Transverse phonons, dispersion

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... 

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]... 

The further solution ejj - n2 = 0 describes the dispersion of another set of strictly transverse phonons which do not degenerate with the ordinary phonons because of the anisotropy of the crystal. 

LiI03 is a uniaxial hexagonal crystal (factor group C6). Vibrations of species A, E, and E2 are allowed in the Raman effect, but only A and E, are infrared-active, therefore polariton dispersion is expected for the transverse phonons of these two species. The phonon and polariton spectra were investigated by Claus 26>27> and Otaguro et al. 28>29). Here we want to show two series of spectra recorded by Claus. 

Figure 9-19 Phonon dispersion relation (angular frequency vs. relative wave vector) for the three-stripe phase of CH4 on the external surface of a bundle. LI, L2, and L3 are longitudinal branches, i.e., molecular motion parallel to the groove. The dotted curve is the result for a ID adsorbate at the same density. The remaining curves correspond to the dispersion relation of transverse modes. (Adapted from Ref. [89].)...

Fig. 3.1. Phonon dispersion curves of diamond along the main symmetry directions calculated from a Born-von Karman model fitted to neutron scattering experimental data (after [50]). The frequencies are expressed in wavenumber v = uj/2itc. Along the [110] directions (E), the modes are neither purely longitudinal nor transverse, and three branches exist for each category. Copyright 1992 by the American Physical Society...

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.)...

Figure 33. Surface phonon dispersion for W(OOl) in the FM portion of the SBZ showing the measured Rayleigh wave (R) and longitudinal (L) modes. The data in the upper panel were obtained at 1200 K, while in the lower panel the data shown by open circles were obtained at 500 K and those represented by closed circles were obtained at 300 K. The edges of the transverse acoustic (TA) and longitudinal acoustic (LA) bulk bands are given by the hatched lines. The vertical lines in the lower panel denote the widths in the energy transfer distributions of these points. (Reproduced from Figs. 10 and 13 of Ref. 110, with permission.)...

Figure 34. Surface phonon dispersion for 2H-TaSe2. The HAS data are shown as solid circles except for weak points which appear in the TOP spectra as hybridized longitudinal modes that are shown as crosses. All the data were obtained at 60 K, well into the low-temperature phase. The calculated striped and shaded regions, corresponding to transverse and longitudinal polarizations respectively, are the slab-adapted bulk phonon bands, while the solid line is a calculation for the Rayleigh wave based on the Dispersive Linear Chain Model (shown schematically in Fig. 35). The open circles at g = 0 are from Raman scattering experiments. (This figure has been corrected from Fig. 23 in Ref. 54.)...

The transition metal niobium attracts a lot of attention of researchers because of its relatively high superconducting transition temperature. It turned out that niobium shows a number of pronounced anomalies in the phonon dispersion, which are also typical for vanadium and tantalum. The anomalies in [00 ], [0 ] directions appear as a crossover of the longitudinal and transverse branches at = 0.7 or = 0.3 as well as additional maxima and minima. 

Fig. 31. Phonon dispersion curves for CcPds al room temperature. Solid symbols are longitudinal modes, open symbols transverse modes. In the [110] direction A, transverse polarized in (110) plane O, transverse polarized in (100) plane. Results of Born-von Karman fit including breathing term are shown as solid and dashed lines (the latter for A data points). Dotted lines if breathing term is neglected (Severing el al. 1988).

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...

Fig. 5.2-58 Surface phonon dispersion curves for Si(lll) 2x1 measured by HATOF. Energies at symmetry points X, 10.2 and 11.1 meV S, 10.5 and ll.bmeV. The flat phonon mode at 10.5 meV is associated with the 2 x 1 reconstruction. The surface mode couples with transverse bulk phonons near the center of the SBZ, giving rise to considerable broadening. The shaded area corresponds to the width ofthe 10.5 meV peak [2.90], The energy of the optical mode (not shown in the figure) is 56.0 meV [2.91]...

Much more interesting is the spatial extent of the [100] transverse forces, and interactions up to the fifth neighbors were included for calculation of the phonon dispersion in Fig. 5.3.3. It is argued in Ref. 39 that the range of these forces is m = 5 2 -the error margin partly resulting from the present use of less precise local potentials. The most sensitive part of the transverse dispersion is the TA-branch a fairly steep slope at the origin (elastic constant becomes a flat dispersion near the... 

Phonon dispersion curves in the directions A, 2, and A ([ 00], [0 ], and calculated using an overlap shell model and compared with the experimental data [3, 4] for r-L ([%, ]) are shown in Fig. 92. The labels II and indicate whether the polarization of the transverse branches In the [0 ] direction Is parallel to [0T1] or to [100], respectively, Zeyher, Kress [8]. Phonon dispersion curves along r-L, r-X, and T-K calculated on a nine-parameter breathing shell model are also in good agreement with the Raman data for r-L [7], cf. [6]. 

Solving this equation for o) and plotting o> versus fc provides a dispersion curve for NaCl as shown in Figure 24.5. Notice that the curve has two branches the upper branch is the phonon dispersion relation for the longitudinal mode and approaches >l at fc = 0 while the lower branch is the dispersion curve for the transverse mode, which approaches wj as fc increases. No frequencies can propagate between < x and o>ij which causes an energy gap in this region. 

Figure 2.17 The phonon dispersion relations for (a) GaN and (b) Si. TA, LA, LO, and TO refer to transverse acoustic, longitudinal acoustic, longitudinal optical and transverse optical phonons, respectively. Each of these represents a particular vibrational mode. Longitudinal modes run along bonds as in Figure 2.16, while for transverse modes the vibration velocity is perpendicular to the bonds. There are two transverse modes because there are two axes perpendicular to a bond direction. Figures after Levinshtein, Rumyantsev, Sergey, and Shur, Reference [5], p. 27 and 184, respectively. This material is used by permission of John Wiley Sons Inc.

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). 

At 795 cm 1 LiI03 has a transverse A phonon. Fig. 5 shows the shift of the Raman line towards lower wave numbers for decreasing angles between the directions of the incident and scattered light. Because the wave vector triangle lies in the isotropic Ary-plane, k is always perpendicular to the optical axis and no directional dispersion is to be expected, therefore the shift of the line is due only to the variation of the magnitude of the wave vector. 

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