Susceptibility Kramers-Kronig

The derivation of Kramers-Kronig relations for the susceptibility was relatively easy, perhaps misleadingly so. With a bit of extra effort, however, we can often derive similar relations for other frequency-dependent quantities that arise in physical problems. Suppose that we have two time-dependent quantities of unspecified origin, which we may call the input X((t) and the output X0(t) the corresponding Fourier transforms are denoted by 9C,(co) and 9Cc(io). If the relation between these transforms is linear,... 

Our derivation of (9.41) follows closely that of Gevers (1946) and is similar to that of Brown (1967, pp. 248-255). Because of the nature of this derivation it should hardly be necessary to do so, but it can be shown directly by integration—more easily than for the Lorentz oscillator—that the real and imaginary parts of the Debye susceptibility satisfy the Kramers-Kronig relations (2.36) and (2.37). 

We get the Kramers-Kronig relation between the real % and the imaginary y parts of the electric susceptibility. 

Interestingly enough, these relations are formally similar to the usual Kramers-Kronig relations between the real and imaginary parts of the generalized susceptibility (except for the evident fact that they hold in the time-domain, and not in the frequency domain). Otherwise stated, at T = 0, the quantities CBA(t) and —ih BA t) must constitute, respectively, the real and imaginary parts of an analytic signal ZBA (f) with only positive frequency Fourier components [38,39]. 

The Kramers-Kronig relation gives the spin susceptibility as the integrated intensity of the ESR absorption [11] ... 

Kramers-Kronig relations links between real and imaginary parts of susceptibilities... 

The real and the imaginary parts of a susceptibility are in certain cases coupled through Kramers-Kronig relations (Butcher and Cotter, 1990) such as (24). 

The real and the imaginary parts of the susceptibility are related by the Kramers-Kronig relations. (A derivation of these equations is given in Kittel, 1976, p. 324 Kittel used analytic continuation of the susceptibility into the complex plane.) One relation is... 

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... 

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]. 

An important characteristic of the optical constants is that they fall in a broad class of generalized susceptibilities with the fundamental property that their real and imaginary parts are connected by Kramers-Kronig (KK) integral relations (Landau and Lifshitz 1992). For the case of linear response, KK relations are model independent since they rely only on the causality principle and analytical properties of the complex susceptibilities. The physical reasoning behind the KK relations is that dissipation of energy of the... 

By using the Kramers-Kronig relation (cu = 0) that coimects the real x and imaginary parts x of the susceptibility, we find... 

Then, we obtain the real part zi(oj) of the susceptibility of oscillator k from the Kramers-Kronig relations... 

We see that the susceptibility separates the elastic part and the viscous dissipative part of the work expended on the system. They show up in the real part and the imaginary part a" (a ) respectively. It is important to note that the two parts are not independent even if they represent quite different physical properties. In fact, they are related by the Kramers-Kronig relations, Eqs. (5.46) and (5.47). 

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