Lorentz cavity

Yh fyT = X)( z) > as cubic and completely disordered lattices (incidentally, the type of arrangements for which in the classical Lorentz cavity-field calculation the contribution of the dipoles inside the small sphere vanishes) these lattice sums vanish for large spherical samples. The sums and... 

Because of the long-range nature of the dipolar interaction, care must be taken in the evaluation of the dipolar field. For hnite systems the sums in Eq. (3.12) are performed over all particles in the system. Eor systems with periodic boundary conditions the Ewald method [57-59], can be used to correctly calculate the conditionally convergent sum involved. However, in most work [12,13] the simpler Lorentz-cavity method is used instead. 

These equations were first derived by Debye,33 107 with the assumption that the relationship between FA and EA was for a Lorentz cavity containing a medium with the macroscopic dielectric constant and no... 

The quantities g and gc describe the shapes of the inclusion and the Lorentz cavity, respectively. For the general case of randomly oriented ellipsoids, Polder... 

More attentive readers noticed other aspects of the Onsager paper. Kirkwood [6] in 1938 remarked that Onsager introduced a real cavity, conceptually quite different from the Lorentz cavity which is just a mathematical device. 

Fig. 7.4 Lorentz model for the local field. Polarization of an ellipsoidal form dielectric sample and appearance of depolarizing field Ei (a), Lorentz cavity field E2 and the field of individual molecules within the cavity E3 (b)...

In order to formulate a theory for the evaluation of vibrational intensities within the framework of continuum solvation models, it is necessary to consider that formally the radiation electric field (static, Eloc and optical E[jc) acting on the molecule in the cavity differ from the corresponding Maxwell fields in the medium, E and Em. However, the response of the molecule to the external perturbation depends on the field locally acting on it. This problem, usually referred to as the local field effect, is normally solved by resorting to the Onsager-Lorentz theory of dielectric polarization [21,44], In such an approach the macroscopic quantities are related to the microscopic electric response of... 

In the relations between the macroscopic susceptibilities y , y and the microscopic or molecular properties a, ft, y, local field corrections have to be considered as explained above. The molecule experiences the external electric field E altered by the polarization of the surrounding material leading to a local electric field E[oc. In the most widely used approach to approximate the local electric field the molecule sits in a spherical cavity of a homogenous media. According to Lorentz the local electric field [9] is... 

Let us first review the basics of the Lorentz theory for polarization. If one assumes that a constant macroscopic field is applied to a homogeneous medium of dielectric constant s, the polarization through the medium will be uniform. However, the polarization of a molecule is not proportional to the macroscopic electric field (created by sources external to the dielectric), but to the local electric field, which contains also the field generated by all the other molecules of the dielectric. To account for the latter, one can separate the medium in a spherical cavity (in which the central molecule and its molecular neighbors reside, see Fig. 1 A) and the rest of the medium, which... 

In the case of a static field, the macroscopic relative permittivity e° has to be used in (82) for the cavity field factor, while the optical relative permittivity extrapolated to infinite wavelength e can be applied to estimate the static polarizability a(0 0) in (84). In this way the Onsager-Lorentz factor for a pure dipolar liquid is obtained (87). 

The polarizability a(-w w) is involved in several linear optical experiments including refractive index measurements. Equation (93) shows that the solute molecule experiences a local field which is larger than the macroscopic field by the cavity field factor/ " and by the reaction field factor f For typical media the magnitude of the productis of the order of 1.3-1.4. In the case of a pure liquid this product simplifies to the Lorentz factor L", (86), and (94) simphfies to (95)... 

If the term in the derivative of the field factor were negligible the expression on the left of this equation would be defined completely in terms of macroscopic measurable quantities. The specifics of the chosen cavity model enter the field factor derivative where Lorentz-Lorenz and Onsager factors may be mixed. The most commonly used procedure is to employ Onsager for the static field and Lorentz factors for the optical fields. For Fj (i = 0,1),... 

Liquid Phase Calculations of the Linear Response. The data in Table 5 for the isotropic polarizability, derived formally via the Lorentz-Lorenz equation (1) from the measured refractive index, shows that the assumption that individual molecular properties are largely retained at high frequency in the liquid is very reasonable. While the specific susceptibilities for the gas and liquid phases differ, once the correction for the polarization of the surface of a spherical cavity, which is the essential feature of the Lorentz-Lorenz equation, has been applied, it is clear that the average molecular polarizabilities in the gas and liquid have values which always agree within 5 or 10%. 

Among the few determinations of of molecular crystals, the CPHF/ INDO smdy of Yamada et al. [25] is unique because, on the one hand, it concerns an open-shell molecule, the p-nitrophenyl-nitronyl-nitroxide radical (p-NPNN) and, on the other hand, it combines in a hybrid way the oriented gas model and the supermolecule approach. Another smdy is due to Luo et al. [26], who calculated the third-order nonlinear susceptibility of amorphous thinmultilayered films of fullerenes by combining the self-consistent reaction field (SCRF) theory with cavity field factors. The amorphous namre of the system justifies the choice of the SCRF method, the removal of the sums in Eq. (3), and the use of the average second hyperpolarizability. They emphasized the differences between the Lorentz Lorenz local field factors and the more general Onsager Bbttcher ones. For Ceo the results differ by 25% but are in similar... 

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