Kronig-Kramers integral transform

ORD and CD are connected through the Kronig-Kramers integral transform. Thus, for the -th transition it follows that ... 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

Impedance is by definition a complex quantity and is only real when 0=0 and thus Z(m) = Z(a>), that is, for purely resistive behavior. In this case the impedance is completely frequency-independent. When Z is found to be a variable function of frequency, the Kronig-Kramers (Hilbert integral transform) relations (Macdonald and Brachman [1956]), which holistically connect real and imaginary parts with each other, ensure that Z" (and 9) cannot be zero over all frequencies but must vary with frequency as well. Thus it is only when Z(linear resistance, that Z(m) is purely real. 

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. 

The derivation of the Kronig-Kramers relationships may be found in Reference 41. The clarity of the presentation there is such as to make any attempt to repeat the proof here seem puerile. We might just mention, however, that the crux of the proof lies in seeing that the principle of causality requires that [% v) — be analytic in the whole lower half of the complex plane. Then, as in the case of Hilbert transforms, the real and imaginary parts of [% (r) —xjl wi not be independent and may be expressed as integral transforms of each other. But even though we shall not go into the details of the proof of (65) and (66), their application to the problem of optical activity requires some comment. 

A critical problem in EIS, as well as any other scientific measurements, is the validation of the experimental data. The use of Kramers-Kronig (KK) transforms has been proposed to assess the quality of the measured impedance data [15]. These integral transforms were derived assuming four basic conditions ... 

Details of the derivation of the harmonic o.scillator dielectric function and of the Kramers-Kronig transformation are described in standard textbooks, such as (Kuzmany, 1990b Kittel, 1976). Eq. 4.8-1 is also well known as the Kramers-Heisenberg dielectric function. The integrated absorption coefficient in Eq. 4.8-5 is very often used in conventional vibronic IR spectroscopy to characterize the concentration of the absorbing species. 

Figure 8(b) shows the difference in refractive indices and a spectrum of the integral part of eq. (5). The possibility of absorption at wavelength shorter than 200 nm was not considered in this calculation. It is clear that the Kramers-Kronig transformation calculated with difference absorption spectra alone does not give correct values in this polymer system. 

Mathematically, integral Kramers-Kronig relations have very general character. They represent the Hilbert transform of any complex function s(co) = s (co) + s"(co) satisfying the condition s (co) = s(—co)(here the star means complex conjugate). In our particular example, this condition is applied to function n(co) related to dielectric permittivity s(co). The latter is Fourier transform of the time dependent dielectric function s(f), which takes into account a time lag (and never advance) in the response of a substance to the external, e.g. optical, electric field. Therefore the Kramers-Kronig relations follow directly from the causality principle. 

It is possible to replace the Kramers-Kronig integration by approximation. If the system can be well approximated by a linear circuit, then it must be Kramers-Kronig transformable. Orazem and coworkers [572, 573] proposed using the Voigt circuit displayed in Fig. 13.2. 

Kramers-Kronig transforms require integration over frequencies from zero to infinity, which, in practice, is difficult to carry out. There exists a Hilbert logarithmic transform [602, 603] that can be used to validate impedance data in a limited frequency range. It is known under the name Z-HIT transform [585] and is used in the Zahner software [604]. The logarithm of a transfer function known for frequencies between tOs and cOo may be written as a function of the phase angle ... 

Ohta, K., and Ishida, I. (1988). Comparison among several numerical integration methods for Kramers-Kronig transformation. Appl. Spectrosc. 42, 952-957. 

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