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On the generic behavior of the electrical permittivity at low frequencies
Physics Letters A 373 (2009) 2793–2795
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
On the generic behavior of the electrical permittivity at low frequencies Mark W. Coffey Department of Physics, Colorado School of Mines, Golden, CO 80401, USA
a r t i c l e
i n f o
Article history: Received 14 February 2009 Received in revised form 11 May 2009 Accepted 27 May 2009 Available online 2 June 2009 Communicated by A.R. Bishop
a b s t r a c t We discuss the complex-valued electrical permittivity and show that generally the real part (ω) is a decreasing function of frequency from ω = 0 to near the location of the first resonance. This behavior results from the dominance of dipolar relaxation, and we use a theory with a general distribution function y (τ ) of relaxation times to describe it. Our result applies to a very wide class of materials, typically up to THz frequencies. © 2009 Elsevier B.V. All rights reserved.
Keywords: Electrical permittivity Dielectric response Distribution function Relaxation time Sum rule
1. Introduction The complex-valued electrical permittivity (ω) is a response function whose real part (ω) includes screening and resonance effects and whose imaginary part (ω) describes energy absorption. In particular, we have for the average energy dissipated per unit of time, w = ω2 /8π , for external radiation of angular frequency ω . It is typically observed experimentally that the real part (ω) is a decreasing function of frequency away from resonance regions. Often transitions from a slowly varying region to another, as a particular relaxation process occurs in the material. Correspondingly, attains a local maximum in the transition interval. In this Letter we are concerned with the generic behavior of the decrease of with frequency. In fact, this behavior is common for a very wide range of materials from dc through the radiofrequency and microwave range [1]. In this frequency range, dipole relaxation dominates, and only at higher frequencies typically do ionic and electronic polarization significantly contribute. Of course, in the most general situation, that we do not attempt to address, various resonances must be included. Other than polarization resonance, geometric resonance happens when an appropriate sample dimension is of the order of one half wavelength of the applied radiation. For microwave frequencies, geometric resonance should not occur for sample dimensions below about 1 mm. We present a general result for the decrease of the function (ω) with ω , away from resonances, that may serve as a sanity check on laboratory
E-mail address: [email protected]. 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.05.069
measurements and as a guide to future theoretical developments. We stress that for a wide range of materials the first resonance does not appear until THz frequencies are reached [6]. We employ a theoretical description at an intermediate level: neither so explicit and restricted as the simple Debye model nor as general and unconstrained as the Kramers–Kronig relations for dielectric response. For theoretical purposes, knowing that there exists a monotonicity region is reassuring for any inversion procedure to recover . We first recall some general properties of the electrical permittivity and mention a simple important practical model including resonances. We then present our main result, its assumptions and open questions. We conclude with a discussion of directions for further research. For real frequencies ω , is an even function, and an odd function. By the principle of causality, either of these functions may be determined from the other, as exhibited by the Kramers– Kronig relations. An explicit expression for the permittivity with physical relevance is given by the extended multipole Debye formula
(ω) = ∞ +
M
Aj
ω2j − ω2 − i γ j ω j =1
,
(1)
where ω j are the resonance frequencies, A j the oscillator strengths and γ j parameters to model loss. With this model, is negative in the range ω j < ω < ω jp , and zero at each plasmon frequency ω jp . Eq. (1) is a very good representation for a large class of local and isotropic media. Underlying this generality is harmonic response with damping. Harmonic response should occur when
2794
M.W. Coffey / Physics Letters A 373 (2009) 2793–2795
oscillations in polarization have small amplitudes and when they remain largely in phase with the applied radiation. Since in practice Eq. (1) seems to be nearly universal, there is probably more to be understood regarding the emergence of effective harmonic response from the microscopic dynamics. By making assumptions on the values of the various parameters in Eq. (1), it is possible to obtain region-by-region descriptions of and . However, in this Letter we adopt a model of a different nature. 2. Main result It is known that the behavior of the orientational polarization of most condensed matter systems in time-dependent fields can, as a good approximation, be characterized with a distribution of relaxation times [3]. As far as typical numerical values for representative materials, for silver we have a relaxation time on the order of 10−19 s, for distilled water on the order of 1 ps, and for quartz 10 days. As expected, a large value of the relaxation time is indicative of an insulator, while very small values are typical of good conductors. We consider a description with distribution function y (τ ), giving the probability distribution of relaxation times in the interval (τ , τ + dτ ). We write
∞
(ω) = ∞ + (s − ∞ ) 0
where the static value gives
1 − i ωτ
dτ ,
(2)
∞
y (τ )
∞ y (τ ) dτ = 1, implying that y (∞) = 0. Furthermore, we have a sum rule [4]
lim
N →∞
1
N
N
1 + ω2 τ
(3)
y (τ )τ 1 + ω2 τ 2
0
y (τ ) d ln τ =
dτ .
(4)
This is indeed a causal model, and Eqs. (3) and (4) show the expected symmetry in ω of the real and imaginary parts. For a single relaxation time τ0 and y a Dirac delta function, y (τ ) = δ(τ − τ0 ), the Debye model is recovered. Another example distribution corresponds to an equal distribution of potential barriers over a range V 0 :
kB T 1
τ
,
τ0 τ τ1 = τ0 e V 0 /k B T ,
(5)
and otherwise 0, where k B is Boltzmann’s constant and T the absolute temperature. This model is suitable for solutions sufficiently dilute that interaction between dipoles may be neglected. Then it is possible to carry out the integrations in Eqs. (3) and (4) to find [5]
1 + ω2 τ02 e 2V 0 /k B T kB T , (6) (ω) = ∞ + (s − ∞ ) 1 − ln 2V 0 1 + ω2 τ02
and
(ω) = (s − ∞ )
kB T V0
(ω) dω = 1.
∞ 0
∞
V0
(8)
(9)
From Eq. (3) we then have
dτ , 2
and
There are fundamental relations governing the distribution y. It is nonnegative everywhere, y (τ ) 0 on τ ∈ [0, ∞) and normalized,
0
0
(ω) = (s − ∞ )ω
ωτ0 for values k B T / V 0 = 0.1,
0
y (τ )
s = (0) and ∞ = (∞). Eq. (2) readily
(ω) = ∞ + (s − ∞ )
y (τ ) = (s − ∞ )
Fig. 1. Plot of [ (ω) − ∞ ]/(s − ∞ ) of Eq. (6) versus 1, and 10.
−1
tan
ωτ0 e
V 0 /k B
T
−1
− tan
ωτ0 .
This model has the property that → s as T → ∞. The quantity [ (ω) − ∞ ]/(s − ∞ ) is plotted versus ωτ0 for various values of k B T / V 0 in Fig. 1.
π (s − ∞ )
(10)
.
Example functions for y may of course be taken in Eqs. (2)–(4) (e.g., [5]). However, we are most interested to maintain this as a general distribution function. Directly from Eq. (3) we have
d (ω) dω
∞ = −2(s − ∞ )ω 0
τ 2 y (τ ) d τ < 0. (1 + ω2 τ 2 )2
(11)
This shows that (within this model) is a decreasing function for all positive ω , with a global maximum only at ω = 0. The result (11) holds without further assumptions on the distribution function y. For instance, is a decreasing function regardless of the average value or variance of y, or whether it is unimodal or not. The first and second derivatives of the imaginary part of from Eq. (4) are given by
d (ω) dω
∞ = (s − ∞ ) 0
τ (1 − ω2 τ 2 ) y (τ ) dτ , (1 + ω2 τ 2 )2
(12)
and
d2 (ω) (7)
2 (1 − ∞ )
dω 2
∞ = −2(s − ∞ )ω 0
τ 3 (3 − ω2 τ 2 ) y (τ ) dτ . (1 + ω2 τ 2 )3
(13)
Several theoretical questions are posed by these equations. We first conjecture that if it is assumed that y has but a single global maximum, then it will follow that d (ω)/dω has only a single zero
M.W. Coffey / Physics Letters A 373 (2009) 2793–2795
2795
and that this zero corresponds to a global maximum. For the latter condition to hold, we then need that the right side of Eq. (13) is negative at the corresponding frequency. It is then to be considered if a similar result may hold by weakening the single global maximum hypothesis for y. From Eq. (12), for sufficiently low frequencies, we have d (ω)/dω > 0, while for sufficiently high frequencies, we have d (ω)/dω < 0. However, further restriction(s) on y is required in order to ensure a unique global maximum in (ω). Although Eq. (12) may be integrated by parts with respect to τ , this alone appears to be insufficient to demonstrate a unique zero. We seem to require additional assumptions on the function y. These considerations indicate that in any theoretical description for the electrical permittivity with general integrands that additional properties of the functions involved are necessary to specify the extremum nature of and .
Now we have two general functions, the distribution y and intensity function h. General considerations [3] constrain the function h: it should depend only upon the product ωτ , h = h(ωτ ), h (∞v ) > 0 for all v > 0, h is normalized on a logarithmic scale, h(ωτ ) d ln τ = 1, h satisfies the functional equation h( v ) = 0 h(1/ v ), and finally h has a maximum at ωτ = 1, and decreases monotonically to zero on both sides of this maximum. In this regard, our model of the previous section should be instructive in how to proceed to this more general context. This should further extend the range of validity of the generic low-frequency behavior of the electrical permittivity that we have described above to even more types of materials.
3. Further discussion
(ω) = ∞
Another direction for future consideration is the use of the factored form of the permittivity in terms of its complex zeros and poles, N
ω − zn . ω − pn
(20)
n =1
Due to their practical utility, it is worth mentioning the Cole– Cole, Cole–Davidson, and Havriliak–Negami formulas for (ω) [3]. These are models that extend the single-relaxation-time Debye model with one or two additional parameters. Each of these models entail the use of different analytic inequalities when determining the nature of the extrema in and . As an illustration, we briefly consider the Cole–Davidson model with parameter β > 0,
(ω) = ∞ + (s − ∞ )
1
(1 − i ωτ )β
(14)
.
Putting φ(ω) = tan−1 ωτ , we have
(ω) = ∞ + (s − ∞ ) cosβ φ cos βφ,
(15)
and
(ω) = (s − ∞ ) cosβ φ sin βφ.
(16)
That approaches a constant value as ω → ∞ explains why there are an equal number of poles and zeros in this equation. Such a dispersion function containing only polar singularities was largely introduced by Berreman and Unterwald [2], and continues to be of interest [7]. The product form (20) possesses a number of properties, and therefore may be of use in further studies of the generic behavior of (ω). Among these, both the zeros and poles must lie in the lower half plane, and the zeros and poles lie symmetrically about the imaginary axis. Additionally, since insulators have only real values of at zero frequency, these materials have a zero on the imaginary axis for every pole on the imaginary axis, and vice versa [2]. When combined with other conditions, it may then be possible to develop other monotonicity results for the electrical permittivity.
For the first derivatives of these functions we have
1
d
(s − ∞ ) dω
=−
Acknowledgements
βτ
[ωτ cos βφ + sin βφ], (17) (1 + ω2 τ 2 )1+β/2
and
1
d
(s − ∞ ) dω
=
βτ 2 2 )1+β/2
(1 + ω τ
[cos βφ − ωτ sin βφ].
(18)
A sufficient but by no means necessary condition for d /dω < 0 is ωτ < tan(π /(2β)). This condition ensures that both the sin and cos terms in Eq. (17) are positive. A generalization of our approach is to make the particular function h( v ) = (2/π ) v /(1 + v 2 ) in Eq. (3) a general function, so that we consider
(ω) = ∞ +
π 2
∞ (s − ∞ )
h(ωτ )
ωτ 0
y (τ ) dτ .
(19)
This work was supported in part by the Electromagnetics Division of NIST Boulder. Useful discussions with J. Baker-Jarvis are gratefully acknowledged. References [1] [2] [3] [4] [5]
J. Baker-Jarvis, M.D. Janezic, C.A. Jones, IEEE Trans. Instrum. Meas. 47 (1998) 338. D.W. Berreman, F.C. Unterwald, Phys. Rev. 174 (1968) 791. C.J.F. Böttcher, P. Bordewijk, Theory of Electric Polarization, Elsevier, 1978. J.D. Jackson, Classical Electrodynamics, John Wiley, 1975. H. Fröhlich, Theory of Dielectrics: Dielectric Constant and Dielectric Loss, Oxford University Press, 1958. [6] N.E. Hill, W.E. Vaughan, A.H. Price, M. Davies, Dielectric Properties and Molecular Behaviour, Van Nostrand Reinhold, 1969. [7] J.L. Ribeiro, L.G. Vieira, Eur. Phys. J. B 36 (2003) 21.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
On the generic behavior of the electrical permittivity at low frequencies Mark W. Coffey Department of Physics, Colorado School of Mines, Golden, CO 80401, USA
a r t i c l e
i n f o
Article history: Received 14 February 2009 Received in revised form 11 May 2009 Accepted 27 May 2009 Available online 2 June 2009 Communicated by A.R. Bishop
a b s t r a c t We discuss the complex-valued electrical permittivity and show that generally the real part (ω) is a decreasing function of frequency from ω = 0 to near the location of the first resonance. This behavior results from the dominance of dipolar relaxation, and we use a theory with a general distribution function y (τ ) of relaxation times to describe it. Our result applies to a very wide class of materials, typically up to THz frequencies. © 2009 Elsevier B.V. All rights reserved.
Keywords: Electrical permittivity Dielectric response Distribution function Relaxation time Sum rule
1. Introduction The complex-valued electrical permittivity (ω) is a response function whose real part (ω) includes screening and resonance effects and whose imaginary part (ω) describes energy absorption. In particular, we have for the average energy dissipated per unit of time, w = ω2 /8π , for external radiation of angular frequency ω . It is typically observed experimentally that the real part (ω) is a decreasing function of frequency away from resonance regions. Often transitions from a slowly varying region to another, as a particular relaxation process occurs in the material. Correspondingly, attains a local maximum in the transition interval. In this Letter we are concerned with the generic behavior of the decrease of with frequency. In fact, this behavior is common for a very wide range of materials from dc through the radiofrequency and microwave range [1]. In this frequency range, dipole relaxation dominates, and only at higher frequencies typically do ionic and electronic polarization significantly contribute. Of course, in the most general situation, that we do not attempt to address, various resonances must be included. Other than polarization resonance, geometric resonance happens when an appropriate sample dimension is of the order of one half wavelength of the applied radiation. For microwave frequencies, geometric resonance should not occur for sample dimensions below about 1 mm. We present a general result for the decrease of the function (ω) with ω , away from resonances, that may serve as a sanity check on laboratory
E-mail address: [email protected]. 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.05.069
measurements and as a guide to future theoretical developments. We stress that for a wide range of materials the first resonance does not appear until THz frequencies are reached [6]. We employ a theoretical description at an intermediate level: neither so explicit and restricted as the simple Debye model nor as general and unconstrained as the Kramers–Kronig relations for dielectric response. For theoretical purposes, knowing that there exists a monotonicity region is reassuring for any inversion procedure to recover . We first recall some general properties of the electrical permittivity and mention a simple important practical model including resonances. We then present our main result, its assumptions and open questions. We conclude with a discussion of directions for further research. For real frequencies ω , is an even function, and an odd function. By the principle of causality, either of these functions may be determined from the other, as exhibited by the Kramers– Kronig relations. An explicit expression for the permittivity with physical relevance is given by the extended multipole Debye formula
(ω) = ∞ +
M
Aj
ω2j − ω2 − i γ j ω j =1
,
(1)
where ω j are the resonance frequencies, A j the oscillator strengths and γ j parameters to model loss. With this model, is negative in the range ω j < ω < ω jp , and zero at each plasmon frequency ω jp . Eq. (1) is a very good representation for a large class of local and isotropic media. Underlying this generality is harmonic response with damping. Harmonic response should occur when
2794
M.W. Coffey / Physics Letters A 373 (2009) 2793–2795
oscillations in polarization have small amplitudes and when they remain largely in phase with the applied radiation. Since in practice Eq. (1) seems to be nearly universal, there is probably more to be understood regarding the emergence of effective harmonic response from the microscopic dynamics. By making assumptions on the values of the various parameters in Eq. (1), it is possible to obtain region-by-region descriptions of and . However, in this Letter we adopt a model of a different nature. 2. Main result It is known that the behavior of the orientational polarization of most condensed matter systems in time-dependent fields can, as a good approximation, be characterized with a distribution of relaxation times [3]. As far as typical numerical values for representative materials, for silver we have a relaxation time on the order of 10−19 s, for distilled water on the order of 1 ps, and for quartz 10 days. As expected, a large value of the relaxation time is indicative of an insulator, while very small values are typical of good conductors. We consider a description with distribution function y (τ ), giving the probability distribution of relaxation times in the interval (τ , τ + dτ ). We write
∞
(ω) = ∞ + (s − ∞ ) 0
where the static value gives
1 − i ωτ
dτ ,
(2)
∞
y (τ )
∞ y (τ ) dτ = 1, implying that y (∞) = 0. Furthermore, we have a sum rule [4]
lim
N →∞
1
N
N
1 + ω2 τ
(3)
y (τ )τ 1 + ω2 τ 2
0
y (τ ) d ln τ =
dτ .
(4)
This is indeed a causal model, and Eqs. (3) and (4) show the expected symmetry in ω of the real and imaginary parts. For a single relaxation time τ0 and y a Dirac delta function, y (τ ) = δ(τ − τ0 ), the Debye model is recovered. Another example distribution corresponds to an equal distribution of potential barriers over a range V 0 :
kB T 1
τ
,
τ0 τ τ1 = τ0 e V 0 /k B T ,
(5)
and otherwise 0, where k B is Boltzmann’s constant and T the absolute temperature. This model is suitable for solutions sufficiently dilute that interaction between dipoles may be neglected. Then it is possible to carry out the integrations in Eqs. (3) and (4) to find [5]
1 + ω2 τ02 e 2V 0 /k B T kB T , (6) (ω) = ∞ + (s − ∞ ) 1 − ln 2V 0 1 + ω2 τ02
and
(ω) = (s − ∞ )
kB T V0
(ω) dω = 1.
∞ 0
∞
V0
(8)
(9)
From Eq. (3) we then have
dτ , 2
and
There are fundamental relations governing the distribution y. It is nonnegative everywhere, y (τ ) 0 on τ ∈ [0, ∞) and normalized,
0
0
(ω) = (s − ∞ )ω
ωτ0 for values k B T / V 0 = 0.1,
0
y (τ )
s = (0) and ∞ = (∞). Eq. (2) readily
(ω) = ∞ + (s − ∞ )
y (τ ) = (s − ∞ )
Fig. 1. Plot of [ (ω) − ∞ ]/(s − ∞ ) of Eq. (6) versus 1, and 10.
−1
tan
ωτ0 e
V 0 /k B
T
−1
− tan
ωτ0 .
This model has the property that → s as T → ∞. The quantity [ (ω) − ∞ ]/(s − ∞ ) is plotted versus ωτ0 for various values of k B T / V 0 in Fig. 1.
π (s − ∞ )
(10)
.
Example functions for y may of course be taken in Eqs. (2)–(4) (e.g., [5]). However, we are most interested to maintain this as a general distribution function. Directly from Eq. (3) we have
d (ω) dω
∞ = −2(s − ∞ )ω 0
τ 2 y (τ ) d τ < 0. (1 + ω2 τ 2 )2
(11)
This shows that (within this model) is a decreasing function for all positive ω , with a global maximum only at ω = 0. The result (11) holds without further assumptions on the distribution function y. For instance, is a decreasing function regardless of the average value or variance of y, or whether it is unimodal or not. The first and second derivatives of the imaginary part of from Eq. (4) are given by
d (ω) dω
∞ = (s − ∞ ) 0
τ (1 − ω2 τ 2 ) y (τ ) dτ , (1 + ω2 τ 2 )2
(12)
and
d2 (ω) (7)
2 (1 − ∞ )
dω 2
∞ = −2(s − ∞ )ω 0
τ 3 (3 − ω2 τ 2 ) y (τ ) dτ . (1 + ω2 τ 2 )3
(13)
Several theoretical questions are posed by these equations. We first conjecture that if it is assumed that y has but a single global maximum, then it will follow that d (ω)/dω has only a single zero
M.W. Coffey / Physics Letters A 373 (2009) 2793–2795
2795
and that this zero corresponds to a global maximum. For the latter condition to hold, we then need that the right side of Eq. (13) is negative at the corresponding frequency. It is then to be considered if a similar result may hold by weakening the single global maximum hypothesis for y. From Eq. (12), for sufficiently low frequencies, we have d (ω)/dω > 0, while for sufficiently high frequencies, we have d (ω)/dω < 0. However, further restriction(s) on y is required in order to ensure a unique global maximum in (ω). Although Eq. (12) may be integrated by parts with respect to τ , this alone appears to be insufficient to demonstrate a unique zero. We seem to require additional assumptions on the function y. These considerations indicate that in any theoretical description for the electrical permittivity with general integrands that additional properties of the functions involved are necessary to specify the extremum nature of and .
Now we have two general functions, the distribution y and intensity function h. General considerations [3] constrain the function h: it should depend only upon the product ωτ , h = h(ωτ ), h (∞v ) > 0 for all v > 0, h is normalized on a logarithmic scale, h(ωτ ) d ln τ = 1, h satisfies the functional equation h( v ) = 0 h(1/ v ), and finally h has a maximum at ωτ = 1, and decreases monotonically to zero on both sides of this maximum. In this regard, our model of the previous section should be instructive in how to proceed to this more general context. This should further extend the range of validity of the generic low-frequency behavior of the electrical permittivity that we have described above to even more types of materials.
3. Further discussion
(ω) = ∞
Another direction for future consideration is the use of the factored form of the permittivity in terms of its complex zeros and poles, N
ω − zn . ω − pn
(20)
n =1
Due to their practical utility, it is worth mentioning the Cole– Cole, Cole–Davidson, and Havriliak–Negami formulas for (ω) [3]. These are models that extend the single-relaxation-time Debye model with one or two additional parameters. Each of these models entail the use of different analytic inequalities when determining the nature of the extrema in and . As an illustration, we briefly consider the Cole–Davidson model with parameter β > 0,
(ω) = ∞ + (s − ∞ )
1
(1 − i ωτ )β
(14)
.
Putting φ(ω) = tan−1 ωτ , we have
(ω) = ∞ + (s − ∞ ) cosβ φ cos βφ,
(15)
and
(ω) = (s − ∞ ) cosβ φ sin βφ.
(16)
That approaches a constant value as ω → ∞ explains why there are an equal number of poles and zeros in this equation. Such a dispersion function containing only polar singularities was largely introduced by Berreman and Unterwald [2], and continues to be of interest [7]. The product form (20) possesses a number of properties, and therefore may be of use in further studies of the generic behavior of (ω). Among these, both the zeros and poles must lie in the lower half plane, and the zeros and poles lie symmetrically about the imaginary axis. Additionally, since insulators have only real values of at zero frequency, these materials have a zero on the imaginary axis for every pole on the imaginary axis, and vice versa [2]. When combined with other conditions, it may then be possible to develop other monotonicity results for the electrical permittivity.
For the first derivatives of these functions we have
1
d
(s − ∞ ) dω
=−
Acknowledgements
βτ
[ωτ cos βφ + sin βφ], (17) (1 + ω2 τ 2 )1+β/2
and
1
d
(s − ∞ ) dω
=
βτ 2 2 )1+β/2
(1 + ω τ
[cos βφ − ωτ sin βφ].
(18)
A sufficient but by no means necessary condition for d /dω < 0 is ωτ < tan(π /(2β)). This condition ensures that both the sin and cos terms in Eq. (17) are positive. A generalization of our approach is to make the particular function h( v ) = (2/π ) v /(1 + v 2 ) in Eq. (3) a general function, so that we consider
(ω) = ∞ +
π 2
∞ (s − ∞ )
h(ωτ )
ωτ 0
y (τ ) dτ .
(19)
This work was supported in part by the Electromagnetics Division of NIST Boulder. Useful discussions with J. Baker-Jarvis are gratefully acknowledged. References [1] [2] [3] [4] [5]
J. Baker-Jarvis, M.D. Janezic, C.A. Jones, IEEE Trans. Instrum. Meas. 47 (1998) 338. D.W. Berreman, F.C. Unterwald, Phys. Rev. 174 (1968) 791. C.J.F. Böttcher, P. Bordewijk, Theory of Electric Polarization, Elsevier, 1978. J.D. Jackson, Classical Electrodynamics, John Wiley, 1975. H. Fröhlich, Theory of Dielectrics: Dielectric Constant and Dielectric Loss, Oxford University Press, 1958. [6] N.E. Hill, W.E. Vaughan, A.H. Price, M. Davies, Dielectric Properties and Molecular Behaviour, Van Nostrand Reinhold, 1969. [7] J.L. Ribeiro, L.G. Vieira, Eur. Phys. J. B 36 (2003) 21.