Kramers—Kronig transforms

The Kramers-Kronig relations hold provided the four above constraints are satisfied and (1) allow the calculation of the imaginary impedance from the real part  

Kendig and Mansfeld used Eq. (232) and supposed that the imaginary impedance is symmetric. They carried out integration between the frequency corresponding to the maximum of the imaginary impedance and infinity, and multiplied the result by 2. However, their method is limited to systems containing one time constant. 

Macdonald et al. and Dougherty and Smedley used a polynomial approximation of the impedance function, followed by analytical integration of the polynomials. However, extrapolation of polynomials over a large frequency range may be unreliable. Haili extrapolated Z as proportional to (0 as (0 0 and as inversely proportional to ro as co  

Esteban and Orazem proposed using Eqs. (230) and (231) simultaneously to calculate the impedance below the lowest measured frequency, cOmin. and to continue the integration procedure to three or four decades of smaller frequency, (Oq- The latter parameter is chosen in such a way that the real impedance goes to a constant value while the imaginary impedance goes to 0 at (Oo. 

This method was further modified by Boukamp, who also used the Voigt circuit but with a fixed distribution of time constants that is, the time constants were defined and the adjustable parameters were Parameters Tj were chosen as equal to the inverse of the experimental 

If the real part of a linear network function of frequency is known over the complete frequency spectrum, it is possible to calculate the imaginary part (and vice versa). There is a relationship between the real and imaginary part of an immittance (or e and e ), given 

KKTs are tools brought to network theory by the work of Kramers (1926) and Kronig (1929) on X-ray optics. Just as the reciprocity theorem, they are purely mathematical rules of general validity in any passive, linear, reciprocal network of a minimum phase shift type. By minimum-phase networks, we mean ladder networks that do not have poles in the right half plane of the Wessel diagram. A ladder network is of minimum phase type a bridge where signal can come from more than one ladder is not necessarily of the minimum-phase type. The transforms are only possible when the functions are finite-valued at all frequencies. With impedance Z = R- -jX the transforms are  

The frequency of integration f is from zero to infinite. The resistance or reactance or modulus of impedance Z must therefore be known for the complete frequency spectrum. Dealing with one dispersion only, the spectrum of interest is limited to that of the dispersion. When the frequency range is limited and the number of measurement points is reduced, some error is committed when obtaining one impedance component out of the other (Riu and Lapaz, 1999). 

Consequently, the area under one dispersion loss peak is independent on the distribution of relaxation times. These equations also represent a useful check for experimental data 

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It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

If i = i — ik] and H2 = ns — are known as a function of wavelength, Eq. 12 can be used to calculate the entire RAIR spectrum of a surface film. Since transmission infrared spectroscopy mostly measures k, differences between transmission and RAIR spectra can be identified. Fig. 6 shows a spectrum that was synthesized assuming two Lorentzian-shaped absorption bands of the same intensity but separated by 25 cm. The corresponding spectrum of i values was calculated from the k spectrum using the Kramers-Kronig transformation and is also shown in Fig. 6. The RAIR spectrum was calculated from the ti and k spectra using Eqs. 11 and 12 and is shown in Fig. 7. 

The two anomalous components of the scattering factor, f" and /, are interrelated through the Kramers Kronig transforms, which have the form... 

Figure 6. Calculated values for n2( —) generated from a Kramers-Kronig transformation of the fit to the two-photon absorption measurement (-).

Figure 29 Spectra of (a) refractive index and (b) extinction coefficient of polystyrene thin film containing SP and PM. The former is based on the observed difference absorption spectra shown in Fig. 28 before and after UV irradiation. The former was calculated from (b) by Kramers-Kronig transformation.

The IR reflectivity measurements were performed on single crystals of 2 0.5 0.3 mm3 in size. A FT-IR Perkin-Elmer 1725X spectrometer equipped with microscope and a helium cryostat was used. Polarized reflectivity spectra (R(ro)) were measured from the conducting plane in two principal directions. Optical conductivity a(co) was obtained by Kramers-Kronig transformation. 

The reflectivity spectra R(E) and the reflectivity-EXAFS Xr(E) = R(E) — Rq(E)]/R()(E) are similar, but not identical, to the absorption spectra and x(E) obtained in transmission mode. R(E) is related to the complex refraction index n(E) = 1 — 8(E) — ifl(E) and P(E) to the absorption coefficient /i(E) by ji fil/An. P and 8 are related to each other by a Kramers-Kronig transformation, p and 8 may be also separated in an oscillatory (A/ , AS) and non-oscillatory part (P0,80) and may be used to calculate Xr- This is, briefly, how the reflectivity EXAFS may be calculated from n(E). which itself can be obtained by experimental transmission EXAFS of standards, or by calculation with the help of commercial programs such as FEFF [109] with the parameters Rj, Nj and a, which characterize the near range order. The fit of the simulated to measured reflectivity yields then a set of appropriate structure parameters. This method of data evaluation has been developed and has been applied to a few oxide covered metal electrodes [110, 111], Fig. 48 depicts a condensed scheme of the necessary procedures for data evaluation. 

Absorption Spectra as Kramers-Kronig-Transformed Reflection Spectra... 

This expression was originally used to estimate the polarization resistance for actively corroding metals whose impedance response was a well-behaved semicircular arc in the complex plane (73), but can be used in certain situations to estimate the equivalent resistance of conversion coated metal surfaces. Equation (9) is derived from the Kramers-Kronig transforms and subject to the conditions that limit their use (74-78). These conditions include... 

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. 

Hie evaluation of the data yields Rjy Nj, and Sjy i.e., the near-range order parameters of the material seen from the absorber atom. XAS permits the evaluation of the near-range order in the vicinity of the atoms of various elements of one specimen if the energies of their absorption edges are different enough and thus are well separated within the spectrum. It should be mentioned that XAS in reflection looks similar to XAS in transmission mode, however it is different and the evaluation of measurements requires the comparison with reflectivity data calculated form transmission EXAFS spectra. These evaluation procedures involving Kramers-Kronig transform are described in the literature [i-v]. 

As both the frequency domain and the time domain methods have disadvantages, Boukamp [87] recommended that both methods be combined using the CNLS-fit procedure, data validation (Kramers-Kronig transformation), and deconvolution. The Kramers-Kronig transformation can be found in Appendix C. 

Urquidi-Macdonald M, Real MS, Macdonald DD (1986) Application of Kramers-Kronig transforms in the analysis of electrochemical impedance data. J Electrochem Soc 133(10) 2018-24... 

Macdonald DD, Urquidi-Macdonald M (1990) Kramers-Kronig transformation of constant phase impedances. J Electrochem Soc 137(2) 515-17... 

Shi M, Chen Z, Sun J (1999) Kramers-Kronig transform used as stability criterion of concrete. Cem Concr Res 29(10) 1685-8... 

Tan GL, DeNoyer LK, French RH, Guittet Ml, Gautier-Soyer M (2004) Kramers-Kronig transform for the surface energy loss function. J Electron Spectrosc Relat Phenom 142(2) 97-103... 

Darowicki K, Kawula J (2004) Validity of impedance spectra obtained by dynamic electrochemical impedance spectroscopy verified by Kramers-Kronig transformation. Pol J Chem 78(9) 1255-60... 

Boukamp BA (1993) Practical application of the Kramers-Kronig transformation on impedance measurements in solid state electrochemistry. Solid State Ionics 62(1-2) 131 11... 

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