10.3 Molecular Crystals as Stage for Novelties
213
relevant (small) order parameter (Sect. 2.2.4). This statement is known as the Landau
condition for the continuous transition. There exists the other condition for transitions
between crystalline phases. Since the formation of a specific superstructure needs
the strongest instability at the specific spatial modulation, this modulation must have
a fixed periodicity. Namely, the period should be “locked.” Unless this is fulfilled,
there is no reason why the resulting phase is not a general incommensurate phase
but a commensurate phase, a kind of superstructures. The condition is known as the
Lifshitz condition.
The Lifshitz condition implies the importance of the periodicity of the modulation
pattern in transitions accompanying the formation of superstructures. The theory of
space group says that irreducible representations of space groups characterize the
periodicity of modulation patterns. A combination of the reciprocal vector (wavevector) and the set of symmetry operations around a point specifies each of them. Thus,
only reciprocal wavevectors with special properties seem to be candidates relevant
to the formation of superstructures. Comprehensive analysis [82] indicates that the
multiplicity, i.e., the size of superstructures relative to the original, is mostly 2, 4, or
8 in non-hexagonal cases. In hexagonal cases, the multiplicity of 3 and 6 additionally
emerge. The possibility of a continuous transition is somewhat limited for the case
of 3 because of the Landau condition.
The multiplicities mentioned above essentially correspond to cases where one,
two, or three of cell constants double by the “softening” of modulations with reciprocal vectors at the Brillouin zone boundary. The appearance of not only 2 but also 4
or 8 originates in the symmetry of structures. Here, we see that a single modulation
and symmetrically equivalents drive the formation of superstructures. In this context, it seems natural, apart from the limited scope of the Landau theory, to ask what
superstructures we can expect while considering only a single modulation pattern
[83]. A trivial family of superstructures is a set of those with a cell constant that is
an integer multiple of the original lattice along a single lattice axis. According to
the Landau theory, the doubled structure may appear through a continuous transition
while others, such as tripled one, emerges only through discontinuous transitions.
We see that the coverage of the reciprocal lattice points in the trivial case discussed above completes only if the fundamental modulation wave and its harmonics
cooperate. The allowance of the participation of the harmonics of the fundamental modulation introduces other possibilities of superstructures driven by a single
modulation [83]. Figure 10.7 shows an example of the relation between reciprocal lattices of the basic structure stable at the high-temperature (HT) side and the
low-temperature (LT) superstructure. The multiplicity is 5 in this case. Upon the
formation of the superstructure, inside the unit cell of the basic structure shown by
black dots, new four reciprocal lattice points (red and green) emerge. Considering
the trivial equivalence of inverted wavevectors, we see that the coverage of all lattice
points of the LT superstructures completes by successive shifts represented by orange
arrows. The successive application is equivalent to harmonics. In this example, the
coverage completes only by the fundamental modulation and the second harmonic.
This easiness originates in the fact that this is the example of such superstructure
with the minimum multiplicity. On the other hand, this example implies that similar
213
relevant (small) order parameter (Sect. 2.2.4). This statement is known as the Landau
condition for the continuous transition. There exists the other condition for transitions
between crystalline phases. Since the formation of a specific superstructure needs
the strongest instability at the specific spatial modulation, this modulation must have
a fixed periodicity. Namely, the period should be “locked.” Unless this is fulfilled,
there is no reason why the resulting phase is not a general incommensurate phase
but a commensurate phase, a kind of superstructures. The condition is known as the
Lifshitz condition.
The Lifshitz condition implies the importance of the periodicity of the modulation
pattern in transitions accompanying the formation of superstructures. The theory of
space group says that irreducible representations of space groups characterize the
periodicity of modulation patterns. A combination of the reciprocal vector (wavevector) and the set of symmetry operations around a point specifies each of them. Thus,
only reciprocal wavevectors with special properties seem to be candidates relevant
to the formation of superstructures. Comprehensive analysis [82] indicates that the
multiplicity, i.e., the size of superstructures relative to the original, is mostly 2, 4, or
8 in non-hexagonal cases. In hexagonal cases, the multiplicity of 3 and 6 additionally
emerge. The possibility of a continuous transition is somewhat limited for the case
of 3 because of the Landau condition.
The multiplicities mentioned above essentially correspond to cases where one,
two, or three of cell constants double by the “softening” of modulations with reciprocal vectors at the Brillouin zone boundary. The appearance of not only 2 but also 4
or 8 originates in the symmetry of structures. Here, we see that a single modulation
and symmetrically equivalents drive the formation of superstructures. In this context, it seems natural, apart from the limited scope of the Landau theory, to ask what
superstructures we can expect while considering only a single modulation pattern
[83]. A trivial family of superstructures is a set of those with a cell constant that is
an integer multiple of the original lattice along a single lattice axis. According to
the Landau theory, the doubled structure may appear through a continuous transition
while others, such as tripled one, emerges only through discontinuous transitions.
We see that the coverage of the reciprocal lattice points in the trivial case discussed above completes only if the fundamental modulation wave and its harmonics
cooperate. The allowance of the participation of the harmonics of the fundamental modulation introduces other possibilities of superstructures driven by a single
modulation [83]. Figure 10.7 shows an example of the relation between reciprocal lattices of the basic structure stable at the high-temperature (HT) side and the
low-temperature (LT) superstructure. The multiplicity is 5 in this case. Upon the
formation of the superstructure, inside the unit cell of the basic structure shown by
black dots, new four reciprocal lattice points (red and green) emerge. Considering
the trivial equivalence of inverted wavevectors, we see that the coverage of all lattice
points of the LT superstructures completes by successive shifts represented by orange
arrows. The successive application is equivalent to harmonics. In this example, the
coverage completes only by the fundamental modulation and the second harmonic.
This easiness originates in the fact that this is the example of such superstructure
with the minimum multiplicity. On the other hand, this example implies that similar