Kramer-Kronig dispersion relation

The real part of n , the dispersive (reactive) part of and the definition of Xy implies a relation between tr yand -/which is known as the Kramers-Kronig relation. 

From these relations we see that the width and shift of the power spectrum and consequently the spectroscopic lines are related through the Kronig-Kramers dispersion relations. Exactly the same arguments apply to the Laplace transform of the time-correlation function, H(/co). The real and imaginary parts, C H(co) and C"//(/(0), are related by Kramers-Kronig dispersion relation. 

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. 

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 Kramers-Kronig dispersion relations between the real and imaginary parts of the dielectric permittivity can be written as follows [11] ... 

Fig. 2. Dispersion of fo + f of europium in EufPhAcAc), at the three L-absorption edges after Lye, Phillips, Kaplan Doniachand Hodgson ). The f"-values were calculated by using the Kramers-Kronig relation (Eq. (17)). The absolute scale (in electrons) relies on values of f and f at CrK, and CuK, which were taken from The International Tables of Crystallography, IV ...

The measurement of the total cross section determines completely the dispersion of f", — The dispersion of the real part is not independent of that of f", for there exists a general relationship between them known as the Kramers-Kronig relation ... 

The dielectric dispersion ( ) and the a.c. conductivity K(frequency range, using the Kramers-Kronig relations [1.4.4.31-321. However, this requires extremely precise data to carry out the required Integrations in practice measurements of H(o) and K([Pg.536]

It is also shown how x(< ) is related to the temporal behaviour of the dielectric polarization follomng the sudden application, or removal, of an electric field. Various forms of the Kramers-Kronig dispersion relations are introduced for y (o>) and x C") aod for a number of functions of The section closes with the ddOnition of the frequency-dependent complex refractive index n() = n(cu) — and a discussion of its relation... 

The quantities n(w) — 1 and k(w) satisfy the dispersion relations [equations (42) and (43)]. Indeed, the dispersion relations were first derived by Kramers and by Kronig for these particular variables. 

One may deduce from equation (32) the well-known Kramers-Kronig dispersion relations which may be written in the following form ... 

The orientational contribution can be computed directly from the integrated line intensity of the absorption band associated with the dispersion by using the Kramers-Kronig relation in the form... 

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]. 

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... 

Scattering theory has very general validity. It is based on the existence of a Schrodinger-type equation and is consistent with causality requirements through the Kramers-Kronig dispersion relations. It is widely used outside atomic physics for the description of all resonance phenomena. There is therefore some advantage in using it in atomic physics it ensures unification and is more transparent to reseachers from other fields. 

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.). 

Chiral molecules occur in pairs related by a symmetry plane, their mirror images cannot be superimposed (enantiomers). Such molecules exhibit optical activity, i. e. they transmit left and right circularly polarized Hght in a different manner. The difference in the refraction indices for left and right circularly polarized light is called optical rotatory dispersion (ORD), the corresponding difference in absorption coef ficients is called circular dichroism (CD). ORD and CD can be related to each other by the Kramers—Kronig transformation. 

K. Krishnan, Applications of the Kramers-Kronig Dispersion Relations to the Analysis of FTIR Specular Reflectance Spectra, Fl lR/IR Notes 51, Biorad Digilab Division, Cambridge, MA, August 1987. 

Polarization and absorption are interconnected by the Kramers-Kronig dispersion relation (cf. e.g. Tauc (1965))... 

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