Scattering tensor

Fig. 2 Raman spectra of a single-crystal of orthorhombic Sg at three different polarizations in which the off-diagonal elements of the Raman scattering tensor are non-zero big, b2g, b g), after [105]. However, Raman intensities of other polarizations like flg components ( 54 cm ) penetrate in the spectra due to optical anisotropy in the crystal...

In crystalline solids, the Raman effect deals with phonons instead of molecular vibration, and it depends upon the crystal symmetry whether a phonon is Raman active or not. For each class of crystal symmetry it is possible to calculate which phonons are Raman active for a given direction of the incident and scattered light with respect to the crystallographic axes of the specimen. A table has been derived (Loudon, 1964, 1965) which presents the form of the scattering tensor for each of the 32 crystal classes, which is particularly useful in the interpretation of the Raman spectra of crystalline samples. 

Accordingly, for a given Raman-active resonance r, the amplitudes of XijM can be expressed in terms of the isotropy and symmetric anisotropy invariants of the corresponding spontaneous Raman scattering tensor, a2 and 7S2, respectively. In the case of frequency-degenerate CRS considered here, the two relevant independent tensor components assume the following form [33, 34] ... 

For a purely isotropic Raman scattering tensor, the depolarization ratio will be zero since the scattered light from such a system must have a polarization parallel to the incident beam. This is evident when equation equation (5.29) is used in equation (5.30) so that [21],... 

The rotational invariants of the scattering tensor, namely a2 and j82 for ordinary Raman scattering, and aG, j8g, and 0 for ROA scattering, are explained in the following section. If we drop the explicit mention of the molecular states between which the molecule transits during the scattering process, then the scattering cross-sections for the off-resonance situation can be written as ... 

Changes of each component of the scattering tensor of the unit cell can be directly observed. The arrangement of the crystal with regard to the spectrometer is described by a nomenclature recommended by Porto et al. (1966). In this system, a(bc)d stands for irradiation in a direction by radiation polarized in b direction, Raman radiation polarized in c direction is observed in d direction. Here, a, b, c, d indicate any crystal axes. 

Polarizers may be employed to measure the scattering tensor of crystals or the depolarization factor of liquids (Hoffmann et al., 1991). 

Finally we mention, that the 6-term contribution to the transition polarizability may give rise to an asymmetric scattering tensor, i. e. Op 7 a p, leading to the phenomenon... 

Shegai TO, Haran G (2006) Probing the Raman scattering tensors of individual molecules. J Phys Chem B 110(6) 2459-2461... 

K. Jurkowska and R. Mierzecki. The influence of intermolecular interactions on the invariants of the Raman scattering tensors, II. The CDCI3 tensors in solutions with acetone, dioxane, benzene and mesythylene. Acta Phys. Polonica, A, 63 701-706 (1983). 

Habashy et al., (1993) and Torres-Verdin and Habashy (1994) developed the extended Born approximation, which replaces the total field in the integral (9.73) not by the background field, like in the Born approximation, but by its projection onto a scattering tensor F (r) ... 

The expression for the scattering tensor is obtained from equation (9.73) by approximating E (r) in the integral by its value at the peak point r = tj of the Green s tensor ... 

The LQL approximation has a similar background to the localized nonlinear (LN) approximation, but there are some important differences. The LN approximation also replaces the total field inside inhomogeneity with a product of the background field and a scattering tensor E (r),... 

However, the scattering tensor is defined explicitly through an integral of the anomalous conductivity. On contrast, the reflectivity tensor in the LQL approximation is determined as the solution of the minimization problem (9.115). Note also that the... 

To proceed further it is necessary to elaborate on the vibronic nature of the states and operators in the scattering tensor. The adiabatic Born-Oppenheimer (ABO) approximation will be employed in which the vibronic states are formed by products of pure vibrational states Ai (Q) with pure electronic states i(Q,r). The latter are referred to some fixed positions of the nuclei and are therefore parametrically dependent on the internuclear distances expressed as a set, Q, of normal coordinates, r is a set of electronic coordinates, and i, u are electronic and vibrational quantum numbers respectively. Thus... 

If this expansion for the transition moments is used then the resonant part of the scattering tensor may be expressed, in terms to the first order in Q, as follows ... 

The matrix elements in the non-resonant part of the scattering tensor may be treated in the same manner. The final expression for Eq. (6), plus its non-resonant counterpart, is... 

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