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A NEW METHOD FOR THE DETERMINATION OF
THE TOTAL STRUCTURE FACTOR OF
AMORPHOUS MATERIALS BY
ENERGY-DISPERSIVE X-RAY DIFFRACTION
(VARIABLE λ-METHOD)
G. Fritsch, A. Lee, C. Wagner
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, supplkment au n012, Tome 46, d&cembre 1985 page C8-131
A NEW METHOD FOR T H E DETERMINATION OF T H E T O T A L STRUCTURE FACTOR
OF AMORPHOUS MATERIALS BY ENERGY-DISPERSIVE X-RAY DIFFRACTION
( V A R I A B L E A-METHOD)
G. ~ r i t s c h * , A . E . Lee and C . N . J . Wagner
Department of Material Science and Engineering, University of California, Los AngeZes, California 90024, U.S.A.
Resume
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Une n o u v e l l e methode e s t propos6e pour determiner l e f a c t e u r de s t r u c t u r e t o t a l de materiaux amorphes 2 l ' a i d e d'une technique d'analyse en energie des rayons X d i f f r a c t & (EDXD, methode de l a longueur d'onde v a r i a b l e ) Cette m6thode e s t fondPe s u r l ' e v a l u a t i o n d i r e c t e du s ~ e c t r e d ' e n e r g i e du faisceau p r i m a i r ea
p a r t i r des donnees de d i f f r a c t i o n sans a p p l i q u e r aucun procede i t e r a t i f . Les f a c t e u r s de s t r u c t u r e de p l u s i e u r s a l l i a g e s amorphes (9e T i Zr, N i Z r e t Ni Co Z r ) obtenus de c e t t e maniere sont comparesa
ceux f o u r n i s par l a methode classique de d i s p e r s i o n a n g u l a i r e (ADXD, methode du 20 v a r i a b l e ) .A b s t r a c t
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A new method was developed f o r t h e determination o f t h e t o t a l s t r u c t u r e f a c t o r o f amorphous m a t e r i a l s usina t h e energy-dispersive X-ray d i f f r a c t i o n technique (EDXD, v a r i a b l e A-method). I t i s based on t h e d i r e c t e v a l u a t i o n o f t h e eneroy spectrum o f t h e primary beam from t h e d i f f r a c t i o n data w i t h o u t a p p l y i n g i t e r a t i v e procedures. The t o t a l s t r u c t u r e f a c t o r f o r several amorphous alloys(BeTiZr, NiZr and NiCoZr) d e r i v e d i n t h i s way i s com- pared t o t h e one obtained from t h e anqular d i s p e r s i v e X-ray d i f f r a c t i o n tech- nique (ADXD, v a r i a b l e 28-method).I
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INTRODUCTIONThe s t a t i c s t r u c t u r e f a c t o r I ( q ) contains important i n f o r m a t i o n on t h e t o p o l o a i c a l and chemical c o r r e l a t i o n s i n a many p a r t i c l e system. This f a c t i s e s p e c i a l l y t r u e f o r a l l o y s where t h e p a r t i a l s t r u c t u r e f a c t o r s have t o be considered. The l a t t e r may be determined by several methods /I/, however, p r e c i s e measurements o f t h e d i f f r a c - t e d i n t e n s i t i e s a r e necessary i n order t o o b t a i n meaninaful r e s u l t s .
I ( ? ) can be evaluated by X-ray d i f f r a c t i o n as a f u n c t i o n - o f t h e l e n g t h o f t h e d i f - f r a c t i o n v e c t o r q = 4nsinB/h. This can be accomplished by v a r y i n g t h e s c a t t e r i n g angle 28 u s i n g monoenergetic r a d i a t i o n (A: wavelength o f t h e X-rays). Such a method i s c a l l e d angular d i s p e r s i v e X-ray d i f f r a c t i o n (ADXD) o r v a r i a b l e 28-method. Here
9
d e f i n e s t h e Bragg- and 20 t h e scattering-angle. Since t h e development o f Ge- and Si-X-ray d e t e c t o r s w i t h h i g h energy r e s o l u t i o n ( rt( 400 eV), a v a r i a t i o n o f t h e wave v e c t o r q i s a l s o p o s s i b l e by measuring t h e i n t e n s i t y as a f u n c t i o n o f t h e energy E a t several f i x e d angles 20. This procedure c o n s t i t u t e s t h e eneray d i s p e r s i v e X-Ray d i f f r a c t i o n (EDXD) o r t h e v a r i a b l e A-method, as t h e wave l e n g t h A i s v a r i e d i n o r - der t o change q.
I n both methods, t h e d i f f r a c t e d i n t e n s i t i e s have t o be c o r r e c t e d f o r absorption, po- l a r i s a t i o n , mu1 t i p l e and Compton-scattering. I n a d d i t i o n , t h e data must be norma- l i z e d w i t h respect t o the s c a t t e r i n a power p e r atom. These c o r r e c t i o n s a r e s t r a i g h t forward, however, i n o r d e r t o perform them p r o p e r l y , the energy spectrum o f t h e p r i - mary beam I o ( E ) must be known.
*permanent address: U n i v e r s i t a t der Bundeswehr M'unchen, Fak. f. BauV/I I,
0-8014
Neubi berg, \!-GermanyC8-132 JOURNAL
DE
PHYSIQUEI 1
-
THEORETICAL CONSIDERATIONSI n a f i r s t approximation the d i f f r a c t e d i n t e n s i t y I(E,8) may be described by /2/:
where ( I ) q = [2/(?1c)l
.
E sine, ( 2E ZE 2 -1
q l = q [ l
-
( ) sin28+
(- ) sinZ811/2.
[l-(-)sin 01 ( 3 )moc
mot
"loCand E 1 = E . [I
-
(+) s i n 2 01 -1.
"locE(E) i s t h e d e t e c t o r e f f i c i e n c y , (E,8) as w e l l as I ~ C ~ ~ ( E , E ' , , B a r e t h e mul- t i p l e s c a t t e r i n g c o n t r i b u t i o n s , I c ( q ) describes t h e C m p t o n - i n t e n s i t y and I a ( q ) =
*
f >' I ( q )+
< f2 > contains t h e f o r m f a c t o r s < f >B as w e l l as < f2 > and t h e s t r u c t u r e f a c t o r I ( q ) . The prime a t q 1 and E ' i n d i c a t e s q u a n t i t i e s s h i f t e d from qand E by t h e Compton-effect. The expressions f o r A and P have been discussed i n de- t a i l elsewhere /2,3/.
We recognize, t h a t E(E)
.
10(E) should be known i n equ. ( 1 ) i n order t o i n t e r p r e t t k e data c o r r e c t l y . The t a s k can be acccomplished as f o l l o w s :I n t r o d u c i n a t h e v a r ' a b l e s q and E i n s t e a d o f 8 and E and i n t h e l i m i t s o f E
2
20 keV as we1 1 as q2
58-I,
we g e t (2)/2/:I n t h i s expression, t h e m u l t i p l e s c a t t e r i n a c o n t r i b u t i o n s have been o m i t t e d f o r c l a - r i t y . They may be i n c o r p o r a t e d i f necessary. The q u a n t i t y Icorr(E,q) can be d e r i v e d d i r e c t l y from t h e measurement by a n a l y z i n ? EDXD-data:
Thus, equ. ( 5 ) gives c(E)-I0(Eki a p a r t from a constant
-,
i f considered a t q = go= const. and i n t h e l i m i t q < 5.
The a n a l y s i s can be performed f o r s e v e r a l qQ-va- lues, p r o v i d i n g E(E)-Io(E)-curves scaled w i t h r e s p e c t t o one another by a m u l t i p l i - c a t i v e constant. This constant i s t h e r a t i o o f two expressions d e f i n e d by t h e c u r l y brackets i n equ. ( 5 ) . F i g . I shows a schematic p l o t o f t h e s i t u a t i o n .The a n a l y s i s o f t h e ADXD-data (E=const.) can now be done i n t h e usual way, however, w i t h an improved Compton c o r r e c t i o n . From equ. ( I ) , we d e r i v e :
I ( 0 ) = r.{P(0).A(O).Ia(q(O))
+
a(E1 (0)).P(E1(0),0).A(E'(B),8).Ic(q1(0)} ( 7 w i t h a ( E 1 (0 ) ) = I 0 ( E 1 ( Q ) ) / I o ( E ) .With r e s p e c t t o t h e a n a l y s i s o f t h e EDXD-data (20=const.), we have:
1
( ) Here, mo denotes t h e e l e c t r o n r e s t mass, Ti = h/(Z~r,) w i t h h t h e Planck constant
and c t h e v e l o c i t y o f l i n h t .
F i g . 1
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Schematic p l o t o f t h e d i f f r a c t e d i n t e n s i t y surface I(E,q) i n t h e q-E-plane. Several c u t s throunh t h i s surface as performed by t h e ADXD (Eo) and EDXD(Bo)-methods a r e i n d i c a t e d . he c u t qo=const. shows t h e eneray spectrum o f t h e primary beam as l o n a as qo < 5 i-1. €Imax and Bmin r e f e r t o t h e a v a i l a b l e anpular ranoe ( h a x = 900C). I n t h i s case, t h e data taken a t f i x e d 28 cover o n l y a l i m i t e d q-range (see Fip. I ) , so t h a t t h e r e s u l t s from several anpular p o s i t i o n s have t o be scaled w i t h respect t o one another. This can o n l y be achieved w i t h a proper knowledge o f c ( E ) . I 0 ( E ) . The q u a n t i t y E(E) should be known from t h e d e t e c t o r s p e c i f i c a t i o n .JOURNAL DE PHYSIQUE
I 1 1
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RESULTSWith t h e h e l p o f a conventional X-ray eouipment, data have been taken f o r several amorphous a l l o y s u s i n g ADXD (20-method) and EDXD (A-method). The d e t e c t o r was a Ge- c r y s t a l w i t h an energy r e s o l u t i o n o f about 400 eV and an e f f i c i e n c y o f E(E) q: 1 over t h e range o f energies i n t e r e s t i n g here. Transmission geometry under nonfocussing con- d i t i o n s was a p p l i e d throughout. The enerFy spectrum o f t h e primary beam I o ( E ) f o r several values o f t h e tube v o l t a p e was d e r i v e d from EDXD-data o f Be37.5Ti62.5 (45, 55, 60 kV) and o f Fe78B13Sig (50 kV) as described above. The r e s u l t s a r e shown i n Fig. 2.
Data were a l s o taken f o r t h e amorphous a l l o y s Be37.5Ti62.5, Be40Ti5oZrlo,Ni35Zr65 as w e l l as N i l 8 C 0 1 7 Z r ~ ~ b o t h i n t h e ADXD- and EDXD-modes. Some o f them a r e reproduced i n Figs. 3 t o 5. - 1 0 2 L 6 8 10 12 11, /A-I F i g . 4 - T h e same as F i g . 3 f o r Ni31jZ.t-65.
I V
-
CDNCLUSIONS 3 2-
i ) I n s p e c t i o n o f F i a . 2 i n d i c a t e s t h a t t h e spectrum o f t h e primary beam I o ( E ) de- creases c o n t i n u o u s l y towards h i p h e r energies. The t o t a l i n t e g r a t e d i n t e n s i t y o f t h e
Bremsstrahlungs-spectrum
i s p r o p o r t i o n a l t o t h e square o f t h e tube v o l - tage, as was p r e d i c t e d /4/.i i ) T h e apreement between t h e I ( q ) - f u n c t i o n s d e r i v e d from ADXD- and from EDXD-data i s convincing. C e r t a i n d e v i a t i o n s (peak heinhts, small q-ranae o f a few o f t h e EDXD-runs) have s t i l l be examined.
u
-
-
1-
F i g . 3-
I ( q ) - d a t a f o r 0 - - Be40Ti50Zr10 as determinedby t h e 20-method (ADXD) and t h e A-method (EDXD).The over- - l - " " " ' " " " l a p o f t h e d i f f e r e n t EDXD-
0 2 L 6 8 10 12 data sets, taken a t various
Fig. 5
-
The same a s Fig. 3 f o r Ni 18C017Zr65i i i ) Since Be(Z=4) d o e s n ' t show up i n t h e a l l o y Be37.5Ti62.5 i n X-ray d i f f r a c t i o n , t h e T i - T i - c o r r e l a t i o n s dominate t h e measured I ( q ) . These c o r r e l a t i o n s s t r e t c h out f a r i n q-space, pointing towards a well defined next neighbour s h e l l ( t e n - dency t o form Ti-cl u s t e r s ) / 5 / . The same i s t r u e f o r t h e Be40Ti50Zr10-al loy. However, t h e o s c i l l a t i o n s show a d i f f e r e n t period i n d i c a t i n g a d i f f e r e n t d i s - tance o f t h e f i r s t s h e l l ( e f f e c t o f s u b s t i t u t i o n o f Ti by Z r ) . In t h e cases o f Ni35Zrb5 and t h e s c a t t e r i n g powers of
Zr
(Z=40) on one hand and of Co(Z=27) a s well a s Ni(Z=28) on t h e o t h e r hand a r e not so d i f f e r e n t , hence both components a r e seen i n I ( q ) . The s u b s t i t u t i o n of Ni by Co d o e s n ' t change t h e o v e r a l l s t r u c t u r e , s i n c e t h e general appearance o f I ( q ) remains t h e same. Therefore, one may be a b l e t o s e p a r a t e I z r z r ( q ) from I ~ i ~ i ( q ) by a n a l y s i n a pre- c i s e d a t a f o r both t h e a l l o y si v ) Covering t h e f u l l range a v a i l a b l e i n t h e E-q-plane with t h e help of t h e EDXD- method f o r many f i x e d 20-positions w i l l give t h e p o s s i b i l i t i e s t o d e r i v e a l a r q e number of I ( q ) - f u n c t i o n s , each of them determined a t a d i f f e r e n t value E=const. In f a c t , i f t h e d e t e c t o r r e s o l u t i o n i s 400 eV and i f t h e energy range amounts from 20 t o 50 keV, we may o b t a i n a s many a s seventy-five i d e n t i c a l I ( q ) - c u r v e s . This multitude may help t o c l a r i f y t h e c o r r e c t i o n s and t o d e r i v e p r e c i s e I ( q ) - d a t a . F i n a l l y , i t should be mentioned t h a t , i f an absorption edge i s p r e s e n t within t h e energy ranne a v a i l a b l e , one may hope t o g e t valuable informations on anomalous d i f f r a c t i o n and e v e n t u a l l y on p a r t i a l s t r u c t u r e f a c t o r s .
ACKNOWLEDGEMENT
This research was sponsored by t h e g r a n t DMR 83-10025 from NSF. The work was per- formed while one of
us
(GF) was on s a b b a t i c a l leave a t UCLA.REFERENCES
/1/ Wagner, C.N.J., Experimental Determination of Atomic S c a l e S t r u c t u r e o f Amorphous Alloys by S c a t t e r i n g Experiments i n Amorphous Alloys, Ed.: F.E. Luborsky,
Butterworth London 1983, 58.
/2/ F r i t s c h , G . and Wagner, C.N.J., Z. f . Physik, t o be published.
/3/ Eaami,T., S t r u c t u r a l Study by Energy Dispersive X-Ray D i f f r a c t i o n i n Glassy Me- t a l s I ; Eds. : H . 4 . GUntherodt,
H .
Beck, Springer Verlag Berlin 1981, 25. /4/ Town Stephenson, S . , The Continuous X-Ray Spectrum i n Handbuch der Physik, Hsg.:S. FlUgge, Vol. XXX, Springer Verlag Berlin 1957.337.
