The Lorentz-Lorenz Equation

The electric field, E, in equation (7.3) is the total field experienced by each dipole and is the sum of the external field, EQ, and an internal field, E(.. This latter field accounts 

Although this result was derived from the special case of an isotropic sphere, the above relationship between the total field and the polarization also applies to anisotropic systems. Furthermore, the volume of the sphere can be made arbitrarily small to envelop a single dipole. 

Combining equations (7.1), (7.2), (7.3), and (7.5) the results for the polarization using both the macroscopic and microscopic models can be equated to yield 

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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 ... 

For white pigments, the hiding power can be expressed through the Lorentz-Lorenz equation (4) as a function of pigment, and medium, nm,... 

The Lorentz-Lorenz equation [2] defines the molar refraction, RD, as a function of the refractive index, density, and molar mass ... 

In this subsection, the connection is made between the molecular polarizability, a, and the macroscopic dielectric constant, e, or refractive index, n. This relationship, referred to as the Lorentz-Lorenz equation, is derived by considering the immersion of a dielectric material within an electric field, and calculating the resulting polarization from both a macroscopic and molecular point of view. Figure 7.1 shows the two equivalent problems that are analyzed. 

The Lorentz-Lorenz equation can be used directly to model the birefringence of a solution of rigid rod molecules subject to an orienting, external field. Figure 7.2 shows a representative molecule, which is modeled as having a uniaxial polarizability of the form... 

The Lorentz-Lorenz equation can be used to express the components of the refractive index tensor in terms of the polarizability tensor. Recognizing that the birefringence normalized by the mean refractive index is normally very small, ( A/i / 1), it is assumed that Aa /a 1, where the mean polarizability is a = (al + 2oc2)/3 and the polarizability anisotropy is Aa = a1-a2. It is expected that the macroscopic refractive... 

This is a simplification of the Lorentz-Lorenz equation. Looyenga showed that the expression (n2— 1 )/(n2 + 2) can, with high accuracy be approximated by the more simple expressions (n213 — 1) for the polymer refraction indices mentioned in Table 10.5, the differences vary from 2.9% (n = 1.35) to 8.8% (n = 1.654). 

Fig. 4 C02 refractive index as a function of pressure at four different temperatures (calculated for X = 546 nm, based on the Lorentz-Lorenz equation [2])...

For semi-polar substances or mixtures of semi-polar substances and non-polar substances the Lorentz-Lorenz equation applies... 

One of the most widely used steric parameters is molar refraction (MR), which has been aptly described as a "chameleon" parameter by Tute (160). Although it is generally considered to be a crude measure of overall bulk, it does incorporate a polarizability component that may describe cohesion and is related to London dispersion forces as follows MR = 4TrNa/By where N is Avogadro s number and a is the polarizability of the molecule. It contains no information on shape. MR is also defined by the Lorentz-Lorenz equation ... 

As implied by the discussion above craze fibril extension ratio or its inverse the fibril volume fraction of the craze is an important parameter of the microstructure. Fibril volume fractions can be measured by several different methods. The refractive index n of the craze can be measured by measuring the critical angle for total reflection of light by the craze surface. Using the Lorentz-Lorenz equation Vf then can be computed from The method is difficult because small variations... 

The calculated moment thus considerably exceeds the experimental value and furthermore represents the dipole as acting in the opposite direction the chlorine is represented as positive and the hydrogen negative. This result is clearly incorrect and Debye has shown that the error is due to the fact that the Lorentz-Lorenz equation is not valid at the small distances considered owing to the non-uniform character of the field. If the internuclear distances were of the order of 5 A, this type of calculation would be permissible. Attempts have been made to calculate the polarizability in a non-uniform electric field by the methods of wave mechanics , but have not yet been successful in producing a theory of the intermediate type of bond. 

Refractive Index Experimental Data for Gas and Liquid. From a measured refractive index it is always possible to extract formally an average linear dipole polarizability, a, from the Lorentz-Lorenz equation,... 

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%. 

The application of the Lorentz-Lorenz equation gives a convincing demonstration of the general similarity of the linear response in gas and liquid but its application in the liquid introduces an approximation which has not yet been quantified. A more precise objective for the theory would be to calculate the frequency dependent susceptibility or refractive index directly. For a continuum model this may lead to a polarizability rigorously defined through the Lorentz-Lorenz equation as shown in treatments of the Ewald-Oseen theorem (see, for example Born and Wolf, plOO),59 but the polarizability defined in this way need not refer to one molecule and would not be precisely related to the gas parameters. 

This is called the Lorentz Lorenz equation, and is used to estimate the molecular refraction Pmfrom the refractive index or Sop. Since the polarizability oCp is often... 

Since values of the molecular polarizability are not available from gas phase measurements for most of the polar solvents considered here, they may be estimated using the MSA in the following way. First, a value of X(,p is calculated by solving equation (4.4.18) using the experimental value of Sop. Then, the parameter is estimated using equation (4.4.21). Finally, the polarizability is found from equation (4.4.19). The results obtained are summarized in table 4.4. They are very close to estimates obtained on the basis of the Lorentz-Lorenz equation (see equation (4.3.21) and table 4.3), usually being a few percent higher. 

According to the Lorentz-Lorenz equation (4.3.21) for the molar refraction at optical frequencies, Y is directly proportional to the molecular polarizability p. The Koppel-Palm equation has also been applied to the analysis of solvent effects on thermodynamic quantities related to the solvation of electrolytes [48, 49]. In the case of the systems considered in table 4.11, addition of the parameter X to the linear equation describing the solvent effect improves the quality of the fit to the experimental data, especially in the case of alkali metal halide electrolytes involving larger ions. The parameter Y is not important for these systems but does assist in the interpretation of other thermodynamic quantities which are solvent dependent [48, 49]. Addition of these parameters to the analysis is only possible when the solvent-dependent phenomenon has been studied in a large number of solvents. 

For this type of isotherm, represents the maximum loading, which correlates with pore volnme among different adsorbents. The other isotherm parameters, and Po [no relation to the terms in Eqnations (14.4) or (14.5)], represent the characteristic parameter of the adsorbent and an affinity coefficient of the compound of interest, respectively. The characteristic parameter, A, defines the shape of the n versns e cnrve. The affinity coefficient, po, adapts the compound of interest to the characteristic cnrve. It is a fndge factoT that has been correlated to the ratio of molar volumes, parachors, or polarizabilities (via the Lorentz-Lorenz equation) of the componnd of interest to that of a reference component (e.g., benzene or n-heptane). These three methods are ronghly eqnivalent in accuracy. The molar volume version is = The only controversy is whether to nse the... 

Spin casting methods are used extensively in the microelectronics industry. We have adopted the best practices available to achieve full density thin films. The full density (po = 1.186 g/cm ) is used for PMMA, justified by measurement of the full refractive index at 632.8 nm n = 1.49) of our spin coated thin films to within = 0.02. The density of PVN was measured by gas pychnometry to be po = 1.34 g/cm somewhat different from the literature [67]. Although there is no literature value for n, the measured n = 1.50 0.03 compares well with the precursor polyvinylalcohol n = 1.52 [68]. An uncertainty in of 0.03 translates into a density uncertainty of 5% via the Lorentz-Lorenz equation, essentially p OC (n -l)/(n +2). Although the samples are likely to be full density, up to 5% porosity may be present (note that... 

In eqn (8.4), k is the rate constant for the hydrolysis of an R substituted ester, and is corresponding constant for the methyl substituted parent, thus all comparisons are made between the substituent and a methyl group. These substituent constant values are used in the same way as the electronic and hydrophobic substituent constants discussed earlier, that is to say they are found in tabulations of substituent constant values, and of course the same problems of missing values apply. In fact, the situation can be even worse for Es as a number of substituents are themselves unstable under the conditions of acid hydrolysis. It has also been argued that this descriptor is not just a measure of steric effects, but that it also includes some electronic information. A number of more or less ingenious fixes were proposed to solve such problems, but a much more popular and generally useful measure of steric effects for both substituents and whole molecules was adopted in the form of molar refraction (MR), as defined by the Lorentz-Lorenz equation [eqn (8.5)]. 

This relationship as such is not well obeyed for most compounds if the static or low-frequency relative permittivity is used, as can be judged from Table 11.1. The relationship can be correctly interpreted by using the relative permittivity due to electronic polarisation in the equation. With this in mind, substitution of the relationship given in Equation (11.10) into the Clausius-Mossotti equation yields the Lorentz-Lorenz equation ... 

It follows from Maxwell s theory of electromagnetic radiation that e = n, where s is the dielectric constant measured at the frequency for which the refractive index is n. Equation (9.8) thus leads immediately to the Lorentz-Lorenz equation... 

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