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# 2 Relaxation functions and dielectric susceptibilities

[123.3.1] Let denote the normalized, i.e. , electrical dipolar polarisation or a similar relaxation function. [123.3.2] Then the complex frequency dependent dielectric susceptibility is , where denotes the Laplace transform of , , , and is the frequency [16, p. 402, eq. (18)]. [123.3.3] Time honoured functional expressions for and are the exponential (Debye) relaxation [6, ch.III,§10]

 (1a) (1b)

or stretched exponential Kohlrausch relaxation [26, 27], revived in [35] (KWW),

 (2a) (2b)

where is the relaxation time and is the stretching exponent. [123.3.4] Remarkably, while formula (2a) for the relaxation function has been used since 1854, formula (2b) for the dielectric susceptibility was discovered only in 2001 and published in [18]. [123.3.5] It is given in terms of inverse Mellin-transforms of -functions, also known as -functions [5]. [123.3.6] A brief definition is can be found in the Appendix below.

[123.4.1] A popular alternative to stretching time is to stretch frequency. [123.4.2] In this case a stretching exponent is introduced into eq. (1b) rather than into eq. (1a). [123.4.3] This leads to the Cole-Cole (CC) relaxation [3]

 (3a) (3b)

where

 (4)

is the Mittag-Leffler function [31]. [123.4.4] It is by now well known, that the relaxation function for Cole-Cole relaxation is intimately related to fractional calculus [15]. [123.4.5] Unfortunately, the Cole-Cole form (3b) exhibits a symmetric -peak, while asymmetric -peaks are observed experimentally for many materials [28]. [123.4.6] Therefore, a second way to introduce the stretching exponent into the Debye function (1b), known as the Cole-Davidson (CD) form, was introduced in [4]

 (5a) (5b)

where

 (6)

denotes the complementary incomplete Gamma function, [123.4.7] Finally, the CC-form and CD-form are combined into the popular Havriliak-Negami (HN) form given as

 (7a) (7b)

[123.4.8] Formula(7a) for the Havriliak-Negami relaxation function was first given in [18].

[123.5.1] The functional forms (1), (3), (5), and (7) are used universally almost without exception to fit broadband dielectric data. [123.5.2] A quantitative comparison of the different forms is shown for propylene carbonate at in Figure 2. [123.5.3] It is found, that all of the functional forms (1), (3), (5), and (7) deviate from the experimetal data at high ferquency or give an unsatisfactory fit. [123.5.4] Therefore, a combination of two or more of these functional forms is routinely used to fit the excess wing in glass forming materials (see e.g. Figure 3.5 in [28, p.66]).