Optical branch, dispersion curve

The dispersion curve for a diatomic chain is given in Figure 8.9. The curve consists of two distinct branches the acoustic and the optic. In the first of these the frequency varies from zero to a maximum cop. The second one has a maximum value of (Oq at q = 0 and decreases to co2 at q - qmax. There are no allowed frequencies in the gap between a>i and a>2. 

The band structure of bulk silicon, with possible optical transitions for (c) absorption and (d) emission of a photon, together with (e) the dispersion curves of phonon branches, is shown on the right. After [Kol5],... 

Fig. 2. Dispersion curves of the optical and acoustical branch in BZ 1 for the linear AB chain...

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. 

Fig. 3. Dispersion curves of the long-wavelength optical phonons, photons, and polaritons in the centre of BZ 1. In order to demonstrate the connection with the dispersion effects in the region 107 < k < 10 cm-1, the branches of an LO and an LA phonon in this region have been added in a different linear scale. The figures correspond to a cubic lattice with two atoms in the unit cell 3S)...

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] ... 

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. ... 

This analysis is based on topological considerations, but other CPs can emerge depending on the actual shape of the dispersion curves in the BZ [50]. Sometimes, the T point is included in the CPs, but when this is done, this can be only for the optical branches. The degeneracy of the TO and LO branches at the T point for the diamond structure. A similar topological degeneracy of the LA and LO branches at the X point (noted L(X)) also exists for this structure. Dispersion curves of phonons in diamond are shown in Fig. 3.1. The curves for silicon and germanium are qualitatively similar. 

The term k in Eqs. 1.235-1.241 is called the wavevector and indicates the phase difference between equivalent atoms in each unit cell. In the case of a one-dimensional lattice, k = L Thus, weusefcrather than Ifcl in this case, andfccantakeany value between —nils, and + 7i/2a. This regionis called the.first Brillouinzone. Figure 1.45 shows aplot of CO versus k for the positive half of the first Brillouin zone. There are two values for each CO that constitute the optical and acoustical branches in the dispersion curves. 

Now we have two (2) phonon dispersion curves, a so-called optical branch and a lower energy acoustical branch. The standing waves are better understood in terms of the actual displacement the atoms undergo ... 

Figure 4. Dispersion reiation o)(q) for the one-dimensionai diatomic ciystai in Fig. 3. The relation is given hy Eq. (8) in Ae text with (top panel) Mq = 2Mh and (lower panel) Me = M for/i (solid lines),/, = 2 (short-dashed lines), and/, = 10/ Oong-dashed lines). In each panel the lower curves which go to zero for 9 = 0 are the acoustic branches of the dispersion relation [negative sign in Eq. (8)] and the upper curves which have finite values for = 0 are the optical branches [positive sign in Eq. (8)]. The ordinate scale for all curves is in units of (/,/Afc) - Note that the gap between the optical and acoustic bands increases as/, increases relative to/ and as the difieience in masses of G and H increases.

Figure 7. Bulk phonon dispersion curves for KBr and RbCl in their <100> and <111> high-symmetry directions. Both crystals have fee lattices and rocksalt structures. Note that the transverse branches, labeled TA (transverse acoustic) and TO (transverse optical), are doubly degenerate in these directions. (Adapted from Fig. 3 of Ref. 32.)...

Brillouin zone. It should be noted that several symmetry-required degeneracies occur which have not been indicated in the figure and that the traditional spec-troscopist s frequency unit of cm (1 cm = 3 X 10 Hz) has been used. The recipe given above applied to this crystal (A = 8, w = 2) gives eight internal mode branches or dispersion curves and 16 external mode branches of which three are translational acoustic, four are rotatory optic (sometimes called librational), and nine are translatory optic. 

Period of the chain is equal to a. Let us suppose the linear relationship between the interaction force between the nearest neighbors and atomic displacement. Every internal motion of the lattice could be represented by the superposition of the mutually orthogonal waves as follows from the lattice dynamic theoiy (see e.g. Bom and Huang 1954 Leibfried 1955). Aiy lattice wave could be described by the wave vector K from the first Brillouin zone in the reciprocal space. Dispersion curve co K) has two separated branches (for 2 atoms in the primitive unit), which could be characterized as acoustic and optic phonons. If we suppose also the transversal waves (along with longimdinal ones), we can get three acoustic and three optical phonon branches. There is always one longitudinal (LA or LO) and two mutually perpendicular transversal (TA or TO) phonons. 

Frequency-dispersion curves are useful for analysing infrared and Raman spectra of polymer chains of Gnite length where chain vibrations associated with various phase different are observed. From the frequency-dispersion curves, the frequency distribution of the optical branches may be calculated, for studying the contributions of chain vibrations to heat capacities and the zero-point energy. The intrachain force field is also used in treating the elastic modulus along the chain axis, for comparison with the experimental data. 

Fig. 6.9. Schematic, representative phonon dispersion curves for the naphthalene crystal. Three acoustic branches and several optical branches can be seen. Note that the shape of the bands is different in the two different paths through the BrUlouin zone, to a and b. This is due to the anisotropy of the crystalline environment. Adapted from data in ref. [13], p. 264.

Problem 2.3. Figure 2.27 represents the dispersion of phonons on the NaF(lOO) surface. Show the dispersion curves corresponding to surface phonons. Which of them are related to (a) acoustical phonon branches (b) optical phonon branches Which of them can be referred to as surface resonance phonons ... 

Problem 2.3. The dispersion curves of surface phonons are shown in Fig. 2.27 by thick solid and dashed lines. Those which are indicated by dashed lines have the character of surface resonances. The branch of acoustical surface phonons (Si) originates at the T-point (ky = 0) where its energy is equal to zero. The dispersion curves S2 and S3 correspond to optical surface phonons. One cannot classify the curve S4 based on this figure. 

In the spectra of CdSe, three modes at 258, 359 and 950 cm are clearly observed. In general, the LO and TO phonons are observed along with the surface modes in polar nanocrystals in resonance Raman spectra and/or surface enhanced Raman spectra [275]. However, LO and TO modes are observed simultaneously only in randomly oriented nanoparticles. Resonance Raman Spectra (RRS) of CdTe nanoparticles give band due to Longitudinal optical (LO) phonons at 170 cm (LO), 340 cm (2LO) and 510 cm (3LO) mode frequency is found to shift due to quantum confinement effect and confined phonons are observed using surface enhanced Raman spectroscopy [275]. Transverse optic (TO) phonon is reported at 145 cm and its position is invariant with decreasing particle size as the dispersion curve for TO phonon branch is almost fiat [275]. In CdSe nanoparticles, LO phonons are reported in the range 180- 200 cm, wheras in ZnSe at 140 cm [Ref. 275 and references therein]. Thus, the Raman spectra observed in the present work well identifies the phonons in these nanoparticles. 

FIGURE 2.8 The dispersion curves for a simple one-dimensional lattice with an optical and an acoustical branch. 

The results are shown in Fig. 2.26. Thus, for the three-dimensional motion of a diatomic chain there is one pair of dispersion curves (one acoustical and one optical branch) for each direction in space. In the three-dimensional motions of a diatomic chain, the transverse directions x and y are equivalent. Consequently, only one transverse optic and acoustic dispersion curve is displayed, as they are degenerate (i.e., have the energy or vibrational motion). 

Figure 9.16 Graphical representation of dispersion curves co(k) for the two-atomic chains. Acoustic (A) and optic (O) branches are presented.

---