Longitudinal Brillouin

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

Bulk silicon is a semiconductor with an indirect band structure, as schematically shown in Fig. 7.12 c. The top of the VB is located at the center of the Brillouin zone, while the CB has six minima at the equivalent (100) directions. The only allowed optical transition is a vertical transition of a photon with a subsequent electron-phonon scattering process which is needed to conserve the crystal momentum, as indicated by arrows in Fig. 7.12 c. The relevant phonon modes include transverse optical phonons (TO 56 meV), longitudinal optical phonons (LO 53.5 meV) and transverse acoustic phonons (TA 18.7 meV). At very low temperature a splitting (2.5 meV) of the main free exciton line in TO and LO replicas can be observed [Kol5]. 

Fig. 26. Temperature dependence of the hyperfine magnetic field and of the longitudinal relaxation parameter in NpSns. The J = 1/2 Brillouin curve for the temperature dependence of is shown. (Gal et al, )...

Recently Maret et al. (JL) have observed the longitudinal acoustic mode in oriented DNA fibres and films in a Brillouin scattering experiment. They observed the largest acoustic velocity for the driest samples, smallest for wet samples and at all times the observed velocity was larger than that for water itself. We have assumed that the velocities for propagation parallel to the helix axis are characteristic of acoustic modes in the DNA double helix and have used these values along with previously refined valence force field parameters (2,3) to fit the non-bonded force constants for the double helix. 

Both A - and Ei-modes are Raman and IR active. The two nonpolar E2-modes E and E are Raman active only. The Bi-modes are IR and Raman inactive (silent modes). Phonon dispersion curves of wurtzite-structure and rocksalt-structure ZnO throughout the Brillouin Zone were reported in [106-108]. For crystals with wurtzite crystal structure, pure longitudinal or... 

The agreement between fee bulk modulus deduced from Brillouin scattering measurements and fee ADX results is very good. The determination of fee elastic moduli by ultrasonics was made by fee measurement of surface acoustic wave velocities on thin films [22], The second ultrasonics experiment was made on sintered powder, by measuring fee longitudinal and transverse sound velocity at ambient and under uniaxial compression. From feat, fee bulk modulus and its pressure derivative were deduced, but this result seems to be quite imprecise. The ultrasonics experiment on thin films gives rise to a very small difference in fee bulk modulus (5%), but fee ADX or Brillouin determination should be utilised for preference. 

A Brillouin spectrum of commercial Mylar film is shown in Figure 8. The longitudinal (L) and transverse (T) Brillouin peaks are easily seen. 

Figure 8. Brillouin spectrum of commercial Mylar film showing longitudinal (L), transverse (T), and unknown ( ) Brillouin peaks...

Commercial Mylar film is partially crystalline and mechanically anisotropic. The longitudinal Brillouin splitting is observed to be greater than that for quenched amorphous PET and to be different for films oriented parallel and perpendicular to the film-roll edge. Longitudinal Brillouin splittings for Mylar and for amorphous PET are plotted vs. temperature in Figure 11. The apparent value of Tg also is elevated for the Mylar film from 70° to 80 °C. 

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. ... 

The G-L functional (23) has to be combined with appropriate cut-off wave-lengths. These wave-lengths need not be the same in d longitudinal and A = = d—d transverse directions. Let us denote them with Qf[ and Q ), respectively. The cut-offs describe that portion of the Brillouin zone to which harmonic part of the expansion (23) applies. As will be shown below and as argued previou-usiy5.i2.i9> question 0f cut-offs is quite important and thus requires some discussion. 

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