Kramers Kronig

Once the imaginary part of the dielectric function is known, the real part can be obtained from the Kramers-Kronig 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. 

Dielectric constants of metals, semiconductors and insulators can be detennined from ellipsometry measurements [38, 39]. Since the dielectric constant can vary depending on the way in which a fihn is grown, the measurement of accurate film thicknesses relies on having accurate values of the dielectric constant. One connnon procedure for detennining dielectric constants is by using a Kramers-Kronig analysis of spectroscopic reflectance data [39]. This method suffers from the series-tennination error as well as the difficulty of making corrections for the presence of overlayer contaminants. The ellipsometry method is for the most part free of both these sources of error and thus yields the most accurate values to date [39]. 

Circular dicliroism has been a useful servant to tire biophysical chemist since it allows tire non-invasive detennination of secondary stmcture (a-helices and P-sheets) in dissolved biopolymers. Due to tire dissymmetry of tliese stmctures (containing chiral centres) tliey are biaxial and show circular birefringence. Circular dicliroism is tlie Kramers-Kronig transfonnation of tlie resulting optical rotatory dispersion. The spectral window useful for distinguishing between a-helices and so on lies in tlie region 200-250 nm and hence is masked by certain salts. The metliod as usually applied is only semi-quantitative, since tlie measured optical rotations also depend on tlie exact amino acid sequence. 

In the same section, we also see that the source of the appropriate analytic behavior of the wave function is outside its defining equation (the Schibdinger equation), and is in general the consequence of either some very basic consideration or of the way that experiments are conducted. The analytic behavior in question can be in the frequency or in the time domain and leads in either case to a Kramers-Kronig type of reciprocal relations. We propose that behind these relations there may be an equation of restriction, but while in the former case (where the variable is the frequency) the equation of resh iction expresses causality (no effect before cause), for the latter case (when the variable is the time), the restriction is in several instances the basic requirement of lower boundedness of energies in (no-relativistic) spectra [39,40]. In a previous work, it has been shown that analyticity plays further roles in these reciprocal relations, in that it ensures that time causality is not violated in the conjugate relations and that (ordinary) gauge invariance is observed [40]. 

It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

If i = i — ik] and H2 = ns — are known as a function of wavelength, Eq. 12 can be used to calculate the entire RAIR spectrum of a surface film. Since transmission infrared spectroscopy mostly measures k, differences between transmission and RAIR spectra can be identified. Fig. 6 shows a spectrum that was synthesized assuming two Lorentzian-shaped absorption bands of the same intensity but separated by 25 cm. The corresponding spectrum of i values was calculated from the k spectrum using the Kramers-Kronig transformation and is also shown in Fig. 6. The RAIR spectrum was calculated from the ti and k spectra using Eqs. 11 and 12 and is shown in Fig. 7. 

The dynamic mechanical experiment has another advantage which was recognized a long time ago [10] each of the moduli G and G" independently contains all the information about the relaxation time distribution. However, the information is weighted differently in the two moduli. This helps in detecting systematic errors in dynamic mechanical data (by means of the Kramers-Kronig relation [54]) and allows an easy conversion from the frequency to the time domain [8,116]. 

II electronic states, 638-640 vibronic coupling, 628-631 triatomic molecules, 594-598 Hamiltonian equations, 612-615 pragmatic models, 620-621 Kramers doublets, geometric phase theory linear Jahn-Teller effect, 20-22 spin-orbit coupling, 20-22 Kramers-Kronig reciprocity, wave function analycity, 201 -205 Kramers theorem ... 

Kramers-Kronig relationships, 7 338 Kramers-Kronig (K-K) transformation, 14 231... 

It is known that measnring the absorption coefficient (and thns the extinction coefficient) over the whole freqnency range, 0 < real part of N(co) - that is, the normal refractive index ( >) - can be obtained by nsing the Kramers-Kronig relationships (Fox, 2001). This is an important fact, because it allows us to obtain the frequency dependence of the real and imaginary dielectric constants from an optical absorption experiment. 

The two anomalous components of the scattering factor, f" and /, are interrelated through the Kramers Kronig transforms, which have the form... 

Lucarini, V., Saarinen, J. J., Peiponen, K. E., and Vartiainen, E. M. 2005. Kramers-Kronig relations in optical material research. Berlin Springer. 

Although the Kramers-Kronig relations do not follow directly from physical reasoning, they are not devoid of physical content underlying their derivation are the assumptions of linearity and causality and restrictions on the asymptotic behavior of x> As we shall see in Chapter 9, the required asymptotic behavior of x is a physical consequence of the interaction of a frequency-dependent electric field with matter. 

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,... 

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