Longitudinal-acoustic branch

According to Eq. (II.7), co = 0 for k = 0 in the center of BZ 1. With these values Eqs. (II. 1)—(II.3) lead to the relation UA = UB. This means that both sets of atoms vibrate with the same amplitude and in phase (because they have the same sign). A translation of the whole chain results which corresponds to an acoustical wave with X = °°. This is called a longitudinal acoustical branch (LA). 

Fig. A.5-22 BaTiOs. Phonon dispersion relation determined by neutron scattering along the [100] direction in the cubic phase, v is the phonon frequency. LA, longitudinal acoustic branch TA, transverse acoustic branch TO, transverse optical branch. The frequency of the TO branch is lower (softer) at 230 °C than at 430 " C, indicating mode softening...

The temperature dependence of the phonons has also been measured in SmBe by Alekseev (1993) and by Alekseev et al. (1993b). They reported an unusual increase of the longitudinal acoustic branches, but a softening of the longitudinal optic branches with decreasing temperature. In addition, they found an unexpected peak at 20meV at high temperatures, and this was interpreted as a resonant gap mode. This mode apparently disappears at low temperature. 

Figure 10 Experimental X-ray inelastic scattering spectra for methane hydrate (a). The inset shows the crossing of the localized methane vibrational band with the framework longitudinal acoustic branch. Comparison to theoretical prediction obtained from lattice dynamics calculations. Phys. Rev. B, 83, 241403(R) (2011).

Indium nitride has twelve phonon modes at the zone centre (symmetry group Cev), three acoustic and nine optical with the acoustic branches essentially zero at k = 0. The infrared active modes are Ei(LO), Ei(TO), Ai(LO) and Ai(TO). A transverse optical mode has been identified at 478 cm 1 (59.3 meV) by reflectance [6] and 460 cm 1 (57.1 meV) by transmission [24], In both reports the location of a longitudinal optical mode is inferred from the Brout sum rule, giving respective values of 694 cm 1 (86.1 meV) and 719 cm 1 (89.2 meV). Raman scattering of single crystalline wurtzite InN reveals Ai(LO) and E22 peaks at 596 cm 1 and at 495 cm 1 respectively [25],... 

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]. 

Wurtzite ZnO structure with four atoms in the unit cell has a total of 12 phonon modes (one longitudinal acoustic (LA), two transverse acoustic (TA), three longitudinal optical (LO), and six transverse optical (TO) branches). The optical phonons at the r point of the Brillouin zone in their irreducible representation belong to Ai and Ei branches that are both Raman and infrared active, the two nonpolar 2 branches are only Raman active, and the Bi branches are inactive (silent modes). Furthermore, the Ai and Ei modes are each spht into LO and TO components with different frequencies. For the Ai and Ei mode lattice vibrations, the atoms move parallel and perpendicular to the c-axis, respectively. On the other hand, 2 modes are due to the vibration of only the Zn sublattice ( 2-low) or O sublattice ( 2-high). The expected Raman peaks for bulk ZnO are at 101 cm ( 2-low), 380 cm (Ai-TO), 407 cm ( i-TO), 437 cm ( 2-high), and 583 cm ( j-LO). 

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. ... 

Of the three acoustic phonon branches, in each case that with the largest slope is the branch of the longitudinal acoustic phonons (LA), while the two others are the transverse acoustic branches (TA). From this we can derive immediately the longitudinal sound velocity... 

Some interesting and important conclusions were drawn by separating the phonon spectrum in accordance with the polarization of the oscillations [15]. The whole spectrum was divided into six branches, each of which has an almost Gaussian form of the distribution curve g( ). For cubic crystals, these six branches consist of three acoustical branches (one branch of longitudinal and two branches of transverse waves) and three optical branches (one longitudinal and two transverse waves). The acoustical vibrations can be compared with the vibrations of atoms in a unit cell, and the optical vibrations with mutual oscillations of the sublattices in relation to one another. The curves of the density distribution of oscillations in each [Pg.180]

Fig. 9. Magnon dispersion for Tb in the ferromagnetic phase along the three principal axis directions. Symbols along the top of the diagram label the directions in the conventional notation shown in fig. 4. Dashed curves labelled PH are [longitudinal acoustic (LA), transverse acoustic (TA), and transverse optic (TO)] phonon branches which interact with the magnons. (After Mackintosh and Bjerrum-Moller 1972.)...

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)...

Fig. 6.5 Vibrations of a linear atomic structure. Note TO transverse vibrations of then optical branch LO longitudinal vibrations of the optical branch TA transverse vibrations of the acoustic branch LA longitudinal vibrations of the acoustic branch. The frequency and energy of vibrations normally increase in the sequence LA < TA < LO < TO at each wavelength...

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