Three-phonon

Important contributions on the Raman side came after 1970 which could make use of the well estabhshed assignment of the fundamentals in interpreting the spectra [97-99, 134]. The most complete studies on two- and three-phonon processes have been carried out by Harvey and Butler (Raman, only gxg combinations assigned) [100] and by Eckert (Raman and IR) [109]. 

We first consider the vibrational relaxation that can be induced by aijQqiqj (three-phonon processes) or Qq qi (four-phonon processes). In the three-phonon processes there are two accepting modes, while in the four-phonon processes there are three accepting modes. To calculate the rate of vibrational relaxation, we use... 

Fig. 2. The dependence of the rates yk (in r units) of two-phonon (k = 2), three-phonon (k = 3) and four-phonon (k = 4) transitions on the dimensionless interaction parameters wk.

The shift transformation method yields expressions for the vibronic state at a point on the trough of minimum-energy points (see Ref. [29] and references therein). We know that Q i, (> and Q are vibrations and therefore, the unitary shift transformation operator U will only act on these three phonon coordinates. The generalised vibronic states allowing for phonon excitations can be written in... 

Charged point defects on regular lattice positions can also contribute to additional losses the translation invariance, which forbids the interaction of electromagnetic waves with acoustic phonons, is perturbed due to charged defects at random positions. Such single-phonon processes are much more effective than the two- or three phonon processes discussed before, because the energy of the acoustic branches goes to zero at the T point of the Brillouin zone. Until now, only a classical approach to account for these losses exists, which has been... 

Equation (5), VER involves a higher-order anharmonic coupling matrix element, which gives rise to decay via simultaneous emission of several phonons nftjph (multiphonon emission). In the ACN case, three phonons must be emitted simultaneously via quartic anharmonic coupling (or four phonons via fifth-order coupling, etc.). 

In the gas phase, the asymmetric CO stretch lifetime is 1.28 0.1 ns. The solvent can provide an alternative relaxation pathway that requires single phonon excitation (or phonon annihilation) (102) at 150 cm-1. Some support for this picture is provided by the results shown in Fig. 8. When Ar is the solvent at 3 mol/L, a single exponential decay is observed with a lifetime that is the same as the zero density lifetime, within experimental error. While Ar is effective at relaxing the low-frequency modes of W(CO)6, as discussed in conjunction with Fig. 8, it has no affect on the asymmetric CO stretch lifetime. The DOS of Ar cuts off at "-60 cm-1 (108). If the role of the solvent is to open a relaxation pathway involving intermolecular interactions that require the deposition of 150 cm-1 into the solvent, then in Ar the process would require the excitation of three phonons. A three-phonon process would be much less probable than single phonon processes that may occur in the polyatomic solvents. In this picture, the differences in the actual lifetimes measured in ethane, fluoroform, and CO2 (see Fig. 3) are attributed to differences in the phonon DOS at 150 cm-1 or to the magnitude of the coupling matrix elements. 

With regards to the second feature of real crystals mentioned earlier, there are different types of anharmonicity-induced phonon-phonon scattering events that may occur. However, only those events that result in a total momentum change can produce resistance to the flow of heat. A special type, in which there is a net phonon momentum change (reversal), is the three-phonon scattering event called the Umklapp process. In this process, two phonons combine to give a third phonon propagating in the reverse direction. 

The first process shown above (Fj) is third order and by far the most commonly cited relaxation mechanism in pure crystals. It corresponds to a three-phonon process whereby the initial phonon either splits into two new phonons of lower energy (first term, down conversion) or interacts with a higher energy phonon (second term, up conversion). The down conversion process leads to a finite linewidth at 0 K, whereas the up conversion process gives zero contribution at low temperature. 

V is the speed of sound, 6 is the Debye temperature, tc is the total phonon-scattering rate, w is the phonon-scattering rate due to three phonon normal processes. In this model, two additional scattering mechanisms of phonon (by point defects and by charge carriers) are considered. 

The shape of the two-phonon spectrum is given by the convolution of the one-phonon spectrum with itself, g(a )ext g((i))ext- The shape of the three-phonon spectrum, n = 3, is obtained by convolving the one- and two-phonon spectra, and so on. The weight of each contribution is given by... 

We apply Eq. (2.72) and obtain the weights of the phonon orders given in Table 2.7. Approximately 98% of the total intensity is found distributed across only the first three phonon orders. The zero-phonon line, or band origin, remains the strongest feature but, at this value of 0, the first order event is quite strong. If we had chosen 0 = 9 A, the first order contribution would have been the stronger. 

Similar relationships can be obtained for three-phonon processes. At low temperature, n (iv, T) is much smaller than unity and the above expression tends to unity for summation processes, and to zero for the difference processes, which are, therefore, not observed at low temperature. At higher temperature, the absorption intensity increases for both processes [45], at a difference with the one-phonon process, which is temperature-independent. 

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... 

There is only one known acceptor in diamond, responsible for the p-type conductivity of the lib diamonds. For some time, it was assumed that this acceptor was aluminium [49], but it has been suggested [43] and finally shown conclusively [38] that boron was indeed responsible for the p-type conductivity and the spectroscopic properties of type lib blue diamonds. Natural lib diamonds had been identified ca. 1954 (see Sect. 2.11), and synthetic lib diamonds were obtained at the beginning of the 1960s [80]. Boron is commonly introduced as a dopant in synthetic diamonds and its ionization energy ) is 370 meV [177]. The discrete acceptor spectrum of B extends approximately 70 meV below ) and is superimposed on the two- and three-phonon spectra of Cdiam- Boron acceptor absorption lines are observed at 305, 347 and 363 meV ( 2780, 2800, and 2930 cm 1) at RT, giving phonon-assisted transitions near 464 and 504meV (see [140], and references therein). 

The stretching frequency of the P—0 bond is 1140—1300 cm and that of the B—O bond 1310—1380 cm-i. The energy of these phonons is quite sufficient to match the gap between the levels of the donor and the acceptor, without the necessity of the cooperation of more than two or three phonons. 

For three-phonon interactions one distinguishes two types of collisions normal processes (N processes), in which the total momentum is conserved and the direction of flow does not change (these processes lead to infinite thermal conductivity) and Umklapp processes (U processes), in which the sum of the wave vectors is not conserved and changes sharply, leading to a finite thermal resistivity of a crystal. In U processes the following conditions are fulfilled ... 

All other terms that are not related to the above terms by permutation of the Indices are zero. As noted before, < gives the optical phonon frequency y gives the amplitudes for three-phonon processes and a and 3 give the amplitudes for the bare four-phonon processes. 

It is important to note at this point that the LA phonon lifetime against spontaneous decay varies as oT, i.e, high frequency LA phonons are predicted to decay much more rapidly than low frequency phonons. In simple systems near T 0, the spontaneous three-phonon decay of LA phonons, described above, should dominate the observed temporal evolution of nonequilibrium phonon distributions (Orbach and Vredevoe, 1964). 

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