Havriliak-Negami empirical formula

The peak of the dielectric loss of this process reflects its viscoelastic nature by obeying the time-temperature superposition principle, wherein the peak is shifted to higher temperatures for shorter times (higher frequencies) and vice versa. This process has been described by the Havriliak-Negami empirical formula [106, 108]... 

Various empirical formulae have been given for fitting data that deviate from the simple semi-circular Cole-Cole plot. The most general of these is the Havriliak and Negami formula... 

The specific case a=l, 1 gives the Debye relaxa tion law, P= 1, a 1 corresponds to the so-called Cole-Cole equation, whereas the case a= 1, P 1 corresponds to the Cole-Davidson formula. Recently, some progress in the understanding of the physical meaning of the empirical parameters (a, P) has been made (7, 8). Using the conception of a self-similar relaxation process it is possible to understand thenature of a nonexponential relaxation of the Cole-Cole, Cole-Davidson, or Havriliak-Negami type. 

These relationships are known as the Debye formulae. The Debye process has a relaxation time distribution, which is symmetrical around /niax= niax/2n and has a full width at half-maximum of 1.14 decades in frequency for the dielectric loss. In most cases, the half width of measured loss peaks is much broader than the predicted by eqn [26] and in addition, their shapes are asymmetric and with a high-frequency tail. This is the non-Debye (or nonideal) relaxation behavior found in many glass formers. In the literature, several empirical model funaions, mostly generalization of the Debye function, have been developed and tested which are able to describe broadened and/or asymmetric loss peaks. Among these empirical model functions, the most important are the Kohlrausch-Williams-Watts (KWW), Cole-Cole (CC), Cole-Davidson (CD), and the Havriliak-Negami (HN) function. The HN function, with two shape parameters, is the most commonly used funaion in the frequency domain. 

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