A new graphical
representation
for dielectric
data
YamZhen Weia) and S. Sridhar
Department of Physics, Northeastern University, Boston, Massachusetts 02115
(Received 16 April 1993; accepted 4 May 1993)
A method of graphically representingcomplex dielectric data is described,which is particularly
useful for situations where the Cole-Cole representation is unsuitable. The representation
emphasizes the orientational dipolar contribution to the conductivity by plotting
u;=WE”(E’ -E, ) vs o$= WE&‘. We discuss the utility of the representation for conducting
dielectrics and for studying multirelaxation processes,particularly modes with smaller amplitudes (absorption) at high frequencies. For a Debye relaxation with dc conductivity, this
representationleads to a semicircle, whereasthe Cole-Cole plot shows a divergence.Even in the
absenceof conductivity, this representation is useful particularly at high frequencies, where it
enablesthe identification of multirelaxation processesnot apparent in the Cole-Cole representation. A particularly striking example is the clear observation of two relaxation processesin
dielectric data on 1-propanol. We also show that in this ox representation, all the well-known
functional forms such as Cole-Cole and Cole-Davidson, approach an asymptotic slope at high
frequencies.Applications of the ax representationto analysis of dielectric spectra taken up to 20
GHz are discussedfor pure glycerol and LiCl/propanol solutions.
I. INTRODUCTION
A widely used graphical representation of frequencydependent complex dielectric functions E(w ) = E’(0)
-i&‘(o)
for various materials is the well-known ColeCole plot, ’ in which en is plotted on the vertical axis
against E’. The Cole-Cole plot is particularly useful for
materials which possessone or more well separatedrelaxation processeswith comparable magnitudes and obeying
the Debye or Cole-Cole functional forms. For instance for
a Debye relaxation processthe Cole-Cole plot reducesto a
semicircle. However, when the material also possessesa
conductivity, the Cole-Cole representation becomes less
useful, becausethe presenceof a dc conductivity leads to a
divergence of E” at low frequencies.
During an extensive study of the dielectric properties
of ionic solutions at GHz frequencies,2-4we have found a
new representation to be useful, which is based upon the
orientational part of the complex conductivity defined as
cr,=o(o) --iwe,e, . In this new representation, which we
call the ax representation, we plot a; =weJ E’--E, ) vs
c~=oe,e”. For a Debye relaxation with a dc conductivity,
this representation yields a semicircle, whereas the ColeCole plot leads to a divergence.A specific example of data
for CsCl/H,O is discussed.The representation is useful in
the absenceof static conductivity also, particularly at high
frequencies.It enablesthe identification of multirelaxation
processeswhich are not apparent in the Cole-Cole representation. As an example, we show very clearly the presence of two relaxation processesin 1-propanol up to 20
GHz. We also show that, in the ax representation, all the
well-known functional forms, such as Cole-Cole, ColeDavidson, etc., approach an asymptotic slope at high frequencies. To our knowledge the utility of this new repre‘)Present address: Department of Pathology, Deaconess Hospital/
Harvard Medical School, Boston, MA, 02215.
sentation has not been recognized before, probably due to
the lack of high frequency data for which it is particularly
suited.
Below we discuss several aspectsof the aX representation for single and multirelaxation processes,and its applications.
II. SINGLE RELAXATION
PROCESSES
A. Sing!e-Debye
with dc conductivity
process
In many cases, such as dilute to moderately concentrated solutions of alkali halides in water, the dielectric
spectrum at room temperature can be very well described
by a dc conductivity along with a single Debye relaxation.2’3The functional form of E(W) is
Eo--Em -00
E(W)=e, + -&j
1-t-i07
WE,*
(1)
Hence the conductivity is
a(o) =im”E(W) =u’+iu”.
(2)
Define
1
sz dEo--E,
T
’
u, =uo+
(3)
%(Eo-E,)
=oo+s,
T
(4)
u;%f f
(5)
u+oE”(E’--E,).
(6)
ai and a; are found to be related in the same way as are E’
and 8’ for single Debye relaxation,
[up
(u(-j+wz)]2+u;2=
(6/2)2
(7)
0021-9606/93/99(4)/3119/6/$6.00
@ 1993 American Institute of Physics
J. Chem. Phys. 99 (4), 15 August 1993
3119
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Y.-Z. Wei and S. Sridhar: Representation
3120
-i
for dielectric data
c
1
=x
b
60
3
9 40
b”
20
0
0
20
40
60
80
100
a;(nm)-’
=~2;
20
FIG. 1. $-a;:
plot for CsCl&O data and corresponding fits; 0:
c=O.66 mM, e,,=70.5, e-=5.2, ~=8.1 ps, and a0=7.6 (am)-‘;
f:
c=l.l mM, ~,,=68.0, E, =4.8, ~=7.8 ps, and cq,= 12.0 (am)-‘; and 0:
c=2.5 mM, .s,,=57.6, e-=4.5, 7=7.0 ps, and u,=24.6 (am)-‘. Inset is
the corresponding Cole-Cole plot.
0
60
u,l40 (nm)-’
80
100
120
FIG. 2. o!; -0; plot for dol+Cole dispersion E,,= 55, E, = 5, T= 8.85 ps,
and o,,= 10 (am)-’ with a=0 (solid line, pure Debye), a=O.l (dashed
line) and Q= l/6 (dot-dash line). The dotted lines are the asymptotic
behavior of a=O. 1 and a= l/6 cases, which form angles of r/20 and
?r/12 with respect to the ai axis.
which is the equation for a circle centered at (0, + 6/2, 0)
with radius 6/2. When ax =0, there are two x intercepts at
x1=63,
(8)
x2=(T, .-
(9)
Also
(Ef-F)2+ [&‘-~+~tan(~a/2)]2
(Eo-%J2
-4 cos2( %-a/i) .
f-=-s
6
2
%l(Eo-E,)
(10)
2r
The above expressions suggest that plotting
WE”(E’-e, ) vs WE& gives a semicircle for a conducting
Debye material, which is similar to the dielectric spectrum
presentedin the Cole-Cole plot of a pure Debye response
without dc conductivity. The low frequency x intercept
gives the solution dc conductivity o. and the radius of the
semicircle is proportional to the magnitude of the susceptibility and inversely proportional to the relaxation time.
This is indeed the case, as shown in Fig. 1, in which the
data for several solutions of CsC1/H20 are shown, where
the x intercepts give the values of the dc conductivity o.
=7.6 (am)-’ for c=O.66 M, ~~=12.0 (am)-’ for
c= 1.1 M and ao=24.6 (nm) -’ for c=2.5 M, along with
the theoretical form IQ. (7). The improvement over a
Cole-Cole plot, shown in the inset of Fig. 1, is apparent.
If o,=O, Eq. ( 12) gives a circle of radius
centered at
(
E0-E,
2
Cii-6,
Y- --tan?
2
1
in the Cole-Cole plot. This is not the case when there is a
contribution from the conductivity ao#O. In the ai-a $
plot, the curve is not a semicircle in any case since
w( E’-E, ) is not zero at high frequencies. However, the
curve approachesan asymptotic slope tan( r/2) when the
frequency is significantly higher than the characteristic relaxation frequency of the spectrum, which can be seen
from
lim W(&---E,) =sin F (eo-e,)~“,
( )
a-m
6. Characteristic
forms
slope for non-Debye
dielectric
For Col+Cole function with a conductivity term
The equation relating E’and 8 is
(12)
(13)
lim ~r?‘=cos f
(Eo-E,)~~“+~o.
(14)
( 1
The angle against the a;l axis constructed by the asymptotic line is ra/2. Curves of different a values 0, 0.1 and
l/6, with eo=55, l ,=5, 7=8.5 ps and a,=10 (am)-‘,
are plotted in Fig. 2. The dotted asymptotic lines construct
angles ?r/20 and r/12 relative to the x axis, respectively.
o-00
J. Chem. Phys., Vol. 99, No. 4, 15 August 1993
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Y.-Z. Wei and S. Sridhar: Representation
data
for dielectric
3121
If the dielectric spectrum is of the Cole-Davidson5
form
E=E’-i&E”+(p+;;;)Q
.
(4
1
(15)
or the Havriliak-Negami6 biparametric function
(16)
in which O<cr<l and O<fl<l,
power law description7
2.5
or the Hill-Jonscher’s
da) =e, + (G3-~,b%w,o~),
where
F(m,n,wr)=(l+ior)“-’
2F1 1-~,l-~zm;2-~n;-
1.0
b
I
,
I
,
I
I
I
I
I
,
I
I
I
,
I
,
0.6
1
l+ior
(17)
and 2F1 is the Gaussian hypergeometric function, the curve
in the a; -ai plane will also approach a straight line at
frequencies much higher than the characteristic frequency
f,= (27rr) -’ of the material, since their high frequency
asymptotic behavior E’ and en are wVP, u-~(‘-@, and
w ‘- ‘, respectively.
-i
z
V
0.6
=b. 0.4
0.2
0.0
0
’ ’ ’ ’ ’ ’ ’ ’ ’ 1 ’ ’ ’ ’ 1 ’ ’ ’ ’
0.6
1.5
2
Ox’ (*In)-:
III. ENHANCEMENT IN HIGHER FREQUENCY
FEATURES FOR MULTIRELAXATION
SPECTRA
FIG. 3. (a) Cole-Cole plot for pure 1-propanol data (0 ) and ColeXole
(CC) fit (dashes), two Debye rehxxation time fit (dot-dash) and Debye
+ CC fit (solid line). (b) c$-ukplot for pure 1-propanol data (0), CC,
Debye +CC and two Debye fit.
Another interesting feature about the ox representation
is that it can enhance the characteristics of Debye-type
processeswith shorter relaxation times probed by the dielectric measurements. From the expression of the radius,
I$. ( lo), we see that if the amplitudes of the susceptibility
~~~~~~~~~~~ -~20=~20-~2zm, the shorter the relaxation
time is, the bigger the radius becomes. This can be also
understood from the point that what are plotted are the
dielectric values multiplied by frequency. The higher the
frequency is, the bigger the enlargement on (E’-E, ) and
E” that results. As a consequence, higher frequency features which may be suppressed in the Cole-Cole plot due
to having a small amplitude, become obvious in a a; - a;(
plot.
It has been long speculated,sP9
that two relaxations give
a better fit to the dielectric spectrum of 1-propanol at frequencies less than 20 GHz, than single Debye, Cole-Cole,
or Cole-Davidson fits. This conclusion was arrived at
based on both theoretical modelling and the degree of the
consistency between data and fits, since more parameters
always fit better numerically. However, there is an inherent
uncertainty in identifying the number of relaxation modes
simply by fitting various function forms to a dielectric
spectrum using either a Cole-Cole plot or plotting versus
frequency. In contrast, here we are able to visualize the two
relaxation modes graphically through the ox plot.
The dielectric spectra of an anhydrous I-propanol solution, handled under dry nitrogen, were taken via the quasicontinuous dielectric measurement method from 45 MHz
to 20 GHz.“*” As shown in the Cole-Cole plot of our
dielectric data and fits in Fig 3 (a), one can not easily
distinguish between a Cole-Cole fit (dashed line), a two
Debye relaxation time fit (dot-dash line) and the combination of Debye and Co&Cole fit (solid line). There is no
apparent signature of the existence of a shorter relaxation
process. However, the curve for 1-propanol data in the
a;-$,
plot shown in Fig. 3(b) shows the unmistakable
evidence of the existence of only one other relaxation process present in the frequency window. Another observation, with the help of the higher frequency enhancement
effect of the yX plot, is that the mismatch between the
shorter relaxatton process and its Debye fit is big. As can
be seen clearly from Fig. 3 (b), the shorter relaxation process is better described by a Cole-Cole form with (Y= 0.13
(solid line) than by two Debye relaxations (dot-dash
line).
There is a proposed third relaxation model2 in
1-propanol which was located around 80 GHz by Barthel
et al. I3 through fitting the spectrum to the superposition of
three Debye relaxations. Their results for the two modes
below 20 GHz are basically consistent with ours; differences between e. and r1 values are less than 2%. Our r2
value 9.7 ps is close to their measurements on ethanol, but
much shorter than the value of 15.1 ps for 1-propanol. This
discrepancy.might due to the existence of the third relaxation process at much higher frequencies.
Another example is pure glycerol at room temperature.
Its dielectric spectrum seems to be a skewed arc in the
Cole-Cole plot as shown in the inset of Fig. 4, which appears to be a typical example of the Cole-Davidson form.i4
J. Chem. Phys., Vol. 99, No. 4, 15 August 1993
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3122
Y.-Z. Wei and S. Sridhar: Representation
for dielectric data
1.50
1.25
‘;
1.00
2
Txo.75
b
0.50
0.25
0.0 I
0
-I
1.5
O&S
a;
2
0.00
,
0
Q.
L
0
0.26
FIG. 4. IT!!--d, plot for pure glycerol data ( 0 ) and Cole-Davidson fit
(dashed line). Inset: Cole-Cole plot of data (0) and Cole-Davidson fit
“(dashed line).
Our data can be visually fitted to a Cole-Davidson dispersion with /?=0.66. However the $--a; plot in Fig. 4
shows that there is a significant disagreementbetween the
data and the Cole-Davidson fit. The data can also be fitted
to the superposition of two Cole-Cole relaxations with
same order of magnitudes in standard deviation. The issue
of which relaxation model best representsthe real dipolar
dynamics in aqueous solutions of glycerol and 1-propanol
will be addressed elsewhere. Here a study of LiCVlpropanol solutions is presented.
IV. DYNAMICS OF LiCI/l-PROPANOL
0.6
0.75
1
1.26
1.5
ai (nm)-’
(nm)~l
SOLUTIONS
The electrodynamic theory for ionic solutions,by Hubbard and Onsager and later improvements,‘5P’6which successfully related the decrease in solvent static dielectric
constant to the solution conductivity,3”7P’8 are not restricted to a single relaxation model of the Debye type for
the solvent. Ibuki et al. I9 derived the explicit expressions
which can be applied to systems with two relaxation times.
The differential equation derived from the equation of motion for the dielectric fluid has the same form as that for a
single relaxation with the only difference being observedin
the coefficients. As discussed above, pure l-prop&o1 can
be regardedas a well characterized solvent. It is therefore
interesting to study the dynamics of ions dissolved in
1-propanol, similar to our earlier studies2’3on aqueous
ionic (alkali-halide) solutions. At sufficiently high ion concentrations, the static dielectric constant of the solution is
expected to be decreased by the increasing conductivity
due to the kinetic polarization effect. The new dielectric
data representation also allows us to observe the effect of
the ions on the solvent dynamics in more detail.
LiCl/l-propanol solutions were prepared by dissolving
anhydrous LiCl in anhydrous 1-propanol. The dielectric
spectra of LiWl-propanol solutions, with concentrations
FIG. 5. d!--o; plot for 0.5 mol % Lb/l-propanol
data (+) and two
Cole-Cole fit (solid line) with 6, =2.7, A,= 15.5, 1-,=329 ps, a,=0.02,
As= 1.4, TV= 10.3 ps, crs=O.l7, and as=O.O42 (l/am). For comparison,
data for pure propanol (0) are also shown. Inset: Cole-Cole plot of data
and the fit.
of 0.5 and 1.1 mol %, were studied. A concentration of 1.1
mol % (moles of LiCl/moles of I-propanol) is close to the
solubility limit.
The results for 0.5 and 1.1 mol % solutions are presented as a ax representation in Fig. 5 and as a 0’ vs IY
plot in Fig. 6, respectively. Also plotted are the fits to the
combination of two Cole-Cole (solid lines)
E(W)=E, +
Fitting parametersfor two Debye and two Cole-Cole combinations for I-propanol and LiCl/l-propanol are listed in
Table I. The static dielectric constant is
eO=e, +A1+A2.
(19)
From the fitting parameters listed in Table I, we observe that the higher solution conductivity induces larger
decreasein the static dielectric constant. The major contribution to the kinetic polarization is from the low frequency relaxation mode, which decreasesA,, while A2 remains basically unchangedwhen the ionic concentration is
increased.Addition of ions slightly raises the reorientation
time of both relaxation modes, but the effects on the higher
frequency mode seem to be more significant.
V. DISCUSSION
A. The influence
of the choice of E,
It is apparent that, similar to the role that o. plays for
a meaningful Cole-Cole curve in the presence of dc conductivity, the ax representation for data of dielectric mea-
J. Chem. Phys., Vol. 99, No. 4, 15 August 1993
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Y.-Z. Wei and S. Sridhar: Representation
1.0
?A
16
-
0.6
10
0.6
10
E’
0.4
WE”E’ 4 co ;
OE”E” --* ITo +
E,(Eo-6,)
7
(23)
which gives, for the LiC1/H20 data shown in Fig. 6, the
asymptotic behavior when w -+ 00 of u’=0.408 (am) -’
located by an arrow, with E, =2.7, A= 14.2, r= 365 ps,
and ao=0.064 l/am. Therefore, if the spectrum is of Debye form, an inaccurate E, will cause an asymptotic behavior to a vertical line (it may go to - 03 if the proposed
value of E, subtracted from 6’ is too big).
1;
a00
16
w+co
1;
‘;
6
b
e”
6
da
z
IIi;
Ii
3123
for dielectric data
xx
/’/
0.2
2. Sfngle Cole-Cole
t
I
0.4
0.3
0.2
0.1
0.6
c+>
FIG. 6. Verification of the existence of two relaxations for the 1.1 mol %
LiCVl-propanol sample, when plotting d’=~e,e’ vs u’=we,,e” directly,
instead of subtracting the effect of E, on the data representation. The
dashed line is a Debye curve with e-=2.7, A=14.2, r=365 ps, and
ae=O.O64 (l/am),
which approaches the vertical line a’=0408
(am)-’ indicated by the arrow, when frequency is much higher than the
characteristic frequency. The dash-dot line is a Cole curve with E, =2.7,
A= 14.2, r=365 ps, us=O.!l64 (l/am) and a=0.03, which represents
the low frequency part of the data but curves up at high frequencies. The
solid line is the two Cole-Cole fit with e-=2.7, A1=14.2, ~,=365 ps,
a,=0.03, As= 1.5, rs= 12.2 ps, a,=0.20 and ue=O.O64 (l/am), which
fits the data set. Inset: Cole-Cole plot of the data and the fits from which
the differences of the two kinds of fits, Debye and Cole-Cole, are not
distinguishable.
surements requires a parameter E, which has to be input
separately. We discuss the influence of the choice of this
parameter in various casesbelow.
1. Single Debye with a dc conductivity
For a spectrum such as represented by Eq. ( 1), the
effects of a0 on E” and E’are
0-o
Co’03
E’+Eo;
E’d,
En+ co,
(20)
E” + 0,
(21)
;
which gives the divergence at W-POas shown by the line in
the inset of Fig. 6. When plotting uN=c&
vs 19=OG&
for the Debye model, and noticing that E, is not subtracted
from E’, we observe
o-+0
oEg’-o;
WEUE”-+ao,
(22)
TABLE I. Two Debye (2DB) and two Cole-Cole (ZCC!) fits for LiCl/
1-propanol solutions. E, =2.7.
c
(mol %)
0
0.5
1.1
Fit
type
E.
A,
$1
2DB 20.9 16.9 329
2CC 20.9 16.9 332
2DB 19.6 15.3 333
2CC 19.6 15.6 329
2DB 18.4 13.9 373
2CC 18.4 14.2 365
a,
0
0
0
0.02
0
0.03
A,
&
1.3 8.8 0
1.3 9.7 0.13
1.3 13.5 0
1.3 10.3 0.17
1.4 20.4 0
1.5 12.2 0.20
0
0
0.042
0.042
0.064
0.064
form with dc conductivity
According to Eqs. (13) and (14), by moving the E,
term from the left to the right side of Eq. ( 13), when
w>wc= l/r, the asymptotic slope s in u” - cr’plane would
be (not applicable to the case of a = 0)
d(m,e’)
S=d(u&‘)
Em
&-cr
=tan 7 +
(24)
a
cos(7Ta/2)(Eg-EE,)
*
( )
Since 0 <a < 1, the slope due to E, , increaseswith increasing frequency, which would give a tail curving upwards as
shown by the dot-dash line for a=0.03 in Fig. 6.
3. Other empirical
dielectric
functions
As discussed above, the asymptotic behavior of the
susceptibility x(w) =E(o) -E, , expressedby other widely
used empirical forms, approachesa straight line as that of
Cole-Cole form. The asymptotic slope in the complex
plane of conductivity would present a similar form as Eqs.
(13) and (14) except that a different constant would appear in the place of the sin(ra/2) and cos(ra/2) terms.
Therefore, the effect of not subtracting the induced polarization E, , or assuming it is zero, would also give a tail
curving upwards.
4. Presence of a second relaxation process at higher
frequencies well separated from a main single
Debye process
When a second o,~ relaxation process with relatively
narrow bandwidth occurs where w,~<w, to certain extent,
the upwards tendency causedby E, would be depressedby
the downward curvature contributed from the second relaxation process.The 1.1 mol % LiCl/l-propanol data presented in Fig. 6 is a typical example. The curvature after
the “kink” still manifests the existence of two distinct relaxation processesfor the LiCl/l-propanol solutions. Such
a plot for the pure glycerol data also demonstrates the
existence of a second relaxation process in the frequency
range, but its characteristics are not as clear as that given
by the a; - ai plot in Fig. 4.
In practice, the value of E, is usually not well known
and can be varied to produce a meaningful curve. If the
data do not access high frequencies, then the representa-
J. Chem. Phys., Vol. 99, No. 4, 15 August 1993
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3124
Y.-Z. Wei and S. Sridhar: Representation
tion is insensitive to this parameter, but it is still able to
provide insight to the existenceof possible relaxation processeswith shorter time scales.
B. Comparison to other frequently used dielectric
representations
1. Dielectric modulus representation
Another representation that is often used is in terms of
the electric modulus” defined as
1
(251
E(W) *
The low frequency divergence in E”, caused by the existence of a dc conductivity ao, appears as a conductivity
relaxation peak with half-width of 1.14 decades, which
shifts to higher frequency with increasing conductivity.
The solvent retardation time rM which occurs in the electric modulus formalism is smaller by a factor e-/e0 than
the dielectric relaxation time r8, which can thus lead to a
dielectric relaxation peak in the modulus plot at higher
frequencies. However, the conductivity relaxation peak
shifts to higher frequencies and often emerges with the
solvent relaxation, when the solution conductivity’* increases.
&f=M’+jM”=-
2. Re(E) versus Re (a) representation
The real part of the dielectric constant plotted vs the
real part of the conductivity is also used to highlight the
frequency data by enhancing the dielectric 10~s.~~‘
It~ is
able to eliminate the divergencedue to the dc conductivity
and does not require an estimation of E, . However, it is
useful for characterizing a single Debye dispersion only,
for which this representation gives a straight line and the
slope is related to the relaxation time.
C. LiCVl-propanol
analysis
The speculation, that the dielectric function for liquid
alcohols possessmore than one relaxation process,and the
main relaxation at lower frequency is usually characterized
by the simple Debye relaxation form, is visibly confirmed,
for the llrst time to our knowledge, through the new ax
representation.The low-frequency dispersion is attributed
to the dipole relaxation of clusters of alcohol molecules
hydrogen bonded with each other, and the dispersions at
higher frequency region are ascribed to the internal rotation of hydroxyl groups and/or the reorientation of free
alcohol molecule.
The nature of ionic effects on the relaxation dynamics
of alcohols are still not very clear. Addition of lithium
perchloride, decreasesthe dielectric relaxation time of the
solvent l-propanol.21In contrast, LiCl in glycerol increases
the dielectric relaxation time of the solution,20which seems
to be the case from our measurements.
for dielectric data
In summary, the a;-$, plot depicts the contribution
from the dynamics of the dipole moment to the conductivity, inaddition to the dc conductivity. The plot is in the
we,x(w) plane, where x(w) representsthe dielectric susceptibility. For a medium with a frequency independent dc
conductivity, similar to the Cole-Cole plot, instead of subtracting away the contribution of a dc conductivity from
the imaginary part of the dielectric constant as neededfor
a meaningful Cole-Cole plot, subtracting a proper value of
E, is required to clearly observe the characteristics of a
spectrum. In the ai - 0;: plane, the low frequency intercept
indicates the value of the dc conductivity oo. For Debye
type with dc conductivity, it yields a semicircle. For all
other commonly used empirical dielectric functions, the
curve approachesan asymptotic slope. The angle formed
betweenthe asymptotic line of the Cole-Cole curve and the
o;Caxis is ra/2. The ax representation enhancesthe high
frequency features of a spectrum. It makes it possible to
visualize multirelaxation processesgraphically. Thus it is a
useful graphical representationfor dielectric spectra taken
up to high frequencies, where many relaxations with
smaller absorption relative to the main relaxation process
are present, for media with or without dc conductivity. We
also point out that data which seemto be Cole-Cole, ColeDavidson or Havriliak-Negami type according to a ColeCole plot, may possesstwo or more relaxation times which
can be possibly demonstrated using the ax representation.
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