Radiated fields polarizability tensor

In inelastic Raman scattering a photon loses (or gains) one quantum of rotational or vibrational energy to (or from) the molecule. The process involves the electric field of the radiation inducing an electric dipole in the molecule and so depends on the polarizability tensor of the molecule. (A (second-order) tensor is a physical quantity with nine components.) The induced electric dipole D is proportional to the electric field E ... 

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)... 

It is interesting to note that formula (64) is applicable to two cylindrically symmetric particles, such as between two nanotubes [43, 61], if fhe applied field is the only source of radiation. If fhis is fhe case, only the principal axis (diagonal elements) of the polarizability tensors contribute, corresponding to the component aligned in the same direction as the laser polarization. 

A more striking example is that of symmetry, which has been discussed by Reid and Richardson (1983a). If we adopt the mechanism of the inhomogeneous dielectric, each polarizability tensor a for ligand L at leads to the contribution (o< ) induced dipole moment produced by the electric field E of the radiation field. The potential at a 4f electron at tj is given by... 

Polarizability Tensor. If the molecule m which an electric dipole y is induced is isotropic, then the relation between y and the electric field ector of the incident radiation is simply given by... 

This generalizes the expression for the energy of - E acquired by an atom of polarizability a in an external electric field E. For the Fourier component eE(io) exp(-io)t) of the incident radiation field, the frequency-dependent atomic polarizability tensor has the form... 

The fundamental equation (1) describes the change in dipole moment between the ground state and an excited state jte expressed as a power series of the electric field E which occurs upon interaction of such a field, as in the electric component of electromagnetic radiation, with a single molecule. The coefficient a is the familiar linear polarizability, ft and y are the quadratic and cubic hyperpolarizabilities, respectively. The coefficients for these hyperpolarizabilities are tensor quantities and therefore highly symmetry dependent odd order coefficients are nonvanishing for all molecules but even order coefficients such as J3 (responsible for SHG) are zero for centrosymmetric molecules. Equation (2) is identical with (1) except that it describes a macroscopic polarization, such as that arising from an array of molecules in a crystal (10). 

The coeflScients a, P, and y are the second, third, and fourth rank tensors and are referred to as the polarizability, first hyperpolarizability, and second hyperpolarizability, respectively. The hyperpolarizability terms are responsible for the nonlinear response of the molecule to impinging radiation. These coefiBcients are not very large, and the associated nonlinear optical effects are usually studied by taking advantage of the high optical field obtainable with laser beams. 

The classic absorption, scattering, reflection, or refraction, the intensity of the light reaching the detector is proportional to the intensity of the incident radiation. When one or more laser beams propagating in materials are large enough, the induced polarization fields are proportional to the product of the incident fields. The polarization p. induced in an atom or a molecule by an external field E can be written as Eq. (1). Where the vectors of p- and E are related by the tensors a, 3, and 7, which are often referred to as the polarizability, hyperpolarizability, and second hyperpolarizability, respectively. Similarly, the polarization induced in a medium by an optical field, can be represented by a power series in the optical fields [Eiq. (2)] where X " is the nth-order susceptibility. 

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