BOND LENGTH DETERMINATION FOR MULTICOMPONENT SYSTEMS - NEW OPPORTUNITIES IN EXAFS DATA ANALYSIS

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BOND LENGTH DETERMINATION FOR MULTICOMPONENT SYSTEMS - NEW

OPPORTUNITIES IN EXAFS DATA ANALYSIS

Y. Babanov, V. Shvetsov

To cite this version:

Y. Babanov, V. Shvetsov. BOND LENGTH DETERMINATION FOR MULTICOMPONENT SYS- TEMS - NEW OPPORTUNITIES IN EXAFS DATA ANALYSIS. Journal de Physique Colloques, 1986, 47 (C8), pp.C8-37-C8-42. �10.1051/jphyscol:1986805�. �jpa-00225989�

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BOND LENGTH DETERMINATION FOR MULTICOMPONENT SYSTEMS - ^NEW

OPPORTUNITIES IN EXAFS DATA ANALYSIS

Y.A. BABANOV and V.R. SHVETSOV

I n s t i t u t e o f Metal Physics, Academy o f Sciences o f the U S S R , U r a l S c i e n t i f i c Center, 620219 Sverdlovsk GSP-170, U . S . S . R .

Abstract

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^Anew method of determining p a r t i d interatomic d i s t a n c e s i n multicomponent systems from extended X-ray ab- s o r p t i o n f i n e s t r u c t u r e data i s presented. The method i s based upon the r e g u l a r procedure of solving a Fredholm i n t e g r a l equation of t h e f i r s t kind. The e f f e c t i v e n e s s of the method has been t e s t e d on model examples of two c l o s e l y spaced coor- d i n a t i o n spheres and c r y s t a l l i n e CuZr2. The p a r t i a l i n t e r a t o - mic d i s t a n c e s of amorphous Cug Z r 7 have been i n v e s t i g a t e d by EXUS using t h e synckcotron r a a i a t l o n of a MPP-4 s t o r s g e rjng. The s t r u c t u r a l r e s u l t s a r e compared with previous expe- rlmental and t h e o r e t i c a l i n v e s t i g a t i o n s .

E X U S spectroscopy is used t o probe t h e l o c a l atomic arrangement around a s p e c i f i c atomic s p e c i e s i n amorphous a l l o y s , c a t a l y s t s , b i o l o g i c a l molecules, s o l u t i o n s , etc./?/. The conventional procedure of e x t r a c t i n g s t r u c t u r a l information i n r e a l space i s based on Fou- r i e r transforming t h e normalized o s c i l l a t o r y p a r t of t h e X-ray ab- s o r p t i o n c o e f f i c i e n t . It i s well known that the peaks i n t h e Fourier transform axe s h i f t e d t o lower r from t h e p o s i t i o n s of t h e corresponding peaks i n t h e p a r t i a l r a d i a l d i s t r i b u t i o n f u n c t i o n (RDF). Be- s i d e s , Hayes e t a l . have shown t h e r e a l space c o n t r i b u t i o n of each s h e l l about a n excited atom t o be long-range and s h a r p l y o s c i l l a t o - ry, The c o n t r i b u t i o n s made by two c l o s e l y spaced s h e l l s i n t e r f e r e s t r o n g l y /2/. As a r e s u l t , f a l s e and unresolved peaks a r i s e . The me- thod proposea /3/ eliminates t h e aforementioned drawbacks i n the processing of experimental data.

The method i s based upon t h e regulariza-tion procedure of solving a l?redholm i n t e g r a l equation of t h e f i r s t k h d . Features p e c u l i a r t o the method axe high r e s o l u t i o n f o r c l o s e l y spaced coordination spheres and high accuracy i n t h e determination of interatomic d i s - tances.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986805

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C8-38 JOURNAL DE PHYSIQUE

According t o theory, the ESOS can be described a s a n e l e c t r o n d i f f r a c t i o n process where t h e e l e c t r o n source i s t h e absorbing atom. The outgoing photoelectron wave and a s m a l l p a r t of i t , which i s backscattered from t h e surrounding atom j , i n t e r f e r e with t h e e x c i t e d atom. This s c a t t e r i n g i s s p e c i f i e d by not only the atomic amplitude f a s is the case i n X-ray d i f f r a c t i o n , but a l s o the phase pj. j I n addition, the s c a t t e r i n g on t h e p o t e n t i a l of t h e cen- tral atom e x h i b i t s a phase s h i f t 2 S i . These s c a t t e r i n g characte- r i s t i c s a r e involved i n the i n t e g r a l operator Aij which, a s it a c t s on t h e p a r t i a l RDF gij, generates a c o n t r i b u t i o n of a s p e c i f i c atomic p a i r t o t h e normalized o s c i l l a t o s y p a s t x i . If n elements a r e present i n t h e system, t h e n

here k i s t h e wave number of a photoelectron, c t h e concentrati- on, 3\ t h e mean f r e e path of a photoelectron, p. j t h e mean atomic density, and v i j ( k ) = 2 Si(k)+ p j ( k ) t h e t o t a l phase s h i f t , which i s a pecular "mark" f o r a s p e c i f i c p a i r of atoms.

The present n o t e suggests using t h e dependence of t h e i n t e g r a l operator Aij on t h e atomic s c a t t e r i n g c h a r a c t e r i s t i c s tabulated i n /4/. To determine t h e p a r t i a l interatomic d i s t a n c e s , a r e g u l a r i z a t i - on method of solving ill-posed inverse problems /5/ i s applied. It allows t h e i n t e r f e r e n c e of t h e c o n t r i b u t i o n s by t h e c l o s e l y spaced s h e l l s t o be removed and t h e r e s o l v i n g a b i l i t y of t h e E g U S teem- que t o be improved d r a s t i c a l y .

'Phe Iikhonov f u n c t i o n a l f o r (1) has the form

min

( I I ~ = ~

^A^{i j}^h^{i j -}

uJI

^t

^f

^j.1 (djllhijl12 + pj(l&hijl12$ ⁹ where hij (r)=gi. ( r ) - l ,

t a k y i ( k ) / f , ( d a n d a amplitude, & j and p j

t h e functions ui( k) include experimental da- .symptotes Aid, f o ( k ) i s t h e mean s c a t t e r i n g

are r e g u l a r i z a t i o n parameters, and 11 2112 i s the square of t h e norm of t h e f u n c t i o n 2. S e t t i n g t h e first v a r i a - t i o n s of t h e f u n c t i o n a l (2) with r e s p e c t t o hij ( j = l ,

.. .

, n ) equal t o zero, w e o b t a i n n r e g u l a r i z e d i n t e g r a l equations, each of which is equivalent t o ( 1 )

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t e r s H . and

J + J y 6 i s t h e Kronecker d e l t a symbol. The i n t e g r a l o p e r a t o r ( A . . A . .+B.) i s p o s i t i v e and t h u s can be inverted. Its i n -

LJ ZJ 3

v e r s e i s denoted by Cij. Then (3) may be r e w r i t t e n a s

The essence of t h e procedure w i l l be c l e a r from a n example. Let a h y p o t h e t i c a l c r y s t a l l i n e Cu-Zr a l l o y have t h e Pollowing s h o r t - range order: Cu atoms a r e surrounded by Cu atoms spaced a d i s t a n c e of 2.50 apaxt and by Z r atoms (2.80 1 a p a r t ) , and Z r atoms a r e surrounded by Cu atoms (2.80 a p a r t ) and Z r a t o m (3.10 a p a s t ) . The p a r t i a l RDFs d e s c r i b i n g t h e atomic d i s t r i b u t i o n in t h e c a s e of thermal motion a r e presented i n Fig. l (curves 1 and 8). For t h e s e RDFs t h e s p e c t r a of Gu and Z r were calculated.

Eroblem Determine t h e f n t e r a t o m i c d i s - t a n c e s Cu-Cu and Cu-Zr from t h e x C u ( k ) data, and t h e Zr-Cu and Zr-Zr d i s t a n c e s from -the 7 Zr(k) data.

Having p r o p e r l y prepared t h e d a t a f o r t h e spectrum of Cu, we a c t on them by t h e inver- s e o p e r a t o r The r e s u l t of t h e solu- t i o n of t h e i n v e r s e problem is d e p i c t e d i n t h e Fig. 1 (curve 3). The p o s i t i o n of t h e f i r s t peak c o i n c i d e s p r a c t i c a l l y with t h e most probable Cu-.Cu d i s t a n c e ( curve 1 ).

Since t h e phase s h i f t ?U (k) f o r a Cu-Zr p a i r does not coxrespond t o t h e phase information of t h e o p e r a t o r Cm-Cu, t h e second peak on curve 3 i s s h i f t e d by 0.1 1 1. Applying t h e o p e r a t o r CCu_Zr t o t h e s a m e x C u ( k ) spec-

Pig. 1.

trum , we determine t h e Cu-Zr d i s t a n c e (curve 4). The r e s u l t s ob- t a i n e d by processing t h e x z r ( k ) s p e c t m ( curves 5 and 6 1 permit t h e proper sequence of c o o r d i n a t i o n spheres t o be determined unam- biguously. For t h i s purpose t h e Cu-Zr d i s t a n c e i s taken t o be t h e r e f e r e n c e d i s t a n c e , For comparison, t h e f i g u r e 1 p r e s e n t s Fourier transformation modules of t h e same model s p e c t r a of Cu ( curve 2 ) and Zr (curve 7).

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C8-40 JOURNAL DE PHYSIQUE

Another example

-

c r y s t a l l i n e and amorphous CuZr2.

The EXAPS experiments were performed a t Nuclear Physics I n s t i t u - t e (Novosibirsk) using the synchrotron r a d i a t i o n of t h e VEPP-4 sto- rage r i n g , K-spectra of Cu and Z r were recorded a t t h e room tempe- r a t u r e . The experimental conditions and t h e preliminary processing were described i n /6/r

The X-ray s c a t t e r i n g d a t a of Nevitt and Downey / 7 / f o r c r y s t a l - l i n e & Z r 2 ( a t e t r a g o n a l phase ) have been used t o generate the (k) s p e c t r a with the phase s h i f t s and backscattering f a c t o r s tabulated i n /4/ f o r pure copper and pure zirconium. Pig. 2 p r e s e n t s experimental ( d o t s ) and c a l c u l a t e d ( s o l i d l i n e ) 3( (k) f o r Cu K edge (curves 1 ) and f o r Z r K edge (curves 2). Be must n o t i c e some d i s - crepancy between t h e c a l c u l a t e d and experimental data. The r e s u l t s of t h e s o l u t i o n of t h e inverse problem f o r experimental s p e c t r a a r e depicted i n t h e Pig. 3 ( curves 1 and 2 f o r Cu spectrum, curves 3 and 4 f o r Z r spectrum). The interatomic d i s t a n c e s obtained from X- r a y d i f f r a c t i o n a r e shown by dash l i n e s . Our r e s u l t s a r e compared with t h e experimental data from l i t e r a t u r e (Table 1). The values of

interatomic d i s t a n c e s f o r c r y s t a l l i n e h Z r 2 agree with X-ray crys- t a l l o g r a p h i c data.

Pig. 2

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l i a b l y . All r e s u l t s may be devided i n t o two groups: l ) t h e statis- t i c a l d i s t r i b u t i o n of both kinds of atoms; 2 ) t h e strong chemical i n t e r a c t i o n a Cu atom with a Z r atom. In t h e l a s t case the atomic distance between u n l i k e atoms i s smaller than t h e mean value of the d i s t a n c e s between l i k e atoms.

Fig. 4 p r e s e n t s experimental (h) of amorphous C U ~ f o r Cu ~ Z ~ ~ ~ K edge (curve 1) and f o r Z r K edge ( curve 2 ). P a r t i a l RDPs a r e

shown i n Fig. 5 ( curves l and 2 f o r Cu edge, curves 3 and 4 f o r Z r edge ). !Phe dash l i n e s correspond t o t h e most probable interatomic distances, whose numerical values a r e given in Table 2,

Swnmarizing the experimental r e s u l t s we can s t a t e :

1) The Cu-Cu d i s t a n c e is much s h o r t e r than the c r y s t a l d i s t a n c e and i s absent i n t h e corresponding & Z r 2 c r y s t a l l i n e s t r u c t u r e . 2) The Cu-Zr d i s t a n c e i s p r a c t i c a l l y t h e same a s i n c-CuZr2.

3 ) The interatomic d i s t a n c e s agree with t h e Goldschmidt r a d i i . Thus, the proposed method i s characterized by high r e s o l u t i o n f o r c l o s e l y spaced coordination spheres and by high accuracy i n determining interatomic distances. In t h i s context t h e method has v a s t prospects of a p p l i c a t i o a i n s t r u c t u r a l i n v e s t i g a t i o n s conven- t i o n a l f o r t h e EXAFSo

Fig. 4

, , ^C..^.^.⁴^.^.^...^,..^,^{. . . .}^{, , -}^.^...⁶ ^.^..,^.J'^.ⁱ_K^.^..--⁸

_6')

,

^-

^, ^,

^-

¹⁰

;

^,¹²

,l

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CS-42 JOURNAL DE PHYSIQUE

The a u t h o r s wish t o thank S.V.Vonsovskii, V.V.Vwin, A.L.Ageev, N. V. Ekshov f o r h e l p f u l discussions and G.N.Ku1 ipanov, M.A. Sheromov, N. V. Bausck f o r supporting t h e EXXE'S experiments.

REFERENCES

/ l / P.A.Lee, P.H.Citrin, P.E.Eisenberger and B.M.Kincaid, Rev.Mod.

Phys. , 52 (1981 1 769.

/2/ T.N.Hayes, P.W.Sen and S.H.Hunter, J.Phys.C, 2 (1976) 4357.

/3/ Yu.A.Babanov and V.R. Shvetsov, phys.stat.so1. (B), lJl(1985) K1 /4/ ^B.-K. ICeo and P.A. Lee, J.Arner. Chem. Soc., 101 ( 1979) 2815.

/5/ A.N. Tikhonov and V. Ya. Arsenin, Solution of 111-Posed Problems, W,N.Winston and Sons, Washington (D. C. ) and John IYLlley and Sons, New York / Toronto / London 1977.

/6/ N.V.Ershov, Yu.A.Babanov and V.R. Galakhov, phys, stat. sol. (b) ,

117 (1983) 749.

/7/ M.V.Nevitt and J.W.Downey, AIME Trans., 224 (1962) 195.

/ 8 / A. Sadoo, Y. Calvayrac , A. Q u i t y , M. Harmelin and A.M.Planlr, J.Non-Cryst.So1. , (1984) 109.

/9/ R.Harris and L. J.Lewis, Phys.Rev., (1982) 4997.

/ l Q/ R.Haensel, P. Rabe , ^G.Tolkiehn and A.Werner, Proc.NAT0 Adv. Stu- dy I n s t i t u t e , Liquid and Amorphous Metals, ed. E,Lusher ( Rei- d e l , Dordrecht, 1981) 467.

/ l l/ P.Lamparter, S. Steeb and E. G r u l a t h , Z.NaturPorsch. ,S (1983) 121 0.

'Pable 1 - Interatomic d i s t a n c e s of t h e c r y s t a l l i n e CuZr2 (8).

---"---m---

&-Zr Cu-Cu Zr-Cu Z r - Z r Z r - Z r References

...

X-ray d i f f r a c t i o n 2.896 3.220 2.896 3.039 3.220 /7/

2.90 3.23 2.89 3.04 3.23 model*

MMS _2.89 _3.22 **

2.89 3.04 3.23 exp.

2.87 3.16 2.85 3.09 3.20 /a/

...

e r r o r : * -TO.OI X , ** 70.02 X,

Table 2 - Interatomic d i s t a n c e s of t h e amorphous C U ~ (2). ~ Z ~ ~ ~

...

Cu-Cu Cu-Zr Zr-Cu Z r - Z r Alloy References

...

computer model 2.54 2.71 2.71 3.12 Cu33Zr67 / 9 /

MUS 2.54 2.71 2.71

:::;

&33zY67 /8/

lc&iPs 2.47 2.74 2.74 3.14 cu46zr54 /lo/

n e u t r o n d i f f r a c t i o n 2.59 2.77 2.77 3.28 &57Zr43 /l?/

MU'S 2.53 2.86 2.86 3.26 &33zr67 t h i s work