Phonons, transverse

There are differences between photons and phonons while the total number of photons in a cavity is infinite, the number of elastic modes m a finite solid is finite and equals 3N if there are N atoms in a three-dimensional solid. Furthennore, an elastic wave has tliree possible polarizations, two transverse and one longimdinal, in contrast to only... 

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... 

Observation of absorption bands due to LO phonons in RAIR spectra of thin, silica-like films deposited onto reflecting substrates demonstrates an important difference between RAIR and transmission spectra. Berreman has shown that absorption bands related to transverse optical (TO) phonons are observed in transmission infrared spectra of thin films obtained at normal incidence [17]. However, bands related to LO phonons are observed in transmission spectra of the same films obtained at non-normal incidence and in RAIR spectra. Thus, it is possible for RAIR and transmission spectra of thin films of some materials to appear very different for reasons that are purely optical in nature. For example, when the transmission infrared spectrum of a thin, silica-like film on a KBr disc was obtained at normal incidence, bands due to TO phonons were observed near 1060,790,and450cm [18]. 

CuZn. We have investigated the phonon dispersion of the B2 phase. Our result compares well with the experimental findings marked as diamonds in Fig. 7. Similar to the fee FcsNi phase, a soft transversal mode is detected in bcc CuZn. This [110]... 

T.R. Finlayson, D. Donovan, J.Z. Larese, and H.G. Smith, Studies of transverse phonon modes in premartensitic indium-thalliun. alloys. Mater. Sci. Forum 27/28 108 (1988). 

The overall specific heat of a polymer is given by a combination of the various contributions to the specific heat of longitudinal and transversal phonons. At temperatures below 1K, the linear contribution due to the TLS must be added. 

When metals have Raman active phonons, optical pump-probe techniques can be applied to study their coherent dynamics. Hase and coworkers observed a periodic oscillation in the reflectivity of Zn and Cd due to the coherent E2g phonons (Fig. 2.17) [56]. The amplitude of the coherent phonons of Zn decreased with raising temperature, in accordance with the photo-induced quasi-particle density n.p, which is proportional to the difference in the electronic temperature before and after the photoexcitation (Fig. 2.17). The result indicated the resonant nature of the ISRS generation of coherent phonons. Under intense (mJ/cm2) photoexcitation, the coherent Eg phonons of Zn exhibited a transient frequency shift similar to that of Bi (Fig. 2.9), which can be understood as the Fano interference [57], A transient frequency shift was aslo observed for the coherent transverse optical (TO) phonon in polycrystalline Zr film, in spite of much weaker photoexcitation [58],... 

Note that dra(t)/dt = [H,ra]=(l/ma)[pa-qaA(ra)] and, consequently, the first term in (69) represents the kinetic energy of the system of particles in the presence of the transverse electromagnetic field. Note the analogy between this representation and the dynamical solute-solvent coupling of section 2.6 where the optical phonons are equivalent to electromagnetic photons of low frequency (the acoustical phonons are related to sound waves). 

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]. 

The calculated Rayleigh mode (SJ, the lowest lying phonon branch, is in good agreement with the experimental data of Harten et al. for all three metals. Due to symmetry selection rules the shear horizontal mode just below the transverse bulk band edge can not be observed by scattering methods. The mode denoted by Sg is the anomalous acoustic phonon branch discussed above. Jayanthi et al. ascribed this anomalous soft resonance to an increased Coulomb attraction at the surface, reducing the effective ion-ion repulsion of surface atoms. The Coulomb attraction term is similar for all three metals... 

At high temperatures above Tb 617 K PMN behaves Hke all other simple perovskites. The dynamics of the system is determined by the soft transverse optical (TO) phonon which exhibits a normal dispersion and is imderdamped at all wave vectors. Below Tb, in addition to the soft mode—which becomes overdamped—a new dielectric dispersion mechanism appears at lower frequencies which can be described by a correlation time distribution function /(t). 

According to the concept of the displacive-type ferroelectric phase transition [10], an increase in the dielectric constant corresponds directly to the softening of the IR-active transverse phonon. When the crystal can be regarded as an assembly of the vibrators of normal coordinates, the soft phonon... 

Figure 7.17 Crystal unit cell structure of Bi and the orientation of the (longitudinal) and (transverse) phonon motions. Reproduced from Ref. [42] with permission from Nature Publishing Group.

Dislocations move when they are exposed to a stress field. At stresses lower than the critical shear stress, the conservative motion is quasi-viscous and is based on thermal activation that overcomes the obstacles which tend to pin the individual dislocations. At very high stresses, > t7crit, the dislocation velocity is limited by the (transverse) sound velocity. Damping processes are collisions with lattice phonons. 

We did not differentiate between the various modes of vibration (longitudinal, transversal, acoustical, optical) for the sake of simplicity. The vibrational states in a crystal are called phonons. Figure 5-2 illustrates the collective, correlated transversal vibrational motion of a linear elastic chain of particles. 

Figure 5-2. Phonon in a crysial, schematic. Transversal motion of a linear atomic chain, described by two different waves with wavelength >2a and wavelength <2a, a = (average) distance of atoms.

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