Dispersion curves, acoustic phonon

Figure 3 Acoustic phonon dispersion curves and phonon density of states of NbC and ZrC (21). Depressed parts, as indicated by arrows, are formed on the dispersion curve of NbC, which moves the state density of the compound to the low-energy side.

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-... 

Figure 2. Phonon pictnre of the origin of the incommensurate phase transition in qnartz. The two plots show the a dispersion curves for the transverse acoustic mode (TA) and the soft optic RUM, at temperatures above (left) and close to (right) the incommensurate phase transition. The RUM has a frequency that is almost constant with k, and as it softens it drives the TA mode soft at an incommensurate wave vector owing to the fact that the strength of the coupling between the RUM and the acoustic mode varies as k. ...

Figure 3. Schematic Illustration of dispersion curves of an acoustic phonon, a band electron, and the SWAP. The Incident phonon -q Is scattered as q2. (Reproduced with permission from reference 5. Copyright 1985 Nljhoff.)...

Collins, D.R., W.G. Stirling, C.R.A. Catlow, and G. Rowbotham. 1993. Determination of acoustic phonon dispersion curves in layer silicates by inelastic neutron scattering and computer simulation techniques. Phys. Chem. Miner. 19 520-527. 

Fig. 10. (a) A ID lattice of rigid diatomic molecules. The dispersion curve for acoustic phonons that result from translations runs from zero frequency to a cut-off termed the Debye frequency. The vibron has no dispersion. Adding flexibility to the molecules introduces dispersion in the vibron state and narrows the gap between vibrons and phonon as shown at right, (b) A 3D lattice of flexible naphthalene molecules. The 12 phonons overlap significantly with the two or three lowest energy vibrations, termed doorway modes. Doorway modes are coupled to both phonons and higher frequency vibrations associated with bond breaking. Adapted from ref. [91]. 

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 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.)...

Fig. 6.11 Calculated exciton band structure of the T] state of anthracene for three directions in the reciprocal lattice. The experimentally-determined numerical values for the triplet resonance interactions are given in the text (from [19]). The dashed curve is a typical dispersion relation for acoustic phonons in anthracene [28].

Fig. 30. Mixed-mode dispersions in ferromagnetic PrAlj at T = 4.4 K for the [001] direction. Full points represent experimental data from Purwins et al (1976), full curves are calculations by Aksenov et al. (1981), TA stands for transverse acoustic phonon, and and are exdton branches with different polarizations.

Figure 1. The phonon dispersion curve for quartz at high pressure, showing the vibrational frequencies in the (110) direction and the softening of the acoustic mode at k (0.3333, 0.3333,0).

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. 

Here, the simple dispersion (oiq) v g was used for the acoustic phonons (if whole molecules vibrate with respect to each other, like the change of the stacking distance in a stack one speaks of acoustic phonons, because they have a long wavelength comparable to those of acoustic waves) and the sound velocity v, is determined by the relation o, = Ci/p), where c, is the longitudinal elastic constant and p is the mass density. One should remark that this very simple linear dispersion relation (o q) = v,g is not necessarily correct. With the help of the FG method described in Section 9.1 one can obtain more accurate dispersion curves. Equation (9.48) can now be used to calculate the charge carrier mobilities and free paths, defined in this case hy p= e xlm ) and A = (t ), respectively, where... 

Figure 21.1 Phonon dispersion curves. (o+ (f) optical, (o q) acoustic, co q) monoatomic (from Eq. 31). The intersections with the straight line m = cq are discussed in Sections 21.4 and 21.6. Parameters a = 0.6, n = 2, dc = 1.3.

Fig. 54. The longitudinal acoustic phonon dispersion relations along the [100] axis of a- and y-Ce (Stassis 1.0 1988). The splitting of the dispersion curve for a-Ce is the result of mode mixing.

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