Dielectric function damped oscillators

If V is localized, say, near the origin, then for locations far from the origin, this behaves like j 2kFr)/r2, which means as cos(2kFr)/ r3. These damped oscillations of frequency 2kF are the Friedel oscillations, which always arise when an electron gas is perturbed the frequency of oscillation comes from the kink in the dielectric function at 2kF. We see the Friedel oscillations (in planar rather than in spherical geometry) for the electron gas at a hard wall [Eq. (12) et seq.] and for the electron density at the surface of a metal. 

In the region between to, and to, which for SiC is between about 800 and 1000 cm-1, the reflectance is high not because of large k but because of small n. If n = 0, the normal incidence reflectance is nearly 100% only for the undamped oscillator (y = 0) is the reflectance actually 100%, but solids like SiC approach this rather closely. If the damping constant y in (9.20) is set equal to zero, the real part of the dielectric function becomes... 

In order to investigate solids or polymer systems with free carriers by IR spectroscopy, it is very convenient to measure the reflectivity instead of absorbance or transmittance. Thus, the problems to be discussed in this context are usually described by a linear response formalism. In its simplest form, this means that the response function (dielectric function) s(u ) of a damped harmonic oscillator is used to describe the interaction between light and matter. The complex form of this function is... 

As an example, it has been pointed out that the Hamaker and Lifshitz theories assume (exphcitly and implicitly, respectively) that intensive physical properties of the media involved such as density, and dielectric constant, remain unchanged throughout the phase—that is, right up to the interface between phases. We know, however, that at the atomic or molecular level solids and liquids (and gases under certain circumstances) exhibit short-range periodic fluctuations they are damped oscillating functions. Conceptually, if one visualizes a hquid in contact with a flat solid surface (Fig. 4.8a), one can see that the molecules (assumed to be approximately spherical, in this case) trapped between the surface and the bulk of the liquid will have less translational freedom relative to the bulk and therefore be more structured. That structure will (or may) result in changes in effective intensive properties near the surface. 

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]). 

Moreover, in considering the effects of the size in the optical response of a metallic nanoparticle, we must put in evidence that in the case of particles with dimensions comparable or smaller than the mean free path of its oscillating electrons (i.e. for gold and silver particles of radius o < 10 nm) the dielectric function of the particles becomes strongly size-dependent and the additional surface damping must be considered for a correct treatment of the resonances intensity. 

Simple Spectral Method [23] In the simple spectral method, a model dielectric response function is used. It combines a Debye relaxation term to describe the response at microwave frequencies with a sum of terms of classical form of Lorentz electron dispersion (corresponding to a damped harmonic oscillator model) for the frequencies from IR to UV ... 

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