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Submitted on 1 Jan 1985
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INTENSITY CORRELATIONS IN
MICRODIFFRACTION FROM ”AMORPHOUS”
MATERIALS
A. Howie, C. Mcgill, J. Rodenburg
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C 9 , supplément au n012, Tome 46, décembre 1985
A . Howie, C . A . McGill and J.M. Rodenburg
Cavendish Laboratory, MadingZey Roud, Cambridge, CB3 OHE, U.K.
Abstract - Using 500-atom c l u s t e r s r e p r e s e n t i n g various amorphous s t r u c t u r e s , we have computed d i f f r a c t e d i n t e n s i t i e s and i n t e n s i t y angular c o r r e l a t i o n s which might be observed i n STEM. Appreciable d i f f e r e n c e s were found between s t r u c t u r e s with t h e same RDF. Preliminary experiments a r e reported.
1
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INTRODUCTIONThe r e a l space image which a microscope provides is p a r t i c u l a r l y well s u i t e d f o r s t r u c t u r a l s t u d i e s of a l 1 but t h e most highly disordered media and, folowing Abbe theory, depends f o r its formation on t h e a b i l i t y of l e n s e s t o recombine d i f f r a c t e d waves preserving ampli tude and phase c o r r e l a t i o n s . I n e l e c t r o n l e n s e s t h i s a b i l i t y is confined by s p h e r i c a l a b e r r a t i o n and o t h e r problems t o a small angular range o of o r d e r 1 0 - ~ rad. Consequently, although with e l e c t r o n s of s h o r t wavelength A t h e r e s o l u t i o n a t t a i n a b l e i n conventional transmission e l e c t r o n microscopy is about 0.2 nm, t h e images a r e e s s e n t i a l l y two-dimensional p r o j e c t i o n s of t h e s t r u c t u r e . T h i s p r o j e c t i o n e f f e c t was one of t h e main problems which bedevilled e f f o r t s t o o b t a i n s t r u c t u r a l information about amorphous m a t e r i a l s by high r e s o l u t i o n e l e c t r o n microscopy ( f o r a review s e e / 1 / ) . The images obtained were c h a r a c t e r i s e d by broken up patches of f r i n g e s i n b r i g h t f i e l d o r by a speckled appearance of small b r i g h t s p o t s i n dark f i e l d and showed many f e a t u r e s a t t h e l i m i t of instrumental r e s o l u t i o n . These e f f e c t s a r i s e from t h e overlap of t h e p r o j e c t e d images from i n d i v i d u a l atoms and, except i n t h e very t h i n n e s t samples t 5 0.5 nm, a r e l i k e l y t o be dominated by u n i n t e r e s t i n g and purely s t a t i s t i c a l overlaps between atoms widely s e p a r a t e d i n t h e beam d i r e c t i o n whose l a t e r a l p o s i t i o n s a r e not s t r o n g l y c o r r e l a t e d . The second major d i f f i c u l t y i n t h e microscopy of amorphous m a t e r i a l s is t o f i n d some e f f i c i e n t way of e x t r a c t i n g u s e f u l and q u a n t i t a t i v e s t r u c t u r a l d a t a from t h e images. Large numbers of t h e s e can be examined q u a l i t a t i v e l y o r even s e m i - q u a l i t a t i v e l y /2/
f o r s i g n i f i c a n t non-random f e a t u r e s , d e t a i l e d and q u i t e l a b o r i o u s comparisons can be made w i t h image computations f o r s p e c i f i c atomic configurations. However t h e l o c a l f l u c t u a t i o n s i n order which microscopy can d e t e c t have not a s yet provided any simple q u a n t i t a t i v e d a t a which can be used t o supplement t h e r a d i a l d i s t r i b u t i o n function.
C9-60 J O U R N A L D E PHYSIQUE
II
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SCANNING TRANSMISSION ELECTRON MICROSCOPYA p o s s i b l e s o l u t i o n t o both of t h e s e problems is o f f e r e d by t h e scanning t r a n s m i s s i o n e l e c t r o n microscope (STEM) where a small s c a t t e r i n g volume is defined by a f i n e l y focussed i n c i d e n t e l e c t r o n probe of diameter 0.5 nm. M i c r o d i f f r a c t i o n p a t t e r n s /3/ and energy l o s s s p e c t r a provide l o c a l information and s i g n a l s from a v a r i e t y of o t h e r d e t e c t o r s can generate images a s t h e probe is scanned over t h e sample. I n p a r t i c u l a r , using e i t h e r small a p e r t u r e o r annular d e t e c t o r s , dark f i e l d images can be obtained a s shown i n f i g . 1 a t a mean s c a t t e r i n g angle 8
=
0.1 rad.Fig. 1
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Schematic STEM geometry showing f i e l d emission source F, probe forming l e n s L and specimen S. Detectors 1 and 2 a r e placed on an annulus a t a s c a t t e r i n g a n g l e 8 with azimuthal s e p a r a t i o n @.This is a much l a r g e r angle than is a v a i l a b l e i n t h e conventional e l e c t r o n microscope. I t has been shown /4/ t h a t overlap e f f e c t s a r e g r e a t l y reduced i n t h e annular d e t e c t o r image being confined t o p a i r s of atoms separated along t h e beam d i r e c t i o n by a d i s t a n c e Az i A/8 s i n 2 ( 8 / 2 ) . Following t h e work of Kam /5/ and t h e o p t i c a l c o r r e l a t i o n s t u d i e s of Clark e t a l / 6 / , i t is a l s o e v i d e n t t h a t u s e f u l q u a n t i t a t i v e s t r u c t u r a l d a t a might be obtained by examining c o r r e l a t i o n s between t h e s i g n a l i n t e n s i t i e s received by two or more dark f i e l d d e t e c t o r s .
I I I
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COMPUTATIONSTo explore a s simply a s p o s s i b l e t h e c o r r e l a t i o n e f f e c t s l i k e l y t o be observed i n t h i n amorph~us fi l m s , we have c a l c u l a t e d , f o r a c l u s t e r of N 500 atoms under plane wave i n c i d e n t i l l u m i n a t i o n . t h e normalized d i f f r a c t e d i n t e n s i t y given by
The i n t e n s i t y was evaluated a t
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104 p o i n t s on a sphere of r a d i u s K = 41~sin(8/;')/X with t h e s c a t t e r i n g angle 8 chosen t o match a d i f f u s e maximum i n t h e averagud d i f f r a c t i o n p a t t e r n . The c o r r e l a t i o n f u n c t i o n considered is given byseparated by a fixed angle $, the intensity product being averaged over al1 possible
orientations of the cluster. We are making the assumption that this procedure
adequately simulates the actual experiment where the convergent illumination probe in the STEM defines a scattering volume containing N atoms and the product of the signals received in two detectors (see fig. 1 ) is averaged over time as the probe is
scanned across the sample. The azimuthal separation angle L$ between the detectors
can from a scattering diagram readily be shown to be related to
IJI
above byFig. 2 shows the correlation function C($) for two models of amorphous Ge, the
polyhedral (PT) model
/7/
and the Polk continuous random network (CRN) model /8/.The difference in C($) for these,two models is substantial despite the fact that they have quite similar radial distribution functions J(R) (also shown in fig. 2).
Fig. 2
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Correlation function C($) and R.D.F. J(R) for the Polk CRN mode1(continuous line) and the PT model (broken line).
Using the analysis of Kam /5/, the correlation function C($) can be analysed in
terms of Legendre polynomials Pl(cos($)) with coefficients which can be obtained by expanding I(K) in spherical harmonics. Because of the need to satisfy Friedel's law, only even values of 1 occur.
IV
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PRELIMINARY EXPERIMENTAL RESULTSSince Our STEM is not currently fitted with multiple dark field detectors, we have attempted to collect data by using the Grigson post-specimen deflection scanning coils. These allow the diffraction pattern, and in particular any annular Segment of it to be recorded sequentially by scanning it over a detector placed on the
instrumental axis. Apart from convenience, two advantages of this procedure are
C9-62 JOURNAL DE PHYSIQUE
Fig. 3 shows a t y p i c a l a n n u l a r s c a n o b t a i n e d i n t h i s way. A slow background i n t e n s i t y v a r i a t i o n is a p p a r e n t , p o s s i b l y due t o a s l i g h t e l l i p t i c i t y i n t h e Grigson s c a n . T h i s need n o t be a s e r i o u s problem s i n c e < I ( $ ) > can b e a c c u r a t e l y measured by a v e r a g i n g o v e r a l a r g e number of s c a n s from d i f f e r e n t p o i n t s on t h e specimen. The i n d i v i d u a l s c a n s c a n t h e n be c o r r e c t e d by d i v i d i n g by <I($)> and t h e a u t o c o r r e l a t i o n f u n c t i o n computed by F o u r i e r methods. E x p e r i e n c e s o f a r i n d i c a t e s t h a t t h i s is
q u i t e a f e a s i b l e p r o c e d u r e , however o b s e r v a t i o n s o f t h e probe p o s i t i o n on t h e specimen i n d i c a t e s i g n i f i c a n t movement, by a b o u t 0 . 5 nm, is o c c u r r i n g d u r i n g t h e Grigson scan. We hope t o e l i m i n a t e t h i s e f f e c t e i t h e r by improved s c r e e n i n g o r by i n t r o d u c t i o n of a compensating d e f l e c t i o n o f t h e i n c i d e n t beam.
Fig. 3
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Grigson a n n u l a r s c a n s I ( @ ) f o r t h e f i r s t d i f f u s e r i n g i n amorphous Ge. Upper c u r v e f o r f o c u s s e d p r o b e , lower c u r v e f o r d e f o c u s s e d probe. Both s c a n s have t h e same a v e r a g e i n t e n s i t y and a r e p l o t t e d on t h e same s c a l e .We thank D r P.H. G a s k e l l f o r v a l u a b l e d i s c u s s i o n s and f o r k i n d l y s u p p l y i n g t h e c l u s t e r c o o r d i n a t e s . F i n a n c i a l s u p p o r t from SERC and from VG S c i e n t i f i c is g r a t e f u l l y acknowledged.
REFERENCES
Howie, A . , J. Non-Cryst. S o l i d s
2
(1978) 41.Krivanek, O.L., G a s k e l l , P.H. and Howie, A . , N a t u r e
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(1976) 454.Rodenburg, J . M . , ( s e e p a p e r i n t h e s e p r o c e e d i n g s ) .
Gibson, J.M. and Howie, A . , Shemica S c r i p t a f i (1979) 109. Kam, Z., Micromolecules (1977) 927.
C l a r k , N . A . , Ackerson, B . J . and Hurd, A . J . , Phys. Rev. L e t t . (1983) 1459. G a s k e l l P.H., P h i l . Mag. (1975) 211.
