Kramers-Kronig relations application

The fimdamental constraints are that the system be stable, in the sense that perturbations to the system do not grow, that the system responds linearly to a perturbation, and that the system be causal in the sense that a response to a perturbation cannot precede the perturbation. The Kramers-Kronig relationships were foimd to be entirely general with application to all frequency-domain measiuements that could satisfy the above constraints. Bode extended the concept to electrical impedance and tabulated various extremely useful forms of the Kramers-Kronig relations. ... 

Following the discussion of equation (22.74), the necessary conditions for application of the Kramers-Kronig relations ate that equation (22.72) be satisfied and that... 

In principle, the Kramers-Kronig relations can be used to determine whether the impedance spectrum of a given system has been influenced by bias errors caused, for example, by instrumental artifacts or time-dependent phenomena. Although this information is critical to the analysis of impedance data, the Kramers-Kronig relations have not found widespread use in the analysis and interpretation of electrochemical impedance spectroscopy data due to difficulties with their application. The integral relations require data for frequencies ranging from zero to infinity, but the experimental frequency range is necessarily constrained by instrumental limitations or by noise attributable to the instability of the electrode. 

It is sometimes said that a finite impedance is needed in order for application of the Kramers-Kronig relations to an electrochemical system. Yet, blocking electrodes do not have a finite impedance. Do the Kramers-Kronig relations apply for blocking electrodes If so, how can they be applied ... 

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. 

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. 

P. Agarwal, M. E. Orazem, and L. H. Garcia-Rubio [1995] Application of Measurement Models to Impedance Spectroscopy. III. Evaluation of Consistency with the Kramers-Kronig Relations, J. Electrochem. Soc. 142, 4159 168. 

R. L. Van Meirhaeghe, E. C. Dutoit, F. Cardon, and W. P. Gomes [1976] On the Application of the Kramers-Kronig Relations to Problems Concerning the Frequency Dependence of Electrode Impedance, Electrochim. Acta 21, 39 3. 

Figure 6.9 An example of the application of the Kramers Kronig relations, (a) Reflectance spectrum measured for Si N(b) calculated n(v) spectrum, and (c) calculated k(v) spectrum. (Source Reproduced from Ref. [21 with permission from the Society of Applied Spectroscopy, 2013).

An is the contribution to the refractive index due to the solute whose concentration is C (mol/L), c is the speed of light, and s(v ) is the molar extinction coefficient. An is the difference between the refractive indices of the solution and the solvent. An is essentially the isotropic contribution to the refractive index by the solute whose absorption spectrum determines s v ). A common application of the Kramers-Kronig relation appears in light scattering, especially when one is interested in the effects of scattering on absorbance measurements. 

It is also shown how x(< ) is related to the temporal behaviour of the dielectric polarization follomng the sudden application, or removal, of an electric field. Various forms of the Kramers-Kronig dispersion relations are introduced for y (o>) and x C") aod for a number of functions of The section closes with the ddOnition of the frequency-dependent complex refractive index n() = n(cu) — and a discussion of its relation... 

K. Krishnan, Applications of the Kramers-Kronig Dispersion Relations to the Analysis of FTIR Specular Reflectance Spectra, Fl lR/IR Notes 51, Biorad Digilab Division, Cambridge, MA, August 1987. 

The Kronig-Kramers relationships are a very general set of integral transforms that find wide application in phjreical problems. They are intimately related to Hilbert transforms which, subject to certain integrability and analyticity conditions, allow the real and imaginary parts of a complex function f(z) = u iv to >t expressed as a pair of transform mates. This property follows from the fact that u and v are not completely independent when / z) is analytic in the whole upper half of the complex plane. 

---