Kramers-Kronig relations effects

However, in contrast to the cases of complex elastic modulus G and dielectric constant e, the imaginary part of the piezoelectric constant, e", does not necessarily imply an energy loss (Holland, 1967). In the former two, G"/G and e"/e express the ratio of energy dissipation per cycle to the total stored energy, but e"/e does not have such a meaning because the piezoelectric effect is a cross-coupling effect between elastic and electric freedoms. As a consequence, e" is not a positive definite quantity in contrast to G" and e". In a similar way to e, however, the Kramers-Kronig relations (Landau and Lifshitz, 1958) hold for e ... 

Let us finally notice that, investigating the dispersion theory for the effective third-order nonlinear susceptibility of nanocomposite media, Peiponen et al. established that Kramers-Kronig relations are not valid for whereas they are valid for other nonlinear processes such as frequency conversion [95]. 

The absorption constant k and the refractive index n are related via the Kramers-Kronig relations. Hence, an increase of the optical density, either due to an increase of the oscillator strength or a spectral red shift of the transition, leads to an increase of the refractive index. Since n is larger than k by two orders of magnitude and usually decreases more slowly with increasing distance from the absorption maximum, it yields the main contribution to the optical susceptibility. Therefore, the effect of... 

When discussing the general aspects of FTNMR, we have to remember that all principal statements about Fourier methods have been introduced for a strictly linear system (mechanical oscillator) in Chapter 1. In Chapter 2, on the other hand, we have seen that the nuclear spin system is not strictly linear (with Kramer-Kronig-relations between absorption mode and dispersion mode signal >). Moreover, the spin system has to be treated quantummechanically, e.g. by a density matrix formalism. Thus, the question arises what are the conditions under which the Fourier transform of the FID is actually equivalent to the result of a low-field slow-passage experiment Generally, these conditions are obeyed for systems which are at thermal equilibrium just before the initial pulse but are mostly violated for systems in a non-equilibrium state (Oberhauser effect, chemically induced dynamic nuclear polarization, double resonance experiments etc.). 

This effect occurs especially in the infrared range, because narrower absorption bands effect a stronger anomaly of n. As a consequence, IR reflection spectra differ severely from the corresponding transmission spectra. The Kramers—Kronig relation can be used to analyze reflection spectra and to relate them to transmittance data. 

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. 

The FR is according to eq. (19.29) a dispersive effect and the CD is an absorptive effect, being proportional to the difference of the absorption indices for right hand and left hand circularly polarized light. Just as in the case of refractive index and absorption index or real and imaginary part of the dielectric function, a Kramers-Kronig relation connects also the FR and CD... 

Further, we need the Kramers-Kronig relation for the Green function of the effective medium. 

In fact, these properties are not specific to the Debye-process, but have a deeper basis which extends their validity. According to the Kramers-Kronig relations, J and J" are mutually dependent and closer inspection of the equations reveals that it is impossible, in principle, to have a loss without a simultaneous change in J. Both effects are coupled, the reason being, as mentioned above, the validity of the causality principle. 

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. 

In the same section, we also see that the source of the appropriate analytic behavior of the wave function is outside its defining equation (the Schibdinger equation), and is in general the consequence of either some very basic consideration or of the way that experiments are conducted. The analytic behavior in question can be in the frequency or in the time domain and leads in either case to a Kramers-Kronig type of reciprocal relations. We propose that behind these relations there may be an equation of restriction, but while in the former case (where the variable is the frequency) the equation of resh iction expresses causality (no effect before cause), for the latter case (when the variable is the time), the restriction is in several instances the basic requirement of lower boundedness of energies in (no-relativistic) spectra [39,40]. In a previous work, it has been shown that analyticity plays further roles in these reciprocal relations, in that it ensures that time causality is not violated in the conjugate relations and that (ordinary) gauge invariance is observed [40]. 

This equation effectively connects the alternating and after-effect solutions provided the response is linear. We may now make use of the Kramers-Kronig dispersion relations [66,67] to rewrite equations (C.ll) and (C.12) as... 

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. 

Rigorously, ORD and CD spectra are related through the Kronig-Kramers theorem, a well-known general relationship between refraction and absorption, i.e. nL - nR is determined by eL - % for A from zero to infinity [128], (The analogous relationship between refraction and reflection applies to cholesteric liquid crystals.) Hence, in order to maximize ORD in the transparent region, Cotton effects, associated with exciton coupling (both intramolecular and intermolecular), have... 

The Cotton effects can be either positive or negative. ORD and CD are related phenomena and data obtained by each method can be transformed to the other by means of the Kronig-Kramers relations [e.g., Ref. (18)]... 

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