4.2 Diffraction and Modern Definition of Crystals
71
odicity belongs to the category of crystals defined by IUCr [1]. Diffractions with
m = 0 are termed as satellite reflections in contrast to main reflections (m = 0).
Although the sinusoidal modulation (Eq. 4.9) gives only the first-order satellite reflections, the details of structural modulation determine how higher-order satellite reflections can be detectable. The strength of higher-order components in the modulation
governs those of satellite reflections.
It is noteworthy that the position of the modulation wave relative to the underlying
(regular) lattice does not alter the physical situation in the incommensurate phases as
a whole if the crystal is sufficiently large (infinite in the ideal sense). This property
brings about the possibility of the sliding of the modulation wave without any energy
penalty.
4 Indeed, this type of collective dynamics is characteristic of incommensurate
phases [10–12]. Interestingly, the charge-carrying modulation wave (charge density
wave) was proposed as a possible mechanism of superconductivity in the early days
of its study [13].
Another important class of materials adopted as crystals by the IUCr definition is
quasicrystals [6]. They possess fivefold or tenfold symmetry, which is incompatible
with translational symmetry. Mathematically, quasicrystals are projections of sixdimensional regular crystals onto the three-dimensional space. Adequately designed
block copolymers indeed exhibit the quasicrystalline order [14]. However, we give
no further discussion of the quasicrystals in this book because no examples have
been identified for matter composed of small molecules.
4.3 Molecular Shape and Crystal Structure
4.3.1 Within Landau Theory
We pointed out that the molecular shape is crucial in crystallization. Here, attempted
is a possible inclusion of the effects into the Landau theory of weak crystallization
[15]. We assume that a molecule is a cylinder with a length larger than its diameter.
Then, there are two characteristic lengths and two characteristic wavevectors in
the reciprocal space, accordingly. Although we have as yet no rigorous proof, it
is intuitively acceptable that the larger length scale (in the direct space) is much
more important for establishing isotropic, i.e., cubic organization upon aggregation.
Consequently, our expansion of free energy is in terms of this shorter wavevector.
Assuming the cubic symmetry, we can describe groups of wavevectors that can
form equilateral triangles. The possible combinations are {1, 1, 0}, {2, 1, 1},{2, 2, 0},
{3, 2, 1}, {4, 2, 2}, {4, 3, 1}, {5, 3, 2}, ... in the order of indexes of wavevectors. If
we repeat the calculation by assuming equilateral triangles, the superior stability of
the BCC structure results. However, this structure implicitly assumes, in reality, the
4 This argument applies to smooth modulation waves, such as a sinusoidal one. If the structural
modulation on each site is discrete, the energy is equal before and after its shift, but the activation
energy is necessary.
71
odicity belongs to the category of crystals defined by IUCr [1]. Diffractions with
m = 0 are termed as satellite reflections in contrast to main reflections (m = 0).
Although the sinusoidal modulation (Eq. 4.9) gives only the first-order satellite reflections, the details of structural modulation determine how higher-order satellite reflections can be detectable. The strength of higher-order components in the modulation
governs those of satellite reflections.
It is noteworthy that the position of the modulation wave relative to the underlying
(regular) lattice does not alter the physical situation in the incommensurate phases as
a whole if the crystal is sufficiently large (infinite in the ideal sense). This property
brings about the possibility of the sliding of the modulation wave without any energy
penalty.
4 Indeed, this type of collective dynamics is characteristic of incommensurate
phases [10–12]. Interestingly, the charge-carrying modulation wave (charge density
wave) was proposed as a possible mechanism of superconductivity in the early days
of its study [13].
Another important class of materials adopted as crystals by the IUCr definition is
quasicrystals [6]. They possess fivefold or tenfold symmetry, which is incompatible
with translational symmetry. Mathematically, quasicrystals are projections of sixdimensional regular crystals onto the three-dimensional space. Adequately designed
block copolymers indeed exhibit the quasicrystalline order [14]. However, we give
no further discussion of the quasicrystals in this book because no examples have
been identified for matter composed of small molecules.
4.3 Molecular Shape and Crystal Structure
4.3.1 Within Landau Theory
We pointed out that the molecular shape is crucial in crystallization. Here, attempted
is a possible inclusion of the effects into the Landau theory of weak crystallization
[15]. We assume that a molecule is a cylinder with a length larger than its diameter.
Then, there are two characteristic lengths and two characteristic wavevectors in
the reciprocal space, accordingly. Although we have as yet no rigorous proof, it
is intuitively acceptable that the larger length scale (in the direct space) is much
more important for establishing isotropic, i.e., cubic organization upon aggregation.
Consequently, our expansion of free energy is in terms of this shorter wavevector.
Assuming the cubic symmetry, we can describe groups of wavevectors that can
form equilateral triangles. The possible combinations are {1, 1, 0}, {2, 1, 1},{2, 2, 0},
{3, 2, 1}, {4, 2, 2}, {4, 3, 1}, {5, 3, 2}, ... in the order of indexes of wavevectors. If
we repeat the calculation by assuming equilateral triangles, the superior stability of
the BCC structure results. However, this structure implicitly assumes, in reality, the
4 This argument applies to smooth modulation waves, such as a sinusoidal one. If the structural
modulation on each site is discrete, the energy is equal before and after its shift, but the activation
energy is necessary.