Optical branch

The Debye model is more appropriate for the acoustic branches of tire elastic modes of a hanuonic solid. For molecular solids one has in addition optical branches in the elastic wave dispersion, and the Einstein model is more appropriate to describe the contribution to U and Cj from the optical branch. The above discussion for phonons is suitable for non-metallic solids. In metals, one has, in addition, the contribution from the electronic motion to Uand Cy. This is discussed later, in section (A2.2.5.6T... 

Finally, in second order, the Raman feature at — 3180 cm" observed in Co- and Ni/Co-catalyzed single-wall nanotube corresponds to a significantly downshifted 2 x fjg mode, where E g represents the mid-zone (see Figs, la and lb) frequency maximum of the uppermost optic branch seen in graphite at 3250 cm". ... 

In the molecular approximation used in (14) only the L = 3W — 6 (W is the number of atoms) discrete intramolecular vibrations of the molecular complex in vacuo are considered. In general these vibrations correspond to the L highest optical branches of the phonon spectrum. The intermolecular vibrations, which correspond to the three acoustical branches and to the three lowest optical branches are disregarded, i.e., the center of mass and - in case of small amplitudes - the inertial tensor of the complex are assumed to be fixed in space... 

The method delineated in the preceding sections can readily be extended to the case of two atoms in the chain. An illustration of the diatomic chain model is given in Figure 8.8. The two atoms are characterized by having different masses m and m2. The two equations of motion obtained (one for each type of atom) must be solved simultaneously, giving two solutions for the angular frequency known as the acoustic and optic branches... 

The optical branch takes a non-zero limiting value at q = 0 ... 

Close to this limit the displacements of the two types of atom have opposite sign and the two types of atom vibrate out of phase, as illustrated in the lower part of Figure 8.10. Thus close to q = 0, the two atoms in the unit cell vibrate around their centre of mass which remains stationary. Each set of atoms vibrates in phase and the two sets with opposite phases. There is no propagation and no overall displacement of the unit cell, but a periodic deformation. These modes have frequencies corresponding to the optical region in the electromagnetic spectrum and since the atomic motions associated with these modes are similar to those formed as response to an electromagnetic field, they are termed optical modes. The optical branch has frequency maximum at q = 0. As q increases slowly decreases and... 

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). 

The bias observed between experimental measurements and Kieffer s model predictions is due to the relative paucity of experimental data concerning cutoff frequencies of acoustic branches, and also to the assumption that the frequencies of the lower optical branches are constant with K and equivalent to those detected by Raman and IR spectra (corresponding only to vibrational modes at K = 0). Indeed, several of these vibrational modes, and often the most important ones, are inactive under Raman and IR radiation (Gramaccioli, personal communication). The limits of the Kieffer model and other hybrid models with respect to nonempirical computational procedures based on the equation of motion of the Born-Von Karman approach have been discussed by Ghose et al. (1992). 

In this approximation co becomes independent of k, therefore this branch of the dispersion curve is horizontal in the center of the BZ 1. This is the optical branch. 

At this point a decision must be made as to whether atom A or atom B has the greater mass. If co+ is to denote the optical branch throughout BZ 1, atom A must have the greater mass mA >mB. 

As regards the notation acoustical and optical for phonons, what is important is the ratio between the vibrational amplitudes in the acoustical and optical branches. The amplitude ratio UA/UB for the different values of co and k can be calculated from Eqs. (II.1) to (II.3). We thus justify the use of the terms acoustical and optical branches as follows. 

The relations derived up to this point are not sufficient to solve the problem of the dispersion of the optical branch for small k values. The retardation effect must also be taken into account. Because of the finite velocity of electromagnetic waves, the forces at a certain point of time and space in a crystal are determined by the states of the whole crystal at earlier times. A precise description of the dispersion effect therefore requires the introduction of Maxwell s equations. With a harmonic ansatz for P and P which is analogous to Eq. (II.15) they lead to the relation... 

The positive roots of Eq. (1 75) are plotted in the positive half of the Brillouin zone as shown in Fig. 1 -38a. It may be observed that the upper curve, which is called the optical branch, represents frequencies occurring in the optical... 

AIN exists in two types the hexagonal (wurtzite structure) and the cubic (zincblende structure). The former is more stable, and has been investigated in more detail. The wurtzitic AIN has two formula units per unit cell (4 atoms per cell) and 9 optical branches to the phonon dispersion curves [1] ... 

Figure 2.6-2 Variation of the frequencies by the incorporation of a tetraatomic molecule with two degenerate vibrational states ( ) in a crystal lattice, a spectrum of the free molecule, R = rotations, T = translations b static influence of the crystal lattice. The degenerate states split, the free rotations change into librations L c dynamic coupling of the vibrations of molecules within a primitive unit cell with z = 2 molecules. Each vibrational level of a molecule splits into z components and 3 z - 3 translational vibrations TS and 3 z librations L appear d dependence of the vibrational frequencies on the wave vector k of the coupled vibrations of all unit cells in the lattice. The three acoustic branches arise from the three free translations with = 0 (for k 0) of the unit cell all vibrations of the unit cells with / 0 (for k 0) give optical branches .

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. 

This band called iTOLA because it is attributed to a combination of two intra-valley phonons the first from the in-plane transverse optical branch (iTO) and the second phonon from the longitudinal acoustic (LA) branch, iTO-t-LA, where the acoustic LA phonon is responsible for the large dispersion that is observed experimentally [69]. 

We wish to extend our personal thanks to the many persons associated with the warm core ring experiments and with the AOL project. In particular, we are indebted to Jack L. Bufton and the Instrument Electro-Optics Branch for the loan of the frequency-doubled YAG laser. We also thank Wayne E. Esaias and the Oceanic Processes Branch of NASA Headquarters for their assistance and encouragement in various aspects connected with these experiments. 

The dispersion curves are conveniently labelled in Fig. 5.1, the transverse acoustic (TA) and longitudinal acoustic (LA) branches are seen rising from the Brillouin zone centre at zero energy transfer. The optical branches (TO, LO) lie fairly flat across the zone in the energy range about 150 to 300 cm. ... 

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