Dielectric response Kramers-Kronig relations

The blocked or restricted motion of the charges in a body of matter appears to the experimenter as a polarization response. The continuous or semicontinuous motion of such charges through the sample appears as a conductive response. The two processes are linked over the entire frequency range by the Kramers-Kronig relations. This is well discussed in Frohlich s book on dielectrics. 

Mathematically, integral Kramers-Kronig relations have very general character. They represent the Hilbert transform of any complex function s(co) = s (co) + s"(co) satisfying the condition s (co) = s(—co)(here the star means complex conjugate). In our particular example, this condition is applied to function n(co) related to dielectric permittivity s(co). The latter is Fourier transform of the time dependent dielectric function s(f), which takes into account a time lag (and never advance) in the response of a substance to the external, e.g. optical, electric field. Therefore the Kramers-Kronig relations follow directly from the causality principle. 

The quantities Cp, hp, ks, and T denote the velocity of light in a vacuum, Planck s constant, Boltzmann s constant, and the temperature, respectively. The parameters Sj n represent the values of the imaginary dielectric response function s(iif) at the imaginary frequencies i-f , which can be computed from the imaginary part of the dielectric permittivity by means of a Kramers-Kronig relation ... 

This is called the Kramers-Kronig (KK) relationship, from which the dielectric function e = ej + e2 can be derived [3.25]. Since e is also a linear response function, ej and 2 are again related by the KK relationship, thus the information contained in the dielectric function can be examined by concentrating on one of the two components of the dielectric function. We choose to work with 2(m) because it is what optical (X-ray) absorption spectroscopy measures and can be directly related to the atomic polarisability Im[a(o )] that appeared in (3.5). 

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