A UNIFYING SCHEME OF INTERPRETATION OF X-RAY ABSORPTION SPECTRA BASED ON THE MULTIPLE SCATTERING THEORY

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X-RAY ABSORPTION SPECTRA BASED ON THE MULTIPLE SCATTERING THEORY

C. Natoli, M. Benfatto

To cite this version:

C. Natoli, M. Benfatto. A UNIFYING SCHEME OF INTERPRETATION OF X-RAY ABSORP- TION SPECTRA BASED ON THE MULTIPLE SCATTERING THEORY. Journal de Physique Colloques, 1986, 47 (C8), pp.C8-11-C8-23. �10.1051/jphyscol:1986802�. �jpa-00225986�

JOURNAL DE PHYSIQUE

C o l l o q u e C 8 , s u p p l 6 m e n t au n o 1 2 , Tome 47, d h c e m b r e 1 9 8 6

A UNIFYING SCHEME OF INTZRPRETATION OF X-RAY ABSORPTION SPECTRA BASED ON THE MULTIPLE SCATTERING THEORY

C.R. N A T O L I ' ( ~ ) and M . BEN FAT TO**(^)

'LURE (Laboratoire CNRS, CEA, MEN), Universite Paris-Sud, F-91405 Orsay, France

" ~ a b o r a t o i r e de Mineralogie-Cristallographie, Universite Paris V 1 et VII, 4, Place Jussieu, F-75252 Paris Cedex 05, France

On a n a l y s e l e s d i f f 6 r e n t . s schkmas t h k o r i q u e s p e r m e t t a n t d e c a l c u l e r l e s s p e c t r e s d ' a b s o r p t i o n d e r a y o n s X 3 p a r t i r d e s couches p r o f o n d e s e t on montre q u ' i l s s o n t B q u i v a l e n t s . Le c a d r e u n i f i a n t e s t donnk p a r l a t h k o r i e d e l a d i f f u s i o n m u l t i p l e . En p a r t i c u l i e r on montre que La f o r m u l a t i o n b a s k e s u r les f o n c t i o n s d e Green permet, s o u s c e r t a i n s c o n d i t i o n s , d 1 6 c r i r e l e c o e f f i c i e n t d l a b s o r p t i o n comme une s 6 r i e de t e r m e s q u i o n t une s i g n i f i c a t i o n physique d i r e c t e . On d i s c u t e l e s c o n d i t i o n s s o u s l e s q u e l l e s l e d6veloppement e s t p o s s i b l e e t e n consequence o n propose un schkma d1 i n t e r p r g t a t i o n d e s s p e c t r e s d l a b s o r p t i o n ?I l a f o i s u n i f i 6 e t coh6rent. On donne e n f i n une formule approchke d e s p r o p a g a t e u r s q u i permet l e c a l c u l r a p i d e du terme g 6 n 6 r a l e d l o r d r e n d e l a s 6 r i e .

ABSTRACT

The v a r i o u s schemes f o r c a l c u l a t i n g i n n e r s h e l l s X-ray a b s o r p t i o n s p e c t r a a r e reviewed and shown t o b e m u t u a l l y e q u i v a l e n t . The u n i f y i n g framework is provided by t h e m u l t i p l e s c a t t e r i n g (MS) t h e o r y . I n p a r t i c u l a r t h e f o r m u l a t i o n based on t h e Green' s f u n c t i o n approach a l l o w s one under c e r t a i n c o n d i t i o n s t o w r i t e t h e a b s o r p t i o n c o e f f i c i e n t a s a sum o f a n i n f i n i t e number o f terms which have a d i r e c t p h y s i c a l meaning. The c o n d i t i o n s under which t h i s expansion is p o s s i b l e is d i s c u s s e d and a s a consequence a u n i f y i n g scheme o f i n t e r p r e t a t i o n o f X-ray a b s o r p t i o n s p e c t r a is proposed. F i n a l l y an approximate formula f o r t h e r a p i d e v a l u a t i o n of t h e n - t h o r d e r t e r m o f t h e expansion is given.

INTRODUCTION

S t a r t i n g from t h e g e n e r a l e x p r e s s i o n f o r t h e X-ray a b s o r p t i o n c r o s s - s e c t i o n o f a c l u s t e r o f atoms

where E is t h e photon e n e r g y , i t s p o l a r i z a t i o n v e r s o r and 6 is t h e d i p o l e t r a n s i t i o n o p e r a t o r ( a t o m i c u n i t s a r e used t h r o u g h o u t ) , t h e r e a r e b a s i c a l l y t h r e e d i f f e r e n t a p p r o a c h e s f o r e v a l u a t i n g t h i s q u a n t i t y :

("permanent address : L.N.F. dell' I.N.F.N., E.P. 13, 1-00044 Frascati (Italy)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986802

1

-

E = E

-

0

s t u d y ( [ l , 21).

+ + - + +

2 - The Green1 s f u n c t i o n approach, whereby o(E)-E I m (ginl p.D G p.D

I

3 - The band s t r u c t u r e approach f o r p e r i o d i c systems, where t h e s c a t t e r i n g s t a t e s a r e r e p l a c e s by Bloch s t a t e s s o t h a t t h e sum o v e r t h e f i n a l p h o t o e l e c t r o n s t a t e s becomes a n i n t e g r a l over t h e a p p r o p r i a t e B r i l l o u i n zone [41.

We s h a l l show i n t h e f o l l o w i n g t h a t i n t h e framework o f t h e m u l t i p l e s c a t t e r i n g (MS) t h e o r y a l l t h r e e a p p r o a c h e s a r e n u m e r i c a l l y e q u i v a l e n t . Only t h e i r language i g d i f f e r e n t , a c c o r d i n g t o t h e d i f f e r e n t p o i n t s o f view t a k e n t o d e s c r i b e t h e p h o t o a b s o r p t i o n p r o c e s s .

It w i l l t u r n o u t however t h a t t h e e x p r e s s i o n f o r t h e a b s o r p t i o n c r o s s s e c t i o n o b t a i n e d by t h e Green1 S f u n c t i o n approach is t h e most s u i t a b l e f o r t a c k l i n g s t r u c t u r a l problems. I n p a r t i c u l a r we s h a l l show t h a t , under c e r t a i n c o n d i t i o n s , t h e i n n e r s h e l l X-ray a b s o r p t i o n s p e c t r o s c o p y p r o v i d e s a s t r a i g h t f o r w a r d , d i r e c t means f o r o b t a i n i n g s t r u c t u r a l i n f o r m a t i o n about h i g h e r o ~ d ' e r c o r r e l a t i o n f u n c t i o n s i n t h e systems under s t u d y . I n t h i s s e n s e g e o m e t r i c a l i n f o r m a t i o n c o n c e r n i n g bonding a n g l e s and p o s i t i o n a l c o r r e l a t i o n s around t h e a b s o r b i n g atom c a n come w i t h i n e x p e r i m e n t a l r e a c h . The f i e l d o f a p p l i c a t i o n t h a t opens up i n t h i s way is extremely r e a c h and it is now time t o e x p l o i t a l l t h e p o t e n t i a l i t y o f t h e t e c h n i q u e .

The s c a t t e r i n g approach

I n t h i s approach t h e sum o v e r t h e continuum o f t h e f i n a l s t a t e s is performed f i r s t . The energy c o n s e r v i n g d e l t a f u n c t i o n s e l e c t s one p a r t i c u l a r f i n a l s t a t e g;

normalized t o one s t a t e p e r Rydberg :

where 6- = (g+)* ( n e g l e c t i n g s p i n ) , i n o r d e r t o impose t h e p h y s i c a l boundary c o n d i t i o n s f o r t h e p h o t o a b s o r p t i o n p r o c e s s [ l ] and gin is a n i n n e r s h e l l c o r e s t a t e .

I t is u s e f u l t o t r e a t b o t h t h e a t o m i c c a s e and t h e c l u s t e r c a s e :

Assuming t h e atomic p o t e n t i a l t o b e o f t h e m u f f i n - t i n t y p e , t h e a n g u l a r momentum L=(l,m) is conserved i n t h e s c a t t e r i n g p r o c e s s . I n t h e e x t e r n a l r e g i o n , where V ( r ) = 0, t h e s o l u t i o n o f t h e Schrbdinger e q u a t i o n is :

+ 3

$ i ( r , E ) = ~ ~ ( f ) ^{+ i} tlHL(r) where

+ ^-t

J~(:) = jl(kr)YL(F), HL(r) = h ; ( k r ) ~ ~ ( ~ ) , N ~ ( ; ) = nl(kr)YL(F)

k = JE and j l ( x ) , n l ( x ) , h+ = j ( X ) + i n ( X ) a r e t h e usual Bessel, Neumann and Hankel

1 1 1 +

functions. J r e p r e s e n t s t h e incoming wave, H t h e s c a t t e r e d wave. This s o l u t i o n is

L L3

t o be matched smoothly t o t h e s o l u t i o n C I R L ( r ) = C R (r)yL(;) of t h e Schrodinger 1 1.

equation i n s i d e t h e muffin-tin sphere o f r a d i u s p , which 1s r e g u l a r a t t h e o r i g i n . One f i n d s , d e f i n i n g W [ f , g ] = f g r - g f r , where f r = - d d r f

s o t h a t

3 3

a ^SC ( E ) = 4n2 a ( E + I ~ ) I (RJ P.D I ^$in)l I tll '

q

where we have e x p l i c i t l y f a c t o r i z e d t h e d e n s i t y of t h e f i n a l s t a t e s k/n coming from t h e normalization t o one s t a t e per Rydberg. For s i m p l i c i t y we assume t h a t t h e d i p o l e o p e r a t o r 3 D s e l e c t s only one f i n a l s t a t e , a s f o r K-edge absorption. The g e n e r a l i z a t i o n only adds complication t o the formulas.

We assume again t h a t t h e p o t e n t i a l is of t h e muffin-tin type. I n t h i s case L is not conserved, so t h a t now one can d e s c r i b e asymptotically t h e p h y s i c a l s i t u a t i o n a s an incoming L p a r t i a l wave J _L (? ) , r e f e r r e d t o t h e c e n t e r of t h e c l u s t e r where t h e absorbing atom is l o c a t e d , p m s a s e t of outgoing waves having a l l 0 L values, emanating from each s i t e j l o c a t e d a t 8. with amplitude B ~ ( L )

J +

+ ³ + 3 + +

$L ( r . ~ ) = J,(:,) + i I B ~ ( L ) HL(rJ) ( P . = r

-

- - _{j L} ^{J J}

I n s i d e t h e muff i n - t i n sphere j , i n analogy with t h e atomic c a s e , t h e s o l u t i o n which matches smoothly with t h e e x t e r n a l s o l u t i o n is given by

- 3

I ^{B:(&) RJ} ( r . ) . Since t h e i n i t i a l s t a t e is confined a t s i t e o and assuming again

L ^{L J}

K-shell photoabsorption, we f i n d

. 3

A s in t h e atomic c a s e , R;(r) is t h e s o l u t i o n o f t h e Schrbdinger equation i n s i d e sphere j t h a t matches smoothly t o J ( r . ) c o t g 3 :6

-

L J L j

r a d i u s p and ^6; is t h e 1 wave phase s h i f t of t h e p o t e n t i a l i n s i d e sphere j. However j

s i n c e now t h e angular momentum is not conserved and we a r e c a l c u l a t i n g t o t a l c r o s s

s e c t i o n s , we have t o add up incoherently a l l amplitudes B;(L) squared r e l a t i n g t o d i f f e r e n t L incoming waves C21.

The s c a t t e r i n g amplitudes BJ (L) s a t i s f y t h e following e q u a t i o n s J -

where t: is t h e t-matrix of t h e atom l o c a t e d a t s i t e i , :G: is t h e amplitude of propagation o f a s p h e r i c a l wave o f angular momentum L emanating from s i t e i f o r a r r i v i n g a t s i t e j with angular momentum L' and J" is t h e amplitude o f t h e incoming wave J ~ ( : ~ ) when r e f e r r e d t o s i t e i. With t h e h e l p of t h e reexpansion theorem C51 one LL f i n d s -

L' + +

where CL ⁼ do Y ~ ( Q ) Y ; , (Cl)YL,,(~) a r e t h e Gaunt c o e f f i c i e n t s and f ^{= Ri} - R . . I t

i i i j J

is u s e f u l f o r the f u t u r e t o d e f i n e GLL, = 0.

Eq. ( 4 ) h a s a simple p h y s i c a l meaning. I t t e l l s t h a t t h e t o t a l L wave s c a t t e r e d amplitude a t s i t e i is t h e sum of t h e s c a t t e r e d wave due t o t h e i n c i d e n t

~ ~

6 ~ )

-

h e r e t o s i t e i with amplitude G::, and f i n a l l y g e t s c a t t e r e d a t s i t e i with amplitude tl. i

By introducing t h e m a t r i c e s i j

t

:

T a = (TaILLt = 6ij _LL'

i i o

and t h e v e c t o r s $(L) = BL (L), j ( L ) = JLL we can w r i t e Eq. ( 4 ) a s

+ + -

( I

-

The s c a t t e r i n g approach is u s e f u l i n discussing shape resonances. I n t h i s c a s e it happens t h a t only one s c a t t e r i n g amplitude B:(&) f o r a p a r t i c u l a r +L becomes b i g a t a C e r t a i n energy, a l l t h e o t h e r amplitudes with L ^{t +L} being n e g l i g i b l e . T h i s means t h a t t h e L wave incoming from i n f i n i t y ( i n a time reversed p i c t u r e ) can e a s i l y overcome t h e c e n t r i f u g a l b a r r i e r , p e n e t r a t e t h e c l u s t e r p o t e n t i a l and a t t a i n a -t-

s i z a b l e amplitude B ~ ( $ ) R ; ( ; ~ ) a t t h e atomic c o r e of t h e photoabsorber. An example is the l+ = 3 resonance i n diatomic molecules (NZ. 02) C61.

The Green' s f u n c t i o n approach

In t h i s approach one transforms Eq. ( 1 ) a s :

where ( E

-

(v2 ^{+ E} - V C:)) G+ , (: ? I ) = 6 , (:

where V(:) = 1 V.(:) is t h e c o l l e c t i o n o f t h e muff i n - t i n p o t e n t i a l s . S i n c e is a c o r e s t a t e l o c a l i z e d a t s i t e o. Eq. j ( 5 ) shows t h a t we need c a l c u l a t e t h e Green's f u n c t i o n only f o r r and r t i n s i d e t h e muffin-tin sphere l o c a t e d a t o.

The s o l u t i o n f o r G+ i n t h i s c a s e is [3] :

where, a s before, a t t h e muffin-tin r a d i u s p

+ + +

RL(r)

-

-

-

I n s e r t i o n of Eq. ( 7 ) i n t o Eq. ( 6 ) g i v e s

+ + + + + +

uGF (E) = 4 n ( E + I o ) a I m k {C(RL1 p.D ] llin)I2 tl

-

I

I

When t h e p o t e n t i a l is r e a l , R and SL a r e r e a l s o t h a t L

U GF ( E ) = 4n ( E + I ~ ) C X k [ ( R J $.S

I

Due t o t h e o p t i c a l theorem

( tll = I m tl Eq. ( 8 ) reduces t o Eq. (2)

b) Elustec-pas:

Again we quote t h e r e s u l t o f r e f C31 + + +

G =

-

T ~ E ~

^(?h) -

where now

With t h i s s o l u t i o n

where t h e s u p e r s c r i p t o i n q0 reminds t h a t t h e c o r e i n i t i a l s t a t e in is l o c a t e d a t s i t e

0.

Again f o r r e a l p o t e n t i a l o + + 0

o GF ( E ) = 4 ^W ( c ⁺ I o ) a k [(RL1 p.D I Uin)I2 I m [(I

-

Using Eq. ( 4 ) it is p o s s i b l e t o prove t h e g e n e r a l i z a t i o n of t h e ~ p t i c a l theorem v a l i d f o r t h e atomic case C71 :

-1 00

1 1 ~ : ( L ) I ~ = I ~ C ( I - T ~ G ) T a l L L

- L

which a l l o w s u s t o recover Eq. (3).

We s h a l l s e e t h a t Eq. ( 9 ) is very u s e f u l f o r analysing the photoabsorption c r o s s s e c t i o n i n terms o f m u l t i p l e s c a t t e r i n g events.

Band s t r u c t u r e approach

I n an i n f i n i t e r e g u l a r l a t t i c e ( f o r s i m p l i c i t y we assume a l l s i t e s t o be e q u i v a l e n t ) , the KKR method [8] w r i t e s t h e Bloch f u n c t i o n a s :

with t h e same d e f i n i t i o n of RL(;) a s above, n l a b e l l i n g t h e band indices.

P.o c o e f f i c i e n t s a (<) s a t i s f y t h e homogeneous equations L

i 6

where tl = e 1 s i n is t h e u s u a l 1 wave atomic t-matrix, common t o a l l g i t e s ,

s i n c e now t h e second sum is independent o f t h e i n i t i a l s i t e o.

A non t r i v i a l s o l u t i o n of Eq. (11) demands t h a t Det 1

I

-

I

which determines the band d i s p e r s i o n € = c n ( < ) . Correspondingly Eq. ( 1 1 ) provides n +

a L ( q ) . Using t h e expression ( 1 0 ) f o r t h e f i n a l s t a t e s wavefunctions. Eq. ( 1 ) g i v e s

where v is the volume of t h e u n i t p r i m i t i v e c e l l . I t is now a matter of labour t o show t h a t :

where now s i t e o is any s i t e i n t h e l a t t i c e . This l a s t r e l a t i o n e s t a b l i s h e s t h e sought equivalence of t h e band approach t o t h e o t h e r methods.

The multiple s c a t t e r i n g s e r i e s

For s i m p l i c i t y we have assumed up t o now a b s o r p t i o n from K-shell core s t a t e s . For unpolarized absorption t h e g e n e r a l i z a t i o n t o a n i n i t i a l c o r e s t a t e I of 1 angular momentum is s t r a i g h t f o r w a r d (9). For t h e atomic -absorotion we f i n d :

where

1;'

1;

having introduced the absorption c o e f f i c i e n t a ( € ) = nabo(E) and t h e d e n s i t y n of t h e photoabsorber i n t h e medium. ab

For a c l u s t e r , remembering Eq. ( g ) , we have :

where now a. 1 ( E ) i n d i c a t e s t h e atomic absorption c o e f f i c i e n t of t h e photoabsorber and

is a s t r u c t u p e f a c t o r c a r r y i n g the information about t h e environment. Notice t h a t , i f G=O (absence o f environment), then X 1 ^{( E )} = 1 .

The f a c t o r i z a t i o n between atomic absorption and s t r u c t u r e f a c t o r is p o s s i b l e only i f t h e p o t e n t i a l is r e a l . For a complex p o t e n t i a l t h e more general expression Eq. (9a) should be used, s i n c e now R~(:) and sL(;) a r e complex. The physical i n t e r p r e t a t i o n of the theory becomes more involved i n t h i s case. In t h e following we s h a l l only d i s c u s s t h e r e a l case.

A s it is, Eq. ( 1 3 ) is not very u s e f u l f o r g e t t i n g some p h y s i c a l i n s i g h t i n t o t h e photoabsorption process. However i f one can perform t h e matrix i n v e r s i o n by s e r i e s

then t h e p h y s i c a l meaning of t h e process becomes t r a n s p a r e n t . In t h i s c a s e

with

1 1 1

xn ( E ) = -

-

sinz6: m

ayd x0 1 ( E ) = 1 , X: ( E ) = 0 , s i n c e G is off-diagonal i n t h e s i t e indices. Clearly X,(€) r e p r e s e n t s t h e p a r t i a l c o n t r i b u t i o n of o r d e r n t o t h e photoabsorption c o e f f i c i e n t of t h e c l u s t e r under study, coming from a l l processes where t h e photoelectron emanating from t h e absorbing s i t e o is s c a t t e r e d n-l times by t h e surrounding atoms b e f o r e escaping t o f r e e space a f t e r r e t u r n i n g t o s i t e o. I n o t h e r words only closed p a t h s beginning from and ending t o t h e photoabsorbing s i t e a r e p o s s i b l e , This l a s t c o n d i t i o n is due t o t h e f a c t t h a t one is c a l c u l a t i n g t o t a l c r o s s s e c t i o n s and t h a t t h e i n i t i a l s t a t e is l o c a l i z e d a t s i t e o. I t is t h i s p e c u l i a r i t y t h a t e n t a i l s t h e s i t e s p e c i f i c i t y of t h e X-ray absorption spectroscopy and makes it a unique t o o l f o r studying s t r u c t u r a l problems and f o r probing 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 i n condensed m a t e r i a l s . I n photoelectron d i f f r a c t i o n where t h i s c o n d i t i o n is n o t operating, t h e i n t e r p r e t a t i o n o f t h e experimental d a t a becomes more complicated.

The development i n Eq. (15) is nothing e l s e t h a t t h e f a m i l i a r MS expansion with s p h e r i c a l wave propagators. For example, using Eq. (16) and Eq. (5). one f i n d s

i z o j * i 1'1" 1"

' ^{l W l * "}

$1 If i:'

where we have introduced 3-j and 6-j symbols a s defined i n t h e l i t e r a t u r e [ l 0 1 and the ltreducedn Hankel f u n c t i o n < ( p ) :

I t is p o s s i b l e t o w r i t e down more cumbersome e x p r e s s i o n s f o r t h e h i g h e r o r d e r t e r m s x n ( c ) (103) u s i n g t h e (3n-3)-j symbols. However 1 their p r a c t i c a l u ? e f u l n e s s d e c r e a s e s w i t h i n c r e a s i n g o r d e r . I t is much e a s i e r t o g e n e r a t e them by u s i n g a MS program t h a t a l r e a d y c a l c u l a t e s t h e m a t r i x (I-T G) and c a n perform t h e m a t r i x i n v e r s i o n e i t h e r e x a c t l y o r v i a t h e s e r i e s expansion Eq. ( 1 4 ) . For a p p l i c a t i o n t o d a t a a n a l y s i s we wish t o remark t h a t t h e f u n c t i o n a l e x p r e s s i o n o f t h e q u a n t i t i e s x n ( f ) 1 is q u i t e s i m p l e , d e s p i t e t h e complexity o f t h e i r d e f i n i t i o f i . I n f a c t a l i t t l e r e f l e c t i o n shows t h a t :

1 1 P

x n ( E ) = I A n ( k , R ") s i n ( k

P- ^{i j}

where t h e sum is o i L r a l l p o s s i b l e p a t h s pn o f o r d e r n d e f i n e d above and Fltot is t h e P. n

c o r r e s p o n d i n g p a t h l e n g t h . T h i s form f o l l o w s from t h e f a c t t h a t each G::, c a r r i e s a f a c t o r e i k R i j independent o f L , L' , c o n t a i n e d i n the Hankel f u n c t i o n ( s e e Eq. ( 5 ) ) t h a t c a n be f a c t o r i z e d . By d e f i n i n g a new m a t r i x

and p u t t i n g

1 Pn 1 I

An ( k , R . .) exp {i(,(k,

1J

Z Z Z ...

^Z

m j,*o j.*j, jnm1 L1..L,,.l

where t h e i n d i c e s jk r u n o v e r t h e p a r t i c u l a r p a t h pn, we a r r i v e a t t h e e x p r e s s i o n n-l

(181, with R t o t = 1 R .

.

s e r i e s converges, one c a n always f i t a n e x p e r i m e n t a l spectrum w i t h a s e r i e s o f EXAFS l i k e f u n c t i o n s .

I t is o b v i o u s l y o f p r a c t i c a l importance t o f i n d approximate e x p r e s s i o n s f o r t h e SW p r o p a g a t o r s :,G: t h a t would a l l o w a r a p i d computation o f t h e a m p l i t u d e and phase f u n c t i o n s d e f i n e d i n Eq. (19). We have found t h a t t h e s i m p l e approximation

g e n e r a l l y r e p r o d u c e s q u i t e w e l l t h e e x a c t EXAFS x2( E ) term 1 ( s i n g l e - s c a t t e r i n g ) b o t h i n a m p l i t u d e and phase, b u t f a i l s t o r e p r o d u c e (sometimes by a f a c t o r o f two) t h e a m p l i t u d e of t h e e x a c t X 1 ( E ) term ( d o u b l e s c a t t e r i n g ) . F i g s . 1 and 2 i l l u s t r a t e t h i s comparison i n t h e c a s e of Mn04 t e t r a h e d r a l c l u s t e r . The u s u a l 3 FW approximation is o b t a i n e d by p u t t i n g

fill

t h e h i g h e s t e n e r g i e s . The r e a s o n is t h a t t h e phase c o r r e c t i o n goes l i k e cr11,/(29) - ^lmax >> 1 , s i n c e p = kR

-

1

-

ENERGY(eV1 ENERGY leV1

F i g u r e 1 : Exact x2 1 s i g n a l and phase F i g u r e 2 : Same a s Fig. 1 f o r x3 ¹ s i g n a l and phase f u n c t i o n 4 ¹

f u n c t i o n ( d o t t e d c u r v e s ) f o r MnO,, 3'

c l u s t e r , compared with S.W. approxi- mation (Eq. 20) ( f u l l l i n e s ) and P.W.

approximation (dot-dashed l i n e s ) .

The importance o f , b e i n g a b l e t o d e t e c t t h e X 1 ( E ) s i g n a l s i n experimental d a t a comes from t h e f a c t t h a t they provide i n f o r m a t i o n about t h e n-th n 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 g (Slo..

.

1 n

q u a n t i t y < X ( E ) > , where t h e b r a c k e t s i n d i c a t e t h e c o n f i g u r a t i o n a l average with r e s p e c t t o t h e d i s t r i b u t i o n of t h e p o s i t i o n s d. around t h e r e f e r e n c e ceriter

so

( p h o t o a b s o r b e r ) . I n o t h e r words

m n-l

X E ) = l ⁺

I 1

...

That one can a c t u a l l y d e t e c t terms o t h e r than < x 2 ( c ) > i n X-ray a b s o r p t i o n 1 s p e c t r a , h a s been proved p o s s i b l e i n some p a r t i c u l a r c a s e s [121. The r e a l c h a l l e n g e

is t o deconvolute Eq. (21) t o o b t a i n t h e f u n c t i o n s gn. Future e f f o r t should be put i n t o t h i s kind of a n a l y s i s .

The question o f convergence of t h e MS s e r i e s : d i s c u s s i o n and conclusions The i n t e r p r e t a t i o n o f t h e X-ray a b s o r p t i o n s p e c t r a i n terms o f MS pathways of t h e photoelectron i n t h e f i n a l s t a t e is meaningful1 only i f t h e r e is numerical equivalence between t h e two s i d e s of Eq. ( 1 4 ) . This implies t h a t t h e expansion on t h e r.h.s. must converge t o t h e 1.h.s. r e l a t i v e t o some convergence c r i t e r i u m . From matrix theory one knows t h a t a b s o l u t e convergence ( r e l a t i v e t o a s u i t a b l y defined matrix norm) is ensured i f p(TaG) < 1 .

This c r i t e r i u m is extremely u s e f u l s i n c e a b s o l u t e convergence e n t a i l s t h e property t h a t terms o f order n i n t h e s e r i e s higher than a c e r t a i n no (which can be very low in favorable c a s e s ) do not c o n t r i b u t e appreciably t o t h e sum. Now p(TaG) is a continuous f u n c t i o n of t h e . photoelectron wave number k = JE, which goes t o zero a s k goes t o i n f t n i t y ( s i n c e

I til

The implication of t h e above c o n s i d e r a t i o n s a r e immediate. A t extremly high e n e r g i e s , where

I

-

e n e r g i e s , where s t i l l

I til

3 i

happen depending on t h e behavior of t h e phase s h i f t s 61 and t h e photoelectron damping and t h e i r i n t e r p l a y . The s p e c t r a l r a d i u s p(TaG) may continue t o r i s e , a s t h e energy approaches t h e edge from above, s o a s t o reach one o r s t a y very near t o i t (normal s i t u a t i o n ) . In such a c a s e very many p a t h s c o n t r i b u t e t o t h e shape of t h e absorption c o e f f i c i e n t o r an i n f i n i t e number o f them, depending on whether p(TaG) is l e s s o r g r e a t e r than one. This is the region of t h e shape resonances where t h e ~ ~ a t t e r i n g power of t h e environment is s t r o n g enough t h a t it can s c a t t e r t h e photoelectron many times. I t might be adequate t o c a l l i t f u l l m u l t i p l e s c a t t e r i n g (FMS) region. One h a s only a g l o b a l information i n t h i s case. However a r a t h e r unexpected s i t u a t i o n may a l s o occur. P(T,G) may s t a y near one a t some intermediate e n e r g i e s and then decrease a s t h e energy decreases toward the edge. This s i t u a t i o n is encountered i n t h e Cupper K-edge spectrum, where i n t h e f i r s t 50 eV above t h e edge t h e EXAFS s i g n a l x ~ ( E ) 1 alone is capable of reproducing t h e experimental spectrum and t h e e x a c t band c a l c u l a t i o n [ l 31. However d e v i a t i o n s begin t o show up i n t h e energy range 50 i 200 eV [ l 1 I . This behavior is due t o t h e p e c u l i a r i t y of the r e l e v a n t atomic phase s h i f t s t h a t a r e small a t low energy and c r o s s *l2 (1 tll

-

-

Summarizing we can say t h a t a t l e a s t i n p r i n c i p l e any X-ray a b s o r p t i o n spectrum c o n t a i n s a l l t h r e e r e g i o n s mentioned above. Their o r d e r with i n c r e a s i n g energy and t h e i r energy e x t e n t a r e abviously system dependent. The only f e a t u r e common t o a l l systems is t h a t i n t h e l i m i t o f high energy t h e IMS s t r u c t u r e should continuously merge i n t o t h e SS r e g i o n and f i n a l l y reduce t o a pure atomic absorption.

The experimental s i t u a t i o n p r e s e n t s a d d i t i o n a l complicating f a c t o r s some of which however have a s i m p l i f y i n g e f f e c t on t h e shape of t h e absorption spectrum, with a corresponding l o s s o f informational c o n t e n t . I t is c l e a r t h a t t h e f i n i t e c o r e h o l e l i f e t i m e , t h e l i m i t e d experimental r e s o l u t i o n , t h e damping o f t h e photoelectron i n t h e f i n a l s t a t e ( e x t r i n s i c l o s s e s ) , t h e thermal and c o n f i g u r a t i o n a l d i s o r d e r , when p r e s e n t , a l l conjure up t o reduce t h e s i z e of p(TaG) a t such a p o i n t t h a t sometimes only the SS term s u r v i v e s a s t h e dominant s i g n a l . There a r e a l r e a d y i n d i c a t i o n s t h a t i n some c r y s t a l l i n e m a t e r i a l s ( S i , Al) l i f e t i m e e f f e c t s a l o n e a r e s u f f i c i e n t t o make t h e s e r i e s convergent i n t h e whole energy range except perhaps 10-15 eV near t h e edge C141. The use o f t h e Fourier transform technique i n following t h e o r g a n i z a t i o n of c r y s t a l l i n e order with annealing temperature i n amorphous t h i n f i l m s of Ge grown on a s u b s t r a t e f i n d s i t s r a t i o n a l e i n t h i s kind of c o n s i d e r a t i o n s [15]. On t h e o t h e r hand shake-up and shake-off processes of i n t r i n s i c o r i g i n tend t o add f e a t u r e s t o t h e spectrum t h a t modify t h e expected one e l e c t r o n shape. I n t h i s c a s e t h e a n a l y s i s i n terms of MS p a t h s should be done a f t e r t h e removal of t h e s e e x t r a f e a t u r e s . We have found a n example of t h i s s i t u a t i o n i n analysing t h e Mn04 c l u s t e r [12].

I n any c a s e a c a r e f u l t h e o r e t i c a l assessment o f a l l these e f f e c t s is h i g h l y d e s i r a b l e and work is i n progress. I n p a r t i c u l a r t h e a p r i o r i g a r a n t e e t h a t t h e MS s e r i e s is convergent g i v e s confidence t h a t one can parametrize t h e experimental d a t a by a s e r i e s of f u n c t i o n s of t h e t y p e shown i n Eq. (18) with a well defined expression f o r A: and @n. 1 This p o i n t is e s s e n t i a l i f one wish t o address t h e problem of t h e determination of t h e g n ( i i ) ' S f o r n>2. Otherwise a l t e r n a t i v e ways f o r analysing photoabsorption d a t a must be devised. 1.l

Acknowledgments

We g r a t e f u l l y acknowledge t h e h o s p i t a l i t y of LURE-ORSAY and Lab. Min. Crist.

PARIS V 1 t h a t h a s allowed u s t o work i n a f r i e n d l y and s t i m u l a t i n g atmosphere.

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