Kramers-Kronig calculation

The peak shifts and shape changes evident in Fig. 3.20 are completely accounted for by the Kramers-Kronig calculation indicating that the frequency and band shape differences are a consequence of the sampling technique. 

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 1/to4 high frequency limit for R can be useful in determining optical constants from Kramers-Kronig analysis of reflectance data (see Section 2.7). Reflectances at frequencies higher than the greatest far-ultraviolet frequency for which measurements are made can be calculated from (9.15) and used to complete the Kramers-Kronig integral to infinite frequency. 

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.

Figure 13.8 Ilustration of the validity of the Kramers-Kronig relations for the piezoelectric relaxation in Sm-modified lead titanate ceramics. The imaginary component is calculated from the real using numerical method and Kramers-Kronig relations and compared with experimentally determined data.

In Sections 1.7 and 4.8.3, we have studied the dielectric relaxation phenomena and dielectric spectroscopy, respectively. In dielectric spectrometry, the methodology allowed us to measure the capacity and, consequently, the real part of the complex dielectric constant. The imaginary part of the complex dielectric constant was calculated, in this case, with the help of the Kramers-Kronig... 

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. 

Figure 3.12. Simulation of the b-polarized (0-0) reflectivity of the anthracene crystal using the bulk reflectivity amplitude derived from a Kramers-Kronig analysis (Section II.C). The total reflectivity is calculated from the scheme of Fig. 3.11 and (3.24)-(3.25) for various values of the nonradiative broadening parameter 7% Comparison with spectra of our best crystals gives the value / ° = 3cm 1 for T = 1.7 K.

With an appropriate extrapolation of the data beyond the highest and lowest measured energies, we also calculated ft by Kramers Kronig (KK) analysis [32]. The agreement between the optical constants determined by the 7Z and T inversion procedures and those obtained from KK analysis is very good (see Fig. 2.3). 

Since almost all equations used in impedance methods are derived assuming linearity, it is important to have some means of verifying this supposition. The Kramers-Kronig relations2 link Z with Z" and allow the calculation of values for Z" at any frequency from a knowledge of the full frequency spectrum of Z, and vice versa. 

This seems trivial, but is the very important result that, if one measures y (ft)), one can calculate y2(ft)), and conversely put differently, the Kramers-Kronig relations show that the absorptive and dispersive properties of a medium are not independent of each other. An experimental difficulty is that one must truncate the integrations at some maximum measured frequency ft) this may lead to considerable error. 

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]. 

For a linear system, the phase and amplitude of the impedance relate to each other. Consequently, if we know the frequency dependence of the phase we can calculate the amplitude of the impedance as a function of frequency. Similarly, we can deduce the frequency dependence of the phase from that of the amplitude of the impedance. The calculation can be achieved by the Kramers-Kronig (K-K) transforms. This is a useful check on the validity of a measured impedance spectrum. For information on K-K transforms, see Appendix C. 

Luckily, the real and imaginary parts of the complex dielectric permittivity are not independent of each other and are connected by means of the Kramers-Kronig relations [11]. This is one of the most commonly encountered cases of dispersion relations in linear physical systems. The mathematical technique entering into the Kramers-Kronig relations is the Hilbert transform. Since dc-conductivity enters only the imaginary component of the complex dielectric permittivity the static conductivity can be calculated directly from the data by means of the Hilbert transform. 

The absorption is very intense, so direct measurement of absorption spectra requires very thin films. When the sample surface is smooth enough, the absorption can be calculated by Kramers-Kronig inversion of the reflectance, which can be done rather accurately for a well-isolated transition [113]. A strongly scattering sample must be studied by attenuated total internal reflectance. 

The dependence of the reflection coefficient on n and k allows these quantities to be calculated from the reflection spectrum by use of the Kramers-Kronig relations. It follows from Equation (9.17) that an intense absorption gives a high reflectivity, e.g. the reflection spectrum of PDA-TS,... 

Another limitation on acoustic properties is expressed by the Kramers-Kronig (KK) relations, which are general relations between the real and imaginary parts of a complex function. These relations were originally derived for optics but can be applied in many other areas as well. The essence of the relations is that the real and imaginary parts of the function are not independent of each other but one may be calculated from an integral of the other. As applied to complex modulus, the specific form of the relations is given elsewhere in this book (J. Jarzynski, A Review of the Mechanisms of Sound Attenuation in Materials). 

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