Ph.D. thesis (pdf) - dirac

5.3. Spectral shape 83
The Havriliak-Negami (HN) function,
ϕ HN (ω) =
1
[1 + (iωτ HN ) α ] γ , (5.3.2)
gives a power law with exponent (−αγ) in the high-frequency limit and a power law
of exponent α in the low frequency-limit of its imaginary part.
The HN function reduces to Cole-Davidson (CD) one when α = 1. (In the case of
a CD function we follow the convention and refer to the γ above as β CD .) The CD
spectrum has the same general characteristics as the KWW one: a high-frequency
power law with exponent given by β CD and a low-frequency power law with exponent
one. However, the shape of the two functions is not the same. The CD function is
narrower for a given high frequency exponent (given β) than the KWW function (see
figure 5.10 a)). The best overall correspondence between the CD-function and the
KWW-function has been determined by Lindsey and Patterson [1980]. (see figure
5.10 b)).
0
0
−0.5
−0.5
φ′′(ω)
−1
φ′′(ω)
−1
a)
−1.5
−2
−2 0 2
ω
CD
KWW
b)
−1.5
−2
−2 0 2
ω
CD
KWW
0
−0.5
φ′′(ω)
−1
c)
−1.5
−2
−2 0 2
ω
AAC
KWW
Figure 5.10: Log-log plots of the different showing the loss of different fitting functions
a) KWW-function with β KWW = 0.5 CD-function with β CD . Dashed lines
illustrate the high frequency power-law. b) KWW-function with with β KWW = 0.5
and the corresponding CD-function according to Lindsey and Patterson [1980] giving
β CD = 0.367. Dashed lines illustrate the high frequency power law. c) KWWfunction
with β KWW = 0.5 and the corresponding AAC-function.
No good correspondence exists in general between the HN and the KWW functions.
First of all because the former involves two adjustable shape parameters and the
latter only one (plus in both cases a parameter for the intensity and one for the time
scale). The KWW function always has a slope of one at low frequencies while the HN