Kronig-Kramers relations

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]. 

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,... 

The real and imaginary parts of the complex refractive index satisfy Kramers-Kronig relations sometimes this can be used to assess the reliability of measured optical constants. N(oj) satisfies the same crossing condition as X(w) N (u) = N( — u). However, it does not vanish in the limit of indefinitely large frequency lim JV(co) = 1. But this is a small hurdle, which can be surmounted readily enough by minor fiddling with JV(co) the quantity jV(co) — 1 has the desired asymptotic behavior. If we now assume that 7V( ) is analytic in the top half of the complex [Pg.28]

A medium is said to be circularly dichroic—it absorbs differently according to the state of circular polarization of the light—if kL — kR 0 it is circularly birefringent, which is manifested by optical rotation, if nL — nR = 0. Optical rotation and circular dichroism are not independent phenomena, but are connected by Kramers-Kronig relations ... 

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). 

We are now in a position to better understand and, we hope, appreciate, the sometimes mysterious Kramers-Kronig relations. 

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. 

Piezoelectric relaxation and Kramers-Kronig relations in a modified lead titanate composition... 

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. 

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