124
6 Melting of Molecular Crystals
6.2 Plastic Crystals and Liquid Crystals
6.2.1 Mean-Field Theory of Ising Model
6.2.1.1 Ising Model
Although the Ising model was originally a concise model of magnetism [22, 23],
it has served as one of the essential models in the study of phase transitions and
critical phenomena. Noteworthy is that a two-dimensional version of it assuming
short-ranged interaction is the first model that has been proven analytically to exhibit
a continuous phase transition [24]. In this section, we analyze the model in the lowest
level approximation to prepare for the understanding of some phase transitions based
on microscopic models. Readers familiar with such treatments may skip this section.
Suppose an ensemble of a vast number of “spins” (σ = ±1) on sites of a regular
lattice. Here, the regular lattice means only the following: All sites are equivalent and
have z neighboring sites. We always assume the simple cyclic boundary condition
to eliminate the effect of the periphery of the system.
4 Assuming that the interaction
between spins work only when two spins are on neighboring sites, the energy of the
ensemble is written as
E = −
1
2
J
i
j∈{NN} i
σ i σ j ,
(6.4)
where the second sum runs over z sites surrounding the i th site (the nearest neighbors). Here, we assume J > 0, which favors the parallel alignment of neighboring
spins. The most stable state with the minimum energy is the ferroic state where all
spins have the same state either of σ = ±1. Thus, the average of σ over all spins
becomes
m = =σ
=
n U − n D
n U + n D
= ±1,
(6.5)
where ·· is an average over all spins calculated using the numbers of Up (σ = 1, ↑)
and Down (σ = −1, ↓) spins, n U and n D . Obviously, N = n U + n D . On the other
hand, at sufficiently high temperatures, the entropy of the ensemble should tend to
N k B ln 2 according to Boltzmann’s principle (Eq. 2.8). In this state,
m = 0
(6.6)
is expected.
4 The detailed description is necessary for some cases with J < 0, which is not considered in this
book.
6 Melting of Molecular Crystals
6.2 Plastic Crystals and Liquid Crystals
6.2.1 Mean-Field Theory of Ising Model
6.2.1.1 Ising Model
Although the Ising model was originally a concise model of magnetism [22, 23],
it has served as one of the essential models in the study of phase transitions and
critical phenomena. Noteworthy is that a two-dimensional version of it assuming
short-ranged interaction is the first model that has been proven analytically to exhibit
a continuous phase transition [24]. In this section, we analyze the model in the lowest
level approximation to prepare for the understanding of some phase transitions based
on microscopic models. Readers familiar with such treatments may skip this section.
Suppose an ensemble of a vast number of “spins” (σ = ±1) on sites of a regular
lattice. Here, the regular lattice means only the following: All sites are equivalent and
have z neighboring sites. We always assume the simple cyclic boundary condition
to eliminate the effect of the periphery of the system.
4 Assuming that the interaction
between spins work only when two spins are on neighboring sites, the energy of the
ensemble is written as
E = −
1
2
J
i
j∈{NN} i
σ i σ j ,
(6.4)
where the second sum runs over z sites surrounding the i th site (the nearest neighbors). Here, we assume J > 0, which favors the parallel alignment of neighboring
spins. The most stable state with the minimum energy is the ferroic state where all
spins have the same state either of σ = ±1. Thus, the average of σ over all spins
becomes
m = =σ
=
n U − n D
n U + n D
= ±1,
(6.5)
where ·· is an average over all spins calculated using the numbers of Up (σ = 1, ↑)
and Down (σ = −1, ↓) spins, n U and n D . Obviously, N = n U + n D . On the other
hand, at sufficiently high temperatures, the entropy of the ensemble should tend to
N k B ln 2 according to Boltzmann’s principle (Eq. 2.8). In this state,
m = 0
(6.6)
is expected.
4 The detailed description is necessary for some cases with J < 0, which is not considered in this
book.