Kramers-Kronig formula

For analyzing photochromic materials we make use of the Kramers-Kronig formulas (27), which express a very general relation between the real and imaginary parts of the dielectric constant of any material ... 

The Kramers-Kronig formula relates the stress relaxation modulus to dynamic quantities (Kronig, 1926 Kramers, 1927)... 

The Einstein relation (159) or the expression (157) of the dissipative part ffiep(m) of the mobility constitute another formulation of the first FDT. Indeed they contain the same information as the Kubo formula (156) for the mobility, since p(co) can be deduced from 9ftep(oo) with the help of the usual Kramers-Kronig relations valid for real co [29,30]. Equation (156) on the one hand, and Eq. (157) or Eq. (159) on the other hand, are thus fully equivalent, and they all involve the thermodynamic bath temperature T. Note, however, that while p(oo) as given by Eq. (156) can be extended into an analytic function in the upper complex half-plane, the same property does not hold for D(co). 

Some different forms of the Kramers-Kronig relations are presented in Table 22.1. Equations (22.55) and (22.56), called the Plemelj formulas, are obtained directly from consideration of causality in a linear system. Equations (22.57) and (22.58) are mathematically equivalent to equations (22.59) and (22.60), respectively, because... 

Remember A.3 The derivation of the Kramers-Kronig relations in Section 22.1 makes use of Cauchy s Integral Formula far evaluation cf a function at a singularity, given as Example A.2. 

However ATR spectra are not identical to the transmittance spectra. It is known that ATR spectra include not only absorption but also refractive index. The intensity shift can be readily corrected by Eq. (2.1), but the shift in the wavelength due to the refractive index cannot be corrected. In order to separate the contributions of the absorption and refractive indices, Kramers-Kronig transformation (KKT) and Fresnel formulas should be used [19], The experimentally observed reflectivity R... 

---