Optical dispersion, lattice vibrations

Figure 2.38 illustrates that in the case of an ionic solid the optical mode of the lattice vibration resonates at an angular frequency, co0, in the region of 1013Hz. In the frequency range from approximately 109-10nHz dielectric dispersion theory shows the contribution to permittivity from the ionic displacement to be nearly constant and the losses to rise with frequency according to... 

The dispersion relationships of lattice waves may be simply described within the first Brillouin zone of the crystal. When all unit cells are in phase, the wavelength of the lattice vibration tends to infinity and k approaches zero. Such zero-phonon modes are present at the center of the Brillouin zone. The variation in phonon frequency as reciprocal k) space is traversed is what is meant by dispersion, and each set of vibrational modes related by dispersion is a branch. For each unit cell, three modes correspond to translation of all the atoms in the same direction. A lattice wave resulting from such displacements is similar to propagation of a sound wave hence these are acoustic branches (Fig. 2.28). The remaining 3N-3 branches involve relative displacements of atoms within each cell and are known as optical branches, since only vibrations of this type may interact with light. 

Here is the momentum transferred from the neutron and x is one of the reciprocal lattice vectors of the palladium lattice. Thus, the incoherent scattering sees all the vibration modes but the coherent scattering selects one particular phonon for a particular experimental value of Q. It is now clear that both the incoherent and coherent one-phonon scattering will depend on the shape of the optical dispersion curves and hence will be influenced by hydrogen-hydrogen interactions. Indeed, one of the first observations of inelastic scattering from a hydride [10] interpreted the shape of the optical peak in terms of a frequency distribution broadened by H-H interactions. 

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]. 

The motion of atoms in the lattice can be depicted as a wave propagation (phonon). By dispersion we mean the variation in the wave frequency as reciprocal space is traversed. The propagation of sound waves is similar to the translation of all atoms of the unit cell in the same direction hence the set of translational modes is commonly defined as an acoustic branch. The remaining vibrational modes are defined as optical branches, because they are capable of interaction with light (see McMillan, 1985, and Tossell and Vaughan, 1992, for more exhaustive explanations). 

Fig. 5.2 A schematic energy diagram J2(K) of the internal and the external molecular vibrations in molecular crystals. Q is the frequency, hS2 the energy and K is the magnitude of the wavevector in a particular direction, e.g. in the direction a. (C = 0 is the centre and K = itja the boundary of the Brillouin zone, with the lattice constant a. P is the usual notation for the centre of the Brillouin zone. MSi is a low-frequency internal molecular oscillation with a small or vanishing dispersion const.). MSi is a high-frequency internal molecular oscillation. All together, there are 3N-6 internal modes N is the number of atoms per molecule. OP is an optical phonon in which whole molecules are excited to carry out translational or libration oscillations whose frequencies are...

The cosine curve duplicates the results and in its middle part represents the upper LO curve in Fig.2.7. We note that the optical vibration at q = 0 of the diatomic chain becomes an acoustic vibration at q = n/d of the mono-atomic chain. The upper part AB of the sine curve in Fig.2.7 can be obtained by folding out the LO branch AC, or equivalently, by translation of the LO branch CD through 2-n/a, that is, by a reciprocal lattice vector t = M. This is called an Umptapp process. The dispersion relation (2.43) can of course be obtained if we start directly with the Hamiltonian of the monoatomic chain which follows easily from (2.3) and then solving the resulting equations of motion by assuming a solution of the form... 

In a non-exhaustive literature search, a brief account is given here on the study of phonon and its vibrations. Corso et al. did an extensive study of density functional perturbation theory for lattice dynamics calculations in a variety of materials including ferroelectrics [93]. They employed a nonlinear approach to mainly evaluate the exchange and correlation energy, which were related to the non-linear optical susceptibility of a material at low frequency [94], The phonon dispersion relation of ferroelectrics was also studied extensively by Ghosez et al. [95, 96] these data were, however, related more with the structure and metal-oxygen bonds rather than domain vibrations or soliton motion. In a very interesting work, a second peak in the Raman spectra was interpreted by Cohen and Ruvalds [97] as evidence for the existence of bound state of the two phonon system and the repulsive anharmonic phonon-phonon interaction which splits the bound state off the phonon continuum was estimated for diamond. 

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