Bulk dielectric function

Hovel H, Fritz S, Hilger A, Kreibig U, VoUmerM. Width of cluster plasmon resonances—bulk dielectric functions and chemical interface damping. Phys Rev B 1993 48 18178-18188. 

The notion of homogeneity is not absolute all substances are inhomogeneous upon sufficiently close inspection. Thus, the description of the interaction of an electromagnetic wave with any medium by means of a spatially uniform dielectric function is ultimately statistical, and its validity requires that the constituents—whatever their nature—be small compared with the wavelength. It is for this reason that the optical properties of media usually considered to be homogeneous—pure liquids, for example—are adequately described to first approximation by a dielectric function. There is no sharp distinction between such molecular media and those composed of small particles each of which contains sufficiently many molecules that they can be individually assigned a bulk dielectric function we may consider the particles to be giant molecules with polarizabilities determined by their composition and shape. 

A particle is subdivided into a small number of identical elements, perhaps 100 or more, each of which contains many atoms but is still sufficiently small to be represented as a dipole oscillator. These elements are arranged on a cubic lattice and their polarizability is such that when inserted into the Clausius-Mossotti relation the bulk dielectric function of the particle material is obtained. The vector amplitude of the field scattered by each dipole oscillator, driven by the incident field and that of all the other oscillators, is determined iteratively. The total scattered field, from which cross sections and scattering diagrams can be calculated, is the sum of all these dipolar fields. 

We must again emphasize, even more strongly than we did at the beginning of this chapter, that surface plasmons and surface phonons are not examples of the failure of the bulk dielectric function to be applicable to small particles. Down to surprisingly small sizes—exactly how small is best stated in specific examples, as in Sections 12.3 and 12.4—the dielectric function of a particle is the same as that of the bulk parent material. But this dielectric function, which is the repository of information about elementary excitations, manifests itself in different ways depending on the size and shape of the system. 

There is one clear exception to the rule that bulk dielectric functions tend to be applicable to very small particles in metal particles smaller than the mean free path of conduction electrons in the bulk metal, the mean free path can be dominated by collisions with the particle boundary. This effect has been... 

As an example of extinction by spherical particles in the surface plasmon region, Fig. 12.3 shows calculated results for aluminum spheres using optical constants from the Drude model taking into account the variation of the mean free path with radius by means of (12.23). Figure 9.11 and the attendant discussion have shown that the free-electron model accurately represents the bulk dielectric function of aluminum in the ultraviolet. In contrast with the Qext plot for SiC (Fig. 12.1), we now plot volume-normalized extinction. Because this measure of extinction is independent of radius in the small size... 

The shapes of the absorption band cease to be independent of size for particles smaller than about 26 A, which suggests that the bulk dielectric function is inapplicable. Indeed, the broadening and lowering of the absorption peak can be explained by invoking a reduced mean free path for conduction electrons (Section 12.1). Thus, the major features of surface modes in small metallic particles are exhibited by this experimental system of nearly spherical particles well isolated from one another. But when calculations and measurements with no arbitrary normalization are compared, some disagreement remains. Measurements of Doremus on the 100-A aqueous gold sol, which agree with those of Turkevich et al., are compared with his calculations in Fig. 12.18 the two sets of calculations are for optical constants obtained... 

In the examples presented here, the extension to the Lindhard RPA [23] suggested by Mermin [24] is used for the bulk dielectric function. This allows one to use non-zero values of the electron gas damping, keeping the number of electrons in the system constant. We want to emphasize that this description incorporates both single-particle excitations (creation of electron-hole pairs) and collective excitations (bulk and surface plasmons). 

The linear optical properties follow directly from x - For example, the bulk dielectric function (or relative permittivity), e cu), is... 

The dipolar response of a large sodium sphere can be calculated from the experimental bulk dielectric function as discussed in the context of Eq. (6). The calculated width is To = 0.19 eV. This asymptotic experimental width is due to a structure in the dielectric function which is caused by an interband transition [55]. The collective oscillation can decay by exciting a single electron to a higher electronic band. The same mechanism occurs in the damping of the bulk plasmon. In this case, the width can be well correlated with the strength of the pseudopotential (see Figure 9 of Ref. [48]). 

Under typical scattering conditions the intensity of electron scattering may, with one degree of precision or another, be related to the bulk dielectric function e(q,AE) with possible additional spectral structure due to genuine surface effects in reflection mode. In the q- 0 limit the bulk and surface dielectric functions take the form... 

Moreover the dielectric function of the void metallic nanoparticles here presented, are corrected by including the surface damping term (see Sec. 2.3] into the bulk dielectric function. 

In Section 3.1.1 we have discussed the phase changes on reflection. For the two-phase model of a crystal-vacuum interface which does not take into ac-coimt any transition layer, the complex bulk dielectric function of the crystal... 

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