Kronig-Kramer equation

The Kronig-Kramers relation is of fundamental importance for optics and for physics in general13). Here, these equations do not seem practical because of the integration of the wavelength from 0 to oo. However, these are very useful for calculating the molar ellipticity magnitude from the observed ORD curve 14). 

Absorption is one consequence of a (vibrational) transition, the spectral behaviour of the refractive index reflects the same phenomenon. Within the spectral interval of an absorption band the refractive index follows a dispersion curve. Kramers-Kronig integral equations (for applications in optics see Caldwell and Eyring, 1971 Hopfe et al., 1981 ... 

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... 

Anomalous ORD and CD both originate from light absorption by a chiral species and as such contain the same information. A mathematical equation, the Kronig-Kramers transform, relates one to the other over the wavelength range of the absorption, namely, [ (/I)] = -2/71 - X )dX. When the... 

Requirements (22.71) and (22.75) place well-defined constraints on the evaluation of the Kramers-Kronig integral equations. 

Qsp were applied for the validation of electrochemical impedance data. Agarwal et al. described an approach that eliminated problems associated with direct integration of the Kramers-Kronig integral equations and accoimted explicitly for stochastic errors in the impedance measurement. 

Since o(co) is causal, the scattering rate and the mass enhancement are not independent, and are connected through the Kramers-Kronig integral equation ... 

If both components of G are known at a single frequency, both components of J can be very simply calculated by equations 27 to 30 of Chapter 1. On the other hand, if one component is known over the whole frequency range, the other can be obtained from it by mechanical analogs of the Kronig-Kramers relations. " ... 

A Kronig—Kramers transform of the two CD bands observed in cyclohexane accounts for the observed positive ORD spectrum. In contrast, a third large and negative ORD band centered at 155.5 nm is responsible for the negative ORD spectrum observed in HFIP. In the latter solution as well as in benzene, the ORD spectrum was found to fit the Drude one term equation with Xq=150 nm. 

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). 

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