Eigen-Coordinates for the Havriliak-Negami Function

APPENDIX B: Eigen-Coordinates for the Havriliak-Negami Function If reliable values of the fitting parameters ε∞, θ and σ can be calculated in the result of procedure described in Appendix A, then one can obtain the desired linear basic relationship for the Havriliak-Negami (HN) function χHN (ω) described in Eqn. (1). At first, we define the susceptibility modulus M (ω) and loss angle δ (ω) of the χHN (ω) by the relationships χ HN ( ω) = Re χ( ω) − j I m χ( ω) ≡ M ( ω) exp[ − j δ( ω) ] .

(B1)

From definition (B1), taking into account the initial HN-expression in Eqn. (1), we have

(

M ( ω) = ⎡⎢1 + 2C ν ω ν + ω / ω p ⎣

)

2ν

⎤ ⎥⎦

−

β 2

,

⎡ S νω ν ⎤ . δ( ω) = β tan −1 ⎢ ν ⎥ 1 C + ω ν ⎣ ⎦

B2a)

(B2b)

with C ν ≡ cos( πν / 2) / ω νp , S ν ≡ sin ( πν / 2) / ω νp .

(B2c)

From expressions (1) and (B1), one can obtain

[ ] [ M ( ω) ]

1 + C νω ν = χ s

1/ β

S ν ω ν = [χ s ]

1/ β

cos[δ( ω) / β] ,

(B3a)

[M ( ω) ]−1/ β sin[δ(ω) / β] .

(B3b)

−1/ β

Here and below χ s ≡ Δε = ε s − ε ∞ . Taking the logarithm of both parts of the last relationships and then applying the operator ω ⋅ d/dω(...), we obtain νC ν ω ν ⎛ δ( ω ) ⎞ 1 1 x ( ω) − tan ⎜ ⎟, ν = − β β ⎝ β ⎠ 1 + C νω

(B4a)

⎛ δ( ω) ⎞ νβ = − x ( ω) + cot ⎜ ⎟ y ( ω) , ⎝ β ⎠

(B4b)

where x ( ω) ≡ ω

d ln[M ( ω) ], dω

y ( ω) ≡ ω

d δ( ω) . dω

Then, taking into account the relationship, which follows from Eqn. (B2b),

(B4c)

S νω ν ⎛ δ( ω ) ⎞ , tan ⎜ ⎟ = ⎝ β ⎠ 1 + C νω ν

(B5)

we have from equations (B4a) and (B4b)

−

⎡ δ( ω ) ⎤ = tan ⎢ ⎥, y ( ω) + νβ cot ( πν / 2) ⎣ β ⎦

(B6a)

⎡ δ( ω ) ⎤ y ( ω) = tan ⎢ ⎥. x ( ω) + νβ ⎣ β ⎦

(B6b)

x ( ω)

Finally, we get from the last equations (B6a) and (B6b) the following formula x 2 ( ω) + y 2 ( ω) = − νβx ( ω) − νβ cot( πν / 2) y ( ω) .

(B7a)

Equation (B7a) suggests that one can present the HN-function as a set of straight lines or equivalently by a complete circle whose center is, in general, located in the third quadrant of the [x (ω), y (ω)] coordinates. This geometrical circle can be described by the formula

[x(ω) + (βν / 2)]2 + [ y(ω) + (βν / 2) ⋅ cot (πν / 2)]2 = [(βν / 2) ⋅ csc(πν / 2)]2 .

(B7b)

Integrating the relationship (B7a) in the limits ωm ≤ ω