Ripplon

Rayleigh Scattering of the Phonons Due to the Elastic Component of Ripplon-Phonon Interaction... 

Figure 12. Tunneling to the alternative state at energy e. can be accompanied by a distortion of the domain boundary and thus populating the ripplon states. The doubled circles denote atomic tunneling displacements. The dashed hne signifies, say, the lowest energy state of the wall, and the dashed circles correspond to the respective atomic displacements. An alternative wall s state is shown by dash-dotted lines the corresponding alternative sets of atomic motions are coded by dash-dotted lines. The domain boundary distortion is diown in an exagerated fashion. The boundary does not have to lie in between atoms and is drawn this way for the sake of argument its position in fact is not tied to the atomic locations in an a priori obvious fashion.

Figure 14. Tunneling to the alternative state at energy can be accompanied by a distortion of the domain boundary and thus populating the ripplon states. All transitions exemplified by solid lines involve tunneling between the intrinsic states and are coupled linearly to the lattice distortion and contribute the strongest to the phonon scattering. The vertical transitions, denoted by the dashed lines, are coupled to the higher order strain (see Appendix A) and contribute only to Rayleigh-type scattering, which is much lower in strength than that due to the resonant transitions.

In order to compute the heat capacity of the ripplons on top of the structural transitions, we will need to consider the (classical) density of the inelastic states in more detail than in the previous section. The density of states (e) = was derived earlier taking as the reference state the generic... 

As is clear from Eq. (42), the approximation amounts to Ending the effective temperature so as to populate the ripplonic states to match the excitation energy CO. The expression for the curvature, Eq. (43), appropriately involves the corresponding heat capacity of the excitations. 

Figure 19. The predicted low T heat conductivity. The no coupling case neglects phonon coupling effects on the ripplon spectrum. The (scaled) experimental data are taken from Smith [112] for a-Si02 (AsTj/ScOd 4.4) and from Freeman and Anderson [19] for polybutadiene (ksTg/Hcao — 2.5). The empirical universal lower T ratio l /l 150 [19], used explicitly here to superimpose our results on the experiment, was predicted by the present theory earlier within a factor of order unity, as explained in Section lllB. The effects of nonuniversaUty due to the phonon coupling are explained in Section IVF.

We approximate the phonon coupling effects by replacing in our spectral sums in Eqs. (41)-(43) and (46)-(49) the discrete summation over different ripplon harmonics by integration over Lorentzian profiles ... 

Finally, we return to the specific heat. The effects of the phonon coupling on the ripplon spectrum can be taken into account in the same fashion as in the conductivity case. Here we replace the discrete summation in Eq. (38) by integration over the broadened resonances, as prescribed by Eq. (57). The bump, as shown in Fig. 15, is also predicted to be nonuniversal depending on Tg/oio-The predicted bump for Tg/(Od = 2 seems to match well the available data for... 

Figure 21. A low-energy portion of the energy level structure of a tunneling center is shown. Here e < 0, which means that the reference, liquid, state structure is higher in energy than the alternative configuration available to this local region. A transition to the latter configuration may be accompanied by a distortion of the domain wall, as reflected by the band of higher energy states, denoted as ripplon states.

Here e is the new value of the energy splitting, the co, are the ripplon frequencies, and the A,- are tunneling amplitudes of transitions that excite the corresponding vibrational mode of the domain wall. Those amplitudes will be discussed in due time for now, we repeat, the expression above will be correct in the limit Ai/Ha>i —> 0. Finally, the renormalized value e was used in the denominator. While, according to Feenberg s expansion [118], including e in the resolvent is actually more accurate, we do it here mostly for convenience. 

Equation (61) raises another interesting point. According to that equation, the values of both the bare and the effective classical energy bias of a transition—e and e, respectively—are limited from above by the lowest ripplon frequency (C02)- (Note that this is only realized in the e < 0 case, discussed in this section.) This is unimportant at low temperamres. But what happens at higher T, near this... 

Figure 23. This caricature demonstrates the predicted phenomena of energy level crossing in domains whose energy bias is comparable or larger than the vibronic frequency of the domain wall distortions. The vertical axis is the energy measured from the bottom state the horizontal axis denotes temperature. The diagonal da ed line denotes roughly the thermal energies. A tunneling center that would become thermally active at some temperature Tq will not possess ripplons whose frequency is less than To.

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