MEASUREMENTS OF MAGNETIZATION

DENSITIES IN RARE-EARTH COMPOUNDS

J. Boucherle, D. Givord, J. Schweizer

To cite this version:

J. Boucherle, D. Givord, J. Schweizer. MEASUREMENTS OF MAGNETIZATION DEN-SITIES IN RARE-EARTH COMPOUNDS. Journal de Physique Colloques, 1982, 43 (C7),pp.C7-199-C7-214. <10.1051/jphyscol:1982728>. <jpa-00222335>

HAL Id: jpa-00222335

https://hal.archives-ouvertes.fr/jpa-00222335

Submitted on 1 Jan 1982

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°12, Tome 43, décembre 1982 page C7-199

MEASUREMENTS OF MAGNETIZATION DENSITIES IN RARE-EARTH COMPOUNDS

J.X. Boucherle , D. Givord and J. Schweizer '

+DRF/VN, CEN-G, 85X, 28041 Grenoble Cedex, Fronce

*Laboratoire Louis Néel, CNRS, 166X, S8042 Grenoble Cedex, France

**ILL, 156X, 3804S Grenoble Cedex, France

Résumé.- Les neutrons polar isés sont un ou t i l unique pour étudier l ' é t a t é l ec t ro ­nique des t e r r e s rares et de leurs composés. Les neutrons à courte longueur d'onde sont particulièrement adaptés vu les problèmes d 'ext inct ion et vu aussi l 'extension spat ia le du facteur de forme dans l 'espace réciproque. C'est à t ravers la densité d'aimantation, qui e l l e , e s t directement "vue" par les neutrons, que l 'on détermine l ' é t a t fondamental de la t e r r e r a r e . On le compare ensuite avec les r é su l t a t s de calculs provenant de modèles où sont inclus champ c r i s t a l l i n et échange. Ces comparaisons ont montré la nécessi té dans cer tains cas d ' inc lure dans le mécanisme d'échange des termes d 'ordre supérieur qui vont au delà de l'approximation du champ moléculaire. Un moment magnétique dû à la polar isa t ion des électrons 5d par le moment 4f a aussi été mis en évidence. I l a pour origine des interact ions avec le spin des électrons 4f, mais aussi avec leur o rb i t e . Dans le cas des t e r res rares anormales où ce t te aimantation 5d joue un rôle primordial, les expériences de neutrons polar isés sont essen t ie l les pour comprendre leurs propr ié tés .

Abstract.- Polarized neutrons are a unique tool for studying the electronic state of rare earth metals and rare earth compounds. Due to extinction problems and also due to the large extension of rare earth form factors in reciprocal space, hot neutrons are most suitable. Through the magnetization density which is directly measured with neutrons, the ground state of the rare earth is directly determined. It can be compared to results of calculations performed with models including crystal field interactions and exchange interactions. Such comparisons show the need in certain cases of high order terms in exchange mechanism which go beyond the molecular field approximation. The existence of a 5d moment, polarized by the localized 4f magnetization is also demonstrated.The study of such conduction electron moments is expected to be helpful in the understanding of abnormal rare earth behaviour in metals and rare earth compounds.

1. Introduction

The lanthanide series provides a large field of investigation for the study of magnetism. The magnetic properties of the rare earth ions are due to the inner incomplete 4f shell, whose electrons are tightly bound inside outer closed shells. An important associated screening by outer electrons makes the influence of the environment rather weak. As a result, the Coulomb correlation energy of the 4f electrons requires a localized description, even in metals, and the ground state is given by Hund's rules. Also, the perturbation of the crystalline electric field (CEF) is small compared to the spin-orbit coupling. Thus the ions are described in terms of a total angular momentum J.

Among the methods used for a study of rare earth magnetism, the direct obser­vation of the spatial distribution of the magnetization density is very meaningful. This microscopic approach can provide a direct picture of the state of the 4f ion which results from competition between CEF and magnetic interactions. Furthermore the magnetization density measurement is the only way to reach arid to analyze the additional contribution which is due to the conduction electrons. This density, present in metals and alloys, is more delocalized than that of the 4f one, and mediates indirect magnetic interactions.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982728

C7-200, JOURNAL DE PHYSIQUE

2 . 4f form f a c t o r : Measurements and formalism

Neutron sca t te r ing allows t h e measurement of t h e magnetic s t r u c t u r e f a c t o r s ~!.c($) which a r e t h e Four ies components of the magnetization densi ty. For accurate measurements of small FM(H), polar ized neutrons a r e needed : t h e e s s e n t i a l advantage of t h i s technique a r i s e s from t h e in te r fe rence term occurring between nuclear and magnetic sca t te r ing . Moreover, t h e measurement of the f l ipp ing r a t i o R (R = I+/I-), between sca t te red i n t e n s i t i e s f o r the two possible nputroq polar i - za t ion of t h e beam, y i e l d s d i r e c t l y t h e value of t h e r a t i o y = FM(H)/FN(H) without any normalization fac tor .

When considering measurements of r a r e e a r t h form f a c t o r s , espec ia l ly i n a l l o y s o r compounds, some s p e c i f i c experimental d i f f i c u l t i e s emerge :

i ) Due t o t h e high l o c a l i z a t i o n of t h e magnetization dens i ty , t h e form f a c t o r is g r e a t l y expanded i n rec iproca l space. I n order t o get a s u f f i c i e n t s e t of observations, measurements of Bragg r e f l e c t i o n s a t l a r g e s i n B / A values a r e needed which can only be obtained with neutrons of short wavelength.

i i ) The accuracy of polarized neutron experiments i s e s s e n t i a l l y l imited by ex t inc t ion phenomena. Moreover, rare-ear th compounds a r e o f t e n b r i t t l e and it is not possible t o reduce ex t inc t ion e f f e c t s by induced mechanical s t ra in ing of t h e samples. To overcome t h i s d i f f i c u l t y , megsurements of f l ipp ing r a t i o s a r e performed a t several short wavelengths (between 0.9 A and 0 . 4 1) which minimizes ex t inc t ion e f fec t s . Furthermore ex t rapola t ion of t h e d a t a t o X = 0 permits t h e determination of ex t inc t ion f r e e values.

i i i ) Another l i m i t a t i o n i n t h e accuracy of t h e measurements of t h e magnetic s t r u c t u r e f a c t o r s FN(H) = yFN(8) a r i s e s from t h e lack of exact knowledge of t h e huclear s t ruc ture . I n ~ a r t i c u l a r , it is e s s e n t i a l t o determine t h e s c a t t e r - ing length of t h e rare-eakth nucleus, value which may be a f fec ted by t h e proximity of resonant absorption peaks, and by nuclear po la r iza t ion :

-t-t b(A) = bo + Ab' (A) + i A ~ " ( A ) + f N BIo ( 1 )

Besides the value of bo which of ten i s not well known, there a r e ex t ra terms which a r e wavelength dependent. These can be calculated from absorpt ion d a t a , using t h e Breit-Wigner [ I ] formulae. Moreover, the very l a r g e hyperfine f i e l d s a t t h e rare-ear th nuc le i , which o r i g i n a t e from t h e la rge e lec t ron ic o r b i t a l contr ibut ions, can po+larize, even a t 4.2K, a s izeab le par t fN of t h e nuclear moment i f t h e nuclear s p i n I d i f f e r s from zero. This ~ o l a r i z a t i o n leads t o coherent s c a t t e r i n g a r i s i n g from t h e nuclear pseudomagnetic term.

I n summary, measurements a t d i f f e r e n t and shor t wavelengths appear t o be very important i n order t o complete t h e measurements and t o t r e a t ex t inc t ion . It i s worth not icing a l s o , t h a t f o r important magnetic elements such a s samarium and gadolinium, absorpt ion is considerably reduced a t short wavelengths. This allows measurements on r e l a t e d compounds t o be performed with na tura l elements. The f a c i l - i t i e s ava i lab le a t t h e IJ.L, thanks t o the hot source, have stimulated numerous s tud ies .

b) _F_p_n"-a_:ism of atomic 4f form f a c t o r s

The local ized character of t h e 4f e lec t rons permits atomic ca lcu la t ions t o be used f o r t h e i r descr ip t ion . I n t h e case of rare-ear th ions, t h e magnetic sca t te r ing involves both spin and o r b i t contr ibut ions. The spin-orbit coupling being la rge compared t o CEF in te rac t ions , t h e ion must be described i n terms of a t o t a l angular momentum 3. The perturbing e f f e c t of a c r y s t a l f i e l d o r a magnetic f i e l d i s t o remove the 2J+1 degeneracy of t h e ground mul t ip le t , The wave-functions a r e given as

-f With t h i s hypothesis, t h e ca lcu la t ion of t h e magnetic amplitudes FM may be + performed by t h e tensor-operator method [2,3], using Racah algebra. The vector FM i s given by i t s spher ica l components FG, F M ~ , FM-,.

K" ,. In t h i s formula Y (H) depends on t h e two angles 0 and Q which represent Q"

t h e d i r e c t i o n of t h e s c a t t e r i n g vec tor 3. A(K1',K') and B(K",K1) represent respec- t i v e l y t h e o r b i t a l and t h e sp in contr ibut ion t o t h e s c a t t e r i n g . They can be ca lcu la ted using tabulated c o e f f i c i e n t s f o r t h e r a r e ea r ths [41 and t h e r a d i a l i n t e g r a l s

Considering t h a t t h e magnetic amplitude i s a vec tor , t h e cross-section f o r polar ised neutrons i s

t h e usual r a t i o 2

~ % ( ? i ) / s i n o

Y = may only be defined ($1

when t h e following r e l a t i o n i s f u l f i l l e d :

2 rMo (3, = s i n 0 LFM, + 2 / W) I J This i s t h e case, o r nearly, i n most of t h e experimental s i t u a t i o n s E51.

3 . Electron, sp in and magnetization d e n s i t i e s

We examine i n t h i s sec t ion , t h e s o r t of r e s u l t s which neutron experiments can p r o v i d e i n t h e case of 4f magnetism. As t h e o r b i t a l momentum L general ly d i f f e r s

from zero, d i f f e r e n t o r b i t a l s a r e involved. In add i t ion , t h e o r b i t a l moment being unquenched, a n important con t r ibu t ion t o the moment appears and i ts l o c a l i z a t i o n d i f f e r s from t h a t of a spin densi ty. Also, t h e l a r g e value of t h e spin-orbit coupling implies t h a t t h e spin-density is a combination of d i f f e r e n t spin-states . Thus, i n general , t h e magnetization densi ty d i f f e r s from t h e sp in dens i ty and the l a t t e r , unl ike t h e case of 3d e lec t rons , i s a l s o d i f f e r e n t from t h e e lec t ron ic densi ty. However, i n the case of gadolinium where t h e 4f s h e l l i s half f i l l e d , these 3 d e n s i t i e s a r e equivalent.

a ) c$~_e-of-anS:sta_ts-~-Gado-Iif!i1!rn The 4f s h e l l is h a l f - f i l l e d , containing seven e lec t rons , and the o r F t a l

moment i s zero. The magnetization dens i ty is spher ica l ly symmetrical ($ ( r ) = R(r ) ) . The measurement of t h e magnetic form f a c t o r allows a d i r e c t determination of the 4f e lec t ron d i s t r i b u t i o n . The comparison of experimental form f a c t o r s with r a d i a l i n t e g r a l s <jo(H) > calculated i n atomic models c o n s t i t u t e s a t e s t f o r these models.

Ce3 +

P r 3 +

Nd3+

Pm3+

Sm3+

Eu3 +

Gd3+

Tb 3 +

„ 3 + Dy

3+ Ho

r. 3 +

Er

Tm3 +

Yb3+

f1

f2

f3

f*

f5

f6

f7

f8

f9

f10

f "

£12

f13

L

3

5

6

6

5

3

0

3

5

6

6

5

3

S

1/2

1

3/2

2

5/2

3

7/2

3

5/2

2

3/2

1

1/2

J

5/2

4

9/2

4

5/2

0

7/2

6

15/2

8

15/2

6

7/2

g J

6/7

4/5

8/11

3/5

2/7

S j J

2.14

3.20

3.27

2 .4

0.71

no moment

2

3/2

4/3

5/4

6/5

7/6

8/7

7.0

9 .0

10.0

10.0

9.0

7.0

4 .0

C2 d i p

1.600

1.644

1.803

2.263

5.422

-

0

0.370

0.533

0.613

0.652

0.667

0.667

s a t u r a t e d ion

magnet. sp in

0 0

P 0

P 0

P P

0 P

P

P P

P P

P 0

0 0

0 0

e l e c t r o n

0

0

0

P

P

0

0

0

P

P

P

I n gadolinium metal C61, t h e r a d i a l dependence of t h e 4f dens i ty appears t o be s i g n i f i c a n t l y expanded with respect t o c l a s s i c a l Hartree-Fock ca lcu la t ions 171. A good agreement can only be achieved using r e l a t i v i s t i c methods [81 (Dirac-Fock). The expansion of t h e 4f r a d i a l dens i ty has been explained a s an i n d i r e c t r e l a t i - v i s t i c e f f e c t , t h e r e l a t i v i s t i c contract ion of the core e lec t rons r e s u l t i n g i n a more e f f e c t i v e screening of nuclear charge. These r e l a t i v i s t i c ca lcu la t ions 191 appear t o be t h e bes t adapted t o a l l t h e o ther s t u d i e s of r a r e e a r t h form fac tors where a comparison have been performed.

A l a rge o r b i t a l contr ibut ion t o the magnetic moment i s associated with an open 4f s h e l l . The o r b i t a l magnetization dens i ty which corresponds t o a moment created i n s i d e t h e e lec t ron o r b i t a l s i s more local ized than t h e sp in densi ty.

As a f i r s t approximation both these d e n s i t i e s can be assumed t o be spher ica l ly symmetrical (dipole approximation). Considering only t h e spher ica l term i n t h e r e l a t i o n ( 3 ) , leads t o

where the values of the c o e f f i c i e n t s c2 which charac te r ize each ion a r e tabulated i n t a b l e I.

The o r b i t a l contr ibut ion d r a s t i c a l l y changes t h e s p a t i a l extension of t h e magnetization densi ty compared t o t h e case of sp in densi ty only. Therefore t h e form f a c t o r s f o r r a r e e a r t h elements with an o r b i t a l moment a r e more expanded i n t h e rec iproca l space than t h e Gd form f a c t o r . Also, t h e form f a c t o r s a r e d i f f e r e n t from l i g h t and heavy rare-earths. This property i s i l l u s t r a t e d i n Figure 1 where rare-ear th form f a c t o r s measured i n d i f f e r e n t RA112 a l l o y s appear. For R elements of t h e f i r s t half t h e form f a c t o r i s more expanded than f o r elements of t h e second ha l f . I n f a c t , when J = L-S, t h e t o t a l magnetization dens i ty r e s u l t s from a p a r t i a l cance l la t ion between t h e o r b i t a l dens i ty (more local ized) and the spin dens i ty ( l e s s loca l ized) .

This behaviour is emphasized i n the case of samarium where S = 512 and L = 5. The values of t h e spin and o r b i t a l moments a r e very c lose : t h e form f a c t o r exh ib i t s a maximum f o r a value of s i n 0 / h which is d i f f e r e n t from zero [ l o , 11 1 (Figures Id and 2) . I n Sm Co5 1121 a t 4.2K t h e Sm moment has a dominant o r b i t a l character and reaches 0.38 ". An i n t e r e s t i n g property i s observed b-hich e s s e n t i a l l y r e s u l t s from a mixing of higher J mul t ip le t s i n t o t h e ground mul t ip le t : t h e form f a c t o r i s l a r g e l y temperature dependent. When excited leve l s , presenting a dominant sp in character , become populated, t h e pealc shape of t h e Sm form f a c t o r is enhanced. A t 300 K, t h e Sm moment has decreased t o 0.04 VB. A cross-over temperature, a t which t h e sp in moment exact ly cancels the o r b i t a l moment, has been calculated t o occur a t 350K. A s t h e coupling between the cobal t moment (which i s almost a spin moment) and t h e r a r e e a r t h sp in i s always antiferromagnetic below t h i s temperature SmCog i s a ferromagnet and above t h i s temperature, it i s a ferrimagnet.

c> Anisotropies of t h e sp in and magneLization d e n s i t i e s The angular dependence of the 4f e lec t rons must be described, a s already

mentioned, on t h e IJM > b a s i s , t h e elements of which a r e not spher ica l ly symmetrical. I n t h e simple case of only one e lec t ron i n t h e 4f s h e l l (Ce3+ ion) the aspher ic i ty i s extreme and is i l l u s t r a t e d below. The 1513 H > wave funct ion r e s u l t s from t h e coupling of a n o r b i t a l moment 2, = 3 and of a sp in s = 1/2 leading t o a t o t a l moment J = j = 512

The corresponding angular p robabi l i ty of occupation is shown a s a polar diagram i n f i g u r e 3 . I n the case of cerium it i s equivalent t o t h e angular p a r t of t h e magne- t i z a t i o n densi ty. Very d i f f e r e n t angular dependences a r e associated with t h e

JOURNAL DE PHYSIQUE

SIN(THETAl/LAH (A-11

Fig. 1: Rare-earth magnetic amplitudes in RAR2 compounds (ref. 17,18,19,26,36)

Fig. 2 : Samarium magnetic amplitudes a t 4.2K and 300K i n SmCo5 (ref . 12).

The dashed l i n e corresponds t o t h e sa tu ra ted s t a t e 15/2,5/2> and t h e f u l l l i n e t o t h e l e v e l scheme.

Fig . 3 : Polar diagrams f o r t h e angular p robab i l i ty of occupation ( f u l l l i n e ) and angular sp in dens i ty (dashed l i n e s ) of t h e 1 5 / 2 , ~ > b a s i s s t a t e s of ~ e 3 + .

C7-206 JOURNAL DE PHYSIQUE

d i f f e r e n t s t a t e s , and consequently d i f f e r e n t form f a c t o r s (Fig. 4). The sp in dens i t i es which a r e a l s o represented i n Figure 3 a r e , a s a lready mentioned, d i f f e r e n t from e lec t ron and magnetization dens i t i es .

With la rge exchange in te rac t ions , t h e r a r e ea r th ions a r e i n t h e i r saturated s t a t e IJJ>. The dens i ty exh ib i t s a x i a l symmetry, and i s e i t h e r elongated (prolate) o r contracted (oblate) along t h e symmetry a x i s . I n such a case, t h e form f a c t o r is dependent only on t h e angle O between t h e z-axis and t h e s c a t t e r i n g vector . The cerium form f a c t o r s associated with t h e s t a t e 15/2,5/2> f o r O = 90" and O = 30" a r e very d i f f e r e n t and d i f f e r a l s o from t h e spher ica l form f a c t o r (Fig. 5 ) . The associated magnetization densi ty is o b l a t e (see f i g . 3a).

I 0.00 0.20 0.LO 0.60 0.80 1.00

SIN(THETA )/LAM ( A - I )

3+ Fig. 4 : Ce form f a c t o r s (0 = ) f o r t h e d i f f e r e n t (5/2,~> bas i s s t a t e s a s well a s f o r t h e d i p o l e approximation.

3+ Fig. 5 : Ce form f a c t o r s of t h e 15/2,5/2> b a s i s s t a t e f o r d i f f e r e n t azimuthal 0 angles a s well a s f o r the dipole approximation.

The a s p h e r i c i t i e s of the magnetization dens i ty f o r t h e IJJ> s t a t e s of t h e rare- e a r t h elements obtained by considering only second order t e rns a r e summarized i n Table I. They a r e compared t o those of t h e sp in densi ty and t o those of t h e e lec t ron densi ty a s deduced from the s ign of t h e Stevens coef f ic ien t s . The a s p h e r i c i t i e s of spin and magnetization d e n s i t i e s may d i f f e r due t o the inf luence of t h e orbitalmoment. Theaspher ic i t i es of sp in and e lee t ron d e n s i t i e s a r e t h e same f o r l i g h t rare-ear ths andopposi te forheavy rare-ear ths . A s a f i r s t approximation t h e spin densi ty r e f l e c t s t h e e lec t ron densi ty, but f o r heavy r a r e ea r ths where the 4f s h e l l i s more than ha l f - f i l l ed it is represen ta t ive of unpaired electrons.

4. 4f form f a c t o r and 4f wave funct ion

I n t h i s sec t ion we g ive some examples t o show how t h e 4f form f a c t o r i s connected t o t h e 4f e lec t ron wave-function which, i n tu rn , r e f l e c t s exchange and CEF in te r - act ions.

A t low temperatures, the observed form f a c t o r i s c h a r a c t e r i s t i c of the ground s t a t e which is t h e only s t a t e populated. In rare-ear th metals where t h e c r y s t a l l i n e symmetry i s hexagonal, dominant exchange in te rac t ions determine the ground s t a t e t o be t h a t of t h e saturated ion I JJ > a s already mentioned.

The magnetization dens i ty and there fore t h e associated form f a c t o r have a cyl in- d r i c a l symmetry with respect t o t h e magnetization ax is . The magnetic amplitudes measured i n t h e equa tor ia l plane exhib i t a smooth decrease versus s i n B / X [13,141. The aspher ic i ty of t h e magnetization dens i ty was shown d i r e c t l y i n t h e case of Tm metal 1141 (where i t is ob la te ) by measurements of Bragg r e f l e c t i o n s out of t h e equatorial plane. It i s worth not icing, although never experimentally measured, t h a t i n such a case, due t o t h e strong spin-orbit coupling, t h e magnetization dens i ty would r e o r i e n t a t e along any d i r e c t i o n under an applied magnetic f i e l d high enough t o overcome magnetocrystalline anisotropy. This i s unl ike t h e case of 3d magnetism, where very la rge c rys ta l - f i e ld e f f e c t s determine t h e 3d o r b i t a l s and lead t o a quenching of t h e o r b i t a l moment. The magnetic moment f r e e l y o r i e n t a t e s along an applied f i e l d without modification of t h e spin-density.

When t h e exchange in te rac t ions a r e weak, t h e s t a t e s of t h e 4f e lec t rons a r e e s s e n t i a l l y determined by c r y s t a l f i e l d e f fec t s . I n TmSb, where exchange in te rac t ions can be neglected, t h e ground s t a t e i s a non-magnetic s ing le t 1151. A magnetic moment of 1.2 UB is induced by a n applied magnetic f i e l d of 12.5 kOe. As expected i n cubic symmetry f o r a low f i e l d , t h i s value does not depend on the f i e l d d i rec t ion . However, t h e magnetization densi ty exh ib i t s 4-fold or 2-fold symmetry depending on whether t h e f i e l d is appl ied along 11001 o r [110]. This leads, a s observed by LANDER e t a l . 1161 t o d i f f e r e n t an i so t rop ies of t h e Tm form f a c t o r f o r d i f f e r e n t f i e l d direc- t ions .

I n many cases, c r y s t a l f i e l d and exchange in te rac t ions a r e competing. The l a r g e r the c r y s t a l l i n e e l e c t r i c f i e l d compared t o t h e exchange in te rac t ions , t h e more c lose ly w i l l t he symmetry of t h e magnetization densi ty r e f l e c t t h e c r y s t a l symmetry. A t y p i c a l example i s tha t of cerium where t h e r e i s only one e lec t ron on t h e 4f s h e l l and t h e magnetization dens i ty may be highly anisotropic . The form fac tor measured i n theLavesphase compound CeAR2 1171 a t 1.5K under an applied f i e l d of 46 kOe p a r a l l e l t o [110] i s shown i n Fig. l a . The points a r e considerably sca t te red and reveal a p ro jec t ion of t h e magnetization densi ty elongated along the 2-fold ax i s i n t h e plane perpendicular t o the f i e l d d i r e c t i o n . Such a shape is c h a r a c t e r i s t i c of a r7 - l ike s t a t e , although t h e q u a n t i t a t i v e ana lys i s shows t h a t mixing with t h e r8 s t a t e of higher magnetization occurs, induced by exchange in te rac t ions .

Analysis of t h e magnetic amplitudes a t low temperature allows a determination of t h e 4f ground s t a t e . Such s tud ies a r e simpler i n cubic compounds such a s RAR2 or RZn where only R atoms a r e magnetic. Except a t low s i n B / A values,-where conduction e lec t ron po la r iza t ion i s present (see next sec t ion) , it may be assumed t h a t only 4f magnetism cont r ibu tes t o t h e observed magnetic amplitudes. A reduct ion of t h e 4f moment compared t o t h a t of t h e saturated s t a t e I J J > appears systematical ly . This gives evidence f o r CEF mixing of d i f f e r e n t I JM> s t a t e s . The associated symmetry of t h e ground s t a t e wave funct ion is re f lec ted i n t h e s e l e c t i o n r u l e s which deter- mines i t : bM = n, where n i s t h e order of symmetry of t h e magnetization ax is . For example, i n t h e cubic Laves phase NdAX2 [181, the easy d i r e c t i o n of magnetization is t h e 4-fold a x i s , t h e ground s t a t e wave funct ion may, a p r i o r i , be w r i t t e n a s :

where t h e a 's a r e t h e only unknown parameters t o be determined from t h e experiment, a s f a r a s tfZe rad ica l i n t e g r a l s <jg> can be taken from atomic calculat ions. The wave funct ion which f i t s bes t the da ta 1s :

The o v e r a l l agreement between experimental and calculated magnetic amplitudes i s excel lent (Fig. l c ) . The Nd moment deduced from these measurements reaches 2.61UB-

C7-208 JOURNAL DE PHYSIQUE

I n RFe (cubic Laves phase) and RCo5 (hexagonal CaCu5 type s t ruc ture ) compounds, 5d and 4f magnetism coexist . I n t h i s case, ana lys i s of t h e da ta is more complex s ince the 3d magnetic con t r ibu t ion must be subtracted from t h e 4f contr i - bution. The Sm form f a c t o r measured i n SmCo5 r121 a t 4.2K i s shown i n Figure 2. The peak shape of t h e form f a c t o r , c h a r a c t e r i s t i c of t h e Sm ion i s a s already mentioned even more pronounced than f o r the s t a t e 1512,512 > of maximum magnetization when considering only t h e ground mul t ip le t . As i n SmAi2 [I91 and SmZn [201, exchange in te rac t ions , increase the sp in contr ibut ion t o the magnetization moment through mixing with t h e exci ted mul t ip le t s [I 1 , 121. The wave funct ion f i t t e d by consider- ing only t h e f i r s t exci ted mul t ip le t J = 712 may be w r i t t e n a s :

I n NdCo5 or HoFe2, s ince exchange i n t e r a c t i o n s a r e much la rger than CEF i n t e r - act ions, t h e rare-ear th form f a c t o r is expected t o be t h a t of t h e sa tura ted ion. I n f a c t , a s shown i n Fig. 6 , t h e ex t rapola t ion t o s i n O/X = 0 leads i n HoFe2 [21][22] t o a 4f moment of 9.15 V B , t h e ground s t a t e wave funct ion being

I t i s worth not icing t h a t recent NMR measurements i n several W e 2

i n t e r - meta l l i cs [231 a l s o revealed a reduced value of t h e ground s t a t e 4f moment.

Similar ly i n NdCo5 [241, t h e Nd moment a t 4.2K i s 2.8 PB instead of 3.27 UB f o r the sa tura ted ion giving evidence f o r a ground s t a t e which r e s u l t s from mixing between d i f f e r e n t I JM > s t a t e s .

Fig. 6 : Holmium magnetic amplitudes i n HoFe2 ( re f . 21, 22).

Exchange and CEF in te rac t ions a c t i n g on the rare-ear th ions determine t h e 4f ground s t a t e . The exchange in te rac t ions a r e usua l ly t rea ted i n t h e molecular f i e l d formalism a s equivalent t o a magnetic f i e l d :

= -gJuB gex 3 where %ex i s t h e exchange f i e l d . Using t h e Stevens operator formalism, t h e CEF in te rac t ions may be w r i t t e n as :

m where the number of operators 02 involved depends on t h e symmetry of t h e environment and t h e B; a r e considered a s parameters.

I n the RAR2 compounds 1251 , cubic c r y s t a l f i e l d parameters were obtained by neutron spectroscopy i n t h e paramagnetic s t a t e , and exchange i n t e r a c t i o n s from magne- t o s t a t i c measurements i n t h e ordered s t a t e . The 4f ground s t a t e calculated by diagonal izat ion of t h e t o t a l Hamiltonian, including CEF and exchange in te rac t ions , i s , f o r example i n NdAR2, i n c lose agreement with the ground s t a t e corresponding t o t h e measured form f a c t o r a t 4.2K. This shows t h e CEP and exchange in te rac t ions t o be cor rec t ly described i n t h e above formalism. On t h e o ther hand, t h e compound HoAR2 i n order t o account f o r the Ho ground s t a t e measured i n t h e polar ised neutron e x p e r b e n t ( F i g . l f ) [26] would need l a r g e r values of CEF parameters than deduced from neutron spectroscopy. Such a discrepancy suggests tnar: higher-ordrer terms a r e involved i n t h e exchange in te rac t ions . Similar behaviour i s known t o occur i n RZn [27] a l loys . The Tm form fac tor i n TmZn [261 was in te rpre ted by considering quadrupolar pa i r in te rac t ions :

then, magnetization measurements i n t h e ordered s t a t e , sp in wave exc i ta t ions and Tm form f a c t o r could be accounted f o r by considering a unique s e t of CEF and exchange in te rac t ions [ 291.

A s a l ready described t h e r e a r e cases where t h e ground s t a t e is not I J J > even when exchange in te rac t ions l a rge ly overcome CEF in te rac t ion . To account f o r t h e moment reduction i n IioFe2 [21,22] and NdCo5 [24] would need i n the molecular f i e l d formalism huge values of CEF parameter, which i s u n r e a l i s t i c . Although i t has always been assumed t h a t t h e molecular f i e l d model is an exce l len t approximation f o r interpre- t i n g magnetic p roper t i es , espec ia l ly when la rge 4f-3d in te rac t ions a r e involved, the r e s u l t s of form f a c t o r measurements show t h a t a s f a r a s the r a r e ea r th ground s t a t e i s concerned such an approximation i s not va l id . Here a l s o higher order terms a r e involved i n exchange in te rac t ions ; but i n HoFe2 and NdCog, the exchange i n t e r - ac t ions involve 3d electrons f o r which t h e o r b i t a l contr ibut ion t o the magnetization i s small. Then biquadrat ic sp in in te rac t ions a r e very un l ike ly t o be so la rge . But t h e magnetic in te rac t ions occur through t h e conduction band, and s imi la r propert ies may be due t o a f a i l u r e of t h e RKKY theory when t h e band magnetization i s large. The magnetic ropert ties of t h e compound GdPIg [29] , t h e magnetic s t r u c t u r e of which i s canted [30] , can be quant i t a t ive ly understood when introducing biquadrat ic pa i r i n t e r a c t ions phenomenologically [31].

C7-210 JOURNAL DE PHYSIQUE

5. Po la r iza t ion of t h e conduction e lec t rons

a ) Evidence f o r a conduction e lec t ron po la r iza t ion

I n most of t h e above cases, where t h e experimental magnetic amplitudes have been compared t o those calculated f o r a 4f e lec t ron configurat ion, a s i g n i f i c a n t discrepancy was present f o r t h e low angle r e f l e c t i o n s . This shows t h a t there e x i s t s i n t h e c e l l a magnetization dens i ty which has not a 4f character and p a r t i c u l a r l y , which is not a s local ized around t h e nuclei . This i s a l s o apparent i n r e a l space where some magnetization dens i ty , f a r from t h e Nd nuclei ,can be seen, a s , f o r example, i n t h e p ro jec t ion along t h e [001] a x i s of NdAR2 (Fig. 7a).

Fig. 7 : Project ion of t h e magnetization densi ty (p3/i2) f o r d i f f e r e n t r a r e e a r t h Laves phases a ) t o t a l densi ty i n NdAL2, ex t ra densi ty i n b) NdAx2, c ) CeAR2, d) Ho Fe2.

One can then consider t h e ex i s t ence of 2 con t r ibu t ions t o t h e magnetic amplitudes : one due t o t h e 4f e l e c t r o n s and another one r e s u l t i n g from t h e pola- r i z a t i o n of t h e conduction e l ec t rons . It i s p o s s i b l e however t o s e p a r a t e these 2 c o n t r i b u t i o n s without making any assumption on t h e exac t shape of t h e 4f e l e c t r o n form f a c t o r . Such a sepa ra t ion i n i t i a t e d by NOON e t a l . i n m e t a l l i c Gd 161, i s based on t h e d i f f e r e n t l o c a l i z a t i o n s expected f o r t h e 2 types of magnet izat ion : a d i s t i n c - + t i o n can be made between a f i r s t magnet izat ion mloc( r l , ve ry loca l i zed around t h e r a r e e a r t h n u c l e i , and a second magnet izat ion mextra(r), more d i f f u s e :

+ m(r) = rn ~ O C (:> + mextra(z)

Consequently t h e magnetic s t r u c t u r e f a c t o r s a r e t h e sum of 2 magnetic con t r ibu t ions :

FII = F + F %oc Mextra

where FM i s assumed t o vanish ve ry r a p i d l y wi th s i n 6/h due t o t h e poor e x t r a

l o c a l i z a t i o n of mextra(f). The FM con t r ibu t ions t o t h e f i r s t Bragg r e f l e c t i o n s l o c a r e determined a s those which g i v e a magnet izat ion d e n s i t y mloc(?), a s c l o s e t o

zero a s p o s s i b l e between t h e r a r e e a r t h atoms.

Table I1 d i s p l a y s t h e va lues of F M ~ ~ ~ determined f o r t h e f i r s t Bragg r e f l e c - t i o n s of NdA9.2, and compares them wi th t h e s t r u c t u r e f a c t o r s FM ca lcu la ted by

4f consider ing t h e 4f e l e c t r o n c o n t r i b u t i o n only and t h e wave f u n c t l o n a s determined previously ( s e c t i o n 4.b). The two methods, t h e f i r s t based on t h e l o c a l i z a t i o n c r i t e r i o n only , and t h e second assuming t h e knowledge of t h e 4f form f a c t o r , g i v e ve ry s i m i l a r r e s u l t s .

Table I1 : Comparison of t h e magnetic s t r u c t u r e f a c t o r s of t h e f i r s t Bragg r e f l e c t i o n s f o r NdAL2 : observed s t r u c t u r e f a c t o r Fw,~,, l oca l i zed s t r u c t u r e f a c ~ o r s

and 4f e l e c t r o n s t r u c t u r e f a c t o r F ~ ~ ~ c a l c u l a t e d f o r I J'Nd > = 0.88 1 912 > YM&5 1 112 > - 0.11 1 - 712 >.

The conduction e l e c t r o n c o n t r i b u t i o n s a r e then obta ined by simple s u b t r a c t i z n . I n t h e case of NdAL2 t h e s e con t r ibu t ions become ve ry small above s i n 6 /A = 0.35 A-I.

b) WhiGh-~lgGt~2~~-~2!!L~if!~-tg-tO-thg-g~t25-~2g:gti~~tjon The conduction e l e c t r o n magnet izat ion d e n s i t y mextr3($) i s obtained by a simple

Four ie r t ransform of t h e s t r u c t u r e f a c t o r s : F Thls d e n s i t y i s represented Mextra'

i n F igure 7 f o r 3 compounds having t h e same c r y s t a l s t r u c t u r e : NdAR2 , CeAL2 and

HoFe2. I n the case of NdAR2, it i s a l s o cornpared t o t h e t o t a l magnetization densl ty. As expected, some dens i ty m,tra(r) i s present f a r from the r a r e e a r t h nuc le i . This l o c a l i z a t i o n is f a r l e s s than t h a t of t h e 4f moment a s seen i n f igures 7a and 7b.

The s p a t i a l extension of the ex t ra magnetization dens i ty is very s imi la r f o r t h e d i f f e r e n t compounds. For t h e 3 compounds of f i g u r e 7, i t i s s t r i k i n g . This implies t h a t f o r t h e d i f f e r e n t compounds, the same e l e c t r o n i c s h e l l i s ~ o l a r i z e d by t h e 4f moment. One can then compare t h i s magnetization densi ty t o t h a t r e s u l t i n g from atomic ca lcu la t ions , and p a r t i c u l a r l y t o t h a t of t h e 5d she l l . However the 5d e lec t rons a r e very s e n s i t i v e t o t h e i o n i c i t y of the r a r e e a r t h atom : according t o Desclaux [ 3 2 ] , the 5d electrons a r e almost f r e e i n t h e neu t ra l atom R0 while i n t h e t r i p o s i t i v e ion R3+ they a r e f a i r l y local ized. The experimental 5d form f a c t o r i s intermediate between these 2 extreme cases (Figure 8) . It is there fore possible t o consider t h e conduction electrons which a r e polarized by the 4f moments a s essen- t i a l l y atomic 5d electrons. But, f o r these e lec t rons , t h e ex t ra p o t e n t i a l r e s u l t i n g from the ion iza t ion of the r a r e e a r t h atom i s p a r t i a l l y screened.

Figure 8 : 5d magnetic amplitudes f o r H ~ A R ~ measured a t 4.2K and calculated f o r 3+

H: and Ho ( r e f . 22, 32).

3) Sign and magnitude of t h e 5d po la r iza t ion

In a l l t h e cases of rare-ear th metals o r rare-ear th compounds where t h e magnetization dens i ty has been invest igated, with the s o l e exception of cerium compounds, t h e magnetization of t h e conduction electrons i s p a r a l l e l t o the sp in S of t h e 4f she l l . It r e s u l t s i n a negative magnetization f o r the f i r s t half of t h e rare-ear th where J = L-S, and a ~ o s i t i v e magnetization f o r t h e second half where J=L+S. This underlines the r o l e of t h e 4f sp in a s a dr iving f o r c e t o po la r ize t h e conduction e lec t rons and suggests t o sca le t h e 5d magnetization t o t h e spin :

= a i Sz >. As t h i s simple law does not f i t t h e experimental da ta , it has been proposed by BELOXIZSKY,NIEZ and LEVY 1333 t o include t h e 4f o r b i t i n t h e dr iving mechanism : M5d = a < SZ > + B < L z > . Table I11 shows t h a t s a t i s f a c t o r y agreement can be obtained f o r t h e RAE2 s e r i e s .

Table 111 : Magnitude of t h e conduction e lec t ron po la r iza t ion (u*).

sin 8 -- A

Figure 9 : Magnetic amplitude measured a t d i f f e r e n t temperatures i n CeSn3 (ref .35).

The s i t u a t i o n is reversed i n cerium compounds where the 5d magnetization is p a r a l l e l t o the 4f moment although J = L-S. I n t h i s case, t h e exchange mechanism with conduction e lec t rons i s d i f f e r e n t . Schrieff e r and Wolf [341 have shown t h a t besides t h e normal contact i n t e r a c t i o n which is p o s i t i v e f o r a l l t h e rare-ear th atoms, t h e r e e x i s t s f o r cerium a resonant exchange, negative and la rger than t h e normal exchange. This negat ive exchange, which explains r e s i s t i v i t y anomalies and Kondo behaviour i n cerium compounds, is d i r e c t l y confirmed by neutron d i f f r a c t i o n .

I n addi t ion, t h e d i f f e r e n t values of t h e 5d po la r iza t ion a r e associated with d i f f e r e n t behaviour, observed i n several cerium compounds. For instance, i n t h e Kondo system CeAQ2, t h e 5d po la r iza t ion represents 20 X of t h e 4f moment. On t h e other hand, a t low temperature i n CeSn3 1351 , t h e 5d pola r iza t ion i s equal t o t h e 4f moment (Figure 9 ) , which confirmed t h e mixed valence character of t h i s compound. However, t h i s po la r iza t ion decreases a t higher temperatures and above 40K it has p r a c t i c a l l y disappeared. Such a thermal v a r i a t i o n i s not f u l l y understood f o r t h e moment. Other s tud ies of these phenomena should help f o r the understanding of t h e complex magnetic p roper t i es i n both Kondo and mixed valence systems.

JOURNAL DE PHYSIQUE

References

[l] B.N. Brockhouse, Canad. J. Phys. 31 (1953) 440. 123 D.E. Johnston, Proc. Phys. Soc. - 38(1966) 37. 131 S.W. Lovesey, D.E. Rimmer, Rep. Prog. Phys. 32 (1969) 333. [41 G.H. Lander, T.O. Brun, J. Chem. Phys., 53 (1970) 1367. [51 J. Schweizer, 'E lec t ron and Magnet izat ion D e n s i t i e s i n Molecules and c r y s t a l s '

Ed. P. Becker (Plenum Press ) (1980) p. 501. C61 R.M. Moon, W.C. Koehler, J.W. Cable, H.R. c h i l d , Phys. Rev. ~5 (1972) 997. [ 7 1 M. Blume, A.J. Freeman, R.E. Watson, J. Chem. Phys. 37 (1962)-1245. [8] A.J. Freeman, J .P . Desclaux, Tnt. J. Magn. 3 (1972) =I. [91 A . J . Freeman, J .P . Desclaux, J.f:ag.~ag.l1at.-12 (1979) 11. [lo] W.C. Koehler, R.Y. Moon, Phys. Rev. L e t t , 2971572 ) 1468. [ll] J . X . Boucherle, D. Givord, J . L a f o r e s t , P. liorin."The r a r e e a r t h s i n modern

sc ience and technologyl',Ed. G . J . McCarthy, J.J. Rhyne, H.B. S i l b e r (Plenum Press ) 2 (1980) 261.

1121 D. GivoFd, J. L a f o r e s t , J. Schweizer, F. Tasse t , J. Appl. Phys. (1979) 2008. [I31 0. Steinswoff , G. Shirane, R. Nathans, M. Blume, H.A. Alpe r in , S.J. P i c k a r t

Phys. Rev. 161 (1967) 499. [I41 T.O. Brun, G.H. Lander, Phys. Rev. L e t t e r s , 2 (1969) 1295. I151 B.R. Cooper, 0. Vogt,Phys. Rev. (1970) 2040. [I61 G.H. Lander, T.O. Brun, 0. Vogt, Phys. Rev. (1973) 1988. El71 B. Barbara, M.F. Rossignol, J . X . Boucherle, J. Schweizer, J.L. Buevoz,

J. Appl. Phys. - 50 (1979) 2300. [I81 J . X . Boucherle, J. Schweizer, J. Mag. Mag. Mat. 24 (1981) 308. [I91 J . X . Boucherle, D. Givord, J. Lafo res t , J. ~ c h w e E e r , F. Tasse t , J. d e Phys.

40 C5- (1 979) C5-321. [20] Givord, P. Morin, D. Schmit t , J. Mag.Mag.Mat. 15-18 (1980) 261. [21] H. Fuess, D. Givord, A.R. Gregory, J. Schweizer, J. Appl. Phys. 50 (1979) 2000. [22] J.X. Boucherle, J. Schweizer, D. Givord, A. Gregory, J. Appl. ~ h s . 53 (1982)

1 950. I231 R.A.B. Devine, Y. Ber th ie r , J. Mag.Mag.Mat. 25 (1981) 135. 1241 J . M . Almeda, D. Givord, R. Lemaire, Q. Lu, S.B. Palmer, F. Tasse t ; t h i s i s s u e . [251 M. Rossignol , t h e s i s , Univ. Grenoble (1980). [261 J . X . Boucherle, J. Schweizer, J. Appl. Phys. 53 (1982) 1953. 1271 P. Morin, D. Schmit t , Phys. Rev. B23 (1981) 5936. [281 D. Givord, P. Morin, D. Schmit t , ~ ~ ~ p p l . Phys. 53 (1982) 2445. [291 D. Givord, P. Morin, D. Schmit t , t o be r301 P. Morin, 3. P i e r r e , D. Schmit t , D. Givord, Phys. L e t t . 65A (1978) 156. [311 B. Kim, P.M. LSvy, J. Mag.Mag.Mat. 27 (1982) 257. [321 J.P. Desclaux, p r i v a t e communication. 1331 E. Belor izky, J.J. Niez, J . X . Boucherle, J. Schweizer, P.M. Levy,

J. Mag.Mag.Mac. 15-18 (1980) 303. [341 J.R. S c h r i e f f e r , P.A. Wolf, Phys. Rev. 149 (1966) 491. 13-53 C. S t a s s i s , C.K. Loong, B.M. Harmon, S . ~ L ~ U , R.M. Moon, J. A p p l . phys.

50 (1979) 7567. 1361 TS. Abel l , J . X . Boucherle, R. Osborn, B.D. Rainford, J. S c h c ~ e i z ~ r ,

ICN (1982) (Kyoto) t o b e publ ished.