The Kramers-Kronig relations

An electrical system with linear properties does not generate harmonics in response to the perturbation signal, and the response to two or more superimposed excitation signals is equal to the sum of the two responses obtained by excitation independently. With electrochemical systems this linearity is possible to a good approximation for perturbations rather less than the thermal potential (lcBT/e) = 25 mV at 298K. 

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. 

Although these have been applied to electrical systems for over 40 years, only recently have they been applied to electrochemical systems3. 

Lancaster, Dc and ac circuits, Oxford Physics Series, Oxford University Press, 1973. 

Network and feedback amplifier design, van Nostrand, New York, 1945, Chapter 4. 

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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. 

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]. 

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 proof is lengthy, although straightforward, and will be omitted here, but it can be shown by direct substitution and integration that x and x" satisfy the Kramers-Kronig relations (2.36) and (2.37). 

Our derivation of (9.41) follows closely that of Gevers (1946) and is similar to that of Brown (1967, pp. 248-255). Because of the nature of this derivation it should hardly be necessary to do so, but it can be shown directly by integration—more easily than for the Lorentz oscillator—that the real and imaginary parts of the Debye susceptibility satisfy the Kramers-Kronig relations (2.36) and (2.37). 

First, we note that the consequence of no absorption (e" = 0) at all frequencies is that the integral in (9.44) vanishes and e = 1. Optically, such a material does not exist there is no way that it can be distinguished from a vacuum by optical means. The Kramers-Kronig relations also tell us that it is a contradiction to assert that either the real or imaginary parts of the dielectric function can be independent of frequency the frequency dependence of the one implies the frequency dependence of the other. These consequences of the Kramers-Kronig relations are almost trivial, but it is disturbing how often they are blithely ignored. 

The optical constants n and k are not independent if k varies strongly, so must n. Either n must be measured by some other method or a theory of optical constants that couples them together properly must be used, such as the oscillator model (9.25) or the Kramers-Kronig relations [(2.49), (2.50)]. 

Figure 6.10 Optical properties of CdS (a) experimental reflectance spectrum of single crystals of CdS (b) refractive index n of CdS obtained from data given in (a) through the Kramers-Kronig relation.

This method is based on the Kramers-Kronig relation (1), saying that both s and e" carry the same information about relaxation processes and are related by a Hilbert transformation ... 

However, in contrast to the cases of complex elastic modulus G and dielectric constant e, the imaginary part of the piezoelectric constant, e", does not necessarily imply an energy loss (Holland, 1967). In the former two, G"/G and e"/e express the ratio of energy dissipation per cycle to the total stored energy, but e"/e does not have such a meaning because the piezoelectric effect is a cross-coupling effect between elastic and electric freedoms. As a consequence, e" is not a positive definite quantity in contrast to G" and e". In a similar way to e, however, the Kramers-Kronig relations (Landau and Lifshitz, 1958) hold for e ... 

Yet it is meaningful to consider the frequency dependence of the separate components because data are always obtained in a too-limited frequency range for the Kramers—Kronig relations to be practically useful. 

Thus if a functional form is chosen for K ooi), K"n((o) and Ku(t) can be determined from the Kramers-Kronig relations. Moreover, the parameters in the functional form, Af j co), can be related to the moments p2 , in addition to the friction constant H(0), so that these parameters can thereby be determined. 

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 13.8 Ilustration of the validity of the Kramers-Kronig relations for the piezoelectric relaxation in Sm-modified lead titanate ceramics. The imaginary component is calculated from the real using numerical method and Kramers-Kronig relations and compared with experimentally determined data.

No attempt is made to summarize conductivity data. Conductivity increases similarly in several major steps symmetrical to the changes of the dielectric constant. These changes are in accord with the theoretical demand that the ratio of capacitance and conductance changes for each relaxation mechanism is given by its time constant, or, in the case of distributions of time constants, by an appropriate average time constant and the Kramers-Kronig relations. 

We get the Kramers-Kronig relation between the real % and the imaginary y parts of the electric susceptibility. 

We have indicated in Section I that the optical properties of the crystal are characterized by the transverse dielectric tensor ex(k, cu) (1.79). The real and imaginary parts of this tensor being related by the Kramers-Kronig relations resulting from the linearity, ex(k, co) is itself determined by its imaginary part. In what follows, we assume that an eigendirection of c1 is excited, and we consider t"(k, oj) and the optical conductivity cuc (k, a>) under the common denomination of optical absorption . In fact, it is the conductivity that determines the absorption by the crystal of the energy of the plane wave (see Appendix A). 

Figure 2.11. The integration contour in the complex plane allowing one to generalize the Kramers-Kronig relations to the modulus and phase of the reflectivity amplitude.

We must remark that the broadening is in general nonlorentzian and asymmetric, particularly when the optical transition occurs at the boundary of the excitonic band (as in the case of the anthranene crystal). Lastly, the analyticity of the CPA method assures that (4.118) satisfies the Kramers-Kronig relations, and that the total oscillator strength of the transition, redistributed on the two bands, is conserved. 

To derive the Kramers-Kronig relation, the contour C, shown in Fig. 2.23, follows the real ("x") axis, except for a hump over the "pole" at x t (defined as the point where [z — f] 1 becomes infinite) and a semicircle in the upper... 

Rearranging yields a compact form of the Kramers-Kronig relations ... 

This seems trivial, but is the very important result that, if one measures y (ft)), one can calculate y2(ft)), and conversely put differently, the Kramers-Kronig relations show that the absorptive and dispersive properties of a medium are not independent of each other. An experimental difficulty is that one must truncate the integrations at some maximum measured frequency ft) this may lead to considerable error. 

Luckily, the real and imaginary parts of the complex dielectric permittivity are not independent of each other and are connected by means of the Kramers-Kronig relations [11]. This is one of the most commonly encountered cases of dispersion relations in linear physical systems. The mathematical technique entering into the Kramers-Kronig relations is the Hilbert transform. Since dc-conductivity enters only the imaginary component of the complex dielectric permittivity the static conductivity can be calculated directly from the data by means of the Hilbert transform. 

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. 

This is the first case for which the Kramers-Kronig relation has been denia onstrated experimentally in the solid state. If the uncorrected CB spectrum wai used, the Kramers-Kronig relation did not hold. 

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