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STRUCTURE OF LARGE ANGLE [001] TWIST BOUNDARIES
K. Milkove, P. Lamarre, F. Schmückle, M. Vaudin, S. Sass
To cite this version:
K. Milkove, P. Lamarre, F. Schmückle, M. Vaudin, S. Sass. DIFFRACTION STUDIES OF THE ATOMIC STRUCTURE OF LARGE ANGLE [001] TWIST BOUNDARIES. Journal de Physique Colloques, 1985, 46 (C4), pp.C4-71-C4-84. �10.1051/jphyscol:1985406�. �jpa-00224655�
JOURNAL DE PHYSIQUE
Colloque C4, suppldinent au n04, Tome 46, avril 1985 page C4-71
DIFFRACTION S T U D I E S OF T H E ATOMIC S T R U C T U R E O F LARGE ANGLE [001] T W I S T BOUNDARIES
K.R. Milkove, P . Lamarre, F. ~ c h m i c k l e + , M.D. Vaudin and S.L. Sass Dept. of Materials Science and Engineering, Come22 University, Bard HaZZ, Ithaca, N.Y. 14853, U.S.A.
'BBC-~orschungszentrwn, 5405 Baden-DaettwiZ, SwitzerZand
Resume - Cet a r t i c l e r e c a p i t u l e l e s r e s u l t a t s de l ' a p p l i c a t i o n des techniques d e d i f f r a c t i o n a 1 'etude de l a s t r u c t u r e atomique des j o i n t s de g r a i n s . La
determination de l a s t r u c t u r e de l a p r o j e c t i o n d ' un j o i n t [001] a grand angle de t o r s i o n e s t d e c r i t e . L ' i n f l u e n c e de l a n a t u r e du metal c.f.c. e t de l a l i a i s o n s u r l a s t r u c t u r e du j o i n t e s t 6tudiee. S'appuyant s u r l e s
r e s u l t a t s des etudes de d i f f r a c t i o n , l e s auteurs proposent des
g 6 n e r a l i s a t i o n s concernant l e s s t r u c t u r e s des j o i n t s [001] a grands angles de t o r s i o n .
A b s t r a c t - The r e s u l t s o f t h e a p p l i c a t i o n o f d i f f r a c t i o n techniques t o study t h e atomic s t r u c t u r e of g r a i n boundaries are reviewed. The d e t e r m i n a t i o n o f t h e p r o j e c t e d s t r u c t u r e of a l a r g e angle [ O O l ] t w i s t boundary i s described.
The i n f l u e n c e o f f.c.c. metal t y p e and bonding type on boundary s t r u c t u r e i s examined. G e n e r a l i z a t i o n s a r e made concerning t h e s t r u c t u r e o f l a r g e angle [001] t w i s t boundaries based on t h e r e s u l t s o f t h e d i f f r a c t i o n studies.
I. I n t r o d u c t i o n
D i f f r a c t i o n techniques have been used e x t e n s i v e l y t o determine t h e atomic
s t r u c t u r e o f t h r e e dimensionally p e r i o d i c s t r u c t u r e s . I n r e c e n t years i t has a l s o been p o s s i b l e t o use these techniques t o o b t a i n i n f o r m a t i o n on t h e atomic
s t r u c t u r e o f i n t e r n a l i n t e r f a c e s , such as g r a i n boundaries /I/. The d i f f i c u l t y i n a p p l y i n g d i f f r a c t i o n t o g r a i n boundaries comes from t h e l a c k o f p e r i o d i c i t y along t h e d i r e c t i o n normal t o t h e i n t e r f a c e , since t h e boundary s t r u c t u r e i s o n l y two dimensionally p e r i o d i c ( i n t h e plane o f t h e i n t e r f a c e ) . Two d i f f r a c t i o n
approaches have been devised f o r s t u d y i n g g r a i n boundaries which wi 11 be described below.
Before d i s c u s s i n g these approaches, i t i s important t o examine t h e r e c i p r o c a l l a t t i c e associated w i t h t h e g r a i n boundary. A l l o f t h e d i f f r a c t i o n observations were made on manufactured b i c r y s t a l s (see reference 2 f o r d e t a i l s ) . A t y p i c a l b i c r y s t a l w i t h a small angle [001] t w i s t boundary a t i t s midplane i s shown s c h e m a t i c a l l y i n Fig. l ( a ) , and t h e r e c i p r o c a l l a t t i c e o f t h i s boundary i s shown i n Fig. l ( b ) , where t h e H and K axes l i e p a r a l l e l t o t h e boundary, and t h e L-axis i s normal t o t h e boundary plane. The p e r i o d i c s t r a i n f i e l d associated w i t h t h e g r a i n boundary g i v e s r i s e t o e x t r a r e f l e c t i o n s , which a r e i n t h e form o f r e c i p r o c a l l a t t i c e rods ( r e l r o d s ) since t h e boundary i s n o t p e r i o d i c along t h e z - d i r e c t i o n /3,4/. R e f l e c t i o n i n t e n s i t i e s i n t h e L=O plane a r e r e l a t e d t o t h e s t r u c t u r e o f t h e boundary p r o j e c t e d o n t o t h e i n t e r f a c e plane w h i l e t h e p r o f i l e s of t h e r e l r o d s a r e r e l a t e d t o t h e s t r u c t u r e along t h e d i r e c t i o n normal t o the
i n t e r f a c e . The two approaches mentioned above i n v o l v e ( 1 ) measurements made on t h e r e f l e c t i o n s i n t h e L=O plane, which can be used t o o b t a i n t h e boundary Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985406
s t r u c t u r e p r o j e c t e d o n t o t h e i n t e r f a c e , and ( 2 ) measurements on t h e i n t e n s i t y p r o f i l e s along t h e r e l r o d s , which can be used t o o b t a i n i n f o r m a t i o n on t h e behavior o f the s t r u c t u r e along t h e d i r e c t i o n normal t o t h e i n t e r f a c e .
Fig. I - ( a ) B i c r y s t a l c o n t a i n i n g a small angle [001] t w i s t boundary w i t h m i s o r i e n t a t i o n 8. ( b ) Schematic three-dimensional r e c i p r o c a l l a t t i c e f o r t h e g r a i n boundary i n (a).
This paper w i l l discuss t h e a p p l i c a t i o n of d i f f r a c t i o n techniques t o study (1) t h e p r o j e c t e d s t r u c t u r e o f l a r g e angle [001] t w i s t boundaries, ( 2 ) t h e i n f l u e n c e o f f.c.c. metal t y p e on boundary s t r u c t u r e and, ( 3 ) t h e i n f l u e n c e o f bonding t y p e ( i o n i c , m e t a l l i c , c o v a l e n t ) on boundary s t r u c t u r e .
The study discussed i n t h i s s e c t i o n i s concerned w i t h determining t h e boundary s t r u c t u r e p r o j e c t e d o n t o t h e i n t e r f a c e plane s i n c e t h e r e a r e a number o f experimental observations i n t h e L=O plane (see Fig. l ( b ) ) , which o n l y c o n t a i n
i n f o r m a t i o n on t t i e x,y coordinates o f t h e atoms. The f i r s t time d i f f r a c t i o n techniques were used t o determine t h e p r o j e c t e d s t r u c t u r e o f a g r a i n boundary, i t was advantageous t o choose a boundary w i t h a small number o f atoms i n t h e u n i t c e l l . The boundary t h a t was s t u d i e d was t h e c=5 (0=36.87') [001] t w i s t boundary i n Au. I n t h i s case t h e number o f atoms i n each (001) plane i n t h e u n i t c e l l i s 5 and s i n c e t h e p r o j e c t e d p o s i t i o n o f each atom i s determined by i t s x,y
coordinates, t h e number o f coordinates t o be determined p e r plane i s 10. I f t h e number o f planes associated w i t h t h e boundary ( c o n t a i n i n g atoms w i t h l a r g e displacements from p e r f e c t c r y s t a l p o s i t i o n s ) i s 4 ( 2 above and 2 below t h e boundary), t h e maximum number o f coordinates t h a t must be determined i s 40. For t h e a c t u a l boundary under consideration, however, t h e number o f v a r i a b l e s i s much smaller. Fig. 2(a,b) shows t o p and s i d e views, r e s p e c t i v e l y , o f t h e u n i t c e l l o f t h e E=5 t w i s t boundary, w i t h 2 atomic planes above and below t h e boundary plane.
The s t r u c t u r e shown i s i n t h e coincidence s i t e c o n f i g u r a t i o n ; t h a t i s , a t t h e corners o f t h e u n i t c e l l t h e two c r y s t a l s a r e i n t h e c o r r e c t s t a c k i n g f o r a f.c.c.
s t r u c t u r e . F o l l o w i n g Bristowe and Crocker /6/, i t i s seen t h a t t h i s u n i t c e l l has t h e symmetry elements shown i n Fig. 2(c). The e x p e r i m e n t a l l y determined s t r u c t u r e f a c t o r r u l e s showed t h a t t h e symmetry o f t h e boundary s t u d i e d was t h a t g i v e n i n Fig. 2(c). This symmetry determines t h e minimum number o f atoms whose p o s i t i o n s must be l o c a t e d i n o r d e r t o completely c h a r a c t e r i z e t h e u n i t c e l l contents.
Examination o f Fig. 2(a) shows t h a t f o r each (001) plane i n t h e u n i t c e l l one atom i s f i x e d a t e i t h e r (0,O) o r (1/2,1/2) by t h e t r a n s l a t i o n s t a t e o f t h e boundary.
The remaining 4 atoms i n each plane a r e symmetry r e l a t e d by t h e 4 - f o l d axis.
Determining t h e x,y coordinates o f one o f these atoms i n each plane, f i x e s t h e p o s i t i o n s o f t h e o t h e r t h r e e atoms. I t i s a l s o seen t h a t t h e atoms i n t h e upper and l o w e r c r y s t a l s a r e r e l a t e d by t h e t w o - f o l d screw axes l y i n g i n t h e i n t e r f a c e plane. Therefore, i f i t i s assumed t h a t t h e g r a i n boundary i s 4 atomic l a y e r s t h i c k , i t i s necessary t o determine t h e x,y coordinates o f o n l y 2 atoms (marked A,B i n Fig. 2 ( a ) ) i n o r d e r t o determine t h e atomic s t r u c t u r e p r o j e c t e d onto t h e boundary plane.
Unrelaxed f 5
0
A 0 .
Ave [ZOO]
la1 I ~ I
_I Perpendicular to grain boundary TWO-fold axis Four-fold axis
In the grain boundary plane
- TWO-fold axis
- Two-fold screw axis
Fig. . 2 - ( a ) Schematic diagram o f t h e unrelaxed c = 5 p r i m i t i v e u n i t c e l l p r o j e c t e d onto t h e g r a i n boundary plane, (b) Side view o f t h e same u n i t c e l l w i t h two planes above and be1 ow t h e boundary shown, ( c ) t h e symmetry elements o f t h e s t r u c t u r e shown i n (a).
The assumption t h a t s i g n i f i c a n t atomic r e l a x a t i o n s f o r a z=5 boundary a r e l i m i t e d t o a r e g i o n which i s o n l y 4 atomic l a y e r s t h i c k (20.8 nm f o r Au) i s based on experimental considerations. The w i d t h o f t h e s t r a i n e d r e g i o n f o r a c=377
(0=23.8') t w i s t boundary i n g o l d was e x p e r i m e n t a l l y determined, from t h e i n t e n s i t y p r o f i l e s o f v a r i o u s r e l r o d s , t o be 20.8 nm /4/. The i n t e n s i t y p r o f i l e s f o r r e l r o d s f o r t h e c=5 (0=36.g0) boundary were q u a l i t a t i v e l y observed t o be broader suggesting t h a t t h i s boundary i s t h i n n e r than t h e e = 23.8' boundary. Thus t h e assumption t h a t s i g n i f i c a n t r e l a x a t i o n s are r e s t r i c t e d t o a f o u r - l a y e r r e g i o n i n t h e v i c i n i t y o f t h e i n t e r f a c e plane i s q u i t e reasonable.
The d e t e r m i n a t i o n o f t h e p r o j e c t e d z=5 boundary s t r u c t u r e proceeds by t h e standard r e l i a b i l i t y f a c t o r approach t h a t has been used f o r c r y s t a l s t r u c t u r e
determinations /7/. I n t h i s method t h e x and y coordinates f o r t h e two
independent atoms l a b e l l e d A and B i n Fig. 2(a) a r e scanned i n small increments over an area c o v e r i n g t h e range o f atomic displacements which generate a l l p o s s i b l e g r a i n boundary c o n f i g u r a t i o n s . For each c o n f i g u r a t i o n t h e magnitude o f t h e s t r u c t u r e f a c t o r s a r e c a l c u l a t e d and compared t o e x p e r i m e n t a l l y observed s t r u c t u r e f a c t o r s u s i n g t h e d e f i n i t i o n o f t h e r e l i a b i l i t y f a c t o r , R, given below,
where F O ~ = t h e e x p e r i m e n t a l l y observed s t r u c t u r e f a c t o r o f t h e j t h . r e f l e c t i o n . j
F~~~ = t h e c a l c u l a t e d s t r u c t u r e f a c t o r o f t h e j t h r e f l e c t i o n . j
W. = w e i g h t i n g f a c t o r (OsW 4).
J 3-
The s e t of xA, yA, x g 9 yB coordinates which leads t o the smallest value f o r R , provides the best f i t t o t h e d i f f r a c t i o n observations. Thus the determination of t h e c=5 s t r u c t u r e becomes a search f o r minimum values f o r t h e function
R(xA, yA, xB, yB). Such a search y i e l d s t h e c=5 s t r u c t u r e shown i n Fig. 3.
Fig. 3 - Atomic displacements f o r the projected c=5 CSL u n i t c e l l . This s t r u c t u r e had t h e smallest R ( ~ 0 . 1 5 ) .
Examination of Fig. 3 reveals several interesting points about the structure.
Note t h a t the displacement associated w i t h atom A ( f i r s t plane from t h e boundary) is much larger than the displacement associated w i t h atom B (second plane from boundary). This rapid decrease i n magnitude f o r t h e atomic relaxations i n planes away from the boundary i s a s expected f o r a l a r g e angle boundary. In order t o understand the o r i g i n of t h e relaxations, consider the forces acting on atom A.
The atoms i n t h e lower crystal a c t t o hold atom A i n i t s unrelaxed position in order t o preserve f.c.c. stacking. The atoms in the upper crystal a c t t o displace atom A, with t h e nearest atoms having the l a r g e s t e f f e c t . In the unrelaxed
configuration, atoms A and A1 a r e 23% c l o s e r than t h e nearest neighbor distance i n a perfect f.c.c. c r y s t a l . Thus t h e observation t h a t in t h i s s t r u c t u r e these atoms move a p a r t i s physically q u i t e reasonable. Examination of t h e boundary s t r u c t u r e shows t h a t i t e x h i b i t s symmetry related displacements which can be interpreted a s local rotations about ' 0 ' elements. For [ O O l ] twist boundaries, the ' 0 ' elements consist of l i n e s perpendicular t o the boundary, some of which pass through coincidence s i t e s /8/. The degree of rotation i s large ( ~ 2 0 " ) . Since t h e magnitude of t h e rotation i s se/2, small regions of median f.c.c. s t r u c t u r e a r e produced.
I t i s i n t e r e s t i n g t o compare the atomic displacements found i n the present study, w i t h predictions from computer modeling f o r t h e same boundary. A number of central pairwise p o t e n t i a l s were employed, a l l of which were empirical and some of which had been matched t o data relevant t o gold. The projected atomic
displacements f o r t h e Z=5 boundary calculated f o r t h e d i f f e r e n t p o t e n t i a l s a r e shown i n Fig. 4 f o r the atoms labeled C ( f i r s t plane) and D (second plane) i n Fig. 3. The displacements determined by t h e d i f f r a c t i o n analysis a r e a l s o presented. I t was shown t h a t t h e values of t h e r e l i a b i l i t y f a c t o r f o r the computer generated s t r u c t u r e s a r e much higher than t h a t f o r the d i f f r a c t i o n s t r u c t u r e . I t i s c l e a r l y seen t h a t the s t r u c t u r e determined from the d i f f r a c t i o n analysis has an atomic displacement in the f i r s t plane away from the boundary which i s a t l e a s t twice as large a s those predicted from computer modeling. I t i s i n t e r e s t i n g t o see t h a t the direction of t h e atomic displacement in t h e f i r s t plane i s approximately the same f o r a l l structures. As expected, in a,ll cases the magnitude of the displacements of the atoms i n the second plane i s smaller than in t h e f i r s t plane.
P 5 D i s p l a c e m e n t s
FIRST PLANE Y W N D PLANE
ATOM C ATOM D
DIFFRACTION 1
F i g . 4 - Comparison o f t h e a t o m i c displacements f r o m t h e d i f f r a c t i o n a n a l y s i s and computer modeling u s i n g d i f f e r e n t i n t e r a t o m i c p o t e n t i a l s . D e t a i l s on t h e p o t e n t i a l s a r e g i v e n i n r e f e r e n c e ( 5 ) . The m e t a l s t o w h i c h t h e p o t e n t i a l s correspond a r e g i v e n i n b r a c k e t s .
0.03 CSL UNIT CELL -
LATTICE PARAMETER
111. The I n f l u e n c e o f F.C.C. M e t a l Type on G r a i n Boundary S t r u c t u r e
I n a r e c e n t computer m o d e l i n g s t u d y /9/ i t was p r e d i c t e d t h a t t h e a t o m i c s t r u c t u r e o f a e=22.6Z0 (c=13) [001] t w i s t boundary s h o u l d be d i f f e r e n t f o r d i f f e r e n t f.c.c.
metals. I n o r d e r t o check t h i s p r e d i c t i o n , t h e s t r u c t u r e o f t h e c=13 (e=22.6") [001] t w i s t boundary c o n t a i n e d i n Au, Ag, Cu and P t b i c r y s t a l s was s t u d i e d u s i n g X-ray t e c h n i q u e s . S p e c i f i c a l l y , X-ray d i f f r a c t i o n p a t t e r n s showing t h e same r e g i o n s o f r e c i p r o c a l space were o b t a i n e d and compared f o r t h e f o u r m e t a l s . The i n t e n s i t i e s o f t h e r e f l e c t i o n s i n t h e s e p a t t e r n s a r e p r o p o r t i o n a l t o t h e square o f t h e s t r u c t u r e f a c t o r s o f t h e s e r e f l e c t i o n s . I t i s expected t h a t i f t h e boundary s t r u c t u r e s a r e s i g n i f i c a n t l y d i f f e r e n t , t h e n t h e r e l a t i v e i n t e n s i t i e s o f t h e g r a i n boundary r e f l e c t i o n s observed f r o m t h e d i f f e r e n t m e t a l s w i l l a l s o be d i f f e r e n t . The specimens o f Au, Ag and Cu were produced by UHV d e p o s i t i o n o f t h i n s i n g l e c r y s t a l s o n t o p r e - o r i e n t e d c l e a v e d NaCl s i n g l e c r y s t a l s u b s t r a t e s f o l l o w e d by h o t i s o s t a t i c p r e s s i n g i n UHV t o f o r m t h e b i c r y s t a l s c o n t a i n i n g t h e [ O O l ] t w i s t boundary. The P t s i n g l e c r y s t a l s were produced by e l e c t r o n beam d e p o s i t i o n o n t o c l e a v e d NaCl i n an i o n pumped vacuum system o p e r a t i n g a t l o m 6 T o r r . The P t s i n g l e c r y s t a l s were p r e s s u r e s i n t e r e d i n a i r a t 600°C t o f o r m a b i c r y s t a l . I t i s expected t h a t t h e P t b i c r y s t a l s w i l l n o t be as c l e a n as t h o s e o f Au, Ag and Cu.
Examples o f t h e e x p e r i m e n t a l o b s e r v a t i o n s a r e p r e s e n t e d i n t h e f o r m o f s e t s o f X-ray d i f f r a c t i o n p a t t e r n s t a k e n i n t h e v i c i n i t y o f t h e f.c.c. r e f l e c t i o n s , w i t h a p p r o x i m a t e l y t h e same d i f f r a c t i o n geometry f o r a l l metals. F i g . 5 shows p a t t e r n s t a k e n i n t h e v i c i n i t y o f t h e 200 f.c.c. r e f l e c t i o n s , w i t h g r a i n boundary
r e f l e c t i o n 6,6,0 on t h e Ewald sphere. ( F o r t h e c o o r d i n a t e system used f o r
i n d e x i n g see r e f . 9. ) The i n d i c e s o f t h e g r a i n boundary r e f l e c t i o n s a r e i n d i c a t e d on t h e p a t t e r n s u s i n g o n l y t h e H,K n o t a t i o n , e x c e p t f o r r e f l e c t i o n 6,6,0 where L i s e x a c t l y zero. F o r t h e o t h e r r e f l e c t i o n s , L has s m a l l d e v i a t i o n s f r o m zero.
Several g r a i n boundary r e f l e c t i o n s a r e c l e a r l y v i s i b l e . These can be compared f o r t h e f o u r m e t a l s and i t i s seen t h a t t h e r e l a t i v e i n t e n s i t i e s o f t h e r e f l e c t i o n s a r e q u i t e s i m i l a r . F o r example, l i s t i n g t h e observed r e f l e c t i o n s i n d e c r e a s i n g o r d e r o f i n t e n s i t y y i e l d s t h e sequence 6,6; 4,4; 2,6; 3,5 f o r a l l f o u r m e t a l s .
Fig. 5 - X-ray d i f f r a c t i o n p a t t e r n s from t h e 200 r e g i o n o f r e c i p r o c a l space w i t h t h e 6,6,0 r e f l e c t i o n on t h e Ewald sphere. E=13 (8=22.6') [001] t w i s t boundary.
Fig. 6 - X-ray d i f f r a c t i o n p a t t e r n s from t h e 220 r e g i o n o f r e c i p r o c a l space. c=13 (8=22.6') [001] t w i s t boundary.
Fig. 6 shows p a t t e r n s taken i n t h e v i c i n i t y o f t h e 220 f.c.c. r e f l e c t i o n s f o r Au, Ag and Pt, w i t h g r a i n boundary r e f l e c t i o n s 11,1,0 and 11,1,0 on t h e Ewald sphere, w h i l e f o r Cu, r e f l e c t i o n s 11,l and 10,4 have an equal L d e v i a t i o n , 10.061, from t h e Ewald sphere. It i s seen t h a t t h e r e f l e c t i o n s 10,O; 10,4; 11,l a r e i n a decreasing o r d e r o f i n t e n s i t y f o p Au, Ag and Cu, w h i l e f o r P t r e f l e c t i o n 10,O i s q u i t e s t r o n g and r e f l e c t i o n s 11,l and 10,4 a r e approximately equal i n i n t e n s i t y .
A d d i t i o n a l observations i n d i f f e r e n t regions o f r e c i p r o c a l space a l s o demonstrated t h a t t h e r e l a t i v e i n t e n s i t i e s o f t h e g r a i n boundary r e f l e c t i o n s are i n t h e same sequence f o r Au, Ag and Cu, w h i l e P t showed d i f f e r e n t behavior. An extensive a n a l y s i s o f these observations showed t h a t t h e Z=13 [001] t w i s t boundary has t h e same symmetry and s i m i l a r s t r u c t u r e i n Au, Ag and Cu, and t h a t t h e symmetry and s t r u c t u r e i s d i f f e r e n t f o r t h e Z=13 boundary i n Pt. The symmetry o f t h e Z-13 boundary s t r u c t u r e i n Au, Ag and Cu i s as shown i n Fig. 2 ( c ) , which i s t h a t o f t h e space group P42i2 / l o / . The space group o f t h e P t boundary s t r u c t u r e i s P42'2
, , n ,
The p o s s i b i l i t y o f i m p u r i t y segregation a t t h e boundaries used i n t h e study by Budai, Donald and Sass /11/ was t h e m o t i v a t i o n f o r t h e present work. The r e s u l t s o f Budai e t al.. on Au and Ag were completely reproduced i n t h e present study, which confirms t h e i r conclusion t h a t t h e E=13 [ O O l ] t w i s t boundary s t r u c t u r e i s s i m i l a r i n Au and Ag. T h i s agreement between t h e r e s u l t s from c l e a n and p o s s i b l y impure boundaries, leads t o t h e suggestion t h a t a small v a r i a t i o n i n boundary composition may n o t i n f l u e n c e t h e boundary s t r u c t u r e i n t h e p a r t i c u l a r case under i n v e s t i g a t i o n .
The conclusion o f t h e present work disagrees w i t h t h e computer modeling p r e d i c - t i o n s o f Bristowe and Sass t h a t t h e boundary s t r u c t u r e should be d i f f e r e n t i n Au and Ag C.13 [001] t w i s t boundaries. I n a more r e c e n t study, Wolf /12/, using pseudopotentials, has p r e d i c t e d t h a t t h e Z=13 boundary s t r u c t u r e should be t h e same i n Au, Ag and Cu, i n agreement w i t h t h e experimental r e s u l t s r e p o r t e d here.
The c a l c u l a t e d s t r u c t u r e f a c t o r s f o r t h e boundary s t r u c t u r e s p r e d i c t e d by Wolf do not, however, agree w i t h t h e observed s t r u c t u r e f a c t o r s . I n p a r t i c u l a r , W o l f ' s s t r u c t u r e s p r e d i c t t h a t r e f l e c t i o n 11,l has h i g h e r i n t e n s i t y than r e f l e c t i o n 10,4, which i s t h e opposite o f t h e experimental observations f o r Au, Ag and Cu i n Fig. 6. Wolf has suggested t h a t t h e reason t h e computer modeling p r e d i c t i o n s do n o t agree w i t h t h e experimental observations i s because t h e computer c a l c u l a t i o n s must be performed a t constant density, w h i l e i t i s known t h a t t h e r e i s a l o c a l expansion a t t h e E=13 boundary i n Au /13/.
t
I n s e c t i o n I 1 i t was noted t h a t t h e important atomic displacements i n t h e Z=5 boundary i n v o l v e d r o t a t i o n s about 0-elements t o achieve l o c a l regions o f f .c.c.
s t r u c t u r e . I n an attempt t o produce a boundary s t r u c t u r e t h a t agreed w i t h t h e experimental observations, a s i m i l a r type o f displacement f i e l d was used f o r t h e Z=13 boundary. Such an approach has been described by Brokman and B a l l u f f i /14/.
When t h i s was done, i t was found t h a t t h e observed i n t e n s i t y sequence o f g r a i n boundary r e f l e c t i o n s c o u l d be reproduced by t h e use o f l a r g e l o c a l r o t a t i o n s about 0-elements, as noted f o r t h e Z=5 boundary. I n f a c t , i t was r o t a t i o n s g i v i n g l o c a l regions o f median f.c.c. s t r u c t u r e t h a t r e s u l t e d i n t h e c o r r e c t o r d e r o f t h e i n t e n s i t i e s o f r e f l e c t i o n s 10,4 and 11,l. I n a previous combined computer modeling-X-ray d i f f r a c t i o n study by Bristowe and Sass, i t was a l s o shown t h a t t h e b e s t match t o t h e X-ray observations was g i v e n by l a r g e r o t a t i o n s about
0-elements. (See Fig. 4 ( c ) o f r e f . 9.)
Based on t h e r e s u l t s from t h e Z=5 and E=13 boundary s t u d i e s taken together, i t i s tempting t o speculate t h a t t h e i m p o r t a n t atomic displacements i n [ O O l ] t w i s t boundaries i n f.c.c. metals i n v o l v e l a r g e r o t a t i o n s about 0-elements t o produce small regions o f median f.c.c. s t r u c t u r e . More extensive s t u d i e s are r e q u i r e d t o check t h e v a l i d i t y o f t h i s suggestion.
I V . The I n f l u e n c e o f Bonding Type on Grain Boundary S t r u c t u r e
The previous s t u d i e s were concerned p r i m a r i l y w i t h t h e s t r u c t u r e o f t h e g r a i n boundary p r o j e c t e d o n t o t h e i n t e r f a c e plane. I n t h i s s e c t i o n a d i f f r a c t i o n approach i s described which can be used t o study t h e s t r u c t u r e o f g r a i n boundaries along t h e d i r e c t i o n normal t o t h e i n t e r f a c e . T h i s technique i s s e n s i t i v e t o d e v i a t i o n s i n plane spacing, composition and atomic d e n s i t y from t h a t i n t h e p e r f e c t c r y s t a l . A d i f f r a c t i o n a n a l y s i s i s performed i n o r d e r t o r e l a t e r e a l i s t i c model s t r u c t u r e s o f t h e boundary t o d i f f r a c t i o n e f f e c t s . The d i f f r a c t i o n
technique i s then used t o study t h e i n f l u e n c e o f bonding t y p e ( i o n i c , m e t a l l i c ,
c o v a l e n t ) on boundary s t r u c t u r e . T h i s was done by u s i n g e l e c t r o n d i f f r a c t i o n t o study t h e s t r u c t u r e along t h e d i r e c t i o n normal t o t h e i n t e r f a c e o f COO11 t w i s t boundaries w i t h t h e same m i s o r i e n t a t i o n ' i n N i O ( i o n i c bonding), Au ( m e t a l l i c bonding) and Ge ( c o v a l e n t bonding).
Lamarre and Sass /13/ obtained d i f f r a c t i o n i n f o r m a t i o n f r o m an edge-on [001] t w i s t boundary i n a Au b i c r y s t a l . They i n t e r p r e t e d t h e i r r e s u l t s i n terms o f l o c a l changes i n plane spacing normal t o t h e boundary. For s i m p l i c i t y , Lamarre and Sass represented t h e g r a i n boundary as a u n i f o r m t h i n c r y s t a l w i t h a plane spacing d i f f e r e n t from t h a t i n t h e neighboring p e r f e c t c r y s t a l s . This model i s c l e a r l y t o o simple, and was considered t o be t h e f i r s t s t e p towards a more r e a l i s t i c r e p r e s e n t a t i o n o f t h e g r a i n boundary. I n t h i s paper t h e previous work i s extended b y u s i n g a more r e a l i s t i c model o f t h e boundary displacement f i e l d and c a r r y i n g o u t a d i f f r a c t i o n a n a l y s i s based on t h i s model. As an i n t r o d u c t i o n t h e t h i n c r y s t a l model o f a g r a i n boundary used by Lamarre and Sass i s reviewed.
I n t h e t h i n c r y s t a l model t h e [001] t w i s t boundary a t t h e midplane o f a b i c r y s t a l i s envisaged as a t h i n c r y s t a l , 2N planes t h i c k w i t h a (002) plane spacing, d,., t h a t i s d i f f e r e n t from t h e spacing, d,, i n t h e p e r f e c t c r y s t a l surrounding t h e boundary r e g i o n (Fig. 7(a)). I t i s considered more l i k e l y t h a t db i s g r e a t e r than do, although t h i s assumption does n o t a f f e c t t h e r e s u l t s o f t h e analysis. It i s suggested t h a t a l o c a l expansion normal t o t h e i n t e r f a c e w i l l g i v e r i s e t o e x t r a s c a t t e r i n g e f f e c t s i n t h e COO11 d i r e c t i o n i n r e c i p r o c a l space a t a d i s t a n c e l / d h
-
from t h e o r i g i n , as shown i n Fig. 7(b). It i s a l s o suggested t h a t t h e g r a i n boundary r e f l e c t i o n w i l l be elongated by an amount p r o p o r t i o n a l t o 1/W, where W i s t h e thickness o f t h e g r a i n boundary region. The l i m i t a t i o n s of t h i s approach a r e t h a t : ( 1 ) i t . i s p h y s i c a l l y u n r e a l i s t i c f o r t h e s t r a i n f i e l d surrounding t h e i n t e r f a c e t o c o n t a i n a sharp step, and i t i s more l i k e l y t o decrease smoothly t o z e r o w i t h i n c r e a s i n g d i s t a n c e f r o m t h e i n t e r f a c e , and ( 2 ) d i f f r a c t i o n from t h e t h i n c r y s t a l r e g i o n cannot be t r e a t e d independently from t h e p e r f e c t c r y s t a l d i f f r a c t i o n .
Real Space Reciprocal Space Fig. 7 - ( a ) ~ i c r y i t a l
Perfset BoundPry Perfect c o n t a i n i n g g r a i n boundary a t i t s
Crystal Region. Crystal midplane. The spacing o f t h e
atom planes p a r a l l e l t o t h e
do 4 4 boundary i s d i n t h e p e r f e c t
?I- 1)- I t c r y s t a l and do i n t h e g r a f n boundary r e g i bn.
(b) Suggested d i f f r a c t i o n
- e f f e c t s along t h e [001]
d i r e c t i o n i n r e c i p r o c a l space
Boundary 000 ~~~~ due t o t h e g r a i n boundary i n Plane
z =o (a).
The d i f f r a c t i o n a n a l y s i s i n t h e present paper i s based on t h e k i n e m a t i c a l theory o f s c a t t e r i n g , which i s more c o r r e c t l y a p p l i e d t o t h e i n t e r p r e t a t i o n o f X-ray d i f f r a c t i o n r e s u l t s than e l e c t r o n d i f f r a c t i o n r e s u l t s where dynamical e f f e c t s may be important. The a n a l y s i s w i l l be used o n l y t o r e l a t e peak p o s i t i o n s and widths to.boundary s t r u c t u r e , and t h e r e f o r e t h i s approach i s considered v a l i d f o r t h e i n t e r p r e t a t i o n o f t h e e l e c t r o n d i f f r a c t i o n observations.
The advantage o f t h e t h i n c r y s t a l model i s t h a t i t p r o v i d e s a simple way t o v i s u a l i z e t h e d i f f r a c t i o n e f f e c t s due t o changes i n plane spacing a t a g r a i n boundary and t h e d i f f r a c t i o n problem can be solved a n a l y t i c a l l y /15/; t h e disadvantage i s t h a t i t i s t o o simple an approximation t o t h e s t r u c t u r e o f t h e g r a i n boundary and t h e r e f o r e cannot be used t o p r e d i c t a c c u r a t e l y t h e d i f f r a c t i o n
e f f e c t s f r o m a g r a i n boundary. I t seems reasonable t o e x p e c t t h a t t h e s t r a i n f i e l d normal t o an [001] t w i s t boundary w i l l decrease i n magnitude w i t h i n c r e a s i n g d i s t a n c e f r o m t h e boundary plane. I n t h i s paper an e x p o n e n t i a l l y decaying s t r a i n f i e l d i s used.
F o r t h e e x p o n e n t i a l model t h e f r a c t i o n a l p o s i t i o n s o f t h e d i f f r a c t i n g p l a n e s a l o n g t h e z - d i r e c t i o n normal t o t h e i n t e r f a c e a r e g i v e n b y
T h i s s t r a i n f i e l d i s c h a r a c t e r i z e d by a maximum s t r a i n , 6, between t h e two p l a n e s e i t h e r s i d e o f t h e i n t e r f a c e , and a number o f planes, N, w i t h i n which t h e s t r a i n on each s i d e o f t h e boundary f a l l s t o 6 / e . Summing t h e d i s p l a c e m e n t terms i n e q u a t i o n s ( 1 ) we f i n d t h a t t h e f r a c t i o n a l r i g i d body d i s p l a c e m e n t R i s g i v e n by:
w h i c h t e n d s t o N6 as N i n c r e a s e s .
An example o f a computer generated i n t e n s i t y p r o f i l e c a l c u l a t e d f o r an e x p o n e n t i a l l y d e c a y i n g s t r a i n f i e l d w i t h 6=0.15, N=1.45, and R=0.452 d,, i s p l o t t e d a l o n g t h e L - d i r e c t i o n i n F i g . 8 as a b r o k e n l i n e . The i n t e n s i t y p r o f i l e f o r t h e t h i n c r y s t a l boundary i s shown a s a s o l i d l i n e i n F i g . 8 f o r 6=0.15 and R=0.45 do, and so t h e two p r o f i l e s can be compared on t h e b a s i s o f equal maximum expansion and r i g i d body displacement, b u t a d i f f e r e n t f o r m o f s t r a i n f i e l d decay.
The 002 r e f l e c t i o n i s asymmetric i n b o t h p r o f i l e s . However, f o r L v a l u e s l e s s t h a n c1.8 t h e i n t e n s i t y i s g r e a t e r f o r t h e t h i n c r y s t a l model t h a n t h e e x p o n e n t i a l model. F o r L v a l u e s between ~ 1 . 8 and 2 t h e r e v e r s e i s t r u e . T h i s b e h a v i o r i s even more o b v i o u s i n t h e case o f t h e 004 r e f l e c t i o n . T h i s can b e e x p l a i n e d a t l e a s t q u a l i t a t i v e l y b y t h e n a t u r e o f t h e s t r a i n f i e l d . I n t h e e x p o n e n t i a l model t h e r e a r e many i n t e r p l a n a r spacings w h i c h d i f f e r o n l y v e r y s l i g h t l y f r o m d, and
"
o n l y one i n t e r p l a n a r s p a c i n g w h i c h has s u f f e r e d t h e maximum s t r a i n . Thus t h e i n t e n s i t y f o r t h e e x p o n e n t i a l model i s r e l a t i v e l y l a r g e f o r t h e p a r t o f t h e p r o f i l e t h a t corresponds t o s m a l l s t r a i n s (i.e., c l o s e t o 002). I n t h e t h i n c r y s t a l model t h r e e i n t e r p l a n a r spacings a r e s t r a i n e d by 6 and a l l t h e r e s t a r e u n s t r a i n e d , p r o d u c i n g more e x t r a i n t e n s i t y i n t h e h i g h e r s t r a i n p a r t o f t h e p r o f i l e (i.e., f a r f r o m 002).
F i g . 8 - I n t e n s i t y p r o f i l e s : s o l i d l i n e f o r t h i n c r y s t a l model, 6 = 0.15, N = 2 ; broken 1 i ne f o r e x p o n e n t i a l model, 6 = 0.15, N = 1.45.
The a n a l y s i s up t o t h i s p o i n t has concentrated on t h e i n f l u e n c e o f changes i n plane spacing along t h e d i r e c t i o n normal t o t h e i n t e r f a c e on diffraction e f f e c t s from t h e boundary. However, t h e d i f f r a c t i o n technique described i n t h i s s e c t i o n i s s e n s i t i v e n o t o n l y t o changes i n plane spacing a t t h e boundary, b u t a l s o t o any d e v i a t i o n i n s t r u c t u r e away from t h a t o f t h e p e r f e c t c r y s t a l s n e i g h b o r i n g t h e boundary. Therefore, i t i s necessary t o examine t h e i n f l u e n c e o f segregation, p o i n t defects, rumpling o r r e c o n s t r u c t i o n o f t h e atom planes, and a t i l t component i n t h e boundary. An extensive treatment o f d i f f r a c t i o n from boundaries, which i n c l u d e s these e f f e c t s , w i l l be published elsewhere /15/.
I n o r d e r t o study t h e d i f f r a c t i o n e f f e c t s f r o m t h e boundary described above u s i n g e l e c t r o n d i f f r a c t i o n , i t i s necessary t o examine edge-on [ O O l ] t w i s t boundaries.
The procedures t o produce these specimens w i l l be o n l y b r i e f l y summarized here.
Specimens c o n t a i n i n g a [ O O l ] t w i s t boundary i n Au i n t h e edge-on o r i e n t a t i o n were produced by e p i t a x i a l growth on a NaCl b i c r y s t a l s u b s t r a t e c o n t a i n i n g an edge-on [ O O l ] t w i s t boundary; t h e s u b s t r a t e was obtained by h o t pressing t o g e t h e r two cleaved NaCl s i n g l e c r y s t a l s /13/. B i c r y s t a l s c o n t a i n i n g a [ O O l ] t w i s t boundary i n NiO were produced by h o t p r e s s i n g t o g e t h e r two cleaved NiO s i n g l e c r y s t a l s a t t h e d e s i r e d m i s o r i e n t a t i o n /16/. A s l a b c o n t a i n i n g t h e edge-on boundary was then c u t from t h i s b i c r y s t a l w i t h a diamond saw and i o n t h i n n e d t o produce a specimen s u i t a b l e f o r e l e c t r o n microscopy. B i c r y s t a l s c o n t a i n i n g a .[001] t w i s t boundary i n Ge were grown from t h e m e l t u s i n g two p r e o r i e n t e d seed c r y s t a l s . The e l e c t r o n microscopy specimen was obtained i n t h e same manner as f o r N i O .
The experiment t o observe t h e d i f f r a c t i o n e f f e c t s described u s i n g Fig. 7 i n v o l v e s t h e d e t e c t i o n o f weak g r a i n boundary s c a t t e r i n g which i s i n t h e v i c i n i t y o f m a t r i x r e f l e c t i o n s o f t h e t y p e OOL. To make t h e r e q u i r e d observations u s i n g e l e c t r o n d i f f r a c t i o n i t i s necessary t o t i l t t h e Ewald sphere by small increments, i n o r d e r t o e x p l o r e t h e r e g i o n o f r e c i p r o c a l space i n t h e v i c i n i t y o f t h e m a t r i x
r e f l e c t i o n s . D i f f r a c t i o n p a t t e r n s were taken u s i n g a Siemens 102 e l e c t r o n microscope operated a t 125 kV, w i t h a well-defocused second condenser l e n s and exposure times o f 30 t o 900 seconds. The o r i e n t a t i o n o f t h e Ewald sphere was changed i n small steps (0.1' - 0.25') by v a r y i n g t h e d i r e c t i o n o f t h e i n c i d e n t e l e c t r o n beam u s i n g t h e d a r k - f i e l d beam d e f l e c t i o n c o i l s .
Fig. 9(a-c) shows t h r e e d i f f r a c t i o n p a t t e r n s taken from a (e=22°)[001] t w i s t boundary i n NiO. These p a t t e r n s a r e p a r t o f a l o n g s e r i e s recorded w i t h d i f f e r e n t i n c i d e n t beam d i r e c t i o n s as described above, and a s t r e a k displaced away from 002 towards 000 i s c l e a r l y v i s i b l e . S i m i l a r experiments were performed on t h e same type o f t w i s t boundary i n Au and Ge, and a c h a r a c t e r i s t i c d i f f r a c t i o n p a t t e r n f o r each m a t e r i a l i s shown i n Fig. 10, t o g e t h e r w i t h Fig. 9(b) from NiO f o r
comparison.
The s t r e a k s i n t h e d i f f r a c t i o n p a t t e r n s i n Fig. 10(a-c) show s i g n i f i c a n t d i f f e r e n c e s f o r t h e t h r e e m a t e r i a l s . The NiO s t r e a k extends from L%1.3 t o L-2.4.
The end o f the Au s t r e a k near 000 i s a t L%1.7. The o t h e r end i s obscured w i t h i n t h e s t r o n g 002 m a t r i x r e f l e c t i o n b u t i t s maximum e x t e n t i s t o L'~2.15. The Ge s t r e a k i s very s h o r t which poses considerable problems i n r e c o r d i n g i t s complete l e n g t h due t o t h e s a t u r a t i o n o f t h e f i l m by t h e d i f f u s e s c a t t e r i n g around t h e 004 m a t r i x r e f l e c t i o n . However, t h e s t r e a k v i s i b l e i n Fig. 10(c) extends from L=1.96 t o 2.06 which i s a v e r y s l i g h t asymmetry i n t h e o p p o s i t e d i r e c t i o n t o t h a t observed f o r NiO and Au. A q u a l i t a t i v e comparison among t h e e x p e r i m e n t a l l y observed streaks shows t h a t they go from l o n g e s t t o s h o r t e s t and most t o l e a s t asymmetric i n t h e o r d e r NiO,.Au, Ge.
Based on t h e d i f f r a c t i o n a n a l y s i s presented above u s i n g an e x p o n e n t i a l l y decaying s t r a i n f i e l d , d i f f r a c t i o n p r o f i l e s f o r v a r i o u s values o f 6 and N were c a l c u l a t e d i n an attempt t o match t h e experimental observations i n Fig. 10. I t was assumed t h a t t h e s t r e a k s were due s o l e l y t o a change i n plane spacing i n t h e boundary region. F o r t h e NiO streaks a good match was obtained f o r 6 = 0.4 + 0.2 and a boundary w i d t h o f 2do + do. For t h e Au streaks a match was found f o r 6 between 0.1 and 0.2 and a boundary w i d t h o f 3d0 + do. I n t h e case o f Ge t h e small
I F i a . 9 - E l e c t r o n d i f f r a c t i o n p a i t e r n s f r o m a l o n g s e r i e s t a k e n f r o m a 22' [001] t w i s t boundary i n NiO. The beam o r i e n t a t i o n changes i n s t e p s o f 0.5' between p a t t e r n s . The arrows i n d i c a t e s t r e a k s i n t h e v i c i n i t y o f t h e m a t r i x
I r e f l e c t i o n s .
F i g . 10 - E l e c t r o n d i f f r a c t i o n p a t t e r n s showing t h e
c h a r a c t e r i s t i c g r a i n boundary d i f f r a c t i o n s t r e a k s f r o m 22' [001] t w i s t b o u n d a r i e s i n ( a ) NiO,
( b ) Au, ( c ) Ge.
asymmetry i.n t h e s t r e a k was n o t c o n s i d e r e d t o be d e f i n i t i v e e v i d e n c e o f a c o n t r a c t i o n a t t h e boundary, s i n c e c a l c u l a t i o n s showed t h a t s m a l l asymmetries of t h i s t y p e c o u l d be caused by e i t h e r r e c o n s t r u c t i o n o r s e g r e g a t i o n o f p o i n t d e f e c t s t o t h e boundary. The s h o r t n e s s o f t h e s t r e a k i n d i c a t e d a r e l a t i v e l y l a r g e
boundary w i d t h o f t h e o r d e r o f 5do t 2do.
I f these observations a r e considered t o be r e p r e s e n t a t i v e o f t h e e f f e c t o f bonding on boundary s t r u c t u r e , then t h e y can be used t o make p r e d i c t i o n s about t h e
behavior o f t h e s t r u c t u r e o f t h e g r a i n boundaries. Thus, along t h e d i r e c t i o n normal t o a [ O O l ] t w i s t boundary i t i s expected t h a t f o r m a t e r i a l w i t h : ( 1 ) i o n i c bonding t h e r e w i l l be a l a r g e expansion, (2) m e t a l l i c bonding t h e r e w i l l be a small expansion, and ( 3 ) c o v a l e n t bonding t h e r e w i l l be l i t t l e i f any change i n plane spacing. The boundary w i d t h s a l s o appear t o change w i t h m a t e r i a l type, from a r e l a t i v e l y narrow boundary i n t h e case o f an i o n i c m a t e r i a l t o a w i d e r boundary f o r a c o v a l e n t m a t e r i a l .
The problems associated w i t h t h i s technique come under t h r e e main headings: ( 1 ) r e c o r d i n g t h e f u l l e x t e n t o f t h e s t r e a k and i t s i n t e n s i t y p r o f i l e ; (2) assessing t h e e f f e c t s o f t h e dynamical n a t u r e o f e l e c t r o n d i f f r a c t i o n on t h e k i n e m a t i c a l l y c a l c u l a t e d d i f f r a c t i o n p r o f i l e s ; and ( 3 ) determining t h e c o n t r i b u t i o n o f t h e a d d i t i o n a l s t r u c t u r a l e f f e c t s l i s t e d above t o t h e observed d i f f r a c t i o n patterns.
Progress has been made on t h e s o l u t i o n o f a l l t h r e e problems. However, a t t h i s stage i t i s o n l y p o s s i b l e t o r e p o r t q u a l i t a t i v e r e s u l t s . I n o r d e r t o check t h e p r e d i c t i o n s from t h e d i f f r a c t i o n study, f o r example, t h e l a r g e v a l u e o f 6 f o r t h e NiO boundary, i t w i l l be necessary t o use o t h e r experimental techniques, such as h i g h r e s o l u t i o n e l e c t r o n microscopy.
Acknowledgements
The research on t h e f.c.c. metals was supported by t h e N a t i o n a l .Science Foundation under g r a n t DMR-79-16331. The work on NiO was supported by t h e Department o f Energy under Contract No. DE-AC02-81-ER10956. A d d i t i o n a l support was provided by t h e M a t e r i a l s Science Center o f C o r n e l l U n i v e r s i t y . The X-ray d i f f r a c t i o n experiments were performed a t t h e C o r n e l l High Energy Synchrotron Source. We a r e pleased t o thank Dr. N.L. Peterson, Argonne N a t i o n a l Laboratory, Argonne, I l l i n o i s f o r a l l o w i n g t h e use o f t h e f a c i l i t i e s i n h i s l a b o r a t o r y f o r growing t h e NiO s i n g l e c r y s t a l s .
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