Frequency dependent dielectric function

The simplest example is that of tire shallow P donor in Si. Four of its five valence electrons participate in tire covalent bonding to its four Si nearest neighbours at tire substitutional site. The energy of tire fiftli electron which, at 0 K, is in an energy level just below tire minimum of tire CB, is approximated by rrt /2wCplus tire screened Coulomb attraction to tire ion, e /sr, where is tire dielectric constant or the frequency-dependent dielectric function. The Sclirodinger equation for tliis electron reduces to tliat of tlie hydrogen atom, but m replaces tlie electronic mass and screens the Coulomb attraction. 

The optical properties of metal nanoparticles have traditionally relied on Mie tlieory, a purely classical electromagnetic scattering tlieory for particles witli known dielectrics [172]. For particles whose size is comparable to or larger tlian tire wavelengtli of the incident radiation, tliis calculation is ratlier cumbersome. However, if tire scatterers are smaller tlian -10% of tire wavelengtli, as in nearly all nanocrystals, tire lowest-order tenn of Mie tlieory is sufficient to describe tire absorjDtion and scattering of radiation. In tliis limit, tire absorjDtion is detennined solely by tire frequency-dependent dielectric function of tire metal particles and the dielectric of tire background matrix in which tliey are... 

The optical constants of a metal are determined to a large degree by the free electrons. According to the Drude model, the contribution of the free electrons to the frequency-dependent dielectric function is expressed as follows (16) ... 

Fig. 6 The Cole-Cole plot of the contribution from water to the frequency dependent dielectric function. The reduced real part [c (u )] is plotted against the reduced imaginary part (c (u>)]. Note the non-Debye character in the micellar solution.

The response of a pure, homogeneous medium to the applied fields may be characterized quite generally by a complex frequency-dependent dielectric function e(o>), which can be written in terms of its real and imaginary parts as... 

Where a is the polarizability, e is the frequency dependent dielectric function [4J], and V the volume of the dipole. The radius of each sphere is calculated using a/R=1.612 42, where a is the spacing between the particles, 40 nm... 

If the nanocrystal permittivity is greater than the host permittivity, snc > Shost, the screening factor S describes depression of the decay due to screening of the radiation field inside the nanocrystal. Frohlich was the first to note that for a metal nanocrystal with the complex frequency-dependent dielectric function... 

The optical properties of dispersions of spherical particles can be predicted by Mie theory. This theory provides expressions for the extinction cross section of spherical particles with a frequency dependent dielectric function e = e -I- ie", embedded in a medium of dielectric function Sm, as [142-144]... 

As shown below, the dispersion reiation is a function of the frequency dependent dielectric function, (o)). When the dielectric function is an even function of co, the dispersion relation is even as well, and /q = 0 by symmetry. 

Figure 8.3. The real part of the complex frequency-dependent dielectric function [e (co)] of aqueous myoglobin solution for different concentrations. Concentrations are (from top to bottom) 161, 99, and 77 mg/mL at 293.15 K. The symbols denote experimental results while the solid line is a fit to the theory of dynamics exchange model developed by Nandi and Bagchi. Adapted with permission from J. Phys. Chem. A, 102 (1998), 8217-8221. Copyright (1998) American Chemical Society.

The QM/classical models introduced earUer are able to deal with this situation, by differentiating between fast and slow — or dynamical and inertial — environment responses. In particular, such a nonequihbrium formulation has been largely and successfully used in QM/continuum models where the different components of the polarization can be accounted for introducing a frequency-dependent dielectric function. 

FIGURE 15.49 The frequency-dependent dielectric function for highly oriented polyacetylene doped with perchlorate from Kramers-Rronig analyses of the reflectance spectra, (a) Light polarized perpendicular to polyacetylene chain direction, (b) Light polarized parallel to polyacetylene chain directions. (From Miyamae, T., Shimizu, M., and Tanaka, J., Bull. Chem. Soc. Jptu, 67, 2407, 1994. Reprinted from the Chemical Society of lapan. With permission.)... 

A wide variety of molecular properties can be accurately obtained with ADF. The time-dependent DFT implementation " yields UV/Vis spectra (singlet and triplet excitation energies, as well as oscillator strengths), frequency-dependent (hyper)polarizabilities (nonlinear optics), Raman intensities, and van der Waals dispersion coefficients. Rotatory strengths and optical rotatory dispersion (optical properties of chiral molecules ), as well as frequency-dependent dielectric functions for periodic structures, have been implemented as well. NMR chemical shifts and spin-spin couplingsESR (EPR) f-tensors, magnetic and electric hyperfme tensors are available, as well as more standard properties like IR frequencies and intensities, and multipole moments. Relativistic effects (ZORA and spin-orbit coupling) can be included for most properties. 

In the dispersion interaction, the Lifshitz theory also treats the electrolyte solution like a continuum, characterized by its frequency-dependent dielectric function, but devoid of any structure. 

This difficulty could be avoided by applying linear response theory, which is widely used in solid-state physics to determine directly the dynamic matrix, polarization, and frequency-dependent dielectric functions, as well as phonon dispersion curves in the harmonic approximation. This method has the great advantage that it requires only a band structure at the equilibrium geometry of the solid (chain), i.e., one does not have to determine a potential hypersurface. However, since this theory has not yet been applied to polymers and involves a rather complicated formalism, we cannot enter into details here but refer the reader to standard solid-state physical works and an application to a simple solid (Si). ... 

If a mechanical or an electric field is applied to a polymer sample and remains suSiciently small, then the reaction, as given by the deformation and the polarization respectively, can be described by linear equations. We shall deal first with the linear viscoelasticity, which can be specified by various mechanical response functions, and then with the linear dielectric behavior, as characterized by the time- or frequency dependent dielectric function. 

In order to investigate the origin and the physical properties of SPPs, we consider a metal-dielectric interface described by the plane z = 0 (see Fig. 1.6). The local frequency-dependent dielectric function is supposed to change in a stepwise manner from the dielectric with e z) = (for z > 0) to the metal with c(z) =... 

The molecule is a classical oscillating charge density (usually a point dipole) and the metal nanoparticle is a continuous body characterized by the frequency-dependent dielectric function (see Chapter 1). This is by far the most common description of the metal-molecule electrodynamic coupling problem in the literature. Notably, sometimes even the metal nanoparticle is reduced to a polarizable dipole. Depending on the phenomenon under study, this may be acceptable or results in an oversimplification [50]. 

Before proceeding with the detailed derivations, it is worth mentioning two simple expressions which capture much of the physics. The first, known as the Drude model, refers to the frequency-dependent dielectric function for the case where only transitions within a single band, intersected by the Fermi level, are allowed ... 

This section considers reports of the frequency-dependent dielectric functions, primarily for polymers in nondilute solutions. By analogy with the treatment of the storage and loss moduli in Chapter 13, the two-parameter temporal scaling approach in that chapter leads to expectations for the dynamic dielectric and dielectric loss functions and their frequency dependences, including for the dynamic dielectric function... 

In this section we shall consider how the results obtained above for reflection from a plane surface are modified in the case of rough surfaces. We shall assume that the amplitude of surface roughness, (ry), can be treated as small and thus the solutions of Maxwell s equations can be expanded as a Taylor series in it (Maradudin and Mills 1975). We suppose for simplicity that above the surface z = (ry) is vacuum, while below it is the isotropic medium with a complex frequency-dependent dielectric function e = e co). The total dielectric function can then be written as... 

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