Chemistry Reference
In-Depth Information
:
¼
k
B
T
8p
L
d
u
2
p
KB
ðrÞ
ln
(6)
The weak logarithmic divergence with the sample size
L
is known as the
Landau-Peierls instability. As a result, for sufficiently large
L
the fluctuations
become of the order of the layer spacing, which means that the layer structure
would be wiped out. However, for samples in the millimolar range and typical
values of the elastic moduli
K
10
11
N and
B
10
7
N/m
2
, the layer displace-
p
u
2
ment amplitude s
¼
hi
does not exceed 0.5-0.7 nm. For a typical smectic
period
d
0.2; the smectic layers are
still well defined. Nevertheless, the displacements are large compared to those of a
typical 3D crystal for which:
3 nm this gives relative displacements s/
d
¼
k
B
T
p
aC
:
u
2
ðrÞ
(7)
10
10
N/m
2
and a lattice size
For a typical value of the elastic modulus
C ¼
a ¼
0.5 nm, this leads to s
0.02 nm and s/
d
0.04.
The pair density correlation function - the quantity essentially measured in an
X-ray experiment - is defined as:
Gð
r
Þ¼
r
ð
h
r
Þ
r
ð
0
Þ
i
r
ð
h
r
Þ
i
r
ð
h
0
Þ
i;
(8)
where the brackets indicate an average. As a result of the Landau-Peierls instability
the correlation function shows a slow algebraic decay
G
(
r
)~
r
.Writing
q
0
¼
2
p
/
d
,
the exponent
is given by:
q
0
k
B
T
8p
K
p
:
¼
(9)
The resulting order is referred to as quasi-long-range order. It provides a mar-
ginal case between true long-range positional order and short-range order. These
various types of order are illustrated in Fig.
8
.
The scattered intensity
I
(
q
) is proportional to the structure factor
S
(
q
), the
Fourier transform of the correlation function
G
(
r
), and thus reflects the nature of
the correlations in the system. In the case of long-range order, the correlation
function
G
(
r
) remains constant as
r!1
. As a result, the Bragg reflections are
nominally delta functions,
S
(
q
)~d(
q q
n
) at each reciprocal lattice vector
q
n
,
graphs). In practice, the central part of the X-ray peak takes the form of a Gaussian
due to the finite size of the ordered domains (grains) and/or the resolution of
the setup. Short-range order is represented by an exponentially decaying function
G
(
r
) ~ exp(
r
/x), in which x is the correlation length. The resulting lineshape is a
Lorentzian (Fig.
8
, lower graphs).
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