Frequency and Time Dependent

XII. Appendix F Frequency and Time Dependent Nonlinear Polarizabilities... 

Figure 1.31 shows the frequency and time dependence of the voltage transformation matrix of an untransposed vertical single-circuit line. It is clear from the figure that the frequency dependence is greater than the time dependence of the transformation matrix. The maximum deviation from the average value is about 10% for the time dependence and about 30% for the frequency dependence. Also, the frequency and time dependencies are much greater in the vertical line case than in the horizontal line case. 

The relaxation spectrum H is independent of the experimental time t and is a fundamental description of the system. The exponential function depends upon both the experimental time and the relaxation time. Such a function in the context of this integral is called the kernel. In order to describe different experiments in terms of a relaxation spectrum H or retardation spectrum L it is the kernel that changes. The integral can be formed in time or frequency depending upon the experiment being modelled. The inclusion of elastic properties at all frequencies and times can be achieved by including an additional process in the relaxation... 

The remarkable stability of the capacitance of the SIKO against variations in bias, temperature, frequency and time of operation is a consequence of the superior properties of its ONO dielectric. In contrast to aluminum and tantalum capacitors, the SIKO is a symmetrical device. It shows no significant voltage dependence of the capacitance, as the high s ceramic capacitors do. Only polymeric capacitors show a lower dependence of capacitance on bias than a SIKO. 

A chemical relaxation technique that measures the magnitude and time dependence of fluctuations in the concentrations of reactants. If a system is at thermodynamic equilibrium, individual reactant and product molecules within a volume element will undergo excursions from the homogeneous concentration behavior expected on the basis of exactly matching forward and reverse reaction rates. The magnitudes of such excursions, their frequency of occurrence, and the rates of their dissipation are rich sources of dynamic information on the underlying chemical and physical processes. The experimental techniques and theory used in concentration correlation analysis provide rate constants, molecular transport coefficients, and equilibrium constants. Magde" has provided a particularly lucid description of concentration correlation analysis. See Correlation Function... 

Microbial metabolism results in an increase in both conductance and capacitance causing a decrease in impedance and a consequent increase in admittance. In the Rapid Automated Bacterial Impedance Technique (RABIT) system, the admittance was plotted against time to provide results (Bolton, 1990). The final electrical signal is frequency- and temperature dependent and it has a conductive and a capacitive component. At present, impedance instruments are able to detect 10 —10 bacteria/ml (Ivnitski et ah, 2000). Several commercially available systems are operated... 

All the system response curves in frequency and time domains were calculated numerically from equations that are much too involved to reproduce in detail here. Transfer functions in Laplace transform notation are easily defined for the potentiostat and cell of Figure 7.1. Appropriate combinations of these functions then yield system transfer functions that may be cast into time- or frequency-dependent equations by inverse Laplace transformation or by using complex number manipulation techniques. These methods have become rather common in electrochemical literature and are not described here. The interested reader will find several citations in the bibliography to be helpful in clarifying details. 

The dispersion interaction between an atom and a metal surface was first calculated by Lennard-Jones in 1932, who considered the metal as a perfect conductor for static and time-dependent fields, using a point dipole for the molecule [44], Although these results overestimate the dispersion energy, the correct l/d3 dependence was recovered (d is the metal-molecule distance). Later studies [45 17] extended the work of Lennard-Jones to dielectrics with a frequency-dependent dielectric constant [48] (real metals may be approximated in this way) and took into account electromagnetic retardation effects. Limiting ourselves to small molecule-metal distances, the dispersion interaction of a molecule characterized by a frequency-dependent isotropic polarizability a embedded in a dielectric medium with permittivity esol (note that no cavity is built around the molecule) reads ... 

The dielectric characteristics of barium titanate ceramics with respect to temperature, electric field strength, frequency and time (ageing) are very dependent on the substitution of minor amounts of other ions for Ba or Ti. 

Since T2 is readily determined from time-domain CARS with high accuracy (<2%), a combined analysis of frequency- and time-domain data was proposed and demonstrated (45), plotting the spontaneous Raman data in normalized frequency units, Aa> x T2 (note abscissa scale of Fig. 7b). In this way the bandshape only depends on the ratio rc/T2, and only this ratio has to be deduced from the wings of the Raman band. With respect to the experimental uncertainties (ordinate value of the baseline, overlap with neighboring combination tones), the approach is more reliable than the determination of two quantities, rc and T2, from the spectroscopic data. 

This chapter concentrates on the results of DS study of the structure, dynamics, and macroscopic behavior of complex materials. First, we present an introduction to the basic concepts of dielectric polarization in static and time-dependent fields, before the dielectric spectroscopy technique itself is reviewed for both frequency and time domains. This part has three sections, namely, broadband dielectric spectroscopy, time-domain dielectric spectroscopy, and a section where different aspects of data treatment and fitting routines are discussed in detail. Then, some examples of dielectric responses observed in various disordered materials are presented. Finally, we will consider the experimental evidence of non-Debye dielectric responses in several complex disordered systems such as microemulsions, porous glasses, porous silicon, H-bonding liquids, aqueous solutions of polymers, and composite materials. 

---