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BOND-ANGLE DISTRIBUTION AND THE LOCAL
FIELD ACTING ON RARE-EARTH IONS IN
AMORPHOUS SOLIDS
G. Czjzek
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C9, supplément au n012, Tome 46, décembre 1985 page C 9-113
BOND-ANGLE D I S T R I B U T I O N AND THE LOCAL F I E L D ACTING ON RARE-EARTH IONS I N AVORPHOUS SOLIDS
Kernforschungszentrurn KarZsruhe, I n s t i t u t für NukZeare Festkorperphysik Postfach 3640, 0-7500 KarZsruhe, F.R.G.
Résumg
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Des sondes l o c a l e s d o n t l e s 2 r o - r i 6 t é s dé2endent de l a d i s t r i b u t i o n d e s c h a r g e s d a n s l ' e n v i r o n ï ï e n t oeuvent r e n d r e d e s i n f o r m a t i o n s s u r l a s t r u c t u r e d e s amorphes q u i s o n t a l l i é e s à l a d i s t r i b u t i o n d e s a n g l e s d e l i a i s o n , un s u p 2 l é n e n t i n p o r t a n t aux d i s t r i b u t i o n s r a d i a l e s d é r i v é e s d e s rnésures d e d i f f r a c t i o n . L a r e l a t i o n e n t r e l e s p a r a m è t r e s d e s t r u c t u r e e t l e s q u a n t i t é s o b s e r v a b l e s p a r e x p é r i e n c e , comme l a d i s t r i b u t i o n d e s n i v e a u x d ' é n e r g i e e t l a s u s c e p t i b i l i t é magnétique, e s t é l a b o r é e n u t i l i - s a n t un f o r m a l i s l i e q u i p e r n e t l ' é v a l u a t i o n d e s r é s u l t a t s e x p é r i - mentaux s a n s s u p p o s i t i o n d ' u n modèle d e l a s t r u c t u r e .A b s t r a c t
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Local p r o b e s whose p r o p e r t i e s de-end uoon t h e d i s t r i - b u t i o n of c h a r g e s i n t h e i r e n v i r o n n e n t c a n p r o v i d e s t r u c t u r a l i n f o r m a t i o n a b o u t amorphous s o l i d s which i s r e l a t e d t o bond a n g l e d i s t r i b u t i o n s , a n i m p o r t a n t supplement t o r a d i a l d i s t r i b u - t i o n s d e r i v e d £rom d i f f r a c t i o n t e c h n i q u e s . The r e l a t i o n between s t r u c t u r a l p a r a m e t e r s and e x p e r i n e n t a l l y o b s e r v a b l e q u a n t i t i e s such a s e n e r g y l e v e l d i s t r i b u t i o n s and ~ a g n e t i c s u s c e - t i b i l i t i e s i s worked o u t i n ternis of a formalism which a l l o w s a model-free e v a l u a t i o n o f t h e e x p e r i n e n t a l r e s u l t s .1
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INTRODUCTION. LOCAL MULTIPOLE FIELDSI n t h e l > a s t few y e a r s , t h e t e c h n i q u e & o f d i f f r a c t i o n measurements f o r t h e d e t e r m i n a t i o n of amorphous s t r u c t u r e s have been r e f i n e d enornous- ly.Thcy y i e l d q u i t e d e t a i l e d i n f o r ~ a t i o n a b o u t p a r t i a l ? a i r c o r r e l a t i o n f u n c t i o n s which a r e e s s e n t i a l f o r Our knowledge a b o u t c h e m i c a l o r d e r i n g i n anornhous n a t e r i a l s . I n s p i t e o f t h i s , fundamental q u e s t i o n s con- c e r n i n g t h e t o p o l o g y of l o c a l atomic arrangements remain open. I t h a s been q o i n t e d o u t by Hafner / 1 / t h a t a c l a r i f i c a t i o n o f t h e s e problems c a n n o t b e a c h i e v e d on t h e b a s i s o f r a d i a l p a i r c o r r e l a t i o n f u n c t i o n s a l o n e . Higher o r d e r c o r r e l a t i o n f u n c t i o n s , a s f o r example bond-angle d i s t r i b u t i o n s s h o u l d b e a c c e s s i b l e t o e x p e r i n e n t a l i n v e s t i g a t i o n . I n t h i s r e p o r t , a p r e l i m i n a r y a c c o u n t o f an i n v e s t i g a t i o n o f t h e u s e of l o c a l p r o b e s , mainly of r a r e e a r t h i o n s , f o r t h e d e r i v a t i o n of ex- ? e r i m e n t a l i n f o r m a t i o n a b o u t h i g h e r o r d e r c o r r e l a t i o n f u n c t i o n s r e l a - t e d t o bond-angle d i s t r i b u t i o n s i s ? r e s e n t e d .
The s t a t e of an i s o l a t e d probe
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an i o n o r an atomic n u c l e u s-
i s de- t e r m i n e d by some i n t e r n a 1 n o t e n t i a l of s p h e r i c a l symmetry. Hence, t h e s t a t e i s c h a r a c t e r i z e d by an a n g u l a r momentum J ( n u c l e a r s p i n 1). I n a s o l i d , t h e probe l o c a t e d a t n o s i t i o nf
i s f u r t h e r m o r e s u b j e c t t o an e l e c t r o s t a t i c p o t e n t i a l which o r i g i n a t e s £rom a l 1 c h a r g e s zi a t p o s i - t i o n s P i i n t h e s u r r o u n d i n g s èJOURNAL DE PHYSIQUE
The t a s k of determining t h e s t a t e s of t h e probe under t h e a c t i o n of
( 1 ) i s most e a s i l y achieved by expansion of Ir-ri 1-l i n terms o f s?he- r i c a l harmonics, and bv i n t r o d u c t i o n of o p e r a t o r e q u i v a l e n t s using t h e Wigner-Eckhart t h e o r e n / % / . I n t h i s way, t h e e x p r e s s i o n f o r VLF i s transformed t o
The q u a n t i t i e s 02 a r e o p e r a t o r s ( " m u l t i p o l e o p e r a t o r s " ) i n t h e a n g u l a r momentun space of t h e l o c a l probe, w i t h m a t r i x elernents /3/
where <J,k,Pl,q ~ J , M ' > a r e Clebsch-Gordan c o e f f i c i e n t s ( n o t a t i o n a s i n A. Messiah / 4 / ) .
1
The l o c a l m u l t i p o l e f i e l d (LPIF) parameters ( ) ,:B i n t h e n o t a t i o n c u s t o m a r i l y used Lor r a r e e a r t h i o n s / 2 / a r e g i v e n by
with
For t h e meaning of t h e o t h e r f a c t o r s i n eq. ( 4 ) , c h a r a c t e r i z i n g t h e probe i o n s , t h e r e a d e r i s r e f e r r e d t o t h e l i t e r a t u r e on c r y s t a l f i e l d e f i e o t s . f o r example r e f
.
/ 5/.
For t h e Stevens f a c t o r EJ),
t h e u s u a l n o t a t i o n i s aJ,B J r
y J f o r k = 2 , 4 , 6 , r e s p e c t i v e l y .The p a r a n e t e r s A 2 d e f i n e d by t h e sums i n eq. (5) a r e t h e q u a n t i t i e s r e l e v a n t f o r s t r u c t u r a i i n v e s t i g a t i o n s . For i o n s enbedded i n a c r y s - t a 1 l a t t i c e , each of them h a s t h e same ~ r e c i s e l y d e f i n e d v a l u e f o r a l 1 e q u i v a l e n t l a t t i c e s i t e s . I n an amorphous s o l i d , i t s v a l u e i s a g a i n p r e c i s e l y d e f i n e d f o r a g i v e n s i t e , b u t v a r i a t i o n s i n t h e l o c a l en- vironments from s i t e t o s i t e l e a d t o some n r o b a b i l i t y d i s t r i b u t i o n of t h e s e q u a n t i t i e s f o r t h e ensemble of i o n s i n t h e n a t e r i a l s t u d i e d . The r e l a t i o n between l o c a l s t r u c t u r e s , i . e . t h e d i s t r i b u t i o n of charge p o s i t i o n s i n t h e environnent of t h e nrobes, of t h e q u a n t i t i e s ,:A and of t h e r e s u l t i n g p h y s i c a l n r o p e r t i e s of t h e probe i o n s w i l l be
t r e a t e d i n t h e following s e c t i o n s .
Before t u r n i n g t o t h i s p r i n c i p a l s u b j e c t of t h e r e p o r t , however, some remarks concerning t h e p a r t i c u l a r c h o i c e of r a r e e a r t h i o n s and n u c l e i a s s t r u c t u r a l probes appear i n p l a c e s i n c e t h e p o t e n t i a l introduced
1
i n eq. ( 1 ) i s q u i t e u n i v e r s a l , a c t i n g on any i o n i n a s o l i d . The s e l e c t i o n i s based on 2 c o n d i t i o n s . F i r s t l y , t h e probe n u s t be w e l l l o c a l i z e d . I f it were n o t , t h e e x t e n s i o n of t h e r e g i o n sampled woulci n o t Le well d e f i n e d . A c t u a l l y , t h e t r a n s f o r m a t i o n of eq. ( 1 ) t o t h e e x p r e s s i o n s i n eq. ( 2 ) éhrough ( 5 ) would n o t be v a l i d f o r extended probes. In a d d i t i o n , a l o c a l i z e d probe has d i s c r e t e energy l e v e l s corresponding t o p r e c i s e l y d e f i n e d s t a t e s .
The c o n d i t i o n of l o c a l i z a t i o n obviously i s met by a t o n i c n u c l e i , and it holds i n g e n e r a l f o r t h e 4f e l e c t r o n s of r a r e e a r t h i o n s u n l e s s we happen t o rneet a c a s e of f l u c t u a t i n g valence.
The second c o n d i t i o n concerns t h e r a t i o between t h e s t r e n g t h of t h e i n t e r a c t i o n with t h e environment, -Eext, and t h e s t r e n g t h of i n t e r n a 1 i n t e r a c t i o n s , Eint, which a r e neasured Sy t h e energp d i f f e r e n c e be- tween t h e ground s t a t e and e x c i t e d s t a t e s of t h e i s o l a t e d probe. Only i f t h i s r a t i o i s s i g n i f i c a n t l y s m a l l e r t h a n 1 , t h e s t a t e r e s u l t i n g £rom t h e i n t e r a c t i o n w i t h t h e e x t e r n a l p o t e n t i a l i s d e s c r i b e d i n a simple way by ground s t a t e p r o p e r t i e s of t h e probe such a s r a d i a l mo- ments. Otherwise, t h e f i n a l s t a t e c o n t a i n s admixtures of e x c i t e d probe S t a t e s , and a r a t h e r d e t a i l e d knowledge of i n t e r n s l i n t e r a c t i o n s i s r e q u i r e d f o r understanding t h e ensuing p r o p e r t i e s of t h e probes.
For n u c l e i ,
-
4xt
i s t y p i c a l l y a iew t i n e s 13-6 eV whereas Eint anounts t o 1 0 j eV o r more. Thus, t h i s r a t i o i s very small. For r a r e e a r t h i o n s , although t h e s i t u a t i o n i s n o t q u i t e a s f a v o u r a b l e , with EeXt%1c2 eV, and Sint varying £rom about 3 . 2 eV f o r l i g h t r a r e e a r t h s t o &out 1 eV f o r heavy r a r e e a r t h s , t h e r a t i o i s s u f f i c i e n t l y small f o r t h e n e g l e c t of e x c i t e d s t a t e admixtures t o be a r e a s o n a b l e a p p r o x i n a t i o n . Then, because of p a r i t y c o n s e r v a t i o n , only even v a l u e s of k have t o be taken i n t o account i n eq. ( 2 ) . For n u c l e i , only k = 2 l e a d s t o n e a s u r a b l e e f f e c t s . Higher terms escape d e t e c t i o n because of t h e smail s v a t i a l e x t e n t of t h e n u c l e a r charge d i s t r i 3 u t i o n . An i n v e s t i g a t i o n of t h e d i s t r i b u t i o n of n u c l e a r quadrupole i n t e r a c t i o n s i n amorphous so- l i d s h a s been d e s c r i b e d i n r e f . !6/.Mere, t h e approaches employed i n r e f . /6/ w i l l be extended t o t h e use of r a r e e a r t h i o n s a s s t r u c t u r a l probes i n amorphous i n a t e r i a l s . For r a r e e a r t h i o n s , t h e terms f o r k = 2 , 4 , and 6 have t o be considered. This complicates t h e a n a l y s i s , b u t a t t h e same time it augments t h e inforination t h a t can be e x t r a c t e d .
II
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DISTRIBUTION OF THE LMF PAP.ETERS, INVARIANT FUNCTIONS The t a s k of d e a l i n g with t h e d i s t r i b u t i o n of a l 1 q u a n t i t i e sAZ
a t f i r s t s i g h t a>pears formidable, s i n p l y because c f t h e l a r g e number of parameters. For each k, t h e r e a r e ( 2 k + l ) parameters :A f o r q running through t h e sequence -k,'-k+l,...,k. For t h e 3 v a l u e s of k t o be con- s i d e r e d f o r r a r e e a r t h i o n s , t h i s makes a t o t a l of 27 independent a r - guments of a g e n e r a l p r o b a b i l i t y d i s t r i b u t i o n PA (.
.:A. .).
C9-116 JOURNAL DE PHYSIQUE
which had shown t h a t f o r a s t r u c t u r e corresponding t o t h e model of dense random packing of hard s p h e r e s t h e e f f e c t s of t h e terms f o r k=2 by f a r outv~eigh a l 1 o t h e r s . The inagnetic p r o p e r t i e s deterrnined e x p e r i - mentally f o r s e v e r a l anorphous a l l o y s were q u i t e adequately d e s c r i b e d by t h i s i~iodel /8/.
Since t h e approach of t h e random a n i s o t r o p y model i s Sased on a s p e c i - f i c s t r u c t u r a l model, i t i s n o t s u i t e d f o r an unbiased i n v e s t i g a t i o n of anorphous s t r u c t u r e . A mosel-independent a l t e r n a t i v e i s proposed here. The only c o n d i t i o n f o r i t s a p p l i c a b i l i t y i s t h e i s o t r o p y on Che average of t h e m a t e r i a l under i n v e s t i g a t i o n .
This average i s o t r o p y i s guaranteed f o r a l a r g e c l a s s of aiaorphous so-
lies
such a s amorphous s o l i d s and semiconductors. I n t h e c a s e of c l a s - s i c a l window 9-lasses, it is j u s t t h i s i s o t r o p y which ma!ces t h e n use- £ u l f o r o p t i c a l i n s t r u m e n t s .For a n isotropie m a t e r i a l , t h e d i s t r i b u t i o n f u n c t i o n PA (..A:. .) must be independent of t h e c h o i c e of t h e c o o r d i n a t e system. It must be a
s c a l a r f u n c t i o n , i n v a r i a n t under any r o t a t i o n . Since t h e q u a n t i t i e s
$
t r a n s f o r a under r o t a t i o n s a s t h e components of a t e n s o r of kth o r d e r , t h e requirement OF i n v a r i a n c e r e s t r i c t s t h e form of t h e f u n c t i o n P A ( . .A:. .).
Zlementary i n v a r i a n t f u n c t i o n s of t h e q u a n t i t i e s
$
can be c o n s t r u c t e i l by t h e r u l e s of c o u p l i n g of a n g u l a r momenta. These f u n c t i o n s a r e homo- geneous forms i n :A of degree N = 2 , 3 , ..
.
.
Çome simple examples a r e :The i n v a r i a n c e of t h e f u n c t i o n PA (
.
.A:. .) i m p l i e s t h a t i t can b e ex- pressed i n terms of t h e i n v a r i a n t f u n c t i o n s Sx:By s u b s t i t u t i o n of eq. (5) i n t o t h e e q u a t i o n s (6), t h e f u n c t i o n s SN
can 3 e e::pressed i n terms of t h e c o o r d i n a t e s of t h e c h a r g e s cau- s i n g t h e l o c a l f i e l d s . I n accordance w i t h t h e i n v a r i a n c e of SNI éhese e x p r e s s i o n s can be transformed t o f u n c t i o n s of r a d i a l d i s t a n c e s ri ana of bond a n g l e s
o i j
where cosOij = ~ o s 0 ~ . c o s 0 . + s i n 0 ~ * s i n 9 * c o s ( $ ~ - $ J ~ ) .I j
For N=3, t h e c a l c u l a t i o n i s sornewhat n o r e t e d i o u s . I quote sone r e s u l t s using t h e a b h r e v i a t i o n s y i j = c o s û i j
,
and i$*Q t o i n i i c a t e sums o v e r 3 i n d i c e s frorn which a l 1 terins w i t h 2 o r 3 equal index v a l u e s a r e oinit- t e d .The d e t a i l s of t h e s e e x p r e s s i o n s a r e n o t of s p e c i a l importance. How- e v e r , t h e e x p r e s s i o n s shown i l l u s t r a t e t h a t t h e degree N o f t h e inva- r i a n t f u n c t i o n s SN i s an i n d i c a t o r of t h e l e v e l of c o r r e l a t i o n involved
-.
i n t h e p r o b a b i l i t y d i s t r i b u t i o n P S ( S N ) . I f t h e c e n t r a l i o n a s o r i g i n of t h e l o c a l c o o r d i n a t e systern i s taken i n t o account, t h e l e v e l of c o r r e l a t i o n i s ( N + l ) . Thus, t h e d i s t r i b u t i o n f u n c t i o n s Ps(Ss) involve t r i p l e c o r r e l a t i o n s , s i m i l a r t o t h e bond-angle d i s t r i b u t i o n f (
ei
) de- f i n e d by Hafner /1/. PS(S3) i n v o l v e s quadruple c o r r e l a t i o n s , e t c .This r e s u l t g i v e s an important c l u e f o r
a
s i m p l i f i c a t i o ni n
handling t h e l a r g e nunber of argunents i n t h e g e n e r a l d i s t r i b u t i i o n f u n c t i o n . For a c r y s t a l , a l 1 f u n c t i o n s S N ( k l , k have some p r e c i s e l y d e f i n e d v a l u e , t h a t i sI f d i s o r d e r i s i n t r o d u c e d g r a d u a l l y , t h e s h a r p d e l t a f u n c t i o n s become smeared o u t , and t h e broadening may be expected t o proceed t h e £ a s t e r , t h e h i g h e r t h e l e v e l of c o r r e l a t i o n . For s t r o n g d i s o r d e r , a s i g n i f i - c a n t s t r u c t u r e w i l l be found only i n t h e dependence of Pç upon t h e a r - guments SN with small v a l u e s N .
JOURNAL
DE
PHYSIQUEOf c o u r s e , t h e c o n d i t i o n s underlying t h e d e r i v a t i o n of eq. ( 8 ) c o r r e s - pond t o t h o s e Pound i n a g a s r a t h e r t h a n i n a densely packed s o l i d . For t h e d i s t r i b u t i o n of e l e c t r i c f i e l d g r a d i e n t s , t h a t i s , f o r LFd p a r a ï e t e r s with k = 2 o n l y , t h e d i s t r i b u t i o n f o r dense random packing was shown i n r e f . / 5 / t o
be
only s l i g h t l y modified compared t o t h a t given i n eq. ( 8 ) . I t i s very w e l l approximated by3
pSRP = [ l + B * S 3 ( 2 . 2 , 2 )
/
C J ~ ] P ; [ S ~ ( ~ ) ]w i t h 181 % 1 0 - ~
For s t r u c t u r a l s t u d i e s of amorphous s o l i d s i n g e n e r a l , information about t r i p l e and quadruple c o r r e l a t i o n f u n c t i o n s corresponding t o p r o b a b i l i t y d i s t r i b u t i o n s of S . > ( k ) and of S 3 ( k l r k 2 , k 3 ) w i l l be of p r i - mary i n t e r e s t .
III
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PHYSICAL PROPERTIES OF AMORPHOUS SOLIDS C O N T A I N I N G 9ARE EARTHIONS
The l o c a l p o t e n t i a l VLF a c t i n g on t h e 4f s h e l l l e a d s t o a p a r t i a l l i f - t i n g of t h e ( 2 ~ + 1 ) - f o i d degeneracy of t h e i o n i c ground s t a t e . Along w i t h t h e v a r i a t i o n s of t h e LW parameters Bg, t h i s r e s u l t s i n a d i s t r i - b u t i o n of ener4y s p l i t t i n g s f o r t h e ensemble of r a r e e a r t h i o n s . The d i s t r i b u t i o n of energy s p l i t t i n g s can be determined experimentally by i n e l a s t i c neutron s c a t t e r i n g / 9 / and by measurements of t h e s p e c i - f i c h e a t / I O / .
Furthermore, t h e magnetic moments of t h e 4f S t a t e s , t h e f i e l d depen- dence of t h e magnetization, and t h e temperature dependence of t h e mag- n e t i c s u s c e p t i b i l i t y a r e s e n s i t i v e i n d i c a t o r s of t h e l o c a l f i e l d s /11,12/.
F i n a l l y , e l e c t r o n s p i n resonance s p e c t r a and t h e h y p e r f i n e i n t e r a c - t i o n s of t h e 4f e l e c t r o n s with t h e n u c l e i of t h e probe i o n s can b e em- ployed i n t e s t i n g t h e d i s t r i b u t i o n of l o c a l f i e l d s a c t i n g on t h e r a r e e a r t h i o n s /13,14/.
A l 1 experimental i n v e s t i g a t i o n s quoted were e v a l u a t e d i n terms O£ t h e random a n i s o t r o p y mode1 /7/, of a s l i g h t l y extended v e r s i o n /15/, o r i n comparison w i t h t h e d i s t r i b u t i o n d e r i v e d i n r e f . / 6 / f o r t h e terms Bg. I n most c a s e s , d e v i a t i o n s £rom t h e r e s u l t s expected f o r a
L
dense random packing s t r u c t u r e were found. Two independent s t u d i e s of amorphous Ce-A1 a l l o y s /10,12/ y i e l d e d evidence f o r important e f f e c t s of higher-order (k=4,6) l o c a l f i e l d s .
The approach d e s c r i b e d i n t h e p r e v i o u s s e c t i o n , employing i n v a r i a n t f u n c t i o n s of t h e LFM parameters, p r o v i d e s t h e p o s s i b i l i t y £or a more g e n e r a l a n a l y s i s of experimental d a t a . The q u a n t i t i e s observable by experiments can b e expressed i n terms of t h e i n v a r i a n t f u n c t i o n s SN. Here, 1 w i l l demonstrate t h i s f o r d i s t r i b u t i o n s of energy l e v e l s and f o r t h e magnetic s u s c e p t i b i l i t y .
1
-
Energy Level D i s t r i b u t i o n sThe energy l e v e l s of an i o n under t h e a c t i o n of t h e l o c a l p o t e n t i a l
F o r g i v e n J I eq. (1 9 ) i s a polynomial e q u a t i o n i n E o f power (2511 )
,
t h e c o e f f i c i e n t s of t h e p o l y n o m i a l s b e i n g l i n e a r cornbinations o f t h e i n v a r i a n t f u n c t i o n s SN d e f i n e d i n s e c t i o n II. Some s i m p l e examples a r e :
The g e n e r a l form is:
w i t h
C9-120 JOURNAL DE PHYSIQUE
2
-
Nagnetic S u s c e p t i b i l i t yThe response of r a r e e a r t h probe i o n s t o an a o p l i e d n a g n e t i c f i e l d
6
i s t r e a t e d along s i m i l a r l i n e s . T assume a d i l u t e d a l l o y with n e g l i - g i à l e exchange i n t e r a c t i o n s between n a g n e t i c i o n s . Onîy t h e i n t e r a c - t i o n w i t h t h e a p p l i e $ g i e l d i s t h e n added t o t h e l o c a l - f i e l d Hamilto- n i a n : ff = VLF-
g J.H. The s o l u t i o n s EX(H) of t h e s e c u l a r e q u a t i o nJ B
depend upon t h e a p p l i e d f i e l d . The znagnetization
-$
( H ) f o r s t a t e h i sgiven by
X
-f -+ -+ and i n t h e l i m i t H -+ O , we have I Y X = x X * H . For t h e bulk s u s c e p t i b i l i t y c o n s i d e r e d h e r e , t h e a n g u l a r d i s t r i b u t i o n of t h e l o c a l c o o r d i n a t e systems with r e s p e c t t o t h e d i r e c t i o n of t h e a p p l i e d f i e l d must b e t a k e n i n t o account. I n t h e c a s e of an i s o t r o p i c s o l i d a s it i s assumed h e r e , t h e average o v e r a random d i s t r i b u t i o n of o r i e n t a t i o n s i s a p p r o p r i a t e .The procedure i s most easiLy o u t l i n e d by a specific-+example4 For J = l , t h e s e c u l a r e q u a t i o n becomes with t h e a b b r e v i a t i o n h = g p H:
J B
The i s o t r o p i c average o v e r t h e l a s t term i s
Cf
s i n c a Tr(Bs)=O. For l a r g e r J I s i m i l a r terms occur with t e n s o r s formzd by products of terms Bg. Then, t h e t r a c e y i e l d s one of t h e i n v a r i a n t f u n c t i o n s . SN.
For t h e c a l c u l a t i o n of t h e s u s c e p t i b i l i t y i n t h e l i m i t
8+0,
t h e expan- s i o n EX ( h ) = 3 ~ ( 0 ) * 6 i s s u b s t i t u t e d i n eq. ( 1 4 ) , and only éerms l i n e a r i n 6 a r e considered. This g i v e sand f o r t h e bulk s u s c e p t i b i l i t y one o b t a i n s
Thus, t h e temperature dependence of t h e magnetic s u s c e p t i b i l i t y i s r e - l a t e d t o t h e d i s t r i b u t i o n of energy l e v % l s which i n t u r n i s determined hy t h e d i s t r i b u t i o n of l o c a l f i e l d s .
I V
-
SUKXARY AND OUTLOOKm e work p r e s e n t e d h e r e i s s t i l l i n p r o g r e s s , Far f r o n b e i n g comple- t e 3 . Thus. it appears premature t o end t h i s r e p o r t Sy "conclusions". F i r s t l y , t h e formalism o u t l i n e d h e r e must be e l a b o r a t e d f o r r e a l i s t i c c a l c u l a t i o n s f o r t h e l a r g e J v a l u e s of r a r e e a r t h i o n s . S p e c i a l a t t e n - t i o n must be devoted t o t h e s h i e l d i n g f a c t o r s O appearing i n eg. ( 4 )
p o t e n t i a l s . They may have c o n s i d e r a b l e v a l u e s , d i f f e r i n g f o r d i f f e r e n t k /5/. Thus, i n c o n t r a s t t o EFG d i s t r i b u t i o n s where (1-0 j u s t i s a s c a l e f a c t o r / 6 / , t h e c o n t r i b u t i o n s of d i f f e r e n t m u l t i p o l e o r d e r s t o t h e LMF d i s t r i b u t i o n s a r e w e i g h t e d d i f f e r e n t l y depending upon t h e v a l u e s o f uk
,
and hence t h e s h a p e of t h e d i s t r i b u t i o n f u n c t i o n s i s m o d i f i e d . For m e t a l l i c m a t e r i a l s , c o n t r i b u t i o n s o f c o n d u c t i o n e l e c t r o n s s t i l l p r e s e n t an u n s o l v e d problem. These q u e s t i o n s w i l l have t o be worked o u t i n connexion w i t h e x p e r i m e n t a l d a t a f o r s p e c i f i c s y s t e n s .I hope t o have g i v e n e v i d e n c e , however, t h a t r a r e e a r t h i o n s c a n b e c o n s i d e r e d a s u s e f u l p r o b e s f o r s t u d y i n g t h e s t r u c t u r e o f a m o r ~ h o u s
s o l i d s , t h a t t h e y c a n y i e l d i n f o r m a t i o n beyon? r a d i a l d i s t r i b u t i o n E u n c t i o n s e x t r a c t e d £rom d i f f r a c t i o n d a t a . The most s i g n i f i c a n t r e - s u l t s c a n b e e x p e c t e d £rom s p e c t r o s c o p i c inethods: i n e l a s t i c n e u t r o n s c a t t e r i n g , e l e c t r o n s p i n r e s o n a n c e , NMR, and i26ssbauer s p e c t r o s c o p j ~ , a s i n t h e s e t h e e x p e r i m e n t a l d a t a a c t u a l l y r e f l e c t t h e d i s t r i b u t i o n f u n c t i o n wliereas r e s u l t s of b u l k rneasureinents a r e deterniined by some i n t e g r a l s a v e r a g i n g o v e r t h e d i s t r i b u t i o n .
ACKNOWLEDGEMENT
Large p a r t s o f t h e work d e s c r i b e d h e r e were performed w h i l e 1 e n j o y e d t h e g e n e r o u s h o s p i t a l i t y of t h e Departinent d e Recherche Fondamentale a t t h e C e n t r e d ' E t u d e s N u c l é a i r e s i n G r e n o b l e , and o f t h e L a b o r a t o i r e d e S p e c t r o m é t r i e Physique a t t h e U n i v e r s i t y of Grenoble. My p a r t i c u l a r t h a n k s g o t o my p e r s o n a 1 h o s t s , Pl. ~ o g é and J. C h a p p e r t a t t h e C e n t r e d.'Etudes N u c l é a i r e s , and t o C . Meyer and F . Hartmann-Boutron a t t h e U n i v e r s i t y .
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