- Email: [email protected]
The M1−xM″xF2 + 2x (M = Ca, Sr, Ba, Pb; M″ = Th, U) solid solutions: electrical properties and clustering processes
SWD STATE Solid State Ionics 78 (1995) 315-333
IQNICS
(M TfieMl-xM’Y2+2*
= Ca, Sr, Ba, Pb; M” = Th, Uf solid solutions: electrical properties and clustering processes J.M. Reau, Xu Yong Jun, J. Senegas, P. Hagenmuller Institut de Chimie de la Mat&e
Condens6e de Bordeaux, Chliteau Briuazac, 33600 Pessac, France
Received 6 February 1995; accepted for publication 8 March 1995
Abstract CM = Ca, Sr, Ba, Pb; M” =Th, U) solid A comparative investigation of elect&al properties of the M,_,M~F,+,, solutions has been carried out. The application of the clustering process model shows that all these solid solntions appear as characterized by very close clustering processes. The electrical properties depend on short-range structure and average ~l~zability of the c&ionic sublattice. Keywords:
Fluorine; Ionic conductivity; Fluorite; Clustering
1. Introduction The fluorite-type M~_fxM~2fnF2+~x ((u = 1, 2) solid solutions disordered at long range are characterized, for very low substitution rates (X < 0.011, by creation of point defect pairs, and, for higher substitution rates, by fo~ation of extended defects, which influence strongly the ~onduc~vi~ of the F- ions El]. For x < 0.01 the replacement for instance of host cations by trivalent substitutional cations involves introduction of charge compensating excess anions into interstitial sites; these extra anions associated with substitutional cations constitute pairs of point defects either of ltlt type when their symmetry is tetragonal or of nnn type when it is trigonal [2,3]. For x > 0.01 more extended defects, i.e. clusters, are formed. Such clusters, labelled nr: n2: n3: pt+ are based on the association of n1 vacancies in the Fr normal positions (l/4, l/4, l/4) of the ~uorite-ape network, n2 FiI (l/2, U, t6: 0.35 < u 5 0.401, rz3 Fg (q, ul, ul: u1 = 0.41) and rz4 F; (uZ, u2, u2: 0.28
< v2 ,< 0.33) interstitial fluoride ions, close to one or several substitutional cations [4-61. Clustering is characterized by significant transfer of fluoride ions from normal positions into interstitial sites with formation of vacancies in the normal Ft sublattice. It results in an enhancement of the F- ion conductivity L71. The optimization criteria of fast fluoride anion transport properties in the M~?$l”2fnF2+an solid solutions (nonstoichiometry, cation polar&ability, size difference between the host and guest cations. . . > are also responsible for the establishment of short range ordering in such long range disordered phases. Clustering has actually an essential influence on transport properties. A clustering process model has been proposed relating in a continuous way the composition dependence of electrical properties and the progressive extension of clustering when x increases. The basis of this model is the detestation of a y(F& =f(x> function representing the (Fi& ions responsible for long range motions in the solid
0167~2738/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDZ 0167-2738(95)00111-S
316
J.M. Reau et al. /Solid
State Zonks 78 (1995) 315-331
solutions: this function is established from the variation of transport properties with substitution rate and anionic dist~bution between normal and ~terstitial sites [6]. Various clustering processes have been detected within the Mf?,M$fF,+, solid solutions. They depend on the nature of the cationic (M, M’) couples
(a) log
-2
Q
tQ.cm)
present [8-111. On the contrary the M~_f,M’$’ Fa+ax solid solutions where the M” substitutional cation is tetravalent have been only partly investigated [12,13]. Application of the clustering process model requires, as a matter of fact, both structural and electrical data relative to the whole composition range, in particular, to the highest substitution rates [6]. Thus cluster-
-1
-3
x80.18
,
0.22
-4
-5
-6
t
-7
a.1.5 Fig. 1. Variation of log CTversus T-l
I
2.0
103/~(~) -
t 2.5
3.0
for some compositions of (a> Bal_JhzF2+2x
3.5 and (bf
Bal_,UzF,+,,
solid SO~I&~I%
J.M. Reau et al. /Solid
State Ionics 78 (1995) 315-331
317
-3
-4
-5
-6
-i
\
I__ 2.0
2.5
3.0
103/T(K) 3.5
Fig. 1. continued.
ing
processes
have been proposed for the CM” = Th, II> and Sr,_,Th,F,,,, Ca1-xM~Fz+2x solid solutions on hand of neutron diffraction data [14-161 and the composition dependence of the electrical properties [17,18]: the progressive transformation with rising x of 4 : 0 : 3 : 3 clusters into 1: 0 : 3 : 0 ones appears as a process valid in the three solid
solutions. The knowledge of the anionic distribution within the Bal_JhxF2+Zx solid solution [16] has incited us to determine the composition dependence of the transport properties and the nature of the clustering process in Ba1_xThxF2+2x. In a final step a comparative study of the clustering processes in the Ml--xThnFz+zx (M = Ca, Sr, Ba, Pb) solid solu-
31s
J.M. Reau et al. /Solid
tions has been achieved. Our goal was to define the influence of the host cation on the clustering processes for the same tetravalent substi~tion~ cation. The enlargement of these results to homologous uranium ( + IV) solid solutions, M, _,U,F, + ax, will be simultaneously discussed using the composition dependence of electrical properties.
State Ionics 78 (1995)
31.5-331
2. Experimentasl The samples belonging to the Bar _$I~F2+ 2X (M” = Th, U) solid solutions used here for electrical measurements are similar to those previously prepared for neutron diffraction experiments [161. The materials have been synthesized from mixtures of
b-4 -2
AF$
(Ba,Th)
(ev)
-4 \
-5 0.
:Sr,Th) Lo-
-6 0.
\(Ba.Thf
0.
x I._
0.10
Fig. 2. Variation
0.20
of (a) log urn0 K and(b)
0.30
x :___ 0.10
A,?& as a function of x for M1_-sM:F2+2x
0.
(M = Ca, Sr, Ba, Pb; M“ = Th, U) solid solutions.
J.M. Reau et al. /Solid
State Ionics 78 (1995) 315-331
BaF, and M”F, fluorides in sealed gold tubes at 850°C during 15 h. XRD analysis of the samples obtained after quenching has confirmed the existence of disordered Bal_xThxF2+2n (0
319
(b> A fast increase of log a,,, k with increasing n, associated with a strong decrease of AE,, is observed for all M,_,MzF,+ 2x solid solutions (M = Ca, Sr, Ba; M” = Th, U) in the composition range (0.05 < x ( x,,,) for the barium and strontium materials and for (0.05 Within both strontium solid solutions x,,, (X,,, = 0.20) is close to xL and apparently does not depend on the nature of the substitutional tetravalent cation, i.e. Th4+ or U4+. On the contrary x,, and xL are clearly separated in the barium solid solutions which show wider existence ranges than the strontium ones. Besides x,, is slightly smaller when M” and is U: xmax = 0.20 and 0.18 for Ba,_,Th,F,+,, Ba, _xU~F2+2x respectively. (d) For a given value of x, whatever the substituting tetravalent cation, the ionic conductivity is all the
3. Results The variation of the conductivity with temperature is shown in Fig. la and b for some Bal_xThxF2+2x and Ba 1_ ,U, F2 + 2x compositions. In the investigated temperature range the temperature dependence of conductivity agrees with an Arrhenius-type law for each composition considered: (+ = a,, exp( - AE,/ kT) with R = 0.98 fitting. The results obtained for compositions are in agreement the Bar-rUxF2+2x with those previously given by Ouwerkerk [19] for single crystals corresponding to x < 0.16. Fig. 2a and b shows respectively the isotherm u-,, x and the variation of the activation energy AE, as a function of x for the Ba,_,M’:F,+,, solid solutions. For the sake of comparison the curves relative to Car _,Th< 6 Fz& ‘“Th” ; G 0.18)Y (o
x <
x $2
2+2x
0.235)
6,05_
(Pb,Th) (Pb,U)
(Sr.Th)
5.85
5.80
(55r.u)
i/-!---e ,
(OG
(Ca,Tb)
[17J8]
(a) Barium and strontium solid solutions show a conductivity maximum associated with an activation energy minimum for a value of the substitution rate x max
(ca.U)
x
f
I
0.05
Fig. 3. Variation MI-xM):Fz+zx DOI.
0.10
,
0115
0.20
I
I
0.25
0,30
-
of the parameter a, as a function of x for (M = Ca, Sr, Ba, Pb; M” = Th,U) solid solutions
320
J.M. Reau et al. /Solid
State Ionics 78 (1995) 315-331
higher as the size of the host cation is larger: o~~,~~,,) ’ @(S&‘) ’ @(Ca,M”)* (e) For a given x the conductivity of the uranic fluorides is higher than that of the thorium compounds when the host cation is Cazf or Sr2’. Conversely, when M is Ba, the highest conductivity is observed for the thorium fluorides. In Fig. 3 we have reported the variation of the unit cell parameter a, as a function of x for the (M = Ca, Sr, Ba; M” = Th, U) solid M1-JG,+,, solutions [20]: whatever the value of x, a higher conductivity is observed for the uranium phases when the fM2+ = M4’ + 2F-) formal substi~tion involves unit celf dilatation (M = Ca, Sr). On the contrary it
appears for the thorium phase when the substitution results in a unit ceil contraction (M = Ba). For a given substi~tion rate one may assume that for charge carrier mobility the broadest bottlenecks occur in the barium phases as they offer larger unit cells than in the strontium and calcium phases. Whatever the host matrix, replacement of Th4+ by U4+, the size of which is slightly smaller, represents only a secondary effect. The comparison of the electrical properties of the U> Ml-XM’: %.+2x (M= Ca, Sr, Ba: M” =Th, phases with those of the Pbl_nThxF2+2x (0 GX G 0.23) [21] arid Pbl_JJXF2+2, (0
f AE,(ev. 1
(Pb,Th) fi tPb,U) -3
0.70 (ca,u)
0.60
-6
-_
1 I
I
1
I
2
Pam')
I
3
Fig. 4. Variation of log qa,, K and AE, as a function of the average polxizability fluorides corresponding to x = 0.18 {M = Ca, Sr, Ba, Pb; M” = Tb, U).
4 (I’,) of the cationic sublattice
for the MI _,M~J?2+,,
J.M. Reau et al. /Solid
State Ionics 78 (1995) 315-331
not the most important criterion for optimization of the transport properties of such materials. The lead fluorides whose unit cells have sizes intermediate between those of strontium and barium phases show the best electrical properties (Figs. 2 and 3). Considering the polarizability of the different cations involved in this study (Table 1) [23], and assuming that for a given material the average polarizability of the cationic sublattice is such as: P, = (1 - x)P, + (X>P,!,, we have plotted in Fig. 4 the variation of log a,,, x and of hE, as a function of P, for the fluorides corresponding to x = 0.18. This substitution rate has been selected as it corresponds in one solid solution to the introduction into the host matrix of the maximum of substitutional cations: xL is actually equal to 0.18 in Ca,_,Th,F,+,, whose composition range is thus the smallest. AE, decreases and log oN, x increases regularly when P, increases. Transport properties of these solid solutions appear as tightly related to the average polarizability of the cationic sublattice. Furthermore, the increase of P, resulting from the replacement of Th4+ by U4+ decreases progressively from calcium to lead: 7.3, 5.5, 3.5 and 2.4% when M is by turns Ca, Sr, Ba and Pb (Table 1). As a consequence the conductivity ratio between uranium and thorium phases decreases from the calcium to the lead series.
4. Clustering
processes
4.1. The Ba, _ .Th,F2 +2x solid solution The application of the clustering process model to (M” = Th, U) and Srl_XCal-xMI:F2+2x solid solutions has shown that the proThxF2+2.X the
Table 1 Polarizability
of the various cations considered
(according
to Ref.
1231) M”+
P(‘?,
Ca” S?+ Ba2+ Pb2+ Th4+ u4+
0.9 1.4 2.4 3.7 to 5 a 2.7 3.2
a The value PCpbz+)= 4 has been selected for the calculations.
321
0 F” 8F.
cation
0°F
Fig. 5. Basic 1: 0 : 3 : 0 cluster generated in a fluorite lattice by the transformation of a M2+(F1)s elementary cube into a M”4+ (F,),Fi polyhedron.
gressive transformation with rising x of 4: 0: 3: 3 clusters into 1 : 0 : 3 : 0 ones was a clustering process valid within these solid solutions [17,18]. Presence in of interstitial fluoride ions of the Bat-xThxF2+2x same type as in M1_XTh,F2+2, (M = Ca, Sr) has been shown by neutron diffraction and the existence of 1: 0: 3: 0 basic clusters (Fig. 5) has been proposed for these thorium solid solutions [16]. The large analogy between the composition dependences of the conductivity observed for Ba, ._,Th,F,+ 2X and occurence of SrI-XTh.XF2+2X and the probable 1:0:3:0 clusters in these materials has incited us to propose for Ba, _JhnFz+ 2x a clustering process close to that of M1_xThxF2+2x (M = Ca, Sr). Let us recall here the main features of the clustering process model [6]. Such a model proposes that within each M~?,M’2+“F 2+rrX solid solution with a X conductivity maximum for x,,, < xL the x, substitution rate for which the number of interstitial fluoride ions responsible for long range motions attains a maximum is close to x,,. This substitution rate corresponds to the progressive transformation of clusters favoring high conductivity into unfavorable clusters. According to the clustering process model the (Fi),,, ions responsible for long range motions can be represented by the y(Fi), function YCFi)m =
&
s
that displays a maximum for x equal to X, and a (y = kxx>line as tangent when x + 0. This clustering process model has been applied to
322
J.M. Reau et aL/Solid
State Ionics 78 (1995) 315-331
%-xTw2+2x on hand of the following cousiderations: - Whatever the value of X, the points representative of iza” are located around the (yFV = 3x) line. The replacement in SrF, of one Sr” cation by one Th”+ tetravalent cation gives rise to one neutral 1: 0 : 3 : 0 basic cluster close to the substituting cation and the transformation from one Sr2+ (F,), elementary cube to one Th4’(F,),Fi polyhedron (Fig. 5). - For small values of x the formation of this cluster involves the supplementary displacement of three Fr ions into F”’ interstitial sites creating thus three new vacancies. The entire extended defect is called the 4 : 0 : 3 : 3 cluster. - For higher values of X, as the steric constraint become stronger and stronger, the linear formation of 4:0:3:3 clusters cannot occur anymore and the clusters then created are more and more of 1: 0 : 3 : 5 type. - The F”’ ions located outside the Th(Fr),Fz polyhedra are the (Fi>,,, ions responsible for long range motions in Srl_rThnF2,.2x, they are represented by the following equation:
determined by neutron diffraction has validity of the clustering process i.e. a progressive Sr1--xThxF2+2Xt with increasing n of 4 : 0 : 3 : 3 clusters clusters [181.
The Bal-XThxF2+2x solid solution offers, as above, a composition dependence of S~I-J-ULZ~ F” fluoride ions in good agreement with the (yr” = 3x) function (Table 2) and a conductivity maximum for x,,~ = 0.20 (Fig. 2a). On the contrary, whatever the value of X, the number of F” ions is smaller in the barium phase than the strontium phase and the [ y,“, = 2x,2x/(x2 -+x,2)] function (k = 2) results in a better agreement between created and experimental values than the [ yFsN= 3x,2x/(x2 + a$)] function. These considerations lead us to propose for BaI-.XThA+2X the following clustering process: progressive transformation with increasing x of 3 : 0 : 3 : 2 clusters into 1: Cl: 3 : 0 clusters. The equations are determined for Ba, _ ,Th,F, + 2X by identifying x, to x,,,: YF”
’
3x,2x YP=Y(Fi)~=~+xz’
(k=3),
yint=
s
- The analytical expressions of the functions representing the F” ions, the sum of interstitial fluoride ions, the sum of vacancies in normal sites, respectively, are determined by YF”7 Yint and y. The good agreement obidentifying’ X, to x,,. served between calculated and experimental values
Table 2 Experimental and caleolated values of Yp. YP*t,Yi”,,and y.
confirmed the proposed for ~~sfo~ation into 1: 0 : 3 : 0
y( Fi),
=z:
,~2’+ 1;
X(75X2 + 5) 25X2+1 ;
YF” =
3x,
X(25X2 + 3)
yiJ=
2cx2+1
-
curves representing these functions are given in Fig. 6. The calculated values of yFft, yFtrP, hnt and y D can be compared in Table 2 with the experimental values determined by neutron diffraction [16]. The validity of the clustering process proposed for is confirmed by the good agreement Ba1-PQ2+2x
The
for some compositions of the Ba, _$hXF2+ 2w solid solution
x
0.05
0.10
0.15
0.20
0.25
0.28
Ref.
%% nWz %” = qF,*), + 8,“), YF!?
0.12 (3)
0.29 (2)
0.46 (2)
0.57 (4)
0.12 (3) 0.15
0.29 (2) 0.30
0.46 (2) 0.45
0.57 (4) 0.60
0.64 (4) 0.14 (6) 0.78 (10) 0.75
0.70 (3) 0.14 (3) 0.84 (6) 0.84
1161 [x61 [I61
n F’ Y,V!
0.12 (2) 0.10
0.16 (2) 0.16
0.16 (2) 0.19
0.14 (3) 0.20
0.19 (2) 0.20
0.20 (4) 0.19
[I61
ilF” + ?zF” YiIl,
0.24 0.25
0.45 0.46
0.62 0.64
0.71 0.80
0.97 0.95
1.04 1.03
[I61
80
0.14 (2)
0.15
0.25 (3) 0.26
0.32 (4) 0.34
0.31(6) 0.40
0.47 (6) 0.45
0.48 (6) 0.47
[I61
YO
JM
Reau et al. /Solid
State Ionics 78 (1995) 315-331
observed between calculated and experimental values (Fig. 6 and Table 2). The difference between the values of no and y. for x = 0.20 can be explained in the following way: ~1~~~ as a matter of fact represents the sum of two groups of F” fluoride ions, the (F”), ions with the (VI, u1, VI.. ul = 0.41) classical coordinates and the (F”>z ions located in a very diffuse (l/2, l/2, l/2) site [16]. These (F”), ions have only been identified for x > 0.20 (Table 2), but it is reasonable to suppose the presence of a small number of (F”), ions for x equal to 0.20: taking into account these extra anions should result in a higher value of II 17. The experimental upper limit for Ba,_,Th,F,+ 2x (x, = 0.29) is higher than the theoretical value for a solid solution based on 1: 0 : 3 : 0 clusters and corre-
Y
1.0
0.6
Fig. 6. Graphic functions versus
representation of the Y,,,, Y,,,,, yin,, and Y, x for the BaI_,Th,F2+ 2x solid solution.
323
sponding
to a long range order of such clusters = 0.251. This theoretical limit should nei[Wtheor ther be exceeded nor even approached. It is the reason why for x > 0.25 condensation of two 1: 0:3:0 clusters into one more dense 2: 0: 6: 0 cluster has been proposed in order to explain the extent of the Bal_-rThxF2+2r range (Fig. 7) [16]: as a matter of fact the theoretical (xL)theor value calculated for a solid solution based on 2 : 0 : 6 : 0 clusters and corresponding to a long range order of these clusters is equal to 0.33. As a consequence we have investigated for the progressive transformation with Bal-rThxF2+2x increasing x of 3 : 0 : 3 : 2 clusters into condensed 2:0:6:0 clusters. As the 1:0:3:0 and 2:0:6:0 clusters are governed by the same equations ( y q = x; of 3 : 0 : 3 : 2 clusters YF” -- 3x), the transformation into 2: 0 : 6 : 0 clusters leads to the same analytical expressions as the transformation of 3 : 0 : 3 : 2 clusters into 1: 0 : 3 : 0 ones and both clustering process hypotheses have the same validity. Considering the transformation of 3 : 0 : 3 : 2 clusters into 2:0 : 6 : 0 clusters in Bal_XTh,F2+2r, the total number of clusters is given by the following expression:
The curve representing that function (Fig. 8) is characterized by the (y = X) line as tangent when x + 0 and the ( y = x/2) assymptot when x -+ ~0. The ( y =x) line corresponds to the transformation of 4:0:3:3 clusters into 1:0:3:0 ones or to only the presence of 1: 0 : 3 : 0 clusters. The (y = x/2) line refers to presence of only 2: 0 : 6 : 0 clusters. The total number of clusters calculated for xL = 0.29, KY,,,,,),,, = 0.191,is smaller than 0.25. This result allows to select as clustering process in the transformation of 3: 0: 3: 2 Bal-Jh.XFZ+ZX clusters into 2 : 0 : 6 : 0 clusters. The transformation of 4: 0: 3 :3 clusters into 2 : 0 : 6 : 0 clusters can also be investigated as clustering process in Sr,_,Th,F,+,, and Ca,_J’h,F,+,,: as a matter of fact the xL values determined for these solid solutions, (xL)(Sr,Thj = 0.225 and W(Ca,Th) = 0.18, are smaller than 0.25 but not very far from that value. According to this clustering
324
J.M. Reau et al. /Solid
process
hypothesis the total number of clusters and Ca,_,Th,F,+,, would be: Sr,-,Th,F,+,,
State Ionics 78 (1995) 315-331
of a number of fluoride ions from normal sites into F”’ interstitial ones smaller in the barium phase than in the strontium and calcium phases (Fig. 9). It results apparently from steric constraints which are all the weaker as the host matrix is widening.
in
X(25X2 + 2) l(
Yclusthotl
(Sr,Th)
=
2;25x2
+
1;
’
4.2. The Pb, _ .Th, F, + 2x solid solution The clustering
process model applies also to the solid solution for which the anionic Pbl-JhXF2+2X distribution between normal and interstitial sites has been determined by neutron diffraction for various compositions [24]. Three series of interstitial fluoride ions respectively of F’, F” and F”’ type have been observed: the coordinates of the F” and F”’ sites are as usual; on the contrary the F’ site (l/2, U, u : u = 0.41) is relatively far from the usual position (l/2, U, u with 0.35 < U < 0.40). The variation with composition of the number of interstitial F’, F” and F”’ ions is given in Fig. 10.
The curves representing these functions are given in Fig. 8. The value of (yc,ust)tOt obtained for each upper limit of the composition range, i.e. = 0.16 for W(Sr,Th) and = 0.15 for (~r)(c~,r~), are close to the value calculated for (q,)ma,rbj. This result pleads in favor of the transformation of 4 : 0 : 3 : 3 clusters into condensed 2 : 0 : 6 : 0 clusters in Sr, _,Th,F2+ 2X and Cal-XThXF2+2X* The defect models suggested for such solid solutions [14] are thus confirmed. Hence the M1_rThxF2+2x (M = Ca, Sr, Ba) solid solutions are characterized by very close clustering processes. For the small values of x the formation of the 1: 0 : 3 : 0 basic cluster involves the displacement
+
0
+
Ba
0
+
0
+
0
+
+
0
+
Ba
Th
I
0
(+I
+
0
above and (0) below elementary Fig. 7. Schematic
representation
cubes.
of the 2 : 0 : 6 : 0 cluster.
J.M. Reau et al. /Solid
State Ionics 78 (1995) 315-331
4.2.1. nFl and nFll Whatever the value of x, nF, and nFff are located respectively around the (yr, = 4x/3) and (yr” = 5x/3) lines. As a consequence the (n,, + n,.) sum is in agreement with the (yr, + yrf8 = 3x) equation. Hence it appears that the sum of F’ and F” ions in plays a role analogous to that of F” Pbl-JkF2+*, ions (ypn =3x) in the other M1_-xThrF2+2x (M = Ca, Sr, Ba) solid solutions. Besides a new refinement computation of the Pb,,,,Th,.,,F2.20 structure, based on a F’ site close to (l/2, 0.41, 0.41), has been proved valid and results in a smaller occupation rate of the F’ site on behalf of the F” site. Another refinement based on the absence of F’ site results in a value of nFff close to the (nrl + n,.) sum determined from the former refinement [16]. It seems that is probably characterized by the Pbr-xThxF~+~n presence of basic clusters analogous to those of the other thorium solid solutions. As well-differentiated F’ sites are not required [16], the basic cluster in
325
Pbl_xTh,Fz+Z, will be called “1:0:3:0”, the (F”)* ions being located in an average (u,, u,, urn) position between both F’ and F” starting sites. The clustering process proposed for the Sri _ x Bi,FZ +x solid solution (0 G n < 0.50), which is also characterized by the presence of three F’, F” and F”’ interstitial fluoride ions, has been set up from analogous considerations [ill.
For each value of x the points representing nr,~ are located around the [ yr,,, = 7x/(70x2 + l)] curve that offers a maximum for x, = 0.119 and the ( y = 7x) line as tangent when x + 0. As in the other thorium solid solutions, the F”’ ions are actually the (Fi)m ions responsible for long range motions in according to the clustering process Phi-nThxFz+zx: model, x, is close to x,, (x,, = 0.125) for which a conductivity maximum is observed in the variation of log a,,, K with x (Fig. 2a).
Fig. 8. Variation with x of the sum of clusters (yCIUSL)totfor the Ml _,M’:F,+
zx (M = Ca, Sr, Ba, F%; M” = Th, U) solid solutions.
J.M. Reau et al. /Solid
326
State Ionics 78 (1995) 315-331
Hence the Pbl_-xThxF2+2x solid solution is characterized by the progressive ~~sformation with inclusters either into creasing x of “8:0:3:7” “1:0:3:0” or into condensed “2:0:6:0” clustens. In fact, the (xr&,rhj value close to 0.25, k)(Pb,Th) = 0.231, leads to select the transformation of “‘8:0:3:7” clusters into “2:O:h:O” ones. The analytical expressions of the sum of interstitial fluo-
ride ions and of the sum of vacancies in normal sites are: x(210x2 + 10)
X(70X2 4 8)
Yint=
70x2+1 ’ yO= 70x2+1 * The curves representing these functions are given in Fig. 10. The values calculated for the different functions characterizing Pb, _XThXF2+ax may be com-
0.
0.
0.
0.
Fig. 9. Graphic representation of the YF” function versus x for the M, _,Th,F2+ 2x (M = Ca, Sr, Ba, Pb) solid solutions.
JM. Reau et al. /Solid
State Ionics 78 (1995) 315-331
pared in Table 3 with experimental values determined by neutron diffraction. The good agreement observed between the calculated and experimental values confirms the validity of the clustering process proposed (Fig. 10 and Table 3). The variation of the total number of clusters in Pbl-xThxF2+2x is: [(
Ycl”sthotlwrh~ = ;I:,“:: 1:;
4.3. The MI _ .U, F2 + 2x (M = Ca, Sr, Ba, Pb) solid solutions
.
It is illustrated by Fig. 8: whatever the value of X, (yclustjtot is smaller in Pbl_-rThxFZ+Zx than in M,_,= ThxF2+2.X (M = Ca, Sr, Ba) and Kyc,ust)totl~b,Th~ 0.14 for x =xr. The smaller value of xs in with respect to other solid solutions Pbr-JkF2+2x resultsinless“8:0:3:7”clustersthan4:0:3:3or 3:0:3:2 clusters and in more “2:0:6:0” clusters (Fig. 11). Finally, for each x-value Pbl_XTh,Fz+z, shows a total number of clusters lower than the other solid solutions. On the contrary, the number of F”’ ions is higher than in M1_XTh,F2+2, (M = Ca, in Pbl-JhxF2+2x Sr, Ba) (Fig. 9). This result cannot be attributed to steric constraints as in the lead phase the unit cells are of size intermediate between those of the baryum and the strontium phases (Fig. 3). The presence of lone pair Pbzf ions in Pbl_XThXFZ+ZX favors in fact the displacement of a larger number of fluoride ions from their normal sites into F”’ positions [25,26].
Table 3 Experimental and calculated values of yr~, Y,v, y,“‘,
yint and y,
327
4.3.1. M = Ca, Sr The transformation with increasing x of 4 : 0 : 3 : 3 clusters into 1: 0 : 3 : 0 clusters has been proposed as clustering process for Ca,_JJ,F,+,, [17] and [18]. As in the homologous thorium Srr-XUXF,,,, solid solutions, the transformation of 4 : 0 : 3 : 3 clusters into condensed 2 : 0 : 6 : 0 ones can also be considered. Consequently, the total number of clusters in that case is given by following expressions (Fig. 8):
The total number of clusters is represented in by the same function as in Srr-.U,F*+,, (M = Sr, Ba). As a matter of fact the Mr-,Th,F2+2, three solid solutions show a conductivity maximum for the same value of x (x,,, = 0.20). The values of (yc,ust)tot obtained for the upper limit of the composition range of M, _JJ,F,+ 2X (M = Ca, Sr) are equal to 2: 0.18 for (~r)(c~,u) and = 0.17 for (~&~,u). They are close to those calculated for the thorium phases. These results plead in
for some compositions of the Pb, -JhxF2+ax
solid solution
x
0.05
0.10
0.125
0.15
0.20
0.23
Ref.
EF’
-
0.067
0.18 (3) 0.167
0.19 (3) 0.20
0.28 (3) 0.267
0.28 (5) 0.307
12Al
yp, = 4x/3
0.14 (4) 0.133
IzF” y,,, = 5x/3
0.12 (2) 0.083
0.19 (4) 0.167
0.22 (3) 0.208
0.25 (3) 0.25
0.34 (3) 0.333
0.4lW 0.383
1241
?zF’+ lzsz
0.12 0.15
0.33 0.30
0.40 0.375
0.44 0.45
0.62 0.60
0.69 0.69
1241
YF’+ y,,, = 3x ilF” YF”
0.37 (8) 0.30
0.37 (8) 0.41
0.40 (8) 0.42
0.41 (10) 0.41
0.40 (10) 0.37
0.31 (10) 0.34
WI
?rF’+ nF” + ?rF”! Yint.
0.49 0.45
0.70 0.71
0.80 0.795
0.85 0.86
1.02 0.97
1.00 1.03
1241
no YO
0.39 (8) 0.35
0.50 (8) 0.51
0.55 (8) 0.545
0.55 (10) 0.56
0.62 (10) 0.57
0.54 (10) 0.57
1241
J.M. Reau et al. /Solid
328
State Ionics 78 (1995) 315-331
favor of the tr~sfo~ation of 4 : 0 : 3 : 3 clusters into 2 : 0 : 6 : 0 clusters in the ur~i~ phases. 4.32. M = Ba shows a Bal-PP2+2, of the electrical properties Ba1_XTh,F2+2x (Fig. 2): conductivity maximum are and = 0.20 respectively for phases. These results induce calcium and strontium solid
composition dependence quite close to that of the x,,,:values of the close, equal to = 0.18 the uranium and thorium us to propose, as for the solutions, the same clus-
tering process in Ba,_,U,F,+,, as in Ba,_,Th,the value of (~~~~.)~n~,u~, the F afax. Considering equations relative to Bar _ ,U,F,, ax9 determ~ed on hand of the clustering process model by identi~ing X, with x,,,., are: 2x Y: = 3ox2 + 1 )
y; = 3x, x(90x2
Yint =
+
5)
3oXa + 1
X(30X2 + 3) 2
YO =
30x2+1
*
The total number of clusters in Bar _ ,I.JXFa + an is given by the expression (Fig. 8>:
1.0
z . Yint
/
0.6
Fig. 10. Graphic functions versus
representation of the Ypr, Y,., YF~jtrYi”* and y D x for the Pbl_-xThxFzi2x solid solution.
Its value (= 0.18) for the upper substitution rate xt equal to 0.285 is smaller than 0.25. This result pleads in favor of the transformation of 3:0:3:2 clusters into 2 : 0 : 6 : 0 clusters as possible clustering process. Variation with x of the number of 3 : 0 : 3 : 2 and 2 : 0 : 6 : 0 clusters can be compared for both Ba 1--rMZF2+2* solid solutions in Fig. 11: whatever the value of X, the number of clusters favoring an e~ancement of the ~nductivity is smaller and, on the contrary, the number of condensed 2:0: 6:0 clusters is higher for uranium phase than for thorium phase. As the size difference between the host and guest cations is slightly larger in Ba,_$JXFz+ ax than the tendency to establishment of in Bal-XThXPa+a,, an order with increasing x is stronger for the uranium phase: consequently the conductivity maximum occurs for a smaller value of x,,, in the uranium phase, a weaker conductivity is observed for the uranium phase, althought U4+ has a larger polarizability than Th4’ (Fig. Za). 4.3.3. M = Pb Whatever the value of x, the electrical properties (M” = Th, U) solid solutions of both Pbr_,M’:F,+,, are very close. In particular a conductivity maximum is observed for the same value x,,, = 0.125 of the substitution rate. As a consequence the clustering process shown in Pbl_JhXFa+zX, i.e. a progressive transformation with increasing x of “8: 0: 3: 7” clusters into “2 : 0 : 6 : 0” clusters, can be considered
J.M. Reau et al. /Solid
329
State Ionics 78 (1995) 315-331
as valid iu Pb, _JJXF2+2x. The application of the clustering process model results in the same analytical expressions for both solid solutions (Figs. 8, 10 and 11). Very close clustering processes characterize the (M = Ca, Sr, Ba, Pb; M” = Th,U) MI-IM;F2+2x
solid solutions. The different compositions corresponding to x equal to 0.06, 0.12 and 0.18 can be located inside a (P,, YFJ,,) diagram where Pa is the average polarizability of the cationic sublattice and y,,,, the function representative of the number of F”’ interstitial ions (Fig. 12). Whatever X, a minimum
(a) y4033
0.20 t
* '3032
' y003?
(Sr,U)
'3032 (Ba,Th) I
=00;7
(Pb,Th)(Pb,U)
04
,
0.20 _y2060
(Pb ,Th) (Pb,U)
(Ba,U) \
Fig.ll.Variationversus xofthenumberofclustersoftype(a)4:0:3:3,3:0:3:2or8:0:3:7and(b)2:0:6:0fortheM~_,M~F~+~, (M = Ca, Sr, Ba, Pb; M” = Th, U) solid solutions.
330
J.M. Rem et al. /SoEd St&e Ionics 78 (19951315-331
0.6
fPb_Th: \ 0.4
9.2
Fig. 12. Variation of YF- as a function of P, for the M1_xM:F2+2x (M = Ca, Sr, Ba, Pb; M” = Th, U) compositions 0.12 and 0.18 (the dotted lines drawn through the reported data are simple guides for the eye).
appears for barium compositions. It results from both presence of polarizable cations and steric constraints: ( *) Cations of higher polar&ability favor a displacement of a larger number of anions from normal positions of the fluorite structure into F”’ sites. ( *) Steric constraints are all the stronger as the unit cell is smaller. As a consequence, 11~~~~ increases from barium to strontium and to calcium compositions. This increase is all the larger as x is higher (Fig. 12).
5. Conclusions
The short range structure within the M,_; (M = Ca, Sr, Ba, Pb; &I” = Th, U) solid MYL,, solutions derives from the same 1: 0 : 3 : 0 basic cluster. Its formation results in transfer of fluoride ions
for x equal to 0.06,
from normal sites into F”’ interstitial sites. The number of such F” fluoride ions increases for stronger steric constraints and higher cation polarizability. As a consequence for the low substi~tion rates clusters of4:0:3:3,4:0:3:3,3:0:3:2~d8:0:3:7types form respectively in the calcium, strontium, barium and lead phases. At rising x-value all these clusters are transformed into condensed 2 : 0 : 6 : 0 clusters. Obviously the M1_xMtF2+2x solid solutions are characterized by very close clustering processes. The electrical properties have been determined as a function of temperature and composition. A comparison of the electrical performance for different Mr--xMI:Fa+tn compositions of the same x-value shows that the higher the average pol~~ability of the cationic sublattice the larger the mobility of the charge carriers and the bigger the fluorine ion conductivity.
J.M. Reau et al./Solid
State Ionics 78 (1995) 315-331
References m J.M. Rtau and P. Hagemnuller, Appl. Phys. A 49 (1989) 3. M C.R.A. Catlow, J. Phys. C 9 (1976) 1845. 131 J. Corish, C.R.A. Catlow, P.W.M. Jacobs and S.H. Ong, Phys. Rev. B 25 (1982) 6425. [41 J.P. Laval, A. Mikou, B. Frit and G. Roult, Solid State Ionics 28-30 (1988) 1300. I51 AK. Cheetham, B.E.F. Fender, B. Steele, R.I. Taylor and B.T.M. Willis, Solid State Commun. 8 (1970) 171. and P. Hagenmuller, Phys. [61 J.M. Riau, M. Wahbi, J. S&gas Status Solidi (b) 169 (1992) 331. t71 S.F. Matar, J.M. RBau, P. Hagenmuller and C.R.A. Catlow, J. Solid State Chem. 52 (1984) 114. P31J.M. Rtau, M.El Omari, J. SBnCgas and P. Hagenmuller, Solid State Ionics 38 (1990) 123. [91 M. El Omari, J.M. Rtau and J. SBnBgas, J. Solid State Chem. 87 (1990) 430. DO1 J.L. Soubeyroux, J.M. R&au, M. Wahbi, J. Sdnegas and Suh Kyung Soo, Solid State Common. 82 (1992) 63. ml J.L. Soubeyroux, J.M. RBau, M. Wahbi, J. Se&gas and Suh Kyung Soo, Solid State Commun. 83 (1992) 259. [I21 N.H. Andersen, K. Clausen and J.K. Kjems, in: TransportStructure Relations in Fast Ion and Mixed Conductors, eds. F.W. Poulsen, N.H. Andersen, K. Clausen, S. Skaarup and O.T. Sorensen (Riso National Lab., Roshilde, 1985) p. 171.
331
[13] M. Ouwerkerk, F.J. Veldkamp and J. Schoonman, Solid State Ionics 22 (1987) 173. 1141 J.P. Laval, A. Mikou, B. Frit and J. Pannetier, J. Solid State Chem. 61 (1986) 359. [U] J.P. Lava& A. Mikou, B. Frit, G. Roult and J. Pannetier, Rev. Chim. Miner. 24 (1987) 165. 1161 A. Abaouz, Thesis (Limoges France 1988). [17] M. El Omari, J.M. R&au, J. S&r&gas, J.P. Laval and B. Frit, J. Solid State Chem. 92 (1991) 312. [18] J.M. RBau, Xu Yong Jun and J. Senegas, Phys. Status Solidi (a) 143 (1994) 51. [19] M. Ouwerkerk, Thesis (Utrecht, The Netherlands, 1986). [20] A. Mikou, Thesis (Limoges, France, 1986). [21] J.M. Reau, A. Rhandour, C. Lucat, J. Portier and P. Hagenmuller, Mat. Res. Bull. 13 (1978) 827. [22] A. Rhandour, Thesis (Bordeaux, France, 1979). [U] C.K. Jorgensen, Top. Curr. Chem. 56 (1975) 38. 1241 J.L. Soubeyroux, J.M. R&au, S. Matar, P. Hagenmuller and C. Lucat, Solid State Ionics 2 (1981) 215. [25] J. Portier, J.M. RBau, S. Matar, J.L. Soubeyroux and P. Hagenmuller, Solid State Ionics 11 (1983) 83. [26] Y. Ito, T. Mukoyama, S. Yoxhikado and K. Koto, Solid State Ionics 67 (1994) 307.
IQNICS
(M TfieMl-xM’Y2+2*
= Ca, Sr, Ba, Pb; M” = Th, Uf solid solutions: electrical properties and clustering processes J.M. Reau, Xu Yong Jun, J. Senegas, P. Hagenmuller Institut de Chimie de la Mat&e
Condens6e de Bordeaux, Chliteau Briuazac, 33600 Pessac, France
Received 6 February 1995; accepted for publication 8 March 1995
Abstract CM = Ca, Sr, Ba, Pb; M” =Th, U) solid A comparative investigation of elect&al properties of the M,_,M~F,+,, solutions has been carried out. The application of the clustering process model shows that all these solid solntions appear as characterized by very close clustering processes. The electrical properties depend on short-range structure and average ~l~zability of the c&ionic sublattice. Keywords:
Fluorine; Ionic conductivity; Fluorite; Clustering
1. Introduction The fluorite-type M~_fxM~2fnF2+~x ((u = 1, 2) solid solutions disordered at long range are characterized, for very low substitution rates (X < 0.011, by creation of point defect pairs, and, for higher substitution rates, by fo~ation of extended defects, which influence strongly the ~onduc~vi~ of the F- ions El]. For x < 0.01 the replacement for instance of host cations by trivalent substitutional cations involves introduction of charge compensating excess anions into interstitial sites; these extra anions associated with substitutional cations constitute pairs of point defects either of ltlt type when their symmetry is tetragonal or of nnn type when it is trigonal [2,3]. For x > 0.01 more extended defects, i.e. clusters, are formed. Such clusters, labelled nr: n2: n3: pt+ are based on the association of n1 vacancies in the Fr normal positions (l/4, l/4, l/4) of the ~uorite-ape network, n2 FiI (l/2, U, t6: 0.35 < u 5 0.401, rz3 Fg (q, ul, ul: u1 = 0.41) and rz4 F; (uZ, u2, u2: 0.28
< v2 ,< 0.33) interstitial fluoride ions, close to one or several substitutional cations [4-61. Clustering is characterized by significant transfer of fluoride ions from normal positions into interstitial sites with formation of vacancies in the normal Ft sublattice. It results in an enhancement of the F- ion conductivity L71. The optimization criteria of fast fluoride anion transport properties in the M~?$l”2fnF2+an solid solutions (nonstoichiometry, cation polar&ability, size difference between the host and guest cations. . . > are also responsible for the establishment of short range ordering in such long range disordered phases. Clustering has actually an essential influence on transport properties. A clustering process model has been proposed relating in a continuous way the composition dependence of electrical properties and the progressive extension of clustering when x increases. The basis of this model is the detestation of a y(F& =f(x> function representing the (Fi& ions responsible for long range motions in the solid
0167~2738/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDZ 0167-2738(95)00111-S
316
J.M. Reau et al. /Solid
State Zonks 78 (1995) 315-331
solutions: this function is established from the variation of transport properties with substitution rate and anionic dist~bution between normal and ~terstitial sites [6]. Various clustering processes have been detected within the Mf?,M$fF,+, solid solutions. They depend on the nature of the cationic (M, M’) couples
(a) log
-2
Q
tQ.cm)
present [8-111. On the contrary the M~_f,M’$’ Fa+ax solid solutions where the M” substitutional cation is tetravalent have been only partly investigated [12,13]. Application of the clustering process model requires, as a matter of fact, both structural and electrical data relative to the whole composition range, in particular, to the highest substitution rates [6]. Thus cluster-
-1
-3
x80.18
,
0.22
-4
-5
-6
t
-7
a.1.5 Fig. 1. Variation of log CTversus T-l
I
2.0
103/~(~) -
t 2.5
3.0
for some compositions of (a> Bal_JhzF2+2x
3.5 and (bf
Bal_,UzF,+,,
solid SO~I&~I%
J.M. Reau et al. /Solid
State Ionics 78 (1995) 315-331
317
-3
-4
-5
-6
-i
\
I__ 2.0
2.5
3.0
103/T(K) 3.5
Fig. 1. continued.
ing
processes
have been proposed for the CM” = Th, II> and Sr,_,Th,F,,,, Ca1-xM~Fz+2x solid solutions on hand of neutron diffraction data [14-161 and the composition dependence of the electrical properties [17,18]: the progressive transformation with rising x of 4 : 0 : 3 : 3 clusters into 1: 0 : 3 : 0 ones appears as a process valid in the three solid
solutions. The knowledge of the anionic distribution within the Bal_JhxF2+Zx solid solution [16] has incited us to determine the composition dependence of the transport properties and the nature of the clustering process in Ba1_xThxF2+2x. In a final step a comparative study of the clustering processes in the Ml--xThnFz+zx (M = Ca, Sr, Ba, Pb) solid solu-
31s
J.M. Reau et al. /Solid
tions has been achieved. Our goal was to define the influence of the host cation on the clustering processes for the same tetravalent substi~tion~ cation. The enlargement of these results to homologous uranium ( + IV) solid solutions, M, _,U,F, + ax, will be simultaneously discussed using the composition dependence of electrical properties.
State Ionics 78 (1995)
31.5-331
2. Experimentasl The samples belonging to the Bar _$I~F2+ 2X (M” = Th, U) solid solutions used here for electrical measurements are similar to those previously prepared for neutron diffraction experiments [161. The materials have been synthesized from mixtures of
b-4 -2
AF$
(Ba,Th)
(ev)
-4 \
-5 0.
:Sr,Th) Lo-
-6 0.
\(Ba.Thf
0.
x I._
0.10
Fig. 2. Variation
0.20
of (a) log urn0 K and(b)
0.30
x :___ 0.10
A,?& as a function of x for M1_-sM:F2+2x
0.
(M = Ca, Sr, Ba, Pb; M“ = Th, U) solid solutions.
J.M. Reau et al. /Solid
State Ionics 78 (1995) 315-331
BaF, and M”F, fluorides in sealed gold tubes at 850°C during 15 h. XRD analysis of the samples obtained after quenching has confirmed the existence of disordered Bal_xThxF2+2n (0
319
(b> A fast increase of log a,,, k with increasing n, associated with a strong decrease of AE,, is observed for all M,_,MzF,+ 2x solid solutions (M = Ca, Sr, Ba; M” = Th, U) in the composition range (0.05 < x ( x,,,) for the barium and strontium materials and for (0.05
3. Results The variation of the conductivity with temperature is shown in Fig. la and b for some Bal_xThxF2+2x and Ba 1_ ,U, F2 + 2x compositions. In the investigated temperature range the temperature dependence of conductivity agrees with an Arrhenius-type law for each composition considered: (+ = a,, exp( - AE,/ kT) with R = 0.98 fitting. The results obtained for compositions are in agreement the Bar-rUxF2+2x with those previously given by Ouwerkerk [19] for single crystals corresponding to x < 0.16. Fig. 2a and b shows respectively the isotherm u-,, x and the variation of the activation energy AE, as a function of x for the Ba,_,M’:F,+,, solid solutions. For the sake of comparison the curves relative to Car _,Th< 6 Fz& ‘“Th” ; G 0.18)Y (o
x <
x $2
2+2x
0.235)
6,05_
(Pb,Th) (Pb,U)
(Sr.Th)
5.85
5.80
(55r.u)
i/-!---e ,
(OG
(Ca,Tb)
[17J8]
(a) Barium and strontium solid solutions show a conductivity maximum associated with an activation energy minimum for a value of the substitution rate x max
(ca.U)
x
f
I
0.05
Fig. 3. Variation MI-xM):Fz+zx DOI.
0.10
,
0115
0.20
I
I
0.25
0,30
-
of the parameter a, as a function of x for (M = Ca, Sr, Ba, Pb; M” = Th,U) solid solutions
320
J.M. Reau et al. /Solid
State Ionics 78 (1995) 315-331
higher as the size of the host cation is larger: o~~,~~,,) ’ @(S&‘) ’ @(Ca,M”)* (e) For a given x the conductivity of the uranic fluorides is higher than that of the thorium compounds when the host cation is Cazf or Sr2’. Conversely, when M is Ba, the highest conductivity is observed for the thorium fluorides. In Fig. 3 we have reported the variation of the unit cell parameter a, as a function of x for the (M = Ca, Sr, Ba; M” = Th, U) solid M1-JG,+,, solutions [20]: whatever the value of x, a higher conductivity is observed for the uranium phases when the fM2+ = M4’ + 2F-) formal substi~tion involves unit celf dilatation (M = Ca, Sr). On the contrary it
appears for the thorium phase when the substitution results in a unit ceil contraction (M = Ba). For a given substi~tion rate one may assume that for charge carrier mobility the broadest bottlenecks occur in the barium phases as they offer larger unit cells than in the strontium and calcium phases. Whatever the host matrix, replacement of Th4+ by U4+, the size of which is slightly smaller, represents only a secondary effect. The comparison of the electrical properties of the U> Ml-XM’: %.+2x (M= Ca, Sr, Ba: M” =Th, phases with those of the Pbl_nThxF2+2x (0 GX G 0.23) [21] arid Pbl_JJXF2+2, (0
f AE,(ev. 1
(Pb,Th) fi tPb,U) -3
0.70 (ca,u)
0.60
-6
-_
1 I
I
1
I
2
Pam')
I
3
Fig. 4. Variation of log qa,, K and AE, as a function of the average polxizability fluorides corresponding to x = 0.18 {M = Ca, Sr, Ba, Pb; M” = Tb, U).
4 (I’,) of the cationic sublattice
for the MI _,M~J?2+,,
J.M. Reau et al. /Solid
State Ionics 78 (1995) 315-331
not the most important criterion for optimization of the transport properties of such materials. The lead fluorides whose unit cells have sizes intermediate between those of strontium and barium phases show the best electrical properties (Figs. 2 and 3). Considering the polarizability of the different cations involved in this study (Table 1) [23], and assuming that for a given material the average polarizability of the cationic sublattice is such as: P, = (1 - x)P, + (X>P,!,, we have plotted in Fig. 4 the variation of log a,,, x and of hE, as a function of P, for the fluorides corresponding to x = 0.18. This substitution rate has been selected as it corresponds in one solid solution to the introduction into the host matrix of the maximum of substitutional cations: xL is actually equal to 0.18 in Ca,_,Th,F,+,, whose composition range is thus the smallest. AE, decreases and log oN, x increases regularly when P, increases. Transport properties of these solid solutions appear as tightly related to the average polarizability of the cationic sublattice. Furthermore, the increase of P, resulting from the replacement of Th4+ by U4+ decreases progressively from calcium to lead: 7.3, 5.5, 3.5 and 2.4% when M is by turns Ca, Sr, Ba and Pb (Table 1). As a consequence the conductivity ratio between uranium and thorium phases decreases from the calcium to the lead series.
4. Clustering
processes
4.1. The Ba, _ .Th,F2 +2x solid solution The application of the clustering process model to (M” = Th, U) and Srl_XCal-xMI:F2+2x solid solutions has shown that the proThxF2+2.X the
Table 1 Polarizability
of the various cations considered
(according
to Ref.
1231) M”+
P(‘?,
Ca” S?+ Ba2+ Pb2+ Th4+ u4+
0.9 1.4 2.4 3.7 to 5 a 2.7 3.2
a The value PCpbz+)= 4 has been selected for the calculations.
321
0 F” 8F.
cation
0°F
Fig. 5. Basic 1: 0 : 3 : 0 cluster generated in a fluorite lattice by the transformation of a M2+(F1)s elementary cube into a M”4+ (F,),Fi polyhedron.
gressive transformation with rising x of 4: 0: 3: 3 clusters into 1 : 0 : 3 : 0 ones was a clustering process valid within these solid solutions [17,18]. Presence in of interstitial fluoride ions of the Bat-xThxF2+2x same type as in M1_XTh,F2+2, (M = Ca, Sr) has been shown by neutron diffraction and the existence of 1: 0: 3: 0 basic clusters (Fig. 5) has been proposed for these thorium solid solutions [16]. The large analogy between the composition dependences of the conductivity observed for Ba, ._,Th,F,+ 2X and occurence of SrI-XTh.XF2+2X and the probable 1:0:3:0 clusters in these materials has incited us to propose for Ba, _JhnFz+ 2x a clustering process close to that of M1_xThxF2+2x (M = Ca, Sr). Let us recall here the main features of the clustering process model [6]. Such a model proposes that within each M~?,M’2+“F 2+rrX solid solution with a X conductivity maximum for x,,, < xL the x, substitution rate for which the number of interstitial fluoride ions responsible for long range motions attains a maximum is close to x,,. This substitution rate corresponds to the progressive transformation of clusters favoring high conductivity into unfavorable clusters. According to the clustering process model the (Fi),,, ions responsible for long range motions can be represented by the y(Fi), function YCFi)m =
&
s
that displays a maximum for x equal to X, and a (y = kxx>line as tangent when x + 0. This clustering process model has been applied to
322
J.M. Reau et aL/Solid
State Ionics 78 (1995) 315-331
%-xTw2+2x on hand of the following cousiderations: - Whatever the value of X, the points representative of iza” are located around the (yFV = 3x) line. The replacement in SrF, of one Sr” cation by one Th”+ tetravalent cation gives rise to one neutral 1: 0 : 3 : 0 basic cluster close to the substituting cation and the transformation from one Sr2+ (F,), elementary cube to one Th4’(F,),Fi polyhedron (Fig. 5). - For small values of x the formation of this cluster involves the supplementary displacement of three Fr ions into F”’ interstitial sites creating thus three new vacancies. The entire extended defect is called the 4 : 0 : 3 : 3 cluster. - For higher values of X, as the steric constraint become stronger and stronger, the linear formation of 4:0:3:3 clusters cannot occur anymore and the clusters then created are more and more of 1: 0 : 3 : 5 type. - The F”’ ions located outside the Th(Fr),Fz polyhedra are the (Fi>,,, ions responsible for long range motions in Srl_rThnF2,.2x, they are represented by the following equation:
determined by neutron diffraction has validity of the clustering process i.e. a progressive Sr1--xThxF2+2Xt with increasing n of 4 : 0 : 3 : 3 clusters clusters [181.
The Bal-XThxF2+2x solid solution offers, as above, a composition dependence of S~I-J-ULZ~ F” fluoride ions in good agreement with the (yr” = 3x) function (Table 2) and a conductivity maximum for x,,~ = 0.20 (Fig. 2a). On the contrary, whatever the value of X, the number of F” ions is smaller in the barium phase than the strontium phase and the [ y,“, = 2x,2x/(x2 -+x,2)] function (k = 2) results in a better agreement between created and experimental values than the [ yFsN= 3x,2x/(x2 + a$)] function. These considerations lead us to propose for BaI-.XThA+2X the following clustering process: progressive transformation with increasing x of 3 : 0 : 3 : 2 clusters into 1: Cl: 3 : 0 clusters. The equations are determined for Ba, _ ,Th,F, + 2X by identifying x, to x,,,: YF”
’
3x,2x YP=Y(Fi)~=~+xz’
(k=3),
yint=
s
- The analytical expressions of the functions representing the F” ions, the sum of interstitial fluoride ions, the sum of vacancies in normal sites, respectively, are determined by YF”7 Yint and y. The good agreement obidentifying’ X, to x,,. served between calculated and experimental values
Table 2 Experimental and caleolated values of Yp. YP*t,Yi”,,and y.
confirmed the proposed for ~~sfo~ation into 1: 0 : 3 : 0
y( Fi),
=z:
,~2’+ 1;
X(75X2 + 5) 25X2+1 ;
YF” =
3x,
X(25X2 + 3)
yiJ=
2cx2+1
-
curves representing these functions are given in Fig. 6. The calculated values of yFft, yFtrP, hnt and y D can be compared in Table 2 with the experimental values determined by neutron diffraction [16]. The validity of the clustering process proposed for is confirmed by the good agreement Ba1-PQ2+2x
The
for some compositions of the Ba, _$hXF2+ 2w solid solution
x
0.05
0.10
0.15
0.20
0.25
0.28
Ref.
%% nWz %” = qF,*), + 8,“), YF!?
0.12 (3)
0.29 (2)
0.46 (2)
0.57 (4)
0.12 (3) 0.15
0.29 (2) 0.30
0.46 (2) 0.45
0.57 (4) 0.60
0.64 (4) 0.14 (6) 0.78 (10) 0.75
0.70 (3) 0.14 (3) 0.84 (6) 0.84
1161 [x61 [I61
n F’ Y,V!
0.12 (2) 0.10
0.16 (2) 0.16
0.16 (2) 0.19
0.14 (3) 0.20
0.19 (2) 0.20
0.20 (4) 0.19
[I61
ilF” + ?zF” YiIl,
0.24 0.25
0.45 0.46
0.62 0.64
0.71 0.80
0.97 0.95
1.04 1.03
[I61
80
0.14 (2)
0.15
0.25 (3) 0.26
0.32 (4) 0.34
0.31(6) 0.40
0.47 (6) 0.45
0.48 (6) 0.47
[I61
YO
JM
Reau et al. /Solid
State Ionics 78 (1995) 315-331
observed between calculated and experimental values (Fig. 6 and Table 2). The difference between the values of no and y. for x = 0.20 can be explained in the following way: ~1~~~ as a matter of fact represents the sum of two groups of F” fluoride ions, the (F”), ions with the (VI, u1, VI.. ul = 0.41) classical coordinates and the (F”>z ions located in a very diffuse (l/2, l/2, l/2) site [16]. These (F”), ions have only been identified for x > 0.20 (Table 2), but it is reasonable to suppose the presence of a small number of (F”), ions for x equal to 0.20: taking into account these extra anions should result in a higher value of II 17. The experimental upper limit for Ba,_,Th,F,+ 2x (x, = 0.29) is higher than the theoretical value for a solid solution based on 1: 0 : 3 : 0 clusters and corre-
Y
1.0
0.6
Fig. 6. Graphic functions versus
representation of the Y,,,, Y,,,,, yin,, and Y, x for the BaI_,Th,F2+ 2x solid solution.
323
sponding
to a long range order of such clusters = 0.251. This theoretical limit should nei[Wtheor ther be exceeded nor even approached. It is the reason why for x > 0.25 condensation of two 1: 0:3:0 clusters into one more dense 2: 0: 6: 0 cluster has been proposed in order to explain the extent of the Bal_-rThxF2+2r range (Fig. 7) [16]: as a matter of fact the theoretical (xL)theor value calculated for a solid solution based on 2 : 0 : 6 : 0 clusters and corresponding to a long range order of these clusters is equal to 0.33. As a consequence we have investigated for the progressive transformation with Bal-rThxF2+2x increasing x of 3 : 0 : 3 : 2 clusters into condensed 2:0:6:0 clusters. As the 1:0:3:0 and 2:0:6:0 clusters are governed by the same equations ( y q = x; of 3 : 0 : 3 : 2 clusters YF” -- 3x), the transformation into 2: 0 : 6 : 0 clusters leads to the same analytical expressions as the transformation of 3 : 0 : 3 : 2 clusters into 1: 0 : 3 : 0 ones and both clustering process hypotheses have the same validity. Considering the transformation of 3 : 0 : 3 : 2 clusters into 2:0 : 6 : 0 clusters in Bal_XTh,F2+2r, the total number of clusters is given by the following expression:
The curve representing that function (Fig. 8) is characterized by the (y = X) line as tangent when x + 0 and the ( y = x/2) assymptot when x -+ ~0. The ( y =x) line corresponds to the transformation of 4:0:3:3 clusters into 1:0:3:0 ones or to only the presence of 1: 0 : 3 : 0 clusters. The (y = x/2) line refers to presence of only 2: 0 : 6 : 0 clusters. The total number of clusters calculated for xL = 0.29, KY,,,,,),,, = 0.191,is smaller than 0.25. This result allows to select as clustering process in the transformation of 3: 0: 3: 2 Bal-Jh.XFZ+ZX clusters into 2 : 0 : 6 : 0 clusters. The transformation of 4: 0: 3 :3 clusters into 2 : 0 : 6 : 0 clusters can also be investigated as clustering process in Sr,_,Th,F,+,, and Ca,_J’h,F,+,,: as a matter of fact the xL values determined for these solid solutions, (xL)(Sr,Thj = 0.225 and W(Ca,Th) = 0.18, are smaller than 0.25 but not very far from that value. According to this clustering
324
J.M. Reau et al. /Solid
process
hypothesis the total number of clusters and Ca,_,Th,F,+,, would be: Sr,-,Th,F,+,,
State Ionics 78 (1995) 315-331
of a number of fluoride ions from normal sites into F”’ interstitial ones smaller in the barium phase than in the strontium and calcium phases (Fig. 9). It results apparently from steric constraints which are all the weaker as the host matrix is widening.
in
X(25X2 + 2) l(
Yclusthotl
(Sr,Th)
=
2;25x2
+
1;
’
4.2. The Pb, _ .Th, F, + 2x solid solution The clustering
process model applies also to the solid solution for which the anionic Pbl-JhXF2+2X distribution between normal and interstitial sites has been determined by neutron diffraction for various compositions [24]. Three series of interstitial fluoride ions respectively of F’, F” and F”’ type have been observed: the coordinates of the F” and F”’ sites are as usual; on the contrary the F’ site (l/2, U, u : u = 0.41) is relatively far from the usual position (l/2, U, u with 0.35 < U < 0.40). The variation with composition of the number of interstitial F’, F” and F”’ ions is given in Fig. 10.
The curves representing these functions are given in Fig. 8. The value of (yc,ust)tOt obtained for each upper limit of the composition range, i.e. = 0.16 for W(Sr,Th) and = 0.15 for (~r)(c~,r~), are close to the value calculated for (q,)ma,rbj. This result pleads in favor of the transformation of 4 : 0 : 3 : 3 clusters into condensed 2 : 0 : 6 : 0 clusters in Sr, _,Th,F2+ 2X and Cal-XThXF2+2X* The defect models suggested for such solid solutions [14] are thus confirmed. Hence the M1_rThxF2+2x (M = Ca, Sr, Ba) solid solutions are characterized by very close clustering processes. For the small values of x the formation of the 1: 0 : 3 : 0 basic cluster involves the displacement
+
0
+
Ba
0
+
0
+
0
+
+
0
+
Ba
Th
I
0
(+I
+
0
above and (0) below elementary Fig. 7. Schematic
representation
cubes.
of the 2 : 0 : 6 : 0 cluster.
J.M. Reau et al. /Solid
State Ionics 78 (1995) 315-331
4.2.1. nFl and nFll Whatever the value of x, nF, and nFff are located respectively around the (yr, = 4x/3) and (yr” = 5x/3) lines. As a consequence the (n,, + n,.) sum is in agreement with the (yr, + yrf8 = 3x) equation. Hence it appears that the sum of F’ and F” ions in plays a role analogous to that of F” Pbl-JkF2+*, ions (ypn =3x) in the other M1_-xThrF2+2x (M = Ca, Sr, Ba) solid solutions. Besides a new refinement computation of the Pb,,,,Th,.,,F2.20 structure, based on a F’ site close to (l/2, 0.41, 0.41), has been proved valid and results in a smaller occupation rate of the F’ site on behalf of the F” site. Another refinement based on the absence of F’ site results in a value of nFff close to the (nrl + n,.) sum determined from the former refinement [16]. It seems that is probably characterized by the Pbr-xThxF~+~n presence of basic clusters analogous to those of the other thorium solid solutions. As well-differentiated F’ sites are not required [16], the basic cluster in
325
Pbl_xTh,Fz+Z, will be called “1:0:3:0”, the (F”)* ions being located in an average (u,, u,, urn) position between both F’ and F” starting sites. The clustering process proposed for the Sri _ x Bi,FZ +x solid solution (0 G n < 0.50), which is also characterized by the presence of three F’, F” and F”’ interstitial fluoride ions, has been set up from analogous considerations [ill.
For each value of x the points representing nr,~ are located around the [ yr,,, = 7x/(70x2 + l)] curve that offers a maximum for x, = 0.119 and the ( y = 7x) line as tangent when x + 0. As in the other thorium solid solutions, the F”’ ions are actually the (Fi)m ions responsible for long range motions in according to the clustering process Phi-nThxFz+zx: model, x, is close to x,, (x,, = 0.125) for which a conductivity maximum is observed in the variation of log a,,, K with x (Fig. 2a).
Fig. 8. Variation with x of the sum of clusters (yCIUSL)totfor the Ml _,M’:F,+
zx (M = Ca, Sr, Ba, F%; M” = Th, U) solid solutions.
J.M. Reau et al. /Solid
326
State Ionics 78 (1995) 315-331
Hence the Pbl_-xThxF2+2x solid solution is characterized by the progressive ~~sformation with inclusters either into creasing x of “8:0:3:7” “1:0:3:0” or into condensed “2:0:6:0” clustens. In fact, the (xr&,rhj value close to 0.25, k)(Pb,Th) = 0.231, leads to select the transformation of “‘8:0:3:7” clusters into “2:O:h:O” ones. The analytical expressions of the sum of interstitial fluo-
ride ions and of the sum of vacancies in normal sites are: x(210x2 + 10)
X(70X2 4 8)
Yint=
70x2+1 ’ yO= 70x2+1 * The curves representing these functions are given in Fig. 10. The values calculated for the different functions characterizing Pb, _XThXF2+ax may be com-
0.
0.
0.
0.
Fig. 9. Graphic representation of the YF” function versus x for the M, _,Th,F2+ 2x (M = Ca, Sr, Ba, Pb) solid solutions.
JM. Reau et al. /Solid
State Ionics 78 (1995) 315-331
pared in Table 3 with experimental values determined by neutron diffraction. The good agreement observed between the calculated and experimental values confirms the validity of the clustering process proposed (Fig. 10 and Table 3). The variation of the total number of clusters in Pbl-xThxF2+2x is: [(
Ycl”sthotlwrh~ = ;I:,“:: 1:;
4.3. The MI _ .U, F2 + 2x (M = Ca, Sr, Ba, Pb) solid solutions
.
It is illustrated by Fig. 8: whatever the value of X, (yclustjtot is smaller in Pbl_-rThxFZ+Zx than in M,_,= ThxF2+2.X (M = Ca, Sr, Ba) and Kyc,ust)totl~b,Th~ 0.14 for x =xr. The smaller value of xs in with respect to other solid solutions Pbr-JkF2+2x resultsinless“8:0:3:7”clustersthan4:0:3:3or 3:0:3:2 clusters and in more “2:0:6:0” clusters (Fig. 11). Finally, for each x-value Pbl_XTh,Fz+z, shows a total number of clusters lower than the other solid solutions. On the contrary, the number of F”’ ions is higher than in M1_XTh,F2+2, (M = Ca, in Pbl-JhxF2+2x Sr, Ba) (Fig. 9). This result cannot be attributed to steric constraints as in the lead phase the unit cells are of size intermediate between those of the baryum and the strontium phases (Fig. 3). The presence of lone pair Pbzf ions in Pbl_XThXFZ+ZX favors in fact the displacement of a larger number of fluoride ions from their normal sites into F”’ positions [25,26].
Table 3 Experimental and calculated values of yr~, Y,v, y,“‘,
yint and y,
327
4.3.1. M = Ca, Sr The transformation with increasing x of 4 : 0 : 3 : 3 clusters into 1: 0 : 3 : 0 clusters has been proposed as clustering process for Ca,_JJ,F,+,, [17] and [18]. As in the homologous thorium Srr-XUXF,,,, solid solutions, the transformation of 4 : 0 : 3 : 3 clusters into condensed 2 : 0 : 6 : 0 ones can also be considered. Consequently, the total number of clusters in that case is given by following expressions (Fig. 8):
The total number of clusters is represented in by the same function as in Srr-.U,F*+,, (M = Sr, Ba). As a matter of fact the Mr-,Th,F2+2, three solid solutions show a conductivity maximum for the same value of x (x,,, = 0.20). The values of (yc,ust)tot obtained for the upper limit of the composition range of M, _JJ,F,+ 2X (M = Ca, Sr) are equal to 2: 0.18 for (~r)(c~,u) and = 0.17 for (~&~,u). They are close to those calculated for the thorium phases. These results plead in
for some compositions of the Pb, -JhxF2+ax
solid solution
x
0.05
0.10
0.125
0.15
0.20
0.23
Ref.
EF’
-
0.067
0.18 (3) 0.167
0.19 (3) 0.20
0.28 (3) 0.267
0.28 (5) 0.307
12Al
yp, = 4x/3
0.14 (4) 0.133
IzF” y,,, = 5x/3
0.12 (2) 0.083
0.19 (4) 0.167
0.22 (3) 0.208
0.25 (3) 0.25
0.34 (3) 0.333
0.4lW 0.383
1241
?zF’+ lzsz
0.12 0.15
0.33 0.30
0.40 0.375
0.44 0.45
0.62 0.60
0.69 0.69
1241
YF’+ y,,, = 3x ilF” YF”
0.37 (8) 0.30
0.37 (8) 0.41
0.40 (8) 0.42
0.41 (10) 0.41
0.40 (10) 0.37
0.31 (10) 0.34
WI
?rF’+ nF” + ?rF”! Yint.
0.49 0.45
0.70 0.71
0.80 0.795
0.85 0.86
1.02 0.97
1.00 1.03
1241
no YO
0.39 (8) 0.35
0.50 (8) 0.51
0.55 (8) 0.545
0.55 (10) 0.56
0.62 (10) 0.57
0.54 (10) 0.57
1241
J.M. Reau et al. /Solid
328
State Ionics 78 (1995) 315-331
favor of the tr~sfo~ation of 4 : 0 : 3 : 3 clusters into 2 : 0 : 6 : 0 clusters in the ur~i~ phases. 4.32. M = Ba shows a Bal-PP2+2, of the electrical properties Ba1_XTh,F2+2x (Fig. 2): conductivity maximum are and = 0.20 respectively for phases. These results induce calcium and strontium solid
composition dependence quite close to that of the x,,,:values of the close, equal to = 0.18 the uranium and thorium us to propose, as for the solutions, the same clus-
tering process in Ba,_,U,F,+,, as in Ba,_,Th,the value of (~~~~.)~n~,u~, the F afax. Considering equations relative to Bar _ ,U,F,, ax9 determ~ed on hand of the clustering process model by identi~ing X, with x,,,., are: 2x Y: = 3ox2 + 1 )
y; = 3x, x(90x2
Yint =
+
5)
3oXa + 1
X(30X2 + 3) 2
YO =
30x2+1
*
The total number of clusters in Bar _ ,I.JXFa + an is given by the expression (Fig. 8>:
1.0
z . Yint
/
0.6
Fig. 10. Graphic functions versus
representation of the Ypr, Y,., YF~jtrYi”* and y D x for the Pbl_-xThxFzi2x solid solution.
Its value (= 0.18) for the upper substitution rate xt equal to 0.285 is smaller than 0.25. This result pleads in favor of the transformation of 3:0:3:2 clusters into 2 : 0 : 6 : 0 clusters as possible clustering process. Variation with x of the number of 3 : 0 : 3 : 2 and 2 : 0 : 6 : 0 clusters can be compared for both Ba 1--rMZF2+2* solid solutions in Fig. 11: whatever the value of X, the number of clusters favoring an e~ancement of the ~nductivity is smaller and, on the contrary, the number of condensed 2:0: 6:0 clusters is higher for uranium phase than for thorium phase. As the size difference between the host and guest cations is slightly larger in Ba,_$JXFz+ ax than the tendency to establishment of in Bal-XThXPa+a,, an order with increasing x is stronger for the uranium phase: consequently the conductivity maximum occurs for a smaller value of x,,, in the uranium phase, a weaker conductivity is observed for the uranium phase, althought U4+ has a larger polarizability than Th4’ (Fig. Za). 4.3.3. M = Pb Whatever the value of x, the electrical properties (M” = Th, U) solid solutions of both Pbr_,M’:F,+,, are very close. In particular a conductivity maximum is observed for the same value x,,, = 0.125 of the substitution rate. As a consequence the clustering process shown in Pbl_JhXFa+zX, i.e. a progressive transformation with increasing x of “8: 0: 3: 7” clusters into “2 : 0 : 6 : 0” clusters, can be considered
J.M. Reau et al. /Solid
329
State Ionics 78 (1995) 315-331
as valid iu Pb, _JJXF2+2x. The application of the clustering process model results in the same analytical expressions for both solid solutions (Figs. 8, 10 and 11). Very close clustering processes characterize the (M = Ca, Sr, Ba, Pb; M” = Th,U) MI-IM;F2+2x
solid solutions. The different compositions corresponding to x equal to 0.06, 0.12 and 0.18 can be located inside a (P,, YFJ,,) diagram where Pa is the average polarizability of the cationic sublattice and y,,,, the function representative of the number of F”’ interstitial ions (Fig. 12). Whatever X, a minimum
(a) y4033
0.20 t
* '3032
' y003?
(Sr,U)
'3032 (Ba,Th) I
=00;7
(Pb,Th)(Pb,U)
04
,
0.20 _y2060
(Pb ,Th) (Pb,U)
(Ba,U) \
Fig.ll.Variationversus xofthenumberofclustersoftype(a)4:0:3:3,3:0:3:2or8:0:3:7and(b)2:0:6:0fortheM~_,M~F~+~, (M = Ca, Sr, Ba, Pb; M” = Th, U) solid solutions.
330
J.M. Rem et al. /SoEd St&e Ionics 78 (19951315-331
0.6
fPb_Th: \ 0.4
9.2
Fig. 12. Variation of YF- as a function of P, for the M1_xM:F2+2x (M = Ca, Sr, Ba, Pb; M” = Th, U) compositions 0.12 and 0.18 (the dotted lines drawn through the reported data are simple guides for the eye).
appears for barium compositions. It results from both presence of polarizable cations and steric constraints: ( *) Cations of higher polar&ability favor a displacement of a larger number of anions from normal positions of the fluorite structure into F”’ sites. ( *) Steric constraints are all the stronger as the unit cell is smaller. As a consequence, 11~~~~ increases from barium to strontium and to calcium compositions. This increase is all the larger as x is higher (Fig. 12).
5. Conclusions
The short range structure within the M,_; (M = Ca, Sr, Ba, Pb; &I” = Th, U) solid MYL,, solutions derives from the same 1: 0 : 3 : 0 basic cluster. Its formation results in transfer of fluoride ions
for x equal to 0.06,
from normal sites into F”’ interstitial sites. The number of such F” fluoride ions increases for stronger steric constraints and higher cation polarizability. As a consequence for the low substi~tion rates clusters of4:0:3:3,4:0:3:3,3:0:3:2~d8:0:3:7types form respectively in the calcium, strontium, barium and lead phases. At rising x-value all these clusters are transformed into condensed 2 : 0 : 6 : 0 clusters. Obviously the M1_xMtF2+2x solid solutions are characterized by very close clustering processes. The electrical properties have been determined as a function of temperature and composition. A comparison of the electrical performance for different Mr--xMI:Fa+tn compositions of the same x-value shows that the higher the average pol~~ability of the cationic sublattice the larger the mobility of the charge carriers and the bigger the fluorine ion conductivity.
J.M. Reau et al./Solid
State Ionics 78 (1995) 315-331
References m J.M. Rtau and P. Hagemnuller, Appl. Phys. A 49 (1989) 3. M C.R.A. Catlow, J. Phys. C 9 (1976) 1845. 131 J. Corish, C.R.A. Catlow, P.W.M. Jacobs and S.H. Ong, Phys. Rev. B 25 (1982) 6425. [41 J.P. Laval, A. Mikou, B. Frit and G. Roult, Solid State Ionics 28-30 (1988) 1300. I51 AK. Cheetham, B.E.F. Fender, B. Steele, R.I. Taylor and B.T.M. Willis, Solid State Commun. 8 (1970) 171. and P. Hagenmuller, Phys. [61 J.M. Riau, M. Wahbi, J. S&gas Status Solidi (b) 169 (1992) 331. t71 S.F. Matar, J.M. RBau, P. Hagenmuller and C.R.A. Catlow, J. Solid State Chem. 52 (1984) 114. P31J.M. Rtau, M.El Omari, J. SBnCgas and P. Hagenmuller, Solid State Ionics 38 (1990) 123. [91 M. El Omari, J.M. Rtau and J. SBnBgas, J. Solid State Chem. 87 (1990) 430. DO1 J.L. Soubeyroux, J.M. R&au, M. Wahbi, J. Sdnegas and Suh Kyung Soo, Solid State Common. 82 (1992) 63. ml J.L. Soubeyroux, J.M. RBau, M. Wahbi, J. Se&gas and Suh Kyung Soo, Solid State Commun. 83 (1992) 259. [I21 N.H. Andersen, K. Clausen and J.K. Kjems, in: TransportStructure Relations in Fast Ion and Mixed Conductors, eds. F.W. Poulsen, N.H. Andersen, K. Clausen, S. Skaarup and O.T. Sorensen (Riso National Lab., Roshilde, 1985) p. 171.
331
[13] M. Ouwerkerk, F.J. Veldkamp and J. Schoonman, Solid State Ionics 22 (1987) 173. 1141 J.P. Laval, A. Mikou, B. Frit and J. Pannetier, J. Solid State Chem. 61 (1986) 359. [U] J.P. Lava& A. Mikou, B. Frit, G. Roult and J. Pannetier, Rev. Chim. Miner. 24 (1987) 165. 1161 A. Abaouz, Thesis (Limoges France 1988). [17] M. El Omari, J.M. R&au, J. S&r&gas, J.P. Laval and B. Frit, J. Solid State Chem. 92 (1991) 312. [18] J.M. RBau, Xu Yong Jun and J. Senegas, Phys. Status Solidi (a) 143 (1994) 51. [19] M. Ouwerkerk, Thesis (Utrecht, The Netherlands, 1986). [20] A. Mikou, Thesis (Limoges, France, 1986). [21] J.M. Reau, A. Rhandour, C. Lucat, J. Portier and P. Hagenmuller, Mat. Res. Bull. 13 (1978) 827. [22] A. Rhandour, Thesis (Bordeaux, France, 1979). [U] C.K. Jorgensen, Top. Curr. Chem. 56 (1975) 38. 1241 J.L. Soubeyroux, J.M. R&au, S. Matar, P. Hagenmuller and C. Lucat, Solid State Ionics 2 (1981) 215. [25] J. Portier, J.M. RBau, S. Matar, J.L. Soubeyroux and P. Hagenmuller, Solid State Ionics 11 (1983) 83. [26] Y. Ito, T. Mukoyama, S. Yoxhikado and K. Koto, Solid State Ionics 67 (1994) 307.