Long-wavelength acoustic phonons

Thus far we have discussed the direct mechanism of dissipation, when the reaction coordinate is coupled directly to the continuous spectrum of the bath degrees of freedom. For chemical reactions this situation is rather rare, since low-frequency acoustic phonon modes have much larger wavelengths than the size of the reaction complex, and so they cannot cause a considerable relative displacement of the reactants. The direct mechanism may play an essential role in long-distance electron transfer in dielectric media, when the reorganization energy is created by displacement of equilibrium positions of low-frequency polarization phonons. Another cause of friction may be anharmonicity of solids which leads to multiphonon processes. In particular, the Raman processes may provide small energy losses. 

The acoustic phonons of (SN) have been studied by inelastic neutron scattering. These investigations are extremely difficult because of the small size and poor quality of the crystals and therefore had to be restricted to the long wavelength region. 

On the other hand, the two modes of CeBeij show a softening of about 2% with respect to the reference line. All other symmetry modes of CeBe j do not show any anomaly. This applies also to the bulk modulus of CeBcij, partly indicative of the behavior of the long-wavelength acoustic phonons as shown in fig. 36. As a function of Q/V the bulk modulus of the RBe j compounds follows a straight line with CeBe j right on it (Mock et al. 1985). However, an even stronger softening of the two F modes is found upon Ce dilution in... 

The elastic constants determine the acoustic phonon dispersion for long wavelengths according to the equations of motion in the continuum limit ... 

Such decoupling in the liquid may be strictly justified only in the long-wave approximation.In this sense, such a procedure is justified for the macroscopic description. However, one should remember that this is the correct method in a number of cases also for short wavelengths. For example, this is the case for phonons in solids. In other cases, such as the electron gas in metals (plasmons), acoustic phonons in quantum liquids and so on, this decoupling may be considered as the self-consistent field method or the random phase approximation (the analog of the superposition approximation in the classical theory of liquids). 

Here, the simple dispersion (oiq) v g was used for the acoustic phonons (if whole molecules vibrate with respect to each other, like the change of the stacking distance in a stack one speaks of acoustic phonons, because they have a long wavelength comparable to those of acoustic waves) and the sound velocity v, is determined by the relation o, = Ci/p), where c, is the longitudinal elastic constant and p is the mass density. One should remark that this very simple linear dispersion relation (o q) = v,g is not necessarily correct. With the help of the FG method described in Section 9.1 one can obtain more accurate dispersion curves. Equation (9.48) can now be used to calculate the charge carrier mobilities and free paths, defined in this case hy p= e xlm ) and A = (t ), respectively, where... 

Figure 2.16 A typical phonon dispersion relation for a one-dimensional lattice of balls connected by springs. On the long-wavelength acoustic branch atoms move as groups in one direction or another with the direction varying over relatively long distances. For the higher-energy optical branch atoms move in opposite directions over very short distances.

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