Clausius-Mossotti theory

When combined with the classical Clausius-Mossotti theory of a dielectric medium, this version of the local response model gives the correct asymptotic form of the polarization potential of a spherical atom, —a(0)/2r4, where... 

When the analysis is carried out in this way it does not yield satisfactory results from a bonding viewpoint, because the values of E and Eq that are obtained do not yield chemically meaningful values of hojp, i.e., the empirical values of cOp do not correlate with N, as required by (9). (In this respect the arbitrariness of the parameters resembles that of the Clausius-Mossotti theory.) However, this approach does represent an improvement over the classical static model because there is established a proportionality between and iVg, the number of valence electrons per cation-anion pair, Z , the anion valence, and N, the cation coordination number, which plays a crucial part in mechanical theories of structures of minerals formally, we have... 

The classical treatment of nonpolar dielectric materials is expressed by the Clausius-Mossotti equation. Polar materials in nonpolar solvents are better handled by Debye s modification, which allows for the permanent dipole of the molecule. Onsager made the next major step by taking into account the effect of the dipole on the surrounding medium, and finally Kirkwood treated the orientation of neighboring molecules in a more nearly exact manner. (See Table 2-1.) The use of these four theoretical expressions can be quickly narrowed. Because of their limitations to nonpolar liquids or solvents, the Clausius-Mossotti and Debye equations have little application to H bonded systems. Kirkwood s equation has great potential interest, but in the present state of the theory of liquids the factor g is virtually an empirical constant. The equation has been applied in only a few cases. 

In 1906, J. C. Maxwell Garnett used the Maxwell Garnett theory, equation (12), for the first time to descibe the color of metal colloids glasses and of thin metal films. Equation (12) can be deviated from the Rayleigh scattering theory for spherical particles [21], or from the Lorentz-Lorenz assumption for the electrical field of a sphere and the Clausius-Mossotti Equation by using the polarizability of an metal particle if only dipole polarization is considered [22]. 

Lorenz-Lorentz theory addressed the issue by extending the approach of Clausius-Mossotti to optical frequency fields (12,13). This extension relies on a spherical cavity (compare with the need for a needle-shaped cavity) and takes into account the effect of other charges. The only thorny issue is that a spherical cavity is not the best choice for anisotropic molecules. Nevertheless, the Lorenz-Lorentz approach has been widely used in studying optical properties of polymers (14). The expression of the local field is given by... 

It can be concluded that remanent polarization and hence the piezoelectric response of a material are determined by Ae this makes it a practical criterion to use when designing piezoelectric amorphous polymers. The Dielectric relaxation strength Ae may be the result of either free or cooperative dipole motion. Dielectric theory yields a mathematical approach for examining the dielectric relaxation Ae due to free rotation of the dipoles. The equation incorporates Debye s work based on statistical mechanics, the Clausius-Mossotti equation, and the Onsager local field and neglects short-range interactions (43) ... 

Semiconductor cluster polarizabilities have been the subject of some very important experimental studies via beam-deflection techniques (Backer 1997 Schlecht et al. 1995 Schnell et al. 2003 Schafer et al. 1996 Kim et al. 2005) while they have been extensively studied using quantum chemical and density functional theory. In this research realm, one of the areas intensively discussed is the evolution of the cluster s polarizabilities per atom (PPA) with the cluster size. The PPA is obtained by dividing the mean polarizability of a given system by the number of its atoms. Such property offers a straightforward tool to compare the microscopic polarizability of a given cluster with the polarizability of the bulk (see O Fig. 20-16) as the latter is obtained by the hard sphere model with the bulk dielectric constant via the Clausius-Mossotti relation ... 

A (r, t)l8Q represents local fluctuations of the relative permittivity (dielectric constant) in the target. The task of classical light-scattering theory has been reduced to solving Equation [34]. The only role of quantum theory, therefore is to calculate the atomic polarizability a - m other words, the microscopic properties of the scattering medium. Once this is achieved, the polarizability can then be related to the dielectric constant through the well-known Clausius-Mossotti relation. 

Continuum models have a long and honorable tradition in solvation modeling they ultimately have their roots in the classical formulas of Mossotti (1850), Clausius (1879), Lorentz (1880), and Lorenz (1881), based on the polarization fields in condensed media [32, 57], Chemical thermodynamics is based on free energies [58], and the modem theory of free energies in solution is traceable to Bom s derivation (1920) of the electrostatic free energy of insertion of a monatomic ion in a continuum dielectric [59], and Kirkwood and Onsager s... 

H. A. Lorentz, Theory of Electrons, Teubner, Leipzig, 1909 (reprinted by Dover, New York, 1952). O. F. Mossotti, Bibl. Univ. Modena, 6, 193 (1847) Mem. Math. Fis. Modena, 24 11, 49 (1850). R. Clausius, Die mechanische Warmetheorie Vol. 11, Braunschweig, 1879. In Ann. Phy., 49, 1 (1916), Ewald showed that for a lattice of polarizable atoms of cubic symmetry, the local field is essentially that of the continuum considered by Lorentz. 

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