Umklapp process

A peculiarity of one-dimensional models is that they may involve so-called Umklapp processes (46) annihilating, for instance, the situation of model 1 b. However, as there are more than two neighboring atoms in three dimensions, we assume that in our example such a process will not occur. 

Umklapp process In the interaction of a continuous wave (photon, electron, etc.) with the lattice, the quasi-momentum of the wave is conserved, modulo a vector in the reciprocal lattice. The introduction of these quanta of momentum leads to the Umklapp process. In many macroscopic treatments the matter is treated as a continuous medium and Umklapp processes are neglected. In our treatment, Umklapp processes are included in the coulombic interactions (calculation of the local field), but implicitly omitted in the retarded interactions, since we dropped the term (cua/c)2 in (1.64). 

For U < 0, the analysis of Emery [16] captures the essence. The attractive interaction favors singlet pairing and the system develops superconducting fluctuations. The charge- or pair-density fluctuations are gapless in the absence of umklapp processes. The spin excitations develop a gap of order U since singlet pairs need to be broken to create spin excitations. 

In the presence of umklapp processes, the charge bosons are no longer free. They are bound by a cosine potential energy term of amplitude g3 /2ir, so the charge excitations have a gap of this magnitude which can be felt at low temperatures. Under those circumstances, only the SDW susceptibility remains singular with ySDW = 1 (see, however, the refinement when umklapp are present in Section IV.B.3.d). 

As pointed out by Kimura [40], when umklapp processes are relevant, the system can differentiate between site (on the molecules) and bond (in... 

Figure 4 Phase diagram of the interacting electron gas showing the competing response functions with positive power law exponent in the absence of umklapp processes. Those in parentheses have a smaller prefactor.

Let me first illustrate the approach with incommensurate systems, that is, those for which the band filling nf (the average number of electrons of a given spin in the conduction band per unit cell) is an irrational number. There are no electronic umklapp processes. 

Highly correlated quasi-one-dimensional conductors that have repulsive interactions and electronic umklapp processes will develop a Hubbard gap in the charge excitations at Tp e v 3 [Eq. (10)] provided that... 

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. 

Electronic transport properties are strongly influenced by a touch of the Fermi sphere with the zone boundary, in the crystalline as well as in the disordered state. Exhaustive reviews on this subject have been given by Massalski and Mizutani [5.35] and Mizutani [5.20], In the same way as sharp zone boundaries in crystalline materials are responsible for umklapp processes, in amorphous systems we can talk in terms of diffuse umklapp processes caused by the pseudo Brillouin-zone boundary. This description was first introduced by Hafner [5.36]. 

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). 

Further simplification can be achieved if the band is not half-filled, in which case the Umklapp processes (gg) can be neglected. The term with g can be incorporated into a Fermi velocity renormalization and it will not be considered any longer. Finally the backward scattering term (gj) will be assumed to be spin independent, glx = % u = Si ... 

Fig. I Shown are diagrammatic representations of the four possible Intrachain Interact ions. The Indices 1 and 2 refer to different sides of the Fermi surface. The umklapp process, g, is only important for a half-filled band and is therefore not considered.

Umklapp process - A process involving the interaction of three or more waves (lattice or electron) in a solid in which the sum of the wave vectors does not equal zero. 

We are In the process of quantifying this picture of charge transfer excitations. Such an approach requires detailed calculations of the full dleletrlc tensor c(Q, 0 > Including the (Important) Umklapp processes using our band structure results as the starting point. 

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... 

The normal process has no effect on the thermal resistance. However, the umklapp process leads to an increase in thermal resistance with temperature given by... 

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... 

Lattice thermal conductivity data below 300 K for representative clathrates are summarized in Fig. 6.4 [35, 40 4], These data, for a representative number of compositions collected from both polycrystalline and single crystalline specimens, allow for a comparison of the effect of guest and framework composition on Kl. While some compositions have lattice thermal conductivities that are characteristic of glasses, others have Kl values that more closely resemble the typical temperature dependence of defect-free crystalline solids in which Umklapp processes [24] produce a monotonically decreasing Kl with increasing temperature above 10 K. 

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