Kramers-Kronig transformation, principle

In principle, CD and ORD yield identical information because they are Kramers-Kronig transforms (eqns [9] and [10]). Thus, it seems to be sufficient to measure either CD or ORD. This is only true from the theoretical point of view because of some practical reasons ... 

For continuous solids with a shiny surface a specular reflection spectrum is obtained. In principle, the absorption spectrum can be derived from this by the Kramers-Kronig transformation. Examples of this are shown for both PP (Figure 4.4(a)) and polyester (Figure 4.4(b)) samples. Equally good results are obtained from both unfilled and carbon-filled samples. The presence of inorganic fillers does not alter the nature of the spectra and any contribution from the filler appears specular. The presence of glass as a filler has little effect on the spectra. The spectra obtained are adequate for qualitative identification, but there are limitations. Band shapes often appear nnsymmetrical and baselines are uneven. When the surface is not shiny the spectra are weaker and may contain a diffuse component. When there are surface species the reflection spectrum may be unrepresentative of the bulk material. No spectra were obtainable from those carbon-filled samples that did not have shiny surfaces. 

If transformations (5.3.7) are used, the complex compliance Z(ito) should be given as an analytical function of u> on the whole complex plane. As the theory of irreversible processes shows, Z(iw) (and, hence, F(r/)) should exhibit some properties resulting from general principles of dynamics (e.g. the principle of causality) and the Kramers-Kronig reciprocal equations l... 

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 beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

It Is interesting to note that the QRD curve resembles the first derivative of the CD curve. ORD and CD are therefore coupled phenomena which in principle are mathematically commutable. It is possible to calculate CD spectrum of a given compound from its ORD spectrum by applying the mathematical relationship known as Kronig-Kramer transform. Interpretation... 

The derivation of the Kronig-Kramers relationships may be found in Reference 41. The clarity of the presentation there is such as to make any attempt to repeat the proof here seem puerile. We might just mention, however, that the crux of the proof lies in seeing that the principle of causality requires that [% v) — be analytic in the whole lower half of the complex plane. Then, as in the case of Hilbert transforms, the real and imaginary parts of [% (r) —xjl wi not be independent and may be expressed as integral transforms of each other. But even though we shall not go into the details of the proof of (65) and (66), their application to the problem of optical activity requires some comment. 

---