[page 2297, §1]

[2297.1.1] Amorphous polymers and supercooled liquids
near the glass transition temperature
exhibit strongly nonexponential response
and relaxation functions in various experiments [10].
[2297.1.2] Dielectric spectroscopy experiments show an asymmetrically
broadened relaxation peak, often called

[2297.2.1] Most theoretical and experimental works use a
small number of empirical expressions such as
the formulae of Cole-Davidson (CD) [2],
Havriliak-Negami (HN) [8] or
Kohlrausch-Williams-Watts (KWW) [13] for fitting
of the asymmetric

(1) |

where

(2) |

where

(3) |

where

[2298.1.1] Dielectric loss spectra very often show a marked excess
contribution at frequencies some decades above the peak
frequency of the

[2298.2.1] Given the objective of introducing an improved phenomenological fit function it is pertinent to list first the traditional formulae against which to compare the new proposal. [2298.2.2] Let me begin with the three-parameter formula for a two-step Debye relaxation

(4) |

where

(5) |

with a sum of two

(6) |

[2298.2.4] Relaxation time distributions with more parameters will give better fits at the expense of introducing more parameters, but here attention will be restricted to fit functions with two or three parameters. [2298.2.5] In Reference [3] Davidson and Cole discussed the two-parameter expression

(7) |

for the normalized susceptibility containing a single
stretching exponent

(8) |

[page 2299, §0]
with two stretching exponents

[2299.1.1] All of the fitting formulae above were defined in the frequency domain. They can be transformed into the time domain using equation (3). [2299.1.2] A widely used fitting formula in the time domain, on the other hand, is the stretched exponential relaxation function

(9) |

with exponent

(10) |

where

(11) |

[page 2300, §0]
convergent for all

[2300.1.1] The main purpose of this short letter is to introduce a simple
three-parameter fit function that seems to work well not only
for fitting to the asymmetric

(12) |

containing a single stretching exponent

[2300.2.1] The results are presented for the broadband
dielectric spectra of glass forming propylene
carbonate reported in [12].
[2300.2.2] At a temperature of

[2301.1.1] Figure 1 shows the results for the real part. [2301.1.2] The data have been displaced in the vertical direction from their original location corresponding to FD in order to better distinguish the quality of the different fits.

[2301.1.3] Clearly the two-step Debye fit is not as good as the other fits in the fitting range. [2301.1.4] Extending the fitting range shows that also the KWW-formula gives not as good agreement as the CD-, HN-, and FD-fits. [2301.1.5] This can also be seen from the fact that the latter fits extend beyond the original fitting range.

[2301.2.1] Figure 2 shows the results for the imaginary part. [2301.2.2] The CD- and HN-fits are seen to be of equal quality. [2301.2.3] They deviate significantly from the experimental data in the excess wing region outside the fitting range. [2301.2.4] Extending the fit range for the CD- and HN-fits would give poorer agreement and systematic deviations around the main peak.

[2301.2.5] Contrary to the CD- and HN-fits the FD-fit extends well beyond the fitting range into the region of the excess wing. [2301.2.6] Extending the fit range in this case would not lower the quality of the fit near the main peak.

[2301.3.1] In summary the present paper has shown that a simple functional form allows to fit an asymmetrically broadened relaxation peak well into the excess wing. [2301.3.2] Similar to the Cole-Davidson or the Kohlrausch susceptibilities but contrary to the Havrialiak-Negami function and contrary to a combination of Cole-Davidson and Cole-Cole fits[11] the new function requires only a single stretching exponent.