Lorentz-Lorenz refraction equation

The Onsager function is dependent on the refractive index of the solvent (n). For supercritical C02 the refractive index is dependent on the fluid density and can be calculated from the Lorentz-Lorenz refraction equation ... 

Pressure-induced changes in the refractive index and the thickness are theoretically rationalized by Tait s and Lorentz-Lorenze s equations. Studies... 

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 ... 

Clausius-Mossotti equation). In this expression, V designates the mole volume and Ae, Be, Cf,... are the first, second, third,... virial dielectric coefficients. A similar expansion exists for the refractive index, n, which is related to the (frequency dependent) dielectric constant as n2 = e (Lorentz-Lorenz equation, [87]). The second virial dielectric coefficient Be may be considered the sum of an orientational and a polarization term, Be = B0r + Bpo, arising from binary interactions, while the second virial refractive coefficient is given by just the polarization term, B = Bpo at high enough frequencies, the orientational component falls off to small values and the difference Be — B may be considered a measurement of the interaction-induced dipole moments [73],... 

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 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... 

If I is oriented along the principal-axis system of the optical indicatrix, then for each of the three components of the refractive index (nlr n2, n3) the anisotropic Lorentz-Lorenz equation is... 

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])...

The linear and non-linear polarizabilities of organic molecules are usually determined from measurements of macroscopic susceptibilities of liquid solutions. Classical examples are the measurements of the refractive index, n, or the relative permittivity of pure organic liquids and their interpretation by the well-known Lorentz-Lorenz and Clausius-Mosotti equations. These... 

Rm is the molar refraction or Lorentz-Lorenz function, and As, Ba, Cb, are the refractivity virial coefficients. The rigorous derivation of the statistical mechanical equations for / m> Ab, and Bb, corresponding to equations (5)—(11), is complicated by the variation of the electric field... 

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... 

To calculate micelle size and diffusion coefficient, the viscosity and refractive index of the continuous phase must be known (equations 2 to 4). It was assumed that the fluid viscosity and refractive index were equal to those of the pure fluid (xenon or alkane) at the same temperature and pressure. We believe this approximation is valid since most of the dissolved AOT is associated with the micelles, thus the monomeric AOT concentration in the continuous phase is very small. The density of supercritical ethane at various pressures was obtained from interpolated values (2B.). Refractive indices were calculated from density values for ethane, propane and pentane using a semi-empirical Lorentz-Lorenz type relationship (25.) Viscosities of propane and ethane were calculated from the fluid density via an empirical relationship (30). Supercritical xenon densities were interpolated from tabulated values (21.) The Lorentz-Lorenz function (22) was used to calculate the xenon refractive indices. Viscosities of supercritical xenon (22)r liquid pentane, heptane, decane (21) r hexane and octane (22.) were obtained from previously determined values. 

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... 

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. 

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... 

From Tables 3, 5, and 6 it is seen that refractions change in some inverse manner with the wave-length A of the light by which they are measured the variations originate, of course, in the refractive indices entering the Lorentz-Lorenz function (1). Since 1827, a number of equations have been developed to describe dispersions of refractive indices n (Wood, 1934, and Partington, 1953, give historical and other details) of these, those due to Cauchy (9) and Sellmeier (10) appear to be best known and most used... 

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)]. 

Measurements of the refractive index, , of a gas at different pressures also provide information on the second virial coefficient since similar equations are obtained to the above, but with replacing e. The Lorentz-Lorenz function is given by ... 

The Lorentz-Lorenz molar refractivity, R, is obtained from Eq. (1), relating the geometric mean refractive index, n, the formula weight, M, and the density,(f. Where the unit cell dimensions and number of formula units per cell areknown, the second equation is useful, avoiding the unnecessary addition of the formula weight. 

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