Transverse acoustic mode

An clastic constant predicted by using LCAO bands and the special points method, compared with experiment also, calculated and experimental values of a zone-boundary transverse acoustic mode frequency, tor, ( ). to be discussed in Chapter 9. 

The calculation of vibration spectra in terms of force constants is similar to the calculation of energy bands in terms of interatomic matrix elements. Force constants based upon elasticity lead to optical modes, as well as acoustical modes, in reasonable accord with experiment, the principal error being in transverse acoustical modes. The depression of these frequencies can be understood in terms of long-range electronic forces, which were omitted in calculations tising the valence force field. The calculation of specific heat in terms of the vibration spectrum can be greatly simplified by making a natural Einstein approximation. 

We could have used particular frequencies instead of the elastic constants to determine the force constants in the model that alternative is of some interest, We have used, for reasons to be discussed, the measured transverse optical frequency at /( = 0 and the transverse acoustical mode at k — Inja to obtain alternate values of C,) and C, which are listed in Table 9-1. The differences from the values given... 

Transverse acoustical mode, /< = In/a Transverse optical mode, /c = 0 0 C ... 

The displacement of atoms in the transverse acoustical mode at the zone boundary. The x-axis is parallel to k in a [100] direction. 

Figure 2. Phonon pictnre of the origin of the incommensurate phase transition in qnartz. The two plots show the a dispersion curves for the transverse acoustic mode (TA) and the soft optic RUM, at temperatures above (left) and close to (right) the incommensurate phase transition. The RUM has a frequency that is almost constant with k, and as it softens it drives the TA mode soft at an incommensurate wave vector owing to the fact that the strength of the coupling between the RUM and the acoustic mode varies as k. ...

J. Tomkinson, S.F. Parker, D.A. Braden B.S. Hudson (2002). Phys. Chem. Chem. Phys., 4, 716-721. Inelastic neutron scattering spectra of the transverse acoustic modes of the normal alkanes. 

Below 600 cm are the V5 in-plane C—C—C bending modes (longitudinal acoustic modes, LAMs), the V9 out-of-plane C—C—C bending modes (transverse acoustic modes, TAMs) and the 12 lattice modes. These comprise in-phase and out-of-phase pairs of librations about the three axes, giving six modes, three optic translation and three... 

At very low energy, many amorphous materials show a Boson peak , so-called because the temperature dependence of its intensity roughly scales with the Bose-Einstein distribution. Amorphous silica is no exception and has a peak at 40 cm. The origin of this has been controversial but in silica it appears to be related to either transverse acoustic modes or torsions of the Si04 tetrahedra with respect to one another [15]. 

The vibrational contribution to the polarizability of poly acetylene amounts to 10% of its electronic counterpart and mainly originates from low-frequency transverse acoustic modes (TAMs) [73]. It tends towards zero in the optical regime. The nature of the frequency dispersion, therefore, constitutes a major difference between electronic and vibrational (hyper)polarizability contributions. The response time is also different the electronic processes are faster and their response time is of the order of 1 fs, whereas vibrational phenomena are slower and have response times ranging between 10 and 100 fs as a function of the associated vibrational normal mode frequency. 

Acoustic Natural Frequencies. The natural frequencies of transverse acoustic modes in a cylindrical shell can be calculated as follows [122] ... 

Tommasini et al have developed a model for calculating the Transverse Acoustic Modes of aU-/ra/w-polyenes and discuss their relevance to the static vibrational polarizability. 

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... 

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