Periodic crystals phonon dispersion

Theoretical calculations of surface phonon dispersion have been carried out in two ways. One method is to use a Green s function technique which treats the surface as a perturbation of the bulk periodicity in the z-direction [34, 35]. The other is a slab dynamics calculation in which the crystal is represented by a slab of typically 15-30 layers thick, and periodic boundary conditions are employed to treat interactions outside the unit cell as the equations of motion for each atom are solved [28, 33, 35, 37]. In the latter both the bulk and the surface modes are found and the surface localized modes are identified by the decay of the vibrational amplitudes into the bulk in the former the surface modes can be obtained directly. When the frequency of a surface mode lies within a bulk band of the same symmetry, then hybridization can take place. In this event the mode can no longer be regarded as strictly surface localized and is referred to as a surface resonance [24]. Figure 8, adapted from Benedek and Toennies [24], shows how the bulk and surface modes develop as more and more layers are taken in a slab dynamics calculation. 

The thermal energy in an insulating crystal takes the form of vibrations of the atoms about their equilibrium positions. The atoms do not vibrate independently, however, since the movements of one atom directly modulate the potential wells occupied by its neighbors. The result is coupled modes of vibration of the crystal as a whole, known as phonons. The phonon dispersion curve for an idealized one-dimensional lattice is sketched in Fig. 4. It is periodic, and it may be seen that there is a definite maximum phonon frequency, known as the Debye cut-... 

We restrict the attention to periodic solids, molecular crystals. The excitations are characterized by wave vectors q, that lie in the first Brillouin zone of the lattice considered. These excitations are not necessarily pure translational phonons, librons or vibrons, in general they will be mixed. Much experimental information has been collected about such excitations, by infrared and Raman spectroscopy and, in particular, by inelastic neutron scattering. Due to the optical selection rules infrared and Raman spectra can only probe the = 0 excitations. By neutron scattering one can excite states of any given q and thus measure the complete dispersion (wave vector dependence) of the phonon and vibron frequencies. 

The molecules treated in this chapter are indeed large systems with complex chemical structures. Moreover, in going from the oligomers to the polymers it becomes necessary to consider the systems as one-dimensional crystals. The optical transitions (both vibrational and electronic) are determined by one-dimensional periodicity and translational symmetry. Collective motions (phonons) that extend throughout the chain need to be considered and are characterized by the wave vector k, and their frequencies show dispersion with k. Lattice dynamics in the harmonic approximation are well developed [12,13J, and vibrational frequency spectroscopy has reached full maturity and has been widely applied in polymer science [8,9,14]. 

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