202
10 Importance of Molecular Crystals
We may deduce the information concerning the energetic relations of possible
states by analyzing the numerical magnitude of relevant entropy in the light of structural details [22]. Further, if the structural feature of the system is restrictive enough,
e.g., quasi-one-dimensional, we may correlate the (averaged or statistical) correlation of molecular states and other physical properties by adequately adopting relevant
models [23, 24].
10.1.3 Dynamics in Quasi-one-dimensional Systems
Although we have not described details, one-dimensional systems have some interesting properties. The impossibility of an ordered state at finite temperatures is a
notable example and has a close connection with a correlated dynamics to discuss.
Suppose a one-dimensional chain of Ising spins interacting via ferromagnetic interaction (J > 0) between the nearest neighbors. Figure 10.2 depicts a state the next
lowest in the energy of the system. A dotted vertical line indicates the location of a
domain “wall” or a “surface.” The energy relative for the ground state is written as n J
using the number of walls, n. It is interesting to see that the energy relies not on the
locations of walls but solely on their number. The flip of one of the spins neighboring
to a wall only shifts it while keeping the energy. Since we have (N − 1) choices of
the domain location for the N -spin system,
2 the entropy of a microcanonical state
with this energy is k B ln(N − 1). Note that the averaged magnetization assigned to
the whole system in this thermodynamic state is 0. A comparison of free energies of
the ground state and the first excited state, F − F GS = J − k B T ln(N − 1), implies
that the latter is more stable at finite temperatures. Thus, the ordered state with finite
magnetization is plausibly impossible in this system. Although the above discussion
is on the ferromagnetic Ising system, the essential characteristics that the number
of the domain wall dominates the system property commonly applies to any purely
one-dimensional systems with a short-ranged interaction(s).
Besides the impossibility of the ordered state, the one-dimensionality brings a
particular property. Spin-flip is more accessible at the domain wall than inside a
domain because it accompanies no energy increment. Furthermore, the flip of a
group of plural spins adjacent to the domain wall shares the property. However, the
energy barrier to overcome will become higher in the latter.
Molecular complexes consisting of two organic components (co-crystals),
phenazine (Phz) and chloranilic or bromanilic acid (H 2 ca or H 2 ba), exhibit the ferroelectricity at low temperatures [25]. Their crystals share the basic structure and
consist of segregated columns of planar molecules. Electronic interaction within a
column is expected from the close stacking of π-orbitals. A strong H-bond connects
a nitrogen atom of Phz and a hydroxyl group of H 2 ca (or H 2 ba) in the neighboring
columns. The structure is thus characterized by one-dimensional chains of alternate
2 A factor 2 additionally appears if we consider the choices of configurations of two terminal spins.
10 Importance of Molecular Crystals
We may deduce the information concerning the energetic relations of possible
states by analyzing the numerical magnitude of relevant entropy in the light of structural details [22]. Further, if the structural feature of the system is restrictive enough,
e.g., quasi-one-dimensional, we may correlate the (averaged or statistical) correlation of molecular states and other physical properties by adequately adopting relevant
models [23, 24].
10.1.3 Dynamics in Quasi-one-dimensional Systems
Although we have not described details, one-dimensional systems have some interesting properties. The impossibility of an ordered state at finite temperatures is a
notable example and has a close connection with a correlated dynamics to discuss.
Suppose a one-dimensional chain of Ising spins interacting via ferromagnetic interaction (J > 0) between the nearest neighbors. Figure 10.2 depicts a state the next
lowest in the energy of the system. A dotted vertical line indicates the location of a
domain “wall” or a “surface.” The energy relative for the ground state is written as n J
using the number of walls, n. It is interesting to see that the energy relies not on the
locations of walls but solely on their number. The flip of one of the spins neighboring
to a wall only shifts it while keeping the energy. Since we have (N − 1) choices of
the domain location for the N -spin system,
2 the entropy of a microcanonical state
with this energy is k B ln(N − 1). Note that the averaged magnetization assigned to
the whole system in this thermodynamic state is 0. A comparison of free energies of
the ground state and the first excited state, F − F GS = J − k B T ln(N − 1), implies
that the latter is more stable at finite temperatures. Thus, the ordered state with finite
magnetization is plausibly impossible in this system. Although the above discussion
is on the ferromagnetic Ising system, the essential characteristics that the number
of the domain wall dominates the system property commonly applies to any purely
one-dimensional systems with a short-ranged interaction(s).
Besides the impossibility of the ordered state, the one-dimensionality brings a
particular property. Spin-flip is more accessible at the domain wall than inside a
domain because it accompanies no energy increment. Furthermore, the flip of a
group of plural spins adjacent to the domain wall shares the property. However, the
energy barrier to overcome will become higher in the latter.
Molecular complexes consisting of two organic components (co-crystals),
phenazine (Phz) and chloranilic or bromanilic acid (H 2 ca or H 2 ba), exhibit the ferroelectricity at low temperatures [25]. Their crystals share the basic structure and
consist of segregated columns of planar molecules. Electronic interaction within a
column is expected from the close stacking of π-orbitals. A strong H-bond connects
a nitrogen atom of Phz and a hydroxyl group of H 2 ca (or H 2 ba) in the neighboring
columns. The structure is thus characterized by one-dimensional chains of alternate
2 A factor 2 additionally appears if we consider the choices of configurations of two terminal spins.