A new statistical thermodynamic theory for substoichiometric fluorite structure compounds and its application

Materials Chemistry 6 (1981) 381 - 400

A NEW STATISTICAL THERMODYNAMIC THEORY F O R SUBSTOICH]OMETRIC F L U O R I T E STRUCTUI~E COMPOUNDS AND ITS APPLICATION I.

AN O R D E R - D I S O R D E R MODEL THEORY F O R THE THERMODYNAMIC FUNCTIONS O F SUBSTOICHIOMETRIC FLUORITE STRUCTURE COMPOUNDS

L. MANES*, E. PARTELI*, C.M. MARl** *

C o m m i s s i o n o f the E u r o p e a n C o m m u n i t i e s , J o i n t R e s e a r c h C e n t r e - E u r o p e a n Institute for Transuranium Elements - P o s t f a c h 2266, D - 7 5 0 0 - K A R L S R U I t E I - West G e r m a n y .

**

I s t i t u t o di E l e t t r o c h i m i c a e Metallurgia - Universitd di Milano - Via Venezian, 21, I - 2 0 1 3 3 M I L A N

- Italy.

Received 16 June 1981; accepted 9 July 1981 Abstract - In this work an attempt is made of constructing the (G, T, x) curves f o r the case of substoichiometric oxides of the fluorite structure on the basis of a statistical model. Considering the fluorite structure as a repetition of oxygen ions tetrahedrically coordinated, the hypothesis is made of a local bond between the reduced cations and the oxygen vacancy in the tetrahedron ("tetrahedral defect"). This "tetrahedral defect" constitutes the building block of the low temperature sub-phases often encountered in these systems; at higher temperatures its packing in the lattice gives rise to ,residual structures". One can express all thermodynamic functions, at given oxygen/metal ratio, as a function of an "~rdering parameter" p, representing larger and larger packings of the tetrahedral defects, in such a way making tiffs model a typical order-disorder treatment. Introducing the formalism of chemical equilibria in the equilibria between the various defect species it is also possible to obtain thermodynamic functions by the solution of a system of pseudo-chemical equilibrium equations, when probability functions f are employed in place of the activities of the defects species. 0390-6035/81/060381-2052.00/0 Copyright © 1981 by CENFOR S.R.L. All rights of reproduction in any form reserved

382 INTRODUCTION Considerations on the thermodynamics of non-stoichiometric compounds In two illuminating papers, J.S. Anderson I and R.J. Thorn 2, (see also 3 in discussing the general thermodynamics of grossly non-stoichiometric compounds) have pointed out that the free energy surfaces of these phases in the (G, T, x) space must contain, at high temperatures, information which permits to predict the low temperature phases which are evidenced by crystallographic studies. This information appears as "residual structures" in the free energy curve, at high temperature, i.e. in zeros of one of the ~Gn/Sx n derivatives, and in practice, in critical points in the G-x or/a-x curves. Experimental studies of thermodynamic functions are not always able to evidence these critical points, but, in the particular system of sub-stoichiometric fluorite-structure oxides of lanthanides and actinides, experimental work has shown peculiar behaviour of the AFi(o2) vs x curves 4' s or the AG(O2) vs x curves, at stoichiometries which are close to the composition of the low-temperature phases 6, 7. In terms of accurate statistical description of these systems, the only possibilities which appear, if one desires to describe free-energy curves and preserve residual structures, are the following: a)

b)

describe mathematically with extreme accuracy the energy of interaction between elementary point defects (in the case of the above mentioned oxides, oxygen vacancies and reduced cations); postulate an association of elementary defects, which necessarily will be extended to some lattice volume, as the building block of the non-stoichiometric phase.

The former way has been very thoroughly pursued by Atlas 8 who developed spacing statistics which discretize a Madelung energy of electrostatic interaction between lattice defects (oxygen vacancies) and electronic defects (heterovalent cations) in fluorite structure oxides. In this work, we follow the latter, and define a building unit extended to the coordination tetrahedron of a vacancy. The partition function for a system so composed and hence the thermodynamic functions will be derived. In a later note, we will show that such a treatment is able to describe suffi-

383 ciently the low temperature phase diagrams of these oxides, the residual structures often encountered in their high temperatures thermodynamics and even to suggest useful information on the nature of their bonding. Some characteristics of substoichiometric fluorite structure oxides The fluorite-structure substoichiometric oxides MeO2. x of the lanthanide and actinide series have been shown (with the remarkable exception of PuO2.x and AmO2-x) to form low temperature subphases with a general empirical formula MenO2n-2, where in takes integral values 9. Recent work by O.T. SCrensen 6, 10 on the oxygen potential at high temperatures (i.e. in the disordered single phase non-stoichiometric region) of CeO2. x shows that stoichiometry regions characterized by well defined (often very high) d In pO2 values of the slope m occur (see Figs. 1 and 2). dlnx It was indeed postulated that the regions are explained by complex defect equilibria, and the anomalously higher values of m (which appear as breaks in the AG(O2) vs In x curves) as indicating "quasi-phase" transitions between different types of extended defects. In tracing a diagram where the compositions in which these phases occur are reported as a function of temperature, this author was able to propose a characteristic "pseudo-phase" diagram for defect equilibria, which has striking similarities for the Ce-O, Pu-O and U-Pu-O systems. We may remark that these anomalies represent the "residual structures" mentioned above. We propose to clarify to some extent, by our'theory, the two facts here described.

BASIC DEFINITIONS OF THE MODEL Definition of the tetrahedral defect In a previous work I 1, a statistical model was presented for the thermodynamic functions of the Pu-O and U-Pu-O systems (treated together). The basic hypothesis of the model are briefly repeated here. It is assumed that, in a fluorite structure substoichiometric oxide MeO2.x, the basis point defects (oxygen vacancy V and reduced cation Me÷a) may be associated in a more

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complex defect which we term "tetrahedral defect" (in the fluorite lattice, an oxygen site is tetrahedricaUy coordinated with the surrounding cations). The "tetrahedral defect" (2 Me ÷~ V) is defined as that unit of the lattice in which the vacancy is locally bound with two reduced cations in its coordina-

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tion tetrahedron. This complex defect is locally neutral with respect to the lattice but it will possess a permanent dipole moment, since the effective charges ze (relative to the lattice) of the constituting point defects do not coincide in position. It may be remarked that the tetrahedral defect requires that at least two re.

386 ducible cations are present in the coordination tetrahedron of an oxygen site: this occurs for all oxygen sites in a MeO2.x system, only for a limited number of oxygen sites, dependent on y, for an Ml.yMeyO2. x system (M, being non-reducible cations such as U +4 in the U-Pu-O and U-Ce-O systems 11).

Type of tetrahedral defects and their packing The tetrahedral defects will gradually fill the fluorite lattice when reducing the oxide. It is supposed that they have an energy of ,ormation: Ef I = E f + / 5

(1)

Ef is the formation energy of a vacancy V in the oxygen sublattice plus 2 reduced cations Me *a , in the case they are independent from each other. The term 8 is a bonding energy of the tetrahedral defect: in general, an endothermic term (8 < 0), representing an attraction between vacancy v and reduced cations Me ÷3. Packing tetrahedral defects of this sort in the fluorite lattice cannot however proceed indefinitely without introducing strains in the lattice. Consider the introduction of a second tetrahedral defect in one of the tetrahedral adjacent to a formed one. For this packir~, a repulsive energy A 2 ~> 0 must be added to E f t . The same will occur for the formation of a third adjacent tetrahedral defect, which will require a repulsive energy A 3/> 0 (in general A 3/> A2), so that: Efj = Efl + Aj

(2)

and Aj+ 1 ~ A j ~ O

(3)

where A 1 =0

(see equation ( 1 ) ) .

The definitions (2) and (3) form the basis for a site exclusion principle. Call an isolated tetrahedral defect a complex of order 1 with formation energy Ell, and two or more continuous tetrahedral defects a complex o f order k, (k = = 2...) with formation energies Elk (equation (2)). This defines a series o f complex

387 es of order j, with j = 1, 2... The total number of complexes of order j in equilibrium in the crystal will be denoted thereon by nj. In order to have an isolated tetrahedral defect (j = 1), all C i oxygen sites, directly bound to one (or more) cations of the tetrahedron, should not be allowed to form other tetrahedral defects of the same order. For a complex of order j = = 2, two tetrahedral defects are packed in these Cl sites: each tetrahedral defect will therefore have, roughly, in a spherical approximation, C1/2 oxygen sites in which a similar tetrahedral defect cannot be lodged. On may say that, for complexes of any order an "envelope constant" Cj 8 exists, given roughly by Cj = C 1/J

(4)

it is important to remark the difference between the concepts of "envelope constante" in 8 and in this work. Whereas in 8 they were away of special discretization of interaction energy between unassociated point defects, here their definition is entirely given by eqns. (2) and (3), which express the progressive straining of the crystal introduced by successive packing of tetrahedral defects. The envelope constant C1 can be found by inspection of the fluorite structure. A "complex 1" will be coordinated to 23 oxygen sites (23 being the number of oxygen sites satisfying the previous condition) a "complex 2" will be coordinated, therefore, to 11,5 oxygen sites and so on.

Dissociation of tetrahedral defects: "free" defects as possible species Tetrahedral defects are assumed to be, at each temperatures partially dissociated into free vacancies V (;2) and free reduced cations Me*3('1) (in upper brackets, the charge of the free point defects relative to the lattice is indicated). The formation energy of one V ~.2) vacancy and two Me.3 (-1) cations is Ef, as already stated. The total number of free vacancies V (;2) in equilibrium in the crystal will be denoted thereon by n o . The total number of free reduced cations will be by electroneutrality 2n o.

388 Conditions of validity of the model

The hypothesis is made that the tetrahedral defects form in the fluorite structure. Therefore, the model is strictly applicable only in the composition and temperature range where the fluorite structure exists (see Fig. 3).

Ce-O 150C

|

,

i

i

1.8

19

I---"

9O(;

30C "C:'

15

16

,1

1.7

2D

o/ce Fig. 3 - The phase-diagram o f the Ce-O system and the range o f validity o f the present model {shaded region).

If extended to other intervals, the Cj's and Aj's should be modified accordingly. Nervertheless, an attempt will be made to extend the model to other ranges than its proper one. This attempt is justified by the fact than it is believed the tetrahedral defects and their different forms of packing to contain the basic information of defect short range ordering in the ~iaorite structure. Also, the model is essentially a single phase model. The search for possible instability regions in the (G, T, x) surfaces, suggesting diphasic regions, is typical however of order-disorder treatments of this sort ]

A POSSIBLE EXPLANATION OF THE MenO2n. 2 FORMULA

If one introduces the tetrahedral defects, defined above, in the lattice, a saturation concentration will exist for any packing of order j.

389 By the definition (4), it is possible to define the saturation concentrations in the lattice of complexes of any order as: 1 Xcj = NM

(max. number of tetrahedral defect j)

1 2NM -~M ( ~ )

2 Cj

(7)

For binary systems the MenO2n. 2 formula can be written in terms of the oxygen to metal ratio O/Me = 2 - x (when taking the formal expression MeO2.x) , as 1 O / M e = 2 (1 -

--

)

(5)

n

Using the definition: x = N /NM, where Nv is the number of vacancies responsible for the oxygen deficien, y x, N M the total number of cations, one can write: 2NM n=~ Nv

(6)

The coefficient n in MenO2n. 2 represents the number of oxygen sites which are associated to one oxygen vacancy in the subphase MenO2n. 2. One may think, therefore, of an MenOzn.2 subphase as the long range reordering of an oxygen deficient fluorite structure MeO2.x, in which a vacancy is, in the average, surrounded by n oxygen-sites in which no vacancy is formed. This correlates the recurring formula MenO2n-2 not with particular oxygen to metal ratio, but rather with determined values of the number of oxygen sites coordinated with a vacancy which are excluded from formation of another oxygen vacancy. In this way, we look for an explanation of the MenO2n.2 formula on the basis of a site-exclusion principle applied to the oxygen sublattice only. If one makes the hypothesis that the site exclusion principle is the one we have exposed above a correspondance should be found between the envelope constants Cj and n in the MenOzn.2 formula. The comparison is made in Table 1. In all the systems listed in Table 1, each intermediate MenO2n-2 phase with n > 12 decomposes peritectoidally (and at successively higher temperature as n decreases) to form a homologue of lower n and the fcc phase ~3' of approximate composition MeOl,83 (n = 12). The last member of the series in Table 1 (Me7012) dissociated instead into the fcc a-phase, and bcc C'-phase for which 1.7 > x ) 1.6 (see Fig. 3). All the crystal structures of the subphases can be thought of as derived from the fluorite structure by progressive distortions. The Cj's evaluated in the present model have the following limitations:

390

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391 a) b)

c)

they are evaluated on the basis of the fluorite structure, and are strictly dependent on its geometry; they are evaluated without paying attention to the conditions of filling of the fluorite lattice: in fact, for instance, packing in an optimal way tetrahedral defects of order 1 without changing the volume of the crystal would leave some oxygen sites unused for purely geometric reasons, and therefore lead to a Ca (average number of oxygen sites associated per tetrahedral defect) closer to 24 rather than 23; the assumption is made that no deformation and/or reorientation of the original tetrahedron occurs when a tetrahedral defect is formed.

Taking into account these limitations, the Cj's listed in Table 1 should describe rather average values of packing, valid at the higher temperatures in the fluorite disordered phase, and not describing the finer structural details. (An analysis of the finer structural details is being made in 12). It is however rewarding that the inspection of Table 1 shows the Cj's to be an average for the group n = 7,9 and to represent well the most stable 3' phase (O/M = 1.833), always present at higher temperature. It is natural at this point, to assume that, at lower temperatures, the limitations a) to c) take a greated importance, causing the peripectoidal decompositions proper to the series. However, the basis definition of the model might include easily the corrections involved, by a more careful definition of the Cj's and the inclusion of the necessary energetic terms in the Aj's (which will be attempted in 12 ).

THE PARTITION FUNCTION OF THE DEFECTS

Generalized partition function of the defects If a stoichiometric crystal is chosen as the reference state, and the defects are considered to be a canonical ensemble, the partition function can be written as: "--= ~A ~'2A exp ( - E A / K T )

(8)

from which the maximum term can be extracted 13 as: Q = ~2(ni) exp ( - E ( n i ) / K T )

(9)

392 which is still def'med by sets of ni, which satisfy the equation: L N v = Z n i = n o + Z nj i j=l

(10)

where n i is the number of defects of type i, with i = O, 1, 2... L, where L is the higher order of complexes considered. In this the implicit assumptions are made, as usual: a) b)

the number of metal ions is constant and the metal lattice is perfect; the total vibrational contribution in the general partition function for the whole crystal is unaffected by the introduction either of tetrahedral defects or by free vacancies and reduced cations (except for the contributions already implicitly introduced in the definition of the 5 and Aj's terms); the total volume of the crystal remains unchanged when defects are in-built.

c)

Expansion of ~ ( n i ) The degeneracy ~2, is given by L ~'~=~"~v~"~e l'l ~'~j j=l

(11)

where f2v [2e describes the lattice and electronic disorders. We write for (2 N M ~2v =

(2 N M - N v ) ! n o ! (N M

f~e =

Z nj)! j=I,L

-

~ n j)! j=I,L

( N u - 2 N v ) ! (2 no)!

(12)

03)

The special forms (12) and (13) discount as unassociated point defects those situations which have the configuration already chosen for the tetrahedral defects. Implicit in this way of counting is also, for instance, that situations such as shown in Fig. 4 are treated as one tetrahedral defect plus one free defect. ~J

=

wj ! (wj-nj)!nj[

Cj nj

(14)

393

<

Me:~" ~

M I e3÷

.

~ j e3~.j

Me 3÷

I-~" - - i

"("k~"M ~3e" ~J(fre~)

Fig. 4 - Two situations in the lattice counted as." tetrahedral defect + point defect a) tetrahedral defect + free reduced cation b) tetrahedral defect + free oxygen vacancy.

where the term C?j takes into account the fact that one vacancy has Cj sites in which to be formed, and wj is given by the formula: 2N M + n j V j -

Y, nrC r r=l,j

wj =

(15)

Cj

as in 6,14 (hence one obtains w I = Xel " NM). A general recurrent formula for wj is also wj=wl-

j-i Y" ( l / r ' n r )

+nl

(15b)

r=l

from which (wj - n j ) =

J j+l

wj + I

(15c)

Remark that, as in 8, 1 3, wj is obtained by considering all lattice points left available for formation oftetrahedral defects, once the less packed tetrahedral defects have removed their quotas. Since a free defect is a possible site of formation for a tetrahedral defect, there is no need, in wj, to subtract the n o free defects which are present.

394

Expansion of the total energy E(ni) The total energy E(ni) is expanded in short-range terms Efj, already defined, and long-range interaction energies e and u: the long range interaction energy e between tetrahedral defects and the long-range interaction energy u between free point defects. With these assumptions, the total energy E(ni) can be expanded as: L

E(ni) = Z n i E f i + ( N v - n o ) 2 e + n o 2 u i=o

(16)

where Efo = Ef (as seen before), or else: E(ni)=EvEf+

L ~ njAj+(Nv-no) 2e+n~u

j=l

(16a)

The thermodynamic function All thermodynamic functions can be obtained from the partition function Q, i.e. from equation (16) and (11). One obtains (Nv - n o = N~): l n ~ ( n i ) = (2NM - N~) In (2NM - N~) + (NM - 2N~) In (NM - 2N¢) + - (2 NM - Nv) In (2NM - Nv) - (NM - 2Nv) In (NM - 2 N v ) - 3no In (no) + +2noin2+

~

j=I,L

(wjlnwj-(wj-nj)ln(wj-nj)-njlnnj+njinCj)

(17)

where the usual Stifling approximations has been applied. To find equilibrium conditionsS,X s, maximize In ~2 with respect to the set of populations ~ni}, and subject to the conditions: ¢l = N v -

E ni=0 i=o., L

~2 = E - E(ni) = 0

(18a) (18b)

This case be done by means of the Lagrange method of undetermined multipliers, giving rise to the system: 8 ~(ha 8ni

~ + t v ~ 1 + 3~02) Nv, E = 0

(19)

395 where a and fl are the undetermined multipliers. As usual 14, one recognizes fl = 1 . By (9) and (18), one f'mds that the system (19) can be written as: kT (20)

Z i - a = 0

where /5 In Q Zi =

/5(In ~ 2 - E ( n i ) / k T ) -

/5 n i

/5 n i

(21)

By solving the system (19) or (20), the set of equilibrium populations {ni} is obtained. There is a convenient way of writing the system (19): I = e "c~

(22)

By somewhat lengthy, but simple algebra, one obtains, for the populations of complexes ni (j :/: 0), and from the equations of the system (20) for j 4:0 yi=j(1/Mj)

II

(Ms/(l + M s ) ) = h j ( l )

(23a)

s=l,j

Mj = a j , I

11 ((1 + Mt)/Mt)t/J', ( j ' :~ L), ML = aL 1 t=(j + I),L

(23b)

Here we have introduced the following def'mitions: /5 + A j aj = (1/Cj) exp - -k T-

(24)

and Yi =

ni Wl

(25)

represent all concentrations with respect to the saturation value for complex 1. In Appendix 1, the relationships (23) are derived. One advantage of (23) is that all yj's are expressed as a function of 1 = e ÷a If one defines: /j = ~j yj the system (20) regroups into:

(26)

396 Zo=a=lnl

(27 a)

/j = h(l) = Zj yj

(27b)

where (A-~)(PZ o = in

2~) 2 + H ~ - V Yo

4 y o3

(28)

We have defined: 2NM , A=-wl 2ewl H = m

kT

,

P-

V-

NM wl 2UWl kT

(29a)

(29b)

Equation (28) is unchanged for whatsoever maximum number L of complexes is taken into account. The reduced system (27) shows that the equilibrium thermodynamics is completely described by the two concentrations Yo and /j. The concentration Yo is the concentration for the "free" vacancies and "reduced cations", i.e. of the electronic and lattice disorder. The concentration ~ is the total concentration of tetrahedral defects, in their different microordered states i= i ... L. If y = Nv/wl one has express all thermodynamic functions, at given oxygen/ /metal ratio, as a function of an ordering parameter p = --

(30)

Y varying from 0 to 1. This makes this model a typical order-disorder treatment, as many times postulated for the thermodynamics of non-stoichiometrie solids 1,2. It is easy to show that the chemical potential of the system is given by: ~s=kT

61nQ [T=kT( - Ef-H~+lnl+b(y,/j)) 8Nv kT

(31)

where use has been made of equations (16), (17) and of the equilibrium system (19),-and of the obvious relation: dni Z = 1 i=o,L d Nv

(32)

397 the term b(y, ~) is defined as: (A - y) (P - 2y) 2 b(y, ~j) = In (A - ~) (P - 2~) 2

(33)

which becomes 0 in the case of/~ ~ y. In this particular case, which is bound to occur at low T, all the information on the system will be contained in the term In 1 which is just a function of ~. This illuminates the necessity of studying this particular part of the chemical potential, which must contain the information on discontinuities such a spinoidal point. This will be done in a subsequent paper. in the alternative case of ~ ~ 0, which is bound to occur at high T, it is easy to show that the chemical potential reduces to that expected for the case of uncorrelated lattice and electronic disorder. Equation (31) and the system (27) permit a very simple way of evaluating the chemical potential/a s and the free energy G, once well definite hypotheses are made on the relevant energetic parameters.

CHEMICAL EQUILIBRIA It seems to us of some interest to try writing the equilibrium equations between the various defect species considered with the formalism of chemical equilibria. The thermodynamic properties o f the defect solid can be obtained by considering the (chemical) reaction: 2 Melatt + Olatt ~

2 Me' + V ' + 1/2 02

(34)

together with a set of (chemical) reactions of the type: [2 Melatt C j O l a t t ] ~ (j=l

[(2 Me V) (Cj - 1) O t a t t ](x) + 1/2 0 2

(35)

... L)

(the term in square brackets being the tetrahedral defect in its different packing, (Cj -- 1) O lattic e the number of oxygen sites excluded, Me a reduced cation). Formally (34) leads to " f2Me' fv'" Ko -

po~/2 f~,le-lat t fO-la t t

(36)

398 and (35) to the set of

fj Kj = ~

fq

po~/2

(37)

The fj do not represent the concentration of the different defect species but rather the probabilities of ffmding them in the proper lattice sites. It is possible to derive the f's bv simple probabilistic considerations, as well as from degeneracies derived in the statistical treatment.

CONCLUSIONS The introduction of the hypothesis of the tetrahedral defect permits to simplify remarkably the description of the thermodynamic partial molar quantities of a sub-stoichiometric fluorite structure compound. From a statistical point of view, what we have done is to state that a quasi-molecular unit (or cluster) is a more suitable "statistical object" than the constituting point defects to achieve this description. The quasi-molecular unit expresses suitably the total energy of the defect system, by parting in a short-range part ("bonding" and "strain" terms) and a long-range part describing interaction at distances greater than characteristic volumes (our envelopes Cj) in which bonding and strain takes place. The advantage is twofold: a) introducing suitable hypotheses on the Cj envelopes, one obtains, from geometric consideration of the fluorite structure only, characteristic compositions of the low temperature subphases, which are attained when thermal motion will no longer be able to sustain the "strain" exothermic contributions; b) one is able to construct surfaces in the (G, x, T) space over a much larger composition range than usually possible with the current interaction hypothesis of point-defect theories, and these surfaces have, already, in-built information about the discountinuity point which will give rise, at low temperature, to sub-phase. The model by Atlas I a, which makes the most valuable effort to extend point-defect interaction theories by describing the spatial discretization of Madelung-type Coulomb interaction between defects, does not contain, in the G-function, information on the critical composition points. It is interesting to note that already in this case the G-curves have critical regions at certain stoichiometry ranges. This is due to the particular form of the entropy, essentially bound to the concept of variation of energy, with packing of defects. But only when description of this packing contains, as in our case, by consideration "external" to the statis-

399 tics, "molecular" information, can the description be really accurate. The statistical treatment of tetrahedral defects results essentially to be an order theory when dissociation of the defects is also taken into account, as it is proper for high temperatures. Conceptually, in fact, one returns to the basic concepts of any order theory 14, having first defined the two species ("free" point defects, tetrahedral defects) which are in an ordered or disordered solution. The problem arises why to choose one particular form of association ("tetrahedral" rather than other ones) for the building block. Rather than attempt a full justification of what is obviously an arbitrary assumption, one may point out hints in this direction and the possible physical meaning of the assumption. The former are given by very interesting discussions on structural and mechanical properties of the fluorite-structure oxides made by Blank 1s, which point out the tetrahedral coordination of an oxygen to be the most relevant feature of the fluorite structure oxides. The latter is evident: the tetrahedral defect means essentially the assumption that the oxide bonding occurs in its largest part in the coordination tetrahedron of an oxygen site, so that, when an oxygen vacancy is formed, it is this tetrahedron as a unit which will give the essential defect form. In any case, the assumption of a "statistica object" is validated by its power of description of the thermodynamic reality. In another paper, we will analyse the G-surface constructed with the formalism here described, and discuss their adequacy to explain the thermodynamic function of these oxides, their phase diagrams, and some differences between them.

A cknowledgemen t The authors wish to thank Dr. H. Blank and Dr. Hi. Matzke for many valuable discussions and Dr. O.T. S~rensen for his constant interest in this work.

Appendix 1

A)

In this appendix we give the derivation of equation (23). The system (20) is constituted by equation (28) and the general equation: ZJ = l n (

nj wj

)+

L ~. (__q. ( W q - n q ) + l n C j ) q =j j Wq

where wj is given by equation (15).

(A1)

400 B)

Starting from Z L and using equation (15c) all equilibria j can be reduced to the form: wj - n i

C)

= Mj nj where Mj is given by equation (23b). From (A2), 1 nj = ~ wj I+Mj

(A2)

(A3)

and using equation (15c) and (22b), equation (23a) is obtained.

REFERENCES 1.

2. 3. 4.

5.

6. 7. 8.

9. 10. 1 I. 12.

13. 14. 15.

J.S. ANDERSON - The chemistry o f extended defects in non-metallic solids, Ed. Le Roy Eyring, M.O'Keeffe, Scottsdale, Arizona, 1969, North Holland, 1970, p. 1. R.J. THORN - l b i d e m , p. 395. I.S. ANDERSON - Problems o f non-stoichiometry, Ed. A. Rabenau, North Holland, Amsterdam, 1970, p. 1. G. DEAN, J.C. BOIVINEAU, P. CHEREAU, J.P. MARCON - Plutonium 1970 and Other Actinides, Proc. Conf. Santa Fe, 1970, Nucl. Metall., Metall. Soc. AIME 17, pt II, 1970, p. 753. P. CHEREAU - Contribution d l'Etude Thermodynamique des Oxides de Plutonium et des Oxides Mixtes Uranium-Plutonium, Th6se de Doctorat, Universit~ de Paris-Sud (Orsay) 1972. O.T. SORENSEN - J. Sol. Stat. Chem., 18, 217, 1976. J. CAMPSERVEUS, P. GERDANIAN - J. Sol. Stat. Chem., 23, 73, 1978. L.M. ATLAS - The chemistry o f extended defects in non-metallic solids, Eds. Le Roy Eyring, M.O'Keeffe, Scotsdale, Arizona, 1969, North Holland, Amsterdam, 1970, p. 425. M.S. JENKINS, R. P. TURCOTTO, L. EYRING - F o i d e m , p. 36. O.T. SORENSEN - Plutonium and other Actinides, Eds. R. Lindner, H. Blank, Baden-Baden, 1975, North Holland, Amsterdam 1976, p. 123. L. MANES, B.N. MANES-POZZI - Ibidem, p. 145. L. MANES, C.M. MARl, I. RAY, O.T. SORENSEN - Thermodynamics o f Nuclear Materials 1979. Vol 1, International Atomic Energy Agency, Vienna, 1980, p. 405. L.M. ATLAS - J. Phys. Chem. Sol., 29, 91, 1968. R. FOWLER, E.A. GUGGENHEIM - Statistical Thermodynamics, University Press, Cambridge, 1960. H. BLANK - Thermodynamics o f Nuclear Materials 1974, Vol. 2, International Atomic Energy Agency, Vienna, 1975, p. 45.