Classical dispersion theory

We now want to study the consequences of such a model with respect to the optical properties of a composite medium. For such a purpose, we will consider the phenomenological Lorentz-Drude model, based on the classical dispersion theory, in order to describe qualitatively the various components [20]. Therefore, a Drude term defined by the plasma frequency and scattering rate, will describe the optical response of the bulk metal or will define the intrinsic metallic properties (i.e., Zm((a) in Eq.(6)) of the small particles, while a harmonic Lorentz oscillator, defined by the resonance frequency, the damping and the mode strength parameters, will describe the insulating host (i.e., /((0) in Eq.(6)). 

We should first correct the wavevector inside the crystal for the mean refractive index, by multiplying the wavevectors by the mean refractive index (1 + IT). This expression is derived from classical dispersion theory. Equation (4. 18) shows us that is negative, so the wavevector inside the crystal is shorter than that in vacuum (by a few parts in 10 ), in contrast to the behaviom of electrons or optical light. The locus of wavevectors that have this corrected value of k lie on spheres centred on the origin of the reciprocal lattice and at the end of the vector h, as shown in Figure 4.11 (only the circular sections of the spheres are seen in two dimensions). The spheres are in effect the kinematic dispersion surface, and indeed are perfectly correct when the wavevectors are far from the Bragg condition, since if D 0 then the deviation parameter y, 0 from... 

From classical dispersion theory we can show that is in practical terms proportional to the number of atoms per cubic centimetre in the flame, i.e. is proportional to analyte concentration. 

For many years no attempt was made to determine the absolute stereochemistry of transition metal complexes, although Kuhn (8) calculated the absolute configuration of (—)n-[Co(C204)3]3- from its anomalous optical rotatory dispersion, using classical coupled oscillator theory. He later (9) extended his theory to other tris-chelated cobalt complexes such as (+)n-[Co(en)3]3+. However, in 1955 Saito (10) showed by anomalous X-ray dispersion that Kuhn s suggested configuration for (+)n-[Co(en)3]3+ was incorrect. 

Dispersional Interaction between Molecules. We still wish to consider briefly energies due to interaction between fluctuating induced electric charge distributions of atoms and molecules. In constrast to electrostatic and induced interactions, these are present even when the molecules do not possess permanent electric moments. These dispersional interactions cannot be dealt with on a classical electrostatics level owing to their relation to London s quantum dispersion theory, they have been termed London dispersional interactions. 

The variation of S with wave-length has also been much investigated, and anomalous dispersion has definitely been found to exist a natural modification of the classical dispersion theory enables us to account for the latter. But since (according to Davisson and Germer) very few facts about these anomalous phenomena are meanwhile available in the case of electron reflection, we shall not trouble to describe these experiments. 

According to the classical dispersion theory, the relationship between the absorption and the number density of the absorbing atoms is given by ... 

Peiponen, K.E., Vartiainen, E.M., and Asakura, M., Dispersion, Complex Analysis and Optical Spectroscopy Classical Theory, New York, Springer, 1998, 130. Peiponen, K.E., Vartiainen, E.M., and Asakura, M., Complex analysis in dispersion theory. Optical Review 4 (1997) 433-441. 

The power to dissolve resins is the foremost requirement of a solvent except in cases involving dispersions in nonaqueous solvents (NADs) or dispersions in water (latices, emulsions, and dispersions). Theories of solvency and solution are covered by Rider in the preceding chapter. The classic books by Hildebrand and Scott (18) and Hildebrand, Prausnitz, and Scott (19) discuss solubility and solutions in considerable depth. The monumental book by Doolittle ( covers both theoretical and applied aspects of solvents. Several chapters in the Mattiello series published in 1941-46 deal with solvents the chapter on lacquer solvents by Bogin... 

Bayliss85 has made a quantitative approach to the study of solvent effects on the absorption spectra. Treating the solvent as a continuous dielectric medium, an expression has been developed for its effect on the Franck" Condon absorption of light, in terms of the polarization fprces of the solvent. The same result was obtained by employing methods based on quantum theory and classical dispersion theory. Bayliss85 has derived the following expression for the frequency shift, Av, caused by the solvent... 

The apparent oscillator strength is proportional to the integrated intensity under the molar absorption curve. To derive the formula, Chako followed the classical dispersion theory with the Lorentz-Lorenz relation (also known as the Clausius-Mosotti relation), assuming that the solute molecule is located at the center of the spherical cavity in the continuous dielectric medium of the solvent. Hence, the factor derived by Chako is also called the Lorentz-Lorenz correction. Similar derivation was also presented by Kortum. The same formula was also derived by Polo and Wilson from a viewpoint different from Chako. 

Equation (B3.3.7) is the fundamental equation of classical dispersion theory [4]. Because Xe is related to the high-frequency dielectric constant by Eq. (3.30), and the high-frequency dielectric constant is the square of the refractive index (Eq. 3.30), it appears that the dielectric constant and refractive index also should be treated as complex numbers. To indicate this, we ll rewrite Eqs. (3.19) and (3.30) using and to distinguish the complex dielectric constant and refractive index from, e and n, the more familiar, real quantities that apply to non-absorbing media ... 

K. E. Peiponen, E. M. Vertiainen, and T, Asakura, Dispersion, Complex Analysis and Optical Spectroscopy. (Classical theory), Springer-Verlag, Berlin, 1999,... 

There are two general theories of the stabUity of lyophobic coUoids, or, more precisely, two general mechanisms controlling the dispersion and flocculation of these coUoids. Both theories regard adsorption of dissolved species as a key process in stabilization. However, one theory is based on a consideration of ionic forces near the interface, whereas the other is based on steric forces. The two theories complement each other and are in no sense contradictory. In some systems, one mechanism may be predominant, and in others both mechanisms may operate simultaneously. The fundamental kinetic considerations common to both theories are based on Smoluchowski s classical theory of the coagulation of coUoids. 

Traditionally, analytical chemists and physicists have treated uncertainties of measurements in slightly different ways. Whereas chemists have oriented towards classical error theory and used their statistics (Kaiser [ 1936] Kaiser and Specker [1956]), physicists commonly use empirical uncertainties (from knowledge and experience) which are consequently added according to the law of error propagation. Both ways are combined in the modern uncertainty concept. Uncertainty of measurement is defined as Parameter, associated with the result of a measurement that characterizes the dispersion of the values that could reasonably be attributed to the measurand (ISO 3534-1 [1993] EURACHEM [1995]). 

At low-conversion copolymerization in classical systems, the composition of macromolecules X whose value enters in expression (Eq. 69) does not depend on their length l, and thus the weight composition distribution / ( ) (Eq. 1) equals 5(f -X°) where X° = jt(x°). Hence, according to the theory, copolymers prepared in classical systems will be in asymptotic limit (/) -> oo monodisperse in composition. In the next approximation in small parameter 1/(1), where (/) denotes the average chemical size of macromolecules, the weight composition distribution will have a finite width. However, its dispersion specified by formula (Eq. 13) upon the replacement in it of l by (l) will be substantially less than the dispersion of distribution (Eq. 69)... 

Dispersions of fine mineral particles can be stabilised by direct electrical charging of the particles or by steric/electrosteric protection from adsorbed polymers. Stabilisation by direct charging is well described by the classical DLVO theory. ... 

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