Kramers-Kronig equations

The most precise values are obtained from experimental measurements. When one of the two anomalous scattering coefficients is measured, the second can be relatively precisely calculated using Kramers - Kronig equation. More details about the anomalous scattering can be found in a special literature, e.g. see J. Als-Nielsen and D. McMorrow, Elements of modern x-ray physics, John Wiley Sons, Ltd., New York (2001). 

In the present work and ei are not readily determinable, particularly since the chemical nature of the polymer is changing. However, for a given relaxation, if it can be assumed that there is a symmetrical distribution of relaxation times ei and can be related by the Kramers-Kronig equation(25), or by the following alternative express ion (25.), ... 

The relationship between rotation and refractive index or between ellipticity and absorption coefficient can be represented, in analogy to dispersion theory, by the model of coupled linear oscillators or by quantum mechanical methods. ORD and CD are related to one another by equations analogous to the Kramers-Kronig equations [34]. 

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]. 

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 ... 

The attenuation may be expressed by making the wavenumber complex (this would be k — ia in eqn (6.12)), and the velocity (= w/k) may also be written as a complex quantity. This in turn corresponds to a complex modulus, so that the relationship v - /(B/p) is preserved indeed the acoustic wave equation may be written as a complex-valued equation, without the need for the extra term in (6.11). Complex-valued elastic moduli are frequency-dependent, and the frequency-dependent attenuation and the velocity dispersion are linked by a causal Kramers-Kronig relationship (Lee et al. 1990). 

The relevance of a recorded impedance spectrum is not clear, as is the case for experiments done with instrumental methods. A number of potentially occurring errors can give rise to a distortion (small or large) of the impedance spectrum, with a certain impact on the interpretation of the data and the curves. A method to analyse the obtained impedance spectra makes use of so-called Kramers-Kronig transformations50,51, which are a set of coupled integral equations that describe the relationship between the real and imaginary part of the impedance. For impedance Z ... 

We show in Figure 13.8 that in the case of a well-behaved piezoelectric relaxation (counterclockwise hysteresis) presented in Figure 13.7, the Kramers-Kronig relations are indeed fulfilled. Closer inspection of the data show that the relaxation curves can be best described by a distribution of relaxation times and empirical Havriliak-Negami equations [19]. It is worth mentioning that over a wide range of driving field amplitudes the piezoelectric properties of modified lead titanate are linear. Details of this study will be presented elsewhere. 

Figure 2.14. Evolution with temperature of the full width y0 at half maximum of the 0-0 absorption peak. Hollow circles represent our results from Kramers-Kronig analysis, (a) Evolution between 0 and 77 K. The solid line was drawn using equation (2.126) and adjusted parameters y, =72cm 1, hfi = 27cm" . The dashed line connects the results of our model (2.127)—(2.130) for six different temperatures, (b) Evolution between 0 and 300 K. The full circles are taken from ref. 62. This summary of the experimental results shows the linear behavior between 30 and 50 K., and the sublinear curvature at temperatures above 200 K. 

This problem is solved using the Kramers Kronig (KK) integral equations, which connect the dispersive (real part) and dissipative (imaginary part) reaction processes, by using the fundamental principle of causality and... 

Since almost all equations used in impedance methods are derived assuming linearity, it is important to have some means of verifying this supposition. The Kramers-Kronig relations2 link Z with Z" and allow the calculation of values for Z" at any frequency from a knowledge of the full frequency spectrum of Z, and vice versa. 

This expression was originally used to estimate the polarization resistance for actively corroding metals whose impedance response was a well-behaved semicircular arc in the complex plane (73), but can be used in certain situations to estimate the equivalent resistance of conversion coated metal surfaces. Equation (9) is derived from the Kramers-Kronig transforms and subject to the conditions that limit their use (74-78). These conditions include... 

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). 

In electromagnetic theory it is shown that the real and imaginary part of e (co) are not independent of each other, but are connected by a pair of integral equations, the Kramers-Kronig relation (e.g. Bohren Huffman 1983). Equation (A3.10) satisfies these relations, i.e. using a Lorentz-Drude model fitted to the laboratory data automatically guarantees that the optical data satisfy this basic physical requirement. 

The dependence of the reflection coefficient on n and k allows these quantities to be calculated from the reflection spectrum by use of the Kramers-Kronig relations. It follows from Equation (9.17) that an intense absorption gives a high reflectivity, e.g. the reflection spectrum of PDA-TS,... 

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... 

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