Kramer-Kronigs relationship

Kramers-Kronig relationships, 7 338 Kramers-Kronig (K-K) transformation, 14 231... 

It is known that measnring the absorption coefficient (and thns the extinction coefficient) over the whole freqnency range, 0 < real part of N(co) - that is, the normal refractive index ( >) - can be obtained by nsing the Kramers-Kronig relationships (Fox, 2001). This is an important fact, because it allows us to obtain the frequency dependence of the real and imaginary dielectric constants from an optical absorption experiment. 

The attenuation may be expressed by making the wavenumber complex (this would be k — ia in eqn (6.12)), and the velocity (= w/k) may also be written as a complex quantity. This in turn corresponds to a complex modulus, so that the relationship v - /(B/p) is preserved indeed the acoustic wave equation may be written as a complex-valued equation, without the need for the extra term in (6.11). Complex-valued elastic moduli are frequency-dependent, and the frequency-dependent attenuation and the velocity dispersion are linked by a causal Kramers-Kronig relationship (Lee et al. 1990). 

INFRAREDTECHNOLOGYANDRAMAN SPECTROSCOPY - INFRARED TECHNOLOGY] (Vol 14) Kramers-Kronig relationship... 

The real and imaginary parts of Eq. (3) are related by the Kramers-Kronig relationship [19] ... 

Since sound speed is a frequency dependent complex quantity, it therefore follows that the characteristic impedance of the media will also be frequency dependent and complex. If the frequency dependence of sound speed is not known, it can be estimated from the attenuation coefficient as follows. For the rubber composites of interest here, usually a A is essentially independent of frequency. Using Kramers-Kronig relationships (5) it can then be shown that ... 

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 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. ... 

A normal mode of vibration in a crystal gives rise to infrared absorption if the atom displacements generate a change in the dipole moment of the molecule or unit cell of atoms under consideration. Usually, one attempts to measure the absorption coefficient of the sample however, because it is often difficult to obtain sufficiently thin samples, it is frequently necessary to measure the reflection spectrum and then to calculate the dielectric parameters, and e where the complex dielectric parameter is e = e - ie. The real and imaginary parts of e are related by the Kramers-Kronig relationship. 

However, model reactions can demcxistrate that the interferometric measurement in combination with application of the Kramer-Kronig relationship [S] allows calculation of the abscnbance-time curves and the determination of the partial photochemical quantum yield. Thus, the method makes it pos-sibile to monitor photoreactions in a solid matrix. As an application the quantum yield of the photoreaction of a 7,7a-dihydrobenzofuran derivative to the (Z)-fulgide is obtained as =0.03 [201]. 

The electric susceptibility can comprise any combination of dipolar, ionic, or electronic polarization processes. This formulation leads to relationships between the real and imaginary parts of the complex electric susceptibility, known as the Kramers-Kronig relationships [28-31] which are very similar to the frequency relations between resistance and reactance in circuit theory [30]. 

Kramers-Kronig relationship, see H. Ibach, H. Liith, Festkdrperphysik, Springer, Berlin 1988, p. 251 (and other texts). 

Riu, P.J., Lapaz, C., 1999. Practical limits of the Kramers-Kronig relationships applied to experimental bioimpedance data. Ann. N.Y. Acad. Sci. 873, 374—380. 

An important requirement for a valid impedance function is that the system be linear. Theoretically, this implies that the real and imaginary components transform correctly according to the Kramers-Kronig relationships (discussed later in this section). Practically, linearity is indicated by the impedance being independent of the magnitude of the perturbation, a condition that is easily (although seldom) tested experimentally. 

The Kramers-Kronig relationships between the real and complex parts of quantities representing physical quantities such as absorbance and impedance have long been known to scientists in such fields as optics and radio engineering, but it has only fairly recently been pointed out [25] that they are a useful adjunct to electrochemical impedance analysis. 

In deriving this result, we have assumed the waveguide to be nonabsorbing. This ignores the intimate relation between dispersiveness and absorption, as expressed by the Kramers-Kronig relationships [2]. However, if we restrict this discussion to comparatively small absorption the correction to Eq. (11-19) due to absorption is a higher-order effect. 

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