Lorentz approximation

The collision terms may be simplified by using the condition that mjM is very small this leads to the Lorentz approximation. If there were no electric field, the equilibrium situation would be one in which the mean kinetic energy of the electrons would be equal to that of the... 

Evaluation for case of general central force law, 36 Lorentz approximation, 46 terms, 28,31,33 Collisions... 

Lorentz approximation, 46 Lorentz condition, 551 Lorentz gauge, 657,664 Lorentz group homogeneous, 490... 

Third, a further simplification of the Boltzmann equation is the use of the two-term spherical harmonic expansion [231 ] for the EEDF (also known as the Lorentz approximation), both in the calculations and in the analysis in the literature of experimental data. This two-term approximation has also been used by Kurachi and Nakamura [212] to determine the cross section for vibrational excitation of SiHj (see Table II). Due to the magnitude of the vibrational cross section at certain electron energies relative to the elastic cross sections and the steep dependence of the vibrational cross section, the use of this two-term approximation is of variable accuracy [240]. A Monte Carlo calculation is in principle more accurate, because in such a model the spatial and temporal behavior of the EEDF can be included. However, a Monte Carlo calculation has its own problems, such as the large computational effort needed to reduce statistical fluctuations. 

F(o>i) are the local field corrections for a"3wave of frequency Generally, one utilizes the Lorentz approximation for the local field in which case (1,4)... 

The measurement of x of solutions can be used to determine the microscopic nonlinearities Y of a solute, provided Y of the solvent is known. This measurement also provides information on the sign of y and (hence x of the molecules if one knows the sign of Y for the solvent (5,7) Under favorable conditions one can also use solution measurements to determine if Y is a complex quantity. The method utilizes two basic assumptions (i) the nonlinearities of the solute and the solvent molecules are additive, and (ii) Lorentz approximation can be used for the local field correction. Under these two assumptions one can write the x of the solution to be... 

We calculated for each solvent the molecular polarizability, P, with the Lorentz approximation ... 

The third-order nonlinear optical susceptibility x was obtained by comparing the measured signals for the sample with that of carbon disulfide as reference under the same experimental condition. The measured x value is 6.2 x 10 esu for the subphthalocyanine at a concentration 1.25 x 10 M. Considering an isotropic media under the Lorenz-Lorentz approximation, the second hyperpolarizability value was found to be 3.0 x 10 esu. Furthermore, A pirc corresponding to the pure subphthalocyanine 17 was calculated to be 6.9 x 10 esu, about four times higher... 

In the above equations, /3 and y" represent the microscopic coefficients at site n which are averaged over molecular orientations 0 and 0 and summed over all sites n. The terms F(a>i) are the local field corrections for a wave of frequency Generally, one utilizes the Lorentz approximation for the local field, where ... 

Here Pe is the electric dipole induced by the electric field in a molecule having mean molecular polarizability y. Then we used the Lorentz approximation for the local field acting on a molecule and found corresponding field induced polarization. From that we have obtained the electric susceptibility of the dielectric (Eq. 7.18) ... 

N Is the number of molecules per unit volume (packing density factor), fv Is a Lorentz local field correction at frequency v(fv= [(nv)2 + 2]/3, v = u) or 2u). Although generally admitted, this type of local field correction Is an approximation vdilch certainly deserves further Investigation. IJK (resp Ijk) are axis denominations of the crystalline (resp. molecular) reference frames, n(g) Is the number of equivalent positions In the unit cell for the crystal point symmetry group g bjjj, crystalline nonlinearity per molecule, has been recently Introduced 0.4) to get general expressions, lndependant of the actual number of molecules within the unit cell (possibly a (sub) multiple of n(g)). 

In Chapter 4 we will consider the so-called classical approximation, in which the electromagnetic radiation is considered as a classical electromagnetic wave and the solid is described as a continnous medium, characterized by its relative dielectric constant e or its magnetic permeability fx. The interaction will then be described by the classical oscillator (the Lorentz oscillator). 

The refractive index is an important quantity for characterizing the structure of polymers. This is because it depends sensitively on the chemical composition, on the tacticity, and - for oligomeric samples - also on the molecular weight of a macromolecular substance. The refractive indices (determined using the sodium D line) of many polymers are collected in the literature. In order to characterize a molecule s constitution one requires knowledge of the mole refraction, Rg. For isotropic samples, it can be calculated in good approximation by the Lorentz-Lorenz equation ... 

To obtain the absorbances at 910 and 967 cm. 1, it was necessary to correct the observed band intensities for the overlapping of adjacent bands. The band at 910 cm."1 for the vinyl group was corrected for the absorbance from the wing of the 967-cm."1 frarw-vinylene band,. and the latter band was corrected for the vinyl band at 995 cm. 1. The Lorentz band shape equation was used to calculate the absorbance in the wings, and in the thicker specimens, successive approximations were necessary. This treatment gave the four equations below, which yielded the concentrations of trans and vinyl groups for the emulsion and sodium polybutadienes listed in Table I. Implicit in these equations is the assumption that the absorptivities are independent of concentration. 

In the Lorentz gas approximation, this term is proportional to the number densities of atoms of type A and B, nA and nB, because the probability of finding an atom of the light species with a speed between vA and vA = dvA is given by the Maxwellian distribution function,... 

Here, instead of the usual delta function d(Eg, — Eg — ftv), which supplies only discontinuous spectral lines, a Lorentz function with a line width 29(normalized to one) is introduced for all vibrational levels labeled by n and m of electronic states j and k which contribute to the transition rate of Eq. (9) [58]. M is the transition operator, in general, pertinent to an electric dipole transition. The nuclear wavefunctions are approximated by products of harmonic oscillator functions... 

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