research article graphical conversion between compliance and...

Research ArticleGraphical Conversion between Complianceand Modulus Permittivity and Electric Modulusand Impedance and Admittance

Masahiro Nakanishi

Department of Applied Physics The Hebrew University of Jerusalem Bergman Building Givat Ram Jerusalem 91904 Israel

Correspondence should be addressed to Masahiro Nakanishi masahironakanishifrontierhokudaiacjp

Received 11 September 2014 Revised 2 November 2014 Accepted 5 November 2014 Published 25 November 2014

Academic Editor Akihide Wada

Copyright copy 2014 Masahiro NakanishiThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Spectrometries probing relaxation and retardation phenomena such as dielectricmechanical and impedance spectroscopies oftenrequire the analyses with both susceptibilities spectra 119911 and its reciprocals 1119911 (eg complex permittivity and electric modulusmechanical compliance and mechanical modulus and impedance and admittance) In the present paper the geometric relationbetween 119911 and 1119911 is derived and the procedure to convert 119911 into 1119911 on a Cole-Cole diagram is proposedThis method helps us torelate them intuitively and yields clearer understanding on their interrelations Moreover it opens the new route for the geometricapproach to derive many mathematical properties of spectra The relation between peak position of 119911 spectrum and that of 1119911spectrum and the shape of spectra are discussed on the basis of this method

1 Introduction

In spectrometries focusing on relaxation or retardationphenomena such as dielectric spectroscopy [1] dynamicmechanical spectroscopy [2] and impedance spectroscopy[3] not only frequency dependent susceptibilities but alsotheir reciprocals are often subject to consideration In dielec-tric spectroscopy for example the complex permittivity isusually employed to discuss the dynamic nature of localizedcharges such as the rotational diffusion of dipoles whileits reciprocal electric modulus is also employed to assessthe dynamics of delocalized charges such as the migrationof ions [4ndash6] This is also the case for the mechanicalspectroscopy Actually both of the mechanical complianceand the mechanical modulus which is the reciprocal of thecompliance are measured and discussed depending of thephenomena of interests [7]

Generally speaking extremely broad frequency rangecan be probed in these techniques it often exceeds 10orders of magnitude in the frequency [1ndash3] Thus obtainedbroad spectra often contain a variety of the dynamic pro-cesses Some of them are compatible with the analysis in

susceptibility whereas the others may be better analyzedin its reciprocal representation In this case one needs toswitch the quantity of consideration from one to the otherdepending on the purpose Therefore the straightforwardmethod which enables us to change from one to the otherhelps to understand the experimental data and makes theirinterrelations clearer

Graphical methods can be a powerful tool to satisfythis demand For example the Smith chart which is indis-pensably used in the field of microwave engineering [8ndash10] graphically relates the complex reflection coefficient andthe load impedance and so forth Of course since thesequantities are related by mathematically rigorous formulaeone can algebraically calculate any quantities once anotherquantity is given Nevertheless the intuitivemanner providesclear perspectives of the analysis Furthermore it can alsoprovide another route to analyze themathematical propertiesof spectra that is geometric method in addition to theconventional algebraic approaches

A Cole-Cole diagram [11 12] is one of themost frequentlyused representations for relaxation spectra Briefly speakingit is a plot of a complex quantity 119911 (or its complex conjugate

Hindawi Publishing CorporationInternational Journal of SpectroscopyVolume 2014 Article ID 538206 7 pageshttpdxdoiorg1011552014538206

2 International Journal of Spectroscopy

depending on the definition) on the complex plane wherecomplex permittivity compliance and impedance are oftenchosen as 119911 Plotting 119911 varying the frequency yields character-istic trajectory reflecting the phenomenon Remarkably someelementally models such as single exponential relaxationfunction in dielectric spectra and the behavior of a parallel RCcircuit in impedance spectra describe circular trajectories asfrequency is varied An analogous diagram is also employedto plot transfer functions and is called Nyquist diagram Theparameters of the transfer function can be determined andthe stability of a system can be analyzed thereby with it

In the present paper I propose a straightforward methodto convert the spectrum of arbitrary complex quantity 119911 intoits reciprocal 1119911 based on the graphical procedure on Cole-Cole diagramThereby interrelation between 119911 (permittivitycompliance and impedance) spectra and 1119911 (moduli admit-tance) can be graphically recognized Moreover it also opensthe route to the geometric method to assess themathematicalproperties of spectra

2 Theory

21 Definition We consider a pair of reciprocally relatedcomplex quantities 119911 and 1119911 Plotting 119911 or its complexconjugate in the complex plane yields a Cole-Cole diagramLoci of 119911 trace a characteristic trajectory on the diagram as aninternal parameter is varied (typically frequency) reflectingthe phenomenon represented by 119911

Such pairs of reciprocally related quantities can be oftenfound in spectroscopic measurements Complex permittivity120576lowast

(120596) and electricmodulus119872lowast120576(120596) are respectivelymeasures

of how easy and how difficult it is to deform a materialelectricallyThe former is defined as the electric displacement119863(120596) for a unit electric field 119864(120596) 119863(120596) = 120576

lowast

(120596)119864(120596)whereas the latter is defined as the electric field generatedwhen a unit electric displacement is imposed 119864(120596) =

119872lowast

120576(120596)119863(120596) Obviously 120576lowast(120596)119872lowast

120576(120596) = 1 meaning that

each is the reciprocal of the other (119911 and 1119911) Similarlymechanical compliance (shear 119869lowast(120596) or tensile 119863lowast(120596)) andmechanical modulus (shear 119866lowast(120596) or tensile 119864lowast(120596)) arereciprocally related as 119869lowast(120596)119866lowast(120596) = 1 and 119863lowast(120596)119864lowast(120596) =1 Impedance 119885lowast(120596) and admittance 119884lowast(120596) also satisfy thereciprocal relation 119885lowast(120596)119884lowast(120596) = 1

From the viewpoint of the similarity of the typicaltrajectories on a Cole-Cole diagram it is convenient todefine generalized compliance119862lowast

119866(120596) representing permittiv-

ity mechanical compliance and impedance whereas electricmodulus mechanical modulus and admittance are repre-sented by generalized modulus 119872lowast(120596) as summarized inTable 1 The generalized modulus is the reciprocal of thegeneralized compliance

119872lowast

(120596) =

1

119862lowast

119866(120596)

(1)

and vice versa Although both 119862lowast119866(120596) and 119872lowast(120596) can be

plotted onto the Cole-Cole diagram as the complex quantity119911 only the former situation is to be described in the followingdiscussion to avoid redundancy

22Modulus Grids Superimposed onto the Cole-Cole DiagramWhen the modulus is decomposed into real and imaginaryparts as119872lowast(120596) = 1198721015840 + 11989411987210158401015840 the real and imaginary parts ofthe corresponding compliance (119862lowast

119866(120596) = 119862

1015840

119866minus 11989411986210158401015840

119866) are

1198621015840

119866=

1198721015840

11987210158402

+119872101584010158402

(2a)

11986210158401015840

119866=

11987210158401015840

11987210158402

+119872101584010158402

(2b)

If the compliances at various frequencies are plotted oncomplex plane with 1198621015840

119866along the horizontal axis and 11986210158401015840

119866

along the vertical axis the well-known Cole-Cole diagramfor compliance 119862lowast

119866(120596) is yielded Now consider how iso-

1198721015840 and iso-11987210158401015840 the lines along which 1198721015840 and 11987210158401015840 are

constant appear in the Cole-Cole diagram Sets of these iso-1198721015840 and iso-11987210158401015840 lines give a coordinate grid for the modulus

119872lowast

(120596) superimposed onto the Cole-Cole diagram for thecompliance 119862lowast

119866(120596) These 1198721015840 and 11987210158401015840 grids enable us to

convert compliance to modulus graphically once a trajectoryon the Cole-Cole diagram is given

To find the iso-1198721015840 line (the trajectory along which1198721015840 isconstant) for given1198721015840 substitute1198721015840 = 1198981015840 and11987210158401015840 = 119905

1in

(2a) and (2b) where1198981015840 is just a constant and 1199051is a parameter

to be eliminated

1198621015840

119866=

1198981015840

11989810158402

+ 1199051

2

(3a)

11986210158401015840

119866=

1199051

11989810158402

+ 1199051

2

(3b)

Equation (3a) yields

(11989810158402

+ 1199051

2

)1198621015840

119866= 1198981015840

(3a1015840)

and dividing (3b) by (3a) yields

1199051= (

11986210158401015840

119866

1198621015840

119866

)1198981015840

(3c)

Substituting (3c) for (3a1015840) to eliminate the parameter 1199051 we

obtain a quadratic equation in 1198621015840119866and 11986210158401015840

119866 Then completing

square leads to an equation for a circle

(1198621015840

119866minus

1

21198981015840

)

2

+ 11986210158401015840

119866

2

= (

1

21198981015840

)

2

(4)

The radius of the circle is 121198981015840 and the center is at (1198621015840119866 11986210158401015840

119866) =

(121198981015840

0) In addition one point of this circle is fixed atthe origin irrespective of the radius As the value of thereal part of the modulus 1198981015840 is decreased the size of thecircle increases Therefore iso-1198721015840 lines on the Cole-Colediagram of compliance appear as circles whose center is onthe horizontal axis with one point fixed at the origin as shownin Figure 1

The iso-11987210158401015840 line (the trajectories along which 11987210158401015840 isconstant) for given11987210158401015840 can be found similarly by substituting

International Journal of Spectroscopy 3

Table 1 Summary of the definitions of compliances and moduli 119863(120596) and 119864(120596) are electric displacement and electric field respectively119890119909119910(120596) and 120590

119909119910(120596) are shear strain and shear stress 119890

119909119909(120596) and 120590

119909119909(120596) are longitudinal strain and longitudinal stress 119881(120596) and 119868(120596) are

voltage and current

Compliance 119862lowast119866(120596) Interrelation Modulus119872lowast(120596)

Permittivity 120576lowast (120596) Electric modulus119872lowast120576(120596)

119863 (120596) = 120576lowast

(120596) 119864 (120596) 120576lowast

(120596)119872lowast

120576(120596) = 1 119864 (120596) = 119872

lowast

120576(120596)119863 (120596)

Shear compliance 119869lowast (120596) Shear modulus 119866lowast (120596)119890119909119910(120596) = 119869

lowast

(120596) 120590119909119910(120596) 119869

lowast

(120596) 119866lowast

(120596) = 1 120590119909119910(120596) = 119866

lowast

(120596) 119890119909119910(120596)

Tensile compliance119863lowast (120596) Tensile modulus 119864lowast (120596)119890119909119909(120596) = 119863

lowast

(120596) 120590119909119909(120596) 119863

lowast

(120596) 119864lowast

(120596) = 1 120590119909119909(120596) = 119864

lowast

(120596) 119890119909119909(120596)

Impedance 119885lowast (120596) Admittance 119884lowast (120596)119881 (120596) = 119885

lowast

(120596) 119868 (120596) 119885lowast

(120596) 119884lowast

(120596) = 1 119868 (120596) = 119884lowast

(120596)119881 (120596)

20

20

10

1000

C998400998400 G

C998400G

M998400998400= 012

M998400998400= 008

M998400998400= 016

M998400998400= 02

M998400=001

M998400= 004

M998400998400= 004

M998400998400= 001

M998400=008

M998400=012

M998400=016

M998400=02

(i)

(ii)

Figure 1 Modulus grid superimposed on a Cole-Cole diagramThegrid is composed of iso-1198721015840 and iso-11987210158401015840 lines along which values of1198721015840 and11987210158401015840 are constant respectively (see Section 22 for the detail)

Orange dash dotted line circles indicate iso-11987210158401015840 lines whereas greendashed circles indicate iso-1198721015840 lines Full line (i) indicates a trajectorycircumscribed to an iso-11987210158401015840 line exhibiting maximum 119872

10158401015840 at thetangent point Full line (ii) indicates a trajectory inscribed to an iso-11987210158401015840 line exhibiting minimum119872

10158401015840 at the tangent point

1198721015840

= 1199052and11987210158401015840 = 11989810158401015840 into (2a) and (2b) Iso-11987210158401015840 lines are

circles

1198621015840

119866

2

+ (11986210158401015840

119866minus

1

211989810158401015840

)

2

= (

1

211989810158401015840

)

2

(5)

with radius 1211989810158401015840 and center (1198621015840119866 11986210158401015840

119866) = (0 12119898

10158401015840

) on thevertical axis Again one point of this circle lies at the originregardless of the radius and the size of the circle increases as11989810158401015840decreases (see Figure 1)The calculated iso-1198721015840 and iso-11987210158401015840 lines form coordinate

grids superimposed on the Cole-Cole diagram of the com-pliance All these circular grid lines pass through the originand increase in size as the value on the grid line decreases

This modulus grid enables us to predict the appearance of themodulus spectrum for a given compliance trajectory on theCole-Cole diagram If the modulus is plotted on Cole-Colediagram instead of compliance similar circular 1198621015840

119866and 11986210158401015840

119866

gridlines are obtained Therefore the following discussion isrelevant in both cases

The Cauchy-Riemann theorem tells us that an arbitrarymapping via an analytic complex function is a conformalmapping in which the angle encompassed by an infinites-imally small triangle is preserved [13] Conversion frommodulus to compliance is also conformal mapping sincethe complex function employed 119891(119911) = 1119911 is analyticexcept at the origin It is obvious that iso-1198721015840 and iso-11987210158401015840lines are orthogonal to one another on the complex planeof 119872lowast According to the Cauchy-Riemann theorem thisrelation is conserved under mapping by the analytic function119891(119911) = 1119911 and therefore iso-1198721015840 and iso-11987210158401015840 curves stillcross orthogonally even on the complex plane of119862lowast

119866 It can be

seen in Figure 1 that two arbitrary intersecting circles alwayscross orthogonally

3 Examples for Applications

With the aid of the circular modulus grids obtained aboveone can obtain graphically the appearance of the modulusspectrum from the trajectory on the Cole-Cole diagramIn this section I will present several applications for themodulus grids

One of the major applications for the modulus grid is tofind a peak in the modulus spectrum As is clear from thecircular shape of the modulus grids maxima or minima of1198721015840 and11987210158401015840 are at points where a circular modulus grid line

is tangent to the trajectory of the compliance (lines (i) and (ii)in Figure 1) Rigorously speaking frequency derivatives of1198721015840or11987210158401015840 are zero at the point where an1198721015840-grid line or an11987210158401015840-grid line is tangent to the compliance trajectory respectivelyMoreover if the trajectory is circumscribed to the circulargrid line (line (i) in Figure 1) the frequency dependenceof 1198721015840 or 11987210158401015840 is concave down whereas if the trajectoryis inscribed to the circular grid (line (ii) in Figure 1) thefrequency dependence of1198721015840 or11987210158401015840 is concave up By usingthese properties of the modulus grids it is possible to find


20

20

15

15

10

1000

5

5

C998400998400 G

Peak in M998400998400

Peak in C998400998400G

DebyeCinfin = 4

ΔC = 16

M998400998400= 01

M998400=005

M998400= 025

120596 = infin 120596 = 0

C998400G

Figure 2 Cole-Cole diagram for a Debye function (119862infin= 4 and

Δ119862 = 16) (Black full circle) An iso-11987210158401015840 circle tangent to thetrajectory of the Debye function is shown as an Orange dash dottedline circle (11987210158401015840 = 01) Two iso-1198721015840 circles tangent to theDebye circleare also depicted as green dashed lines (1198721015840 = 025 and1198721015840 = 005)The left-hand end of the trajectory corresponds to infinite frequencywhereas the right-hand end corresponds to infinitesimal frequency

peaks ofmoduli graphically and even quantitatively in severalspecial situations as shown below

As the simplest example Figure 2 displays a Cole-Colediagram of a Debye function representing the Fourier-Laplace transformation of a single exponential relaxation

119862lowast

119866(120596) = 119862

infin+

Δ119862

1 + (119894120596120591)

(6)

The trajectory of the Debye function describes a semicircleon the Cole-Cole diagram given by

[1198621015840

119866minus (119862infin+

Δ119862

2

)

2

] + 11986210158401015840

119866

2

= (

Δ119862

2

)

2

(7)

Apparently the peak in the compliance spectrum is at thetop of the semicircle for this trajectory To find the peak inthe modulus spectrum a circular11987210158401015840-grid is drawn keepingits center along the vertical axis and a point fixed to theorigin (see the orange dash dotted line in Figure 2) Whenthe grid line is tangent to the semicircle of the trajectory ofthe Debye relaxation the peak in the modulus spectrum isfound at the tangent pointThen the value of the peak height(radius of the circular 11987210158401015840-grid) can be straightforwardlycalculated based on a geometrical analysis As is clear fromFigure 2 the peak in the modulus spectrum is always on theleft-hand side of the semicircle of the trajectory Taking intoaccount that the locus on the Debye semicircle moves fromright to left with increasing frequency we can graphicallyconfirm the well-known statement that the peak is at higherfrequency in the modulus spectrum than the compliance

25

25

20

20

15

15

10

1000

5

5

C998400998400 G

C998400G

C998400G

Peak in M998400998400

at C998400G = C

998400998400G

C998400998400G

ClowastG = 10 minus i120590120596

120596 = infin

120596 = 0

M998400998400= 005

Figure 3 Cole-Cole diagram for 119862lowast119866(120596) = 119862

infinminus 119894120590120596 (119862

infin= 10)

(black full straight line) An iso-11987210158401015840 circle touching the trajectory ofthe compliance is shown as an Orange dash dotted line circle (11987210158401015840 =005) Since both 1198621015840

119866and 11986210158401015840

119866have to coincide with the radius of the

iso-11987210158401015840 circle the tangent point satisfies 1198621015840119866= 11986210158401015840

119866

spectrum Considering1198721015840-grid line which is tangent to thetrajectory of theDebye relaxation two situations are possibleone circumscribes to it and the other inscribes (see greendashed lines in Figure 2) The tangent point of the former isat infinite frequency and indicates that there is a maximumin1198721015840 at this point while that of the latter is at zero frequencyand it is seen that a minimum in1198721015840 shows up there

As a second example let us find an 11987210158401015840-peak for thecomplex compliance composed of the constants 1198621015840

119866and the

11986210158401015840

119866inversely proportional to the frequency (Figure 3)

119862lowast

119866(120596) = 119862

infinminus 119894

120590

120596

(8)

where 119862infinand 120590 are parameters independent of frequency If

the generalized compliance represents complex permittivitythis represents a parallel circuit of a capacitor and resistorwith constant permittivity and dc conductivity If the com-pliance represents mechanical compliance this represents aMaxwell model [2] of a spring and a dashpot connected inseries As obvious from the straight line of the trajectorydrawn on the Cole-Cole diagram shown in Figure 3 thecompliance spectrum exhibits neither maxima nor minimawithin finite frequency ranges On the other hand theexistence of a circular11987210158401015840-grid tangent to the straight line ofthe trajectory indicates that a peak shows up in the modulusspectrum From the geometric restriction on the11987210158401015840-grid itis graphically proved that the peak of the modulus spectrumof (8) occurs at the frequency where 1198621015840

119866= 11986210158401015840

119866

Before proceeding with further examples I will state atheorem that is useful for finding modulus peaks When thetrajectory on the Cole-Cole diagram has a circular shape


0

0

C998400998400 G

B

P

O

O998400

A

C998400G

Figure 4 Proof that the straight line from the top (119861) of thetrajectory circle with center 119874 to the origin (119860) passes through thetangent point (119875) between the iso-11987210158401015840 circle with center 1198741015840 and thetrajectory circle

the peak position of the modulus spectrum can be foundvia a routine procedure irrespective of the position of thecenter of the circle Such circular trajectories are found ina variety of cases not only the Debye function but alsothe Cole-Cole function (CC function) [14] and Van Vleck-Weisskopf function (VW function) [15ndash17] The statement ofthis theorem is as follows

Theorem 1 When the trajectory on the Cole-Cole diagram iscircular the peak of11987210158401015840 appears at the intersection of the circleand the straight line running from the top of the circle to theorigin irrespective of the position of the circle center

Proof The proof which is based on elementary geometryis straightforward Consider a circle centered at arbitraryposition 119874 as drawn in Figure 4 and suppose that anothercircle with center1198741015840 and passing through the origin is tangentto the first circle at 119875 The line segment 1198741015840119874 cuts both circlesat the tangent point 119875 Since 1198601198741015840 119861119874 alternate-interiorangles are equal ang1198601198741015840119875 = ang119861119874119875 Therefore the isoscelestrianglesR1198601198741015840119875 andR119861119874119875 are homologous to one another Itfollows that ang1198741015840119875119860 = ang119874119875119861 Since vertically opposite anglesang1198741015840

119875119860 and ang119874119875119861 are equal line segments 119860119875 and 119875119861 turnout to be a continuous straight line119860119861Therefore the straightline running from 119861 to 119860 crosses circle with center 119874 at thecommon tangent point

Two applications of this theorem are presented as followsAlthough it is less important an analogous theorem for the1198721015840-grid can also be proved ldquoWhen the trajectory on the

Cole-Cole diagram is circular a peak of 1198721015840 appears at theintersection of the circle and the straight line running from

20

10

0

C998400998400 G

20100

C998400G

Peak in M998400998400

Peak in C998400998400G

120596 = infin 120596 = 0

VW

CC

M998400998400= 0146

M998400998400= 0066

Figure 5 Cole-Cole diagram for Cole-Cole (CC) and Van Vleck-Weisskopf (VW) functions which are represented as off-centercircles Dark colored lines and symbols are for the VW functionwhile light colored ones are for the CC function Full curvecircles stand for trajectory and dash dotted circles indicate iso-11987210158401015840contours The left-hand end of a trajectory corresponds to infinitefrequency whereas the right-hand end corresponds to infinitesimalfrequency

the extreme right-hand point on the circle to the originirrespective of the position of the circle centerrdquo

The CC function [14] is given by

119862lowast

119866(120596) = 119862

infin+

Δ119862

1 + (119894120596120591)120573

(9)

where 120573 is a parameter that controls the broadness ofthe spectrum This spectrum function also has a circulartrajectory on the Cole-Cole diagram although the center ofthe circle is in the lower half planeThe trajectory on theCole-Cole diagram is

[1198621015840


Δ119862

2

)]

2

+ [11986210158401015840

119866minus (minus

Δ119862

2

cot(120573120587

2

))]

2

= [

Δ119862

2

csc(120573120587

2

)]

2

(10)

Figure 5 shows an example of such a trajectory (labeledldquoCCrdquo) Invoking the theorem the peak position of themodulus spectrum can be found straightforwardly As shownin Figure 5 by drawing a straight line from the top of the circleto the origin the peak in modulus spectrum is found to beat the crossing point It should be stressed that the peak wasfound without drawing an11987210158401015840-grid


A similar analysis is also possible for the VW functionwhich represents a vibrational mode [15ndash17] The VW func-tion is

119862lowast

119866(120596) = 119862

infin+ (

Δ119862

2

)[

1 + 1198941205960120591

1 + 119894 (120596 + 1205960) 120591

+

1 minus 1198941205960120591

1 + 119894 (120596 minus 1205960) 120591

]

= 119862infin+ int

infin

0

119889119905119890minus119894120596119905

(minus

119889

119889119905

)Δ119862119890minus119905120591 cos (120596

0119905)

(11)

Although the trajectory of a VW function on the Cole-Colediagram slightly deviates from a perfect circle in proximityto the horizontal axis it is approximately circular when thevibrational frequency 120596

0is much faster than the damping

rate 1205960120591 ≫ 1 and | ln(120596120596

119898)| ≪ 1

[1198621015840


Δ119862

2

)]

2

+ [11986210158401015840

119866minus (

Δ119862

2

sdot

(120596119898120591)2

minus 1

2 (120596119898120591)2

minus 1

120596119898120591)]

2

asymp [

Δ119862

2

sdot

(120596119898120591)2

2 (120596119898120591)2

minus 1

120596119898120591]

2

(12)

where120596119898is the peak angular frequency in compliance given

by 120596119898120591 = radic(120596

0120591)2+ 1 Then the approximated circle is

centered in the upper half plane on the Cole-Cole diagramrepresenting the characteristic sigmoidal shape of the 1198621015840

119866

spectrum around the vibrational frequency Again the peakin the modulus spectrum can be found by drawing a straightline from top of the circular trajectory to the origin as shownin Figure 5 (labeled ldquoVWrdquo) Then the peak is found at thepoint where the straight line crosses the circular trajectoryFurthermore it can be clearly seen from the Cole-Colediagram that the real part of the modulus1198721015840 also exhibitsmaxima and minima that are similar to those of the 1198621015840

119866

spectrum since it is possible to draw circular1198721015840-grid linescircumscribed and inscribed to the VW trajectory (thesecircles are not shown in Figure 5) This is not the case forthe CC function as 1198721015840-grid lines cannot be tangent to thefunctionrsquos trajectory within the upper half of the Cole-Colediagram

The circular modulus grids tell us graphically that thepeak of 11987210158401015840 always occurs at a higher frequency than thatof 11986210158401015840119866since circular 11987210158401015840-grid lines that are tangent to the

trajectory always lie on the left-hand side of the trajectoryif it has tangent point This is one of the well-knownrelations between peaks in the modulus and the complianceAdditionally the position of the tangent point relative to thecenter of the circular trajectory depends on the values of119862infin

and Δ119862 With increasing 119862infin

(moving the circle to theright while keeping its size unchanged) the tangent pointapproaches the peak of 11986210158401015840

119866 meaning that the frequency of

11987210158401015840 peak approaches that of the 11986210158401015840

119866peak As Δ119862 is increased

(making the circle bigger while keeping its left-hand endpoint fixed) on the other hand the tangent point approaches

the point where (1198621015840119866 11986210158401015840

119866) = (119862

infin 119862infin) resulting in infinite

frequency Furthermore information about loss tangent canbe also associated with this graphical method On the Cole-Cole diagram the loss tangent tan 120575 = 11986210158401015840

1198661198621015840

119866 corresponds

to the slope of a line from the origin to an arbitrary pointon the trajectory of compliance The maximum of the losstangent is at the point where a straight line running from theorigin is tangent to the trajectory circle Since the straightline used to find the modulus peak (119860119861 in Figure 4) has alower slope than the tangent to the trajectory circle the peakof the loss tangent lies between the peaks of the modulusand the compliance (119875 and 119861 in Figure 4 resp) Thuswe have reproduced several well-known relations betweenthe peaks of modulus and compliance based on graphicalconsiderations It should be noted that if the modulus isplotted as Cole-Cole diagram instead of the compliancetypical trajectories run in the opposite direction from leftto right with increasing frequency Then a tangent point onthe left-hand side of the trajectory means that peak of thecompliance appears at a lower frequency than that ofmoduluspeak which is consistent with the previous results

Although only circular trajectories have been consideredabove (straight line is infinitely large circle) somenoncircularfunctions are also widely employed to reproduce experimen-tal results for example Cole-Davidson (CD) function [18]and Havriliak-Negami (HN) function [19] Reflecting theirasymmetric distribution of the relaxation times trajectoriesof CD and HN functions on Cole-Cole diagram draw asym-metrically skewed arcs In these situations the theorem tofind modulus peak described above is no longer applicablebecause it is limited only to circular trajectories Hence thepresent graphical method enables us to make only qualita-tive analysis for noncircular trajectories By drawing 11987210158401015840-grid lines qualitative relation between peaks of compliancemodulus and loss tangent can be straightforwardly foundMore generally one can predict the qualitative shape of themodulus from a complicated spectrum as explained in whatfollows In Figure 6 superposition of two HN functions isplotted as Cole-Cole diagram (its frequency dependence isalso shown in the inset) As is clear from the diagram the HNprocess at higher frequency exhibits only a shoulder insteadof a peak in the compliance representation By drawing two11987210158401015840-grid lines which tangent to the trajectory (orange dash

dotted lines in Figure 6) appearance of two peaks in themodulus spectrum can be recognized Since tangent line atlower frequency is nearly parallel to the trajectory for largerange it is estimated that the corresponding peak in themodulus is not well resolved (it is peak but close to shoulder)Such qualitative discussion is always possible irrespectively ofthe shape of spectra and provides clear understanding on theinterrelation between processes in the modulus and those ofcompliance

4 Conclusions

General properties of modulus grids (formed from iso-1198721015840and iso-11987210158401015840 lines) on the Cole-Cole diagram for compliancewere studied in detail Both iso-1198721015840 and iso-11987210158401015840 lines are


35

35

30

30

25

25

20

20

15

15

10

10

5

500

C998400998400 G

C998400998400 G

C998400998400G

C998400G

Peak in M998400998400

M998400998400

HN1

HN1

HN2

HN2

120596 (au)

Figure 6 Cole-Cole diagram for the compliance spectrum com-posed of superposition of two Havriliak-Negami (HN) processesThe compliance spectrum and corresponding modulus spectrumare shown in the inset Full bold line stands for the trajectory of thetotal compliance and full thin lines indicate each component (HN1and HN2) Dash dotted circles indicate iso-11987210158401015840 contours

circular with one point fixed at the origin and increasein radius as the values of 1198721015840 and 119872

10158401015840 are decreasedThis modulus grid superimposed on the Cole-Cole diagramallows us to convert graphically from compliance tomodulusMaxima and minima of 1198721015840 and 11987210158401015840 correspond to thetangent points where 1198721015840 and 11987210158401015840 grid lines respectivelymeet the trajectory of compliance tangentially Thereforethe peak of the modulus can be found graphically from thetrajectory When the trajectory is a circle (Debye Cole-Coleor Van Vleck-Weisskopf functions) the exact position of themodulus peak can be found by drawing a straight line fromtop of the circle to the origin and taking the point at which thestraight line intersectswith the circleWith the aid ofmodulusgrids shapes of modulus spectra can be graphically derivedonce the Cole-Cole plot of the compliance has been obtainedThis procedure enables us to relate compliance with modulusmuch easier and makes their interrelation clearer

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author thanks Dr Alexei P Sokolov Dr Yuri Feldmanand Dr Ivan I Popov for reading the manuscript and usefulcomments

References

[1] F Kremer and A Schonhals Broadband Dielectric SpectroscopySpringer Berlin Germany 2002

[2] G Stroble The Physics of Polymers Springer Berlin Germany2nd edition 2002

[3] E Barsoukov and J R Macdonald Impedance SpectroscopyTheory Experiment and Applications John Wiley amp SonsHoboken NJ USA 2005

[4] H Wagner and R Richert ldquoDielectric relaxation of the electricfield in poly(vinyl acetate) a time domain study in the range10minus3-106 srdquo Polymer vol 38 no 2 pp 255ndash261 1997

[5] H Wagner and R Richert ldquoMeasurement and analysis of time-domain electric field relaxation the vitreous ionic conductor 04Ca(NO

3)2minus06 KNO

3rdquo Journal of Applied Physics vol 85 no 3

pp 1750ndash1755 1999[6] M Paluch Z Wojnarowska and S Hensel-Bielowka ldquoHetero-

geneous dynamics of prototypical ionic glass ckn monitored byphysical agingrdquo Physical Review Letters vol 110 no 1 Article ID015702 2013

[7] K Schroter and E Donth ldquoViscosity and shear response at thedynamic glass transition of glycerolrdquo The Journal of ChemicalPhysics vol 113 no 20 pp 9101ndash9108 2000

[8] T Mizuhashi ldquoTheory of four-terminals impedance transfor-mation circuit and matching circuitrdquo Journal of the Institute ofElectrical Communication Engineers of Japan vol 177 pp 1053ndash1058 1937

[9] P H Smith ldquoTransmission line calculatorrdquo Electronics vol 12pp 29ndash31 1939

[10] PH SmithElectronic Applications of the Smith ChartMcGraw-Hill New York NY USA 1969

[11] K S Cole ldquoElectric impedance of suspensions of spheresrdquoJournal of General Physiology vol 12 pp 29ndash36 1928

[12] K S Cole ldquoElectric phase angle of cell membranesrdquo Journal ofGeneral Physiology vol 15 pp 641ndash649 1932

[13] L Ahlfors Complex Analysis McGraw-Hill New York NYUSA 3rd edition 1979

[14] K S Cole and R H Cole ldquoDispersion and absorption indielectrics I Alternating current characteristicsrdquoThe Journal ofChemical Physics vol 9 no 4 pp 341ndash351 1941

[15] J H van Vleck and V F Weisskopf ldquoOn the shape of collision-broadened linesrdquo Reviews of Modern Physics vol 17 no 2-3 pp227ndash236 1945

[16] H Frohlich Theory of Dielectrics Oxford University PressLondon UK 1958

[17] R Kubo M Toda and N Hashitsume Statistical Physics IINonequilibrium Statistical Mechanics Springer Berlin Ger-many 2nd edition 1991

[18] D W Davidson and R H Cole ldquoDielectric relaxation inglycerol propylene glycol and n-propanolrdquo The Journal ofChemical Physics vol 19 no 12 pp 1484ndash1490 1951

[19] S Havriliak and S Negami ldquoA complex plane representationof dielectric and mechanical relaxation processes in somepolymersrdquo Polymer vol 8 pp 161ndash210 1967

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

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International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


Chromatography Research International


Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


depending on the definition) on the complex plane wherecomplex permittivity compliance and impedance are oftenchosen as 119911 Plotting 119911 varying the frequency yields character-istic trajectory reflecting the phenomenon Remarkably someelementally models such as single exponential relaxationfunction in dielectric spectra and the behavior of a parallel RCcircuit in impedance spectra describe circular trajectories asfrequency is varied An analogous diagram is also employedto plot transfer functions and is called Nyquist diagram Theparameters of the transfer function can be determined andthe stability of a system can be analyzed thereby with it

In the present paper I propose a straightforward methodto convert the spectrum of arbitrary complex quantity 119911 intoits reciprocal 1119911 based on the graphical procedure on Cole-Cole diagramThereby interrelation between 119911 (permittivitycompliance and impedance) spectra and 1119911 (moduli admit-tance) can be graphically recognized Moreover it also opensthe route to the geometric method to assess themathematicalproperties of spectra

2 Theory

21 Definition We consider a pair of reciprocally relatedcomplex quantities 119911 and 1119911 Plotting 119911 or its complexconjugate in the complex plane yields a Cole-Cole diagramLoci of 119911 trace a characteristic trajectory on the diagram as aninternal parameter is varied (typically frequency) reflectingthe phenomenon represented by 119911

Such pairs of reciprocally related quantities can be oftenfound in spectroscopic measurements Complex permittivity120576lowast

(120596) and electricmodulus119872lowast120576(120596) are respectivelymeasures

of how easy and how difficult it is to deform a materialelectricallyThe former is defined as the electric displacement119863(120596) for a unit electric field 119864(120596) 119863(120596) = 120576

lowast

(120596)119864(120596)whereas the latter is defined as the electric field generatedwhen a unit electric displacement is imposed 119864(120596) =

119872lowast

120576(120596)119863(120596) Obviously 120576lowast(120596)119872lowast

120576(120596) = 1 meaning that

each is the reciprocal of the other (119911 and 1119911) Similarlymechanical compliance (shear 119869lowast(120596) or tensile 119863lowast(120596)) andmechanical modulus (shear 119866lowast(120596) or tensile 119864lowast(120596)) arereciprocally related as 119869lowast(120596)119866lowast(120596) = 1 and 119863lowast(120596)119864lowast(120596) =1 Impedance 119885lowast(120596) and admittance 119884lowast(120596) also satisfy thereciprocal relation 119885lowast(120596)119884lowast(120596) = 1

From the viewpoint of the similarity of the typicaltrajectories on a Cole-Cole diagram it is convenient todefine generalized compliance119862lowast

119866(120596) representing permittiv-

ity mechanical compliance and impedance whereas electricmodulus mechanical modulus and admittance are repre-sented by generalized modulus 119872lowast(120596) as summarized inTable 1 The generalized modulus is the reciprocal of thegeneralized compliance

119872lowast

(120596) =

1

119862lowast

119866(120596)

(1)

and vice versa Although both 119862lowast119866(120596) and 119872lowast(120596) can be

plotted onto the Cole-Cole diagram as the complex quantity119911 only the former situation is to be described in the followingdiscussion to avoid redundancy

22Modulus Grids Superimposed onto the Cole-Cole DiagramWhen the modulus is decomposed into real and imaginaryparts as119872lowast(120596) = 1198721015840 + 11989411987210158401015840 the real and imaginary parts ofthe corresponding compliance (119862lowast

119866(120596) = 119862

1015840

119866minus 11989411986210158401015840

119866) are

1198621015840

119866=

1198721015840

11987210158402

+119872101584010158402

(2a)

11986210158401015840

119866=

11987210158401015840

11987210158402

+119872101584010158402

(2b)

If the compliances at various frequencies are plotted oncomplex plane with 1198621015840

119866along the horizontal axis and 11986210158401015840

119866

along the vertical axis the well-known Cole-Cole diagramfor compliance 119862lowast

119866(120596) is yielded Now consider how iso-

1198721015840 and iso-11987210158401015840 the lines along which 1198721015840 and 11987210158401015840 are

constant appear in the Cole-Cole diagram Sets of these iso-1198721015840 and iso-11987210158401015840 lines give a coordinate grid for the modulus

119872lowast

(120596) superimposed onto the Cole-Cole diagram for thecompliance 119862lowast

119866(120596) These 1198721015840 and 11987210158401015840 grids enable us to

convert compliance to modulus graphically once a trajectoryon the Cole-Cole diagram is given

To find the iso-1198721015840 line (the trajectory along which1198721015840 isconstant) for given1198721015840 substitute1198721015840 = 1198981015840 and11987210158401015840 = 119905

1in

(2a) and (2b) where1198981015840 is just a constant and 1199051is a parameter

to be eliminated

1198621015840

119866=

1198981015840

11989810158402

+ 1199051

2

(3a)

11986210158401015840

119866=

1199051

11989810158402

+ 1199051

2

(3b)

Equation (3a) yields

(11989810158402

+ 1199051

2

)1198621015840

119866= 1198981015840

(3a1015840)

and dividing (3b) by (3a) yields

1199051= (

11986210158401015840

119866

1198621015840

119866

)1198981015840

(3c)

Substituting (3c) for (3a1015840) to eliminate the parameter 1199051 we

obtain a quadratic equation in 1198621015840119866and 11986210158401015840

119866 Then completing

square leads to an equation for a circle

(1198621015840

119866minus

1

21198981015840

)

2

+ 11986210158401015840

119866

2

= (

1

21198981015840

)

2

(4)

The radius of the circle is 121198981015840 and the center is at (1198621015840119866 11986210158401015840

119866) =

(121198981015840

0) In addition one point of this circle is fixed atthe origin irrespective of the radius As the value of thereal part of the modulus 1198981015840 is decreased the size of thecircle increases Therefore iso-1198721015840 lines on the Cole-Colediagram of compliance appear as circles whose center is onthe horizontal axis with one point fixed at the origin as shownin Figure 1

The iso-11987210158401015840 line (the trajectories along which 11987210158401015840 isconstant) for given11987210158401015840 can be found similarly by substituting




119909119909(120596) and 120590


voltage and current



119863 (120596) = 120576lowast

(120596) 119864 (120596) 120576lowast

(120596)119872lowast

120576(120596) = 1 119864 (120596) = 119872

lowast

120576(120596)119863 (120596)


lowast

(120596) 120590119909119910(120596) 119869

lowast

(120596) 119866lowast

(120596) = 1 120590119909119910(120596) = 119866

lowast

(120596) 119890119909119910(120596)


lowast

(120596) 120590119909119909(120596) 119863

lowast

(120596) 119864lowast

(120596) = 1 120590119909119909(120596) = 119864

lowast

(120596) 119890119909119909(120596)


lowast

(120596) 119868 (120596) 119885lowast

(120596) 119884lowast

(120596) = 1 119868 (120596) = 119884lowast

(120596)119881 (120596)

20

20

10

1000

C998400998400 G

C998400G

M998400998400= 012

M998400998400= 008

M998400998400= 016

M998400998400= 02

M998400=001

M998400= 004

M998400998400= 004

M998400998400= 001

M998400=008

M998400=012

M998400=016

M998400=02

(i)

(ii)





1198721015840


circles

1198621015840

119866

2

+ (11986210158401015840

119866minus

1

211989810158401015840

)

2

= (

1

211989810158401015840

)

2

(5)


119866) = (0 12119898

10158401015840




119866and 11986210158401015840

119866



119866 It can be







20

20

15

15

10

1000

5

5

C998400998400 G

Peak in M998400998400

Peak in C998400998400G

DebyeCinfin = 4

ΔC = 16

M998400998400= 01

M998400=005

M998400= 025

120596 = infin 120596 = 0

C998400G





119862lowast

119866(120596) = 119862

infin+

Δ119862

1 + (119894120596120591)

(6)


[1198621015840


Δ119862

2

)

2

] + 11986210158401015840

119866

2

= (

Δ119862

2

)

2

(7)


25

25

20

20

15

15

10

1000

5

5

C998400998400 G

C998400G

C998400G

Peak in M998400998400

at C998400G = C

998400998400G

C998400998400G


120596 = infin

120596 = 0

M998400998400= 005


infinminus 119894120590120596 (119862

infin= 10)


119866and 11986210158401015840



119866



119866and the

11986210158401015840


119862lowast

119866(120596) = 119862

infinminus 119894

120590

120596

(8)



119866= 11986210158401015840

119866



0

0

C998400998400 G

B

P

O

O998400

A

C998400G








20

10

0

C998400998400 G

20100

C998400G

Peak in M998400998400

Peak in C998400998400G

120596 = infin 120596 = 0

VW

CC

M998400998400= 0146

M998400998400= 0066




119862lowast

119866(120596) = 119862

infin+

Δ119862

1 + (119894120596120591)120573

(9)


[1198621015840


Δ119862

2

)]

2

+ [11986210158401015840

119866minus (minus

Δ119862

2

cot(120573120587

2

))]

2

= [

Δ119862

2

csc(120573120587

2

)]

2

(10)




119862lowast

119866(120596) = 119862

infin+ (

Δ119862

2

)[

1 + 1198941205960120591

1 + 119894 (120596 + 1205960) 120591

+

1 minus 1198941205960120591

1 + 119894 (120596 minus 1205960) 120591

]

= 119862infin+ int

infin

0

119889119905119890minus119894120596119905

(minus

119889

119889119905

)Δ119862119890minus119905120591 cos (120596

0119905)

(11)



rate 1205960120591 ≫ 1 and | ln(120596120596

119898)| ≪ 1

[1198621015840


Δ119862

2

)]

2

+ [11986210158401015840

119866minus (

Δ119862

2

sdot

(120596119898120591)2

minus 1

2 (120596119898120591)2

minus 1

120596119898120591)]

2

asymp [

Δ119862

2

sdot

(120596119898120591)2

2 (120596119898120591)2

minus 1

120596119898120591]

2

(12)


by 120596119898120591 = radic(120596



119866


119866










the point where (1198621015840119866 11986210158401015840

119866) = (119862



1198661198621015840

119866 corresponds




4 Conclusions



35

35

30

30

25

25

20

20

15

15

10

10

5

500

C998400998400 G

C998400998400 G

C998400998400G

C998400G

Peak in M998400998400

M998400998400

HN1

HN1

HN2

HN2

120596 (au)






Acknowledgments


References






3)2minus06 KNO


























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




119909119909(120596) and 120590


voltage and current



119863 (120596) = 120576lowast

(120596) 119864 (120596) 120576lowast

(120596)119872lowast

120576(120596) = 1 119864 (120596) = 119872

lowast

120576(120596)119863 (120596)


lowast

(120596) 120590119909119910(120596) 119869

lowast

(120596) 119866lowast

(120596) = 1 120590119909119910(120596) = 119866

lowast

(120596) 119890119909119910(120596)


lowast

(120596) 120590119909119909(120596) 119863

lowast

(120596) 119864lowast

(120596) = 1 120590119909119909(120596) = 119864

lowast

(120596) 119890119909119909(120596)


lowast

(120596) 119868 (120596) 119885lowast

(120596) 119884lowast

(120596) = 1 119868 (120596) = 119884lowast

(120596)119881 (120596)

20

20

10

1000

C998400998400 G

C998400G

M998400998400= 012

M998400998400= 008

M998400998400= 016

M998400998400= 02

M998400=001

M998400= 004

M998400998400= 004

M998400998400= 001

M998400=008

M998400=012

M998400=016

M998400=02

(i)

(ii)





1198721015840


circles

1198621015840

119866

2

+ (11986210158401015840

119866minus

1

211989810158401015840

)

2

= (

1

211989810158401015840

)

2

(5)


119866) = (0 12119898

10158401015840




119866and 11986210158401015840

119866



119866 It can be







20

20

15

15

10

1000

5

5

C998400998400 G

Peak in M998400998400

Peak in C998400998400G

DebyeCinfin = 4

ΔC = 16

M998400998400= 01

M998400=005

M998400= 025

120596 = infin 120596 = 0

C998400G





119862lowast

119866(120596) = 119862

infin+

Δ119862

1 + (119894120596120591)

(6)


[1198621015840


Δ119862

2

)

2

] + 11986210158401015840

119866

2

= (

Δ119862

2

)

2

(7)


25

25

20

20

15

15

10

1000

5

5

C998400998400 G

C998400G

C998400G

Peak in M998400998400

at C998400G = C

998400998400G

C998400998400G


120596 = infin

120596 = 0

M998400998400= 005


infinminus 119894120590120596 (119862

infin= 10)


119866and 11986210158401015840



119866



119866and the

11986210158401015840


119862lowast

119866(120596) = 119862

infinminus 119894

120590

120596

(8)



119866= 11986210158401015840

119866



0

0

C998400998400 G

B

P

O

O998400

A

C998400G








20

10

0

C998400998400 G

20100

C998400G

Peak in M998400998400

Peak in C998400998400G

120596 = infin 120596 = 0

VW

CC

M998400998400= 0146

M998400998400= 0066




119862lowast

119866(120596) = 119862

infin+

Δ119862

1 + (119894120596120591)120573

(9)


[1198621015840


Δ119862

2

)]

2

+ [11986210158401015840

119866minus (minus

Δ119862

2

cot(120573120587

2

))]

2

= [

Δ119862

2

csc(120573120587

2

)]

2

(10)




119862lowast

119866(120596) = 119862

infin+ (

Δ119862

2

)[

1 + 1198941205960120591

1 + 119894 (120596 + 1205960) 120591

+

1 minus 1198941205960120591

1 + 119894 (120596 minus 1205960) 120591

]

= 119862infin+ int

infin

0

119889119905119890minus119894120596119905

(minus

119889

119889119905

)Δ119862119890minus119905120591 cos (120596

0119905)

(11)



rate 1205960120591 ≫ 1 and | ln(120596120596

119898)| ≪ 1

[1198621015840


Δ119862

2

)]

2

+ [11986210158401015840

119866minus (

Δ119862

2

sdot

(120596119898120591)2

minus 1

2 (120596119898120591)2

minus 1

120596119898120591)]

2

asymp [

Δ119862

2

sdot

(120596119898120591)2

2 (120596119898120591)2

minus 1

120596119898120591]

2

(12)


by 120596119898120591 = radic(120596



119866


119866










the point where (1198621015840119866 11986210158401015840

119866) = (119862



1198661198621015840

119866 corresponds




4 Conclusions



35

35

30

30

25

25

20

20

15

15

10

10

5

500

C998400998400 G

C998400998400 G

C998400998400G

C998400G

Peak in M998400998400

M998400998400

HN1

HN1

HN2

HN2

120596 (au)






Acknowledgments


References






3)2minus06 KNO


























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


20

20

15

15

10

1000

5

5

C998400998400 G

Peak in M998400998400

Peak in C998400998400G

DebyeCinfin = 4

ΔC = 16

M998400998400= 01

M998400=005

M998400= 025

120596 = infin 120596 = 0

C998400G





119862lowast

119866(120596) = 119862

infin+

Δ119862

1 + (119894120596120591)

(6)


[1198621015840


Δ119862

2

)

2

] + 11986210158401015840

119866

2

= (

Δ119862

2

)

2

(7)


25

25

20

20

15

15

10

1000

5

5

C998400998400 G

C998400G

C998400G

Peak in M998400998400

at C998400G = C

998400998400G

C998400998400G


120596 = infin

120596 = 0

M998400998400= 005


infinminus 119894120590120596 (119862

infin= 10)


119866and 11986210158401015840



119866



119866and the

11986210158401015840


119862lowast

119866(120596) = 119862

infinminus 119894

120590

120596

(8)



119866= 11986210158401015840

119866



0

0

C998400998400 G

B

P

O

O998400

A

C998400G








20

10

0

C998400998400 G

20100

C998400G

Peak in M998400998400

Peak in C998400998400G

120596 = infin 120596 = 0

VW

CC

M998400998400= 0146

M998400998400= 0066




119862lowast

119866(120596) = 119862

infin+

Δ119862

1 + (119894120596120591)120573

(9)


[1198621015840


Δ119862

2

)]

2

+ [11986210158401015840

119866minus (minus

Δ119862

2

cot(120573120587

2

))]

2

= [

Δ119862

2

csc(120573120587

2

)]

2

(10)




119862lowast

119866(120596) = 119862

infin+ (

Δ119862

2

)[

1 + 1198941205960120591

1 + 119894 (120596 + 1205960) 120591

+

1 minus 1198941205960120591

1 + 119894 (120596 minus 1205960) 120591

]

= 119862infin+ int

infin

0

119889119905119890minus119894120596119905

(minus

119889

119889119905

)Δ119862119890minus119905120591 cos (120596

0119905)

(11)



rate 1205960120591 ≫ 1 and | ln(120596120596

119898)| ≪ 1

[1198621015840


Δ119862

2

)]

2

+ [11986210158401015840

119866minus (

Δ119862

2

sdot

(120596119898120591)2

minus 1

2 (120596119898120591)2

minus 1

120596119898120591)]

2

asymp [

Δ119862

2

sdot

(120596119898120591)2

2 (120596119898120591)2

minus 1

120596119898120591]

2

(12)


by 120596119898120591 = radic(120596



119866


119866










the point where (1198621015840119866 11986210158401015840

119866) = (119862



1198661198621015840

119866 corresponds




4 Conclusions



35

35

30

30

25

25

20

20

15

15

10

10

5

500

C998400998400 G

C998400998400 G

C998400998400G

C998400G

Peak in M998400998400

M998400998400

HN1

HN1

HN2

HN2

120596 (au)






Acknowledgments


References






3)2minus06 KNO


























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0

0

C998400998400 G

B

P

O

O998400

A

C998400G








20

10

0

C998400998400 G

20100

C998400G

Peak in M998400998400

Peak in C998400998400G

120596 = infin 120596 = 0

VW

CC

M998400998400= 0146

M998400998400= 0066




119862lowast

119866(120596) = 119862

infin+

Δ119862

1 + (119894120596120591)120573

(9)


[1198621015840


Δ119862

2

)]

2

+ [11986210158401015840

119866minus (minus

Δ119862

2

cot(120573120587

2

))]

2

= [

Δ119862

2

csc(120573120587

2

)]

2

(10)




119862lowast

119866(120596) = 119862

infin+ (

Δ119862

2

)[

1 + 1198941205960120591

1 + 119894 (120596 + 1205960) 120591

+

1 minus 1198941205960120591

1 + 119894 (120596 minus 1205960) 120591

]

= 119862infin+ int

infin

0

119889119905119890minus119894120596119905

(minus

119889

119889119905

)Δ119862119890minus119905120591 cos (120596

0119905)

(11)



rate 1205960120591 ≫ 1 and | ln(120596120596

119898)| ≪ 1

[1198621015840


Δ119862

2

)]

2

+ [11986210158401015840

119866minus (

Δ119862

2

sdot

(120596119898120591)2

minus 1

2 (120596119898120591)2

minus 1

120596119898120591)]

2

asymp [

Δ119862

2

sdot

(120596119898120591)2

2 (120596119898120591)2

minus 1

120596119898120591]

2

(12)


by 120596119898120591 = radic(120596



119866


119866










the point where (1198621015840119866 11986210158401015840

119866) = (119862



1198661198621015840

119866 corresponds




4 Conclusions



35

35

30

30

25

25

20

20

15

15

10

10

5

500

C998400998400 G

C998400998400 G

C998400998400G

C998400G

Peak in M998400998400

M998400998400

HN1

HN1

HN2

HN2

120596 (au)






Acknowledgments


References






3)2minus06 KNO


























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



119862lowast

119866(120596) = 119862

infin+ (

Δ119862

2

)[

1 + 1198941205960120591

1 + 119894 (120596 + 1205960) 120591

+

1 minus 1198941205960120591

1 + 119894 (120596 minus 1205960) 120591

]

= 119862infin+ int

infin

0

119889119905119890minus119894120596119905

(minus

119889

119889119905

)Δ119862119890minus119905120591 cos (120596

0119905)

(11)



rate 1205960120591 ≫ 1 and | ln(120596120596

119898)| ≪ 1

[1198621015840


Δ119862

2

)]

2

+ [11986210158401015840

119866minus (

Δ119862

2

sdot

(120596119898120591)2

minus 1

2 (120596119898120591)2

minus 1

120596119898120591)]

2

asymp [

Δ119862

2

sdot

(120596119898120591)2

2 (120596119898120591)2

minus 1

120596119898120591]

2

(12)


by 120596119898120591 = radic(120596



119866


119866










the point where (1198621015840119866 11986210158401015840

119866) = (119862



1198661198621015840

119866 corresponds




4 Conclusions



35

35

30

30

25

25

20

20

15

15

10

10

5

500

C998400998400 G

C998400998400 G

C998400998400G

C998400G

Peak in M998400998400

M998400998400

HN1

HN1

HN2

HN2

120596 (au)






Acknowledgments


References






3)2minus06 KNO


























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


35

35

30

30

25

25

20

20

15

15

10

10

5

500

C998400998400 G

C998400998400 G

C998400998400G

C998400G

Peak in M998400998400

M998400998400

HN1

HN1

HN2

HN2

120596 (au)






Acknowledgments


References






3)2minus06 KNO


























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of










Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

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