Phonons acoustic modes

Molecules vibrate at fundamental frequencies that are usually in the mid-infrared. Some overtone and combination transitions occur at shorter wavelengths. Because infrared photons have enough energy to excite rotational motions also, the ir spectmm of a gas consists of rovibrational bands in which each vibrational transition is accompanied by numerous simultaneous rotational transitions. In condensed phases the rotational stmcture is suppressed, but the vibrational frequencies remain highly specific, and information on the molecular environment can often be deduced from hnewidths, frequency shifts, and additional spectral stmcture owing to phonon (thermal acoustic mode) and lattice effects. 

In addition to the acoustical modes and MSo, we observe in the first half of the Brillouin zone a weak optical mode MS7 at 19-20 me V. This particular mode has also been observed by Stroscio et with electron energy loss spectrocopy. According to Persson et the surface phonon density of states along the FX-direction is a region of depleted density of states, which they call pseudo band gap, inside which the resonance mode MS7 peals of. This behavior is explained in Fig. 16 (a) top view of a (110) surface (b) and (c) schematic plot of Ae structure of the layers in a plane normal to the (110) surface and containing the (110) and (100) directions, respectively. Along the (110) direction each bulk atom has six nearest neighbors in a lattice plane, while in the (100) direction it has only four. As exemplified in Fig. 17, where inelastic... 

In the hydrate lattice structure, the water molecules are largely restricted from translation or rotation, but they do vibrate anharmonically about a fixed position. This anharmonicity provides a mechanism for the scattering of phonons (which normally transmit energy) providing a lower thermal conductivity. Tse et al. (1983, 1984) and Tse and Klein (1987) used molecular dynamics to show that frequencies of the guest molecule translational and rotational energies are similar to those of the low-frequency lattice (acoustic) modes. Tse and White (1988) indicate that a resonant coupling explains the low thermal conductivity. 

The condition for observation is that the phonon coherence length is larger than the layer thickness. Low frequency acoustic modes fulfill this condition because they are an in-phase motion of a large number of atoms and are not strongly influenced by the disorder-instead reflecting the average bulk elastic properties of the materials. 

For the particular case of longitudinal optical modes, we found in Eq. (9-27) the electrostatic electron-phonon interaction, which turns out to be the dominant interaction with these modes in polar crystals. Interaction with transverse optical modes is much weaker. There is also an electrostatic interaction with acoustic modes -both longitudinal and transverse which may be calculated in terms of the polarization generated through the piezoelectric effect. (The piezoelectric electron phonon interaction was first treated by Meijer and Polder, 1953, and subsequently, it was treated more completely by Harrison, 1956). Clearly this interaction potential is proportional to the strain that is due to the vibration, and it also contains a factor of l/k obtained by using the Poisson equation to go from polarizations to potentials. The piezoelectric contribution to the coupling tends to be dominated by other contributions to the electron -phonon interaction in semiconductors at ordinary temperatures but, as we shall see, these other contribu-... 

As far as the theoretical description is concerned, the confined acoustic phonons in an elastic sphere were first theoretically studied by Lamb [29]. He derives two types of confined acoustic modes, spheroidal and torsional modes. The frequencies of these two modes are proportional to the sound velocities in particles and inversely proportional to the particle size. The spheroidal mode is characterized by the quantum number / > 0, while the torsional modes are characterized by / > 1. From the symmetry arguments, Raman-active modes are spheroidal modes with I = 0 and 2. The I = 0 mode is purely radial with spherical symmetry and produces totally polarized spectra, while the Z = 2 mode is quadrapolar and produces partially... 

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. ...

In the framework of the model, the difference Ei — E is within the acoustic phonon bandwidth, and that is why Skinner and Trommsdorff [21] reasonably assumed that the longitudinal acoustic modes could influence the tunneling proton transfer. The interaction potential (22) was chosen in the form of the deformation potential approximation (see Ref. 74)... 

Here a is a dimensionless constant, 5p(R) is the density fluctuation of the medium at the position R (the center of symmetry of the benzoic acid dimer), 0)D is the Debye frequency, and N is the number of acoustic modes, cot = 7 sound k, (bk) is the Bose operator of creation (annihilation of a acoustic phonon with the wave vector k). In the localized representation we have... 

Fig. 3.3. Absorption of InP in the two-phonon absorption region at 300K (full line) and 20K (dashed line). The practically absorption-free domain between groups I and II correspond to the phonon gap between the optic and acoustic modes. The very strong one-phonon TO absorption is at 304cm 1 [97]. Copyright Wiley-VCH Verlag GmbH Co. KGa. Reproduced with permission...

Below the Debye temperature, only the acoustic modes contribute to heat capacity. It turns out that within a plane there is a quadratic correlation to the temperature, whereas linear behavior is observed for a perpendicular orientation. These assumptions hold for graphite, which indeed exhibits two acoustic modes within its layers and one at right angles to them. In carbon nanotubes, on the other hand, there are four acoustic modes, and they consequently differ from graphite in their thermal properties. StiU at room temperature enough phonon levels are occupied for the specific heat capacity to resemble that of graphite. Only at very low temperatures the quantized phonon structure makes itself felt and a linear correlation of the specific heat capacity to the temperature is observed. This is true up to about 8 K, but above this value, the heat capacity exhibits a faster-than-Unear increase as the first quantized subbands make their contribution in addition to the acoustic modes. 

It is easily seen that, for the acoustical modes of lattice phonon vibration, both types of atoms vibrate together (the displacement is in the same direction - which may account for the lower energy required). In contrast, in the higher energy optical modes, each type of atom vibrates together, but in opposite direction to the other, i.e.- the displacement is opposite... 

In parallel with these experimental studies of PL for SQD suspensions, the role of the observed linewidth broadening has been examined. In particular, the linewidth broadening due to acoustic-phonon-assisted transitions is expected [1] to contribute to satellite lines in PL spectra that are downshifted by the acoustic phonon energies. Within the elastic continuum approach [6], the phonon mode frequencies sensitive to the boundary conditions at the SQD surface were calculated. The lowest-order spherical acoustic mode frequency for CdS for different matrix materials differ by as much as a factor of three for a given SQD radius. 

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