Frequency-dependent dielectric

The expression immediately gives an estimate of the enthalpy of adsorption in taking an atom from the gaseous (vacuum) state to a liquid, or to a composite medium like a zeolite, characterised by its measured dielectric frequency dependent response (o>). It is, exactly as for the electrostatic Bom self-energy in taking an ion from vacuum to water ... 

Neumann M, Steinhauser O and Pawley G S 1984 Consistent calculation of the static and frequency-dependent dielectric constant in computer simulations Mol. Phys. 52 97-113... 

The Hamaker constant can be evaluated accurately using tire continuum tlieory, developed by Lifshitz and coworkers [40]. A key property in tliis tlieory is tire frequency dependence of tire dielectric pennittivity, (cij). If tills spectmm were tlie same for particles and solvent, then A = 0. Since tlie refractive index n is also related to f (to), tlie van der Waals forces tend to be very weak when tlie particles and solvent have similar refractive indices. A few examples of values for A for interactions across vacuum and across water, obtained using tlie continuum tlieory, are given in table C2.6.3. 

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 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 ... 

Neumann, M., Steinhauser, O. On the calculation of the frequency-dependent dielectric constant in computer simulations. Chem. Phys. Lett. 102 (1983) 508-513. 

Because of very high dielectric constants k > 20, 000), lead-based relaxor ferroelectrics, Pb(B, B2)02, where B is typically a low valence cation and B2 is a high valence cation, have been iavestigated for multilayer capacitor appHcations. Relaxor ferroelectrics are dielectric materials that display frequency dependent dielectric constant versus temperature behavior near the Curie transition. Dielectric properties result from the compositional disorder ia the B and B2 cation distribution and the associated dipolar and ferroelectric polarization mechanisms. Close control of the processiag conditions is requited for property optimization. Capacitor compositions are often based on lead magnesium niobate (PMN), Pb(Mg2 3Nb2 3)02, and lead ziac niobate (PZN), Pb(Zn 3Nb2 3)03. 

The dielectric medium is normally taken to have a constant value of e, but may for some purposes also be taken to depend for example on the distance from M. For dynamical phenomena it can also be allowed to be frequency dependent i.e. the response of the solvent is different for a fast reaction, such as an electronic transition, and a slow reaction, such as a molecular reorientation. 

Dielectric measurements were used to evaluate the degrees of inter- and intramolecular hydrogen bonding in novolac resins.39 The frequency dependence of complex permittivity (s ) within a relaxation region can be described with a Havriliak and Negami function (HN function) ... 

The fact that the dielectric constant depends on the frequency gives SPFM an interesting spectroscopic character. Local dielectric spectroscopy, i.e., the study of s(w), can be performed by varying the frequency of the applied bias. Application of this capability in the RF range has been pursued by Xiang et al. in the smdy of metal and superconductor films [39,40] and dielectric materials [41]. In these applications a metallic tip in contact with the surface was used. 

The dramatic slowing down of molecular motions is seen explicitly in a vast area of different probes of liquid local structures. Slow motion is evident in viscosity, dielectric relaxation, frequency-dependent ionic conductance, and in the speed of crystallization itself. In all cases, the temperature dependence of the generic relaxation time obeys to a reasonable, but not perfect, approximation the empirical Vogel-Fulcher law ... 

The dielectric constant of a polymer (K) (which we also refer to as relative electric permittivity or electric inductive capacity) is a measure of its interaction with an electrical field in which it is placed. It is inversely related to volume resistivity. The dielectric constant depends strongly on the polarizability of molecules tvithin the polymer. In polymers with negligible dipole moments, the dielectric constant is low and it is essentially independent of temperature and the frequency of an alternating electric field. Polymers with polar constituents have higher dielectric constants. When we place such polymers in an electrical field, their dipoles attempt... 

Treating the free electrons in a metal as a collection of zero-frequency oscillators gives rise51 to a complex frequency-dependent dielectric constant of 1 - a>2/(co2 - ia>/r), with (op = (47me2/m)l/2 the plasma frequency and r a collision time. For metals like Ag and Au, and with frequencies (o corresponding to visible or ultraviolet light, this simplifies to give a real part... 

Relation [1] Is the frequency-dependent analogue of a formula proposed by Chasset and Thirion (2, 3) which has since been applied very frequently to relaxation measurements on cured rubbers. The next three equations are Inspired by similar relations In dielectrics (they are not derived from these) Equation [2] by the Cole-Cole and Equation [3] by the Davidson-Cole relation (15, 16). Both are special cases of the most general Equation [4] which contains five parameters (17). 

To use this equation in evaluating I, one needs a model for e(t ) that is consistent with available experiments on the frequency-dependent dielectric constant. 

In the equation s is the measured dielectric constant and e0 the permittivity of the vacuum, M is the molar mass and p the molecular density, while Aa and A (po2) are the isotope effects on the polarizability and the square of the permanent dipole moment respectively. Unfortunately, because the isotope effects under discussion are small, and high precision in measurements of bulk phase polarization is difficult to achieve, this approach has fallen into disfavor and now is only rarely used. Polarizability isotope effects, Aa, are better determined by measuring the frequency dependence of the refractive index (see below), and isotope effects on permanent dipole moments with spectroscopic experiments. 

It is known that measnring the absorption coefficient (and thns the extinction coefficient) over the whole freqnency range, 0 < real part of N(co) - that is, the normal refractive index ( >) - can be obtained by nsing the Kramers-Kronig relationships (Fox, 2001). This is an important fact, because it allows us to obtain the frequency dependence of the real and imaginary dielectric constants from an optical absorption experiment. 

In the next section, we will develop a simple model to predict the frequency dependence of the relative dielectric constants si and 2 of a given material. At that point, we will be able to determine the measurable optical magnitudes defined in Chapter 1 at any particular wavelength (or frequency) if the relative dielectric constants (and thus n and k) are known at that wavelength. 

Chapter 4, presents details of the absorption and reflectivity spectra of pure crystals. The first part of this chapter coimects the optical magnimdes that can be measured by spectrophotometers with the dielectric constant. We then consider how the valence electrons of the solid units (atoms or ions) respond to the electromagnetic field of the optical radiation. This establishes a frequency dependence of the dielectric constant, so that the absorption and reflectivity spectrum (the transparency) of a solid can be predicted. The last part of this chapter focuses on the main features of the spectra associated with metals, insulators, and semiconductors. The absorption edge and excitonic structure of band gap (semiconductors or insulator) materials are also treated. 

Frequency dependent complex impedance measurements made over many decades of frequency provide a sensitive and convenient means for monitoring the cure process in thermosets and thermoplastics [1-4]. They are of particular importance for quality control monitoring of cure in complex resin systems because the measurement of dielectric relaxation is one of only a few instrumental techniques available for studying molecular properties in both the liquid and solid states. Furthermore, It is one of the few experimental techniques available for studying the poljfmerization process of going from a monomeric liquid of varying viscosity to a crosslinked. Insoluble, high temperature solid. 

In the past, impedance or dielectric studies have been examined as an experimental technique to monitor the flow properties, effects of composition, and the advancement of a reaction during cure [1]. Until a paper by Zukas et al [2], little emphasis had been placed on the frequency dependence except to note the shift in position and magnitude of impedance maxima and minima. Furthermore, most measurements on curing systems reported results in terms of... 

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