68
4 Molecular Crystals
vanish if summed as (1, 1, 0) + (−1, 0, 1) + (0, −1, −1) = (0, 0, 0). Since we are
dealing with the wavevectors, the lattice formed by the vectors is the so-called reciprocal lattice. The corresponding lattice in the direct space is the body-centered cubic
(BCC) structure.
In this treatment, possibilities of phase transitions and of emerging spatial order are
discussed, in contrast to the standard use of the Landau theory, where a specific phase
transition is described. The symmetric phase is the isotropic fluid common for all
cases. For a meaningful comparison of emerging orders, it is necessary to normalize
the free energy by the number of wavevectors in each group. Then, the phase transition
with the highest transition temperature is concluded to occur. Through the analysis,
the transition to the BCC structure was identified to occur at the highest temperature.
Further analysis concerning temperature dependence revealed that the BCC structure
is stable irrespective of temperature as far as the expansion (Eq. 4.3) is assumed.
The above consideration leads to a striking (but rather strange) conclusion that
globular molecules should crystallize into BCC structures. By incorporating slightly
different wavevectors {1, 1, 1} and {2, 0, 0} that are within the experimental width
of the wavevector of density fluctuation in isotropic liquid, the FCC structure can be
stabilized [7]. The formation of wavevector triangles, though they are always mixed
ones, is also essential in this case. Note that the difference in length (mismatch) is
(2 −
√
3)/2 ≈ 0.13 in this case.
4.2 Diffraction and Modern Definition of Crystals
In contrast to naïve understanding, the International Union of Crystallography (IUCr)
adopts the definition of crystals as “any solid having an essentially discrete diffraction
diagram” [1]. This definition is different from the traditional definition assuming
translational symmetries. In this section, we see the relation between modern and
traditional definitions. It is necessary because the modern definition is necessary for
some molecular systems.
We start with a naïve definition of crystals, i.e., a periodic arrangement of structural
units. For simplicity, we consider a one-dimensional periodic array of point scatterers,
of which the number and the period are N + 1 and a, respectively. The extension to
three-dimensional cases is trivial. The density of scatterer is expressed as
ρ(r ) =
j=0,N
δ(r − ja).
(4.4)
Here, we take the finite size of the array into account. When the scattering power of
atoms is f j = f
◦ ,
2 the complex strength of the scattered wave, A(q), is given by
3
2 Generally, the scattering power of scatterers, such as atoms, is a function of scattering vector q.
3 Note that “i” in formulas in this section is the imaginary unit, i.e., i 2 = −1 while “ j” is an integer
index.
4 Molecular Crystals
vanish if summed as (1, 1, 0) + (−1, 0, 1) + (0, −1, −1) = (0, 0, 0). Since we are
dealing with the wavevectors, the lattice formed by the vectors is the so-called reciprocal lattice. The corresponding lattice in the direct space is the body-centered cubic
(BCC) structure.
In this treatment, possibilities of phase transitions and of emerging spatial order are
discussed, in contrast to the standard use of the Landau theory, where a specific phase
transition is described. The symmetric phase is the isotropic fluid common for all
cases. For a meaningful comparison of emerging orders, it is necessary to normalize
the free energy by the number of wavevectors in each group. Then, the phase transition
with the highest transition temperature is concluded to occur. Through the analysis,
the transition to the BCC structure was identified to occur at the highest temperature.
Further analysis concerning temperature dependence revealed that the BCC structure
is stable irrespective of temperature as far as the expansion (Eq. 4.3) is assumed.
The above consideration leads to a striking (but rather strange) conclusion that
globular molecules should crystallize into BCC structures. By incorporating slightly
different wavevectors {1, 1, 1} and {2, 0, 0} that are within the experimental width
of the wavevector of density fluctuation in isotropic liquid, the FCC structure can be
stabilized [7]. The formation of wavevector triangles, though they are always mixed
ones, is also essential in this case. Note that the difference in length (mismatch) is
(2 −
√
3)/2 ≈ 0.13 in this case.
4.2 Diffraction and Modern Definition of Crystals
In contrast to naïve understanding, the International Union of Crystallography (IUCr)
adopts the definition of crystals as “any solid having an essentially discrete diffraction
diagram” [1]. This definition is different from the traditional definition assuming
translational symmetries. In this section, we see the relation between modern and
traditional definitions. It is necessary because the modern definition is necessary for
some molecular systems.
We start with a naïve definition of crystals, i.e., a periodic arrangement of structural
units. For simplicity, we consider a one-dimensional periodic array of point scatterers,
of which the number and the period are N + 1 and a, respectively. The extension to
three-dimensional cases is trivial. The density of scatterer is expressed as
ρ(r ) =
j=0,N
δ(r − ja).
(4.4)
Here, we take the finite size of the array into account. When the scattering power of
atoms is f j = f
◦ ,
2 the complex strength of the scattered wave, A(q), is given by
3
2 Generally, the scattering power of scatterers, such as atoms, is a function of scattering vector q.
3 Note that “i” in formulas in this section is the imaginary unit, i.e., i 2 = −1 while “ j” is an integer
index.