Lorentz-Lorenz Model

It shows that relative permittivity is the ratio of total permittivity of one mole of material with that in vacuum. Typically, RI is measured at visible light region of frequency 10 to 10 Hz. At this frequency, the orientational polarization is almost undetected, leaving the effect of interaction of light with atomic and electronic polarization as the dominating factors in influencing the measured RI. Under this condition the RI is related to relative permittivity as summarized in the Maxwell equation  

Substituting this into the above Equation 5.1 gives the Lorentz-Lorenz equation  

---

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. 

CMLL Clansins-Mossotti/Lorentz-Lorenz (model)... 

The Lorentz-Lorenz model is obtained when the host material is chosen as air or vacuum (eh=l). The EMA equation reduces to ... 

The sPS has been exploited as sensitive material for fiber optic chemical sensors based on reflectance measurements and aimed to detection of chloroform and toluene in water and air environments48 50. The refractive index of sPS thin films is estimated to be about 1.578. The effect of the analyte sorption in the crystalline domain was modeled as an increase in the material density, which in turn leads to an increase in the refractive index according to the Lorentz-Lorenz law ... 

It should be noted that also the refractive index in eq. (17) will change under pressure. This change can be estimated by the Lorenz local-field model, where the refractive index n is related to the atomic polarizability a p and the density of the material p by the Lorentz-Lorenz relation ... 

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... 

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),... 

Empirical models for the induced trace have also been obtained from (nonspectroscopic) measurements of the second virial dielectric coefficient of the Clausius-Mosotti and Lorentz-Lorenz expansions [30]. Excellent surveys with numerous references to the historical as well as the modern dielectric research activities were given by Buckingham [27], Kielich [89], and Sutter [143] in 1972 see also a recent review with a somewhat more spectroscopic emphasis [11]. 

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. 

Among the few determinations of of molecular crystals, the CPHF/ INDO smdy of Yamada et al. [25] is unique because, on the one hand, it concerns an open-shell molecule, the p-nitrophenyl-nitronyl-nitroxide radical (p-NPNN) and, on the other hand, it combines in a hybrid way the oriented gas model and the supermolecule approach. Another smdy is due to Luo et al. [26], who calculated the third-order nonlinear susceptibility of amorphous thinmultilayered films of fullerenes by combining the self-consistent reaction field (SCRF) theory with cavity field factors. The amorphous namre of the system justifies the choice of the SCRF method, the removal of the sums in Eq. (3), and the use of the average second hyperpolarizability. They emphasized the differences between the Lorentz Lorenz local field factors and the more general Onsager Bbttcher ones. For Ceo the results differ by 25% but are in similar... 

For larger particle concentrations, the interactions between particles influence the electromagnetic properties. For interparticle distances much smaller than the wavelength, the Maxwell-Gamett model applies and leads to the Lorentz-Lorenz relation for the effective dielectric constant eff of the composite medium, which takes the form... 

In order to find if the uniform-layer approximation can be used in our gradient-layer model, we simulated the i (6) curves of the layer of 40-nm-diameter particles, calculating n z) from the volume fraction (Eq. 5) and the Lorentz Lorenz formula (Eq. 6) (using the refractive index values 1.450 for silica, 1.333 for water and 1.000 for air), and evaluated these simulated reflectance curves by the uniform-layer method. The results are shown in Fig. 3. It is evident that the uniform-layer method can... 

The results shown in Table 1 and Fig. 7 prove the applicability of our model - a gradient refractive index layer, where the local refractive index is determined from the volume fraction of the constituents by the Lorentz-Lorenz formula, as the fitted layer thickness corresponded to the particle diameter (Table 1) - and the other results were also consistent with the expectations both about the behavior of the particles and the structure of the interfacial layer. 

Lorentz was the first to consider such problems for a reasonably defensible model of induced dipoles derive the local Lorentz field j and from this obtain the venerable Clausius Mossotti (or perhaps more properly Lorentz-Lorenz) formula. As shown schematically in Figure 1 (a) the molecules are assumed to be at sites on a cubic lattice with uniform macroscopic along the z axis. 

A variation of the water adsorption isotherm method was then developed (Yeatman, 1994 Dawnay, 1995) called Molecular Probe Ellipsometry (MPE). In this technique, refractive index readings are made in dry nitrogen, and then in nitrogen saturated with an adsorbate solvent vapor. The porous structure is modelled as an effective medium according to the Lorentz-Lorenz relation, giving a relation between the measured film index f and the indices of the material in the pores and of the solid skeleton, p and s, respectively ... 

Double films formed on InP were studied by applying ex situ ellipsometry after successive etching steps of the thermally formed oxide by an HF solution. From calculations based on a four-phase model (see Appendix A1 for treatment of two and more layers), the existence of an easily etched outer layer with smaller refractive index and a more durable inner layer with larger index was shown, which is consistent with XPS analysis. In the same work, in situ ellipsometry data was shown to be explained by the presence of a liquid layer when a microscopic model based on the Lorentz-Lorenz relation was used. Other ellipsometric studies of double films, such as double films of Si02 and Si3N4 on Si, have been reported. 

Continuum models have a long and honorable tradition in solvation modeling they ultimately have their roots in the classical formulas of Mossotti (1850), Clausius (1879), Lorentz (1880), and Lorenz (1881), based on the polarization fields in condensed media [32, 57], Chemical thermodynamics is based on free energies [58], and the modem theory of free energies in solution is traceable to Bom s derivation (1920) of the electrostatic free energy of insertion of a monatomic ion in a continuum dielectric [59], and Kirkwood and Onsager s... 

An optical three-layer model has proved superior to a one-layer model for the interpretation of the ellipsometric data. The refractive indices of the film and surface layers are determined and it is found that the index for the surface is higher than that for the film core. A Lorenz-Lorentz type treatment of NBF reveals that there are approximately seven water molecules per molecule of surfactants in both NaDoS and NaDoBS films. The optical data obtained by the three-layer model for NBF from NaDoS indicate that the thickness of the aqueous core is zero while that of the adsorption monolayers of surfactant molecules with refractive index 1.365 is 1.8 nm, i.e. the thickness of NBF is 3.6 nm. 

An increase in fractional free volume will reduce the number of polarisable groups per unit volume, and thereby reduce the relative permittivity of the polymer. Quantitatively, the effect may be estimated by means of the Clausius-Mossotti/Lorenz-Lorentz model for dielectric mixing (Bottcher, 1978) ... 

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]. 

---