Phonons in other Molecular Crystals

The limitation to second derivatives in computing the force constants is called the harmonic approximation. The calculation does not take into account either the zero-point oscillation or the anharmonicity of the potentials. It is, strictly speaking, valid only at T = 0. Thermal expansion, thermal conductivity and other nonlinear effects are thus not contained in this model. 

In detail, the solution of the equations of motion requires strict observance of the crystal symmetry and precise equilibrium coordinates of the molecules and their atoms in dependence on the temperature (and the pressure), since only then can the atom-atom potentials be computed correctly. Natkaniec et al. [7] solved the equations of motion numerically at T = 0 K for the perdeuterated naphthalene crystal N-ds (and later for many other molecular crystals). The lattice sum in Eq. (5.14) was limited to 24 neighbouring molecules j, after it had been found that inclusion of additional neighbour shells had no influence on the results. This is a direct consequence of the short range of the van der Waals interaction. 

Qualitatively, the dynamic properties of different polyacene crystals are similar, since both the intermolecular van der Waals forces and also the molecular masses are to first order proportional to the number of C atoms per molecule. Therefore, for example all the sound velocities are of the same order of magnitude (Table 5.4). The same is true for the maximum frequencies of the optical phonons and of the intramolecular osdUations. A clear and general difference is however to be found between smaller and larger molecules in terms of the lowest frequencies of the intramolecular modes with increasing molecular mass, the frequency of the low-energy intramolecular vibrations shift towards lower values. These molecular excitations are either bending or torsional oscillations. This frequency shift can cause the gap and the strict separation of internal and external modes to become less sharp. 

In the following section, the peculiarities of the dynamic properties of a few selected molecular crystals wiU be treated briefly. 

Perylene The two different crystalline phases, a perylene and perylene (see Chap. 2, Fig. 2.12) differ strongly also in their dynamic properties a perylene has four molecules or two dimers per unit ceU. From this, 24 internal modes result, 21 optical and three acoustic. They have also been observed and identified by inelastic neutron diffraction [17] and by Raman scattering [18]. Their spectrum has a width of about 4 THz, again similar to the cases of naphthalene and anthracene. For a perylene, the model treated in Sect. 5.6 again yields satisfactory theoretical dispersion relations. The low-energy internal modes are torsional (twisting) and butterfly modes. Their spectrum overlaps with that of the external modes. 

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