108
5 Lattice Dynamics of Molecular Crystals
of freedom can couple with the molecular twist. Thus, Fig. 5.2 contains 8 = (3 +
1) × 2 branches formed by 3 translational and 1 twisting degrees of freedom per
molecule. The results without the coupling are drawn by a broken line for the twisting
branches and by dot-dash lines for translational branches. It is evident that, even
without the coupling, the twisting branches exhibit a notable dispersion against the
wavevector. The absence of the coupling allows the crossing of branches. When the
coupling is taken into account, their crossing disappears, and a dip in the dispersion
relation appears near b
∗
/2. Remember that the phonon frequencies are square-roots
of eigenvalues of a kind of dynamical matrix. Negative eigenvalues are equivalent
to the instability of an assumed crystal structure. Thus, a dip in a phonon dispersion
relation is, in general, a symptom of some potential lattice instability. The appearance
of the dip in Fig. 5.2 indicates that the presence of the soft twisting degree of freedom
and its coupling produces the lattice instability. The importance of the coupling in
this lattice instability was demonstrated through varying its strength artificially. The
authors of [24] indicated that the crystal structure loses stability for some conditions.
Later, the other group [26] indicated that the negative intramolecular contribution to
the force field for the twisting is essential.
Crystalline bis(4-chlorophenyl)sulfone [BCPS, (ClC 6 H 4 ) 2 SO 2 ] is another example to exhibit a structural phase transition in which intramolecular motional degrees
play an important role. The crystal at room temperature belongs to the space group
C2/c with molecules on the two-fold axis. The crystal undergoes a displacive phase
transition around 220 K to an incommensurate phase having the modulation wavevector k
∗
≈ 0.8c
∗ . The lattice-dynamical calculation incorporating molecular deformations [27] was used to clarify the instability of the crystal structure. An idealized
molecular model is assumed: all bond angles are 2π/3, and benzene rings are regular
hexagons. Only two twisting degrees of freedom are incorporated if applicable. To
separate the twisting degrees of freedom from the translation and rotation of a whole
molecule, they are included in the calculation as symmetric and antisymmetric combinations concerning the twofold symmetry around the axis that coincides with the
bisectors of ∠OSO and ∠CSC. Intermolecular interaction is expressed by sums of
atom-atom potentials discussed in Sect. 1.2.5.2. Before the lattice-dynamical calculations, the lattice energy is minimized while keeping the symmetry of the crystal
structure experimentally known, in order to ensure the structure is minimum in the
lattice energy.
Figure 5.3 shows the phonon dispersion relations calculated under three different
force constants for the twist of chlorophenyl groups. A dip in the dispersion relation
in the lowest-lying mode can be recognized at q ≈ 0.8c
∗ in the result assuming
rigid molecules. This dip can be regarded as a symptom of the potential instability
for the modulation with this wavevector. On the other hand, the crystal is unstable
as indicated by imaginary frequencies for some wave vectors if the intramolecular
restoring force is assumed null. Indeed, with assuming the intramolecular twisting
frequency being 70 cm
−1 , the lattice vibrations have positive frequencies everywhere
in the q space, indicating the stability of the crystal structure. Thus, the intramolecular
twisting degrees of freedom does not cause instability but enhances it in contrast to
the case of crystalline biphenyl.
5 Lattice Dynamics of Molecular Crystals
of freedom can couple with the molecular twist. Thus, Fig. 5.2 contains 8 = (3 +
1) × 2 branches formed by 3 translational and 1 twisting degrees of freedom per
molecule. The results without the coupling are drawn by a broken line for the twisting
branches and by dot-dash lines for translational branches. It is evident that, even
without the coupling, the twisting branches exhibit a notable dispersion against the
wavevector. The absence of the coupling allows the crossing of branches. When the
coupling is taken into account, their crossing disappears, and a dip in the dispersion
relation appears near b
∗
/2. Remember that the phonon frequencies are square-roots
of eigenvalues of a kind of dynamical matrix. Negative eigenvalues are equivalent
to the instability of an assumed crystal structure. Thus, a dip in a phonon dispersion
relation is, in general, a symptom of some potential lattice instability. The appearance
of the dip in Fig. 5.2 indicates that the presence of the soft twisting degree of freedom
and its coupling produces the lattice instability. The importance of the coupling in
this lattice instability was demonstrated through varying its strength artificially. The
authors of [24] indicated that the crystal structure loses stability for some conditions.
Later, the other group [26] indicated that the negative intramolecular contribution to
the force field for the twisting is essential.
Crystalline bis(4-chlorophenyl)sulfone [BCPS, (ClC 6 H 4 ) 2 SO 2 ] is another example to exhibit a structural phase transition in which intramolecular motional degrees
play an important role. The crystal at room temperature belongs to the space group
C2/c with molecules on the two-fold axis. The crystal undergoes a displacive phase
transition around 220 K to an incommensurate phase having the modulation wavevector k
∗
≈ 0.8c
∗ . The lattice-dynamical calculation incorporating molecular deformations [27] was used to clarify the instability of the crystal structure. An idealized
molecular model is assumed: all bond angles are 2π/3, and benzene rings are regular
hexagons. Only two twisting degrees of freedom are incorporated if applicable. To
separate the twisting degrees of freedom from the translation and rotation of a whole
molecule, they are included in the calculation as symmetric and antisymmetric combinations concerning the twofold symmetry around the axis that coincides with the
bisectors of ∠OSO and ∠CSC. Intermolecular interaction is expressed by sums of
atom-atom potentials discussed in Sect. 1.2.5.2. Before the lattice-dynamical calculations, the lattice energy is minimized while keeping the symmetry of the crystal
structure experimentally known, in order to ensure the structure is minimum in the
lattice energy.
Figure 5.3 shows the phonon dispersion relations calculated under three different
force constants for the twist of chlorophenyl groups. A dip in the dispersion relation
in the lowest-lying mode can be recognized at q ≈ 0.8c
∗ in the result assuming
rigid molecules. This dip can be regarded as a symptom of the potential instability
for the modulation with this wavevector. On the other hand, the crystal is unstable
as indicated by imaginary frequencies for some wave vectors if the intramolecular
restoring force is assumed null. Indeed, with assuming the intramolecular twisting
frequency being 70 cm
−1 , the lattice vibrations have positive frequencies everywhere
in the q space, indicating the stability of the crystal structure. Thus, the intramolecular
twisting degrees of freedom does not cause instability but enhances it in contrast to
the case of crystalline biphenyl.