Dispersion curves acoustic branch

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. 

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

Brillouin scattering provides information about the acoustic branches of the dispersion curves of the material under study. The measured frequency shift of the radiation is equal to that of the phonon under consideration (EQN (1)), and its wave vector is deduced from EQN (2), so the sound velocity may be calculated by ... 

As an example. Fig. 3 plots the phonon dispersion curves for three highly S5mimetric directions in the Brillouin zone of the perfect ZnO crystal. Comparison of the theoretical and experimental frequencies shows good agreement for the acoustic branches. The densities of phonon states of the perfect ZnO crystal calculated by integrating over the Brillouin zone are displayed in Fig. 4. Comparison of the results of our calculation and a calcu-... 

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. ... 

To understand the reasons for the differences between the experimental spectra and that predicted by the models, it is necessary to consider the data from which the models are built. The only way to directly measure the dispersion curves of a material is by coherent INS spectroscopy. For hydrogenous polyethylene, this method fails because the background caused by the incoherent scattering from hydrogen completely swamps the coherent signal ( 2.1.1). For perdeuterated polyethylene, the larger coherent and smaller incoherent cross sections of deuterium (see Appendix 1) has allowed the vs acoustic branch to be mapped by coherent INS [11]. 

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. 

In Fig. 5.3 we report the results of the diagonalization of Solid line in Fig. 5.3(a) represents the dispersion curve corresponding to the acoustic branch. It is seen that the dispersion behavior of the acoustic mode is very similar to that of a monatomic system. The k Q limit of the eigenfrequency of this mode is given by [5]... 

Fig. 2 Normalized dispersion relation for pSi superlattiees showing composite data from large set of superlattice samples obtained by Brillouin scattering experiments. The solid dashed) curves are theoretical longitudinal (transverse) modes obtained from the Rytov model. The horizontal hne identifies a locahzed surface mode lying within die phononic bandgap of the bulk longitudinal mode, trapped at the surface of the phononic crystal. Full details of die samples and acoustic branch identification in Parsons and Andrews (2012) (Reprinted with permission from Journal of applied physics by American Institute of Physics, Copyright 2012, American Institute of Physics)...

Figure 3-3. Dispersion curves of crystalline orthorhombic polyethylene with two molecules per unit cell (from [49]). Comparison with Figure 3-1 shows the splitting of the frequency branches and the shape of the acoustical branches at p 0 (see text). Attention should be paid to the fact that the frequencies and the shape of the dispersion curves shown in Figure 3-1 and in this figure may differ because they have been calculated with two different force fields.

Figure 3-6. Example of lattice dynamical calculations on ir-bonded tridimenional crystals with short range interactions. Dispersion curves for cubic diamond along the F — (K) —> X symmetry direction. Experimental points from neutron-scattering experiments dispersion curves from least squares frequency fitting of a six parameters short range valence force field (from [60]). The Raman active phonon is the triply degenerate state indicated with F j near 1300 cm Notice that at k 0 the degeneracy at F is removed because of the lowering of the symmetry throughout the whole BZ. Notice also the three acoustic branches for which v 0 at k F.

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. 

The frequency-dispersion curves of the acoustic branches (Vj and Vg) of polyethylene are shown in Fig. III.1. For the phase difference of 90°, the vibrational frequencies reach maxima and give rise to the frequency-distribution peaks near 550 and 200cm . These peaks were in fact observed by Danner, Safford, Boutin, and Berger (1964), Myers, Donovan, and King (1965) [to be discussed in detail in Section VII]. The vibrational modes of the two acoustic branches for the phase difference of 90° are schematically shown in Fig. III.4. 

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. 

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). 

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