The effect of homovalent substitution on the ionic conductivity of KClxBr1−x mixed crystals

Solid State lonics 28-30 (1988) 1310-1316 North-Holland, Amsterdam

THE EFFECT OF HOMOVALENT ON THE IONIC CONDUCTIVITY

SUBSTITUTION O F KC|.BrI_~ M I X E D C R Y S T A L S

Oivind JOHANNESEN

Institute of lnorganic Chemistry, Universityof Trondheim, 7034 Trondheim-NTH, Norway Received 5 August 1987

The ionic conductivity of KCI,Br~ _, mixed crystals, try has been measured from 780 K to close to their melting points. The maximum conductivity of KClxBr~_xwas never far outside the range of that of 1he component crystals. It is difficult from the conductivity measurements alone to confirm a pronounced increase in the vacancy concentration in mixed crystals due to the effect of homovalent substitution. The conductivity results of KCIxBr~_~have been related to the previous theoretical considerations reported for KBr.,.l~_.,.mixed crystals.

1. Introduction

There has been a general disagreement about the effect of homovalent substitution in mixed alkali halides in the literature. Whereas one group of investigators claims that the mixed crystals can contain a considerably higher vaca~:cy concentration than the component crystals [ 1-5 ], the other group has questioned such an argument on the basis of the lack of experimental evidence [ 6-11 ]. For instance when the conductivity measurements by Schulze [ 12 ] do not show any significant increase in the conductivity of KCIxBr~-x compared to that of KCI and KBr, Lidiard [6] concludes that these results do not support a substantial increase in the vacancy concentration for the mixed crystals. It is therefore of great interest when Shahi and Wagner [ 13 ] report a substantial increase in the ionic conductivity of polycrystalline KBr.,.I~ -x compared to that of pm ~ KBr. These authors suggest that the IIII~III~IUIII L~;tW~II

UI~; IIU~L ;'IIIU U I ~ d U i , J~1:1[ UI-I O i l s

causes a strain field in the lattice which can be relieved by the creation of vacancies. On the basis of the pronounced increase in the ionic conductivity for tl~e KBr-KI system, Shahi and Wagner have questioned th~ ~ondu~iivily measurements on KCl,Br~_, by Schulze [ 12] and suggested tha~ the conductivity should be remeasured [13]. In view of the practical applications of solid elec-

trolytes exhibiting high ionic conductivity a program has been initiated at this institute to further investigate the effect of homovalent substitution in mixed alkali halides. Previous papers have reported the detailed study of KBrxI ~_x [ 14-16 ]. This paper gives the results for KClxBr~ _~.

2. ExperimenNl The mixed crystals were grown by the Czochralski method from KC1 and KBr with 99.999% stated purity with respect to metals. The detailed experimelLtal procedure is given in ref. [16]. Previously Smakula et al. [ 17] found it impossible to grow KClxBh-x mixed crystals without cracks, and on this basis they concluded that the crystals were extremely strained [ 17 ]. We encountered no problems in grow~ng the crystals, and the crystals stayed clear and without cracks when kept under dry. conditions in a ~UVg-UU~

IUI lllUlg

Llli~ll U l l g y ~ i a l g l t L g l ~1 yaLO.l ~ I U v v u l ,

X-ray diffractometer analysis shows that the variation in the lattice constants as a function of composition tbllows Vegard's law [18]. The X-ray analysis does not indicate any phase segregation even at room temperature. The composition of the samples was determined with potentinmetric titration. The electrical conductivity was measured with a standard two-probe ac technique at 1592 Hz (Wayne

0 167-2738/88/$ 03.50 © Elsevier Science Publishers B.V. {No~h-HoHand Physics Publishing Division)

O. Johannesen/lonic conductivity of KClxBrt_~ mixed crystals

131

2

2C "

KCIo 71Br,o 29

557°C

exp (ASs + 2As~ (o_ 5 \al / N~v~.l \ j

trl

12~ kNz

(~r-/s + 2Ah~ '~

X exp\

~

2,

(1)

tO

o I.

to

f

I 0

,,,

AA.

1

,,,1.0 ,v,,,,,,, 2.0

~ 2kHz

,I,/,

3.0

z'(,105).0 Fig. 1. Complex impedance plot showing the arc corresponding to the crystal behaviour of KClo.TtBro.:9 mixed crystal at 557°C. The circuit to which the data have been fitted is included in the

figure. Kerr B 905A). Daring the measurements special precautions were taken to avoid the effects of sublimation and plastic deformation of the samples. A description of the experimental procedure is given in refs. [ 15,16 ]. More thorough measurements based on impedance spectroscopy [ 18 ] applywg a Solartron 1172 and Wayne Kerr 6425 confina that the electrical conductivity at 1592 Hz is the true crystal conductivity for the KCI~Br~ _~ samples in the present study, i.e. for temperatures from 780 ~o 1020 K. A typical impedance plot is shown in fig. ~ for KClo.7~Bro.29 which includes the equivalen: circuit to which the experimental data can be fitted. No depression of the circular arc was observed. The conductivity, a, measured at 1592 Hz is given by a = ( Z ' ) - ~, where Z' refers to the impedance at phase angle ~o= 0. It should be noted that a slight frequency dependence of the conductivity was observed at lower temperatures ever~ for the component crystals [ 18 ].

where the expressions of ax and a~ refer to the Lidiard model for Schottky defects [ 61. The symbols are defined as follows#t: ax is the anion-cation separation distance, Nx is the number of cation (or anion) sites per unit volume, V¢,xis the cation lattice frequency, ASs=Ss~-Ss, t and Z~s=k/s,~-Hs~ where Ss,x and Hs,x refer to the entropy and the enthalpy of formatio~ of Schottky defects (in KBr~I~_x). It is further noted that As¢=s¢.x-s¢,~ and Ahc = h¢,~- hc~, where &,xand h¢,x are the entropy and the enthalpy of cation migration (in KBrxI~_x). When disregarding the pre-exponential factor in eq. (1), i.e. 2

(ax~ Nxv¢,,~exp(ASs+2As~) ka-~/ N~ Vc,, 2k

= l,

(2)

this equation reduces to ax /"AHs + 2Ahc "~ a, - exp,, }-~-~ ) .

(3)

It has also been shown [ l 6] that for ti~e calculation ot cr,/a~ wnm me same s'.~? ,,,,,- . . . . . . . . . . . ax and cr~ are due to the migration of cation vacancies only, it follows that AHs+2Ah¢ - - -T[-T~:( . +2h~x) - 7 - - /-Is~ Tt

_arm( , T; , H s l +2h~,l),

(4)

when log axT~ is constant (i.e. log ax T'~= 10 . 3 ohmcm -~) at each temperature Z; and the apprexima3. Expressiens ef ~ / e ~ in the ~ntfins~c redden In previous papers [ 14-16 ] describing the ionic conductivity, a,., in KBrxI~ _x it has been shown that when crx of KBrxI~ _x and a~ of KBr in a limited intrinsic region are due to the migration of cation vacancies only, the ratio a A . / a ! at constant temperature T can be written as

LIUII

z2 ] bLl2ll I I U I U ~ .

For the K~3r,~,_, system it was further shown [ I~, !6] tha~ a p!ot of the temperatures T; as a function & t h e composition x gave a curve similar to that of the solidus temperature curve, i.e. Tx versus x for the KBr-VG system. Thus, each Tj~ has to satisfy T;.+ 7 = l~., where 7 is a constant, and ~ The subscript x refers to KBr,II _,, while x = ! refers to KBr.

O. Johannesen/lonic conductivity of KClxBr t_ x mixed crystals

1312

AT~ = Ti - T'x= T, - T,,. =aTm ,

(5)

where T~ refers to the melting point of KBr. Combination of eqs. (4) and (5) gives A/-/s + 2Ahc = const. "ATm,

(6)

where it should be emphasized that essentially the same mathematical expression (eq. (6)) can be derived from the entirely empirical expressions of Hs and hc for pure alkali halides reported by Barr and Lidiard [19] and Barr and Dawson [20] (i.e. HsOCTm and h~oc Tin). On the basis of the above considerations it was possible to calculate the enhanced conductivity in the KBr-KI system [16], i.e. ax/at by combining eqs. . . . . gxveg (3), (4) and ( "o ~. ."rl,;~ • ~=exp (Hs t +2he, \ ~k'~t '

aTm/,

(7)

Table i Arrhenius activation energies in the intrinsic region a = (C/T) × exp ( - D/kT).

Sample

D (eV)

KBr singlecrystal KCI singlecrystal KCI singlecrystal KCI singlecrystal KClo.tsBro.s4singlecrystal KClo.32Bro.6ssinglecrystal KClo.TtBro.29 single crystal

2.04 present study a

2.10 presentstudy 2.10 ref. [25l 2.09 ~f. [ 27 l 1.94 presentstudy 1.94 presentstudy 2.00 presentstudy

"~ A similar comparison of D values for KBr has been performed in ref. [16].

to further investigate the validity of the axla~ (or ax/ao) expressions.

4. Ionic conductivity of KCI~Brt_~ in the intrinsic region

or more generally when T[ - T~ # Tt - T~,

(ns,, + 2h .,

aX=exp \ 2k--T~ aTm:. O"!

(8)

Thus, it is possible to calculate ax/a~ of KBrxla_~ wherL knowing the enthalpy terms of the component crystal K33r and the values of T'~andATm(or AT&). A similar mathematical deduction of cL,/~o (ao refers to the conductivity of KI) gives

~o =eXp ( Hs ° + 2h~ °

\ 2k- o arm,

(9)

whc'e Hs,o and h¢,o are the enthalpy terms of pure KI. T6 refers to the temperature of KI when log aT;= log aT'~= constant (i.e. log aTx= l r)'3 ~ - 1 cm -~) and ATm= T o - T x is the difference in temperature between the melting point of K1 and the solidus temperature Tx of KBr,d: ,~.. It is fu~he, noted that when ATm¢:ATm (i.e. 7;,- T, ,-a 7;;- Ti ) it follows that cr,/rz, of KBrd~ ......... at the temperature T is given by

~7=exp ( &rs,o+ 2h~.o aT;.)., ., The ,.onductivity resuhs for KCI,.Br~_,. will be discussed below in terms of zhe sanic mcutcuva~ iltou~t

The purpose of this investigation is to study the effect of homovalent substitution on the ionic conductivity of KCIxBr~-x mixed crystals in the intrinsic region. This is mainly due to the complexity when interpreting the results [ 15,16 ]. Several investigators [3,1!,21-24] have performed measurements on mixed alkali halides in the extrinsic region without emphasizing the problems concerning the impurity and dislocation concentrations, ctc. Additional problems arise when the measurements are based on sintered samples only [ 13], where the effects of grain boundaries and particle sizes may play an important role. One should therefore be careful not to draw too definite conclusions about the effect of homovalent substitutions in mixed crystals on the basis of ionic conductivity measurements in '~he extrinsic region only. A general discussion of the extrinsic and intrinsic regions is given in rcls. i lo The ionic conductivity of KC1,Bru _~ mixed crystals was measured as a function of temperature from 780 to 1020 K. The ionic conductivity measurements of the component crystals agree well with those of Chandra and Rolfe The Arrhenius activation energies in the intrinsic region are given in table 1. Fig. 2 shows the log a T versus 1/T curves from 780 to 945 K. Measurements have been pe~ormed

[25].

O. Johannesen/lonic conductivity of KCl.,.Br~_x mixed crystals i 10-4 50

w

o

K C i o 3 2 Bro.68

+

KCIo.~6 B r o 8 4

i

1313

L){(..

i

i

i

I

i

i

_

918 K

-~

KClo.7~ B r o a ~ ,i

KCI •, K B r

"7 E

10-2

i

m

10-500

__

//

O

>: tx

Et -

t.) 10 - 6 0 0 c) z O o

O

_.1

"7

Eu

I

0

1 BOO K

< o

p-

re

>

ohi .-1 lal

F-

o :D t~

Z

10 - 6 . 5 0

0 0

.J

o re F-

o

10-3

--

,

_L_

0

01

I _ _ L ...... L ..... J- ...... 1...........__.L____L

0.2

03

04

05

06

07

MOLE

10

1.1

12

1.3

1031 T, K

Fig. 2. Electrical c o n d u c t i v i t y o f KCI.,.Br,_, m i x e d crystals (0 ~
on three cr3,stals for each composition and the reucibili~y of the measurements was extremely good. The m a x i m u m increase in the ionic conductivity of KCI.,Br~-x mixed crystals compared to that of the component crystals at each temperature was observed at x = 0 . 3 2 , i.e. for KClo.32Bro.68. Thus, at 800 K the m a x i m u m increase in the ionic conductivity r~f KClo.32Bro.68, 000.3z~!,2, that of pure KCI, 00,, was ~o.s:/~r~ =2.5. At 918 K drO.32/O'l =2.4, which agrees

09

J 10 KCI

KBr

.J w

0.8

I

FRACTION

KCI

I::ig. 3. Electrical conductivity, a,, o f KCI~Br~ , as a function o f mole fraction o f KC1 at 800 and 918 K. @ a n d ~, are the experimental o , at 800 a n d 918 K, respectively, its:, = 2 . 5 4 eV and h,~ =:0.73 eV for KCI. H s . o = 2 . 3 0 eV and h , . ~ = 0 . 6 6 eV for KBr. O and [] are ~r, calculated from eq:'. (7) and (8), respectively. \~hen t r , = 2 . 9 6 4 2 X 10 -7 f~-~ crn -t is k n o w n at 800 K. + and A are o'x calculated f r o m eqs. (9) a n d (10), :respectively, w h e n ~o = 5.221 X I 0 - 7 f~ - , c m - ' is k n o w n at 800 K. " a n d [] are ao.3_, calculated f r o m eqs. (7) a n d ( 8 ) , respectively, w h e n cry= 1.0282X 10 - s ~ - ' c m -~ is k n o w n at 918 K. X is ao3z calculated from eq. ( 9 ) when a o = 1.8496X 10 -5 ~]-t crn-~ is known at 918 K.

reasonablv well with the observed O°0.32/0"1,~2.1 reported by $chulze [ 12]. Fig. 3 shows the variation in the ionic conductivity, ~,, of KCIxBr~_, mixed am~ 9~8 K. It is of great interest to per~brm the same type of analysis of the conductivity data for KC1,Br, _, as for KBr,.I~ _, [ ! 4 - i 6 ] . Thus, when keeping log 00T con,;tant ( i . e . l o g o T = l 0 - : f~ - n c m - ~ ), the plot of the corresponding

temperatures

7.',. v e r s u s c o m p o -

x is compared with the solidus temperature curve, T,, of KC1,Brt _, as reported by BeHanca [ 26 ]. sitions

1314

O. Johannesen/lonic conductivity of KClxBr ~_x mixed crystals

790 °0770~"

I

I

I

I

"~

I

/

I 769 4

f

750

/ 01/

~ 739.6 X ~7 ~J ~- 710

I

I

0 KBr

3~ i

I 02

I 04

I

I 06

I

MOLE FRACTION KCI

O't

/2.54+2X0.73 \ = exp k2k × 800 × 846 × 41.4)

0 I

(Hs.t + 2h¢.t 0.0.32 =exp \ Tk-'TT~ AT.,)

I 08

10 KCI

Fig. 4. The variation of T',. ( ° C ) + 1 9 6 .6°C ( O ) and T~ (°C) + 185°C ( x ) as a function of the mole fraction x of KCI. The solid line refers to the solidus temperature curve of Bellanca

[26]. The results of Bellanca [26] agree well with recent measurements in our laboratory [ 18 ]. Fig. 4 shows the results of the analysis where the open circles refer to the application of the melting point of KCI, Tt, as the reference temperature, i.e. each Tx has been displaced upwards by y = T~- Ti= 1042.8-846= 196.6 K (196.6°C). In a similar manner the crosses in fig. 4 refer to the use of the melting point of KBr, To, as the reference temperature, i.e. each T~, has been moved upwards by y = T o - T ~ = 1 0 1 2 . 8 - 8 2 7 . 8 = 1 8 5 . 0 K (or 185°C). A more complete description of ~his type of analysis is given in refs. [ 14-16 ]. Although there seems t9 be a reasonable agreement between the solidus temperature curve, T, versus x and the "Tx versus x curves" in fig. 4, it should be emphasized that the agreement is generally less than for the similar type of analysis of the ¥..BrAt _~ system. For K B r J t - x the "T~, versus x curves" were nearly coincident with solidus ~emperature curve [ 14-16]. On this basis one should preferably apply the eq. (8) versio:l when calculating cry/cry, i.e. ATm *: A T e . However, referring to fig. 4 it should be ~ressed that when using the melting point of KBr as a reference temperature, the "T~ versus x curve" shows a considerably better agreement with the solidus temperature curve in the limited composition region 0~
=4.1(2.5),

where the values reported by Beniere et al. [27] have been applied for the enthalpy terms of KCI, i.e. Hs.t=2.54 eV and he.t=0.73 eV. The value of ATm = 41.4 K has been estimated from the phase diagram of Bellanca [ 26 ]. The value in parentheses refers to the observed ao.32/0.t value. A similar calculation of 0.o 32/0.t at 800 K applying the eq. (8) version gives 0"0"32 =exp (2"54 + 2 X 0.73

at

\2-~80"0"X8-'~ X ( 8 4 6 - 8 1 5 )

)

=2.9(2.5). The calculation of ao.a2/0.o at 800 K applying the eq. (7) version gives

(Hso+Zh¢o ) ~ k - ~ o ' Arm

ao32 =exp \ o"o

( 2.3+2×0.66 ) =exp \2k---X8-0-0-X8-~.8 × l 1.6 = 1.4(1.4), where the values reported by Brown and Jacobs [ 28 ] have been used as the enthalpy terms of KBr, i.e. Hs,o=2.3 eV and h¢.o=0.66 eV. AT.,= 11.6 K has been estimated from BeUanca's phase diagram [ 26 ]. 1 he agreement between the calculated and the observed cre.32/cr o values agrees well with the comment above about the similarity between the solidus temperature curve and the "T~, versus x curve" for 0 .<.x~<0.32. The results of the detailed calculations of crx/at and at 800 and 918 K are given in table 2. The resuits are further shown in fig. 4 where a comparison

ax/ao

the calculated conductivity data. The results at 918 K will be commented upon beluw. S~ Concluding remarks It should be emphasized that the calculations of

I r ~-x are strongly depencr,icr~(or cr,ic%) of KCi~B

O. Johannesen /Ionic conductivity of KClxBr ~_.,. mixed crystals

1315

Table 2 The calculation of a., la, and axlao of KClxBrt-x at 800 ;.and 918 K.. The AT,. values have been estimated from ref. [ 26]. q'h~. T', value when log aT, =const. (i.e. 10 -x o h m - ' c m - t ) are T~= 827.8 K, T~,.,,= 817.0 K, Ta x~=815.0 K, T67t =824.4 K a n d ~'i =846.0 K. Values in parentheses refer to the observed O'x/O't and a~/ao ratio.

age, ao.t61at ao.~2lat ao.~21at aO.Tl/tl I

(800 K) (800K) (800K) (918 K) (800 K)

Eq. (7)

Eq. (8)

4.1(2.5) 3.4(2.4) -

1.9(1.8) 2.7(2.4) 2.9(2.5) 2.5(2.4) 2.1(1.9)

dent on the choice of the enthalpy values of the component crystals (and for eqs. ( 7 ) and ( 9 ) on the phase diagram). Referring to table 2 it is shown that the calculated a,,/a, (or a,-/ao) agrees well with the observed values at 918 K. However, at this temperature the anion conductivity in KC1 (or KBr) is expected to contribute and our basic assumptio~ of a pure cation conductivity can be questioned. Assuming that the migration enthalpies of cations and anions in KCI (or KBr) are essentially the same, i.e. h¢.~=h~.~ (or hc.o=ha.o), it follows that the original ax/a~ (or ex/ ao) expression is not restricted to ?.he limited intrinsic regions of the cation conductivity, but applies at higher temperatures as well. Thus, the a.,./e~ expression of KCI,Brt _,. becomes =exp\

2kTTI ATm ,

(11)

atlao ao.tdao ao.~.,lao ao.x,.lao

(800 K) (800K) (800K) (918 K) ao.TtlO'o (800 K)

Eq. (9)

Eq. (10)

1.3(1.4) 1.5(1.5) 1.4(1.3) -

0.6(0.6) 1.4(1.4) 1.5(1.5) 1.4(1.3) 1.1 (1.1)

by Shahi and Wagner [13], i.e. a considerable decrease in the enthalpies of migration and fi~rmation (or a substantial increase in the vacancy concentration) compared with the component crystals. The above conductivity measurements of KClxBrt_x rather indicate only a minor decrease in the formation and migration enthalpies of defects in the KCI~Br,_ ,. mixed crystals with respect to that of KCI and KBr. (A more thorough discussion of possible defect clustering in mixed alkali haIide,s is given in ref. [ 16].) A minor change in the enihalpy terms of KC1,. Br,_x to that of KCR and KBr is also in agreemem with. the basic assumption (2). Eq. (2) can alternatively be expressed as

___d, t,~.,exp(ASs+2Asc ) .2.;,-

=1,

~2)

da uc, 1

where h=h¢,~ =ha, t. The assumption h¢,~=h~,t (or h¢,o=h~,o) for the component crystal agrees reasonably well with the anion migration enthalpies obtained from self-diffusion measurements. It is further noted that theoretical calculations by Catlow et al. [29,30] of the migration enthalpies in KCl and KBr seem to confirm that ha ~ h~ ~z. A more detai~ed cov.sideration of

where d~ and d, refer to the lattice parameters. Ref;erring to the maximum increase in the conductivity, ao.3z/at, the X-ray determination of KC l.,.Br t _,- gave d~=6.293 A and do.32=:6.498 A, (as a first approximation dt/dx is assumed to be temperature independent). From a Karo-H~rdy type consideration [32] one may suggest that v,:,o.32~Vc,,(~3.2× 10 t2 s-t). Minor changes in defect concentrations and migration enthalpies cart tentatively be related m minor

halides is given in ref. [16 ]. It is difficult from the slight {ncrease in ~, of KCI.,.Br~_:, compared to that of KCl and K.Br to confirm the suggested effect of homovalent substitution

References

~2 It is beyond the scope of this paper to discuss the applied interionic potentials used m refs. [29,30]. The reader should consult ref. [ 31 ],

[ 1 ] W.E. Wallace and R.A. Flinn, Nature 172 (1953) 631. [ 2 ] J.S. Ivankina. Izvest. Vyssh. Uchebn. Zaved., Fiz. I (1958) 101.

1316

O. Johannesen/lonic conductivity of KCl~Br s_ x mixed crystals

[3] T. Bima Sankaram and K.C~. Ba~sigir, Crystal Lattice Defects 7 (1978) 209. [4] R. Thyagarajan, J. Phys. Chem. Solids 27 (1965) 218. [5] M. Annenkov and V.A. Grisbukov, Izvest. Vyssh. Uchebn. Zaved., Fiz. ! 0 ( ! 967 ) 74. [61 A.B. Lidiard, in: Handbuch der Physik, Vol. 20, ed. S. Hiigge (Springer, Berlin, 1957) p. 246. [7] Y. Havea, Report on the Conference on Defects in Crystalline Solids, Bristol, 1954 (London, 1955) p. 261. [8] J.S. Woilam and W.E. Wallace, J. Phys. Chem. ¢~0 (1956) 1654. [91 J.E. Ambrose and W.E. Wallace, J. Phys. Chem. 63 (1959) 1536. [10] J. Arends, H.W. den Hartog and A.J. Dekker, Phys. Status Solidi 10 (1965) 105. [11] V. Hari Babu, U. Subba Rao and K. Venkata Ramiah, Phys. Status Solidi 28 a (1975) 269. [121 H. Schulze, Thesis (University of GSttingen, 1952). [131 K. Shahi and J.B. Wagner Jr., J. Phys. Chem. Solids 44 (1983) 89. [141 0. Johannesen, in: Proceedings of the 10th International Symposium on Reactivity of Solids, Dijon, France, 1984, eds. P. Barret and L.C. Dufour (Elsevier, Amsterdam, 1985) p. 377. [15] O. Johannesen and M. McKeh,y, Solid State Ionics 17 (1985) 251. [16] O. Johannesen and M. McKelvy, J. Phys. Solids 3 (1986) 265.

[17] A. Smakula, N. Maynard and A. Repucci, J. Appl. Phys. 33 (1962) 453. [181 0. Johannesen, unpublished results. [191 L.W. Ban"and A.B. Lidiard, in: Physical chemistry: an advanced treatise, Vol. 10, eds. H. Eyring, D. Henderson and W. Jost (Academic Press. ~ew York, 1970) p. 15~. [201 L.W. Ban" and D.K. Dawson, Proc. Brit. Ceram. Soc. 19 (1971) 151. [211 U.V. Subba Rao and V. Hari Babu, Crystal Lattice Defects 8 (1978) 21. [221 K.S. Cholokov and V.A. Gfishukov, Izvest. Vyssh. Uchebn. Zaved., Fiz. 6 (1970) 145. [23] E.K. Zavadovskaya, V.A. Gdshukov and K.S. Cholokov, Izvest. Vyssh. Uchebn. Zaved., Fiz. 12 (I 970) 92. [241 R. Mercier, M. Tachez, J.P. Malugani and G. Robert, Solid State lonics 15 (1985) 109. S. Chandra and J. Rolfe, Can. J. Phys. 48 (1970) 412. ::6 ] A. Bellanca, Periodico Mineral l0 (1939) 18. [27] M. Beniere, M. Chemla and E Beniere, J. Phys. Chem. Solids 37 (1976) 525. [281 N. Brown and P.W.M. Jacobs, J. Phys. (Paris) 34 (1973) 437. [291 C.R.A. Catlow, K.M. Diiler and M.J. Norgett, J. Phys. C 10 (1977) 1395. [301 C.R.A. Ca[low, J. Corish, K.M. Diiler, P.W.M. Jacobs and M.J. Norgett, J. Phys. C 12 (1979) 451. [311 R. Eggenhoffner, F.G. Fumi and C.S.N. Murthy, J. Phys. Chem. Solids 43 (1982) 583. [32] A.M. Karo and J.R. Hardly, Phys. Rev. 129 (1963) 2024.