r
+
r
0
)
−
φ(
r
0
)
]
(1.15)
Equations (
1.14
) and (
1.15
) make the analogy to harmonic solids transparent -
the angles
2
{
φ
j
}
correspond to the displacement vectors
u
j
and
[
φ(
r
+
r
0
)
−
φ(
r
0
)
]
corresponds to the displacement correlation
G
(
y
)
. Now the well-known power-law
r
−
η
b
, immediately follows from a logarithmic variation of
the angular displacement correlation
decay of
g
(
r
)
,
g
(
r
)
∝
2
[
φ(
r
+
r
0
)
−
φ(
r
0
)
]
=
(
k
B
T
/
2
π
J
)
ln
(π
r
/
a
0
),
r
a
0
(1.16)
which yields
. Equation (
1.16
) is the analog of (
1.12
).
When one introduces the continuum approximation that corresponds to (
1.14
)
namely
η
b
=
k
B
T
/(
2
π
J
)
J
2
2
d
x
d
y
H
=
[∇
φ
]
(1.17)
one can treat the problem of a semiinfinite half-space (with a free boundary at
x
=
0)
by postulating a von Neumann boundary condition
∂
G
(
r
1
,
r
2
)/∂
x
1
|
x
1
=
0
=
0
(1.18)
for the correlation function
G
(
r
1
,
r
2
)
=
φ(
r
1
)φ(
r
2
)
.Using(
1.17
) and (
1.18
) one
can show [
26
] that
y
1
)
−
η
||
,
||
=
exp
[
i
φ(
r
1
)
]
exp
[
i
φ(
r
2
)
] ∝
(
y
2
−
2
η
b
(1.19)
when both sites
η
b
being the exponent
quoted in (
1.16
). When only one site is in the surface, while the other site is deep in
the bulk, one rather predicts a power law
g
r
1
,
r
2
lie at the surface (
x
1
=
x
2
=
0
)
,
x
−
η
⊥
, where scaling [
27
] yields
(
x
)
∝
η
⊥
=
2. While there are no data available to test the latter relation, Fig.
1.9
provides a test for the analog of the relation
n
||
=
3
η
b
/
η
b
for the surface behavior of
2
the displacement correlation of a solid.
1.5 Effects due to Incommensurate Confinement: Soliton
Staircases and Strain Density Waves
We now focus on the case where the confinement creates a misfit in the system [
12
],
such that we still have a total number of
N
=
n
x
n
y
particles, with a linear dimension
n
y
a
where
a
is the lattice spacing appropriate for an ideal, undistorted trian-
gular lattice, with periodic boundary conditions in the
y
-direction, b
u
t the thickness
D
of the strip no longer has the commensurate value
D
L
y
=
a
√
3
=
n
x
(
/
2
)
but rather
