Phonons Umklapp processes

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. 

In electrically insulating solids, heat is transferred in the form of elastic waves or phonons [1], Anything that affects the propagation of the phonons through the solid affects the thermal conductivity of the solid. In a pure crystalline ceramic, the intrinsic thermal conductivity is limited by the energy dissipated during phonon-phonon collisions or so-called Umklapp processes [15], Commonly, the intrinsic thermal conductivity of solids is described by (5). 

The mean free path A may be determined by many different scattering mechanisms but the dominant one at temperatures not too close to o °K is phonon-phonon scattering, the coupling taking place through the anharmonicity of the lattice vibrations. There are two possible types of phonon-phonon scattering processes normal processes in which total phonon wave vector is conserved, and umklapp processes in which the total wave vector after collision differs from that before collision by a vector of the reciprocal lattice. Since normal processes do not affect the total phonon momentum or energy, they do not contribute to thermal resistance and only umklapp processes need be considered. For an umklapp process to occur between two phonons of wave vectors q and q we must have a relation of the form... 

Apart from phonon-phonon scattering, which is allowed by the anharmonicity of the interatomic potentials, the umklapp or flipover process is another effect that decreases the mean free path of the phonons and the conductivity two short-wavelength waves may interfere to form one long-wavelength wave going back (Figure 4.38). The lattice periodicity contributes to the umklapp process, a reflection of drifting phonons. 

T temperature dependence characteristic of phonon scattering dominated by Umklapp processes in crystalline solids... 

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 ... 

The semiclassical approach to the problem of atom-crystal inelastic scattering is very attractive due to its relative simplicity, analytical nature and wide applicability. This approach allows one to obtain a simple Gaussian approximation (Brako and Newns 1982 Manson 1991) to the dynamic structural feictor of inelastic phonon scattering and the intensities of diffraction peaks (Billing 1975). The effect of umklapp processes on the dynamic structural factor hcis been considered only in the hard-wall approximation (Berry 1975 Bogdanov 1980) or numerically (Manson 1991). 

As expected, the behavior of the thermal conductivity of graphite foam with temperature is similar to that of graphite that is, thermal conductivity decreases as temperature inereases. With increasing temperature, the dominant phonon interaction becomes phonon-phonon scattering (Umklapp processes) and, thus, the observed reduetion of thermal conductivity with increasing temperature. The temperature dependence of thermal conductivity for nuclear-grade graphite H 451 is shown, for reference, in Fig. 4.34. 

The mean free path of phonons is affected by a variety of processes. All of these processes therefore determine the thermal conductivity of ceramics. One major process that influences the mean free path is the Umklapp process. At low temperatures, the mean free path corresponding to this process becomes large. 

The particle aspect of phonons is especially convenient if interactions between phonons and other particles such as photons, electrons, neutrons, or with other phonons are studied. In all these interactions in which one or several phonons are involved, the conservation of energy holds strictly but the conservation of momentum holds only to within a vector x of the reciprocal lattice. Processes with x = 0 are called normal processes while processes for which x + 0 are called Umklapp processes. 

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