Ultrasonic attenuation studies of solid electrolytes

Ultrasonic attenuation studies of solid electrolytes

Solid State lonics 26 (1988) 265-278 North-Holland, Amsterdam ULTRASONIC ATI'ENUATION STUDIES OF SOLID ELECTROLYTES D.P. ALMOND University of Bath, S...

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Solid State lonics 26 (1988) 265-278 North-Holland, Amsterdam

ULTRASONIC ATI'ENUATION STUDIES OF SOLID ELECTROLYTES D.P. ALMOND University of Bath, School of Materials Science, Clam,on Down, Bath BA2 7AY, UK

and A.R. WEST University of Aberdeen, Department of Chemistry, Meston Walk, Aberdeen, AB9 2UE. UK

Received 3 April 1987; accepted for publication7 December 1987

Experimentaland theoretical aspects of the measurementof ultrasonicattenuation in ionic conductorsare reviewed.The conditions under which relaxation peaks associated with ion transport may be observed in ionicaUyconductingsolids are assessed and procedures givenfor extractingcharacteristic hoppingfrequencies.A reviewof literature data on variousbeta aluminasand Agion conductingglassesis given.

1. Introduction

The high ionic conductivity of solid electrolytes is attributable to both a high concentration of mobile ions and the rapidity with which they are able to hop from site to site a't ambient temperature. Typical ionic hopping rates fall in the range 10-1000 MHz in contrast to, the perhaps more familiar, atomic hopping rates of 1-100 Hz of impurity atoms, such as carbon, in metals, such as iron. Mechanical relaxation techniques have a long history of contributions to the study of atomic diffusion in metals. In this field, techniques such as the torsion pendulum have been used to measure internal friction at frequencies of a few Hz. Atomic hopping rates have been obtained from the analysis of internal friction peaks, found when the mechanical vibrational frequency is comparable to the hopping rate. The determination of ionic hopping rates in solid electrolytes would add considerably to the understanding of their conduction mechanisms. Mechanical relaxation techniques offer a means for determining these hopping rates but require the generation of elastic vibrations, in the solid electrolyte, at much higher, MHz frequencies. Alternatively, cor.ventional lower frequency internal friction

techniques might be employed on samples cooled to very low temperatures. Measurements at MHz frequencies can be accomplished by the well-established ultrasonic pulse-echo technique. This technique has been employed in the study of a wide variety of physical phenomena. These include, the measurement of clas:!c censtants; the study of structural, magnetic, ferroelectric and superconductive phase transformations; the study of electron and phonon scattering and the study of superfluidity in liquid helium. In addition, it is used extensively for non-destructive testing and for medical imaging. In recent years, the ultrasonic pulse-echo technique has been applied to solid electrolytes to determine ion hopping rates. The purpose of this review is to introduce the ultrasonic pulse-echo technique as a means of evaluating solid electrolytes. The experimental technique is outlined in section 2. Details of the mechanism of interaction between an elastic wave ~ . u a 1, a.,,.,,,l~. structure containing mobile ions and the information that can be obtained from an ultrasonic study of such a system, are presented in section 3. Finally. in section 4, the results obtained from the small number of ionic conductors studied to date are reviewea Although only a few of the many solid elec-

266

D.P. Almond, A.R. West~Ultrasonic attenuation studies !

200 HzI

trolytes have been studied, the technique has proved successful with conductors as diverse as ceramic betaalumina and Ag + ion conducting glasses.

>

[ ,oo E ,,

I

ooh

Generato, r ,

2. Experimental techniques The experimental techniques that prove suitable for ultrasonic studies of solid electrolytes are identical with those which have been developed for the study of such phenomena as: magnetism, ferroelectricity, electron-phonon scattering, superconductivity, structural phase transitions, dislocations and elastic constants. The techniques facilitate measurements of acoustic response at frequencies as low as 1 MHz and up to several GHz, though by far the most commonly used frequency range falls between 10 and 100 MHz. Two parameters, sound velocity and attenuation, characterise the acoustic properties of a material. Whilst both of these can be obtained, attenuation is probably of most interest in the study of solid electrolytes because it exhibits distinctive peaks which are caused by resonant interactions with ion hopping processes. The magnitude of the sound velocity is governed by the elastic constants of the material, though it shows some change with attenuation, because of a Kramer-Kronig relationship between attenuation and velocity. However, the magnitudes of such changes are very small ( ~ 0.1%) and their detection requires the use of particularly sensitive measuring techniques [ 1-3 ]. By contrast, attenuation phenomena are large and can be monitored using comparatively simple techniques. For these reasons, the following discussion is confined to the measurement ef ultrasonic attenuation. 2.1. E l e c t r o n i c s

The equipment employed in the measurement of ultrasonic attenuation is a pulse-echo system, shown

~-,~,.at~a,.y in

fig. I.

Ultrasonic wave¢ are generated and detected by a piezoelectric tran,sducer attached to the sample surface. The transducer is energised by a pulse of RF energy from a pulsed oscillator which is tuned to the resonant frequency of the transducer. The burst of ultrasound that is produced travels through the sam-

I Pulsed I Oscl:letor ,

Amplifier Receiver

t ~¢~Transducer II s"-,,.I • -

|,

Fig. I. Block diagram of an ultrasonic pulse-echo

system.

pie, is reflected at the back face and returns through the sample to reactivate the transducer. The electrical signal produced by the reactivated transducer is amplified and usually converted to a de form by a detector. The sound wave is then reflected back and forth through the sample a number of times; each time its energy is diminished by the various attenuation processes in the sample. The successive signals generated by the transducer are "echoes" of the initial drive pulse. These echoes are displayed on an oscilloscope. To accomplish this, the oscilloscope and pulsed oscillator are triggered by the same timing pulse, which usually has a repetition frequency of about 200 Hz. A schematic of the resulting oscilloscope display is also shown in fig. 1. It comprises the sequence of pulse echoes, diminishing in amplitude with time. The time interval between the echoes is the time taken for the ultrasonic wave to travel through the thickness of the sample twice. Sonic velocities in solids are typically 5 km/s. Hence, in a 1 cm thick sample the echo spacing will be ~ 4 ~ts. In order to observe distinct pulse-echoes, the drive pulse width must obviously be less than the echo spacing. Pulse width is variable in most commercially pro.... "I equipmem, h,,, has a i. . . . . . ;;";* of n~,,..,. 1 A~.P ~ ~vJt

/lill/lL

L

O.~,.J4Jr U L

~tS.

Measurement of the variation of attenuation with change in environment of the sample, most commonly temperature, is accomplished by monitoring the variation in amplitude of the echoes. The wide variety of meth,~ds which have been devised for

D.P. Almond, 4.R. West~Ultrasonicattenuation studies

monitoring echo height include: visual assessment using a calibrated oscilloscope screen; comparison of echo heights with a signal produced by a calibrated oscillator [4 ]; use of box-car integrators and attenuation recorders to automatically monitor the amplitude of one of the echoes [ 5 ] and ultimately, the digitisation and computer analysis of the complete echo train. Before discussing further the choice of measuring technique it is necessary to explore some of the potential problems that may be encountered in attempting to make ultrasonic attenuation measurements. 2.2. Sample preparation The dimensions of samples suitable for ultrasonic studies are dictated by the necessity to separate the pulse-echoes, mentioned above, and by the size of the transducer. Assuming the pulses are 1 laS wide and using a 2 gts echo spacing as a minimum criterion for the resolution of separate echoes the minimum sample thickness, t is given by

2 t = 2 × 10-6/3,

( 1)

where v is the velocity of sound. This velocity, for longitudinal waves, may be as low as 1 km/s in soft, compliant materials but may exceed 10 km/s in hard, high specific stiffness material such as ceramic eleco trolytes. Hence, for these latter materials a sample thickness of at least 1 cm is required. If this proves to be impractical or if the attenuation through such a length of material proves to be excessive, a thinner sample may be used, bonded to a length of a second material to provide the required time delay between echoes. Suitable "delay rods" can be fabricated from lengths of optical quality silica. The cross-sectional area of the sample must be large eno,,gh to accommodate the transducer. Transducers are typically 6 mm in diameter, though smaller ones may be used, but with an inevitable decrease in signal intensity with decrease in transducer area. In addition, it is necessa~ to recogmse that the acoustic field generated by a transducer diverges, due to acoustic diffraction, and that the angle of divergence, O, is related to the transducer radius, a, by [ 6 ] 0 = arctan (;t / a ),

(2 )

in which 2 is the wave length of sound. At 10 MHz,

267

2 is ~ 0.5 mm. It is desirable to keep 0 to a minimum so as to avoid serious distortions of the echo uain caused by divergent sound waves reflecting from the sample sides and interfering with the central colum:'. of acoustic flux. Evidently there is a case for minimising the divergence by using at large a transducer area as is practical. In addition, the effects of beam divergence can be alleviated by ensuring that the sample area significantly exceeds that of the transducer. In the study of li-alumina ceramics [ 7 ], samples in the form of 1 cm diameter, 1 cm thick cylindrical pellets with 5 mm diameter transducers were found to be satisfactory. The ends of the sample must be cut and polished, flat and parallel to each other. This ensures that all elements of the acoustic field generated by the transducer travel equal distances to the opposite sample face where they are reflected normally. Deviations from flat and/or parallel ends cause variations in the phase of the reflected acoustic waves across the area of the transducer and hence, significant distortions in amplitude and shape of the echo train. These phase changes depend on the path differences caused by the errors in sample geometry and ,.,versely on the wavelength of the ultrasound. For frequencies up te about 100 MHz, conventional metallographic polishing to a 1 lxm finish and the use of simple jigs to produce parallel faces proves to be adequate. At higher frequencies, high quality optical finish lapping techniques become necessary [ 8 ]. 2.3. Transducers and bonding

The most widely used transducers are discs of single crystal quartz. The orientation or "cut" of the transducer determines the type of piezoelectric strain produced. X-cut plates dilate in the direction perpendicular to their surface and hence generate longitudinal waves, whilst Y-cut plates strain in a lateral direction and may be used to generate transverse waves. These discs are mechanical resonators whose thicknesses determine their resonant frequencies. A 10 MHz longitudinal wave transducer, for example, is approximately 0.3 mm thick. Such a transducer can also be used at odd multiples, 30, 50, 70 ... MHz, of the fundamental frequency [6 ] but it must be stressed that, since it is a strongly resonant device, it is largely ineffective when driven off-resonance.

D. t'. Almond, A.R. West~Ultrasonic attenuation studies

268

Hence it is possible to make measurements only at the resonant frequencies pravided by the transducer. Transducer faces are plated with metallic films to facilitate electrical contact. As solid electrolytes are usually electronically insulating it is not possible to use them as the ground plane. So-called "co-axi,,dly plated" transducers must then by used, fig. 2a, in which the plating of the lower surface is extended round to the upper surface to provide access for the ground connection. The mechanical vibrations induced in the transducer are coupled to the sample by a bonding layer, fig. 2b. At MHz frequencies it i~ not possible to gain adequate coupling between transducer and sample by simply holding the transducer against the sample surface. The reason for this is that the inevitable residual roughness of transducer and sample surfaces leaves air gaps between the two solids which severely reflect the ultrasound ~6 ]. This high reflectivity of thin air gaps, such as cracks, accounts for the success of ultrasonic in the field of non-destructive testing. To overcome this problem in attenuation measurements, the transducer must be coupled to the sample by a layer of liquid or solid which is known as the "'bond". A wide variety of glues and oils have been used to form these bonds. One of the require~ents of the bond is that it should have sufficiently !ow viscosity to allow the transducer to settle with its surface parallel to the surface of the sample. The other is that is should continue to wet both transducer and sample surfaces over the range of temperatures required by the experiment. Conspiring against this requirement are: different rates of ~.hermal expansion of transducer, bond and sample materials; the freezing of liquid bonds at reduced temperatures and the significant

Transducer ~Bond

Goldg ~ Platin

\=

(a)

.

(b)

Fig. 2. (a) A coaxially plated transducer; (b) schematic diagram of a transducer bonded to a sample.

changes in sample dimensions that accompany structural phase transitions. It must be admitted that bonding is the "Achilles heel" of the ultrasonic technique. Bonding substances that have proved successful in particular applications include: Nonaq stopcock grease, Durofix and Duco cements, silicone oils, glycerine, epoxy resins and cyanoacrylate resin. The bond is generally made by depositing a small amount of the bonding agent onto the sample and then "wringing" the transducer onto the surface to produce the required thin bonding layer. The success of a particular bond depends on the properties of the sample and the temoerature range concerned. Because of this, the choice of bond is often arrived at by trial and error. Since all the above bonding agents become rigid at low temperatures, there is a risk that the sample may be damaged by differential contraction of the sample and the transducer. This damage may be particularly serious in glasses, which have very different rates of contraction to quartz. In practice, the failure of a bond causes the disappearance of the pulse-echo train and its replacement by ihe free ringing response of the uncoupled resonating transducer. Bonding problems can be overcome ~y the use of thin film transducers deposited onto the surface of the sample. Thin film CdS transducers have proved invaluable in the study ofNa ~-alumina [9,10 ]. CdS is a piezoelectric semiconductor which can be grown in a suitably oriented form by co-evaporation of cadmium and sulphur [ 11 ]. This produces a transducer which is an integral part of the sample, with a broad band frequency response between -,-20 MHz and ~ 300 MHz. Such a transducer would have a thickness of ~ 20 l~m. Prior to the deposition of the CdS layer, it is necessary to metallise the sample surface to provide a ground plane. In the case of 13-alumina, a chromium film was deposited by evaporation to form an adherent underlayer and a gold film was sputtered over the chromium. CdS transducers work excellently down to the lowes'~ temperatures but fall off in performance at high temperatures where, being semiconductors, they become electrically conducting. The orientation of the crystallites within the film is eas'ly controlled, giving transducers which generate either longitudinal or transverse waves, or both at the same time. Because they are thin films, the possibility of sample damage

D.P. Almond, A.R. West~Ultrasonic attenuation studies

by differential contraction is virtually eliminated. The CdS deposition process necessitate~ the sample being held at 180°C, its exposure to cadmmm and sulphur vapour and the formation of an adherent conducting film on its surface. In some sam~!es, the latter may prove impossible and for others the deposition conditions may be intolerable so that although CdS transducers have great merits, they are not suitable for all materials. 2.4. Attenuation measurement

Having explained in some detail the problems associated with the generation of ultrasonic waves in solids, it is appropriate to return briefly to the measurement of ultrasonic attenuation. Firstly, in the study of solid electrolytes, it is not necessary to measure the absolute attenuation of a sample because changes in relative attenuation as a function of temperature are of most interest. This is fortunate because, despite the best of preparative techniques, the attenuation evident in an echo train is due to a mixture of bond losses, diffraction and genuine sample attenuation processes [ 12 ]. It is essential that, during an experiment, true changes in attenuation are distinguished from changes in the echo pattern caused by the transducer bond or contact to the transducer electrode. This distinction can be made only by monitoring at least two of the echoes in the echo train. Because each successive echo is caused by a pulse of sound that has travelled through an additional two thicknesses of sample, it should be a~tenuated by the additional appropriate amount. Hence the attenuation change measured for the second echo should be twice that of the first, etc. Changes in bond, however, will affect the signal produced by each echo equally, allowing the discrimination of the two effects. Of the echo monitoring techniques, mentioned above, all but the box-car integrator allow the simultaneous measurement of two or more echoes. The amplitudes of ultrasonic pulse-echoes are conventionally expressed in decibels with respect to noise. It is common f~r these to have a signal to noise ratio of better than 30 dB. The amplitudes of such echoes can be assessed with an accuracy of ~ 0.1 dB. Finally, attenuation measurements are conventionally expressed in a normalised form as dB/cm or

269

riB/ram (i.e. dB of attenuation per unit length of acoustic path through the samvle).

3. Theory 3.1. Attenuation processes

Numerous processes can give rise to the attenuation of sound waves passing through a solid sample. Many give rise to a general background attenuation which, though important from both theoretical and practical viewpoints, is not central to the present review. Sources of such background attenuation include scattering by: thermal phonons, dislocations, crystal defects, cracks, porosity and grain boundaries. The background attenaation in polycrystalline materials is generally much higher than that found in single crystals. The origin of this attenuation is the scattering of sound as it passes from one randomly oriented grain to another. The elastic anisotropy of the grains produces, in most materials, a significant variation in the speed of sound with crystal orientation. As a result, a beam of sound may be refracted as it travels from grain to grain in a polycrystalline material. The mechanisms of attenuation in polycrystalline materials have been discussed in some detail [13,14] and it is generally accepted that the attenuation a increases with frequency, f, as: a=eOf2+QD3f 4 ,

(3)

where P and Q are parameters determined by the elastic properties of the materials and D is the mean grain diameter. The second term in eq. (3) is due to Rayleigh scattering and it dominates the attenuation at high frequencies. It should be noted that the two attenuatien terms increase with D. As a result, grain diameter and porosity are found to determine the bulk of the background attenuation in polycrystalline samples. Ultimately this attenuation imposes an upper frequency limit on ultrasonic measurements of polycrystalline materials. For example, in a high density, l ~tm grain size samole of volvcrystalline beta-alumina, the background attenuation restricted measurements to frequencies below 100 MHz [ 7 ]. In solid electrolytes, superposed on the background attenuation there may be an attenuation peak caused by the hopping of the mobile ions. This at-

270

D.P. Almond, A.R. West~Ultrasonic attenuation studies

tcnuation peak o~cul.~ where the frequency oi' the sound wave is similar to that of the ion hopping rate: the precise relationship is discussed below. In practice, such ultrasonic attenuation peaks are found by sweeping temperature; the peak in attenuation occurs at the temperature for which the ionic hopping rate matches the frequency of the sound wave. As a sound wave moves through a solid material, a periodic strain is induced in the lattice. Consequently, the energies of the lattice sites occupied by the mobile ions may be altered and their populations modified by ions hopping preferentially to the lower energy sites. This results in a net extraction of energy from the sound wave, which is the observed attenuation. Solids exhibiting such effects are said to be "uaelastic". Probably the best known example of ane!astic phenomena is the Snoek effect, involving the diffusion of carbon in iron. In dilute systems, containing a low concentration of mobile atoms or ions, the attenuation may be described as a Debye-like, single relaxation time process in which the individual atom/ion hops occur independently of each other. In such cases, the attenuation, ~x, for a wave of angular frequency, to, takes the form [ 15 ]: oe= o)zf [ 1 +-~2172].

(4)

The term in square brackets descri0es a Debye peak; it passes through a maximum wl~en the condition toz= 1, is satisfied. The elastic strain relaxation time, z, is closely related to the inverse of the hopping frequency. Hence such Debye peaks may be used to obtain estimates of atom/ion hopping rates. The parameter, A, the relaxation strength, determines the magnitude of the attenuation peak. It is related to the strain dependence of the mobile atom/ion site energy, or deformation potential, B, by [16]" A= NB2/4~pvakT

'

, ¢x

in which N is the number of mobile atoms/ions, v is the velocity of the sound wave, p is the density of the solid, T is absv!ute temperature and k is the Boltzmann constant. The direct relationship between the magnitude of the attevuation and the number of mobile atoms/ions contributing, N, is to be expected. In addition, the attenuation is inversely proportional to

the cube of the sound velocity, v. The velocity of sound is given by:

v=(cip) '/:,

(6)

in which c is the appropriate elastic modulus for the sound wave concerned. For example, for a longitudinal wave propagated along [ 1 0 0 ] in a cubic crystal the elastic modulus would be ct t, but for a shear wave ,. would be c44. From eq. (6), high stiffness materials such as ceramic solid electrolytes have much higher sound velocities than low stiffness materials such as polymers and ion conducting glasses. As a result, much larger attenuation peaks should be found in the ion conducting gl~sses than in ceramic electrolytes such as beta-alumina; this is found to be the case in practice and will be discussed later. In addition to the relaxation strength being sufficiently large, certain crystallographic criteria must be satisfied for an attenuation peak to be observed. As has been mentioned, the mechanism of the interaction between an acoustic wave and the mobile ions depends on modification of the population of crystallographic sites available to the ions (in an ionic conductor). This necessitates the ions being accommodated in more than one crystaUographically distinct site, and the energies of at least two of these being affected differently by the acoustic wave. These requirements give rise to anelastic selection rules for the occurrence of attenuation peaks [ 17 ]. Formally, the rules are obtained by consideration of the effect of the acoustic wave on the symmetry of sites occupied by the ions. These selection rules are discussed in some detail by Nowick and Berry [ 17 ]. Where the requirements for attenuation, as given by the selection rules, are not met, an atteauation peak will not be found, despite the presence of hopping ions which satisfy the toz= i condition. The relaxation time, z, obtained from an acoustic relaxation study is the relaxation time of the mobile ion redistribution process, induced by the sound wave. This net relaxation time takes into account all the possible ion hops into and out of the available sites. The relaxation rate, T-t, will, in general, not equal the ion hopping rate but will be a simple numerical multiple of it. For example, in the Snoek effect for carbon in iron, the multiple is six. In the theo~ of ionic conduction [ 18] similar multiplication factors arise in the relationsMt; between io~;c

271

D.P. Almond, A.R. West/U!trasonic attenuation studies

conductivity and ion hopping rate. These factors are, in general, not equal te those for acoustic relaxation [191. In cases where the crystallog~,'aphy of an ionic conductor and the location of its mobile ions is known, it is possible to determine the exact relationship between the relaxation times. Where details of the site occupancy are not known, as is often the case in solid electrolytes, the relaxation times cannot be identified with specific hopping processes. Nevertheless, they may be regarded as being characteristic relaxation times of the conductor.

3.2. Non Debye-like processes In solid electrolyte materials, experimental attenuation studies give peaks that are much broader than an ideal Debye peak [9,20,21]. Different approaches have been used to characterise such non Debye-like peaks. The traditional approach, especially in electrical relaxation studies, is to treat them as the sum of a number of Debye-like peaks, corresponding to a range of relaxation times arising from a distribution of activation energies, E. The attenuation then takes the form [ 16 ]: t~--Zl

g(E) m2r(E) dE 1 +CO2~'2(E)

" '

(7)

where the distribution function, g(E), may be obtained from the shape of the relaxation peak. This approach depends on the assumption that ion migration may be treated in terms of a set of non-interacting, Debye-like processes. In solid electrolytes, however, mobile ion concentrations are large and conduction mechanisms are thought to be cooperative. The validity of the distribution of relaxation times, or activation energies, approach has been questioned by Jonscher. He has shown [22] that the relaxation phenomena observed in a wide variety of materials exhibit a power-law type of frequency dependence, often spanning mary decades of frequency, and suggests that there is no reasonable justification for attributing such extensive effects to distributions of relaxation times. Jonscher demonstrated that dielectric loss peaks conform to a "universal" response function, which was subsequently shown to be attributable to ma':y-body processcs by Dissado and

Hill [23 ]. The authors [9 ] suggested that the same response function could be used to account for the attenuation peaks obtained in ionic conductors, i.e that ultrasonic attenuation peaks take the form

[( (.0)--m ( CO)l--n].-l

a

tl/r).

+

.

(8)

Relationship to Debye behaviour is clarified if expressed in form (CO )m

aoc(l/T)

1 + (COlt)! + ' ' - " '

where m and n are power-law exponents, which take values between 0 and 1, and COpis a characteristic, thermally activated frequency. Eq. (8) reduces to ,he equation for a single Debye-like process when m = 1, n=O and COp=,[.--I. Recently, Dyre [24,25] has demonstrated that power-law responses can be produced by a simple exponential distribution of relaxation times and Palmer et al. [26 ] have suggested that in strongly interacting glassy materials the power-law, or Kohlrausch, forms of relaxation emerge naturally.

3. 3. Relationship between mechanical and electrical re!~rotian phenomena Electrical conductivity data are usually obtained for a range of icequencies at fixed temperature, in contrast to ultrasonic attenuation data. In solid electrolytes, such conductivity data usually exhibit power law frequency dependences. Those effects specifically attributable to bulk conduction may be described by an equation of the form [ 27]: 0.(co)=Kcolp-,nco,,, +~cop - - , - , , 2 to "2,

(9)

in which COp= r - ~, where ~ is the electrical relaxation time and K is a combination of a number of terms, including carrier concentration. This expression has also been obtained [ 28 ] by a theoretical analysis of ionic conduction involving many-body interactions. In that analysis, the two terms in eq. (9) have the same origin as those in eq. (8) which describe ultrasonic attenuation and dielectric loss phenomena. The simplicity of the two expressions facilitates a particularly straightforward parameterisation of both conductivity and ultrasonic data. Thus the two char-

D.P. Almond, A.R. West~Ultrasonic attenuation studies

27 2

4, Review of ultrasonic attenuation studies of solid electrolytes

ion vibrational frequency, 2 X 1012Hz, obtained from infrared absorption [ 32 ], the ultrasonic attenuation peaks were ~ttributed to sodium ion hopping. The above ultrasonic data were found [ 9,36 ] to be fitted by eq. (8), as shown in fig. 4 for one of the data sets. This fit was obtained by assuming the charac~a~¢~gpping frequency, cop, to have the activation energy 9.16 eV, in line with the established conductivity activation energy [ 31 ]. The best fit values of the power law exponents, m and n, were I and 0.65, respectively. The value of COpat the peak temperature is given by the condition:

4. I. Single crystal sodium beta-alumina

cop

Measurements of the attenuation of longitudinal waves propagating in the conduction plane orientation of a well eharacterised, dried sample of single crystal ~a~umina showed a single broad peak which was displaced to higher temperatures with increasing frequency [ 10 ], fig. 3. The temperature dependence of the relaxation time, z, obtained assuming CO~=1 at the peak, is shown in Arrhenius format as an inset to fig. 3. The data fall on a straight line which indicates an activation energy of 0.14 eV and a prefactor of 3X 10-13 s. Since this activation energy value is similar to that of about 0.16 eV obtained from conductivity measurements [ 31 ] and the prefactor value is similar to the inverse of the sodium

Substituting for m and n gives the ratio co/cop: 2.18; the value of COpis therefore about half the relaxation frequency, z - i, obtained previously assuming ¢oz= 1 at the peak maximum. The attenuation peak in fig. 4 is very different to a single relaxation time Debye peak, shown for comparison as a dashed curve. For this reason, a value of COp,obtained by fitting the entire peak and using eq. (8) is considered to provide a better estimate of the relaxation rate than by assuming a Debye-like process and the cnndition coz= 1 at the peak maximum.

acteristic frequencies, cop, may be compared as may the power law exponents, m, n, nl and n2. It has been shown [29,30] that other commonly used representations of electrical data, such as spectroscopic plots of the imaginary part of the electric modulus, M", or complex impedance plane plots, Z" versus Z', can be quantified using expressions containing the same parameters as in eq. (9).

o[,___m]'"--"

'7/''

I

I

6L

I

(10)

I

I

I

I

~-alumina 235 M,.tz -

-

:0.35

,,,

,ooo/,

/

tx "%%

,,/

//1..,-

~

--

-,.. i

O0

--

/ /

~

r-

-

150 200 TEMPERATURE(K)

o.q~

Fig. 3. Measurements of the attenuation of longitudinal waves propagated along the c-axis direction in a single crystal of Na alumina (from ref. [ 10] ). The inset shows values of relaxation times obtained from ~he data as explained in the text, plotted in an Arrhenius fashion.

0

4

I

I

"1".--

-I_

S 6 7 8 9 10 Inverse Temperature IOOOIT (K -I)

i

Fig. 4. The attenuation of longitudinal waves propagated perpendicular to the c-axis direction in a single crystal of Na IS-alumina plotted against inverse temperature. The solid line through the data is a fit of eq. (8) using the parameters shown in the figure (from ref. [36] ).

D.P. Almond, A.R. West~Ultrasonicattenuation studies

Data for cop obtained by curve fitting and eq. (8), using the ultrasonic data shown in fig. 3 and lower frequency internal friction data [ 33-35] are shown in Arrhenius format in fig. 5. Excellent agreement between the different data sets is obtained. Electrical conductivity data, obtained from a similar single crystal sample of sodium ~alumina were analysed using eq. (9) [ 30 ]. The values obtained for the exponent n~ were dose to zero (0.035) and for n2 were 0.6 above 120 K but increased somewhat at lower temperatures [ 36 ]. There is a remarkable similarity in the values of the mechanical exponent n and the electrical exponent n2. Both correspond to the parts of the respective responses for which to/cop >> 1, indicating a eornmon orion. The values of the mechanical exponent m and the electrical exponent n~ are both close to Debye-like values, 1 and 0 respectively, again indicating a common origin. Values of cop obtained from the conductivity data are also shown for comparison with the mechanical values in fig. 5. The close agreement between the two sets of data indicates that the relaxation times for

I

"

I

12 ~

I

I

i

® mechanical a electrical

X

1

0

I

I

I

!

!

3

2

4

6

8

10

12

10001 T. K'I Fig. 5. An Arrhenius plot of the characteristic hopping frequencies obtained in the analysis of ultrasonic attenuation, internal friction and ac electrical data ofsi,gle crystal Na ~alumina (from ref. [481 ).

27 3

mechanical and electrical ph :nomena are very similar. in addition, the values of top are similar to ion hopping frequencies in ~alumina calculated from an attempt frequency of 2 × 10~2 Hz and an activation energy of 0.16 eV. This agreement ,*~pears to indicate that the mt:!tiplieation factors, discussed earlier, are not large. Simmons et al. [ 34 ] have proposed a microscopic model to explain the mechanical relaxation and electrical conduction in IS-alumina. The resulting multiplicatior factors for the mechanically active mode, which they identified, and for electrical conduction, are 1.58 and 0.67, respectively. These factors are indeed small, as the experimental data, fig. 5, would seem to indicate. The precision of the experimental data are insufficient to test quantitatively the predictions of Simmons et al. [ 34 ]. Simmons et al. considered two interstitialcy hopping mechanisms and whether or not they could give rise to mechanical relaxation. In one, an interstitial Na + ion occupies an anti-Beevers Ross (aBR) site and displaces a regular Na + ion which in turn moves to another aBR site. They cor.cluded that anelastic relaxation would not be expected since the aBR sites are crystallographically equivalent. In the other mechanism, pairs of Na + ions occupy two out of three mid-oxygen (mO) sites. In this case, application of mechanical stress lowers the symmetry of the defect complex ind';cating that anelastic relaxation might be expected to occur. Crystallographic studies on sodium ~alumina indicate very low occupancies of the aBR sites and lend support to the second interstitialcy mechanism, as does the observance of anelastic relaxation phenomena. Samples of ~alumina exposed to moist air exhibited additional, high temperature attenuation effects. Similar effects were found in internal friction studies [34]. These effects have not been explored in detail but merit further study.

4.2. Polycrystalline P/if' aluminas Measurements of the attenuation of lon~tudinal waves in a polycrystaUine sample of composition 70% "-, 30% [3-alumina [ 7 ], are shown in fig. 6a. Values of the relaxation time, z, obtained by assuming o ~ - 1 at the peaks, are shown plotted ~n Arrhenius format in fig. 6b. The attenuation data have also been treated

D.P. Almond, A.R. West~Ultrasonicattenuation studies

274

i

.... I

I

TEMPERATURE IK)

I

D o

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in the same way as the single crystal data referred to above, vcith fitting to eq. (8) [ 37 ]. An activation energy of 0.14 eV and an attempt frequency of 3 × 10 ~2 Hz was obtained, witl, m = 1 and n=0.67. These values are remarkably ~imilar to the single cystal J]-alumina values, despite this being the minority phase ;n t h e Inr~lxrr.rxretmll;n~ emm~l~ , D ~ , P l ] ~ f l ] ItJ~,IbJLIII,IL~- ~ 4 ~ l l l l , , p l ~ ¢ • l&l

Fig. 6. (a ~ Measurements of the attenuation of longitudinal waves in a polycrystalline sample of Na 70% 13"-30% ~alumina; (b) an Arrhenius plot of relaxation times obtained from the data in (a) (from ref. [7] ); (c) measurements of the attenaation of longitudinal waves in a poiycrsstaiiine samlfle of Na > 90% ~"-, < 10% 13-alumina [ 38 ].

ature attenuation peak indicates that anelastic relaxation does not occur in the [~" phase and further supports the conclusion that the attenuation peaks in fig. 6a are solely due to the [3 component of the 70% 13", 30% 13 mixture. In Nal]'i-alumi,~a, the mobile i,,a + ions occupy

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Similar measurements on a sample of almost phase-pure 13"-alumina (of composition > 90% ~", < 10% ~), fig. 6c, [ 38 ] showed no evidence of an attenuation peak in the temperature range in which peaks were found in the 70% ~", 30% 8 sample, fig. 6a. The attenuation was, however, fore : to increase at high temperatures. The absence of a low temper-

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tion plane. Since the equivalence of these sites is unaffected by an applied stress, anelasfic relaxation phenomena are not expected to occur. In this respect, the situation for 13"-alumina is similar to the first interstitialcy model for B-alumina mentioned above, in which some Na + ions occupy aBR sites and for which no anelastic relaxation effects were pre-

D.P. Almond, A.R. West~Ultrasonic attenuation studies

dieted. While this contrasting behaviour proviOes an interesting example of anelastic selection rules, it unfortunately does mean that relaxation rates in 15"-alumina cannot be obtained by mechanical relaxation techniques. Recently, Patel and Nicholson [ 39 ] have reported ultrasonic studies of polycrystalline 15/15"-aluminas containing one or two alkali ions (Ha, K). Their resuits for Na [l/15"-alumina were similar to those shown above for temperatures up to 300 K. However, an additional attenuation peak was found at temperatures above 400 K for frequencies in the range 10.4 to 31 MHz. For this, they assumed toz= 1 at the peak maxima and obtained an activation energy, E, of 0.387 eV and a prefactor, P, of 0.7× 10 t2 Hz. This high temperature peak was attributed to Na + ion interactions with grain boundaries and the lower temperature peak (E=0.183 eV, P = 5 × 1012 Hz) to ion hopping predominantly in the more abundant 15" component. From the arguments presented above, it seems likely that the low temperature peak was in fact due to the 13 component. The assignment of the high temperature peak to a grain bourdary effect is not regarded as being conclusive, since polycrystalline samples are particularly prone to moisture absorption, resulting in a high temperature attenuation peak [ 7,37 ]. The data for a K-~/15"-alumina sample also showed two attenuation peaks moving to higher temperature with increasing frequency. Again, the low temperature peak (E=0.27 eV, P = 2 × 1012 HZ) was attr~ uted to K + ion migration in the bulk and the high temperature peak (E=0.37 eV, P=().8 × 10'2 Hz) to grain br dary effects. The ~ata for a mixed-alkali, N a / K 15/15"-alumi~ia showed the presence of four temperature dependent peaks. These fell into two groups: a low temperature pair and a high temperature pair. The low temperature pair (E=0.239 eV, P = 6 × I 0 '2 Hz and E=0.252 eV) was attributed to the hopping ofNa + and K + ions respectively. It was suggested that the activation energy for the Na ÷ ion hopping had been modified by the presence of the K ÷ ions (the mixed alkali effect) whilst the reverse was not the case. Clearly these complex systems exhibit a range of effects and the ultrasonic technique offers considerable scope for their investigation. The magnitudes of the attenuation peaks in the

275

polycrystalline samples, fig. 6a, are almost an order of magnitude greater than in single crystal 15 material, fig. 3. This is all the more remarkable since apparently only the minority 15component contributes to attenuation in the rJlycrystalline materials. In the absence of studies ~:i?the orientation and mode dependence of attenuation in single crystal samples these effects remain uaexplained. Surface acoustic wa~ze attenuation measurements [40 ] in sputtered thin film samples of Na a-alumina produced an attenuation peak at similar temperatures to those found in bulk specimens [ 10 ], with an activation energy of 0.13 eV. The films were polycry.stalline but were oriented so that the conduction planes of the 15-alumina lay parallel to the film surface; the sound waves were propagated along the conduction planes, as in the single crystal study [ 9,10 ]. A surprising result was that the attenuation peak in the thin film sample showed no broadening from a single relaxation time, Debye peak. The reasons for this difference me not underst, d.

4.3. Agl-Ag oxysalt glasses The AgI-Ag oxysalt glasses are a family of Ag+ k,n ~.onducting solid electrolytes with room temperatare conductivities, 0.01-0.1 mho/cm, co' npamble to that of the [3-aluminas. Two sets of ultrasonic attenuation studies have been reported oa these glasses. Measurements of the attenuation of longitudinal waves, at 1.25 and 10 MHz, for a sample of composition, 3AgI.Ag2MoO4 [21 ] are shown in fig. 7. The two attenuation peaks shown in the figure are in many respects similar to those found in the [baluminas. The peaks appear to be thermally activated and are considerably broader than a Debye peak, shown dashed for comparison. However, the magnitudes of the attenuations are much greater than in the [l-aluminas. The 10 MHz data are incomplete because the attenuation at the peak was so large that the ultrasonic pulse-echo could not be monitored over that part of the temperature range. The amplitude of the 1.25 MHz peak is similar to that found in polycrystalline I]/15"-alumina at about 70 MHz, fig. 6a, and in single crystal fi-alumina at about 200 MHz, fig. 3. Since peak ~ :enuation increases linearly with frequency, eq. (4), the relaxation strength, A, appears to be about two orders of magnitude greater in

276

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this silver glass than in the ~aluminas. It is tempting to attribute this difference to a stronger coupling, B in eq. (5), of the sound wave to the lattice in this ok:~s. However, aceotmt must be taken of the other factors in eq. (5), in particular the velocity of sound, y.

The longitudinal wave velocity in I~-alumina, perpendicular to the c-axis, is 10.5 km/s [47 ] whereas in the silver glass it was found to be about 1.4 km/s [38 ]. Since the sound velocity appears as a cube in eq. (5), this factor alone would alter the relaxation strengths of the two materials by a factor of over 400. Values of the. terms in eq. (5) are listed in table 1 for the silver glass, single cryst,d 13-alumina and polycrystaUive Na 13/13"-alumina. Estimates of mobile ion concentration, N, are taken from [ 41 ]; in the ease of polycrystaUine 13/13"-alumina, it was assumed that only the 30%, 13, component contributed to the at-

tenuation. The derived deformation potentials, B, are comparable in magnitude, confirming that the large differences in attenuation are attributable to other factors in the expression (5) for A. The curve fitted to the 1.25 MHz peak, fig. 7, was obtained using eq. (8) with m=0.8 and n=0.7 and assuming an activation energy of 0.27 eV [21 ]. The xponent n is identical to the value of n2 used in eq. (9) to fit the frequency dependence of the ac conductivity. Estimates of mechanical and electrical relaxation frequencies, COp, were in close agreement [2 ! ]. Again, as in I$-alumina, this evidence strongly suggests that the ultrasonic and electrical techniques have probed the same basic ion hopping process in this ion cor,ducting glass. Measurements of the attenuation of both shear and longitudinal sound waves, with frequencies in the range 5 to 45 MHz, in AgI-Ag borate glasses have been reported by Carini ¢t at. [20,42,43 ]. Several series of glasses, with either variable AgI content and /or variable Ag20:B203 ratio were studied. All the glasses showed a single, broad, thermally activated attenuation peak of amplitude comparable to that found in the AgI-Ag molybdate glasses [ 21 ]. In the series of glasses, x ( A g I ) - ( 1 - x ) (Ag20"B203):x=0.3, 0.5 and 0.7 [20], the temperature of the attenuation peak fell with increasing AgI content (from 288 K for x=0.3 to 202 K for x = 0.7), ia line with a corresponding decrease in activation energy (from 0.36 to 0.24 eV). In addition, the amplitudes of the attenuation peaks increased significantly with AgI content, despite the fact that the total concentration ofAg + ions is about constant for this series. Whilst the authors gave no clear explanation for this effect, it is noted that they report

Table 1 Values of deformation potentials, B, obtained from eq. (5) using densities, p, and sound velocities, v, shown, carrier concentrations, N, from ref. [ 41 ] and the ultrasonic attenuation peaks in the appropriate figures. Conductor Na 13-alumina single Xtal Nal3/13"-alumina polycrystalline 3Agl'Ag:MoO4 glass

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D.P. Almond, A.R. Weo~t/U!trasonicattenuation studies

the sound velocity to decrease from 3.51 km/s to 2.182 km/s as x increases from 0.3 to 0.7. The lower value of the velocity for the x = 0 . 7 composition would be expected to lead to an increase in relative peak attenuation by a factor of about 4, as compared to the experimentally observed value for this ratio of about 3. In the series of glasses, x ( A g I ) - ( l - x ) (Ag20"2B203), x--0, 0.2, 0.4 and 0.6 [42], very similar effects were observed. The broad attenuation peaks were fitted well to eq. (7) using a distribution of relaxation times obtained from a Gaussian distribution of barrier heights. The deformation potential, B, was estimated by making the assumption that all the Ag + ions present contributed to the relaxation loss; a value of 0.33 eV was obtained. In the series of glasses, x ( A g I ) - ( 1 - x ) [y(Ag20) - (l - y ) (B203) ]:0.1 < x < 0 . 7 and 0 . 2 < y < 0 . 5 [43], similar effects were again observed with increasing AgI content, x. However, the ability to alter the total Ag + ion content, through varying y, showed the peak amplitudes to depend linearly on the total Ag+ ion concentration. Data for all the glasses were fitted by the Gaussian distribution of barrier heights medel. The activation energies obtained from the fittings had values in the range 0.251 to 0.528 eV, similar to those obtai,led from electrical conductivity measurements. Attempt frequencies were also obtained from the fitting and had values of about 1 × 1014 Hz. These attempt frequencies and activation energies were then used to estimate the average ion jump distances in the glasses, giving values of about l to 1.5 A. All the deformation potentials were calculated to fall within the range 0.31 to 0.47 eV. It is particularly notable that the deformation potentials obtained for the silver glasses and the crystalline ~aluminas all fall in the comparatively narrow range 0.33 to 0.93 eV. It is arguable, however, that there is considerabie uncertainty in these values since, although N B 2 is obtained experimentally, an independent estimate of N is required in order to obtain B. Carini et al. [20] assumed that all the Ag + ions contribute to N, whilst we have taken only that portion indicated to participate in the conduction process. 4.4. Other ultrasonic attenuation studies

RbAg415 was. studied over the temperature range

277

170 to 370 K and frequency range 15 to 45 MHz [44 ] and over the temperature range 120 to 300 K at 15 MHz [45 ], but no complete anelastic relaxation peak was found over these temperature ranges; relaxation effects ~ere observed at certain temperatures, but these were associated with phase transitions in the RbAg4Is. There have been a number of studies of ultrasonic attenuation in [l-alumina at temperatures < 50 K [46,47 ]. The phenomena observed at these temperatures are characteristic of coupling to a glassy two-level system and are outwith the scope of this review. 5. Conclusions

Ultrasonic attenuation peaks associated with ion hopping have been observed in two different groups of solid electrolytes, the beta aluminas and silver ion glasses. In both cases, the peaks are thermally acti~/ated with a similar activation energy to that observed for electrical conduction. Selection rules for the possible occurrence of attenuation peaks have been discussed and it has been shown that the non-occurrence of peaks in ~"-alumina is consistent with the symmetry equivalence of sites involved in conduction pathways through the ~"-alumina structure. The factors responsible for the magnitude of the attenuation peaks, or the relaxation strength, A, have been discussed. It has been shown that the principal factor responsible for the variation in peak amplitude in different materials is the sound velocity in the direction of propagation, since this appears as an inverse cube term in the expression for `4. Other factors which affect ,4 include, the concentration of mobile ions, temperature, sample density and the deformation potential, B. From the attenuation peaks, relaxation times characteristic of the hopping mechanism may be obtained. These are found to be very similar to those obtained from electrical relaxation measurements and provide a means for separating the effects of cartier concentration and mobility or hopping rate. The ultrasonic attenuation peaks are markedly non Debye-like. Their shapes may be described by a Jonscher universal response function as well as by a traditional distribution of relaxation times or of relaxation energies approach. For a given sample, there is a close similarity between the shapes of the me-

278

D.P. Almond, A.R. West~Ultrasonic attenuation studies

chamcal and electrical relaxation peaks and the values of the power law exponents used to describe them.

References [ 1 ] J.E. May, IRE Natl. Cony. Ree. 6 (1958) 134. [2] E.P. Papadakis, J. Acoust. Soc. Am. 42 (1967) 1045. [ 3 ] H.L McSkimin, J. Acoust. Soc. Am. 33 ( 1961 ) 12. [4] R.L. Roderick and R. Truell, J. Appl. Phys. 23 (1952) 267. [5] B.B. Chick, G.P. Anderson and R. Truell, J. Acoust. Soc. Am. 32 (1960) 186. [ 6 ] J. Blitz, Fundamentals of ultrasonics (Butterworths, London, 1963). [7] D.P. Almond, H. CaUica and A.R. West, Mater. Res. Bull. 16 ( 1981 ) 117. [ 8 ] R. TrueU and W. Oates, J. Acoust. SOe. Am. 35 ( 1963 ) 1382. [9] D.P. Almond and A.R. West, Phys. Rev. Letters 47 ( 1981 ) 431. [ 10] D.P. Almond and A.R. West, Solid State lonics 3/4 ( 1981 ) 73. [ 11 ] J.D. Llewellyn, H.M. Montagu-Pollock and E.R. Dobbs, J. Sci. Instrum. 2 (1969) 535. [ 12 ] R. Truell, C. Elbaum and B.B. Chick, Ultrasonic methods in solid state physics (Academic Press, New York, 1969). [ 13] R.L. Roderick and R. Truell, J. Appl. Phys. 23 (1952) 267. [ 14] H.B. Huntington, J. Acoust. Soc. Am. 22 (1950) 362. [ 15 ] C. Zener, Elasticity and anelasticity of metals (University of Chicago Press, Chicago, IL, 1948). [16] J. Jackle, L. Piche, W. Arnold and S. Hunkliner, J. NonCryst. Solids 20 (1976) 365. [ 17 ] A.S. Nowick and B.S. Bert'3', Anelastic relaxation in crystalline solids (Academic Press, New York, 1972 ). [ 18 ] R.A. Huggins, Diffusion in solids, recent developments, eds. A.S. Nowick and J.J. Burton (Academic Press, London, 1975) p. 445. [ 191J.B. Wachtman Jr., Phys. Rev. 131 (1963) 517. [ 20 ] G. Carini, M. Cutroni, M. Federico and G. GaUi, Sohd State Commun. 44 (1982) 1427. [21 ] D.P. Almond, G.K. Duncan and A.R. West, J. Non-Cryst. Solids 74 (1985) 285. [22] A.K. Jonscher, Colloid Polym. Sci. 253 (1975) 231. [23] L.A. Dissado and R.M. Hill, Nature 279 (1979) 685. [24] J.C. Dyre. Phys. Letters 108A (1985) 457. [25] J.C. Dyre, J. Non-Cryst. Solids 88 (1986) 271. [26] R.G. Palmer, D.L. Stein, E. Abrahams and P.W. Anderson, Phys. Rev. Letters 53 (1984) 958.

[271 D.P. Almond, C.C. Hunter and A.R. West, J. Mater. Sci. 19 (1984) 3236. [281 L.A. Dissado and R.M. Hill, J. Chem. Soc. Faraday Trans. 2, 80 (1984) 291. [291 D.P. Almond and A.R. West, Solid State lonics 11 (1983) 57. [301 D.P. Almond and A.R. West, J. Electroanal. Chem. 186 (1985) 17. [31l M.S. Whittingham and R.A. Huggins, in: NBS Spec. Publ. 364, solid state chemistry, eds. R.S. Roth and S.J. Schneider (Natl. Bur. Standards, Washington, D.C., 1972) p. 139. [ 32 ] S.J. Allen Jr. and J.P. Remeika, Phys. Rev. Letters 33 (1974) 1478. [33] M. Barmatz and R. Farrow, in: 1976 Ultrasonic Symposium Proc. eds. J.deKlerk and B. McAvoy (IEEE, New York, 1976) p. 62. [ 34 ] J.H. Simmons, A.D. Franklin, K.F. Young and M. Linzer, J. Am. Ceram. SOe. 63 (1980) 78. [ 35 ] R.E. Walstedt, R.S. Berg, J.P. Remeika, A.S. Cooper, B.E. Prescott and R. Dupree, in: Fast ion transport in solids, eds. P. Vashishta, J.N. Mundy and G.K. Shenoy (North-Holland, Amsterdam, 1979) p. 355. [ 36 ] D.P. Almond and A.R. West, Solid State lonics 9/10 (1383) 277. [37] D.P. Almond and A.R. West, J. Phys. (Paris) C6 (1981) 187. [ 38 ] D.P. Almond, unpublished results. [39] N.D. Patel and P.S. Nicholson, Solid State lonics 22 (1987) 305. [40 ] K. Nobugai, Y. Nakagiri, F. Kanamaru, M. Tokumara and T. Miyasato, in: 5th Intern. Conf. Phonon Scattering in Condensed Matter (Illinois, USA, 1996). [41 ] D.P. Almond and A.R. West, Solid State Ionics 23 (1987) 27. [42] G. Carini, M. Cutroni, M. Federicc, G. Galli and G. Tripodo, J. NonoCryst. Solids 56 (1983) 393. [43] G. Carini, M. Cutroni, M. Federico, G. GaUi and G. Tripodo, Phys. Rev. B30 (1984) 7219. [ 44 ] M. Nagao and T. Kaneda, Phys. Rev. Bl I (1975) 271 I. [45] L.S. Graham and R. Chang, J. Appl. Phys. 46 (1975) 2433. [46 ] P. Doussineau, R.G. Leisure, A. Levelut and J.-Y. Prieur, J. Phys. Lett. 41 (I980) Lo65. [47 ] P. Doussineau, C. Frenois, R.G. Leisure, A. Levelut and J.Y. Prieur, J. Phys. (Paris) 41 (1980) 1193. [48] D.P. Almond and A.R. West, J. Electroanal Chem. 193 (1985) 49.