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A C.V.M. APPROACH OF MULTIPLET
CORRELATION FUNCTIONS IN SUBSTITIONAL
SOLID SOLUTIONS
D. Gratias, P. Cenedese
To cite this version:
A C,V,M, APPROACH OF MULTIPLET CORRELATION FUNCTIONS I N SUBSTITIONAL S O L I D SOLUT IONS
D.G. G r a t i a s and P . Cenedese
Centre d 'Etudes de Chimie MétaZZurgique (C.N.R. S. ), 15 Rue Georges Urbain, 94400 Vitry sur Seine, France e
Résum4
-
Nous r4sumons une methode thermodynarnlque pour dbtermlner les fonctions de corr4latlon de multiplets dans les solutions solides. à partir des donn4es exp4rlmentales de diffusion diffuse. Une illustration en est donne8 dans le cas des ferrltes LIFe02.Abstract
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A thermodynamlcal method Iç summarlzed for the determinatlon of the multiplets correlation functions In substitutlonal solld solutions starting from the experimental diffuse scattering data. The technique Is exemplifled wlth the study of llthlum ferrlte.1. INTRODUCTION
Short range ordered crystalllne alloys cannot be totally described by standard
crystallographlc concepts slnce the translational symmetry applles only to the 'average alloy'. A full descrlptlon of a statlstical short range ordered alloy would requlre an inflnlte flle of the values of al1 the correlatlon functlons assoclated w i t h t h e i n f i n i t e hierarchy of clusters composing the crystal. A way of by-passlng thls fundamental difflculty consists In uslng numerical slmulatlons for bulldlng a conflguratlon. by trial and error. whlch exhlbits pair correlation functions close to the ones experlmentally determined. T h i s t e c h n i q u e m a k e s n o e x p l i c i t c o n s t r a i n ç o n t h e h i g h o r d e r c o r r e l a t i o n functlons (triplets. multiplets) else than the geometrlc connections whlch frustrate the posslble conflguratlons of the multiplets. In partlcular there are no thermodynamlcal requlrements whlch lnsure the final configuration to correspond to a mlnlmum of some free energy functlonal.
The alm of the present note Is to propose an a l t e r n a t i v e method e s s e n t i a l l y based on thermodynamlcal considerations and whlch leads to the slmultaneous d e t e r m i n a t i o n of : 1. The palr potentlals,
II. The multiplets correlatlon functlon.
The used thermodynamlcal model Is an k i n g Hamlltonlan whlch 1s known to be a good approxlmate fordescribing the coherent phase transformations (essentially order
dlsorder transltlons) ln blnary alloys. The statistlcal treatment Is based on a C. V. M.
C l 1 approxlmate entropy.
Our derlvatlon Is mostly lnspired from P.C. Clapp's t21 earller works on blnary alloys who used a P.V. M. entropy whlch unfortunately does not take optlmally lnto account the overlapps between clusters. Also Clapp's models dld not conslder the Interna1 energy part of the free energy and therefore were not dedlcated to glve the palr potentlals.
II. THE C.V. M. MODEL ( 2 )
The C.V. M. free energy functlonal In the a-cluster approxlmatlon may be wrltten as C21 :
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(21 A more detalled paper wlll be submltted to the Journal de Physique.
JOURNAL
DE
PHYSIQUEwhere
*
p 1s the Boltzmann factorq
are the multipllcltles of the subclusters jthe effective potentlals (In the palr lnteractlon mode1 Vj
=
O for al1 j different from pairs and points)a ai are the C.V. M. coefflclents determlned by the recurslve relation
where 1s the number of L-type clusters sharlng the same ]-type subcluster the pj's are the probablllties assoclated to the 2") configurations of the j-type cluster contalnlng ni points.
*
op's are the occupancy numbers (ap =*
1) on the cluster p := n a
p~ n e p n
n
=
sites of p ciuster, and 'Op> the correspondlng correlation functlon deflned by < a > - TrP j > p
a ~ P 1
( 4 )Once the point and pair correlatlon functlons are flxed the correspondlng V'S become Lagrange multlpliers In the minirnizatlon of the free energy functionnal ( 1 ) . The hlgh order correlatlons functlons and the palr potentlals are solutions of the non llnear system of N slmultaneous equatlons and N unknowns
where the solution vector & of comoonents {<Qp>) 1s lnslde the convex polyhedron S deflned by :
VI
q . 0
( 6 )III. THE CHOICE OF THE BASIC CLUSTER
The cholce of thls cluater 1s determlned by the range of the lnteratomlc lnteractlons In
conjunction with t h e degree of accuracy required for the solution. Although the C. V. M.
approximation 1s not restrlcted to palr wlse lnteractlons only. we conslder essentlally thls speclal case whlch has been proofed to be pertinent for metalllc alloys : for noble metals. lt has been shown E4.51 that both cohesive and orderlng energles are correctly descrlbed wlth pseudo potenttals behaving assymptotlcally as Friedel's osclllatlons. The C.V. M. basic cluster should be as large as possible lnduclng a large number of correlatlon functlons whlch would make the technique untractable. A way to solve the problem 1s to treat In the C.V. M. framework, the flrst few potentlals and to treat ail others In a mean field Hamlltonlan.
T h i s cluster 1s bulld around the central octahedron by addlng four tetrahedra. It lncludes the 4 first neighbourg pairs and Is fully descrlbed by a set of 85 correlatlon functions.
Table I : QTO C.V. M. coefflclents Cequatlon ( 1) 3
Table II. Valldity range of the QTO : T refers to the reduced temperature
:
T/Tc. X1 t o the MC tetrahedron correlatlon functlon : X2 to the QTO one : Y1 t o the MC triangle correlatlon functlon : Y2 t o the QTO one.JOURNAL DE PHYSIQUE
IV. VALlDlTY OF THE METHOD
Mean fleld theories are qulte accurate In the case of long range potentlals. Hence the C.V. M. method has been compared wlthln the most severe mode1 wlth only flrst nelghbour Interactions wlth Monte-Carlo experlments C71.
Correctly extrapolated pair, triangle and tetrahedron correlations functlon have been calculated at varlous composltlons as a functlon of the reduced energy parameter
il
/kT from conventionnal Monte-Carlo slmulatlon ( Metropolls algorlthml forferromagnetlc and antiferromagnetlc coupllngs. These values have been compared wlth the QTO C.V. M. results and are llsted ln Table II.
For T = OD to 1.30. there Is no slgnlflcant dlfferences between the two sets of values.
they correspond to the errors In MC measurements. The C.V. M. QTO 1s fairly accurate ln the ferromagnetlc (pure cooperatlve phenomena) coupllng but b i s g e r d i s c r e p a n c i e s appear beween 'exact' MC data and C.V. M. predlctions for
a n t i f e r r o m a g n e t i c c o u p l i n g ( i n t r o d u c t i o n of f r u s t r a t i o n e f f e c t s ) . Even t h r o u g h h i g h s e r i e s
expanslon, 1t 1s impossible to properly handle such frustration effects. and the results given a r e c e r t a i n l y amonq t h e most a c c u r a t e . When d e a l i n g w i t h l o n g e r ranged
potentlals. C. V. M. gets better results : a large value of the next nearest nelghbour potentlal Jp leads asymptotically to a slmple cublc structure lnteractlng through
J1
=
Jp (cfc).
wlth no frustratlon effects llke ln the ferromagnetic case (thls happens In the TIC carblde).V. APPLICATION OF THE METHOD
The method applylng for any klnd of coherent order-dlsorder transformatlon wlll be lllustrated In the case of llthlum ferrltes LIFe02. The structure undergoes an order- disorder transformatlon at 943 K between the catlonlc specles (Li. Fe) whlch are distributed at the nodes of an fcc sublattice. At higher temperature a short range order remains C8,91 characterized by a dlffuse scatterlng lntenslty locallsed on a smooth surface (see for Instance Sauvage and Parthe Cl01) deflned by :
COS n
&
+ COS n + cos n kz = O (I')The pair potentlals are obtained by Cc. V%. Inversion <rom the 4 first palr correlatlon functlons experlmentally determlned by S. Lefebvre C91.
As already notlced only the 2 flrst potentlals have signlflcant values (VI = 112 meV. Vp
=
70 meV. V3 = -6.3 meV. V4=
-9.7 meV). A very spectacular effect 1s observed on the probablllty dlstrlbutlon : the octahedron conflgiiratlons which do not correspond t o charge c a n c e l l a t i o n h w e negliglble probabilitles (28 among the 116 probablllties are essentlally O). the most favorable conflguratlon al1 correspond to exact neutral balance. Thlsisiruclose relation wlth the partlcular dlstrlbutlon of the dlffuse Intenslty.CONCLUSION
The dlscussed rnethod have been proved to be very efficlent. glving simultaneously the relevant energetic parameters and the hlgher correlation functlons from the sole knowledge of the experimental flrst four palr correlatlon functlons. As polnted out, the knowledge of hlgher correlatlon functlons allow a preclse descrlptlon of short range order which clearly lndlcate whlch klnd of order wlll be reached at low temperatures. Flnally thls method may become an efflclent tool for testlng the quallty of the experimental determlnatlon of multiplet correlation functlons lssued from the development of the new spectroscoplc technlques.
Cl1 Klkuchl R.. Phys. Rev. v. 81. 988 (1951)
C23 Clapp P.C., J. Phys: Chem. Sotids. 30, 2589 (1969) Phys. Rev. 8, v. 4. no2, 255 <19711
C31 Sanchez J. M.
.
Ducastelle F..
Qratlas D. G. Physlca 128A. 334 ( 19841 C41 Hafner J.. Phys. Rev. B. v. 21. no2. 406 (1980)t53 Katsnelson A.A.. Sllonov V. M.. Khwaja A. Farid. Phys. Stat. Sol. B. 91. 11
171 Cenedese P . . Thesis Parls (1983)
(81 Brunel
