Phonons polarization

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. 

This was first investigated by Huang n). Long-wave optical phonons are those with small k values. A polar phonon is an infrared-active phonon. Polar phonons therefore can only be observed in the Raman effect for crystals having no center of symmetry in the elementary cell. For centro-symmetric crystals the rule of mutual exclusion applies infrared-active phonons are forbidden in Raman scattering and vice versa. The elementary cells of NaCl and LiF have a center of symmetry, but GaP has none. The following considerations may therefore be applied to GaP as an example. This crystal has two atoms in the elementary cell and is cubic. It can be treated as an optically isotropic medium. 

Figure 32. Surface phonon dispersion for Nb(OOl). The data are the solid points which were taken at 900 K. Panels a and b correspond to slab dynamics calculations with two different force constant models the calculation in panel b uses the force constants from the bulk phonon fits. The solid lines represent the surface phonons and resonances polarized mainly longitudinally (or parallel), the lines with long dashes represent phonons polarized mainly perpendicularly, and those with short dashes are shear horizontal. (Reproduced from Fig. 6 of Ref. 107, with permission.)...

Here, and N are the phonon polarization and the number of lattice molecules, respectively, while Vq is the Fourier transform of the particle-crystal interaction Vq, and Pq = /dr wo(r)pe i, with wo(r) the Wannier function of the lowest Bloch band [128], The validity of the single-band Hubbard model requires that J,V < A, and the temperature k T < A with A the energy gap to the first excited Bloch band. 

Therefore, the phonon DOS g(E) determined from Eq. (1.37) is the direction-projected density of phonon states g( ), that is, g(E) weighted by the projections of the phonon polarization vectors Cj(q) along s ... 

In Eq. (34), the phonon polarization vectors are complex numbers. However, in diamond-type materials, such as Si and Ge, with two similar atoms within the basis (S = 1,2) a simplification can be made. If we choose the origin to lie midway between the two atoms, we find that 2 1 - - (a/8)(111). By employing time reversal invariance and inversion symmetry, we find that the polarization vectors e . and ef o are related in the... 

This expression can be interpreted as the scattering from two phonons, one of wave-vector q, the other of wave-vector q — q the corresponding phonon amplitudes and projections of the scattering wave-vector onto the phonon polarizations are also involved. 

The frequencies and the broadening parameters of the Fi phonons polarized in the jc-y plane parallel to the GaN [1120] and [1100] directions, extracted from... 

Table 9.4 Best-fit values and standard deviations of the , (TO), (LO), and phonon frequencies, to, and broadening parameters, y, for polarizations along the GaN [1120] (/ = x) and [1100] / = y) directions for an o-plane GaN film. Forthe ] modes, j denotes the phonon polarization, while for the 2 modes, j denotes the polarization of the incident light in the parallel and cross-polarized configurations, respectively.

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