Phonon displacements

If we expand the value of the displacement field c[) in terms of spherical harmonics according to = Jt/cj) t/(cos 0) /(r =0), it is then possible to write down equations of motion for the (/, m) components of both ripplon and phonon displacements ... 

A number of theoretical studies have investigated multiphonon relaxation in solids. Nitzan and Jortner considered a harmonic oscillator coupled to a harmonic lattice the coupling potential was taken to be linear in the vibrational coordinate and of high order in phonon displacements. 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

If the displacements of the atoms are given in terms of the harmonic normal modes of vibration for the crystal, the coherent one-phonon inelastic neutron scattering cross section can be analytically expressed in terms of the eigenvectors and eigenvalues of the hannonic analysis, as described in Ref. 1. 

Thus far we have discussed the direct mechanism of dissipation, when the reaction coordinate is coupled directly to the continuous spectrum of the bath degrees of freedom. For chemical reactions this situation is rather rare, since low-frequency acoustic phonon modes have much larger wavelengths than the size of the reaction complex, and so they cannot cause a considerable relative displacement of the reactants. The direct mechanism may play an essential role in long-distance electron transfer in dielectric media, when the reorganization energy is created by displacement of equilibrium positions of low-frequency polarization phonons. Another cause of friction may be anharmonicity of solids which leads to multiphonon processes. In particular, the Raman processes may provide small energy losses. 

The two terms correspond to different polarization of phonons. The cosine term corresponds to displacements along the rotation axis or the direction tp = 0. The sine contribution arises from the phonons polarized along the line tp = The interaction (6.29) does not change the symmetry of the (p potential, and, in this respect, it is symmetric coupling, as defined in sections 2.3 and 2.5. Nonetheless, the role of the cosine and sine couplings is different. The former ( breathing modes ) just modulate the barrier (6.22), while the latter ( shaking modes ) displace the potential. 

A. Kmmhansl, Competing displacive interactions, phonon anomalies and stmctural transitions which do... 

DSP crystal, a detailed picture of the lattice motion and related displacements was constructed and related to the topochemical postulate and the mechanism of phonon assistance. Holm and Zienty (1972) have measured the quantum yield for the overall polymerization process of a,a -bis(4-acetoxy-3-methoxybenzylidene)-p-benzenediacetonitrile (AMBBA) crystals in slurries and reported it to be 0.7 on the basis of the disappearance of two double bonds ( = 1.4 if assigned on the basis of the number of double bonds consumed). 

The volume integral will give a higher order term in k, so for now, we focus on the surface integral. The displacement due to the phonon is conveniently expanded in terms of the spherical waves e " =... 

Independent of specific theoretical models for the phonon spectrum of a solid matrix, the recoil-free fraction can be given in terms of the y-energy Ej and the mean local displacement of the nucleus from its equilibrium position ([2] in Chap. 1) [5] ... 

The temperature dependence of sod is related to that of the recoil-free fraction /(T) = Qxp[— x )Ey / Hc) ], where (x ) is the mean square displacement (2.14). Both quantities, (x ) and can be derived from the Debye model for the energy distribution of phonons in a solid (see Sect. 2.4). The second-order Doppler shift is thereby given as [20]... 

When Wqi / Wq2 the magnetization recovery may appear close to singleexponential, but the time constant thereby obtained is misleading [50]. The measurement of 7) of quadrupolar nuclei under MAS conditions presents additional complications that have been discussed by comparison to static results in GaN [50]. The quadrupolar (two phonon Raman) relaxation mechanism is strongly temperature dependent, varying as T1 well below and T2 well above the Debye temperature [ 119]. It is also effective even in cases where the static NQCC is zero, as in an ideal ZB lattice, since displacements from equilibrium positions produce finite EFGs. 

Fig. 2.2. Two generation models of coherent optical phonons, (a), (c), (e) impulsive stimulated Raman scattering (ISRS). (b), (d), (f) displacive excitation of coherent phonons (DECP). Graphs (e) and (f) display the time evolution of the driving force (grey areas) and that of the displacement (solid, curves) for ISRS and DECP, respectively...

The classical equation of motion1 describing the coherent phonons for a small nuclear displacement Q is that of a driven harmonic oscillator [9,10,15]... 

Optical detection offers the most conventional technique to time-resolve the coherent phonons. It includes four-wave mixing [8], transient reflectivity [9,10] and transmission [7] measurements, as well as second harmonic generation (SHG) [15,32]. Coherent nuclear displacement Q induces a change in the optical properties (e.g., reflectivity R) of the crystal through the refractive index n and the susceptibility y,... 

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