Maxwell-Garnett

Fig. 3. Reflectivity (a) and optical conductivity spectra (b) of oriented CNTs films along the an and aj directions. Bruggeman (BM) and Maxwell-Garnett (MG) fits (see text and Table 2) are also presented.

Fig. 4. The reflectivity (a) and the optical conductivity (b) in the p direction are similar to the ones along the a directions (Fig. 3). However, the absence of data above 4 eV changes the high energy spectrum of the optical conductivity. These changes are not relevant for the low frequency spectral range. The Maxwell-Garnett (MG) fit is also displayed as well as the intrinsic reflectivity and conductivity calculated from the fit (see Table 2 for the parameters).

We now consider the influence of the various parameters in the Maxwell-Garnett approach. Figure 8 displays the behaviour of a]((o) if we only change the filling... 

Table 2. These parameters refer to the Maxwell-Garnett (Eq.(6)) and Bruggeman calculations with a filling factor f 0.6. N denotes the geometrical factor. All other values are in eV.

The TEM data have been used to simulate, in the frame of the Mie theory and Maxwell-Garnett effective medium approximation [15], the optical absorption spectra of the sample implanted with 5 x lO Au /cm. The results are reported in Figure 8(c). In the first model used to describe... 

Fig. 6. Maxwell-Garnett theory used for the prediction of dielectric constant containing dispersed regions of low dielectric polymer (e = 2.0,0) or air (e = 1.0, )...

The dielectric constant of the pure cyanurate network under dry nitrogen atmosphere at 20 °C is 3.0 (at 1 MHz). For the macroporous cyanurate networks, the dielectric constant decreases with the porosity as shown in Fig. 57, where the solid and dotted lines represent experimental dielectric results together with the prediction of the dielectric constant from Maxwell-Garnett theory (MGT) [189]. The small discrepancies between experimental results and MGT might be due to the error in estimated porosities, which are calculated from the density of the matrix material and cyclohexane assuming that the entire amount of cyclohexane is involved in the phase separation. It is supposed that a small level of miscibility after phase separation would result in closer agreement of dielectric constants measured and predicted. Dielectric constant values as low as 2.5 are measured for macroporous cyanurates prepared with 20 wt % cyclohexane. 

Note that (8.49), which is a generalization of the Maxwell Garnett dielectric function (8.50), is not invariant with respect to interchanging the roles of matrix and inclusions if we make the substitutions e - em, em - t, and f - (1 — /), then eav is not, in general, unchanged. If, therefore, a two-component mixture is to be described by (8.49), a choice must be made as to which component is the matrix and which the inclusions (there may be physical reasons to guide this choice). The limiting values of eav are independent of 0 ... 

The average dielectric function (8.50) was first derived by Maxwell Garnett (1904) subsequently, it has been rederived under various sets of assumptions (see, for example, Genzel and Martin, 1972, 1973 Barker, 1973 Bohren and Wickramasinghe, 1977). 

Our first problem is how to properly calculate the average optical constants when 1% of carbon by volume is uniformly distributed in a nonabsorbing medium. The usual procedure, as in the papers cited above, has been to simply volume-average n and k separately. But optical constants are not, in general, additive, so we have used the Maxwell Garnett expression (8.50). The result is mgr = 1.55 + /0.007, which in this instance is identical, to the number of figures shown, with the result obtained by volume-averaging the refractive indices 1.55 + /0.0 and 1.7 + /0.7. 

We can estimate the magnitude of the shift attributable to interaction between particles by appealing to the Maxwell Garnett theory (Section 8.5). This theory is strictly applicable only to a medium consisting of small particles distributed throughout a volume, whereas the slides consist of a single layer of particles on a surface. Nevertheless, for our limited purposes here the Maxwell Garnett theory is adequate. 

Maxwell Garnett, J. C., 1904. Colours in metal glasses and in metallic films, Philos. Trans. R. Soc., A203, 385-420. 

Treu, J. I., 1976. Mie scattering, Maxwell Garnett theory, and the Giaever immunology slide, Appl. Opt., 15, 2746-2750. 

Quantitative simulation of spectra as outlined above is complicated for particle films. The material within the volume probed by the evanescent field is heterogeneous, composed of solvent entrapped in the void space, support material, and active catalyst, for example a metal. If the particles involved are considerably smaller than the penetration depth of the IR radiation, the radiation probes an effective medium. Still, in such a situation the formalism outlined above can be applied. The challenge is associated with the determination of the effective optical constants of the composite layer. Effective medium theories have been developed, such as Maxwell-Garnett 61, Bruggeman 62, and other effective medium theories 63, which predict the optical constants of a composite layer. Such theories were applied to metal-particle thin films on IREs to predict enhanced IR absorption within such films. The results were in qualitative agreement with experiment 30. However, quantitative results of these theories depend not only on the bulk optical constants of the materials (which in most cases are known precisely), but also critically on the size and shape (aspect ratio) of the metal particles and the distance between them. Accurate information of this kind is seldom available for powder catalysts. 

Hornyak, G. L., Patrissi, C. J., and Martin, C. R., Fabrication, characterization, and optical properties of gold nanoparticle/porous alumina composites the nonscattering Maxwell-Garnett. J. Phys Chem. 101,1548 (1997). 

An effective index for nanoparticles can be obtained, for example, using the Maxwell Garnett EMT [29, 30] such that... 

Maxwell Garnett JC (1906) Colours in metal glasses, in metallic films and in solutions II. Philos Trans R Soc A 205 237-288... 

This chapter is devoted to describe the impact of metallic nanosphere to the multi-photon excitation fluorescence of Tryptophan, and little further consideration to multi-photon absorption process will be given, as the reader can find several studies in [11-14]. In section II, the nonlinear light-matter interaction in composite materials is discussed through the mechanism of nonlinear susceptibilities. In section III, experimental results of fluorescence induced by multi-photon absorption in Tryptophan are reported and analyzed. Section IV described the main results of this chapter, which is the effect of metallic nanoparticles on the fluorescent emission of the Tryptophan excited by a multi-photon process. Influence of nanoparticle concentration on the Tryptophan-silver colloids is observed and discussed based coi a nonlinear generalization of the Maxwell Garnett model, introduced in section II. The main conclusion of the chapter is given in secticHi IV. 

The referred topology, known as Maxwell Garnett geometry [21], is considered to be isotropic macroscopically, and an effective-medium dielectric coistant can be written as... 

Here, E is the inclusion dielectric constant. Equation 3 is known as the Maxwell Garnett result [21]. 

The fascinating optical properties of metal nanoparticles have caught the attention of many researchers from the pioneering and almost parallel works of G. Mie and J.C. Maxwell-Garnett at the beginning of the twentieth century. These original properties, like many other phenomena specifically appearing in matter divided to the nanoscale, are linked with confinement effects, since quasi-free conduction 461... 

Moreover, the derivation of the Maxwell-Garnett expression of with respeet to (Eq. (9)) provides... 

Sipe, J.E., Boyd, R.W. Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model. Phys. Rev. A 43, 1614-1629 (1992)... 

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