Two phonon process

The role of two-phonon processes in the relaxation of tunneling systems has been analyzed by Silbey and Trommsdorf [1990]. Unlike the model of TLS coupled linearly to a harmonic bath (2.39), bilinear coupling to phonons of the form Cijqiqja was considered. In the deformation potential approximation the coupling constant Cij is proportional to (y.cUj. There are two leading two-phonon processes with different dependence of the relaxation rate on temperature and energy gap, A = (A Two-phonon emission prevails at low temperatures, and it is... 

Fig. 16 Raman spectrum of two-phonon processes in single crystalline a- 8g range 500-1000 cm , after [109]. The strong bands in the range 800-950 cm result from combinations of components of the stretching vibrations...

The Orbach process is a two-phonon process that takes place via population of an excited electronic state with energy Eq. The temperature dependence of the relaxation rate is given by... 

The infra-red spectrum of Li-6 hydroxide was discussed by Decius and Lilley 69> who report that the major feature at 3681 400 cm"1 in the LiOH spectrum (which was previously assumed to involve a librational mode) is partly due to the absorption at 3681 290 cm 1 which involves a translator mode in which there is a significant amount of Li7-participation the rest may be a two-phonon process (see Oehler and Gunthard S6>). 

At low temperatures (/3ho)c>l) the contributions from one-phonon, two-phonon processes, etc., can be systematically extracted from the general expression for the rate constant, and the type of the dominant process is determined by the bath spectrum and temperature. The results of Leggett et al. [1987] show that quantum dynamics of a TLS crucially depends on the spectrum of the bath. For sufficiently strong coupling, the bath may dramatically slow the tunneling rate, even to the point of localizing the particle in one of the wells. This strong dependence on the bath spectrum is inherent to the quantum dynamics and does not show up in classical transitions. 

At higher temperatures, the transfer rate of the impurity molecule between sites is enhanced by phonon emission or absorption this can be viewed as a phonon-dressed state transfer. When these off-diagonal processes are included in the analysis, the transition probability can be broken up into phonon processes of different order. One-phonon processes do not contribute, because momentum is not conserved in these events. Among the two-phonon processes, only Raman events contribute. These lead to a I7 dependence of the transition probability (see, for example, Flynn and Stoneham [1970]). This regime takes place when kBT... 

A spectrum of pristine Cfto sublimed in a tube and unexposed to air, O2, or alkali metal was taken (Fig. 2). We have previously shown (15) that exposure of vacuum-grown films to either air or oxygen alters the spectrum, especially in the frequency region of the symmetric mode at 1458 cm . In previous experiments (15) the Si substrate inhibited the identification of modes near 520 cm and 950 cm to 1000 cm . In the present experiments, the Si substrate has been eliminated and we can identify die mode at 522 em and the broad feature between 950 cm and 1000 cm as due to 60- The broad feature may result from a two phonon process involving the 490... 

The similarity between the A C o spectra doped with different alkali metals again indicates that the observed modes are due only to the Cfio molecule, and that there are no observable A-Cfto modes above 100 cm The mode at 270 cm has a resolution-limited FWHM for CSftCfio, is broadened in nd is a doublet in K C q. The high-frequency mode of the doublet may result from a two-phonon process involving a low cnergy-A-Cfrequency modes (1383 cm and 1478 cm ). If this is the case, it is not clear that these modes should be broadened in the same way as the... 

Discussing the relaxation time origin we will also follow [2]. Three mechanisms of relaxation can be accounted here thermal activation over the potential barrier Vo (described with tt), tunnelling through the barrier accompanied by phonon emission (r,), and two phonon process analogous to Raman scattering (tr) ... 

The pseudospin methodology is widely used not only for the description of hydrogen containing ferro- and antiferroelectrics, but also for the study of many other systems with hydrogen bonds. In particular, the pseudospin methodology was applied by Silbey and Trommsdorff [99] for examining the influence of two-phonon process on the rate constant of molecular compounds. In the next sections we will also employ the pseudospin formalism for the investigation of some problems where protons are exemplified by the cooperative behavior. 

In certain cases involving very small energy mismatches two-phonon assisted transfer processes may be more important than the one-phonon processes. Numerous different types of two-phonon processes have been developed and found to predict a variety of different temperature dependences i2), The opposite limit of very large energy mismatch can be treated in a similar way to multiphonon relaxation processes 12,13). fhe matrix element in Eq. (2) is carried out to the Nth power where N is the number of phonons involved in the transfer process. The temperature dependence of the transfer rate for phonon emission processes then becomes... 

For two-phonon processes involving branches t and t and phonons with wave vectors q and q, the temperature-dependent term is ... 

We have mentioned the existence of CPs in the one-phonon density of states, but this can be measured only for compound crystals. The situation is different in multi-phonon absorption because the high-frequency phonons of the BZ boundary are mostly involved (note that in three-phonon processes, the q = 0 zone-centre phonons can also be involved without problem for momentum conservation). For the two-phonon absorption, the density of states is proportional to an integral similar to the one in expression 3.20, with wt replaced by the sum of the two pulsations and of the phonons of the combination. Besides the trivial case where = u> = 0, the condition Vq (wt (q) + wt (q)) = 0 is fulfilled when Vq (wt) = Vq(uv) = 0 or when Vq (wt) = — Vq (wt ). The observed two-phonon absorption is the sum of the contributions of the possible two-phonon processes. Figure 3.2 shows the RT absorption of silicon in the two- and three-phonon absorption region. In the two-phonon region, it is fitted with the two-phonon dispersion curves... 

In a two-phonon process which involves an internal mode of vibration for which the frequency p varies little with wave vector, the combined density of states approximates, at least in a cubic crystal, the density of vibrational states spectrum [see Equation (6)]. This is important for a discussion of the absorption that occurs on the high- and low-frequency sides of many internal-mode absorption lines. 

From the above discussion when dpjdq, or more rigorously, VqPy is zero or has a slope discontinuity, there are likely to be slope discontinuities in the combined density of states, as revealed by infrared and Raman spectra of two-phonon processes. Points in the Brillouin zone where each of the components of VqP = 0 are known as critical points. The intensity of infrared absorption or Raman scattering depends upon quantum mechanical matrix elements which are, in general, not simple to evaluate. However, by using symmetry considerations and group theoretical methods, the various modes can be assigned as infrared or Raman active. 

Anderson, Sun, and Donkersloot (1970) and Cahill and Leroi (1969a) reported a 30 cm" broad absorption peaking at about 80cm. This feature is, presumably, due to two-phonon processes. 

It is a two-phonon process involving the simultaneous absorption of one phonon and emission of another phonon, accompanied by a spin flip with energy that equals the energy difference between the corresponding phonons. When the temperature is much smaller than the Debye temperature, in... 

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