Crystal Umklapp process

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. 

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). 

The unique properties of liquid crystals have also provided opportunity for study of novel nonlinear optical processes. An example involves the ability to modify the pitch of cholesteric liquid crystals. Because a pseudo-wave vector may be associated with the period of pitch, a number of interesting Umklapp type phasematching processes (processes in which wave vector conservation is relaxed to allow the vector addition to equal some combination of the material pseudo-wave vectors rather than zero) are possible in these pseudo-one-dimensional media. Shen and coworkers have investigated these employing optical third harmonic generation (5.) and four-wavemixing (6). 

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