HAL Id: jpa-00225196
https://hal.archives-ouvertes.fr/jpa-00225196
Submitted on 1 Jan 1985
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
RARE EARTH IONS IN RANDOM CRYSTAL FIELDS
A. Guessous, K. Matho
To cite this version:
RARE EARTH I O N S I N RANDOM C R Y S T A L F I E L D S
A. Guessous and K. Matho
Centre de Recherches sur l e s TrGs Basses Tempe'ratures,
C.N.R.S., B.P. 166X, 38042
Grenoble Cedex, Frmce
Resume - La s u s c e p t i b i l i t b magnetique d e s s y s t e m e s amorphes d e t e r r e s r a r e s e s t d 4 c r i t e p a r un modBle oh l e s i o n s d e t e r r e r a r e s o n t soumis
B
un g r a d i e n t d e champ e l e c t r i q u e v a r i a n t d e s i t e en s i t e . L ' h a m i l t o n i e n du modBle c o n t i e n t l a c o m p e t i t i o n e n t r e d e s t e r m e s d e 2" e t 4" o r d r e , c e q u i permet d e r e n d r e compte a u s s i d e s r e s u l t a t s d u c r i s t a l l i n c o r r e s p o n d a n t . Pour l ' a m o r p h e , nous d e t e r m i n o n s une d i s t r i b u t i o n d e s p a r a m e t r e s du modgle. Une a p p l i c a t i o n e s t f a i t e pour l e c a s du systgme CeA13.A b s t r a c t - The m a g n e t i c s u s c e p t i b i l i t y of r a r e e a r t h i o n s i n a n amorphous s y s t e m i s d e s c r i b e d by a model w i t h randomly v a r y i n g e l e c t r i c f i e l d g r a d i e n t s . The h a m i l t o n i a n c o n t a i n s competing t e r m s of second and f o u r t h o r d e r , a l l o w i n g a l s o t o go t o t h e c r y s t a l l i n e l i m i t w i t h w e l l d e f i n e d p o i n t symmetry. I n t h e g e n e r a l d i s o r d e r e d c a s e , we d e t e r m i n e d i s t r i b u t i o n s f o r t h e model parameters. An a p p l i c a t i o n t o t h e CeA13 s y s t e m i s p r e s e n t e d . I
-
INTRODUCTION We a r e s t u d y i n g t h e i n f l u e n c e of d i s o r d e r on d e n s e "Kondo" s y s t e m s . These a r e m e t a l - l i c a l l o y s c o n t a i n i n g m a g n e t i c r a r e e a r t h s , w i t h 4 f - s h e l l s t h a t a r e h y b r i d i z e d t o some e x t e n t w i t h t h e c o n d u c t i o n e l e c t r o n s , forming a "heavy fermion" band.Among v a r i o u s e f f e c t s of d i s o r d e r on t h e symmetry and c o h e r e n c e p r o p e r t i e s of such systems a t low t e m p e r a t u r e , t h e m o d i f i e d e l e c t r i c f i e l d g r a d i e n t s (EFG) a r e e x p e c t e d t o p l a y a key r o l e . T h i s h a s m o t i v a t e d o u r s y s t e m a t i c s t u d y of t h e magnetic s u s c e p t i - b i l i t y of 4 f - e l e c t r o n s o r h o l e s p l a c e d i n a q u a d r u p o l a r f i e l d g r a d i e n t t h a t v a r i e s randomly from s i t e t o s i t e
111.
I n a g e n e r a l i z a t i o n of t h e model, we s u p e r p o s e t h e random EFG's on a s h a r p c r y s t a l f i e l d of h i g h p o i n t symmetry. F o r CeA13 121, t h i s l a t t e r t e r m i s assumed t o b e of h e x a g o n a l symmetry i n t h e p r e s e n t c a l c u l a t i o n . I n a z e r o t h o r d e r a p p r o x i m a t i o n t h e h y b r i d i z a t i o n was n e g l e c t e d . At t h i s l e v e l , o u r model i s g e n e r a l l y r e l e v a n t f o r Kramers i o n slaced
a s l o c a l p r o b e s v i h i n a d i s o r d e r e d o r amorphous m a t r i x . The o b s e r v a b l e q u a n t i t y we c o n s i d e r i s t h e e f f e c t i v e moment a s d e f i n e d v i a t h e i n i t i a l m a g n e t i c s u s c e p t i b i l i t y .I1
-
THE MODELI n t h e d i s o r d e r e d s u b s t a n c e , t h e EFG v a r i e s from s i t e t o s i t e a c c o r d i n g t o t h e d i s - t r i b u t i o n of e l e c t r i c c h a r g e s . These v a r i a t i o n s may b e s m a l l f l u c t u a t i o n s a b o u t a w e l l d e f i n e d a v e r a g e o r , on t h e c o n t r a r y , t h e f l u c t u a t i o n s may b e l a r g e r t h a n t h e mean v a l u e s . The g e n e r a l m u l t i p o l e - e x p a n s i o n f o r t-he EFG w i t h r e s p e c t t o a g i v e n r e - f e r e n c e s i t e / 3 / h a s many i n d e p e n d e n t p a r a m e t e r s . I n o r d e r t o r e d u c e t h e i r number we have a d o p t e d t h e f o l l o w i n g s t r a t e g y : On t h e one hand, second o r d e r ("quadrupolar") t e r m s t e n d t o dominate i n h i g h l y random s t r u c t u r e s / 4 , 5 / . We t h u s r e t a i n t h e most g e n e r a l c o m b i n a t i o n of them c o r ~ v e n i e n t l y e x p r e s s e d by t h e p a r a m e t r i z a t i o n of F e r t - Campbell 161,
g2
= ( 3 + n ) c ( 1 - u ) ~ 212 0; +
7
O2 ( 1 )C8-356 JOURNAL
DE
PHYSIQUE
i n t e r m s of a n o v e r a l l second o r d e r energy C and a symmetry parameter
n ,
(111 S 1 . The l o c a l q u a n t i z a t i o n a x i s f o r t h e S t e v e n s o p e r a t o r s / 3 / 0; i s o r i e n t e d a t random. On t h e o t h e r hand, among a l l p o s s i b l e h i g h e r o r d e r t e r m s Oft: and 0: t h a t can a c t on f - e l e c t r o n s we choose o n l y a r e s t r i c t e d s e t c o m p a t i b l e w i t h a d e f l n i t e l o c a l p o i n t symmetry. T h i s c h o i c e may b e guided by an assumption a b o u t s t r u c t u r a l b u i l d i n g blocks of a g i v e n g l a s s y m e t a l / 7 / o r by t h e p r o x i m i t y of an o r d e r e d compound, t h e s t r u c - t u r e of which i s assumed t o p l a y a r o l e i n t h e l o c a l environment181.
The f o l l o w i n g e x p l i c i t model i s a n i l l u s t r a t i o n of t h a t s t r a t e g y meant t o b e a p p l i e d t o c e 3 + , f o r which t h e l o w e s t Hund's r u l e m u l t i p l e t ( a f t e r s p i n - o r b i t c o u p l i n g ) h a s t o t a l a n g u l a r momentum J = 512 (The S t e v e n s o p e r a t o r s a r e c o n s t r u c t e d w i t h t h e com- p o n e n t s of
3).
I f o t h e r m u l t i p l e t s a r e h i g h enough i n e n e r g y ,%
6 i s o n l y a c o n s t a n t . T h i s assumption r e s t r i c t s t h e energy r a n g e t o 2. 0.1 eV. Having t h e CeA13 system i nmind, which i s hexagonal i n t h e c r y s t a l l i n e form, we t a k e o n l y a s i n g l e f o u r t h o r d e r term,
a4
= BiOi. I n p r i n c i p l e , t h e r e i s n o r e a s o n t o " a l i g n "g2
ands4
by e x p r e s s - i n g them i n t h e same l o c a l r e f e r e n c e frame. However, t h e e f f e c t of t h r e e a d d i t i o n a l E u l e r a n g l e s , n e c e s s a r y t o c h a r a c t e r i z e t h e r e l a t i v e p o s i t i o n of a x e s i s p r a c t i c a l l y i n d i s t i n g u i s h a b l e from t h e e f f e c t ofn
#
1 i nz2,
w i t h r e f e r e n c e s t i l l a l i g n e d . T h i s example shows, how a number of e x p e r i m e n t a l l y u n r e s o l v a b l e p a r a m e t e r s c a n b e absorb- ed i n t o one e f f e c t i v e p a r a m e t e r . It should b e n o t e d t h a tz 2 ,
w i t hn
= 1 and a l i g n e d w i t ha4
i s a l s o a d m i t t e d by t h e hexagonal p o i n t symmetry. The t o t a l H a m i l t o n i a n i sCompetition between
W 2
ands4
o c c u r s f o r v a l u e s of t h e r a t i o C ' of t h e o r d e r o f 1 . For t h e s i x - d i m e n s i o n a l subspace J = 5 / 2 ,aEFG
decomposes i n t o two t h r e e - d i m e n s i o n a l b l o c k s a c c o r d i n g t o e i g e n v a l u e s T = ?I of t h e t i m e - r e v e r s a l o p e r a t o r .These b l o c k s a r e e a s i l y d i a g o n a l i z e d and t h e i r e i g e n v a l u e s denoted El -5 E2 2 E3, i n - d e p e n d e n t l y of T (Kramers d o u b l e t s ) . An e q u a l i t y s i g n o c c u r s o n l y f o r ( o , C 1 ) = ( 1 ,-5/21, (1 ,-116) and ( 1 , l ) . These s p e c i a l p o i n t s i n t h e p a r a m e t e r s p a c e c o r r e s p o n d t o l e v e l c r o s s i n g between two Kramers d o u b l e t s l a b e l e d r e s p e c t i v e l y by (M = f 1 / 2 , & 3 / 2 ) , (M = &1/2,+5/2) and M = ( ? 3 / 2 , + 5 / 2 ) . I t h a s been determined p r e v i o u s l y / 9 / t h a t c-CeA13 i s n o t f a r from t h e p o i n t (n,C1) = (1,-512) w i t h M = f 3 / 2 t h e l o w e s t Kramers d o u b l e t . The e i g e n s t a t e s a r e unambigously l a b e l e d by Ik,.r>, w i t h k = 1 , 2 , 3 ,
t h e energy l a b e l and T = f1 t h e t i m e - r e v e r s a l l a b e l . 111
-
SUSCEPTIBILITYThe c o n t r i b u t i o n from each s i t e t o t h e macroscopic m a g n e t i c , s u s c e p t i b i l i t y i s a ten- s o r q u a n t i t y . The ensemble a v e r a g e o v e r s i t e s i s devided i n t o two c o n s e c u t i v e p a r t s :,
i ) A f t e r t h e i s o t r o p i c "powder a v e r a g e o ' o v e r o r i e n t a t i o n s of t h e l o c a l r e f e r e n c e frames o n l y a s c a l a r s u r v i v e s , g i v i n g t h e macroscopic l o n g i t u d i n a l r e s p o n s e t o a n a p p l i e d f i e l d
8.
Now,aBFG
and t h e l o n g i t u d i n a l and t r a n s v e r s e o p e r a t o r s Jz and J+ =J + i J y a r e d e f i n e d i n t h l s frame. The powder a v e r a g e d s u s c e p t i b i l i t y p e r s i t e of ce3+
;Y
w ~ t h a g i v e n s e t of (Ecf,C,
n)
i sw i t h a n e f f e c t i v e moment f u n c t i o n
-
B(EkrdEk) 2 2peff = g
I
p l ? [ ( k ~ , k ' r ' ) { % l + ( l - \ k l ) B(Ek, -Ek) ( 5 )k , T
k ' , T'
m a t r i x e l e m e n t s
e f f e c t i v e moment f u n c t l o n i s d e f i n e d a s r(T,Ecf ,C,rl) =
Pzff
/
L 2 J ( J + 1 ) ] . Without ac- c i d e n t a l degeneracies t h e f i r s t term i n t h e c u r l y b r a c k e t s of (5) d e f i n e s t h e Curie l i k e c o n t r i b u t i o n , i n p a r t i c u l a r t h e f i n i t e low temperature e f f e c t i v e moment r(O,Ecf,C, ) , whereas t h e second term d e f i n e s t h e van Vleck l i k e c o n t r i b u t i o n , i n p a r t i c u l a r t h e low temperature s l o p e of r ( T ) . With an a c c i d e n t a l degeneracy Ek = E ~ I(confer § II), p a r t of t h e van Vleck c o n t r i b u t i o n becomes C u r i e - l i k e .
0
1
I I- 3 -2.5 - 2 -1.5 -1 -0.5 log ( k B T / U )
Figwze
I
: En;lmbLe a v m g e d neduced n w c e p f i b U y . ( a ) The edde.c.t ad inhomageneaun b.t~~adeYLing -a<C1> Label. I b l The eddecA ad changing ,the avenage.< C ' >
( L a b e l ) . ( c l The eddecX06
changing Xhe pmmeAen(RabeL).
i i ) The second s t e p of t h e ensemble average i s t o assume a d i s t r i b u t i o n p(Ecf,C,q). F i r s t , we have analyzed t h e i n t e r n a l "skewness" of t h e e x c i t a t i o n spectrum. We car- r i e d o u t c o n s t r a i n e d p a r t i a l averages a t f i x e d I- and i n t r i n s i c v a r i a n c e a = 2(E2+E$+
3 1 E$) of t h e l o c a l l e v e l schemes. T h i s was a l r e a d y a p p l i e d t o t h e s p e c i f i c h e a t / l o / . Now, i n f i g u r e s la-c, we show t h e analogous r e s u l t s f o r t h e f u n c t i o n r ( T ) . The ave- r a g e s , done on a computer, a r e over r e c t a n g u l a r d i s t r i b u t i o n s p ( C r ) of t h e r a t i o C ' = C/EcF, c h a r a c t e r i z e d by t h e i r mean < C 1 > and v a r i a n c e a ( C 1 ) . The energy Ecf =
E C f ( o , n ) 1 s r e s c a l e d t o keep t h e i n t r i n s i c v a r i a n c e u c o n s t a n t .
-0 6-
e
3
v
I n f i g u r e l a , t h e mean v a l u e i s < C f > = -512, and I- = 1 , corresponding t o 4 f o l d degeneracy of
aEFG.
For a low v a r i a n c e , a ( C t ) = . I , t h e p l o t of r =x/xCuyie
a s a f u n c t i o n of logT h a s two i n f l e x i o n p o i n t s , corresponding t o s e p a r a t e excitation
e n e r g i e s t h a t a r e a l s o m a n i f e s t i n t h e s p e c i f i c h e a t p l o t s / l o / . As t h e v a r i a n c e i s i n c r e a s e d , t h i s s t r u c t u r e d i s a p p e a r s , g i v i n g way t o a n o v e r a l l broad van Vleck term.
S O 2 - D
@
u
0
The low T e f f e c t i v e Curie moment i s i n v a r i a n t because t h e p(C1) was cen- t e r e d a t t h e degeneracy p o i n t . Figure Ib shows t h e e f f e c t of v a r y i n g < C 1 > , f o r I- = 1
C8-35
8
JOURNALDE
PHYSIQUEand o(C1) = 0.1. Finally, the effect of
n
is shown in figure Ic for <C'> = -512 and o(C1) = 0.1. Again, occurrence of two inflexion points always corresponds to separa- te peaks in the specific heat or, basically, in the excitation spectrum.In an application to polycrystalline and amorphous CeAl3 /lo/ (indices c- and a-) we have relaxed the constraint of constant intrinsic variance o by averaging over a factorized distribution p(C1).q(Ecf), where p and q are each rectangular, characte- rized by <C1>, o(Cf) and (Ecf), o(Ecf) respecfively. In each run, q was kept cons- tant and was optimalized along with the four parameters of the distribution. The re- sults are summarized in figure 2 and table 1. .%
1
! ! 1 ! I I
c-CeAl,! 80
K!
::S ~ ~ ~ l7
! ~set equal;set=; - l .! ! ! ! to zero ;to 1 ;
Tabbe 1 : OpZimaL p a h a m d m
as
d e X m i n e d by &LWing pnocedL(ne. I n t h e ctyn-taLLine cabe, only n h m p v d u e n don Ec6 and C' Wc?he ophOn&zed.F i g w e 2 : Reduced n w c e p a Z b L t 2 y
06
ctyn- ,tal&ne C N 3 and amonphoun CeZ5Ab75. The v d u e n od .the6L.t
p a h a m d m atLe.iM
,tableI .
For c-CeA13, the sharp EFG parameters confirm the previous crystal field model / 9 /
For a-CeA13, the much enlarged average <C7> indicates that the quadrupolar terms dominate.
The energy <Ecf> is also increased and widely distributed. The optimal value for Q was found to be q = 1, but the quality of the fit depends little on q , in contrast to the crystalline case. We anticipate that the averages and variances given in table 1 are fairly independent of the shape of the distributions, e.g. gaussian ins- tead of rectangular. This remains to be checked. We are also working on analytical methods of carrying out the ensemble averaging for Lorentzians.
IV
-
CONCLUSIONWe have outlined a strategy for determining distributions of EFG in random substan- ces containing rare earth ions from magnetic susceptibility or specific heat data.
A
delicate point is, how this can be done in a meaningful way, without introducing a forbiddingly large number of EFG-parameters.
An application was given, concerning the magnetic susceptibility of crystalline and amorphous CeA13.
More generally, our analysis summarized in figure
1
and in reference /lo/ suggests that a plot ofP2ff
as a function of logT may be sensitive to the main features of the excitation spectrum, even if these are fairly spread in energy.REFERENCES
/I/ Guessous, A. and Matho,
K.,
LT 17 Proceedings, p. 163 (North-Holland, 1984). /2/ Guessous, A., Matho, K., Mazuer,J.
and Palleau, J., J. Non-Cryst. Sol.65
(1984) 429.
/ 5 / C z j z e k , G . , F i n k , J . , GGtz, F . , S c h m i d t , H., Coey, J.M.D., R e b o u i l l a t , J . P . and L i e n a r d , A . , Phys. Rev. B
23
(1981) 2513.1 6 1 F e r t , A. and Campbell, I . A . , J . P h y s . F.:Met. P h y s .
5
(1978) L 5 7 ./ 7 / M a u r e r , M . , F r i e d t , J . M . , S = h w T E . , J . P h y s . F:Met. P h y s . 15 (1985-)-1_449. / 8 / F e l s c h , W., K u s h n i r , S.G. a n d Samwer, ~ . > h ~ s i ~ u e
