Total Dielectric Function

If we have some fictitious material that possesses a permanent dipole moment along with an ionic and electronic dielectric fimction, the total real dielectric function could be written as the real part of the electronic contribution (Equation 23.49), plus the real part of the ionic susceptibility (Equation 23.19 with damping terms added), plus the real part of the dipolar susceptibility (Equation 23.34), 

Total dielectric function for a hypothetical material with a static dielectric constant of 20, t —10 s, s(0) = 5 arid e(oo) = 2.523. The damping constant y was set to 0.1 s. Note the resonances in the far infrared and in the near ultraviolet. 

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Noble metals - copper, silver and gold - are monovalent elements with a /cc-like crystallographic structure in the bulk phase under normal conditions. Their dielectric function has been the subject of various experimental investigations in the past [1-6]. A compilation and an analyse of the main results can be found in [7]. The response of noble metals to an electromagnetic excitation in the UV-visible range cannot be described, contrarily to the case of alkalis, by the only behaviour of the quasi-free conduction electrons (sp band), but must include the Influence of the bound electrons of the so-called d bands [8]. Hence, the total dielectric function of noble metals can be written as the sum of two contributions, one due to electronic transitions within the conduction band (intraband transitions) and the other stemming from transitions from the d bands to the conduction one (Interband... 

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... 

In our opinion, the use of and calculations for one-particle Green s functions are uniquely suitable for solid-state systems periodic in any number of dimensions. When faithfully implemented, it satisfies all criteria above. Green s functions offer analytically compact and physically rich tools for representing many properties for extended, periodic systems. They satisfy powerful and elegant relations for quantities such as density of states, lifetimes for excitations, dielectric functions, photo-emission and absorption spectra, total crystal energies, and many more. 

One can easily adjust the values of the dielectric constants D(, and Dj to obtain the experimental values of W, as in Table 4.4. With a choice of = 19.6 and Dj. = 51.0 for water, and D. = 12.5 and Dj. = 31.8 for 50% water-ethanol, we obtain the experimental values of W. We now compute the total correlation function for the two-state model for succinic acid. Here the correlation cannot be computed as an average correlation of the two configurations (see Section 4.5). The total correlation of the equilibrated two-state model is... 

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. 

Figure J5 The most important interband contributions to the imaginary part of the dielectric function of the 2-DL Si/CaF2 MQWs. Solid line total < 2i short dashed line contribution from the transition between the last occupied state and the first unoccupied one long dashed line the same considering the last two occupied states and the first two empty states.

Fluctuations in the dielectric properties near the interface lead to scattering of the EW as well as changes in the intensity of the internally reflected wave. Changes in optical absorption can be detected in the internally reflected beam and lead to the well-known technique of attenuated total reflectance spectroscopy (ATR). Changes in the real part of the dielectric function lead to scattering, which is the main topic of this review. Polarization of the incident beam is important. For s polarization (electric field vector perpendicular to the plane defined by the incident and reflected beams or parallel to the interface), there is no electric held component normal to the interface, and the electric field is continuous across the interface. For p polarization (electric field vector parallel to the plane defined by the incident and reflected beams), there is a finite electric field component normal to the interface. In macroscopic electrodynamics this normal component is discontinuous across the interface, and the discontinuity is related to the induced surface charge at the interface. Such discontinuity is unphysical on the molecular scale [4], and the macroscopic formalism may have to be re-examined if it is applied to molecules within a few A of the interface. 

Dielectric relaxation (DR) experiments measure the collective polarization response of all the polar molecules present in a given system. The DR time provides a measure of the time taken by a system to reach the final (equilibrium) polarization after an external field is suddenly switched on (or off). DR measures the complex dielectric fimction, s(w), that can be decomposed into real and imaginary parts as efca) = s (o) — is" (o) where s (co) and s fo ) are the real (permittivity factor) and imaginary (dielectric loss) parts, respectively. The total dipole moment of the system, at any given time t, M(t) = fift) where N is the total number of dipolar molecules and /Af is the dipole moment vector of the ith molecule. The complex dielectric function e((w) is given by the following relation. 

Many text-books argue that penultimate orbitals should not be taken too seriously because a closed-shell anti-symmetrized Slater determinant is invariant when new linear combinations of the one-electron functions are formed by a unitary transformation. This mathematical truth is rather irrelevant for our purposes, in part because the actual total wave-function is not a well-defined configuration but shows correlation effects which can be ascribed to a first approximation (11, 24) to an internal dielectric screening of the interelectronic repulsion corresponding to admixture of configurations... 

It was shown for the RISM/HNC theory [65, 66] that the dielectric constant which is related to the coefficients of in the low-A expansion of the site-site total correlation functions is determined solely by their renormalized long-range part For the 3D-RISM equation (4.37), the renormalized long-range part of the 3D site total correlation functions h r) takes the form... 

Fermi level that is, we must have k — k/2 < fcp and k + k/2 > kp (or vice versa), since otherwise the occupation numbers are either both 0 or both 1. These considerations indicate that the contributions to the integral will come from electron-hole excitations with total momentum k. At T = 0, the integral over k in the Lindhard dielectric function can be evaluated to yield... 

It is customary to denote the static electronic dielectric function (o> >o) as (oo), which seems strange. The only justification seems to be that (0) is reserved for the total static dielectric constant (ionic + electronic) and the infinity simply means at frequencies far above those where the ionic contributions are no longer significant, but are still much smaller than wq-... 

SPPs can be excited at an interface between two metals. In the Drude model their dielectric functions, ei and 62, are determined by their plasma frequencies, Wpi and u p2- If ojpi > ( p2, then in the frequency range ojp2 < co < cupi one has 62 > 0 and ei < 0, i.e., the condition of the existence of SPP is fulfilled. It is worthwhile to note that, at such frequencies, 2 < 1 and, hence, total internal reflection, necessary for SPP excitation, is realized at an interface between metal 2 and a vacuum without a prism. 

In total, this means that the initial Poisson-Boltzmann equation is modified in two respects (i) the solvent structure is taken into account via the dielectric function, and (ii) the total interactions between ions and the surfaces, modified by the presence of water, are described by the electrostatic potential and the PMF, which can be directly implemented in the interaction term of the PB equation. Figure 3 illustrates these interactions experienced by the ions immersed in solution close to a surface. With these two modifications, a new and truly much improved alternative to the old DLVO theory is available now. It should be noted that no adjustable parameters are used here. 

The same idea was actually exploited by Neumann in several papers on dielectric properties [52, 69, 70]. Using a tin-foil reaction field the relation between the (frequency-dependent) relative dielectric constant e(tj) and the autocorrelation function of the total dipole moment M t] becomes particularly simple ... 

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