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Lecture 6Lecture 61. The dielectric response functions. Superposition

principle.

2. The complex dielectric permittivity. Loss factor.

3. The complex dielectric permittivity and the complex conductivity

4. The Kronig-Kramers relations

5. The dielectric relaxation.

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PHENOMENOLOGICAL THEORY OF LEANER DIELECTRIC IN TIME-DEPENDENT FIELDSThe dielectric response functions. Superposition principle.

A leaner dielectric is a dielectric for which the superposition principle is valid, i.e. the polarization at a time ttoo due to an a electric field with a time-dependence that can be written as a sum E(t)+E’(t),E(t)+E’(t), is given by the sum of the polarization’s P(tP(too)) and P’(tP’(too)) due to the fields E(t)E(t) and E’(t)E’(t) separately. Most dielectrics are linear when the field strength is not too high.

The superposition principle makes it possible to describe the polarization due to an electric field with arbitrary time dependence, with the help of so-called response functions.response functions.

Let us consider the changes of electric field from value EE11 to a value EE22 at a moment t‘ t‘ : )t'S(t(t) 121 )( EEEE (6.1)

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01

00

t> S(t)=

t S(t)

where S S is the unit-step function:

(6.2)

The time-dependent field given by (6.1) can be considered as the superposition of a static field, EE22, and time-dependent field given by:

t' t

P 1

P 2

E 2

E 1

E

P

0

0 t' t

P 1

P 2

E 2

E 1

E

P

0

0

)(1)E-(E(t)E 21 ttS (6.3)

Therefore, we find from the superposition principle that the polarization at times t t t’ t’ due to the field given by eqn.(6.1) is the equilibrium polarization EE22 for the static field EE22 and the response of the field change EE11-E-E22 (see fig. 6.1).

Figure 6.1

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For a linear dielectric this response will be proportional to EE11--EE22, so that the total polarization is given by:

tt ,ttEEEP(t) 212(6.4)

Here (t)(t) is called the step-response function or decay function of the polarization. For simplicity let us rewrite this expression in much convenient form:

)t()(E)t(P(0)P(t) 0 (6.5)

At t=0 (t=t’ in 6.4)

10 )( (6.6)

In principle, both a monotonously decreasing and oscillating behavior of (t-t’)(t-t’) are possible. For high values of tt, PP will approximate the equilibrium value of the polarization connected with the static field EE22. From this it follows that

0 )( (6.7)

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Let us consider the case of block function. For tt11--t<tt<ttt11 the field strength is equal to EE11 , and for t t t11--tt and t> tt> t11 it equal to zero. This block function can be considered as the superposition of two fields with unit-step time dependence:

1111ttSEtttSE)t(E (6.8)

The resulting polarization for t t t t11 can be considered as the superposition of the effects of both unit-step functions:

)tt()ttt(EP(t) 111 (6.9)

An arbitrary time dependence of EE can be approximated by splitting it up in a number of block functions E=EE=Eii for tti i - - t < t t < t t tii.. The effect of one of these block functions is given by (6.9). Since the effects of all block functions may again be superimposed, we have:

iii tttttt

i)()()( EP (6.10)

In the limit increasing the number of block functions (6.10) can be written in the integral form:

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dt'ttt=

=dt't

t'ttt

t

t

)'()'(

)()'()(

E

EP

(6.11)

where, called pulse-response functionpulse-response function of polarization. The equation (6.11) gives the general expression for the polarization in the case of a time-dependent Maxwell field.

)'()'( tttt pp

Let us consider now the time dependence of the dielectric displacement DD for a time dependent electric field EE.

)t(P4E(t)D(t) (6.12)

For the linear dielectrics the dielectric displacement is a linear function of the electric field strength and the polarization, and for those dielectrics where the superposition principle holds for P, it will also hold for D.

Thus, we can write for D analogously to (6.11):

D E( ) ( ' ) ( ' )t t tD

t

t dt' (6.13)

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( ' ) ( ' ) D Dt t t twith

The relation between pp and DD is the following:

D pt t S t t t t( ' ) ( ' ) ( ' ) 1

14 (6.14)

Taking the negative derivative of (6.14) we can get the relation between pp and DD :

D pt tt t

t t( ')( ')

( ')

4

(6.15)

The unit step function in (6.14) implies that there is an instantaneous decrease of the function DD (t-t’), (t-t’), from the value DD (0)=1 (0)=1 to a limit value given by:

t tD t t

'

lim ( ' )( )

1

(6.16)

In contrast the step-response function of the polarization cannot show, in principle, such an instantaneous decrease, since any change of the polarization is connected with the motion of any kind of microscopic particles, that cannot be infinitely fast.

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However, in the case of orientation polarizationorientation polarization we can neglect the time necessary for the intermolecular motions by which the induced polarization adapts itself to the field strength. In this approximation, the induced polarizationinduced polarization is given at any time ttime t:

41 /)E(t)((t)Pin (6.17)

where is the dielectric constant of induced polarization. We can rewrite (6.12) in the following way:

)(4)()( tPtEtD or (6.18)

It is then useful to introduce response functions pporor and pp

oror describing the behavior of the orientation polarization for a time dependent field and to consider there relationship with DD

and DD respectively.

Ds

s

sport t S t t t t( ' ) ( ' ) ( ' )

1 (6.19)

D

s

s

sport t t t t t( ' ) ( ' ) ( ' )

(6.20)

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From (6.19) that now (6.16) no longer holds, but should be changed by:

s

s

D'tt

)'tt(lim

(6.21)

From comparison of (6.19) and (6.20) with (6.14) and (6.15) one can obtain the expressions for the response functions of response functions of the polarizationthe polarization in the case that the time necessary for the intramolecular motion connected with the induced polarization can be neglected.

p

s

s

s

p

ort t S t t t t( ' ) ( ' ) ( ' )

1

11

1(6.22)

p

s

s

s

p

ort t t t t t( ' ) ( ' ) ( ' )

1

1 1(6.23)

As was expected, the assumption that the induced induced polarization follows the electric field without any delaypolarization follows the electric field without any delay leads to the occurrence of a unit-step function in the expression for response function of orientation polarization. From (6.16) it follows:

t tp

s

s

t t

'

lim ( ' ) 1

(6.24)

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The complex dielectric The complex dielectric permittivity. Laplace and Fourier permittivity. Laplace and Fourier Transforms.Transforms.Let us consider the time dependence of the dielectric displacement DD for a time dependent Electric fieldtime dependent Electric field:

D E( ) ( ' ) ( ' )t t tD

t

t dt'

Applying to the left and right parts the Laplace transform and Laplace transform and taking into account the theorem of deconvolutiontaking into account the theorem of deconvolution we can obtain:

),(*)(*)(* ED (6.25)

where

)]([)exp()()(0

* tLdtstts DsDs

(6.26)

(s=s=+i+i; ; 00 and we’ll write instead of ss in all Laplace transforms ii).

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Taking into account the relation (6.20) we can rewrite (6.26) in the following way:

)]([)()(* tLi orps

From another side complex dielectric permittivity can be written in the following form:

)("i)(')(* )("i)(')(*

(6.27)

(6.28)

The equation (6.27) justifies the use of the symbol for the dielectric constant of induced polarization, since for infinite frequency the Laplace transform vanishes, and the expression becomes equal to . The real part of complex dielectric permittivity ’(’()) is associated with real part of Laplace transform of orientation pulse-response function:

0

cos)()(' dttorp (6.29)

and the imaginary part of complex dielectric permittivity ’’(’’()) is associated with the negative imaginary part of the Laplace transform of orientation pulse-response function:

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0

sin)()('' dttorp (6.30)

Let us now reconsider the relationship between time dependent displacement and harmonic electric field:

tiEtEeEtE ootio sincos)( (6.31)

We’ll rewrite in this case the relation (6.25) in the following form:

t sinE"t cosE)(')t(D oo (6.32)

that can be presented as follows:

where

2200 "' ED tg ( ) ="( )

'( )

tg ( ) ="( )

'( )

and

D t D t D t D t( ) cos cos sin sin cos( ) 0 0 0 (6.33)

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From the equation (6.32) it clearly appears that the dielectric

displacement can be considered as a superposition of two superposition of two

harmonic fields with the same frequencyharmonic fields with the same frequency, one in phase with

electric field and another with a phase difference . The

amplitudes of these fields are given by ´́E E oo and ”” E E oo ,

respectively. Calculation of the energy changes during one

cycle of the electric field shows that the field with a phase

difference with respect to the electric field gives rise to

absorption of energy.

2

2

The total amount of worktotal amount of work exerted on the dielectric during one cycle can be calculated in the following way:

20

/2

0

/2

0

20

/2

0

))((''4

1

sincos)(''coscos)(')(4

1

4

1

E

tdttdtE

EdDW

t t

t

(6.34)

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Since the fields E E and DD have the same value at the end of the cycle as at the beginning, the potential energy of the dielectric is also the same. Therefore, the net amount of work exerted by the field on the dielectric corresponds with absorption of energy. Since the dissipated energy is proportional to ”,”, this quantity is called the loss factor. loss factor. From (6.34) we find the average energy dissipation per unit of time:

sinED)(")E(

W ooo

88

2

(6.35)

where called a loss angle.

According to the second law of thermodynamics, the amount of energy dissipated per cycle must be always positive or zero. It means that

0 0 (6.36)

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The complex dielectric The complex dielectric permittivity and complex permittivity and complex conductivityconductivity

The complex dielectric The complex dielectric permittivity and complex permittivity and complex conductivityconductivity In a harmonic field with angular frequency and amplitude EEoo the dissipation of energy per unit of time in a dielectric is given by (6.35):

( )"( )W

E

0 2

8This equation holds for dielectrics that are ideal isolators. However, most of real dielectrics show a certain conductivity , leading, in a first approximation, to an electric current density I I in phase with the electric field:

)t(EI(t) (6.37)

The electric current causes dissipation of energy. According to Joule’s lawJoule’s law, the amount of energy dissipated during the time interval dtdt is given by:

W dt E dt I E 2 (6.38)

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For a harmonic field, the energy dissipation during one cycle amounts to:

W E tdt Et

( ) cos ( )/

0 2 2

0

2

0 2 (6.39)

Hence, the average dissipation of per unit of time due to condition is:

W E

1

20 2 ( ) (6.40)

Comparing (6.40) with (6.35), we see that if we determine ”(”()))) from the absorption of energy in a dielectric we always obtain the sum ”(”()+4)+4// so that we must correct for the contribution 44// due to the conductivity of the dielectric; the reason for this is the equivalence of the current density and the time derivative of the dielectric displacement in Maxwell’s law:

)(c

1curl ID 4=H (6.41)

)("8

)( 2

oEW

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As long as the current density is given by (6.38) there is no problem in separating the effects of conduction and polarization, since is a constant that can be determined from measurements in static fields. However, when it takes a certain time for the current to reach its equilibrium value, the relation between the field and the current density is given by

a pulse-response function II :I E I( ) ( ' ) ( ' )t t t

t

t dt' (6.42)

As for the pulse-response functions of the polarization pulse-response functions of the polarization pp and of dielectric displacement dielectric displacement DD , the pulse-response function of the pulse-response function of the current density current density II is associated with a step-response function step-response function II:

I I( ' ) ( ' )t tt

t t

(6.43)

Analogously to the relation between displacement and the electric field (6.24) after application to the left and right parts of (6.42) the Laplace transform and taking into account the theorem of deconvolution we can obtain:

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)(E)()(I **

where)("i)('][L)( s

I

(6.44)

(6.45)

The quantity ’(’()) gives the part of the current which is in phase with the field and which therefore leads to absorption of energy. Hence, this quantity is comparable with ’’(’’().). The quantity ’’(’’()) gives the part of the current with a phase difference of with respect to the field. Thus, ’’(’’()) is comparable with ’(’())It is possible to combine the dielectric displacement and the electric current by defining a generalized dielectric displacement DD (t): (t):

t

-

)dt'(t'4+(t)=(t) ID D (6.46)

To ensure convergence of the integral, it is necessary that E(t)E(t) approach a limiting value zero for t=-t=- fast enough; this corresponds with the fact that the field has been switched on at some moment in the past. The relation between DD (t) (t) and electric field can be found by substituting (6.13) and (6.42) into (6.46):

D E( ) ( ' ) ( ' )t t tD

t

t dt' I E I( ) ( ' ) ( ' )t t t

t

t dt'

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"')"'()"(4')'()'(= (t)t

-

'

dtdttttEdttttE I

t t

D

D (6.47)

Using (6.43) and the fact that the current step-response function II(0)=1(0)=1, we can find

t

-

t

-

')'()'(

')}'(1){'(4')'()'(= (t)

dttttE

dttttEdttttE I

t

D

D

D

(6.48)

where DD(t),(t), the pulse-response function of the generalized

dielectric displacement, is given by:

)]'(1[4)'()'( tttttt ID D (6.49)

If we’ll again apply Laplace transform to the left and to the right parts of (6.47), we’ll obtain:

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)()()( *** ED (6.50)

)t(L*D

where(6.51)

* *

*

( ) ( )( )

4

i

Making the Laplace transform in (6.51) we’ll get:

(6.52)

Splitting up into real and negative imaginary part we arrive at:

* ( )

' ( ) '( )

"( )

4

"( ) "( )

'( )

4

(6.53)

(6.54)

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The Kramers-Kronig The Kramers-Kronig relationsrelationsThe Kramers-Kronig The Kramers-Kronig relationsrelationsThe Kramers-KronigKramers-Kronig relations are ultimately a consequence of the principle of causality - the fact that the dielectric response function satisfies the condition:

0t for )t( 0 (6.55)

It means that there should be no reaction before action. It means that there should be no reaction before action.

Let us consider again the relations for real and imaginary part of complex dielectric permittivity:

 

' ( ) ( )cos

por t dt

0

' ' ( ) ( ) sin

por t dt

0

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Both ‘(‘()) and “(“()) are derived from the same generating

function pp(t)(t) and that it should be possible in principle to

“eliminate” this function and to express ‘(‘()) in terms of “(“().).

Let us consider the properties of Hilbert transform:

])[()(

)cos(sin

])[()(

)sin(cos

sin

txdtx

txt

txdtx

txtdx

x

xt

])[()(

)cos(sin

])[()(

)sin(cos

sin

txdtx

txt

txdtx

txtdx

x

xt

(6.56)

In this integral we ignore the imaginary contributions arising from integration through the pole at x=x=. The first integral is equal to , the second vanishes so that we can obtain:

( / )sin

cos1

xt

xdx t

( / )sin

cos1

xt

xdx t

(6.57)

This called Hilbert transform

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Let us apply this to the ´() (6.29):

' ( ) ( )sin

( ) sin

1

1 1

por

por

txt

xdxdt

xt dxdt (6.58)

In these manipulations we have extended the integration (6.29) to - which is permissible in view of the causality principle. The second integral in (6.58) is equal to “() in view of (6.30) so that one can finally write :

' ( ) ( / )

"( )

1x

xdx

and similarly:

' ' ( ) ( / )

' ( )

1x

xdx

(6.59)

(6.60)

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' ( ) ( / )

"( )

2 2 20

x x

xdx

' ( ) ( / )"( )

2 2 20

x x

xdx

' ' ( ) ( / )

' ( )

2 2 20

x

xdx

' ' ( ) ( / )' ( )

2 2 20

x

xdx

These are the Kramers-Kronig relationsKramers-Kronig relations which express the

value of either “() or ‘() at a particular value of the

frequency in terms of the integral transform of the other

throughout the entire frequency range (-, ). In view of what

was mentioned above about the even and odd character of

these functions, one may change the range of integration to

(0, ) and thus obtain the one-sided Kramers-Kroning

integrals:

(6.62)

(6.61)