Dispersion relations acoustic modes

Figure 8.7 Experimental dispersion relations for acoustic modes for lead at 100 K [2], Reproduced by permission of B. N. Brockhouse and the American Physical Society.

Perylene The two different crystalline phases, a perylene and perylene (see Chap. 2, Fig. 2.12) differ strongly also in their dynamic properties a perylene has four molecules or two dimers per unit ceU. From this, 24 internal modes result, 21 optical and three acoustic. They have also been observed and identified by inelastic neutron diffraction [17] and by Raman scattering [18]. Their spectrum has a width of about 4 THz, again similar to the cases of naphthalene and anthracene. For a perylene, the model treated in Sect. 5.6 again yields satisfactory theoretical dispersion relations. The low-energy internal modes are torsional (twisting) and butterfly modes. Their spectrum overlaps with that of the external modes. 

We next turn our attention to the dispersion relation of another branch of (u ), presented as dashed line in Fig. 5.3(a). This branch is well separated from the acoustic branch at all wavevectors, and it is apparent from the figure that this mode does not vanish in the A — 0 limit. From an analogy with solid state physics, this mode will be re-... 

The resultant low and high peak frequencies (dispersion relations) of the longitudinal current spectra are shown as solid lines in Figs. 5.6(a) and 5.6(b), respectively. It is seen from Fig. 5.6(a) that the dispersion curve of the low-frequency acoustic mode is very similar to that of monatomic fluids. The dispersion behavior of the optical mode presented... 

Figure 5.6. The dispersion relation of the (a) acoustic and (b) optical modes as evaluated from the peak positions in the longitudinal current spectra (solid lines), by diagonalizing ((u ) (lower dashed lines), and by diagonalizing l) (upper dashed lines) [54].

The dispersion relation for such collective modes was first measured by Rainford and Houmann (1971) for dhcp Pr and later studied in more detail by Houmann et al. (1975) (see also Bak (1975)). Figure 7.31 shows the excitation dispersion relations for the hexagonal sites in Pr at 6.4 K. The hexagonal sites show four branches corresponding to two acoustic and two optic modes (there are two such sites per unit cell) and to excitations from the singlet (0)... 

Fig. 7.31. Dispersion relations for magnetic excitations propagating on the hexagonal sites of dhcp Pr at 6.4 K. The circles represent acoustic-type modes and the squares optic-type modes (after Houmann et al., 1975).

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. 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. 54. The longitudinal acoustic phonon dispersion relations along the [100] axis of a- and y-Ce (Stassis 1.0 1988). The splitting of the dispersion curve for a-Ce is the result of mode mixing.

Dispersion relaKons for NaCl in the reduced zone scheme for the various modes of oscillation longitudinal optical (LO), transverse optical (TO), longitudinal acoustic, and transverse acoustic. The curves were computed from Equation 16.19 using the elastic coefficients for NaCl given in Table 16.1. It should be noted that the a in this figure is the distance between the individual ions. Usually this dispersion relation is shown plotted between —-ir/fl and -ir/fl where a is taken as the distance between ion pairs. 

As we have seen in Sect.2.2.5, the Einstein model represents the optical modes well, especially if their dispersion is weak, while the Debye model is more adapted to the acoustical modes. For this reason, the Debye model is often applied only to the acoustic modes and the Einstein model only to the optic modes. The corresponding approximations for the dispersion relation are shown in Fig.3.9d. For the density of states, we write in this case... 

Surface modes can be clearly identified in the dispersion relations ft>(qn) when they appear in regions where no bulk bands appear. Similar to the identification of surface electronic states, the projected bulk modes form the bulk phonon bands in the surface Brillouin zone, as shown in Figure 9.46. In the bulk case, there are three acoustic phonon bands and 3(S-1) optical phonon bands, with S as the number of atoms in the primitive unit cell of the bulk crystal. Along high-symmetry directions in the bulk, such as the (100) or (111) directions in cubic crystals, the phonons can be classified either as transverse or longitudinal, depending on whether or not their displacements are perpendicular or parallel to the direction of the 3D wave vector. 

The dispersion relation for this doubly degenerate transverse acoustical mode is also shown in Figure 2.9. The relation (2.39) indicates that the dispersion curve... 

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.

The nondimensionalized frequencies are related to linear and angular frequencies by equation 3.36. The conversion factor from linear frequencies in cm to undimension-alized frequencies is chik = 1.4387864 cm (where c is the speed of hght in vacuum). Acoustic branches for the various phases of interest may be derived from acoustic velocities through the guidelines outlined by Kieffer (1980). Vibrational modes at higher frequency may be derived by infrared (IR) and Raman spectra. Note incidentally that the tabulated values of the dispersed sine function in Kieffer (1979c) are 3 times the real ones (i.e., the listed values must be divided by 3 to obtain the appropriate value for each acoustic branch see also Kieffer, 1985). 

The dispersion relationships of lattice waves may be simply described within the first Brillouin zone of the crystal. When all unit cells are in phase, the wavelength of the lattice vibration tends to infinity and k approaches zero. Such zero-phonon modes are present at the center of the Brillouin zone. The variation in phonon frequency as reciprocal k) space is traversed is what is meant by dispersion, and each set of vibrational modes related by dispersion is a branch. For each unit cell, three modes correspond to translation of all the atoms in the same direction. A lattice wave resulting from such displacements is similar to propagation of a sound wave hence these are acoustic branches (Fig. 2.28). The remaining 3N-3 branches involve relative displacements of atoms within each cell and are known as optical branches, since only vibrations of this type may interact with light. 

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