Refractive index Lorentz-Lorenz equation

Keywords Silicone, polysiloxane, refractive index, Lorentz-Lorenz equation, polarizability, free volume, dielectric constant... 

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... 

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])...

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... 

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... 

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... 

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... 

Equations (9.40) can be used to evaluate the refractive indices for the crystals of any polymer with orthorhombic symmetry, provided that the co-ordinates of all the atoms within the crystal structure are known, so that all the bond orientations can be calculated. The value of an for the crystal is substituted into the Lorentz Lorenz equation (9.9) in place of a and the value of n obtained is assumed to be equal to the refractive index for light polarised with the electric vector parallel to OXi. Similar calculations are performed to calculate the indices for light polarised parallel to OX2 and... 

Starting from the Clausius-Mosotti equation (9.7) and assuming that it can be applied to each principal direction Ox, in an orthorhombic crystal, show that the Lorentz-Lorenz equation can be written in a form that leads to the equation , = (1 + 2x,)/(l — x,), where , is the refractive index for light polarised parallel to Ox,- and X,- = q ,-/(3 oF), with a,- equal to the polarisability per unit cell for light polarised parallel to Ox,- and V the volume of the unit cell. 

As discussed in section 9.2.1, the refractive index of a substance depends on the polarisabilities of its basic units. The Lorentz-Lorenz equation derived there as equation (9.9) links the refractive index n and the polari-sability a of the structural units in any isotropic medium. It ean be written in the slightly simpler form... 

Lorentz-Lorenz equation A relation between the polarizability a of a molecule and the refractive index n of a substance made up of molecules with this polarizability. The Lorentz-Lorenz equation can be written in theforma= (3/4tcA/) [(n -l)/(n + 2)l, where JV is the number of molecules per unit volume. The equation provides a link between a microscopic quantity (the polarizability) and a macroscopic quantity (the refractive index). It was derived using macroscopic electrostatics in 1880 by Hendrik Lorentz and independently by the Danish physicist Ludwig Valentin Lorenz also in 1880. Compare ClAUSIUS-MoSSOTTr EQUATION. 

In particles possessing no permanent dipole moment, equation (5), by application of the Maxwell relation between refractive index and dielectric constant, becomes the Lorentz-Lorenz equation ... 

An empirical estimate of the polarizability can be obtained from refractive index measurements on the solution. Here it may be worthwhile to mention that for the purpose of determining the polarizability of the solute molecule the Lorentz-Lorenz equation is sufficient [46]. (See Reference 47 for a description of the procedure for the determination of the polarizability of bovineserum-albumin.) The electronic part of the molecular polarizability (i.e., the part which arises from visible and UV bands) has been shown to be almost independent of the molecular environment [47-50]. 

The refractive index of a compound can be calculated from its molar refraction and molecular volume using the Lorentz-Lorenz equation... 

Figure 4.8 shows the refractive indices and values of these polymers. The refractive index is dependent on the molar refraction [JJ] and molecular volume V, as expressed by the Lorentz-Lorenz equation ... 

The polarizability tensor (p,y) can be converted to the corresponding refractive index tensor (njj) according to the Lorentz-Lorenz equation ... 

The refractive index of binary ionic liquid mixtures can be interpreted in terms of electron polarizability if the density is a linear function of the mixtures composition, the refractive index is also linearly related to the molar refraction by the Lorentz-Lorenz equation. Our data show that the refractive index increases linearly for all ionic liquids with the mole fraction of [C2mim][OAc] (Fig. 9). Linear relationships were also found for [C4mpy][(CN)2N]/[C4mpy][BF4],... 

Table 2 shows that the parachors of pure ionic liquids calculated from density and surface tension data and the parachors estimated by the group contribution method agree very well [122]. Similarly, the molar refraction Rm,est was estimated and used to determine the refractive index R est by (2) (Lorentz-Lorenz equation), which is also compared to Ri,exp in Table 2 [123] ... 

Birefringent components are used for optical polarization devices such as polarizers, waveplates, and beam splitters. Birefringence means that the refractive index of materials is different in different directions. The refiactive index of a material is related to the molecular polarizability via the Lorentz-Lorenz equation as follows ... 

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