Optical polarizability tensor

In the case of axially symmetric molecules the optical polarizability tensor a has the form (86) (similar to electric polarizability), and ... 

As implied by the trace expression for the macroscopic optical polarization, the macroscopic electrical susceptibility tensor at any order can be written in temis of an ensemble average over the microscopic nonlmear polarizability tensors of the individual constituents. 

Table 6-1. C2(l molecular poinl group. The electronic stales of the flat T6 molecule are classified according lo the lwo-1 old screw axis (C2). inversion (/). and glide plane reflection (o ) symmetry operations. The A and lt excited slates transform like translations Oi along the molecular axes and are optically allowed. The Ag and Bg stales arc isoniorphous with the polarizability tensor components (u), being therefore one-photon forbidden and Iwo-pholon allowed.

The fundamental scattering mechanism responsible for ROA was discovered by Atkins and Barron (1969), who showed that interference between the waves scattered via the polarizability and optical activity tensors of the molecule yields a dependence of the scattered intensity on the degree of circular polarization of the incident light and to a circular component in the scattered light. Barron and Buckingham (1971) subsequently developed a more definitive version of the theory and introduced a definition of the dimensionless circular intensity difference (CID),... 

Optical activity in light scattering thus arises from interference between the molecular polarizability and optical activity tensors. 

A simple calculation for urea by Spackman is instructive. Urea crystallizes in an acentric space group (it is a well-known nonlinear optical material), in which the symmetry axes of the molecules coincide with the two-fold axes of the space group. All molecules are lined up parallel to the tetragonal c axis. If the electric field is given by E, and the principal element of the diagonalized molecular polarizability tensor along the c axis by oc , the induced moment along the polar c axis is... 

The stress-optical behaviour of an elastomeric network of PDET is measured over a wide range of elongation ratios and temperatures. Theoretical calculations are carried out with the RIS model. For Act, no reasonable modification of the conformational energies or contributions to the anisotropic part of the polarizability tensor would achieve agreement between theory and experiments. The discrepancy between theoretical and experimental results may be qualitatively explained by intermolecular interactions. Agreement between theory and experiment is only obtained assuming the unlikely value of about + 4.2 kJ mol-1 for E(on). 

Of particular interest in connection with the derivation of the stress-optical law is, that an analogous equation exists for the average polarizability tensor of the single chain 62). This expression is derived from the work of Kuhn and Grun (64) and reads ... 

The polarizability tensor, a, introduced in section 4.1.2, is a measure of the facility of the electron distribution to distortion by an imposed electric field. The structure of the electron distribution will generally be anisotropic, giving rise to intrinsic birefringence. This optical anisotropy reflects the average electron distribution whereas vibrational and rotational modes of the molecules making up a sample will cause the polarizability to fluctuate in time. These modes are discrete, and considering a particular vibrational frequency, vk, the oscillating polarizability can be modeled as... 

For oriented samples, the rotation of the plane-polarized light becomes a tensor - that is, the optical rotation becomes directionally dependent - and includes a contribution from the electric dipole-electric quadrupole polarizability tensor, which is traceless and thus vanishes for freely rotating molecules [30], The term arising from these quadrupolar interactions can be expressed as [30]... 

Early theoretical treatments [3,4] of optically active scattering by molecules did little to arouse renewed interest in a measurement of ROA. The decisive cross-terms between the electric dipole-electric dipole polarizability and the optical activity tensor were missed, and effects predicted for the optical activity tensor alone were too small to be practically useful. It was only after the proper cross-terms were identified [5] that new interest in the measurement of ROA accrued, and that the existence of the phenomenon was finally proved [6,7], It was not long before the first measurement of whole ROA spectra was demonstrated [8],... 

The orientation of the polarizability tensor related to the planar molecule axes (i.e., to the crystal structure) is investigated by measuring the orientation and dispersion of the indicatrix. Usually, the principal axes of polarizability coincide with the directions of the three principal axes of the indicatrix X, Y, and Z. Perpendicular to each of the two circular sections of radius np is an optic axis. The maximum absorption is always parallel to one of the three optical directions. 

A surface is illuminated with a high-intensity laser, and photons are generated at the second-harmonic frequency through non-linear optical process. For many materials only the surface region has the appropriate symmetry to produce a SHG signal. The non-linear polarizability tensor depends on the nature and geometry of adsorbed atoms and molecules. 

Non-linear optical techniques, such as second harmonic generation (SHG), have recently been used as surface probes. Bulk materials with inversion symmetry do not generate second harmonic signals, while surfaces and interfaces cannot have inversion symmetry, so the total SHG signal will come from the surface region for many systems. The components of the non-linear polarizability tensor have been used to determine the orientation of chemisorbed molecules. 

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. 

Similarly to the transition polarizability the vibrational transition optical activity tensors are written as ... 

Translational-orientational fluctuations. Re-writing the expansion (195) of the optical polarizability deviation tensor in a manner adapted to multi-component systems (as done previously for electric polarizability), one obtains from the general formula an additional contribution from binary correlations of axially symmetric molecules, of the form ... 

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