Lorentz local field theory

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... 

From the point of view of theory, the formulae of Table 2.6 are equally applicable to both gas and condensed phase samples, as they include the local field factors, which account for local modifications to the Maxwell fields due to bulk interactions within the Onsager-Lorentz model. 

CPT invariance is a fundamental property of quantum field theories in flat space-time which results from the basic requirements of locality, Lorentz invariance and unitarity [1,2,3,4,5]. A number of experiments have tested some of these predictions with impressive accuracy [6], e.g. with a precision of 10-12 for the difference between the moduli of the magnetic moment of the positron and the electron [7] and of 10-9 for the difference between the proton and antiproton charge-to-mass ratio [8],... 

In traditional Lorentz—Debye theory, the local field that acts on a water dipole is given by E + P/3so, where E is the macroscopic field and P is the polarization [5], In the present case, the local field /ilocal should also include the field Ep generated by the neighboring dipoles of a water molecule. 

The crudest approximation to the density matrix for the system is obtained by assuming that there are no statistical correlations between the elementary excitations (perfect fluid), so that can be written as a simple product of molecular density matrices A. A better approximation is obtained if one does a quantum field theory calculation of the local field effects in the system which in a certain approximation gives the Lorentz-Lorenz correction L(TT) in terms of the refractive index n53). One then writes,... 

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... 

We show below how these inaccuracies in the Lorenz-Lorentz formula can be eliminated. We employ the local field method to modify this formula and apply it to anisotropic organic crystals of complex structure. We discuss below also a variety of their optical properties which were earlier analyzed in a less general form only in the framework of exciton theory. We explain how it has... 

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. 

Lorentz, using classical electrostatic theory, showed that the local field in an isotropic insulator such as a gas, a glass or a crystal with cubic symmetry is uniform everywhere and given by ... 

Finally, (2J + 1) is the degeneracy of the initial state and the expression involving the refractive index n is known as Lorentz s local-field correction. Calculations of transition probabilities within file frame of JO theory are usually made assuming that all Stark sublevels within the ground level are equally populated and that the material under investigation is optically isotropic. The former hypothesis is only reasonable in some cases, e.g., when transitions initiate from non-degenerate states such as Eu( Fo). Otherwise, there is a Boltzmann distribution of the population among the crystal-field sublevels. The second assumption is not valid for uniaxial or biaxial crystals, but, of course, holds for solutions. 

Friedrich (1970) has measured the absolute infrared intensities of the two optic librational modes of the solids of CICN and BrCN. He also calculated the intrinsic intensities of these modes using a dipolar coupling theory referred to in Section IVB.2. Good agreement was obtained between calculated and measured values. It was found that the local field correction [Eq. (4.22)] causes an increase of about 100% in the theoretical intensities for these substances, whereas the Lorentz effective field correction gave rise to an increase of less than 50%. The samples were prepared by deposition of gas on a cold window and the sample thicknesses were measured by an interference fringe technique. 

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... 

The free thermal of the phenomenon electrons in the interstellar medium behaves classically, and it is straightforward to apply the Lorentz theory to the transfer of polarized radiation through this medium. The index of refraction depends on the sense of polarization because the resonant frequency of the electron is moved to greater or lesser values depending on the sense of helicity of the particles around the local field direction. Thus, if the index of refraction of an electron gas 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... 

With regard to dipolar liquids and thdr solutions in non-dipolar solvents, Piekara developed a complete theory of dipolar couplings calculating all three correlation factors Rcn, Pk, and Rs- These studies revealed that nitrobenzene molecules coalesce momentarily into antiparalld aggregates, which then tend to couple mutually into almost parallel pairs. Feterlin and Stuart performed a detailed study of the influence on Ae not orjly of Debye dipole couplings but moreover of anisotropy of the local Lorentz field. 

The principle of local invariance in a curved Riemannian manifold leads to the appearance of compensating fields. Like the electromagnetic field, which is the compensating field of local phase transformation, the gravitational field may be interpreted as the compensating field of local Lorentz transformations. In modern physics all interactions are understood in terms of theories which combine local gauge invariance with spontaneous symmetry breaking. 

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