Brillouin transverse

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]. 

Outside of a small region around the center of the Brillouin zone, (the optical region), the retarded interactions are very small. Thus the concept of coulombic exciton may be used, as well the important notions of mixure of molecular states by the crystal field and of Davydov splitting when the unit cell contains many dipoles. On the basis of coulombic excitons, we studied retarded effects in the optical region K 0, introducing the polariton, the mixed exciton-photon quasi-particle, and the transverse dielectric tensor. This allows a quantitative study of the polariton from the properties of the coulombic exciton. 

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...

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 anions located in centrosymmetrical cavities lie slightly above or below the molecular planes. This structure results in a dimerization of the intermolecular distance (overlap) with a concomitant splitting of the HOMO conduction band into a filled lower band separated from a half-filled upper (holelike) band by a gap at 2k% called the dimerization gap which is shown in Fig. 6 at the point X of the new Brillouin zone. However, on account of the transverse dispersion, this dimerization gap does not lead to a genuine gap in the density of states as shown from the extended-Hiickel band calculation (Fig. 7). The only claim which can be made is that these conductors have a commensurate band filling (3/4) coming from the 2 1 stoichiometry with a tendency towards half... 

A rough, zero-order, estimate of the extent to which a Fermi liquid description would be viable in the normal phase is provided by the scale of given by band calculations. Consider the temperature range Tthermal fluctuations are suffieiently weak to lower the uncertainty on the transverse band wave vector to a range of values l/dj, that is small compared to the size of Brillouin zone. The band wave vector is therefore a good quantum number so the transverse band motion and the curvature of the Fermi surface are coherent. Otherwise, when one has which is large enough for... 

We shall see below that the transport properties above are consistent with the picture of Luttinger chains in (a - b) planes. The exponent a = 0.7 derived from the constant volume transverse data leads to K = 0.22. This value of allows in turn a prediction for the constant volume T-dependence for p . The only scattering process through which electron-electron collisions can contribute to resistivity in this 1-D electron gas occurs when the total momentum transfer is commensurate with a Brillouin zone wave vector. For the situation of a 1/4-filled 1-D band which is likely to apply to (TMTSFljPF as the dimerization can be forgotten in first approximation (A ,[Pg.254]

Applying the concept of the Peierls transition to TTF-TCNQ 2kp is evaluated as 0.295 b. It leads to the band filling up to 29.5% of the first Brillouin zone. The average valences of TTF and TCNQ are expected as + 0.59 and - 0.59, respectively. This is verified by X-ray photoelectron spectroscopy (XPS) [61]. The upper panel of Fig. 15 shows temperature dependence of the transverse component of the wave vector Q of CDW. Generally the transverse components... 

Z) = 0.142 nm is the interatomic distance in graphene. Integration and summation in Eq. (3) is taken over the first Brillouin zone of the CNT which is the set of discrete lines due to the quantization of the transversal quasi-momentum [4]. 

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. 

Taking into account the hexagonal crystalline structure and transverse quantization of the first Brillouin zone, it can easily be obtained that in zigzag CN in the tight-binding approximation the dispersion law of 7t-electrons [3] and matrix elements Rab are as follows ... 

The spectrum of amorphous bisphenol-A polycarbonate showing both longitudinal and transverse Brillouin peaks is shown in Figure 2. 

The longitudinal Brillouin spectrum appears when both the incident and scattered light are polarized vertically with respect to the scattering plane (/vv) The transverse Brillouin spectrum results when the incident polarization is perpendicular to the scattered polarization (Ihv =1vh). 

When Ts < < 10" sec, the transverse phonon velocity is imaginary and no transverse Brillouin peaks are observed. However, shear fluctuations do occur and they can couple with other modes of motion such as molecular reorientation. The spectrum that results is given by ... 

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