Phonon polarization vectors

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... 

In this equation, h is Planck s constant divided by 2tt, V is the crystal volume, T is temperature, fej, is Boltzmann s constant, phonon frequency, is the wave packet, or phonon group velocity, t is the effective relaxation time, n is the Bose-Einstein distribution function, and q and s are the phonon wave vector and polarization index, respectively. 

The e term refers to a unit polarization vector, is the first-order derivative of the susceptibility tensor with respect to the phonon amplitude, also known as the Raman tensor, and rij is the phonon occupation number for the /th mode, given by ... 

The LO phonon is polarized along q, where the direction of as defined by (17.1b). The contribution of longitudinal phonons may be obtained by the projection of the Raman tensor element along the phonon unit vector qy. 

Considering Equation 6.38 again, we need to transform the Hamiltonian expression. Thus, if cos(k) and ss(k) are the frequency and the polarization vector for the classic modes with polarization s and wave vector k, respectively, we can define the phonon creation (aks+) and annihilation ( /,s ) operators as... 

Ii3/2 multiplets. Frequencies and polarization vectors of phonons in the LiYp4 crystal were obtained at 8000 points in the irreducible part of the Brillouin zone using the rigid ion model of lattice dynamics derived on the basis of neutron scattering data. Matrix elements of electronic operators Vds) were calculated with the wave functions obtained from the crystal-field calculation. The inverse lifetimes of the crystal-field sublevels determine the widths of corresponding absorption lines. 

The experimental g(oj) (or g(v)), while showing qualitatively all the features of the calculated g(co), cannot yet be quantitatively compared because the vibrational amplitudes (generally called polarization vectors in neutron-scattering circles) of the various normal modes (phonons) are not yet taken into account. The situation is similar to the features of the vibrational spectra of an isotropic sample compared with the features of the spectra in polarized light of an oriented anisotropic sample. 

Grazing incidence scattering geometry the photon beam with a wave vector k is illuminating the sample at an incidence angle (typically below 1 ).The polarization vector basis of the phonon (ejJ = x,y,z) and the projections of and 6y along the direction of the beam propagation are shown, n denotes the surface normal. 

Thus, from Eqs. 12 and 13, the phonon frequency can be evaluated from the curvature of the calculated energy vs. displacement curve for small displacements. These results can be extended to the case of compounds and to general wave vectors where the lack of symmetry requires the calculation and dlagonalization of the dynamical matrix to obtain the phonon frequencies and polarization vectors. Moreover, this approach allows a detailed investigation of the role of core-core, electron-core, electron kinetic, and electron-electron energies to determine the vibrational frequencies of the solids examined. This kind of information has been valuable in analyzing and understanding phonon anomalies in semiconductors and transition metals. 

Here we have introduce he phonon frequencies UJj (q) and the polarization vectors ej (0,j). n( w) is the Bose ractor, phonon creation (energy loss) correspond to 03. phonon annihilation (energy gain) corresponds to (q)>0. V(q,z) is the 2-dimensional Fourier transform of the potential and Q now is considered in the extended zone scheme. The Debye Waller factor 2W is given by ... 

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. 

In particular, the phonon dispersion relations and polarization vectors can be calculated with reasonable accuracy using force-constant models [59] or the embedded atom method [60-62], In recent calculations of Fe-ph and X for surface states, wave functions obtained from the one-electron model potential [63, 64] have been used. For the description of the deformation potential, the screened electron-ion potential as determined by the static dielectric function and the bare pseudopotential is used, Vq z) = f dz e (z,2/ gy)qy), where (jy is the modulus of the phonon momentum wave vector parallel to the surface, and bare Fourier transform parallel to the surface of the bare electron-ion... 

In the above relation, quantum states of phonons are characterized by the surface-parallel wave vector kg, whereas the rest of quantum numbers are indicated by a the latter account for the polarization of a quasi-particle and its motion in the surface-normal direction, and also implicitly reflect the arrangement of atoms in the crystal unit cell. A convenient representation like this allows us to immediately take advantage of the translational symmetry of the system in the surface-parallel direction so as to define an arbitrary Cartesian projection (onto the a axis) for the... 

The interaction between light and matter can be viewed as the creation of a coherent quantum superposition of initial and final electron states that has an associated polarization [3], as shown in Figure 1. The coherence between states with different wave vector requires an intermediate virtual state and the presence of a coherent phonon. A transition between the initial and final states may occur when the coherence of the system is broken either due to the finite width of an optical wave packet or by scattering from the environment. The transition results in the absorption of a photon and the creation of a hot electron-hole pair. Otherwise, the photon is re-radiated with a different phase and, perhaps, polarisation. 

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