Lorenz-Lorentz expression

We may modify the Lorenz-Lorentz expression, if we note that Mt / pNA is an estimate of an individual molecule s volume. Assuming the molecule is spherical, we may deduce that ... 

For a crystal, the polarizability for a principal direction of the ellipsoid is calculated by using the same expression the refractive index n for that direction is taken to be related to the polarizability by the Lorenz-Lorentz expression,... 

The molar refractivity is obtained experimentally for neutral species, and generally the refractive index at the sodium D line (589 nm), d, is used to obtain the molar refractivity Ru in lieu of the infinite wavelength value Roo- The Lorenz-Lorentz expression is used ... 

The refractive index (at the sodium D-line, 589 nm), n, the range of which for the solvents listed is fairly narrow, from 1.3265 for methanol to 1.550 for nitrobenzene, is also listed in Table 3.5. From the refractive index is derived the molar refraction by means of the Lorenz-Lorentz expressions ... 

To understand how the refractive indices of a liquid crystal depend on temperature and wavelength, it is necessary to consider the molecular basis for optical refraction. For isotropic materials, the refractive index can be related to the molecular polarisability through the Lorenz-Lorentz expression ... 

This equation, called the Lorenz-Lorentz equation, was derived in 1880 by combining the Clausius-Mossotti expression for the local field with the idea of molecular polarizability. 

The molar refractivity is the volume of the substance taken up by each mole of that substance. In SI units, MR is expressed as m /mol. MR is a molecular descriptor of a liquid, which contains both information about molecular volume and polarizability, usually defined by the Lorenz-Lorentz equation [Lorentz, 1880a, 1880b] (also known as the Clausius-Mosotti equation) ... 

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... 

One of the presented structures is a monodispersion of subwavelength inclusions i (spheres) in dielectric host h. Fig. 2.22a. The other is polydispersion. Fig. 2.22b. The first situation can be described by the well-known MaxweU-Gamett model [171], the oldest effective medium model, obtained by the use of Clausius-Mossotti/ Lorenz-Lorentz equation. The other case is polydispersion, described by the implicit Bruggeman expression [172, 173]. 

The refractive power is a value which attempts to correct the effects of temperature, pressure, and concentration of the substance, all of which cause the refractive index, n, to vary with the slightest alteration of the conditions. The most accurate expression for the refmctive power is that of Lorenz and Lorentz, which is... 

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],... 

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

The molar refractivity R of the solvent is derived from the Lorentz-Lorenz expression, and at the D-... 

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... 

An expression for the molar refraction, formulated by Lorentz and Lorenz (1980), that has been widely proved with regard to its additivity 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). 

However, n may be expressed as a function of Vj by means of Lorentz-Lorenz equation ... 

If the term in the derivative of the field factor were negligible the expression on the left of this equation would be defined completely in terms of macroscopic measurable quantities. The specifics of the chosen cavity model enter the field factor derivative where Lorentz-Lorenz and Onsager factors may be mixed. The most commonly used procedure is to employ Onsager for the static field and Lorentz factors for the optical fields. For Fj (i = 0,1),... 

This result may be expressed in the more customary units of cubic meters or cubic nanometers by dividing by dTiSp. Thus, Op is equal to 1.472 x 10 nm at 25°C. When the calculation is repeated at 50°C the result is p = 1.471 x 10 nm. One expects the polarizability to be independent of temperature in a range where the electrons in the molecule remain in the same molecular orbitals. The small change in the polarizability reflects the weakness of the Lorentz-Lorenz model, which is based on continuum concepts. However, the estimated change is small, so that one may assume that the model is reasonably good. 

The refractive index is usually estimated in terms the molar refraction R, which quantifies the intrinsic refractive power of the structural units of the material. Several definitions have been proposed for R. Two of these definitions incorporate both of the key physical factors determining the refractive index, and are thus especially useful. Equation 8.5 expresses n in terms of the molar refraction RT T according to Lorentz and Lorenz [1,2], and Equation 8.6 expresses n in terms of the molar refraction Rgd according to Gladstone and Dale [3]. 

Dispersive power is more constitutive than refractivity—a fact first recognized by Gladstone (1886, 1887) when investigating the quotients (w — nXl) d and M nXl — nX2)jd for additivity. Briihl (1891), using the Lorentz-Lorenz expression for the specific or molecular refractions, considerably extended the subject and prepared lists of atomic dispersions for the a and y hydrogen lines these were later revised by von Auwers and Eisenlohr. Tables such as those produced by the last-named author (1912, 1923) have always included values for Rp — Ra and Ry — Ra. Predicted dispersions are sometimes satisfactory when absorption wavelengths are well away from the visible region (e.g. from Table 2 Ry — Ra for acetyl chloride and pentyl alcohol are 0-44 and 0-64 cm3 the observed differences are 0 48 and 0-64 cm3 2 ... 

The majority of the procedures currently being used in the conformational analysis of solutes (e.g. infra-red absorption differences between conformers, optical rotatory dispersions, NMR proton shifts, etc.) are qualitative and based upon empirical observations and analogies. It is therefore claimed that the present applications of anisotropic polarizabilities, built as they are on the theoretical arguments of Lorentz, Lorenz, Langevin, Born, Gans, Debye, and others, have—where solutes are concerned—advantages both in their foundations and in the quantitatively expressible natures of the conclusions they can provide. 

So far we have just defined another four-component quantity Af, but by now it is not clear whether it properly transforms under Lorentz transformations in order to justify the phrase 4-vector. In order to prove the transformation property of the gauge field, we re-express the inhomogeneous Maxwell equations in Lorenz gauge as given by Eq. (2.138) in explicitly covariant form by employment of the charge-current density and the gauge field A, ... 

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 ... 

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