6.2 Plastic Crystals and Liquid Crystals
125
It has been established that the properties of the ensemble strongly depends on
the dimensionality of the lattice. If the lattice is one-dimensional, no phase transition
occurs at finite temperature. On the other hand, as described previously, the model
can be rigorously analyzed for some cases on two-dimensional lattices with z = 3
(honeycomb lattice), z = 4 (square [24] and Kagomé lattices) and z = 6 (triangular
lattice). All of these show a continuous phase transition between the ferro (m = 0) and
para (m = 0) phases at finite temperature. In the three-dimensional cases, a similar
phase transition is proven to occur [27], but exact expressions of thermodynamic
functions have not been derived.
6.2.1.2 Bragg–Williams Approximation
According to statistical mechanics, thermodynamic properties of a system having an
energy expression of Eq. 6.4 are obtained from the partition function Z (T ) given by
Z (T ) =
exp
⎛
⎝ J
2k B T
i
j∈{NN} i
σ i σ j
⎞
⎠ ,
(6.7)
where the first summation runs over all possible states of the system. Although many
efforts have been devoted to tackling the problem, they have been successful only
in limited cases in one or two dimensions [24]. Some approximation is necessary to
proceed with, accordingly. One of the most primitive approximations is after Bragg
and Williams [28]. By virtue of its simplicity (with many issues to be corrected),
however, it is intuitively applicable to a wide range of problems. The analysis strategy
of the Ising model consists of two steps, constructing an approximate expression of
the Helmholtz energy F(m) as a function of m, which serves as the order parameter,
and determining m as one that minimizes F(m).
The number of states having the same m is given by a binomial coefficient
W =
N
n U
=
N !
n U !n D !
.
(6.8)
Thus, the entropy is given by
S = k B ln
N !
n U !n D !
≈ k B [N (ln N − 1) − n U (ln n U − 1) − n D (ln n D − 1)]
= −N k B
n U
N
ln
n U
N
+
n D
N
ln
n D
N
=
1
2
N k B [2 ln 2 − (1 + m) ln(1 + m) − (1 − m) ln(1 − m)] ,
(6.9)
125
It has been established that the properties of the ensemble strongly depends on
the dimensionality of the lattice. If the lattice is one-dimensional, no phase transition
occurs at finite temperature. On the other hand, as described previously, the model
can be rigorously analyzed for some cases on two-dimensional lattices with z = 3
(honeycomb lattice), z = 4 (square [24] and Kagomé lattices) and z = 6 (triangular
lattice). All of these show a continuous phase transition between the ferro (m = 0) and
para (m = 0) phases at finite temperature. In the three-dimensional cases, a similar
phase transition is proven to occur [27], but exact expressions of thermodynamic
functions have not been derived.
6.2.1.2 Bragg–Williams Approximation
According to statistical mechanics, thermodynamic properties of a system having an
energy expression of Eq. 6.4 are obtained from the partition function Z (T ) given by
Z (T ) =
exp
⎛
⎝ J
2k B T
i
j∈{NN} i
σ i σ j
⎞
⎠ ,
(6.7)
where the first summation runs over all possible states of the system. Although many
efforts have been devoted to tackling the problem, they have been successful only
in limited cases in one or two dimensions [24]. Some approximation is necessary to
proceed with, accordingly. One of the most primitive approximations is after Bragg
and Williams [28]. By virtue of its simplicity (with many issues to be corrected),
however, it is intuitively applicable to a wide range of problems. The analysis strategy
of the Ising model consists of two steps, constructing an approximate expression of
the Helmholtz energy F(m) as a function of m, which serves as the order parameter,
and determining m as one that minimizes F(m).
The number of states having the same m is given by a binomial coefficient
W =
N
n U
=
N !
n U !n D !
.
(6.8)
Thus, the entropy is given by
S = k B ln
N !
n U !n D !
≈ k B [N (ln N − 1) − n U (ln n U − 1) − n D (ln n D − 1)]
= −N k B
n U
N
ln
n U
N
+
n D
N
ln
n D
N
=
1
2
N k B [2 ln 2 − (1 + m) ln(1 + m) − (1 − m) ln(1 − m)] ,
(6.9)