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Modelling frequency-dependent conductivities and permittivities in the

framework of the MIGRATION concept

K. Funke*, R.D. Banhatti

University of Munster, Institute of Physical Chemistry and Sonderforschungsbereich 458, Schlossplatz 4, D-48149 Munster, Germany

Received 12 December 2002; received in revised form 23 April 2003; accepted 12 June 2003

Abstract

Frequency-dependent conductivities and permittivities of ion-conducting materials reflect the dynamics of the mobile ions.Experimentally, the development of the ion dynamics with time can thus be monitored from less than a picosecond up to minutes and

hours. In this paper, we show that the characteristics of frequency-dependent conductivities and permittivities of solid ion conductors are well

reproduced within the framework of the MIGRATION concept. The meaning of the acronym is MIsmatch Generated Relaxation for the

Accommodation and Transport of IONs. In this model treatment, a simple set of rules is introduced in order to describe the essence of the ion

dynamics in terms of a physical picture of the most relevant elementary processes. The rules are expressed in terms of three coupled rate

equations, which form the basis for deriving frequency-dependent model conductivities and permittivities as well as mean square

displacements.

D 2004 Elsevier B.V. All rights reserved.

PACS: 66.30.Dn; 73.61.Jc; 77.22.dKeywords: Ion dynamics; Frequency-dependent conductivities and permittivities; Mean square displacement; Sodium germanate glass

1. Introduction

The frequency-dependent electric and dielectric proper-

ties of ion-conducting materials with disordered structures,

glassy or crystalline, are determined by the hopping

dynamics of the mobile ions. Therefore, valuable informa-

tion on the elementary hopping processes and, in particu-

lar, on correlations between hops is obtainable from a

study of the frequency-dependent conductivity, r(x), and

the frequency-dependent permittivity, e(x) [1]. These two

functions are the constituent parts of the complex conduc-

tivity, r(x) = r(x) + ixe0e(x). Here, x denotes the angular

frequency, while e0 is the permittivity of free space.

In the following, we consider data below microwave

frequencies, i.e., in a frequency regime where r(x) has to be

attributed to the hopping motion of the ions, while vibra-

tional contributions to r(x) may still be neglected. In this

frequency range, vibrations and fast ionic polarisations as

well as even faster (electronic) processes contribute a

constant value, e(l), to e(x), while the remaining part,

e(x) e(l), results from the hopping motion. Sometimes,we will use the notation rHOP(x) = r(x) ixe0e(l).

A surprising experimental observation is made when

conductivity spectra, r(x), of different fast ion conducting

materials, i.e., of structurally disordered crystals and

glassy electrolytes, are compared with each other. In fact,

many of them are found to be virtually identical in shape,

not only for a given material at different temperatures, but

even for different materials. This is most clearly seen

when a scaled conductivity is plotted versus a scaled

angular frequency. This property of scaling is known as

the time temperature superposition principle [212]. In

our notation, rS(xS) , t he s c al ed c on du ct iv it y i s

rS(x) = r(x)/r(0), and the scaled angular frequency is

xS =x/x0. Here, x0 marks the onset of the conductivity

dispersion on the x scale. A quantitative definition will

be given later, see Eq. (13).

Figs. 1 and 2 are, respectively, plots of unscaled and

scaled ionic conductivities of a particular glassy ion con-

ductor, i.e., of a sodium germanate glass of composition

0167-2738/$ - see front matterD 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.ssi.2003.06.003

* Corresponding author. Tel.: +49-251-8323418; fax: +49-251-

8329138.

E-mail addresses: [email protected] (K. Funke),

[email protected] (R.D. Banhatti).

www.elsevier.com/locate/ssi

Solid State Ionics 169 (2004) 18

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0.2Na2O0.8GeO2 [8,13]. Virtually the same scaled curve,rS(xS), as in Fig. 2 has also been obtained for many other

glassy and crystalline fast ion conductors. Therefore, this

master curve is considered representative.

Recently, a model concept was introduced which repro-

duces and explains experimental data such as those ofFigs.

1 and 2. This model, called the concept of mismatch andrelaxation (CMR) [14], is based on the previous jump

relaxation model [15]. Although the CMR is, indeed, very

successful in reproducing conductivity spectra, it does not

yield the proper low-frequency behaviour of the permittiv-

ity, e(x), i.e., a finite dc plateau value, e(0).

In this contribution, we wish to show that a subtle but

physically meaningful modification of the model is suffi-

cient to produce the correct long-time behaviour, while the

dynamics at shorter times remain unchanged. At its new

level, the model is called the MIGRATION concept, the

acronym standing for MIsmatch Generated Relaxation for

the Accommodation and Transport of IONs. In the MIGRA-

TION concept, the hopping motion of the mobile ions is

described in terms of three coupled rate equations (instead

of two in the CMR[14]). The solid lines included in Figs. 1

and 2 result from this model treatment. As the differences

between the CMR and the MIGRATION concept show up at

long times only, they are not visible in these two figures.

This paper is organised as follows. In Section 2, the time-

dependent correlation factor, W(t) [15], is introduced in

order to provide an effective means of expressing frequen-

cy-dependent conductivities and permittivities as well as

ionic mean square displacements. Section 3 is devoted to the

derivation of the MIGRATION concept and to a comparison

of model functions and experimental data. We will inparticular show that our present model treatment is able to

reproduce not only conductivity spectra, but also frequency-

dependent permittivities and time-dependent mean square

displacements. On the other hand, it is not meant to describe

or predict the dependence of ionic transport properties on

the particular structure and composition of the material

under consideration. A brief conclusion is given in the final

Section 4.

2. Frequency-dependent versus time-dependent

functions

Before considering the MIGRATION concept explicitly,

we wish to review the relationships between some relevant

functions. The transition from the frequency-dependent

functions, which are usually determined experimentally, to

time-dependent functions, is most easily performed, if theHaven ratio is close to unity, its own frequency dependence

being negligible [14,16]. This condition, which appears to

be fulfilled in most structurally disordered ionic materials, is

in the following assumed to be valid. In this case, according

to linear response theory [17], the complex conductivity

caused by the hopping motion, rHOP(x), is in a good

approximation proportional to the Fourier transform of the

autocorrelation function of the velocity of the mobile ions,

hv(0)v(t)iHOP. The latter function is characterised by a sharppeak at t= 0, plus a decaying negative contribution at t>0.

While the sharp peak reflects the self-correlation of the

velocity during hops, the decaying negative component

results from a decaying probability for an ensuing back-

ward hop, if the hop performed at t= 0 defines the

forward direction.

At this point, it is useful to introduce the time-dependent

correlation factor, W(t), which is the normalised integral of

hv(0)v(t)iHOP, with W(0) = 1. At the same time, W(t) is alsothe normalised derivative of the mean square displacement

due to hops, hr2(t)iHOP. At long times, W(t) tends to W(l).This value is the fraction of successful hops. While all

hops contribute to the high-frequency plateau of the con-

ductivity, r(l), only the successful ones contribute to the

low-frequency plateau, r(0), cf. Fig. 3.Fig. 1. Conductivity isotherms of 0.2Na2O0.8GeO2 glass. Fits are madeusing the MIGRATION concept.

Fig. 2. Scaled conductivity plot for 0.2Na2O0.8GeO2 glass. Fit is madeusing the MIGRATION concept.

K. Funke, R.D. Banhatti / Solid State Ionics 169 (2004) 182

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The relationship between r(x) and W(t) is

rx

rl 1

Zl

0

dWt

dtcosxtdt: 1

The expression of Eq. (1) tends to W(l) at low frequencies

and to unity at high frequencies. When plotted in loglogrepresentations, r(x)/r(l) and W(t) are approximately (but

not exactly) mirror images of each other, see Figs. 3 and 4.

In view of experimental data such as those of Fig. 2, we

also wish to express r(x)/r(0) in terms of W(t):

rx

r0 1 x

Zl

0

Wt

Wl 1

sinxtdt: 2

This equation results from Eq. (1) by dividing it by W(l),

then replacing d(W(t)/W(l))/dt with d[(W(t)/W(l))1]/dtand finally performing an integration by parts. With the

notations rS(x) = r(x)/r(0) and WS(t) = W(t)/W(l) for the

scaled functions, it is evident from Figs. 3 and 4 that again,in log log representations, rS(x) and WS(t) are approxi-

mately mirror images of each other.

Eq. (2) is the real part of Eq. (3):

rHOPx

r0 1 ix

Zl

0

WSt 1expixtdt: 3

Considering the imaginary part of Eq. (3) and dividing it

by the angular frequency, we obtain a reduced permittivity,

eR(x), which is caused by the hopping motion of the mobile

ions:

eRx e0ex el

r0

Zl

0

WSt 1cosxtdt:

4

This function, eR(x),decays from eR(0) = e0De/r(0)=e0(e(0)e(l))/r(0), to zero, cf. Fig. 5. Here, De is sometimes called

dielectric strength of relaxation. The quantity eR(0) has the

dimension of time and was first introduced by Sidebottom.

Itis, therefore, sometimes called the Sidebottom time, tSB[7,8]:

tSB eR0 e0De

r0: 5

In Figs. 5 and 6, we compare the frequency-dependent

reduced permittivity, eR(x), with a corresponding function

of time, hrLOC2

(t)iR. The latter is defined by

hr2LOCtiR hr2ti 6Dt

6D

Zt0

WStV 1dtV: 6

It is a reduced localised mean square displacement describ-

ing that part of the ionic motion which does not contribute

to macroscopic diffusion, but remains localised. Its long-

time limiting value, hrLOC2 (l)iR, is of particular interest.

According to Eqs. (4) and (6), it is related to De= e(0) e(l)via

hr2LOCliR eR0

e0De

r0 tSB: 7

From Figs. 5 and 6 it is evident that a similar kind of

approximate mirror symmetry as already found for r(x)/

r(l) and W(t) also applies foreR(x) and hrLOC2 (t)iR. Again,

the frequency-dependent function is obtained from the time-

dependent function by first forming the time derivative and

then the cosine Fourier transform.

Fig. 3. Model conductivity spectrum in a log log representation indicating

two possibilities of scaling, cf. Eqs. (1) and (2).

Fig. 4. Time-dependent correlation factor as obtained from the MIGRA-

TION concept, in a loglog representation, again indicating the two

possibilities of scaling.

Fig. 5. Reduced-permittivity spectrum as defined in Eq. (4), in a log linear

representation. Note that the low-frequency limit of this function is the

Sidebottom time, tSB.

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In this section, we have shown that WS(t) 1 acts as aturn-table relating frequency-dependent conductivities and

permittivities as well as mean square displacements to oneanother. Therefore, it may be considered a key function

for modelling and understanding conductivity and permit-

tivity spectra. In the following section, we will show how

W(t), and hence WS(t) 1, are obtained from the MI-GRATION concept and thus become available for deriv-

ing model spectra such as those included in Figs. 13,5

and 6.

3. The MIGRATION concept

The solid lines in Figs. 16 have all been calculatedusing the following equation for WS(t),

dWS=dt x0W2

S lnWSlnWS Nl; 8

as well as the equations of Section 2. The meaning of x0and N(l) will be explained later in this section.

Eq. (8) results from the rate equations which constitute

the MIGRATION concept. These are

Wt

Wt B gt; 9

gt

gt C0WtNt; 10

Nt

Nt C0WtNt Nl: 11

In Eqs. (9) to (11), a dot denotes a differentiation with

respect to time. The equations contain three time-dependent

functions. These are the correlation factor, W(t), the mis-

match function, g(t), and a number function, N(t). The latter

two functions will be explained shortly. Here, it is important

to note that g(t) decays from g(0) = 1 to g(l) = 0 as W(t)

decays from W(0)= 1 to W(l). As a consequence of Eq. (9)

we find

Wl expB; 12

where both W(l) and B depend on temperature.

In Eq. (10),y0 denotes the (temperature-dependent)elementary hopping rate of the mobile ions. Therefore,

x0 C0Wl C0expB 13

is the rate of successful hops, i.e., the random hopping rate.

The equation for dWS/dt which results directly fromthe rate equations still differs from Eq. (8) by featuring

ClnWS instead of lnWS, with C= (N(0) N(l))/B, in theexpression in the brackets. Constructing r(x) with the help

of WS(t) as derived from this equation, we see that x0C

marks the onset of the conductivity dispersion on the

angular frequency scale, indicating the transition from

random to non-random hopping. On the other hand, we

expect this particular angular frequency to be x0 itself, cf.

Eq. (13). Therefore, C must be a temperature-independent

constant of the order of one. For simplicity, we assume C= 1

as a working hypothesis and, consequently, use Eq. (8). Of

course, C can be included in our equations at any time, if

this is necessitated by further experimental results.

At this point, we emphasise thatx0 is the only parameter

that can be determined on the basis of experimental spectra

such as those of Figs. 1 and 2. In addition, experimental

permittivity spectra provide a possibility to determine N(l),

while B is obtained from measurements of high-frequency-

plateau conductivities. It is considered the strength of ourmodel treatment that, irrespective of the particular values of

any parameters, it is able to produce a well-defined scaled

conductivity master spectrum, see Fig. 2. At the same

time, realistic scaled permittivities and mean square dis-

placements are also derived, see below. This is, indeed,

possible on the basis of the above set of simple rate

equations, which are meant to capture the essence of the

ion dynamics in a physical picture.

We now briefly outline the physical concept leading to

Eqs. (9)(11). The central idea of the MIGRATION concept

is the following. After each hop of a mobile ion, mismatch is

created between its own position and the arrangement of its

neighbours. There are two possible ways for the system to

reduce the mismatch. Either the neighbours rearrange or the

central ion hops back into its previous site. This explains

the existence of forwardbackward correlations of succes-

sive hops.

Suppose mismatch is created by a hop of a mobile ion at

time t= 0. Then, at times t> 0, the mismatch function, g(t)

(see Refs. [14,15]), represents a normalised distance be-

tween the actual position of the ion and the position at

which it would be optimally relaxed with respect to the

momentary arrangement of its mobile neighbours. The

function g(t), therefore, varies with time from g(0) = 1 to

Fig. 6. Reduced localised mean square displacement in a loglinear

representation, cf. Eq. (6). Note that the long-time limit of this function is

the Sidebottom time, tSB.


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g(l) = 0, describing the way mismatch decays because the

neighbouring ions rearrange. The negative time derivative,

g(t), is thus the rate of mismatch relaxation on the many-particle route. On the other hand, W(t)/W(t) is the rate ofmismatch relaxation on the single-particle route, with the

ion hopping backwards. Here, the factor 1/W(t) is required,

since we consider only cases where the ion is (still or again)at its new site at time t. Eq. (9), therefore, expresses the

assumption that the rates of relaxation on the single- and

many-particle routes are proportional to each other at all

times. In other words, the tendency of the ion to hop

backwards is assumed to be proportional to the tendency

of its neighbours to rearrange.

In Eq. (10), we consider the rate of decay of g(t). Here it

is important to realise thatg(t) plays the role of a normalised

dipole moment. Its dipole field is the driving force felt by

the mobile ions, inducing their rearrangement and, as a

consequence, the concomitant decay of g(t) itself. As the

rearrangement of the surrounding ion cloud proceeds, the

central dipole becomes increasingly shielded. This means

that two effects occur simultaneously. One is the reduction

of g(t) with time. The other is the shrinking of the effective

volume of the dipole field, i.e., the reduction of the effective

number of mobile neighbours, N(t), which are still available

for the relaxation on the many-particle route.

In Eq. (10), we try to grasp the essence of the time-

dependent processes described in the preceding paragraph.

We first assume that g(t) is proportional to the convolu-tion of the driving force, which is g(t) itself, and the velocity

autocorrelation function of the neighbouring ions. The latter

function is assumed to be the same as for the central ion.

Apart from a factor, the product, g(t)W(t), turns out to be anexcellent approximation for this convolution, the reason

being that g(t) varies with time much more slowly than

W(t) does. Secondly, g(t) will be proportional to theelementary hopping rate, y0, and thirdly, to the time-

dependent effective number of neighbours available for

the relaxation, N(t). We thus obtain Eq. (10). Note that

any unknown but constant factors are included in the

definition of N(t).

The time dependence of N(t) is certainly not easy to

derive in a simple model. Eq. (11), which is used in the

MIGRATION concept and is found to reproduce experi-

mental spectra very well, is equivalent to

dNt Nl=dt

Nt Nl C0WtNt

gt

gt: 14

The equation thus implies that the difference, N(t) N(l),and g(t) decay in the same fashion. Physically, the number

N(l) represents those nearest neighbours of the central

ion, which are not shielded by others and, therefore, will

always experience the dipole field. Comparing model con-

ductivity spectra to experimental ones such as those of Fig.

1, we find that N(l) is much smaller than N(0).

The previous concept of mismatch and relaxation (CMR)

differed from our present treatment in two respects. In the

first place, it did not contain any non-zero value of N(l).

This implied that, in the limit of long times, both g(t) and W(t) varied with time as 1/t instead of decaying expo-nentially. As a consequence, the low-frequency plateaux

observed in experimental permittivities could not be repro-duced. The second difference concerned the product,

g(t)N(t), which was not necessarily regarded as proportional

to g2(t) as in our present treatment, but was allowed to vary

as gK(t). It was, however, shown that for a large number of

ionic materials the optimum value of K was close to 2.0

[14]. Nevertheless, examples were found where the expo-

nent K was clearly larger than 2.0, e.g., 2.6 [14,18]. In this

paper, we wish to keep our equations as simple as possible

and have, therefore, chosen K to be 2.0. In a later publica-

tion, we will present model equations allowing for K larger

than 2.0. It will then be shown that slight variations of K

imply corresponding slight variations of the shapes of the

model spectra. Those minor systematic differences between

experimental and model spectra, which are still visible in the

present paper, can, indeed, be removed by varying the value

of K.

In Fig. 7, we present the experimental permittivities,

e(x), of glassy 0.2Na2O0.8GeO2 [8,13] taken at the sametemperatures as the conductivities of Fig. 1. Note that there

are considerable experimental uncertainties in the permit-

tivity spectra, in particular in the limit of low frequencies,

where unavoidable electrode effects are always present, but

also at higher frequencies. The solid l ines are our

corresponding model spectra as obtained via Eqs. (8) and

(4). In all of these spectra, the same value fore(l) has beenadded to the expression for e(x) e(l). Remarkably, it ispossible to reproduce the frequency-dependent permittiv-

ities measured at different temperatures, and in particular

their low-frequency limiting values, with only one temper-

Fig. 7. Permittivity spectra of 0.2Na2O0.8GeO2 glass at differenttemperatures. Note that the low-frequency data of e(x) are affected by

electrode polarisation effects. At low frequencies, the permittivity has to be

constant. Fits have been made using the treatment of the MIGRATION

concept with N(l) = 0.075, cf. Eqs. (8) and (4).


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ature-independent value of N(l), 0.075. Evidently, N(l)

reflects a temperature-independent structural property, in

accordance with the physical meaning of Eq. (14).

For the purpose of scaling, i.e., for generating scaled

model spectra for both conductivity and permittivity, it is

useful to rewrite Eq. (8) with a scaled time, tS = tx0:

dWS=dtS W2

S lnWSlnWS Nl: 15

Eq. (15) may be regarded as resulting from the following

scaled rate equations. With a suitably scaled mismatch

function, gS(tS) =Bg(tS) = l nWS(tS), and with the dot now

denoting a differentiation with respect to scaled time, these

equations are:

WStS

WStS gStS; 16

gStSgStS

WStSNtS; 17

NtS

NtS WStSNtS Nl: 18

Eqs. (17) and (18) imply that the difference, N(tS) N(l), must be proportional to gS(tS). Here, it is important to

note that any temperature-dependent proportionality factor

between N(tS) N(l) and gS(tS) would destroy the propertyof scaling. However, there is no such factor in the expres-

sion

NtS Nl N0 NlgtS CgStS gStS;

19

since we have put C= 1.

The scaled experimental conductivity spectrum and the

corresponding scaled model spectrum have already been

presented in Fig. 2. Note that the scaled time-dependent

correlation factor, WS(tS), as obtained from Eq. (15), dis-

plays a temperature dependence at short scaled times, since

WS(0) = exp(B) = 1 / W(l). This remains, however, invisible

in Fig. 2, since the sine Fourier transform of Eq. (2) gives

little weight to the short-time values of WS(tS).

In Fig. 8, we present the scaled experimental permittivitiesof glassy 0.2Na2O0.8GeO2, along with the correspondingscaled permittivities derived from the MIGRATION concept.

The scaled dimensionless permittivity function is eS(xS),

with

eSxS eRxSx0

Zl

0

WStS 1cosxStSdtS

20

and xS =x/x0 as in Fig. 2. To obtain the scaled experimental

data, we have used the known values ofe(l), r(0), and x0,

with x0 from the fits ofFig. 1. For the scaled model spectra at

different temperatures, we have calculated temperature-de-pendent functions WS(tS), based on the assumption that B

accounts for the larger part of EDC/kT. Here, EDC is the

activation energy of the dc conductivity. Although there is a

cosine Fourier transform in Eq. (4), it turns out that the model

functions, eS(xS), display only little temperature dependence.

In comparison with the experimental uncertainties involved

in the data, these variations may still be neglected.

In the limit of low frequencies, the scaled permittivity,

eS(xS), tends to tSB,S, which is the scaled Sidebottom time,

tSB,S = tSBx0, cf. Eq. (5). Likewise, the scaled dimensionless

localised mean square displacement,

hr2LOCtSiS hr2LOCtSiRx0

ZtS0

WStSV 1dtSV;

21

cf. Eq. (6), tends to the same dimensionless value, tSB,S, in

the limit of long times. This is seen in Fig. 9. The figure

again includes both experimental data and a model curve.

To obtain the experimental data points in Fig. 9, the data

corresponding to T= 323 K have been interpolated, and a

continuous function has thus been generated. This function

has then been Fourier transformed and integrated, providing

the discrete values plotted in Fig. 9. The model curve in Fig.

9 has been obtained using Eq. (21), with the model function,WS(tS), generated from Eq. (15) for 323 K.

In Fig. 10, both scaled mean square displacements,

hr2tSiS hr2tSix0=6D

ZtS0

WStSVdtSV 22

and hrLOC2 (tS)iS, are plotted versus scaled time in a loglog

representation. In hr2(tS)iS, a transition is observed from asubdiffusive behaviour at short scaled times to ordinary

macroscopic diffusion at long scaled times. The subdiffu-

sive behaviour results from the preference of the ions for

correlated backward hops, while any memory for the

Fig. 8. Scaled permittivity of 0.2Na2O0.8GeO2 glass as defined by Eq.(20). Solid curve obtained from the MIGRATION concept with

N(l) = 0.075.


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previous site is lost in the diffusive long-time regime. To

locate the transition on the time scale, two different times

have been introduced. One is t0 = 1/x0, the other is theSidebottom time, tSB. The two times are not identical, since

the scaled Sidebottom time,

tSB;S x0tSB

Zl

0

WStS 1dtS; 23

cf. Eqs. (4) and (5), is not unity. Fig. 10 includes a graphic

visualisation of the scaled Sidebottom time, tSB,S. At tSB,S,

the line with slope one, representing macroscopic diffusion,

intersects the horizontal line, (hrLOC2(l)iS). In an unscaled

notation, this means that

hr2LOCli 6DtSB: 24

It is worth mentioning that hrLOC2 (l)i is a temperature-

independent quantity. This is immediately seen from Eq.

(24), since D is proportional to x0, while tSB is proportional

to 1/x0. We have thus shown that the long-time mean square

value of the displacement caused by the localised, non-

translational hopping motion is independent of temperature.

This is perfectly in line with an observation made earlier,

when introducing scaled frequency, cf. Figs. 1 and 2.

Indeed, apart from the dynamic processes occurring at very

short times, the only effect of temperature is to speed up or

slow down the hopping motion of the ions, while the basicrules and geometrical details of the mechanism remain

unchanged [19].

4. Conclusion

The MIGRATION concept is based on the previous

CMR model. The latter already provided the possibility to

reproduce and explain experimental conductivity spectra

taken from a wide variety of ionic materials with disordered

structures. On the other hand, the CMR did not yield the

proper low-frequency behaviour of the permittivity, i.e., its

finite dc plateau. In this paper, we have shown that a finite

value of the low-frequency permittivity is, indeed, obtained

if the CMR is modified in a physically meaningful, subtle

way. The modification concerns the long-time behaviour of

the number function, N(t). This function describes the time-

dependent shielding of the dipole field, and hence it is a

measure of the number of mobile neighbours still available

for mismatch relaxation on the many-particle route. In our

present MIGRATION concept, N(t) remains positive in the

limit of long times, reflecting the obvious fact that the

nearest neighbours will never be shielded. Not excluding

the nearest neighbours from the possibility of rearranging,

even when the process of shielding is already quite ad-vanced, is decisive for the initial hop of the central ion

eventually to prove successful. As a consequence, the

localised, non-translational mean square displacement

remains finite at long times, and so does the permittivity

at low frequencies.

Fig. 10. Loglog representation of hr2(tS)iS and hr2LOC(tS)iS of 0.2Na2O0.8GeO2 glass. The model fit is obtained by using Eqs. (21) and (22).

Fig. 9. Log linear representation of the scaled localised mean square

displacement of glassy 0.2Na2O0.8GeO2. The solid line shows the fit fromthe MIGRATION concept, cf. Eq. (21).


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Glasses with different compositions differ in their char-

acteristic low-frequency values, eS(0), requiring different

values of N(l). Correlations between N(l) and the overall

number density of mobile ions will be studied in a forth-

coming paper.

Acknowledgements

It is a pleasure to thank M.D. Ingram and A. Heuer for

many fruitful discussions. M.D. Ingram also helped with a

critical reading of the manuscript. We would like to thank B.

Roling and C. Martiny for the use of their impedance data

on sodium germanate. Financial support from the Fonds der

Chemischen Industrie is hereby acknowledged. Sonderfor-

schungsbereich 458 is funded by the German Science

Foundation, DFG.

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