Raman scattering derived polarizability tensor

The intensity of Rayleigh scattering and the linear Raman effect is governed by the polarizability tensor apa of a molecule and its derivatives with respect to the normal coordinates. When the electric field of the exciting radiation is very high, further terms in the expression for the induced dipole moment 104)... 

The occurrence of Raman scattering is connected to the change in polarizability during the transition of the molecule from one vibrational state to the other. Circular polarization ROA arises from interference of the electric dipole electric dipole polarizability tensor with the electric dipole - magnetic dipole and the electric dipole electric quadrupole optical activity tensors. Due to limited space, no rigorous derivation of the theory will be given here, but only the most important results shall be shown. 

JUU.C KXxy — Xyx, xxz — Xix> Xyz — zy> there are only six independent tensor elements. The intensity of the Raman scattering light on the surface is directly related to the derivative of the polarizability tensor, with respect to the given normal coordinates. In order to determine these tensor elements, it is necessary to consider the system, including the molecule and surface atoms, and to define the Cartesian coordinate axis, z, along the surface normal and others along the surface parallel. 

The intensity of the Raman lines is proportional to the product of the Raman scattering cross section aR, which depends according to (3.12) on the matrix elements atj) of the polarizability tensor and the density Ni of molecules in the initial state. If the cross sections aR have been determined elsewhere, the intensity of the Raman lines can be used for measurements of the population densities N(v, J). Assuming a Boltzmann distribution (3.11a), the temperature T of the sample can be derived from measured values of N(v, J). This is frequently used for the determination of unknown temperature profiles in flames [325] or of unknown density profiles in liquid or gaseous flows [326] at a known temperature (Sect. 3.5). 

The approach considered here to calculate the tensor components for the first derivatives of the polarizability for the interacting diatomic molecules can be applied to larger molecular complexes. Also, it may be useful to calculate the tensor components of higher derivatives of the polarizability. The values of the tensor components of the first derivatives of the polarizability for molecules N2 and O2 may be used to estimate the parameters of the light Raman scattering in the dimers. 

For D D , the reciprocals of these correlation times are raised by an amount [(D /D ) — 1] as indicated in Eq. (7.55). As noted previously, K rriL, ttim) values at a particular temperature are computed from (P2) and (P4). The fourth-rank order parameter P4) cannot be directly measmed from a NMR spectrmn, but may be derived from measurements of the mean square value of a second-rank quantity [7.19-7.22]. In the Raman scattering technique [7.21], the second-rank molecular quantity is the differential polarizability tensor of a localized Raman mode. In fluorescence depolarization [7.19], the average of the product of the absorption and emission tensors is used to determine (P4). Since there is a lack of experimental determination of (P4) in liquid crystals, this may be calculated based on the Maier-Saupe potential... 

Analytical derivative ab initio calculations on Raman scattering activities provide currendy results that are in fairly good accord with the available experimental data. The impact of the theoretical calculations is even greater in view of the difficulties in obtaining full polarizability derivative tensors from experimental measurements. 

Each aij can be expressed as a linear combination of the principal components a, 2 and q 3 of the derived polarizability or Raman tensor of the vibration investigated. The experimental values of the scattered intensities are of the form loSaijUp. ... 

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