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Submitted on 1 Jan 1981
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SIMULTANEOUS CONDENSATION OF SEVERAL
MODES AT STRUCTURAL PHASE TRANSITIONS
P. Toledano, G. Pascoli, M. Coiret
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C6, suppliment au n o 12, Tome 42, dicembre 1981 page C6-746
SIMULTANEOUS CONDENSATION OF SEVERAL MODES A T STRUCTURAL PHASE T R A N S I T I O N S
P. Toledano, G. Pascoli and M. Coiret
Groupe de Physique Thiorique, FacuZti des Sciences, 80000 Amiens, F ~ a n c e
Abstract.-
The s t r u c t u r a l t r a n s i t i o n s in a number of materials has been explained by t h e simultanous condensation of several modes a t the t r a n s i t i o n temperature.The va- r i o u s types of couplings between the order-parameters corresponding t o t h i s mechanism are discussed.
Most of the observed s t r u c t u r a l phase t r a n s i t i o n s have been r e l a t e d t o the con- densation of a s i n g l e normal mode of the crystal.This confirms the basic a s s e r t i o n of the Landau theoryA which s t a t e s t h a t when the space-groups of two phases are group- subgroup related,the t r a n s i t i o n from one t o another phase can be described by a s i n g l e ~ r d e r - ~ a r a m e t e r ( ~ ~ ) . I n other words,the number of phenomenological variables t h a t have t o be introduced i n the description of the t r a n s i t i o n is minimal and cannot be re- duced.However, a small f r a c t i o n of the materials undergoing s t r u c t a l t r a n s i t i o n s does not comply with the preceding scheme.This is, f o r example, t h e case of families of compounds such a s the 3oracites2 ,the I,angbeinites3 and p a r t of the ~ e r o v s k i t e s ~ ; o f
6
i s o l a t e d compounds such a s ~ e n z i l ' o r Bismuth t i t a n a t e :and a l s o of incommensurate
8
systens,such a s Sodium n i t r i t e and Thiourea
.
In these materials,the s t r u c t u r a l changes cannot be interpreted with a s i n g l e OP and several =odes r e l a t e d t o d i s t i n c t OP seem t o be simultaneously involved.& each OP expresses a s e t of atomic displacements (a degree of freedom) associated t o
a
c e r t a i n normal mode of t h e system, the f a c t t h a t several OP are involved s i g n i f i e sthat the displacements r e l a t e d .to one OP brings about an i n s t a b i l i t y in parameters inducing other displacements.Due t o t h e non-linear i n t e r a c t i o n between t h e various parameters,the modifications accompanying t h e t r a n s i t i o n ( symmetry change,dielectric
o r e l a s t i c anomalies..
.
) w i l l require f o r t h e i r description an increasing number of thermodynamic variables.Thus,it can be asked t o w h a t extent a Landau-type theory involving several OP may still have a prsdictive value.In t h i s paper we b r i e f l y dis- cuss t h e case of t r a n s i t i o n s induced by two OP whose c h a r a c t e r i s t i c s have been pre-cised by a number of recent studiesFi3.
As we have i n mind a q u a l i t a t i v e description ,we can r e s t r i c t t o consider only s i n g l e component OP's,namely p and q.The free-energy(F53) of t h e t r a n s i t i o n w i l l con- t a i n , i n addition t o invariants of each OP, mixed invariants expressing t h e coupling between p and q.As it is the lowest degree coupling term wNch determines the essen- tial f e a t u r e s of the t r a n s i t i o n , w e can therefore distinguish two main cases;
1 ) The FE is of the
tm:
-
-4
P = \p2
+
+p+
A$+
B~~~ t B2$+
B?$+
cpZq2
2 ( 1 ) Such a c a s e b encountered when lower couplings of the t y p or p q are for- bidden by symmetry.It i s r e d i z e d , f o r instance, f o r two non-unit one dimensional r e a l irreducible representations(1~) of the space-group Go describing the more symmetric (high temperature ) phase.
When A =B -0, (1) is related t o an experimentally u n r e a l i s t i c s e t of two suc-
3
3-
cessive second order transitions.klhen only one of the sixth-degree coefficients ( l e t say B ) is non-zero, we obtain an asymmetric
F'E
which has been used by Hola-9
kovsky t o describe the simultaneous onset of spontaneous values f o r p and q a t a f i r s t order transition.Moxe precisely ,when one assumes a single temperature-depen- dent coefficient
( A ~
),
the softening of t M s coefficient triggers two phase kransi- tions e i t h e ra t
different o r a t the same temperature.In t h i s l a t t e r case,which is realized f o r s u f f i c i e n t large values of the coupling coefficient C, thk spontaneous2
values of p change the coefficient a t q inducing a t r a n s i t i o n i n q.The s o f t vibra- .tlmal made of the high symmetry phase remains related t o the "primary" @,but the space group of the low symmetxy phase is the intersection of the symmetries deter- mined separately by p and q.In addition t o t h i s remarkable property,other features allow the identification of "triggeredn phase transitions,namely a cmfiex domain structure depending on the possible orientation of two OP having d i f f e r e n t symmetry propertiesfor the mixed behaviour of the d i e l e c t r i c constant(proper below the Curie
6
poht,improper above). Bismuth t i t a n a t e has bee2 propose(: a s an i l l u s t r a t i v e example of the preceding model though the available data are s t i l l incomplete f o r this material.
The phase diagraJll of (1) when A
#
B#
0 has been discussed by Gufan and h i n t 03
3
It displays various s e t s of successive(and possibly simultaneous) transitions with intersecting l i n e s of second and f i r s t order transitions.
2 ) The
FE
5s of the type:4
6 2F=
n,p2+
%p+
A3p t Biq t B2q+
~~q~+
Cpq2C6-748 JOURNAL DE PHYSIQUE
c r y s t d l while t h e coupling b e t w e e n the OP p arid q is l i k e l g t o be a c t i v e both i n its s t a t i c and dynamic behaviours.0n the other hand q u a n t i t a t i v e values show t i t
i n b m 4 2 , t h e t r i g g e r i n g though e s s e n t i a l t o explain t h e sjmnetry c h a r a c t e r i s t i c s of the t r a n s i t i o n influences only weakly t h e other physical properties.
The two FE considered i n 1 ) and 2 ) summarize t h e q u a l i t a t i v e s i t u a t i o n s which may be encountered f o r s t r u c t u r a l t r a n s i t i o n s induced by two 0P.Nwnerica.l. models should a l s o take i n t o account t h e a c t u a l symmetry of the OP and higher degree cou- plings.Besides when dealing with t r a n s i t i o n s towards incommensurate phases other type of coupling,involving t h e space d e r i v a t i v e s of t h e OP ,must be consfdemdrThis brings us t o a t h i r d case.
The onset of intermediate incommensurate phases,such as i n Sodium n i t r i t e o r ,has been a t t r i b u t e d t o the coupling of two d i s t i n c t OP whose transfor- mation properties allow an antisymmetric i n v a r i a n t of the type:
(3)
If w e consider p and q a s the amplitudes of two d i f f e r e n t normal modes of the crys- t a l r ( 3 ) expresses t h e existence of a r e l a t i v e maximum i n the lowest mode due t o t h e i r interaction.Th3.s s i t u a t i o n can be recognised when a commensurate phase ,generally s t a b l e a t lower temperature,is r e l a t e d t o an I R of the high temperature commensu- r a t e phase which does not -tmnsfonn l i k e
( 3 ) . k
S o d i m n i t r l t e and Thiourea tha low temperature commensurate phase iscomected t o a one dimensional I R of the orthorhom- b i c B r i l l o u i n zone c e n t e r s o t h a t t h e construction of an i n v a r i a n t of type(3)
needs t o recourse t o two d i s t i n c tIR
a t t h a t point .The preceding model i s however f a r from being experimentally confirmed as it does not explain the complex s e t of phases which is f o r instance observed i n Thiourea.REFERENGES
i .L.D.Landau and E.M.Lifshitz,Statistical ~ h y s i c s ( ~ e r g a m o n Press, Oxf ord, 1958). 2.A.P.Levanyuk and D.G.Sannikov,Sov.Phys.Solid State,l7,327(1975).
3.Yu.M.Gufan,V.P.Dmitriev,and ~.1.~orgashev,Sov.Phys.0ristal1ogr.24,~2(1979). 4.Y.Cochran and ~,~ia,~hys.Stat.~o1.25,273(1968).
5 . ~
.c.Toledano,Phys. ~ev.~20,114?(1979). 6. ~.~olakovsky,~hys.Stat.Sol~ (b)69,5?5(1975).7.G.Pascoli,These de troisieme c y c l e ( ~ n i v e r s i t y of LSLE I,198i),unpublished. 8.A.P.Levanyuk and D.G.Sannikov,Sov.Phys.Solid S t z t e , t 8 1122(1976).
9. J.Holakovsky,P~.Stat.Sol. (b)56,615(1973).
1O.Yu.M.Gufan and E.S.Larin,Sov.Phys.Solid ~tate,22,2?0(1980). 11.Yu.M.Gufan and V.I.Torgashev,Sov.Phys.Solid ~tate,22,9~51(1980).
i2,P.Toledano and G,Pascoli,in N.Boccara,Symmetries and Broken Symmetries in Condensed Hatter ~ h y s i c s ( I D S ~ ~ , ~ a r i s , 1981).
