Frequency-Dependent Elements

Warburg resistance represents the resistance related to mass transfer in an electrochemical process. The resistance is frequency dependent, and consists of both resistance and capacitance. As discussed in Chapter 3, the impedance of the Warburg resistance (Zw (co)) is written as follows 

The meanings of symbols in Equations 4.7 to 4.11 have been defined in Section 3.2 

The Warburg impedance can be considered as a resistance (Rf=craf 2) connected with a capacitance (C = aof 11) in series. Since the impedance of the 

For n = 1-e, where 0 e 0.2, the CPE corresponds to distortion of the capacitance due to electrode surface roughness or distribution/accumulation of charge carriers. For n = 0.5 e, where 0 0.1, the CPE is related to diffusion, with deviations from Fick s second law. For n = 0 e, where 0 e 0.2, the CPE represents distorted resistance. For n 0, the CPE describes inductive energy accumulation. Therefore, the CPE is a generalized element. Several factors can contribute to the CPE surface roughness, varying thickness or composition, non-uniform current distribution, and a distribution of reaction rates (non-homogeneous reaction rates on the electrode surface) [3], 

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In the previous Section 3.5.4.1 we learned that capacitors and inductors (coils) are frequency-dependent elements. For low frequencies, it is easier to pass inductors, and for high frequencies, to pass capacitors. RC filtershave the disadvantage that for DC sieving a drop in potential occurs. Therefore it is suitable to combine passive filters with amplifiers. Nowadays, this idea can be easily realized with IC s (Fig. 3-25). 

In addition to the frequency-dependent elements, a complete seismograph or accelerograph also has the frequency-independent elements, the generator constant, and the digitizer gain which has to be multiplied with the normalization constant. 

As the wetting front advances at speed U, the solid undergoes a strain cycle at a variety of frequencies, /, the local frequency depending on the distance of the element of solid from the contact line at the moment under consideration. The solid the furthest from the contact line, yet still perturbed by the presence of the three-phase line, is at a distance of ca. to and thus feels a strain cycle at frequency [//to. At the other extreme, near the lower cutoff at x = 8, the frequency is ca. [7/8. The latter frequency will be dominant, since it is in the direct vicinity of the three-phase line that the solid is strained the most. As a consequence, and using Eq. (10), we can define the rate at which work is being done as ... 

But when considered over a wide range of frequencies, the properties of a real electrode do not match those of the equivalent circuits shown in Fig. 12.12 the actual frequency dependence of Z and a does not obey Eq. (12.21) or (12.22). In other words, the actual values of R and or R and are not constant but depend on frequency. In this sense the equivalent circuits described are simplified. In practice they are used only for recording the original experimental data. The values of R and Cj (or R and C ) found experimentally for each frequency are displayed as functions of frequency. In a subsequent analysis of these data, more complex equivalent circuits are explored which might describe the experimental frequency dependence and where the parameters of the individual elements remain constant. It is the task of theory to interpret the circuits obtained and find the physical significance of the individual elements. 

Under potentiostatic conditions, the photocurrent dynamics is not only determined by faradaic elements, but also by double layer relaxation. A simplified equivalent circuit for the liquid-liquid junction under illumination at a constant DC potential is shown in Fig. 18. The difference between this case and the one shown in Fig. 7 arises from the type of perturbation introduced to the interface. For impedance measurements, a modulated potential is superimposed on the DC polarization, which induces periodic responses in connection with the ET reaction as well as transfer of the supporting electrolyte. In principle, periodic light intensity perturbations at constant potential do not affect the transfer behavior of the supporting electrolyte, therefore this element does not contribute to the frequency-dependent photocurrent. As further clarified later, the photoinduced ET... 

A constant phase element (CPE) rather than the ideal capacitance is normally observed in the impedance of electrodes. In the absence of Faradaic reactions, the impedance spectrum deviates from the purely capacitive behavior of the blocking electrode, whereas in the presence of Faradaic reactions, the shape of the impedance spectrum is a depressed arc. The CPE shows power law frequency dependence as follows129 130... 

To obtain Raman spectra one needs the trajectories of the pq tensor elements of the chromophore s transition polarizability. Actually, for the isotropic Raman spectrum one needs only the average transition polarizability. This depends weakly on bath coordinates and this, together with the weak frequency dependence of the position matrix element, was included in our previous calculations [13, 98, 121]. For the VV and VH spectra, others have implemented... 

It should also be mentioned that capacitors were then added in parallel with the resistors in equivalent circuit elements because the frequency-dependent experimental electrical impedance data had a component that was 90° out of phase with the resistor. 

Ramsey obtained ay, by first-order and op by second-order perturbation theory (76) variational treatments give similar results (14, 71, 73). The term wp is sometimes called the second-order paramagnetic term and sometimes the high-frequency term (14), because of the dependence of the (temperature-independent) paramagnetism in molecules on the high-frequency matrix elements of the orbital moments (91). 

We showed in Section 2.3 that the real and imaginary parts of the electric susceptibility are connected by the dispersion relations (2.36) and (2.37). This followed as a consequence of the linear causal relation between the electric field and polarization together with the vanishing of x(<°) in the limit of infinite frequency to. We also stated that, in general, similar relations are expected to hold for any frequency-dependent function that connects an output with an input in a linear causal way. An example is the amplitude scattering matrix (4.75) the scattered field is linearly related to the incident field. Moreover, this relation must be causal the scattered field cannot precede in time the incident field that excited it. Therefore, the matrix elements should satisfy dispersion relations. In particular, this is true for the forward direction 6 = 0°. But 5(0°, to) does not have the required asymptotic behavior it is clear from the diffraction theory approximation (4.73) that for sufficiently large frequencies, 5(0°, to) is proportional to to2. Nevertheless, only minor fiddling with S makes it behave properly the function... 

This work attempts to model a semiconductor/molten salt electrolyte interphase, in the absence of illumination, in terms of its basic circuit elements. Measurement of the equivalent electrical properties has been achieved using a newly developed technique of automated admittance measurements and some progress has been made toward identification of the frequency dependent device components (1 ). The system chosen for studying the semiconductor/ molten salt interphase has the configuration n-GaAs/AlCl3 1-... 

In the expression of the eigenvalues, the wavevector- and frequency-dependent transport coefficients are present. As mentioned before, these are defined by the U matrix elements. Thus in the definition of all these transport coefficients there appears a structure of the form... 

Finally, the circuit is solved. It can be done in two ways. Either a corresponding physical circuit is actually built from these electrical components and its transfer function (H) is measured, or the transfer functions of the individual elements and of the entire circuit are represented by explicit mathematical equations and the transfer function is calculated. There are always certain eigenvalues in these solutions that aid in the assignment of physical meaning to the calculated parameters. The transfer function is defined as the frequency-dependent ratio of the output voltage to the input voltage. It is a complex variable for an AC excitation signal. 

The coefficients <5, / , and y relate to the independent tensor elements, and are frequency-dependent material parameters. For cubic media, all four nonvanishing tensor elements are independent and another term, fE co) VjEj(a>), appears in this expression. This additional term gives rise to an anisotropy in the SH radiation and will be discussed in more detail later in this review. 

The electrochemical impedance of a real electrode is frequently represented by an equivalent circuit containing constant phase element (CPE) showing power-law frequency dependence as follows... 

Our present focus is on correlated electronic structure methods for describing molecular systems interacting with a structured environment where the electronic wavefunction for the molecule is given by a multiconfigurational self-consistent field wavefunction. Using the MCSCF structured environment response method it is possible to determine molecular properties such as (i) frequency-dependent polarizabilities, (ii) excitation and deexcitation energies, (iii) transition moments, (iv) two-photon matrix elements, (v) frequency-dependent first hyperpolarizability tensors, (vi) frequency-dependent polarizabilities of excited states, (vii) frequency-dependent second hyperpolarizabilities (y), (viii) three-photon absorptions, and (ix) two-photon absorption between excited states. 

The basic postulate of the model is that the frequency-dependent polarizability of each electron density volume element is given by... 

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