Concentration dielectric function

The calculations were subsequently extended to moderate surface charges and electrolyte concentrations.8 The compact-layer capacitance, in this approach, clearly depends on the nature of the solvent, the nature of the metal electrode, and the interaction between solvent and metal. The work8,109 describing the electrodesolvent system with the use of nonlocal dielectric functions e(x, x ) is reviewed and discussed by Vorotyntsev, Kornyshev, and coworkers.6,77 With several assumptions for e(x,x ), related to the Thomas-Fermi model, an explicit expression6 for the compact-layer capacitance could be derived ... 

Gruen and Marcelja considered that the electric and polarization fields are not proportional in the vicinity of a surface and that while the electric field has the ion concentrations as its source, the source of the polarization field is provided by the Bjerrum defects. The coupled equations for the electric and polarization fields were derived through a variational method. Attard et al.14 contested the Gruen—Marcelja model because, to obtain an exponential decay of the repulsion, the nonlocal dielectric function was assumed to have a simple monotonic dependence upon the wavelength (eq 33 in ref 13). This was found to be inconsistent with the exact expression for multipolar models.14 In addition, the characteristic decay length for polarization (denoted in eq 18, ref 13) is inversely proportional to the square of the (unknown) concentration of Bjerrum defects in ice. While at large concentrations of Bjerrum defects the disordered ice becomes similar to water and the traditional Poisson—... 

The early treatments of the electroreflectance effect concentrated on the case of uniform electric fields and zero thermal broadening and was therefore suitable only for very lightly doped samples at very low temperatures. The optical properties of a solid are contained in the dielectric function. This function is complex, with the imaginary part only non-zero if the material actually absorbs light. The imaginary part of the dielectric function, e2, can be written for a single band-to-band transition as [186]... 

Details of the derivation of the harmonic o.scillator dielectric function and of the Kramers-Kronig transformation are described in standard textbooks, such as (Kuzmany, 1990b Kittel, 1976). Eq. 4.8-1 is also well known as the Kramers-Heisenberg dielectric function. The integrated absorption coefficient in Eq. 4.8-5 is very often used in conventional vibronic IR spectroscopy to characterize the concentration of the absorbing species. 

Equation 23 can be significantly simplified in the limit of dilute media, that is, for low metal concentrations, since in this case the local field factor is the same for all inclusions and is given by Eq. (4). D. Ricard et al. have proposed a straightforward perturbation method to get the expression of x e// iii dilute medium approximation [79]. The optical Kerr effect in the material amounts to modifying the effective dielectric function under the action of the applied field Eq as... 

For weak electrolytes, at small concentrations, the electrical conductivity is almost proportional to the electrolyte concentration. For higher concentrations, the degree of dissociation decreases. Consequently, the electrical conductivity reaches a maximum as a function of the electrolyte concentration. For strong electrolytes this maximum also exists, because when the electrolyte concentration is increased, the ionic interaction becomes stronger. This is illustrated for a NaOH solution in Fig. 3.6. In this figure the influence of the electrolyte temperature on the conductivity can be seen. The temperature mainly influences the viscosity of the electrolyte, which influences the drift velocity of the ions. Other effects are the changes in the dielectric function of water and the degree of dissociation of the electrolyte. 

This is called the Kramers-Kronig (KK) relationship, from which the dielectric function e = ej + e2 can be derived [3.25]. Since e is also a linear response function, ej and 2 are again related by the KK relationship, thus the information contained in the dielectric function can be examined by concentrating on one of the two components of the dielectric function. We choose to work with 2(m) because it is what optical (X-ray) absorption spectroscopy measures and can be directly related to the atomic polarisability Im[a(o )] that appeared in (3.5). 

In contrast to metals, most studies have concentrated on insulators and semiconductors where the optical structure readily lends itself to a straightforward interpretation. Within certain approximations, the imaginary part of the dielectric function for semiconducting or insulating crystals is given by... 

The critical volume concentration for the blends used here lies in the region of 8 vol,%. In agreement with direct current measurements by the four-point method, this was determined in the test design used here by analysing the complex dielectric function (DF) (7 = 260(0 at 5 Hz (see Figure 11.110). 

Fig. 7.15 Hydrogen concentration dependence of (a) the complex ( -2) and (b) the real (ei) part of the dielectric function for YH - Similar dependences are observed in Lai-zYzH alloys although the trihydrides Lai-zYzHs with z< 0.67 remain cubic while for z> 0.86 they are hexagonal. (From Van Gogh et al. (2001), Ref [56].)...

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 dielectric function of a metal can be obtained considering the dielectric response of a plasma sea of electrons with electron concentration N. The optical properties of metals are in fact determined mainly by the response of free electrons. The role of the crystal lattice can be reduced to the modification of the electron mass to give the effective mass m instead of the free electron mass me, and to the development of states of high energy for optical transitions. 

Besides such advantages, the interpretation of LDS data is not as straightforward as in BDS, since here the capacitance does not depend linearly on the dielectric function. In fact, the asymmetry of tip shape causes the applied field to be highly asymmetric, concentrated near the tip and departing in radial direction from it. Therefore, a suitable model correlating the system capacitance to the dielectric function must be assumed. Computer simulation validates the following simple model for tip/sample distance much smaller than the tip radius, z R [51]... 

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