Phonon propagator

The presence of isotopic impurities causes clear effects in the vibrational spectra. Almost all modes studied so far show frequency shifts on S/ S substitution [81, 107]. The average shift of the internal modes is ca. 0.6 cm , and of the external modes it is 0.1-0.3 cm (Tables 3, 4 and 5). Furthermore, the isotopomers which are statistically distributed in crystals of natural composition can act as additional scattering centers for the phonon propagation. Therefore, in such crystals the lifetime of the phonons is shortened in comparison with isotopically pure crystals and, as a conse-... 

Discontinuities in the lattice such as vacancies, impurities, or grain boundaries also act to scatter phonon propagation, hence a lower thermal conductivity is expected in solids containing these defects at cryogenic temperatures. Whichever mecha-... 

Film conductances are also often defined for the impedance to thermal conduction when two solid conductors are placed in mechanical contact. A significant contact resistance is often observed when, on a microscopic scale, heat transfer involves an air-gap between the materials. Under such conditions, phonon propagation must be replaced by the kinetic interaction amongst gaseous atoms and then back to phonon heat transfer in the next solid. Fibrous and foam insulation axe effective thermal insulators because of the numerous contact resistances involved in the transfer of heat. 

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. 

Fig. 1 (a) Self-energy in diagrammatic representation, (b) Expansion series for the vertex E up to g. Thick solid, thick dashed, and thin dashed lines indicate, respectively, the electron Green s function, the dressed phonon, and the bare phonon propagators... 

The thermal Green s function, or phonon propagator, that is used in lattice dynamics theory is defined as... 

The Fourier representation of the anharmonic propagator is obtained from Eqs. (73) and (72), by using Wick s theorem (69), together with the Fourier representation of the harmonic phonon propagators. The time integrations can then explicitly be performed, yielding the condition 2,- w, = 0 and a factor /3 at every interaction vertex. 

AH matrices are of dimension 6Z and the harmonic propagator matrix G(0)(q m) is diagonal. The problem of calculating the phonon propagators thus reduces to the calculation of the self-energy matrices S(q ioif) that contain all anharmonic information. It is not difficult to demonstrate that the self-energy matrix is a Hermitian function of oif from which it follows that its analytic continuation in the complex frequency plane, in the neighborhood of the real axis, has the form... 

Direct. Of the vast number of phonons propagating through the solid, a small number have a frequency identical to the Larmor frequency of a nucleus in the sample. It is, therefore, possible for these phonons to act directly on such a nucleus causing, if the nucleus is in an energized state, the emission of energy and the return of the nucleus to its ground state they can, of course also cause absorption transitions. 

In the case of a cubic crystal with three independent coefficients, C has the following values for phonons propagating along the [110] direction ... 

Because of symmetry constraints at the surface, phonons propagating in a high-symmetry direction must be polarized either in the sagittal plane... 

Using the pseudofermion representation for and a similar one for Hq we can now calculate the phonon propagator renormalized by this interaction ... 

Fig. 59. (a) The Dyson equation [eq. (AlO)] for the phonon propagator, (b) The RPA approximation for the quadrupolar susceptibility with respect to the quadrupolar interaction Kp, see eq. (A7). Circles represent the quadrupolar operators Op. 

Together with eq. (A7) this defines the dispersion of collective quadrupolar excitations that is given by the zeros of the denominator in eq. (A7). The renormalized phonon propagator is now obtained by Dyson s equation (fig. 59a)... 

The peculiarity of the physics of conducting polymers is that they can be considered one-dimensional crystals, since intermolecular forces (yan der Waals type) are much weaker than the intramolecular forces (coyalent forces). This concept is immecfiately translated in eq. 7, by the fact that the k yector has only one component for 1-D crystals, and phonons propagate only longitudinally along the chain backbone [3,4]. 

An important ingredient of the AM model is that all IRAVs are interconnected and contribute to the same phonon propagator [65]. It has been therefore defined as the pinned, many-phonon-propagator, A)i(w) = Do(< ))/[l — apDo(<<))], where ap is the polaron-vibrational pinning parameter and Do(w) is the bare phonon propagator. The latter is given [65] by Dofw) = S do, (w)> and do,nM = idnj, where o , and A are the bare phonon frequencies,... 

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