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NARROW BAND SOLIDS
Y. Baer, W.-D. Schneider
To cite this version:
Y. Baer, W.-D. Schneider. BIS VERSUS OTHER SPECTROSCOPIES IN NAR- ROW BAND SOLIDS. Journal de Physique Colloques, 1987, 48 (C9), pp.C9-967-C9-980.
�10.1051/jphyscol:19879174�. �jpa-00227289�
JOURNAL DE PHYSIQUE
Colloque C9, supplgment au n012, Tome
48,
decembre 1987B I S VERSUS OTHER SPECTROSCOPIES I N NARROW BAND S O L I D S
Y. BAER and W.-D. SCHNEIDER
I n s t i t u t d e Physique, Universit.5 de NeuchStel, CH-2000 Neuchdtel
.
SwitzerlandA b s t r a c t
B r e m s s t r a h l u n g i s o c h r o m a t s p e c t r o s c o p y (BIS) is t h e most d i r e c t method a l l o w i n g u s t o p r o b e t h e u n o c c u p i e d s t a t e s i n s o l i d s . When a p p l i e d t o b r o a d bands, . t h e BIS s p e c t r a a r e commonly i n t e r p r e t e d w i t h i n a s i n g l e p a r t i c l e framework. However, when t h e c o r r e - l a t i o n o f t h e o u t e r m o s t s t a t e s becomes s o s t r o n g t h a t t h e Koopmans' a p p r o x i m a t i o n l o - o s e s g r a d u a l l y its v a l i d i t y t h e f i n a l s t a t e s c a n no l o n g e r b e i n t e r p r e t e d i n terms o f i n i t i a l d e n s i t y o f s t a t e s . The s i t u a t i o n is p a r t i c u l a r l y i n t e r e s t i n g i n t h e A c t i n i d e s and L a n t h a n i d e s where t h e f - s t a t e s c a n be v e r y weakly h y b r i d i z e d w i t h t h e band s t a t e s . The many-body n a t u r e o f t h e h y b r i d i z e d f - s t a t e s is r e f l e c t e d by t h e d u a l i t y o f t h e i r e x c i t a t i o n s : a t low e n e r g y t h e y form a s p e c t r u m showing s i m i l a r i t i e s w i t h an e x t r e m e l y narrow m e t a l l i c band w h e r e a s a t h i g h e n e r g y t h e y h a v e a l o c a l i z e d cha- r a c t e r r e m i n i s c e n t o f p u r e l y a t o m i c f i n a l s t a t e s .
The e q u i v a l e n c e between p h o t o e m i s s i o n and BIS is d e m o n s t r a t e d by t h e m i r r o r p r o p e r t y between c o r r e s p o n d i n g e l e c t r o n and h o l e c o n f i g u r a t i o n s . However, t h r e s h o l d s p e c t r o s - c o p i e s l i k e X-ray a b s o r p t i o n o r e l e c t r o n e n e r g y l o s s a p p l i e d t o c o r r e l a t e d s y s t e m s a r e markedly d i f f e r e n t from BIS - s i n c e t h e deep h o l e m o d i f i e s l o c a l l y t h e d e n s i t y o f f i n a l s t a t e s above EF.
1. INTRODUCTION
The a n a l y s i s o f t h e B r e m s s t r a h l u n g e m i t t e d by t h e d e c e l e r a t i o n o f m o n o e n e r g e t i c e l e c - t r o n s p r o p a g a t i n g i n a s o l i d is nowadays an e s t a b l i s h e d t e c h n i q u e t o p r o b e t h e unoc- c u p i e d e l e c t r o n i c s t a t e s . F o r a s u f f i c i e n t l y h i g h e n e r g y o f t h e i m p i n g i n g e l e c t r o n s , t h e d i f f e r e n t methods o f measurement a r e u s u a l l y e q u i v a l e n t . I n t h e most c o n v e n i e n t t e c h n i q u e , t h e i n t e n s i t y o f t h e e m i t t e d p h o t o n s is d e t e c t e d a t a f i x e d wave l e n g h t w h i l e t h e e n e r g y o f t h e e l e c t r o n beam is swept, f o r t h i s r e a s o n
i t
is c a l l e d Brem- s t r a h l u n g i s o c h r o m a t s p e c t r o s c o p y ( 0 1 s ) [ I ] . D i s r e g a r d i n g p r o b l e m s r e l a t e d t o t h e background [ 2 ] , t h i s t e c h n i q u e p r o b e s t h e d e n s i t y o f t h e u n o c c u p i e d s t a t e s w e i g h t e d by t h e m a t r i x e l e m e n t s and i t can be c o n s i d e r e d a s an i n v e r s e X-ray p h o t o e m i s s i o n s p e c t r o s c o p y [3-51. I n t h e low e n e r g y r e g i o n , t h e development o f e n e r g y - s e l e c t i v e de- t e c t o r s a t-
10 eV [61 and t h e u s e of UV-monochromators [7,8] h a s opened t o t h i s t e c h n i q u e t h e r i c h s p e c t r u m o f p o s s i b i l i t i e s e x p l o i t e d i n UV-photoemission [9,10].Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19879174
I n t h e p r e s e n t review we s h a l l c o n s i d e r t h e high-energy l i m i t o f i n v e r s e photoemis- s i o n (BIS) where t h e unique r e l e v a n t p a r a m e t e r is t h e energy o f t h e e l e c t r o n i n t h e f i n a l s t a t e . Fig. 1 shows f o r Pd and Ag an e a r l y a t t e m p t [ 5 ] t o compare d i r e c t l y BIS s p e c t r a t o a c a l c u l a t e d d e n s i t y of s t a t e s DOS { l l ] i n a broad energy r a n g e above EF. I t is i n t e r e s t i n g t o o b s e r v e t h a t t h e spectrum of Ag is a n e a r l y p e r f e c t r e p l i - c a of t h e spectrum o f Pd, provided t h a t it is s h i f t e d by 3.5 eV t o a c c o u n t f o r t h e f i l l i n g o f t h e empty d-symmetry s t a t e s which g i v e r i s e t o t h e l e a d i n g peak a t EF i n t h e Pd spectrum. I f a background i n c r e a s i n g smoothly towards h i g h e r e n e r g i e s and an i m p o r t a n t l i f e t i m e broadening a r e t a k e n i n t o a c c o u n t , a c o n v i n c i n g agreement between t h e o r y and experiment is observed. An e x t e n s i o n o f t h i s t y p e o f s t u d y [12] h a s con- f i r m e d t h a t a t l e a s t q u a l i t a t i v e l y t h e broad BIS s t r u c t u r e s s p r e a d f a r above EF a r e c l e a r l y dominated by t o t a l DOS e f f e c t s . I n a measurernent o f Cu performed up t o 400 eV above EF, f i n e s t r u c t u r e s superimposed on t h e f r e e - e l e c t r o n behaviour have been observed. These have an o r i g i n q u i t e s i m i l a r t o t h e mechanism r e s p o n s i b l e f o r EXAFS [13]. For t h e 4d and 3d t r a n s i t i o n m e t a l s where c o r r e l a t i o n e f f e c t s a r e e x p e c t e d t o become i n c r e a s i n g l y i m p o r t a n t , a BIS s t u d y [14] h a s shown t h a t t h e computed d e n s i t i e s of empty s t a t e s a r e i n f a i r agreement w i t h t h e measured s p e c t r a .
ENERGY
Fig. 1 Comparison o f t h e BIS s p e c t r a o f Pd and Ag [ 5 ] w i t h a c a l c u l a t e d DOS of Pd [ I l l .
2. SIGNATURE OF THE LOCALIZATION IN OUTER LEVEL SPECTRA
I t is now i n t e r e s t i n g t o jump t o t h e extreme s i t u a t i o n met i n t h e l a n t h a n i d e s e r i e s where t h e 4f s h e l l is g r a d u a l l y f i l l e d . I n most c a s e s t h e s e s t a t e s have such a s m a l l h y b r i d i z a t i o n w i t h t h e band s t a t e s t h a t t h e y can be c o n s i d e r e d a s atomic l i k e . In o r d e r t o c a l c u l a t e t h e i r e x c i t a t i o n s
it
is s u f f i c i e n t t o c o n s i d e r t h e Hamiltonian of a c e l l c o n t a i n i n g o n l y one l a n t h a n i d e atom ( s i n g l e - i m p u r i t y model) :The f i r s t t e r m is t h e t o t a l e n e r g y o f a l l o c c u p i e d band s t a t e s ( i n d e x k), t h e s e c o n d term is t h e e n e r g y o f t h e f e l e c t r o n s ( i n d e x m) assumed t o have n o m u t u a l i n t e r a c t i o n and t h e t h i r d term i s t h e f i n t r a - s h e l l Coulomb c o r r e l a t i o n e n e r g y . The two l a s t t e r m s a c c o u n t f o r t h e e n e r g y o f o n e c o r e l e v e l ( i n d e x c ) . They remain c o n s t a n t (nc
=
1) i n a l l o u t e r l e v e l e x c i t a t i o n s and c a n be i g n o r e d f o r t h e moment. Even i n t h i s a t o i n i c approach o f t h e f s t a t e s i n s o l i d s , t h e e n e r g y r e q u i r e d f o r any p o p u l a t i o n c h a n g e is d e e p l y i n f l u e n c e d by t h e band s t a t e s i n which t h e y a r e embedded. The e n e r g y p a r a m e t e r s~ f ,
E ~ ,U f f and U f c o f t h e H a m i l t o n i a n ( 1 ) i n c l u d e f u l l y t h e scre- e n i n g by t h e band s t a t e s and t h e y c a n b e c a l c u l a t e d w i t h i n t h e r e n o r m a l i z e d atom scheme [15-171. The Koopmans a p p r o x i m a t i o n is no l o n g e r v a l i d f o r t h e f e x c i t a t i o n s showing a d i s c r e t e s p e c t r u m which c a n n o t be deduced from a ground s t a t e c a l c u l a t i o n . F i g . 2 v i s u a l i z e s f o r XPS and BIS s p e c t r a o f a m e t a l l i c r a r e e a r t h t h e r e l a t i o n s h i p between t h e c o n t i n u o u s s p e c t r u m of t h e e x t e n d e d s t a t e s and t h e d i s c r e t e e n e r g i e s i n - v o l v e d i n t h e p o p u l a t i o n c h a n g e s o f t h e f s h e l l [18]. The t o t a l e n e r g y o f t h e ground s t a t e , assumed t o have t h e c o n f i g u r a t i o n f n , is most c o n v e n i e n t l y c h o s e n a s o r i g i n o f t h e e x c i t a t i o n s p e c t r u m a t EF. The d e c o m p o s i t i o n o f t h e p r o c e s s e s i n t o two s t e p s shows t h a t when t h e p o s i t i o n o f t h e f p e a k s a r e r e f e r e n c e d t o EF, A- a n d A+ a r e t h e minimum m a n y - e l e c t r o n e n e r g i e s r e q u i r e d t o modify t h e p o p u l a t i o n o f t h i s s h e l l , I n f a c t r e a l s p e c t r a d i s p l a y c o m p l i c a t e d m u l t i p l e t l e v e l s a c c o u n t i n g f o r t h e symmetry- a l l o w e d f i n a l s t a t e s . T h i s is a c l a s s i c a l a p p r o a c h o f a t o m i c p h y s i c s which h a s p r o v e n t o be v e r y u s e f u l and is commonly e x p l o i t e d [ I B ] . E x a c t l y t h e same f o r m a l i s m c a n b e u s e d f o r t h e a d d i t i o n and s u b t r a c t i o n o f an f e l e c t r o n . I f one c o n s i d e r s i n XPS t h eF i g . 2 S c h e m a t i c r e p r e s e n t a t i o n o f t h e XPS and BIS p r o c e s s e s i n L a n t h a n i d e s [ 1 8 ] .
t r a n s i t i o n 4 f n + 4fn-1 f o r t h e occupied o r b i t a l s , i n BIS one can e x p r e s s a s i m i - l a r t r a n s i t i o n 4f14-n + 4f14-(,+I) f o r t h e h o l e s . In both c a s e s one is d e a l i n g w i t h t h e same
(LS)
i n i t i a l s t a t e and (L'S') f i n a l s t a t e s r e l a t e d by t h e same f r a c t i o - n a l p a r e n t a g e c o e f f i c i e n t s . This symmetry f o r n=
5 is i l l u s t r a t e d i n Fig. 3 by t h e s p e c t r a of Sm and Dy [ 1 9 ] . I n t h e two s p e c t r a t h e o r i g i n o f t h e energy s c a l e h a s been chosen a t t h e p o s i t i o n o f t h e lowest l e v e l and t h e l e n g t h s o f t h e v e r t i c a l b a r s a r e p r o p o r t i o n a l t o t h e squared c o e f f i c i e n t s of f r a c t i o n a l parentage. I n t h e BIS spectrum t h e d e t a i l s o f t h e l e v e l s a r e washed o u t by t h e s h o r t e r f i n a l s t a t e l i f e t i m e broade- ning t h a n i n XPS b u t t h e s i m i l a r i t y of t h e two s p e c t r a is s u f f i c i e n t l y s t r i k i n g t o demonstrate t h i s symmetry p r o p e r t y .5 4 3 2 1 0
ENERGY (eV)
Fig. 3 XPS spectrum o f Sm and BIS spectrum o f Dy d i s p l a y i n g t h e m i r r o r p r o p e r t y bet- ween c o r r e s p o n d i n g e l e c t r o n and h o l e c o n f i g u r a t i o n s 1191.
U n t i l now we have examined two c l e a r - c u t s i t u a t i o n s : i) I n t h e c a s e of extended band s t a t e s , t h e XPS and BIS s p e c t r a form a continuous f u n c t i o n a c r o s s t h e Fermi l e v e l . I t is c l o s e l y r e l a t e d t o t h e s i n g l e - p a r t i c l e d e n s i t y of t h e occupied and empty s t a t e s . i i ) For t h e p o p u l a t i o n v a r i a t i o n s induced by XPS and BIS i n a l o c a l i z e d and uncom- p l e t e f s h e l l , two d i s c r e t e s e t s of
F-'
and fW1 l e v e l s a r e observed w e l l sepa- r a t e d on both s i d e s o f EF. The c o n t r a s t between t h e s e two s i t u a t i o n s is i l l u s t r a t e d i n Fig. 4 [ 2 0 ] by t h e s p e c t r a of U and Nd m e t a l s which c o n t a i n t h e same number of oc- c u p i e d 5f and 4 f s t a t e s . The c r o s s - s e c t i o n f o r t h i s symmetry is much l a r g e r t h a n t h o s e f o r t h e band s t a t e s s o t h a t t h e s e s p e c t r a account mainly f o r f e x c i t a t i o n s . I n uranium t h e 5 f s t a t e s a r e a l r e a d y s t r o n g l y c o r r e l a t e d b u t t h e i r j o i n t XPS-BIS spec- trum has t h e appearance o f a c o n t i n u o u s band a c c o u n t i n g f o r 14 s t a t e s . In Nd t h e 5d and 6 s s t a t e s form a l s o a c o n t i n u o u s band a c r o s s EF but t h e two s e t s of d i s c r e t e 4 f 2 and 4 f 4 m u l t i p l e t s i n d i c a t e d by t h e v e r t i c a l b a r s a r e pushed away from EF. A d e l o c a l i z a t i o n w i t h i n t h i s 4 f s h e l l is i m p o s s i b l e s i n c ei t
would r e q u i r e t h e t r a n s f e r o f o n e e l e c t r o n from t h e f s h e l l o f one atom t o t h e f s h e l l of a n o t h e r atom. A s shown i n Fig. 2 , t h e minimum energy f o r such a t r a n s i t i o n is t h e Coulomb c o r r e l a t i o n energyU f f
=
A-+
A+ which i n t h e p a r t i c u l a r c a s e o f Nd c o r r e s p o n d s t o 6.5 eV, t h e e n e r g y s e p a r a t i o n o f t h e 'I,, and 3 ~ 4 terms i n t h e s p e c t r a o f F i g . 4. I t is beyond t h e s c o p e o f t h i s r e v i e w t o d i s c u s s t h e r e s o n a n c e phenomena o c c u r i n g a t t h e t h r e s h o l d i o n i z a - t i o n o f c o r e l e v e l s . For t e c h n i c a l r e a s o n s t h e s e Fano r e s o n a n c e s commonly u s e d i n p h o t o e m i s s i o n have been seldom o b s e r v e d i n i n v e r s e p h o t o e m i s s i o n [21].-8 -6 -4 - 2 0 2 4 6 8
I
>-
X P S'=
U)Z
z
..
- 8 -6 -4 - 2 0 2 4
6 8
ENERGY (e V )
Fig. 4 Combined XPS-€315 s p e c t r a o f U and Nd [20].
3 . THE DUALITY OF THE EXCITATIONS I N WEAKLY HYBRIDIZED f SYSTEMS
The i n t e r e s t i n g problem now is t o i n v e s t i g a t e t h e s i t u a t i o n on t h e b o r d e r - l i n e b e t - ween l o c a l i z a t i o n and d e l o c a l i z a t i o n . When t h e c o r r e l a t i o n among band s t a t e s o f a m e t a l is g r a d u a l l y i n c r e a s e d , t h e band w i d t h d e c r e a s e s and t h e d e n s i t y o f o c c u p i e d a n d u n o c c u p i e d s t a t e s a c c u m u l a t e s a r o u n d EF. Nhen g o i n g t o t h e l i m i t o f v e r y l a r g e c o r r e l a t i o n ( i . e . l o c a l i z a t i o n ) t h i s p i c t u r e would y i e l d an i n f i n i t e l y narrow band o f i n f i n i t e l y l a r g e i n t e n s i t y . T h i s is o b v i o u s l y n o t what is o b s e r v e d i n t h e example o f Nd ( F i g . 4). We w i l l show t h a t i n t h e i n t e r m e d i a t e s i t u a t i o n s , t h e s p e c t r a d i s p l a y s i m u l t a n e o u s l y f e a t u r e s a c c o u n t i n g f o r i o n i c l i k e and band l i k e e x c i t a t i o n s , i n o t h e r words a s u p e r p o s i t i o n o f t h e two t y p e s o f s p e c t r a shown i n F i g . 4.
S t a r t i n g from t h e l o c a l i z e d d e s c r i p t i o n o f an f - s t a t e , t h e f i r s t mechanism which c a n l e a d t o a d e l o c a l i z a t i o n is a n h y b r i d i z a t i o n w i t h t h e band s t a t e s . T h i s r e q u i r e s t o c o m p l e t e t h e H a m i l t o n i a n ( 1 ) w i t h t h e new.term
i n d u c i n g t h e mixing o f f and band s t a t e s . The e s s e n t i a l c o n s e q u e n c e o f t h i s new term is t h e f o r m a t i o n of many-body s t a t e s w i t h f c o u n t s which a r e no l o n g e r i n t e g r a l num-
b e r s s i n c e they i n v o l v e a t l e a s t two f c o n f i g u r a t i o n s . Gunnarsson and SchSnhammer (GS) have f i r s t developed t r a c t a b l e model c a l c u l a t i o n s [22,23] t o d e s c r i b e a l l spec- t r o s c o p i c d a t a w i t h i n t h i s s i n g l e - i m p u r i t y model. I n t h i s approach t h e h y b r i d i z a t i o n i s c h a r a c t e r i z e d by a s i n g l e a d j u s t a b l e parameter, t h e h y b r i d i z a t i o n s t r e n g t h A. I n t h e ground s t a t e o f t h e s o l i d , t h e energy l o w e r i n g r e s u l t i n g from t h e h y b r i d i z a t i o n i s g i v e n b y 6. T h i s mechanism r e q u i r i n g t o c r e a t e h o l e s i n t h e band can o n l y be e f f i - c i e n t i n t h e narrow energy r e g i o n o f t h e o r d e r o f 6 below EF. The s p e c t r a l func- t i o n s f o r XPS and BIS show t h a t around EF t h e s p e c t r a c o n t a i n a narrow peak (FWHM
-
6 ) 1241 which can be v i s u a l i z e d as r e f l e c t i n g t h e extended m e t a l l i c c h a r a c t e r t h a t the f s t a t e s have a c q u i r e d by h y b r i d i z a t i o n . A t l a r g e r energy s e p a r a t i o n s on b o t h s i d e s o f EF t h e s p e c t r a l i n t e n s i t y accounts f o r i n c r e a s i n g l y more i o n i c f i n a l s t a t e s w i t h f count m o d i f i c a t i o n s approaching
+
1. I n t h e l i m i t A + 0, t h e h e i g h t s of t h e peak near EF d i v e r g e s w h i l e i t s i n t e g r a t e d i n t e n s i t y vanishes. T h i s s p e c t r a l weight i s t r a n s f e r e d t o t h e more atomic l i k e f i n a l s t a t e s which g r a d u a l l y narrow and f i n a l l y a r e s i m p l y t h e l o c a l i z e d m u l t i p l e t s . T h i s model p r o v i d e s an a c c e s s i b l e des- c r i p t i o n o f t h e s p e c t r a e v o l u t i o n from band t o l o c a l i z e d m u l t i p l e t s .4. EXCITATION SPECTRA OF f STATES WEAKLY HYBRIDIZED WITH BAND STATES
The most c o n v i n c i n g v e r i f i c a t i o n s o f t h e p r e d i c t i o n s deduced from t h e s i n g l e - i h p u r i t y model have been o b t a i n e d i n Ce-systems [16,25-291 where the parameters s a t i s f y always t h e c o n d i t i o n A
<<
c f d e f i n i n g t h e Kondo regime. I n t h i s case t h e f p o p u l a t i o n n f i s s l i g h t l y s m a l l e r t h a n one. Fig. 5 shows two t y p i c a l j o i n t XPS-BIS s p e c t r a [30]. I n t h e case o f CeN, t h e h y b r i d i z a t i o n i s r a t h e r i m p o r t a n t (6=
15 meV, nf=
0.86) [ 3 1 ] and a very s p e c t a c u l a r peak o f low-energy e x c i t a t i o n s i s observed from b o t h s i d e s of EF ( n o t i c e t h e d i f f e r e n t i n t e n s i t y c a l i b r a t i o n o f t h e two spectra). I n t h e XPS spectrum t h e dominating emission s t r u c t u r e around 3 eV o r i g i n a t e s e s s e n t i a l l y from N 2p d e r i v e d s t a t e s , w h i l e the more atomic l i k e p a r t o f t h e 4 f e x c i t a t i o n spectrum can n o t be discerned. I n t h e BIS spectrum, t h e t o t a l f c o n t r i b u t i o n i s r o u g h l y a f a c t o r 14 l a r g e r t h a n i n t h e XPS spectrum and t h e broad peak between 3 and 8 eV can be a t - t r i b u t e d t o t h e unresolved 4 f 2 m u l t i p l e t s . A r a t h e r s i m i l a r s i t u a t i o n i s met i n t h e s p e c t r a o f CeSi2 b u t w i t h a s m a l l e r h y b r i d i z a t i o n s t r e n g t h ( 6=
3 meV, nf=
0.97) [32]. The consequence o f the decrease o f t h i s parameter can be most c l e a r l y reco- g n i z e d i n t h e BIS spectrum where t h e i n t e n s i t y o f t h e peak a t EF i s weak when com- pared w i t h t h e dominating s t r u c t u r e accounting f o r t h e atomic l i k e f i n a l s t a t e s .ENERGY (eV1
Fig. 5 Combined XPS-BIS s p e c t r a o f CeN and CeSiq [30].
The n a t u r e o f t h e y + a phase t r a n s i t i o n i n Ce m e t a l has been f o r many years a sub- j e c t o f controversy. I n photoemission s p e c t r a w i t h moderate r e s o l u t i o n , a two-peak f e a t u r e has revealed very e a r l y the t w o f o l d o r i g i n o f t h e observed f i n a l s t a t e s [ 3 3 ] b u t such s p e c t r a have n o t p r o v i d e d a very c o n c l u s i v e e x p l a n a t i o n o f the n a t u r e o f t h e t r a n s i t i o n . The use o f t h e s i n g l e - i m p u r i t y model ( G S model) has allowed us t o make a break-through i n t h i s puzzle. With a moderate r e s o l u t i o n t h e consequence o f t h e elec- t r o n i c s t r u c t u r e change concomitant t o t h e phase t r a n s i t i o n i s b e s t observed i n t h e BIS s p e c t r a [34] as shown i n Fig. 6. I n y-Ce, p r a c t i c a l l y no f s p e c t r a l weight i s ob- served i n t h e range o f low-energy e x c i t a t i o n s j u s t above EF. The whole i n t e n s i t y i s c o n c e n t r a t e d on t h e f2 m u l t i p l e t s ( v e r t i c a l bars) which f i t very w e l l t o t h e e x p e r i - mental shape when the l i f e t i m e broadening i s taken i n t o account. T h i s means t h a t t h e h y b r i d i z a t i o n i s very weak (6
=
5 meV) so t h a t t h e i n i t i a l s t a t e has a n e a r l y i n t e - g r a l f count ( n f = 0.98). I n t h e a-phase, an i m p o r t a n t f r a c t i o n o f t h e i n t e n s i t y i s t r a n s f e r e d t o t h e low-energy f i n a l s t a t e s w i t h dominant f1 c h a r a c t e r , i n d i c a t i n g an i n c r e a s e o f t h e h y b r i d i z a t i o n (6=
26 meV) which lowers t h e i n i t i a l f p o p u l a t i o n (nf=
0.88). T h i s q u a l i t a t i v e d i s c u s s i o n based on BIS s p e c t r a has been f u l l y con- f i r m e d by o t h e r s p e c t r o s c o p i c r e s u l t s , i n p a r t i c u l a r h i g h - r e s o l u t i o n UV-photoemission s p e c t r a [35].I 1 I I 1 1 I I
I
0 -1 -2 -3 -4 -5 -6 -7
- ENERGY
[eV)
Fig. 6 BIS s p e c t r a o f a-Ce and y-Ce [34].
The v a l i d i t y o f t h e s i n g l e - i m p u r i t y model i s n o t l i m i t e d t o t h e Kondo regime encoun- t e r e d i n Ce-systems b u t i t p r e d i c t s a l s o o t h e r s i t u a t i o n s r e s u l t i n g from d i f f e r e n t values o f t h e parameters. Simple examples o f t h e mixed-valent regime can be found i n Yb compounds. One can i n v o k e t h e m i r r o r p r o p e r t y o f r a r e e a r t h elements l o c a t e d sym- m e t r i c a l l y w i t h r e s p e c t t o t h e middle o f t h i s s e r i e s (see Fig. 3 and r e l a t e d discus- s i o n ) and s i m p l y use t h e model c a l c u l a t i o n performed f o r Ce. I n YbA13, Mossbauer ab- s o r p t i o n experiments have shown t h a t Yb has a temperature-independent f r a c t i o n a l va-
l e n c e o f about 2.7 [36]. T h i s p e c u l a r i t y can o n l y be e x p l a i n e d by a s t r o n g m i x i n g o f t h e 4 f 1 3 and 4 f 1 4 c o n f i g u r a t i o n s . The s p e c t r o s c o p i c measurements c o n f i r m t h i s des- c r i p t i o n and t h e i r a n a l y s i s y i e l d s t h e corresponding s i n g l e - i m p u r i t y parameters (371.
I n F i g . 7 t h e XPS spectrum shows two d i s t i n c t s e t s o f f i n a l s t a t e s : between EF and 2eV t h e two main peaks which o r i i n a t e from t h e 4f+P2 and 4 f & j 2 uncoupled s t a t e s and
1 '
between 5 eV and 11 eV t h e 4 f m u l t i p l e t s . At f i r s t s i g h t one may imagine t h a t an i n t e r p r e t a t i o n i n terms o f a s i m p l e s u p e r p o s i t i o n o f atomic l i k e f 1 3 and f12 f i n a l s t a t e s i s s a t i s f a c t o r y . However, h i g h r e s o l u t i o n measurements [37] show t h a t t h e f i r s t peak maximum i s l o c a t e d a t 240 meV and n o t a t EF where i t s h o u l d be i n an atomic model. The low-energy t a i l o f t h i s peak d i e s o f f s l o w l y , i t i s c u t by EF and i s observed as a weak f s i g n a l i n t h e B I S spectrum (Fig. 7). I n o r d e r t o s i m u l a t e these d a t a one has t o choose i n t h e c a l c u l a t i o n 6
=
240 meV and E T=
-200 meV ( n f= 13.5). These values are q u i t e d i f f e r e n t from those found i n Ce systems. For t h e c a l c u l a t i o n o f t h e XPS f s p e c t r a l f u n c t i o n a s i n g l e l e v e l broadened by a L o r e n t z i a n has been used t o s i m u l a t e t h e fl* m u l t i p l e t s . I n t h e BIS s p e c t r a l f u n c t i o n , t h e band assumed i n t h e c a l c u l a t i o n has been added so t h a t t h e f s i g n a l accounting f o r about 0.5 s t a t e s corresponds t o t h e i n t e n s i t y above t h e dashed l i n e .
These few examples have been b r i e f l y presented i n o r d e r t o i l l u s t r a t e t h a t BIS i s as i m p o r t a n t as XPS i n t h e study o f t h e narrow bands o f many-body s t a t e s r e s u l t i n g from a weak h y b r i d i z a t i o n . These two complementary techniques o f f e r an unique o p p o r t u n i t y t o i n v e s t i g a t e t h e f u l l spectrum o f these e x c i t a t i o n s . However, one should be aware o f t h e f a c t t h a t t h e v a l i d i t y o f t h e p r e s e n t d i s c u s s i o n i s o n l y q u a l i t a t i v e . We have n o t considered t h e i m p o r t a n t consequences o f t h e low-energy atomic l i k e e x c i t a t i o n s ( s p i n - o r b i t s p l i t t i n g , c r y s t a l f i e l d s p l i t t i n g ) , o f t h e f o r m a t i o n o f a magnetic o r d e r and o f t h e temperature. They g i v e r i s e t o more s u b t i l e m a n i f e s t a t i o n s r e l a t e d t o t h e unconventional low-energy p r o p e r t i e s o f these systems (heavy-fermions). T h e i r d i r e c t o b s e r v a t i o n [31, 32, 35, 37, 381 r e q u i r e s a h i g h e r r e s o l u t i o n (< 20 meV) which c o u l d be achieved u n t i l now o n l y w i t h UV-photoemission.
Calc.
-
V).-
C5
n
L m
- >- t,
10 5 EF EF 5
ENERGY [ e V I
A
1 1 1 1 1 1 1 1 v I 1 1 1 1 1 1 1 1 1
Y bAt3
x5
T = 2 5 K 3
. ,,
..7-
..<:-* BIS
la1
..A,*.-
.. . - . ..
5..,+-.-- (bl
..S.."&J~
xps - --. i. i :
A,
1 1 1 ~ 1 1 1 1 ~ 1 1 1A 1 ~~ 1 1 1 1 ~ 1 1 1 1
v
Fig. 7 XPS (a) and BIS (b) s p e c t r a o f YbA13. ( c ) and (d) model c a l c u l a t i o n s [37].
5. BIS SPECTRA OF f LEVELS VERSUS CORE LEVEL SPECTRA
Core l e v e l s p e c t r o s c o p i e s a r e a l s o commonly used t o probe t h e ground-state e l e c t r o n i c s t r u c t u r e of s o l i d s . I n p a r t i c u l a r , s p e c t r a i n the range o f t h e t h r e s h o l d i o n i z a t i o n o f core l e v e l s a r e o f t e n i n t e r p r e t e d i n terms o f unoccupied DOS modulated by t h e ma- t r i x elements and broadened by the c o r e h o l e l i f e t i m e . Besides t h e edge s i n g u l a r i t i e s [ 3 9 ] which can be very i m p o r t a n t , t h i s i n t e r p r e t a t i o n appears t o be c o r r e c t i n s i m p l e metals. For i n c r e a s i n g c o r r e l a t i o n among t h e o u t e r l e v e l s , t h e s i t u a t i o n becomes un- c e r t a i n and i n t h e extreme case o f weak h y b r i d i z a t i o n considered here, t h e s p e c t r a can no l o n g e r be i n t e r p r e t e d i n t u i t i v e l y as accounting f o r t h e unperturbed i n i t i a l s t a t e s .
5.1 The z e r o band-width approximation
A simple approach showing c l e a r l y t h e d i f f e r e n c e between a l l spectroscopies i s p r o v i - ded by t h e zero band-width l i m i t o f t h e GS model [40]. I t i m p l i e s t h a t a l l eigenva- l u e s ~k o f t h e band s t a t e s i n t h e H a m i l t o n i a n ( 1 ) have t h e same value which i s f o r convenience chosen t o be zero. The consequence i s t h a t t h e continuum o f b a s i s s t a t e s r e s u l t i n g from t h e band i s now reduced t o t h e number o f t h e r e l e v a n t c o n f i g u r a t i o n s b u t t h e h y b r i d i z a t i o n term ( 2 ) i s f u l l y taken i n t o account. F o r t h e p r e s e n t discus- s i o n i t i s s u f f i c i e n t t o consider t h e c o n f i g u r a t i o n s fO, f1 and f2 so t h a t t h e eigen- values o f t h e H a m i l t o n i a n a r e t h e s o l u t i o n s o f an e q u a t i o n o f t h e t h i r d degree. The r e s u l t o f such a c a l c u l a t i o n f o r t y p i c a l parameters o f l i g h t r a r e e a r t h s i s shown i n F i g . 8 as a f u n c t i o n o f E F ( i n t h i s model A i s r e p l a c e d by V ) . The s t r a i g h t dashed l i n e s a r e t h e eigenvalues o f t h e u n h y b r i d i z e d H a m i l t o n i a n ( 1 ) and t h e f u l l l i n e s those o f t h e complete H a m i l t o n i a n (1)
+
(2). The lowest l i n e corresponds t o t h e ground s t a t e Eg and t h e d i s c r e t e e x c i t a t i o n energies a r e t h e d i f f e r e n c e s between a l l eigenvalues and Eg. I t i s a s i m p l e m a t t e r t o c a l c u l a t e t h e observed i n t e n s i t i e s by p r o j e c t i n g t h e s t a t e c r e a t e d suddenly i n t h e considered spectroscopy on t h e s e t o f f i n a l e i g e n s t a t e s [40]. An approach t o a l l types o f s p e c t r a can be o b t a i n e d from t h i s approximation b u t we s h a l l c o n c e n t r a t e on .%IS and on XPS, EELS and XAS c o r e l e v e l spectra. A s e r i e s o f systems d i s p l a y i n g a simple systematic behaviour w i l l be consi- dered.3d:
at=
E~+Uf,
+20 - Uff = 7 eV
w"
I
.-
W
"
.-
W
-
'D m-
r 1 I I I
-20
-10 0+I0
4 f :
Eflev)
F i g . 8 Energy eigenvalues c a l c u l a t e d i n t h e zero bandwidth l i m i t as a f u n c t i o n o f
E f f o r photoemission processes [40].
5.2 BIS s p e c t r a
I n t h e f i r s t row o f F i g . 9 a r e r e p r e s e n t e d t h e u n h y b r i d i z e d f e i g e n v a l u e s S ( E t ) o f t h e d i f f e r e n t s y s t e m s which a r e o r d e r e d v e r t i c a l l y i n s u c h a way t h a t t h e e n e r g y se- p a r a t i o n between t h e f1 and f 0 c o n f i g u r a t i o n s d e c r e a s e s g r a d u a l l y from 1 4 eV i n Ba t o 0.5 eV i n CeD2 [ 2 8 ] . I n t h e z e r o band-width l i m i t , t h e bands a l s o r e p r e s e n t e d i n t h e f i g u r e c a n b e s i m p l y assumed t o be c o n c e n t r a t e d on t h e f 0 l e v e l . Except i n Ce02, t h e c o n f i g u r a t i o n s a r e t o o f a r from e a c h o t h e r t o g i v e rise t o a s i z a b l e m i x i n g i n t h e g r o u n d s t a t e ( n f
-
O), e v e n f o r s i z a b l e h y b r i d i z a t i o n s t r e n g t h s A. T h i s s i t u a t i o n i s r e c o g n i z e d i n F i g . 8 by t h e f a c t t h a t f o r E+->
5 eV, t h e e i g e n v a l u e s a r e v e r c l o s e t o t h e i r a s y m t o t i c u n h y b r i d i z e d b e h a v i o u r . T h i s means t h a t an a t o m i c l i k e ft;
a p p r o a c h is a s a t i s f a c t o r y a p p r o x i m a t i o n o f t h e ground s t a t e . The BIS s p e c t r a a r e an- t i c i p a t e d t o show a p r a c t i c a l l y p u r e f1 l e v e l a n d t h e XPS s p e c t r a t o c o n t a i n no f i n - t e n s i t y . I n a g r e e m e n t w i t h t h i s p r e d i c t i o n , t h e BIS f s p e c t r a c a l c u l a t e d w i t h coup- l i n g and shown i n t h e s e c o n d row t r a c e f a i t h f u l l y t h e p o s i t i o n s o f t h e u n c o u p l e d f1 l e v e l s from Ba t o La. I n t h e e x p e r i m e n t a l BIS s p e c t r a t h e f s i g n a l s h a v e b e e n h a t c h e d i n o r d e r t o d i s t i n g u i s h them from band s t a t e s . The s p i n - o r b i t s p l i t t i n g i s t o o s m a l l t o be r e s o l v e d and s i n g l e f1 p e a k s a r e o b s e r v e d w i t h q u i t e d i f f e r e n t w i d t h s d e p e n d i n g o n t h e l i f e t i m e and i n s t r u m e n t a l b r o a d e n i n g s .
The s i t u a t i o n i n Ce02 is a l r e a d y v e r y d i f f e r e n t . With ~f = 0.5 eV, t h e f s t a t e s a r e s t r o n g l y mixed w i t h t h e v a l e n c e band s t a t e s . An f ground s t a t e p o p u l a t i o n nf
=
0.45 is found a n d t h e BIS f p e a k is pushed somewhat away from t h e uncoupled f 1 p o s i t i o n . A t h i g h e r e n e r g i e s t h e model c a l c u l a t i o n p r e d i c t s a weak f 2 s i g n a l which is h a r d l y d i s c e r n e d from c o n d u c t i o n band s t a t e s i n t h e e x p e r i m e n t . We have p r e s e n t e d t h e GS model f o r m e t a l s b u t it is a s i m p l e m a t t e r t o show t h a t it c a n be u s e d w i t h o u t res- t r i c t i o n s f o r i n s u l a t o r s (LaF3, La203, Ce02) p r o v i d e d t h a t t h e d e f i n i t i o n o f t h e pa- r a m e t e r s is c o n s i s t e n t w i t h t h e e n e r g y r e f e r e n c e o f t h e s p e c t r a [ 4 1 ] .5.3 XPS c o r e l e v e l s p e c t r a
I n o r d e r t o c a l c u l a t e t h e XPS e i g e n v a l u e s c o n t a i n i n g a c o r e h o l e , o n e h a s t o c h o o s e nc = 0 i n t h e H a m i l t o n i a n (1) s o t h a t a l l s o l u t i o n s a r e s h i f t e d by E,. F u r t h e r - more, i f c f is r e p l a c e d by e f
+
Uf,
t h e H a m i l t o n i a n is f o r m a l l y n o t m o d i f i e d a n di t
a c c o u n t s f o r t h e same number 'of o c c u p i e d o u t e r l e v e l s . With t h e s e e n e r g y t r a n s l a t i o n s one c a n t a k e a d v a n t a g e o f t h e same s o l u t i o n s a s t h o s e found w i t h o u t c o r e h o l e . T h i s means t h a t i n Fig. 8 o n e h a s t o u s e t h e t o p and r i g h t axes. The e s s e n t i a l d i f f e r e n c e is t h a t f o r a s y s t e m w i t h a ground s t a t e c h a r a c t e r i z e d by c f , t h e s e ex- c i t e d s t a t e s a r e f o u n d a l o n g a v e r t i c a l l i n e a tE C ~ =
E~+
U f,
where t h e l o w e s t s t a t e i s no l o n g e r t h e ground s t a t e . S i n c e U f,
t h e Coulomb e & r g y between an f e l e c - t r o n and a c o r e h o l e , is a l w a y s o f t h e g r d e r o f -10 eV i n t h e s e s y s t e m s , t h e si- t u a t i o n which is e n c o u n t e r e d is now d r a s t i c a l l y d i f f e r e n t . I n t h e f i r s t row o f t h e r i g h t p a n e l ( F i g . 9 ) a r e g i v e n t h e u n c o u p l e d f i n a l s t a t e p o s i t i o n s which c a n be found i n t h e e n e r g y r a n g e -12 eV<
E C<
4 eV o f F i g . 8. The i n t e n s i t y d i s t r i b u t i o n o f t h e s e l e v e l s o b t a i n e d by a GS c a l c u f a t i o n a r e d i s p l a y e d a s f u l l l i n e s i n t h e s e c o n d row. I n Ba, t h e i n i t i a l s t a t e m i x i n g is s o weak t h a t a n e a r l y p u r e f: component is found. P r o c e e d i n g t o LaF3 and La203, t h e f i n a l s t a t e h y b r i d i z a t i o n becomes i m p o r t a n t s o t h a t t h e two p e a k s c o n t a i n now a c o m p a r a b l e amount o f f g and f f c h a r a c t e r and t h e i r e n e r g y s e p a r a t i o n is l a r g e r t h a n f o r t h e u n c o u p l e d l e v e l s . I n La m e t a l , t h e c o u p l i n g s t r e n g t h is s o weak t h a t t h e d o m i n a t i n g peak and t h e s h o u l d e r a c c o u n t f o r n e a r l y p u r e f: and f k c o n f i g u r a t i o n s , r e s p e c t i v e l y . The s i t u a t i o n becomes more com- p l i c a t e d i n Ce02 which h a s a f r a c t i o n a l o c c u p a t i o n nf=
0.45 i n t h e ground s t a t e . The s h i f t by Ufc o f t h e uncoupled l e v e l s a l l o w s now an i m p o r t a n t m i x i n g o f f f and f g , g i v i n g r l s e t o a continuum w i t h two i n t e n s e maxima, w h i l e t h e f g component r e m a i n s p u r e a t h i h e r e n e r g i e s . From Ba t o La, t h e s e e x a m p l e s d e m o n s t r a t e t h a t t h ej . . .
p r a c t i c a l l y p u r e f i n l t i a l c o n f i g u r a t i o n r e v e a l e d by BIS c a n g i v e r i s e t o s t r o n g l y h y b r i d i z e d e x c i t a t i o n s i n XPS c o r e l e v e l s . F o r many y e a r s c o n v e n t i o n a l i n t e r p r e t a - t i o n s i g n o r i n g t h e s e mechanisms have o b s c u r e d t h e XPS s t u d i e s of s u c h s y s t e m s .
Fig. 9
3d CORE LEVEL EXCITATION SPECTRA MODEL CALCULATION I EXPERIMENT
I".I'"~I'"ENERGY ( eV 1
Excitbtion spectra of the outer and core levels of Ba, LaF3, La203, La and Ce02 calculated without coupling (s(E~)), with the GS-model and measured by XPS, BIS and EELS [28].5.4 EELS c o r e l e v e l s p e c t r a
The e l e c t r o n energy l o s s s p e c t r a (EELS) near t h e t h r e s h o l d energy o f t h e 3d c o r e l e v e l s a r e reproduced i n t h e l a s t row o f Fig. 9 and t h e i r c a l c u l a t e d s p e c t r a l func- t i o n s a r e represented as d o t t e d l i n e s t o g e t h e r w i t h t h e XPS model c a l c u l a t i o n s . F o r i n t e r m e d i a t e energies (1 keV) o f t h e i m p i n g i n g e l e c t r o n beam, t h e induced t r a n s i t i o n s do n o t obey s t r i c t s e l e c t i o n r u l e s b u t i t can be shown [42,43] t h a t t h e observed s p e c t r a account m a i n l y f o r t r a n s i t i o n s t o 4 f f i n a l s t a t e s . F o r m a l l y t h i s i s equiva- l e n t t o a BIS process b u t i n t h i s case t h e a d d i t i o n a l e l e c t r o n p o p u l a t i n g t h e o u t e r l e v e l s i s o r i g i n a t i n g from a c o r e s h e l l . For t h i s t y p e o f t r a n s i t i o n s , t h e f: s t a t e i s v e r y u n l i k e l y t o be r e a l i z e d so t h a t i t must be e l i m i n a t e d from t h e s e t o f b a s i s s t a t e s and eigenvalues d i f f e r e n t from those used f o r XPS (Fig. 8) a r e o b t a i n e d [40].
One has t o n o t i c e t h a t t h e same s i t u a t i o n occurs i n BIS f o r which we used a b u s i v e l y t h e r e p r e s e n t a t i o n o f F i g . 8. T h i s had no consequences s i n c e t h e h y b r i d i z a t i o n was n e g l i g i b l e . For EELS, t h e uncoupled f i n a l s t a t e s f i and f: can i n p r i n c i p l e mix b u t they have a t o o l a r g e energy s e p a r a t i o n and remain pure. Except i n Ce02, t h e EELS process y i e l d s i n t e n s i t y o n l y on t h e atomic l i k e
fk
f i n a l s t a t e which i s represented by a L o r e n t z i a n i n t h e model c a l c u l a t i o n . I n t h i s approach, t h e m u l t i p l e t s p l i t t i n g o f t h e 3d94f1 s t a t e r e v e a l e d by t h e measurements has n o t been taken i n t o account. I n o r d e r t o f a c i l i t a t e t h e comparison w i t h t h e c a l c u l a t i o n s , t h e c e n t e r o f g r a v i t y o f t h e m u l t i p l e t s i s i n d i c a t e d by an arrow which is a l s o drawn i n t h e XPS spectra.I n Ce02, t h e EELS spectrum corresponds t o a more complicated m i x i n g o f 3 d 9 4 f i and 3d94f$ f i n a l s t a t e m u l t i p l e t s . The X-ray a b s o r p t i o n s p e c t r a i n v o l v i n g t h e 3d o r 4d s h e l l correspond t o t h e same t y p e o f t r a n s i t i o n s b u t they l o o k q u i t e d i f f e r e n t s i n c e they show o n l y t h e d i p o l e allowed m u l t i p l e t terms [18].
5.5 LIII X-ray a b s o r p t i o n edges
On t h e b a s i s o f an i n t u i t i v e model, t h e LIII X-ray a b s o r p t i o n edges (2p3/2 + 5 d) have been promoted as a s t a n d a r d t o o l f o r d e t e r m i n i n g t h e valence i n r a r e e a r t h mate- r i a l s [44]. I t i s i n t e r e s t i n g t o c o n t r a s t t h i s approach w i t h the p r e d i c t i o n o f t h e s i n g l e - i m p u r i t y model. For a g i v e n photon energy, one p a r t i c u l a r a b s o r p t i o n process p o p u l a t i n g an extended 5d s t a t e o f energy E above the t h r e s h o l d can be considered i n f i r s t approximation as a photoemission process. I t leaves t h e s o l i d i n an e i g e n s t a t e o f energy E r e f e r e n c e d t o EF. I n an XPS process from t h i s c o r e l e v e l p o p u l a t i n g f r e e - e l e c t r o n s t a t e s , t h e p r o b a b i l i t y t o c r e a t e t h e same f i n a l s t a t e i s g i v e n by t h e i n t e n s i t y o f t h e XPS spectrum o r t h e c a l c u l a t e d s p e c t r a l f u n c t i o n
pgPS
(E). Since i n t h e t h r e s h o l d process t h e e x c i t e d e l e c t r o n occupies a s t a t e w i t h a pronounced band c h a r a c t e r , t h e i n f l u e n c e o f t h e DOS and o f t h e m a t r i x elements i s taken i n t o account by a f u n c t i o n w * ( E ) . T h i s s i t u a t i o n i s d e p i c t e d i n Fig. 10 showing t h a t t h e energy c o n s e r v a t i o n r e q u i r e s hv=
E+
E. The a b s o r p t i o n i s then t h e sum o f a l l a l l o w e d t r a n - s i t i o n s absorbing t h e photon o f energy hvW i t h i n t h i s formalism [28] an LIII edge i s s i m p l y an XPS c o r e l e v e l spectrum washed o u t by t h e c o n v o l u t i o n w i t h t h e f u n c t i o n w2(s). I t does n o t o f f e r any s t r a i g h t f o r w a r d p o s s i b i l i t y t o determine t h e valence. The agreement between experiment and t h e o r y f o r t h e s p e c t r a o f CeCo2 compared i n F i g . 10 [ 2 8 ] i s s u f f i c i e n t l y c o n v i n c i n g t o demon- s t r a t e t h a t t h i s rough model i s based on t h e c o r r e c t concepts. However, t h i s approach i g n o r e s t h e f a c t t h a t t h e e j e c t e d e l e c t r o n s have a very low energy and a r e l i k e l y t o occupy s t a t e s l o c a l l y m o d i f i e d by t h e presence o f the deep hole. A model has been proposed t o i n c l u d e t h i s e f f e c t which appears t o be p a r t i c u l a r l y i m p o r t a n t i n i n s u l a - t o r s [45].
As f o r a l l o t h e r c o r e l e v e l spectroscopies, t h e X-ray a b s o r p t i o n edges a r e deeply i n - f l u e n c e d by t h e Coulomb energy Ufc and i t i s much more hazardous and d i f f i c u l t t o e x t r a c t d i r e c t l y from them t h e ground s t a t e parameters t h a n from XPS and BIS s p e c t r a o f t h e o u t e r l e v e l s .
F i g . 10 Upper p a r t : Schematic r e p r e s e n t a t i o n o f a b s o r p t i o n processes a t c o r e e x c i - t a t i o n t h r e s h o l d energies. Lower p a r t : C a l c u l a t e d and measured LIII ab- s o r p t i o n edge f o r CeCop [28].
ACKNOWLEDGEMENTS
We acknowledge t h e f i n a n c i a l support o f t h e Swiss N a t i o n a l Science Foundation and t h e f r u i t f u l c o l l a b o r a t i o n o f E. Wuilloud, F. Patthey, J.-M. Imer and B. D e l l e y .
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