Kramers-Kronig consistency

Use a measurement model on Kramers-Kronig-consistent data set to get a model for data. 

E. Shiles, T. Sasaki, M. Inokuti, and D. Y. Smith, "Self-consistency and sum-mle tests in the Kramers-Kronig analysis of optical data Applications to aluminum," Phys. Rev. B, 22, 1612-28 (1980). 

In Section I, the spectra of e"(ai) consist of Dirac 5 peaks (1.79). In a real crystal these peaks are broadened by static disorder, thermal fluctuations, and excitation-relaxation processes. Discarding for the moment the static disorder, we focus our attention on broadening processes due to lattice phonons, which may be described alternatively in terms of fluctuations of the local energies of the sites, or in terms of exciton relaxation by emission and absorption of phonons. These two complementary aspects of the fluctuation-dissipation theorem64 will allow us to treat the exciton-phonon coupling in the so-called strong and weak cases. The extraordinary (polariton) 0-0 transition of the anthracene crystal will be analyzed on the basis of these theoretical considerations and the semiexperimental data of the Kramers-Kronig analysis. 

The traditional way is to measure the impedance curve, Z(co), point-after-point, i.e., by measuring the response to each individual sinusoidal perturbation with a frequency, to. Recently, nonconventional approaches to measure the impedance function, Z(a>), have been developed based on the simultaneous imposition of a set of various sinusoidal harmonics, or noise, or a small-amplitude potential step etc, with subsequent Fourier- and Laplace transform data analysis. The self-consistency of the measured spectra is tested with the use of the Kramers-Kronig transformations [iii, iv] whose violation testifies in favor of a non-steady state character of the studied system (e.g., in corrosion). An alternative development is in the area of impedance spectroscopy for nonstationary systems in which the properties of the system change with time. 

A final comment seems to be pertinent. In most cases actual measurements are not made at the frequencies of interest. However, one can estimate the corresponding property at the desired frequency by using the time (fre-quency)-temperature superposition techniques of extrapolation. When different apparatuses are used to measure dynamic mechanical properties, we note that the final comparison depends not only on the instrument but also on how the data are analyzed. This implies that shifting procedures must be carried out in a consistent manner to avoid inaccuracies in the master curves. In particular, the shape of the adjacent curves at different frequencies must match exactly, and the shift factor must be the same for all the viscoelastic functions. Kramers-Kronig relationships provide a useful tool for checking the consistency of the results obtained. 

The potentials and currents were measured and controlled by a Solatron 1286 potentiostat, and a Solatron 1250 frequency response emalyzer was used to apply the sinusoidal perturbation and to calculate the transfer function. The impedance data analyzed in this section were taken after 12 hours of immersion and were found by the methods described in Chapter 22 to be consistent with the Kramers-Kronig relations. 

A distinction is drawn in equation (21.1) between stochastic errors that are randomly distributed about a mean value of zero, errors caused by the lack of fit of a model, and experimental bias errors that are propagated through the model. The problem of interpretation of impedance data is therefore defined to consist of two parts one of identification of experimental errors, which includes assessment of consistency with the Kramers-Kronig relations (see Chapter 22), and one of fitting (see Chapter 19), which entails model identification, selection of weighting strategies, and examination of residual errors. The error analysis provides information that can be incorporated into regression of process models. The experimental bias errors, as referred to here, may be caused by nonstationary processes or by instrumental artifacts. 

Bias errors are systematic errors that do not have a mean value of zero and that cannot be attributed to an inadequate descriptive model of the system. Bias errors can arise from instrument artifacts, parts of the measured system that are not part of the system under investigation, and nonstationary behavior of the system. Some types of bias errors lead the data to be inconsistent with the Kramers-Kronig relations. In those cases, bias errors can be identified by checking the impedance data for inconsistencies with the Kramers-Kronig relations. As some bias errors are themselves consistent with the Kramers-Kronig relations, the Kramers-Kronig relations cannot be viewed as providing a definitive tool for identification of bias errors. 

The impedance response of low-impedance systems may include the finite impedance behavior of wires and connectors. These may be considered, from the perspective of model identification, as yielding artifacts in the mezisured response. Such artifacts may be simply resistive, but may also exhibit a capacitive or even an inductive frequency dependence. Such artifacts will be generally consistent with the Kramers-Kronig relations. 

The use of measurement models to identify consistency with the Kramers-Kronig relations is equivalent to the use of Kramers-Kronig transformable circuit analogues. An important advantage of the measurement model approach is that it identifies a small set of model structures that are capable of representing a large... 

The Kramers-Kronig relations are extremely general emd have been applied to a wide variety of research areas. In the field of optics, for which the validity of the Kramers-Kronig is not in question, the relationship between real and imaginary components has been exploited to complete optical spectra. In other areas where data cannot be assumed to satisfy the requirements of the Kramers-Kronig relations, the equations presented in Table 22.1 have been used to check whether real and imaginary components of complex variables are internally consistent. Failure of the Kramers-Kronig relations is assumed to correspond to a failure within the experiment to satisfy one or more of the constraints of linearity, stability, or causality. 

At any frequency co the expectation of the observed impedance E (Zob(it ))/ defined in equation (3.1), is equal to the value consistent with the Kramers-Kronig relations, i.e.,... 

Example 23.2 demonstrates the utility of the error analysis for determining consistency with the Kramers-Kronig relations. In this case, the low-frequency inductive loops were foimd to be consistent with the Kramers-Kronig relations at frequencies as low as 0.001 Hz so long as the system had reached a steady-state operation. The mathematical models that were proposed to account for the low-frequency features were based on plausible physical and chemical hypotheses. Nevertheless, the models are ambiguous and require additional measurements and observations to identify the most appropriate for the system under study. 

P. Agarwal, M. E. Orazem, and L. H. Garcia-Rubio, "Measurement Models for Electrochemical Impedance Spectroscopy 3. Evaluation of Consistency with the Kramers-Kronig Relations," Journal of The Electrochemical Society, 142 (1995)4159-4168. 

This part provides a conceptual understanding of stochastic, bias, and fitting errors m frequency-domain measurements. A major advantage of frequency-domain measurements is that real and imaginary parts of the response must be internally consistent. The expression of this consistency takes different forms that are known collectively as the Kramers-Kronig relations. The Kramers-Kronig relations and their application to spectroscopy measurements are described. Measurement models, used to assess the error structure, are described and compared with process models used to extract physical properties. 

Scattering theory has very general validity. It is based on the existence of a Schrodinger-type equation and is consistent with causality requirements through the Kramers-Kronig dispersion relations. It is widely used outside atomic physics for the description of all resonance phenomena. There is therefore some advantage in using it in atomic physics it ensures unification and is more transparent to reseachers from other fields. 

It is equivalent to a series of a number of RC parallel combinations in series with the solution resistance, which is also known as the Voigt model. If a sufficient number of terms is included, the model should be able to fit impedance data of any stationary electrochemical cell. Since the model is consistent with the Kramers-Kronig relations, any failure to fit experimental data using this model indicates a violation of the Kramers-Kronig relations. 

Here i is the imaginary unit. The first term fo is the nonresonant term which is equal to the atomic number of the element. The second and the third factor are energy dependent and show strong variation only in the vicinity of the adsorption edge. The imaginary part /" is directly related to the adsorption ctoss section of the X-rays. Both / and /" are coimected to each other by the Kramers-Kronig relation. From this, it can be derived that the intensity consists of three different terms ... 

The mathematical theory of the frequency-domain methods consists of beautiful applications of matrix and complex analysis. The general matrix rate equations have been derived for the monomolecular photochemical processes, and matrix analysis is used in deriving the general solution for the temporal concentrations of the excited species in the presence of an arbitrary functional form of excitation. The sinusoidal excitation and dual-phase lock-in detection of the emission lead to a signal which can be effectively treated as a complex number. For instance, the Kramers-Kronig relation, better known fi om the solid-state physics, can be used for checking the internal consistency of data. 

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