Macroscopic polarizability

If one assumes fibre symmetry of the sample (like in uniaxial deformation), transverse isotropy for the molecular units (for instance ai U2 = 03), and additivity of polarizabilities, it is easily shown [9] that the difference in macroscopic polarizabilities along and perpendicular to the fibre axis of the sample is simply ... 

It remains now to relate the molecular quantities ccq and p to the macroscopic polarizability or dielectric constant, which can be measured experimentally. This is a very difficult task and will not be carried out in a rigorous fashion here. Rather, we start our discussion with an approximate equation, given by Debye, which describes the complex dielectric constant in terms of molecular properties. We rationalize the form of the equation through the Clausius-Mosotti equation and then show how e (o)) and s"((o) can be derived from this expression. Additional factors that were not included in Debye s original work, such as the effect of the reaction field and orientation correlation-which are important in condensed phases-will also be discussed... 

The calculation of the intrinsic birefringence for a given molecular organization is almost impossible at present. It involves both molecular and macroscopic polarizability and the form of the Lorenz-Lorentz or internal fields relation relevant to an anisotropic molecular organization. The orientation parameter is given by ... 

Making the usual assumption that the birefringence arising from the deformation of swollen elastomer is a result of the orientation of optically anisotropic segments, then the mean macroscopic polarizability remains constant, (a, + + Oj) = 3o where... 

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. 

The susceptibility tensors give the correct relationship for the macroscopic material. For individual molecules, the polarizability a, hyperpolarizability P, and second hyperpolarizability y, can be defined they are also tensor quantities. The susceptibility tensors are weighted averages of the molecular values, where the weight accounts for molecular orientation. The obvious correspondence is correct, meaning that is a linear combination of a values, is a linear combination of P values, and so on. 

We define the concentration of fluctuation domains at any instant by the symbol N. In addition, we assume that the polarizability associated with one of these domains differs from the macroscopic average value for the substance... 

The theoretical methods can be divided into two fundamental groups. The so-called continuum models are characterized by assuming that the medium is a structureless and polarizable dielectricum described only by macroscopic physical constants. On the other hand there are the so-called discrete models. The main advantage of... 

In Eq. (6) Ecav represents the energy necessary to create a cavity in the solvent continuum. Eel and Eydw depict the electrostatic and van-der-Waals interactions between solute and the solvent after the solute is brought into the cavity, respectively. The van-der-Waals interactions divide themselves into dispersion and repulsion interactions (Ed sp, Erep). Specific interactions between solute and solvent such as H-bridges and association can only be considered by additional assumptions because the solvent is characterized as a structureless and polarizable medium by macroscopic constants such as dielectric constant, surface tension and volume extension coefficient. The use of macroscopic physical constants in microscopic processes in progress is an approximation. Additional approximations are inherent to the continuum models since the choice of shape and size of the cavity is arbitrary. Entropic effects are considered neither in the continuum models nor in the supermolecule approximation. Despite these numerous approximations, continuum models were developed which produce suitabel estimations of solvation energies and effects (see Refs. 10-30 in 68)). 

Birefringence is one of the simplest methods for the characterization of molecular orientation in polymers. The polarizability of a structural unit is usually not equivalent in all directions, leading to three independent refractive indices along its principal axes. In an isotropic sample, a single averaged macroscopic refractive index is observed whereas birefringence or trirefringence is observed... 

Fig. 2.2 Self-Consistent Reaction Field (SCRF) model for the inclusion of solvent effects in semi-empirical calculations. The solvent is represented as an isotropic, polarizable continuum of macroscopic dielectric e. The solute occupies a spherical cavity of radius ru, and has a dipole moment of p,o. The molecular dipole induces an opposing dipole in the solvent medium, the magnitude of which is dependent on e.

Most of biological reactions take place in a highly polarizable medium which contains mobile polar water molecules, reorientable polar groups, and mobile ions. For macroscopic media, the energetics of an electric charge distribution placed in a vicinity of a polarizable medium can be described by means of the classical dielectric theory159. 

Nonintuitive Light Propagation Effects In Third-Order Experiments. One of the first tasks for a chemist desiring to quantify second- and third-order optical nonlinear polarizability is to gain an appreciation of the quantitative manifestations of macroscopic optical nonlinearity. As will be shown this has been a problem as well for established workers in the field. We will present pictures which hopefully will make these situations more physically obvious. 

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 constant a is the molecular polarizability, and the macroscopic constant X(1) is known as the linear susceptibility. The molecular polarizability is related... 

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